Subgroups — C₆₇×#61284 #30740535

Order 17152.

1882 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3506	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3355	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3354	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3341	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3339	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3334	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3096	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3077	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3058	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2913	C₂	1	2	yes	maximal, complemented
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#131×C₆₇	C₂²	1	2	yes	—
16	#559×C₆₇	—	2	4	no	—
17	#556×C₆₇	—	2	2	no	—
18	#555×C₆₇	C₂²	1	4	yes	complemented
19	#553×C₆₇	C₂²	1	8	yes	complemented
20	#550×C₆₇	—	2	2	no	—
21	#547×C₆₇	C₂²	1	4	yes	complemented
22	#538×C₆₇	—	2	4	no	—
23	#535×C₆₇	C₂²	1	4	yes	—
24	#533×C₆₇	C₂²	1	2	yes	—
25	#532×C₆₇	C₂²	1	1	yes	characteristic
26	#525×C₆₇	—	2	4	no	—
27	#520×C₆₇	C₂²	1	4	yes	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
29	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	#499×C₆₇	C₂²	1	2	yes	complemented
32	#499×C₆₇	C₂²	1	2	yes	complemented
33	#498×C₆₇	C₂²	1	2	yes	complemented
34	#482×C₆₇	C₂²	1	8	yes	complemented
35	#482×C₆₇	C₂²	1	8	yes	complemented
36	#481×C₆₇	C₂²	1	8	yes	complemented
37	#480×C₆₇	C₂²	1	8	yes	complemented
38	#478×C₆₇	C₂²	1	8	yes	complemented
39	#476×C₆₇	C₂²	1	4	yes	complemented
40	#474×C₆₇	C₂²	1	4	yes	complemented
41	#469×C₆₇	C₂²	1	8	yes	—
42	#466×C₆₇	C₂²	1	4	yes	—
43	#466×C₆₇	C₂²	1	4	yes	—
44	C₄×D₈×C₆₇	C₂²	1	2	yes	—
45	C₄×D₈×C₆₇	C₂²	1	2	yes	—
46	C₄×D₈×C₆₇	C₂²	1	2	yes	—
47	C₄×D₈×C₆₇	C₂²	1	2	yes	—
48	#436×C₆₇	C₂²	1	2	yes	complemented
49	#436×C₆₇	C₂²	1	2	yes	complemented
50	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
51	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
52	#428×C₆₇	C₂²	1	4	yes	—
53	#428×C₆₇	C₂²	1	8	yes	complemented
54	#427×C₆₇	C₂²	1	2	yes	—
55	#427×C₆₇	C₂²	1	2	yes	—
56	#427×C₆₇	C₂²	1	4	yes	complemented
57	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
58	C₂×#107×C₆₇	C₂²	1	4	yes	complemented
59	#423×C₆₇	C₂²	1	1	yes	characteristic
60	#418×C₆₇	C₂²	1	4	yes	—
61	C₂×#102×C₆₇	C₂²	1	2	yes	—
62	#408×C₆₇	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
64	C₂×#41×C₆₇	—	2	4	no	—
65	C₂×#41×C₆₇	C₂³	1	2	yes	—
66	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₆₇	—	2	2	no	—
68	C₂×D₈×C₆₇	—	2	4	no	—
69	C₂×D₈×C₆₇	C₂³	1	2	yes	—
70	C₂×D₈×C₆₇	C₂³	1	2	yes	—
71	#131×C₆₇	—	2	8	no	—
72	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
73	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
74	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
78	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
80	#127×C₆₇	—	2	4	no	—
81	#127×C₆₇	D₄	1	1	yes	characteristic
82	#126×C₆₇	—	2	8	no	—
83	#125×C₆₇	C₂³	1	4	yes	—
84	#124×C₆₇	—	2	4	no	—
85	#124×C₆₇	—	2	8	no	—
86	#124×C₆₇	D₄	1	2	yes	—
87	#123×C₆₇	—	2	8	no	—
88	#123×C₆₇	—	2	8	no	—
89	C₂²⋊Q₈×C₆₇	—	2	2	no	—
90	C₂²⋊Q₈×C₆₇	—	2	2	no	—
91	C₂²⋊Q₈×C₆₇	—	2	4	no	—
