Subgroups — C₆₇×#60877 #30740128

Order 17152.

1786 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3508	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3503	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3241	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3228	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3228	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3227	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3226	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#3096	C₂	1	4	yes	maximal, complemented
13	C₆₇×#3081	C₂	1	4	yes	maximal, complemented
14	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
15	#559×C₆₇	—	2	1	no	—
16	#559×C₆₇	—	2	1	no	—
17	Q₈²×C₆₇	—	2	1	no	—
18	#557×C₆₇	—	2	1	no	—
19	#554×C₆₇	—	2	2	no	—
20	#554×C₆₇	—	2	1	no	—
21	#550×C₆₇	—	2	1	no	—
22	#550×C₆₇	—	2	1	no	—
23	D₄×Q₈×C₆₇	—	2	2	no	—
24	D₄×Q₈×C₆₇	—	2	1	no	—
25	#546×C₆₇	—	2	2	no	—
26	#546×C₆₇	—	2	2	no	—
27	#544×C₆₇	C₂²	1	2	yes	—
28	#544×C₆₇	C₂²	1	4	yes	complemented
29	#540×C₆₇	C₂²	1	4	yes	—
30	#535×C₆₇	C₂²	1	2	yes	—
31	#533×C₆₇	C₂²	1	1	yes	characteristic
32	#533×C₆₇	C₂²	1	1	yes	characteristic
33	#533×C₆₇	C₂²	1	2	yes	complemented
34	#532×C₆₇	C₂²	1	2	yes	—
35	#532×C₆₇	C₂²	1	2	yes	—
36	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
38	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
39	#520×C₆₇	C₂²	1	2	yes	—
40	#518×C₆₇	C₂²	1	2	yes	—
41	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
42	#501×C₆₇	C₂²	1	4	yes	complemented
43	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	#500×C₆₇	C₂²	1	2	yes	complemented
46	#499×C₆₇	C₂²	1	4	yes	complemented
47	#498×C₆₇	C₂²	1	1	yes	characteristic, complemented
48	#498×C₆₇	C₂²	1	1	yes	characteristic, complemented
49	#498×C₆₇	C₂²	1	2	yes	complemented
50	#490×C₆₇	C₂²	1	4	yes	—
51	#486×C₆₇	C₂²	1	2	yes	—
52	#486×C₆₇	C₂²	1	2	yes	—
53	#486×C₆₇	C₂²	1	4	yes	—
54	#486×C₆₇	C₂²	1	2	yes	—
55	#486×C₆₇	C₂²	1	2	yes	—
56	#482×C₆₇	C₂²	1	8	yes	complemented
57	#482×C₆₇	C₂²	1	8	yes	complemented
58	#481×C₆₇	C₂²	1	8	yes	complemented
59	#480×C₆₇	C₂²	1	8	yes	complemented
60	#478×C₆₇	C₂²	1	8	yes	complemented
61	#478×C₆₇	C₂²	1	8	yes	complemented
62	#476×C₆₇	C₂²	1	8	yes	complemented
63	#474×C₆₇	C₂²	1	8	yes	complemented
64	#433×C₆₇	C₂²	1	4	yes	complemented
65	#431×C₆₇	C₂²	1	2	yes	complemented
66	#431×C₆₇	C₂²	1	2	yes	complemented
67	#428×C₆₇	C₂²	1	4	yes	—
68	#427×C₆₇	C₂²	1	4	yes	—
69	#424×C₆₇	C₂²	1	2	yes	—
70	#418×C₆₇	C₂²	1	4	yes	—
71	C₂×#102×C₆₇	C₂²	1	4	yes	—
72	#408×C₆₇	C₂²	1	2	yes	—
73	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
74	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
75	C₂×#41×C₆₇	—	2	2	no	—
76	C₂×#41×C₆₇	—	2	2	no	—
77	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₆₇	—	2	1	no	—
79	C₂²×Q₈×C₆₇	—	2	2	no	—
80	C₂²×D₄×C₆₇	—	2	1	no	—
81	C₂²×D₄×C₆₇	—	2	1	no	—
82	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
83	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
84	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
85	C₄⋊Q₈×C₆₇	—	2	2	no	—
86	C₄⋊Q₈×C₆₇	—	2	4	no	—
87	C₄⋊Q₈×C₆₇	—	2	1	no	—
88	C₄⋊Q₈×C₆₇	—	2	1	no	—
89	C₄⋊Q₈×C₆₇	—	2	2	no	—
90	C₄⋊Q₈×C₆₇	—	2	2	no	—
91	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