92	#121×C₆₇	—	2	4	no	—
93	#121×C₆₇	—	2	8	no	—
94	#121×C₆₇	D₄	1	2	yes	—
95	#121×C₆₇	D₄	1	2	yes	—
96	#121×C₆₇	C₂³	1	8	yes	—
97	#121×C₆₇	C₂³	1	8	yes	—
98	#120×C₆₇	—	2	8	no	—
99	C₄×Q₈×C₆₇	—	2	4	no	—
100	C₄×Q₈×C₆₇	—	2	2	no	—
101	C₄×Q₈×C₆₇	—	2	2	no	—
102	C₄×D₄×C₆₇	—	2	8	no	—
103	C₄×D₄×C₆₇	—	2	4	no	—
104	C₄×D₄×C₆₇	—	2	8	no	—
105	C₄×D₄×C₆₇	—	2	2	no	—
106	C₄×D₄×C₆₇	—	2	2	no	—
107	C₄×D₄×C₆₇	C₂³	1	4	yes	—
108	C₄×D₄×C₆₇	C₂³	1	4	yes	—
109	C₄×D₄×C₆₇	C₂³	1	2	yes	—
110	C₄×D₄×C₆₇	C₂³	1	2	yes	—
111	#117×C₆₇	—	2	4	no	—
112	#117×C₆₇	C₂³	1	8	yes	—
113	#117×C₆₇	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	#107×C₆₇	C₂³	1	2	yes	—
119	#107×C₆₇	C₂³	1	2	yes	—
120	#107×C₆₇	C₂³	1	8	yes	—
121	#107×C₆₇	C₂³	1	2	yes	—
122	#107×C₆₇	C₂³	1	2	yes	—
123	#107×C₆₇	C₂³	1	2	yes	—
124	#107×C₆₇	C₂³	1	2	yes	—
125	#106×C₆₇	C₂³	1	4	yes	—
126	#106×C₆₇	C₂³	1	8	yes	—
127	#106×C₆₇	C₂³	1	4	yes	—
128	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
129	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
130	#102×C₆₇	C₂³	1	8	yes	—
131	#102×C₆₇	C₂³	1	8	yes	—
132	#102×C₆₇	C₂³	1	4	yes	—
133	#102×C₆₇	C₂³	1	4	yes	—
134	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
135	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
136	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
140	C₂⁴×C₆₇	—	2	2	no	—
141	#41×C₆₇	—	2	4	no	—
142	#41×C₆₇	—	2	4	no	—
143	#41×C₆₇	—	8	8	no	—
144	C₂×Q₈×C₆₇	—	2	2	no	—
145	C₂×Q₈×C₆₇	—	2	2	no	—
146	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₆₇	—	2	8	no	—
148	C₂×D₄×C₆₇	—	2	8	no	—
149	C₂×D₄×C₆₇	—	2	2	no	—
150	C₂×D₄×C₆₇	—	2	4	no	—
151	C₂×D₄×C₆₇	—	2	4	no	—
152	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
153	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
154	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
155	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
156	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	8	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	8	4	no	—
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
166	D₈×C₆₇	—	2	4	no	—
167	D₈×C₆₇	—	2	4	no	—
168	M₁₆×C₆₇	—	2	4	no	—
169	M₁₆×C₆₇	—	2	8	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	—	2	8	no	—
172	C₂×C₈×C₆₇	#41	1	4	yes	complemented
173	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
174	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
175	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	2	8	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	2	4	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
187	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
189	C₂²⋊C₄×C₆₇	—	2	4	no	—
190	C₂²⋊C₄×C₆₇	—	2	8	no	—
191	C₂²⋊C₄×C₆₇	—	2	4	no	—
192	C₂²⋊C₄×C₆₇	—	2	4	no	—
193	C₂²⋊C₄×C₆₇	—	2	8	no	—
194	C₂²⋊C₄×C₆₇	—	2	8	no	—
195	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
196	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
197	C₄²×C₆₇	—	2	8	no	—
198	C₄²×C₆₇	—	2	4	no	—
199	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
202	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
203	C₂³×C₆₇	—	2	4	no	—
204	C₂³×C₆₇	—	4	4	no	—
205	C₂³×C₆₇	—	8	2	no	—
206	C₂³×C₆₇	—	8	2	no	—
207	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
208	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₆₇	—	2	4	no	—
210	Q₈×C₆₇	—	4	4	no	—
211	Q₈×C₆₇	—	8	2	no	—
212	D₄×C₆₇	—	2	8	no	—
213	D₄×C₆₇	—	2	4	no	—
214	D₄×C₆₇	—	2	4	no	—
215	D₄×C₆₇	—	2	4	no	—
216	D₄×C₆₇	—	2	4	no	—
217	D₄×C₆₇	—	4	2	no	—
218	D₄×C₆₇	—	4	4	no	—
219	D₄×C₆₇	—	8	8	no	—
220	D₄×C₆₇	—	8	8	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	4	1	no	—
228	C₂×C₄×C₆₇	—	4	8	no	—
229	C₂×C₄×C₆₇	—	8	4	no	—
230	C₂×C₄×C₆₇	—	8	2	no	—
231	C₂×C₄×C₆₇	—	8	8	no	—
232	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
233	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
239	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
240	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
241	C₈×C₆₇	—	2	4	no	—
242	C₈×C₆₇	—	2	1	no	—
243	C₈×C₆₇	—	2	1	no	—
244	C₈×C₆₇	C₂×#41	1	2	yes	—
245	C₈×C₆₇	C₂×#41	1	2	yes	—
246	C₂²×C₆₇	—	2	1	no	—
247	C₂²×C₆₇	—	2	1	no	—
248	C₂²×C₆₇	—	2	4	no	—
249	C₂²×C₆₇	—	2	4	no	—
250	C₂²×C₆₇	—	2	4	no	—
251	C₂²×C₆₇	—	4	4	no	—
252	C₂²×C₆₇	—	4	4	no	—
253	C₂²×C₆₇	—	4	4	no	—
254	C₂²×C₆₇	—	8	8	no	—
255	C₂²×C₆₇	#547	1	2	yes	—
256	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
257	C₄×C₆₇	—	2	1	no	—
258	C₄×C₆₇	—	2	1	no	—
259	C₄×C₆₇	—	2	4	no	—
260	C₄×C₆₇	—	2	1	no	—
261	C₄×C₆₇	—	4	4	no	—
262	C₄×C₆₇	—	4	4	no	—
263	C₄×C₆₇	—	4	4	no	—
264	C₄×C₆₇	#547	1	2	yes	—
265	C₄×C₆₇	#576	1	2	yes	—
266	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
267	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
268	#61284	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
269	C₂×C₆₇	—	2	2	no	—
270	C₂×C₆₇	—	4	4	no	—
271	C₂×C₆₇	—	8	4	no	—
272	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
273	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
274	C₂×C₆₇	#3584	1	1	yes	characteristic
275	#3506	C₂×C₆₇	1	2	yes	complemented
276	#3355	C₂×C₆₇	1	4	yes	complemented
277	#3354	C₂×C₆₇	1	4	yes	complemented
278	#3341	C₂×C₆₇	1	2	yes	complemented
279	#3339	C₂×C₆₇	1	4	yes	complemented
280	#3334	C₂×C₆₇	1	2	yes	complemented
281	#3160	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#3096	C₂×C₆₇	1	4	yes	complemented
283	#3077	C₂×C₆₇	1	2	yes	complemented
284	#3058	C₂×C₆₇	1	2	yes	complemented
285	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
286	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
287	#2913	C₂×C₆₇	1	2	yes	complemented
288	C₆₇	#61284	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
290	C₂×#131	C₂²×C₆₇	1	2	yes	—
291	#559	—	2	4	no	—
292	#556	—	2	2	no	—
293	#555	C₂²×C₆₇	1	4	yes	complemented
294	#553	C₂²×C₆₇	1	8	yes	complemented
295	#550	—	2	2	no	—
296	#547	C₂²×C₆₇	1	4	yes	complemented
297	#538	—	2	4	no	—
298	#535	C₂²×C₆₇	1	4	yes	—
299	#533	C₂²×C₆₇	1	2	yes	—
300	#532	C₂²×C₆₇	1	1	yes	characteristic
301	#525	—	2	4	no	—
302	#520	C₂²×C₆₇	1	4	yes	—
303	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
304	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
305	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
306	#499	C₂²×C₆₇	1	2	yes	complemented
307	#499	C₂²×C₆₇	1	2	yes	complemented
308	#498	C₂²×C₆₇	1	2	yes	complemented
309	#482	C₂²×C₆₇	1	8	yes	complemented
310	#482	C₂²×C₆₇	1	8	yes	complemented
311	#481	C₂²×C₆₇	1	8	yes	complemented