92	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
93	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
96	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
98	#127×C₆₇	—	2	2	no	—
99	#127×C₆₇	—	2	1	no	—
100	#127×C₆₇	—	2	1	no	—
101	#127×C₆₇	C₂³	1	2	yes	—
102	#127×C₆₇	C₂³	1	2	yes	—
103	#125×C₆₇	—	2	2	no	—
104	#125×C₆₇	—	2	4	no	—
105	#125×C₆₇	C₂³	1	4	yes	—
106	#125×C₆₇	C₂³	1	2	yes	—
107	#125×C₆₇	C₂³	1	2	yes	—
108	#124×C₆₇	—	2	2	no	—
109	#124×C₆₇	—	2	2	no	—
110	#124×C₆₇	—	2	2	no	—
111	#123×C₆₇	—	2	4	no	—
112	#123×C₆₇	—	2	4	no	—
113	C₂²⋊Q₈×C₆₇	—	2	1	no	—
114	C₂²⋊Q₈×C₆₇	—	2	1	no	—
115	C₂²⋊Q₈×C₆₇	—	2	1	no	—
116	C₂²⋊Q₈×C₆₇	—	2	1	no	—
117	C₂²⋊Q₈×C₆₇	—	2	2	no	—
118	C₂²⋊Q₈×C₆₇	—	2	2	no	—
119	C₂²⋊Q₈×C₆₇	—	2	4	no	—
120	C₂²⋊Q₈×C₆₇	—	2	4	no	—
121	C₂²⋊Q₈×C₆₇	—	2	4	no	—
122	#121×C₆₇	—	2	2	no	—
123	#121×C₆₇	—	2	2	no	—
124	#121×C₆₇	—	2	4	no	—
125	#121×C₆₇	—	2	4	no	—
126	#121×C₆₇	C₂³	1	4	yes	—
127	#121×C₆₇	C₂³	1	4	yes	—
128	#120×C₆₇	—	2	4	no	—
129	C₄×Q₈×C₆₇	—	2	2	no	—
130	C₄×Q₈×C₆₇	—	2	1	no	—
131	C₄×Q₈×C₆₇	—	2	1	no	—
132	C₄×Q₈×C₆₇	—	2	1	no	—
133	C₄×Q₈×C₆₇	—	2	1	no	—
134	C₄×Q₈×C₆₇	—	2	2	no	—
135	C₄×Q₈×C₆₇	—	2	2	no	—
136	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
137	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
138	C₄×D₄×C₆₇	—	2	4	no	—
139	C₄×D₄×C₆₇	—	2	2	no	—
140	C₄×D₄×C₆₇	—	2	2	no	—
141	C₄×D₄×C₆₇	—	2	2	no	—
142	C₄×D₄×C₆₇	—	2	2	no	—
143	C₄×D₄×C₆₇	—	2	2	no	—
144	C₄×D₄×C₆₇	D₄	1	2	yes	—
145	C₄×D₄×C₆₇	D₄	1	2	yes	—
146	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
147	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
148	C₄×D₄×C₆₇	C₂³	1	4	yes	—
149	C₄×D₄×C₆₇	C₂³	1	4	yes	—
150	#117×C₆₇	C₂³	1	4	yes	—
151	#117×C₆₇	C₂³	1	2	yes	—
152	#117×C₆₇	C₂³	1	2	yes	—
153	#117×C₆₇	C₂³	1	1	yes	characteristic
154	#117×C₆₇	C₂³	1	2	yes	—
155	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
156	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
158	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
159	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
160	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
161	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
162	#107×C₆₇	C₂³	1	4	yes	—
163	#107×C₆₇	C₂³	1	8	yes	—
164	#107×C₆₇	C₂³	1	4	yes	—
165	#106×C₆₇	C₂³	1	4	yes	—
166	#106×C₆₇	C₂³	1	8	yes	—
167	#106×C₆₇	C₂³	1	4	yes	—
168	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
169	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
170	#102×C₆₇	C₂³	1	8	yes	—
171	#102×C₆₇	C₂³	1	8	yes	—
172	#102×C₆₇	C₂³	1	8	yes	—
173	#102×C₆₇	C₂³	1	8	yes	—
174	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
175	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
176	#97×C₆₇	C₂³	1	2	yes	—
177	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
178	C₄×C₈×C₆₇	C₂³	1	2	yes	—
179	C₄×C₈×C₆₇	C₂³	1	2	yes	—
180	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
181	C₂⁴×C₆₇	—	2	2	no	—
182	#41×C₆₇	—	2	4	no	—
183	#41×C₆₇	—	8	4	no	—
184	#41×C₆₇	—	8	4	no	—
185	C₂×Q₈×C₆₇	—	2	1	no	—
186	C₂×Q₈×C₆₇	—	2	1	no	—
187	C₂×Q₈×C₆₇	—	2	1	no	—
188	C₂×Q₈×C₆₇	—	2	1	no	—
189	C₂×Q₈×C₆₇	—	2	2	no	—
190	C₂×Q₈×C₆₇	—	2	4	no	—
191	C₂×Q₈×C₆₇	—	2	2	no	—
192	C₂×Q₈×C₆₇	—	8	2	no	—
193	C₂×Q₈×C₆₇	—	8	4	no	—
194	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
195	C₂×D₄×C₆₇	—	2	4	no	—
196	C₂×D₄×C₆₇	—	2	4	no	—
197	C₂×D₄×C₆₇	—	2	2	no	—
198	C₂×D₄×C₆₇	—	2	2	no	—
199	C₂×D₄×C₆₇	—	2	2	no	—
200	C₂×D₄×C₆₇	—	2	2	no	—
201	C₂×D₄×C₆₇	—	2	2	no	—
202	C₂×D₄×C₆₇	—	2	8	no	—
203	C₂×D₄×C₆₇	—	8	2	no	—
204	C₂×D₄×C₆₇	—	8	2	no	—
205	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
206	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
208	C₂²×C₄×C₆₇	—	2	4	no	—
209	C₂²×C₄×C₆₇	—	2	2	no	—
210	C₂²×C₄×C₆₇	—	2	4	no	—
211	C₂²×C₄×C₆₇	—	2	4	no	—
212	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
213	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
214	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
215	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
216	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
217	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
218	M₁₆×C₆₇	—	2	4	no	—
219	C₂×C₈×C₆₇	—	2	4	no	—
220	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
221	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
222	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
223	C₄⋊C₄×C₆₇	—	2	2	no	—
224	C₄⋊C₄×C₆₇	—	2	1	no	—
225	C₄⋊C₄×C₆₇	—	2	4	no	—
226	C₄⋊C₄×C₆₇	—	2	2	no	—
227	C₄⋊C₄×C₆₇	—	2	4	no	—
228	C₄⋊C₄×C₆₇	—	2	2	no	—
229	C₄⋊C₄×C₆₇	—	2	1	no	—
230	C₄⋊C₄×C₆₇	—	2	1	no	—
231	C₄⋊C₄×C₆₇	—	2	1	no	—
232	C₄⋊C₄×C₆₇	—	2	2	no	—
233	C₄⋊C₄×C₆₇	—	2	2	no	—
234	C₄⋊C₄×C₆₇	—	2	4	no	—
235	C₄⋊C₄×C₆₇	—	2	2	no	—
236	C₄⋊C₄×C₆₇	—	2	4	no	—
237	C₄⋊C₄×C₆₇	—	2	2	no	—
238	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
239	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
240	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
241	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
242	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
243	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
244	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
245	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
246	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
247	C₂²⋊C₄×C₆₇	—	2	2	no	—
248	C₂²⋊C₄×C₆₇	—	2	4	no	—
249	C₂²⋊C₄×C₆₇	—	2	2	no	—
250	C₂²⋊C₄×C₆₇	—	2	4	no	—
251	C₂²⋊C₄×C₆₇	—	2	4	no	—
252	C₂²⋊C₄×C₆₇	—	2	4	no	—
253	C₂²⋊C₄×C₆₇	—	2	4	no	—
254	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
255	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
256	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
257	C₄²×C₆₇	—	2	2	no	—
258	C₄²×C₆₇	—	2	4	no	—
259	C₄²×C₆₇	—	2	2	no	—
260	C₄²×C₆₇	C₂×D₄	1	2	yes	—
261	C₄²×C₆₇	C₂×D₄	1	2	yes	—
262	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
263	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
264	C₄²×C₆₇	C₂⁴	1	2	yes	—
265	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
266	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
267	C₂³×C₆₇	—	2	4	no	—
268	C₂³×C₆₇	—	4	4	no	—
269	C₂³×C₆₇	—	8	2	no	—
270	C₂³×C₆₇	—	8	2	no	—
271	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
272	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
273	Q₈×C₆₇	—	2	2	no	—
274	Q₈×C₆₇	—	4	2	no	—
275	Q₈×C₆₇	—	4	4	no	—
276	Q₈×C₆₇	—	4	2	no	—
277	Q₈×C₆₇	—	4	2	no	—
278	Q₈×C₆₇	—	4	2	no	—
279	Q₈×C₆₇	—	8	4	no	—
280	Q₈×C₆₇	—	8	2	no	—