312	#480	C₂²×C₆₇	1	8	yes	complemented
313	#478	C₂²×C₆₇	1	8	yes	complemented
314	#476	C₂²×C₆₇	1	4	yes	complemented
315	#474	C₂²×C₆₇	1	4	yes	complemented
316	#469	C₂²×C₆₇	1	8	yes	—
317	#466	C₂²×C₆₇	1	4	yes	—
318	#466	C₂²×C₆₇	1	4	yes	—
319	C₄×D₈	C₂²×C₆₇	1	2	yes	—
320	C₄×D₈	C₂²×C₆₇	1	2	yes	—
321	C₄×D₈	C₂²×C₆₇	1	2	yes	—
322	C₄×D₈	C₂²×C₆₇	1	2	yes	—
323	#436	C₂²×C₆₇	1	2	yes	complemented
324	#436	C₂²×C₆₇	1	2	yes	complemented
325	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
326	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
327	#428	C₂²×C₆₇	1	4	yes	—
328	#428	C₂²×C₆₇	1	8	yes	complemented
329	#427	C₂²×C₆₇	1	2	yes	—
330	#427	C₂²×C₆₇	1	2	yes	—
331	#427	C₂²×C₆₇	1	4	yes	complemented
332	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×#107	C₂²×C₆₇	1	4	yes	complemented
334	#423	C₂²×C₆₇	1	1	yes	characteristic
335	#418	C₂²×C₆₇	1	4	yes	—
336	C₂×#102	C₂²×C₆₇	1	2	yes	—
337	#408	C₂²×C₆₇	1	2	yes	—
338	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×#41	—	2	4	no	—
340	C₂×#41	C₂³×C₆₇	1	2	yes	—
341	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
342	C₂³×C₄	—	2	2	no	—
343	C₂×D₈	—	2	4	no	—
344	C₂×D₈	C₂³×C₆₇	1	2	yes	—
345	C₂×D₈	C₂³×C₆₇	1	2	yes	—
346	#131	—	2	8	no	—
347	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
348	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
349	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
350	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
352	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
353	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
354	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
355	#127	—	2	4	no	—
356	#127	D₄×C₆₇	1	1	yes	characteristic
357	#126	—	2	8	no	—
358	#125	C₂³×C₆₇	1	4	yes	—
359	#124	—	2	4	no	—
360	#124	—	2	8	no	—
361	#124	D₄×C₆₇	1	2	yes	—
362	#123	—	2	8	no	—
363	#123	—	2	8	no	—
364	C₂²⋊Q₈	—	2	2	no	—
365	C₂²⋊Q₈	—	2	2	no	—
366	C₂²⋊Q₈	—	2	4	no	—
367	#121	—	2	4	no	—
368	#121	—	2	8	no	—
369	#121	D₄×C₆₇	1	2	yes	—
370	#121	D₄×C₆₇	1	2	yes	—
371	#121	C₂³×C₆₇	1	8	yes	—
372	#121	C₂³×C₆₇	1	8	yes	—
373	#120	—	2	8	no	—
374	C₄×Q₈	—	2	4	no	—
375	C₄×Q₈	—	2	2	no	—
376	C₄×Q₈	—	2	2	no	—
377	C₄×D₄	—	2	8	no	—
378	C₄×D₄	—	2	4	no	—
379	C₄×D₄	—	2	8	no	—
380	C₄×D₄	—	2	2	no	—
381	C₄×D₄	—	2	2	no	—
382	C₄×D₄	C₂³×C₆₇	1	4	yes	—
383	C₄×D₄	C₂³×C₆₇	1	4	yes	—
384	C₄×D₄	C₂³×C₆₇	1	2	yes	—
385	C₄×D₄	C₂³×C₆₇	1	2	yes	—
386	#117	—	2	4	no	—
387	#117	C₂³×C₆₇	1	4	yes	—
388	#117	C₂³×C₆₇	1	8	yes	—
389	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
390	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
391	C₂×C₂²⋊C₄	—	2	2	no	—
392	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
393	#107	C₂³×C₆₇	1	2	yes	—
394	#107	C₂³×C₆₇	1	2	yes	—
395	#107	C₂³×C₆₇	1	2	yes	—
396	#107	C₂³×C₆₇	1	8	yes	—
397	#107	C₂³×C₆₇	1	2	yes	—
398	#107	C₂³×C₆₇	1	2	yes	—
399	#107	C₂³×C₆₇	1	2	yes	—
400	#106	C₂³×C₆₇	1	4	yes	—
401	#106	C₂³×C₆₇	1	4	yes	—
402	#106	C₂³×C₆₇	1	8	yes	—
403	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
404	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
405	#102	C₂³×C₆₇	1	4	yes	—
406	#102	C₂³×C₆₇	1	4	yes	—
407	#102	C₂³×C₆₇	1	8	yes	—
408	#102	C₂³×C₆₇	1	8	yes	—