281	D₄×C₆₇	—	2	4	no	—
282	D₄×C₆₇	—	2	2	no	—
283	D₄×C₆₇	—	2	8	no	—
284	D₄×C₆₇	—	4	4	no	—
285	D₄×C₆₇	—	4	4	no	—
286	D₄×C₆₇	—	4	4	no	—
287	D₄×C₆₇	—	8	2	no	—
288	D₄×C₆₇	—	8	2	no	—
289	D₄×C₆₇	—	8	4	no	—
290	D₄×C₆₇	—	8	4	no	—
291	C₂×C₄×C₆₇	—	2	2	no	—
292	C₂×C₄×C₆₇	—	2	4	no	—
293	C₂×C₄×C₆₇	—	2	2	no	—
294	C₂×C₄×C₆₇	—	2	4	no	—
295	C₂×C₄×C₆₇	—	2	4	no	—
296	C₂×C₄×C₆₇	—	2	2	no	—
297	C₂×C₄×C₆₇	—	2	2	no	—
298	C₂×C₄×C₆₇	—	2	2	no	—
299	C₂×C₄×C₆₇	—	2	4	no	—
300	C₂×C₄×C₆₇	—	2	2	no	—
301	C₂×C₄×C₆₇	—	8	4	no	—
302	C₂×C₄×C₆₇	—	8	4	no	—
303	C₂×C₄×C₆₇	—	8	2	no	—
304	C₂×C₄×C₆₇	—	8	4	no	—
305	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
306	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
307	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
308	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
312	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
313	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
314	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
315	C₈×C₆₇	—	2	2	no	—
316	C₈×C₆₇	—	2	2	no	—
317	C₈×C₆₇	—	2	4	no	—
318	C₂²×C₆₇	—	2	1	no	—
319	C₂²×C₆₇	—	2	1	no	—
320	C₂²×C₆₇	—	2	2	no	—
321	C₂²×C₆₇	—	2	2	no	—
322	C₂²×C₆₇	—	2	2	no	—
323	C₂²×C₆₇	—	4	4	no	—
324	C₂²×C₆₇	—	4	4	no	—
325	C₂²×C₆₇	—	4	4	no	—
326	C₂²×C₆₇	—	16	4	no	—
327	C₂²×C₆₇	#536	1	2	yes	—
328	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
329	C₄×C₆₇	—	2	2	no	—
330	C₄×C₆₇	—	2	2	no	—
331	C₄×C₆₇	—	2	1	no	—
332	C₄×C₆₇	—	2	2	no	—
333	C₄×C₆₇	—	2	1	no	—
334	C₄×C₆₇	—	4	2	no	—
335	C₄×C₆₇	—	4	4	no	—
336	C₄×C₆₇	—	4	2	no	—
337	C₄×C₆₇	—	4	2	no	—
338	C₄×C₆₇	—	4	2	no	—
339	C₄×C₆₇	#536	1	2	yes	—
340	C₄×C₆₇	#576	1	2	yes	—
341	C₄×C₆₇	#577	1	2	yes	—
342	C₄×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
343	#60877	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
344	C₂×C₆₇	—	2	2	no	—
345	C₂×C₆₇	—	4	2	no	—
346	C₂×C₆₇	—	8	4	no	—
347	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
348	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
349	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
350	#3508	C₂×C₆₇	1	1	yes	characteristic
351	#3503	C₂×C₆₇	1	1	yes	characteristic, complemented
352	#3241	C₂×C₆₇	1	8	yes	complemented
353	#3228	C₂×C₆₇	1	2	yes	complemented
354	#3228	C₂×C₆₇	1	2	yes	complemented
355	#3227	C₂×C₆₇	1	2	yes	complemented
356	#3226	C₂×C₆₇	1	2	yes	complemented
357	#3165	C₂×C₆₇	1	1	yes	characteristic, complemented
358	#3160	C₂×C₆₇	1	1	yes	characteristic, complemented
359	#3138	C₂×C₆₇	1	1	yes	characteristic, complemented
360	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
361	#3096	C₂×C₆₇	1	4	yes	complemented
362	#3081	C₂×C₆₇	1	4	yes	complemented
363	#2922	C₂×C₆₇	1	1	yes	characteristic, complemented
364	C₆₇	#60877	1	1	yes	characteristic, direct product factor, Sylow
365	#559	—	2	1	no	—
366	#559	—	2	1	no	—
367	Q₈²	—	2	1	no	—
368	#557	—	2	1	no	—
369	#554	—	2	2	no	—
370	#554	—	2	1	no	—
371	#550	—	2	1	no	—
372	#550	—	2	1	no	—
373	D₄×Q₈	—	2	2	no	—
374	D₄×Q₈	—	2	1	no	—
375	#546	—	2	2	no	—
376	#546	—	2	2	no	—
377	#544	C₂²×C₆₇	1	2	yes	—
378	#544	C₂²×C₆₇	1	4	yes	complemented
379	#540	C₂²×C₆₇	1	4	yes	—
380	#535	C₂²×C₆₇	1	2	yes	—
381	#533	C₂²×C₆₇	1	1	yes	characteristic
382	#533	C₂²×C₆₇	1	1	yes	characteristic
383	#533	C₂²×C₆₇	1	2	yes	complemented
384	#532	C₂²×C₆₇	1	2	yes	—
385	#532	C₂²×C₆₇	1	2	yes	—
386	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
387	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
388	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
389	#520	C₂²×C₆₇	1	2	yes	—
390	#518	C₂²×C₆₇	1	2	yes	—
391	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
392	#501	C₂²×C₆₇	1	4	yes	complemented
393	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
394	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
395	#500	C₂²×C₆₇	1	2	yes	complemented
396	#499	C₂²×C₆₇	1	4	yes	complemented
397	#498	C₂²×C₆₇	1	1	yes	characteristic, complemented
398	#498	C₂²×C₆₇	1	1	yes	characteristic, complemented
399	#498	C₂²×C₆₇	1	2	yes	complemented
400	#490	C₂²×C₆₇	1	4	yes	—
401	#486	C₂²×C₆₇	1	2	yes	—
402	#486	C₂²×C₆₇	1	2	yes	—
403	#486	C₂²×C₆₇	1	4	yes	—
404	#486	C₂²×C₆₇	1	2	yes	—
405	#486	C₂²×C₆₇	1	2	yes	—
406	#482	C₂²×C₆₇	1	8	yes	complemented
407	#482	C₂²×C₆₇	1	8	yes	complemented
408	#481	C₂²×C₆₇	1	8	yes	complemented
409	#480	C₂²×C₆₇	1	8	yes	complemented
410	#478	C₂²×C₆₇	1	8	yes	complemented
411	#478	C₂²×C₆₇	1	8	yes	complemented
412	#476	C₂²×C₆₇	1	8	yes	complemented
413	#474	C₂²×C₆₇	1	8	yes	complemented
414	#433	C₂²×C₆₇	1	4	yes	complemented
415	#431	C₂²×C₆₇	1	2	yes	complemented
416	#431	C₂²×C₆₇	1	2	yes	complemented
417	#428	C₂²×C₆₇	1	4	yes	—
418	#427	C₂²×C₆₇	1	4	yes	—
419	#424	C₂²×C₆₇	1	2	yes	—
420	#418	C₂²×C₆₇	1	4	yes	—
421	C₂×#102	C₂²×C₆₇	1	4	yes	—
422	#408	C₂²×C₆₇	1	2	yes	—
423	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
424	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
425	C₂×#41	—	2	2	no	—
426	C₂×#41	—	2	2	no	—
427	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
428	C₂²×Q₈	—	2	1	no	—
429	C₂²×Q₈	—	2	2	no	—
430	C₂²×D₄	—	2	1	no	—
431	C₂²×D₄	—	2	1	no	—
432	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
433	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
434	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
435	C₄⋊Q₈	—	2	2	no	—
436	C₄⋊Q₈	—	2	4	no	—
437	C₄⋊Q₈	—	2	1	no	—
438	C₄⋊Q₈	—	2	1	no	—
439	C₄⋊Q₈	—	2	2	no	—
440	C₄⋊Q₈	—	2	2	no	—
441	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
442	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
443	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
444	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
445	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
446	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
447	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
448	#127	—	2	2	no	—
449	#127	—	2	1	no	—
450	#127	—	2	1	no	—
451	#127	C₂³×C₆₇	1	2	yes	—
452	#127	C₂³×C₆₇	1	2	yes	—
453	#125	—	2	2	no	—
454	#125	—	2	4	no	—
455	#125	C₂³×C₆₇	1	2	yes	—
456	#125	C₂³×C₆₇	1	2	yes	—
457	#125	C₂³×C₆₇	1	4	yes	—
458	#124	—	2	2	no	—
459	#124	—	2	2	no	—
460	#124	—	2	2	no	—
461	#123	—	2	4	no	—
462	#123	—	2	4	no	—
463	C₂²⋊Q₈	—	2	1	no	—
464	C₂²⋊Q₈	—	2	1	no	—
465	C₂²⋊Q₈	—	2	1	no	—
466	C₂²⋊Q₈	—	2	1	no	—
467	C₂²⋊Q₈	—	2	2	no	—
468	C₂²⋊Q₈	—	2	2	no	—
469	C₂²⋊Q₈	—	2	4	no	—
470	C₂²⋊Q₈	—	2	4	no	—
471	C₂²⋊Q₈	—	2	4	no	—
472	#121	—	2	2	no	—
473	#121	—	2	2	no	—
474	#121	—	2	4	no	—
475	#121	—	2	4	no	—