409	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
410	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
411	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
412	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
413	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
414	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
415	C₂⁴	—	2	2	no	—
416	#41	—	2	4	no	—
417	#41	—	2	4	no	—
418	#41	—	8	8	no	—
419	C₂×Q₈	—	2	2	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
428	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
429	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
430	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
431	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	8	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	8	4	no	—
437	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
439	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
441	D₈	—	2	4	no	—
442	D₈	—	2	4	no	—
443	M₁₆	—	2	4	no	—
444	M₁₆	—	2	8	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	8	no	—
447	C₂×C₈	#41×C₆₇	1	4	yes	complemented
448	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
449	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
450	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
451	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
452	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
460	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
461	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
462	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
463	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
471	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
472	C₄²	—	2	8	no	—
473	C₄²	—	2	4	no	—
474	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
475	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
476	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
477	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
478	C₂³	—	2	4	no	—
479	C₂³	—	4	4	no	—
480	C₂³	—	8	2	no	—
481	C₂³	—	8	2	no	—
482	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
483	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
484	Q₈	—	2	4	no	—
485	Q₈	—	4	4	no	—
486	Q₈	—	8	2	no	—
487	D₄	—	2	8	no	—
488	D₄	—	2	4	no	—
489	D₄	—	2	4	no	—
490	D₄	—	2	4	no	—
491	D₄	—	2	4	no	—
492	D₄	—	4	2	no	—
493	D₄	—	4	4	no	—
494	D₄	—	8	8	no	—
495	D₄	—	8	8	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	8	no	—
504	C₂×C₄	—	8	4	no	—
505	C₂×C₄	—	8	2	no	—
506	C₂×C₄	—	8	8	no	—
507	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
508	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
509	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
510	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
513	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
514	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
515	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
516	C₈	—	2	4	no	—
517	C₈	—	2	1	no	—
518	C₈	—	2	1	no	—
519	C₈	C₂×#41×C₆₇	1	2	yes	—
520	C₈	C₂×#41×C₆₇	1	2	yes	—
521	C₂²	—	2	1	no	—
522	C₂²	—	2	1	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	2	4	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	8	8	no	—