476	#121	C₂³×C₆₇	1	4	yes	—
477	#121	C₂³×C₆₇	1	4	yes	—
478	#120	—	2	4	no	—
479	C₄×Q₈	—	2	2	no	—
480	C₄×Q₈	—	2	1	no	—
481	C₄×Q₈	—	2	1	no	—
482	C₄×Q₈	—	2	1	no	—
483	C₄×Q₈	—	2	1	no	—
484	C₄×Q₈	—	2	2	no	—
485	C₄×Q₈	—	2	2	no	—
486	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
487	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
488	C₄×D₄	—	2	4	no	—
489	C₄×D₄	—	2	2	no	—
490	C₄×D₄	—	2	2	no	—
491	C₄×D₄	—	2	2	no	—
492	C₄×D₄	—	2	2	no	—
493	C₄×D₄	—	2	2	no	—
494	C₄×D₄	D₄×C₆₇	1	2	yes	—
495	C₄×D₄	D₄×C₆₇	1	2	yes	—
496	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
497	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
498	C₄×D₄	C₂³×C₆₇	1	4	yes	—
499	C₄×D₄	C₂³×C₆₇	1	4	yes	—
500	#117	C₂³×C₆₇	1	2	yes	—
501	#117	C₂³×C₆₇	1	2	yes	—
502	#117	C₂³×C₆₇	1	4	yes	—
503	#117	C₂³×C₆₇	1	1	yes	characteristic
504	#117	C₂³×C₆₇	1	2	yes	—
505	C₂×C₄⋊C₄	—	2	1	no	—
506	C₂×C₄⋊C₄	—	2	2	no	—
507	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
508	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
509	C₂×C₂²⋊C₄	—	2	4	no	—
510	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
511	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
512	#107	C₂³×C₆₇	1	4	yes	—
513	#107	C₂³×C₆₇	1	4	yes	—
514	#107	C₂³×C₆₇	1	8	yes	—
515	#106	C₂³×C₆₇	1	4	yes	—
516	#106	C₂³×C₆₇	1	4	yes	—
517	#106	C₂³×C₆₇	1	8	yes	—
518	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
519	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
520	#102	C₂³×C₆₇	1	8	yes	—
521	#102	C₂³×C₆₇	1	8	yes	—
522	#102	C₂³×C₆₇	1	8	yes	—
523	#102	C₂³×C₆₇	1	8	yes	—
524	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
525	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
526	#97	C₂³×C₆₇	1	2	yes	—
527	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
528	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
529	C₄×C₈	C₂³×C₆₇	1	2	yes	—
530	C₄×C₈	C₂³×C₆₇	1	2	yes	—
531	C₂⁴	—	2	2	no	—
532	#41	—	2	4	no	—
533	#41	—	8	4	no	—
534	#41	—	8	4	no	—
535	C₂×Q₈	—	2	1	no	—
536	C₂×Q₈	—	2	1	no	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	—	2	1	no	—
539	C₂×Q₈	—	2	2	no	—
540	C₂×Q₈	—	2	4	no	—
541	C₂×Q₈	—	2	2	no	—
542	C₂×Q₈	—	8	2	no	—
543	C₂×Q₈	—	8	4	no	—
544	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
545	C₂×D₄	—	2	4	no	—
546	C₂×D₄	—	2	4	no	—
547	C₂×D₄	—	2	2	no	—
548	C₂×D₄	—	2	2	no	—
549	C₂×D₄	—	2	2	no	—
550	C₂×D₄	—	2	2	no	—
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	8	no	—
553	C₂×D₄	—	8	2	no	—
554	C₂×D₄	—	8	2	no	—
555	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
556	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
557	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
558	C₂²×C₄	—	2	4	no	—
559	C₂²×C₄	—	2	2	no	—
560	C₂²×C₄	—	2	4	no	—
561	C₂²×C₄	—	2	4	no	—
562	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
563	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
564	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
565	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
566	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
567	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
568	M₁₆	—	2	4	no	—
569	C₂×C₈	—	2	4	no	—
570	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
571	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
572	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	4	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	4	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	2	no	—
586	C₄⋊C₄	—	2	4	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
589	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
590	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
591	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
592	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
593	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
594	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
595	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
596	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
597	C₂²⋊C₄	—	2	2	no	—
598	C₂²⋊C₄	—	2	4	no	—
599	C₂²⋊C₄	—	2	2	no	—
600	C₂²⋊C₄	—	2	4	no	—
601	C₂²⋊C₄	—	2	4	no	—
602	C₂²⋊C₄	—	2	4	no	—
603	C₂²⋊C₄	—	2	4	no	—
604	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
605	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
606	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
607	C₄²	—	2	2	no	—
608	C₄²	—	2	4	no	—
609	C₄²	—	2	2	no	—
610	C₄²	C₂×D₄×C₆₇	1	2	yes	—
611	C₄²	C₂×D₄×C₆₇	1	2	yes	—
612	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
613	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
614	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
615	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
616	C₄²	C₂⁴×C₆₇	1	2	yes	—
617	C₂³	—	2	4	no	—
618	C₂³	—	4	4	no	—
619	C₂³	—	8	2	no	—
620	C₂³	—	8	2	no	—
621	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
622	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
623	Q₈	—	2	2	no	—
624	Q₈	—	4	2	no	—
625	Q₈	—	4	4	no	—
626	Q₈	—	4	2	no	—
627	Q₈	—	4	2	no	—
628	Q₈	—	4	2	no	—
629	Q₈	—	8	4	no	—
630	Q₈	—	8	2	no	—
631	D₄	—	2	4	no	—
632	D₄	—	2	2	no	—
633	D₄	—	2	8	no	—
634	D₄	—	4	4	no	—
635	D₄	—	4	4	no	—
636	D₄	—	4	4	no	—
637	D₄	—	8	2	no	—
638	D₄	—	8	2	no	—
639	D₄	—	8	4	no	—
640	D₄	—	8	4	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	4	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	4	no	—
645	C₂×C₄	—	2	4	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	4	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	8	4	no	—
652	C₂×C₄	—	8	4	no	—
653	C₂×C₄	—	8	2	no	—
654	C₂×C₄	—	8	4	no	—
655	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
656	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
657	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
658	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
659	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
660	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
661	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
662	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
663	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
664	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
665	C₈	—	2	2	no	—
666	C₈	—	2	2	no	—
667	C₈	—	2	4	no	—
668	C₂²	—	2	1	no	—
669	C₂²	—	2	1	no	—
670	C₂²	—	2	2	no	—
671	C₂²	—	2	2	no	—
672	C₂²	—	2	2	no	—
673	C₂²	—	4	4	no	—
674	C₂²	—	4	4	no	—
675	C₂²	—	4	4	no	—
676	C₂²	—	16	4	no	—
677	C₂²	#536×C₆₇	1	2	yes	—