530	C₂²	#547×C₆₇	1	2	yes	—
531	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
532	C₄	—	2	1	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	4	no	—
536	C₄	—	4	4	no	—
537	C₄	—	4	4	no	—
538	C₄	—	4	4	no	—
539	C₄	#547×C₆₇	1	2	yes	—
540	C₄	#576×C₆₇	1	2	yes	—
541	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
542	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
543	C₂	—	2	2	no	—
544	C₂	—	4	4	no	—
545	C₂	—	8	4	no	—
546	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
547	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
548	C₂	C₆₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #532×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3506	C₂	semidirect
2. C₆₇×#3355	C₂	semidirect
3. C₆₇×#3354	C₂	semidirect
4. C₆₇×#3341	C₂	semidirect
5. C₆₇×#3339	C₂	semidirect
6. C₆₇×#3334	C₂	semidirect
7. C₆₇×#3160	C₂	semidirect
8. C₆₇×#3096	C₂	semidirect
9. C₆₇×#3077	C₂	semidirect
10. C₆₇×#3058	C₂	semidirect
11. C₂×C₄×D₈×C₆₇	C₂	semidirect
12. C₆₇×#2931	C₂	semidirect
13. C₆₇×#2913	C₂	semidirect
18. #555×C₆₇	C₂²	semidirect
19. #553×C₆₇	C₂²	semidirect
21. #547×C₆₇	C₂²	semidirect
29. #500×C₆₇	C₂²	semidirect
30. #500×C₆₇	C₂²	semidirect
31. #499×C₆₇	C₂²	semidirect
32. #499×C₆₇	C₂²	semidirect
33. #498×C₆₇	C₂²	semidirect
34. #482×C₆₇	C₂²	semidirect
35. #482×C₆₇	C₂²	semidirect
36. #481×C₆₇	C₂²	semidirect
37. #480×C₆₇	C₂²	semidirect
38. #478×C₆₇	C₂²	semidirect
39. #476×C₆₇	C₂²	semidirect
40. #474×C₆₇	C₂²	semidirect
48. #436×C₆₇	C₂²	semidirect
49. #436×C₆₇	C₂²	semidirect
50. C₈×D₄×C₆₇	C₂²	semidirect
51. C₈×D₄×C₆₇	C₂²	semidirect
53. #428×C₆₇	C₂²	semidirect
56. #427×C₆₇	C₂²	semidirect
58. C₂×#107×C₆₇	C₂²	semidirect
172. C₂×C₈×C₆₇	#41	semidirect
268. #61284	C₆₇	direct (N × H)
275. #3506	C₂×C₆₇	semidirect
276. #3355	C₂×C₆₇	semidirect
277. #3354	C₂×C₆₇	semidirect
278. #3341	C₂×C₆₇	semidirect
279. #3339	C₂×C₆₇	semidirect
280. #3334	C₂×C₆₇	semidirect
281. #3160	C₂×C₆₇	semidirect
282. #3096	C₂×C₆₇	semidirect
283. #3077	C₂×C₆₇	semidirect
284. #3058	C₂×C₆₇	semidirect
285. C₂×C₄×D₈	C₂×C₆₇	semidirect
286. #2931	C₂×C₆₇	semidirect
287. #2913	C₂×C₆₇	semidirect
288. C₆₇	#61284	direct (N × H)
293. #555	C₂²×C₆₇	semidirect
294. #553	C₂²×C₆₇	semidirect
296. #547	C₂²×C₆₇	semidirect
304. #500	C₂²×C₆₇	semidirect
305. #500	C₂²×C₆₇	semidirect
306. #499	C₂²×C₆₇	semidirect
307. #499	C₂²×C₆₇	semidirect
308. #498	C₂²×C₆₇	semidirect
309. #482	C₂²×C₆₇	semidirect
310. #482	C₂²×C₆₇	semidirect
311. #481	C₂²×C₆₇	semidirect
312. #480	C₂²×C₆₇	semidirect
313. #478	C₂²×C₆₇	semidirect
314. #476	C₂²×C₆₇	semidirect
315. #474	C₂²×C₆₇	semidirect
323. #436	C₂²×C₆₇	semidirect
324. #436	C₂²×C₆₇	semidirect
325. C₈×D₄	C₂²×C₆₇	semidirect
326. C₈×D₄	C₂²×C₆₇	semidirect
328. #428	C₂²×C₆₇	semidirect
331. #427	C₂²×C₆₇	semidirect
333. C₂×#107	C₂²×C₆₇	semidirect
447. C₂×C₈	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3506	2
C₆₇×#3355	2
C₆₇×#3354	2
C₆₇×#3341	2
C₆₇×#3339	2
C₆₇×#3334	2
C₆₇×#3160	2
C₆₇×#3096	2
C₆₇×#3077	2
C₆₇×#3058	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#2931	2
C₆₇×#2913	2
#61284	67

Subgroups — C67×#61284 #30740535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61284 #30740535