678	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
679	C₄	—	2	2	no	—
680	C₄	—	2	1	no	—
681	C₄	—	2	2	no	—
682	C₄	—	2	1	no	—
683	C₄	—	2	2	no	—
684	C₄	—	4	2	no	—
685	C₄	—	4	4	no	—
686	C₄	—	4	2	no	—
687	C₄	—	4	2	no	—
688	C₄	—	4	2	no	—
689	C₄	#536×C₆₇	1	2	yes	—
690	C₄	#576×C₆₇	1	2	yes	—
691	C₄	#577×C₆₇	1	2	yes	—
692	C₄	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
693	C₂	—	2	2	no	—
694	C₂	—	4	2	no	—
695	C₂	—	8	4	no	—
696	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
697	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
698	C₂	C₂×#577×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3508	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×#41×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#3503	C₂	semidirect
3. C₆₇×#3241	C₂	semidirect
4. C₆₇×#3228	C₂	semidirect
5. C₆₇×#3228	C₂	semidirect
6. C₆₇×#3227	C₂	semidirect
7. C₆₇×#3226	C₂	semidirect
8. C₆₇×#3165	C₂	semidirect
9. C₆₇×#3160	C₂	semidirect
10. C₆₇×#3138	C₂	semidirect
11. C₂×#486×C₆₇	C₂	semidirect
12. C₆₇×#3096	C₂	semidirect
13. C₆₇×#3081	C₂	semidirect
14. C₆₇×#2922	C₂	semidirect
28. #544×C₆₇	C₂²	semidirect
33. #533×C₆₇	C₂²	semidirect
37. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
42. #501×C₆₇	C₂²	semidirect
43. #500×C₆₇	C₂²	semidirect
44. #500×C₆₇	C₂²	semidirect
45. #500×C₆₇	C₂²	semidirect
46. #499×C₆₇	C₂²	semidirect
47. #498×C₆₇	C₂²	semidirect
48. #498×C₆₇	C₂²	semidirect
49. #498×C₆₇	C₂²	semidirect
56. #482×C₆₇	C₂²	semidirect
57. #482×C₆₇	C₂²	semidirect
58. #481×C₆₇	C₂²	semidirect
59. #480×C₆₇	C₂²	semidirect
60. #478×C₆₇	C₂²	semidirect
61. #478×C₆₇	C₂²	semidirect
62. #476×C₆₇	C₂²	semidirect
63. #474×C₆₇	C₂²	semidirect
64. #433×C₆₇	C₂²	semidirect
65. #431×C₆₇	C₂²	semidirect
66. #431×C₆₇	C₂²	semidirect
343. #60877	C₆₇	direct (N × H)
351. #3503	C₂×C₆₇	semidirect
352. #3241	C₂×C₆₇	semidirect
353. #3228	C₂×C₆₇	semidirect
354. #3228	C₂×C₆₇	semidirect
355. #3227	C₂×C₆₇	semidirect
356. #3226	C₂×C₆₇	semidirect
357. #3165	C₂×C₆₇	semidirect
358. #3160	C₂×C₆₇	semidirect
359. #3138	C₂×C₆₇	semidirect
360. C₂×#486	C₂×C₆₇	semidirect
361. #3096	C₂×C₆₇	semidirect
362. #3081	C₂×C₆₇	semidirect
363. #2922	C₂×C₆₇	semidirect
364. C₆₇	#60877	direct (N × H)
378. #544	C₂²×C₆₇	semidirect
383. #533	C₂²×C₆₇	semidirect
387. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
392. #501	C₂²×C₆₇	semidirect
393. #500	C₂²×C₆₇	semidirect
394. #500	C₂²×C₆₇	semidirect
395. #500	C₂²×C₆₇	semidirect
396. #499	C₂²×C₆₇	semidirect
397. #498	C₂²×C₆₇	semidirect
398. #498	C₂²×C₆₇	semidirect
399. #498	C₂²×C₆₇	semidirect
406. #482	C₂²×C₆₇	semidirect
407. #482	C₂²×C₆₇	semidirect
408. #481	C₂²×C₆₇	semidirect
409. #480	C₂²×C₆₇	semidirect
410. #478	C₂²×C₆₇	semidirect
411. #478	C₂²×C₆₇	semidirect
412. #476	C₂²×C₆₇	semidirect
413. #474	C₂²×C₆₇	semidirect
414. #433	C₂²×C₆₇	semidirect
415. #431	C₂²×C₆₇	semidirect
416. #431	C₂²×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3508	2
C₆₇×#3503	2
C₆₇×#3241	2
C₆₇×#3228	2
C₆₇×#3227	2
C₆₇×#3226	2
C₆₇×#3165	2
C₆₇×#3160	2
C₆₇×#3138	2
C₂×#486×C₆₇	2
C₆₇×#3096	2
C₆₇×#3081	2
C₆₇×#2922	2
#60877	67

Subgroups — C67×#60877 #30740128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60877 #30740128