Subgroups — C₆₇×#61851 #30741102

Order 17152.

1704 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3548	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3388	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3385	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3381	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3368	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3368	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3362	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3241	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3235	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3228	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#3126	C₂	1	4	yes	maximal, complemented
13	C₆₇×#3112	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#3103	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2985	C₂	1	1	yes	maximal, characteristic, complemented
16	#562×C₆₇	—	2	2	no	—
17	Q₈²×C₆₇	—	2	1	no	—
18	Q₈²×C₆₇	C₂²	1	2	yes	—
19	#557×C₆₇	—	2	1	no	—
20	#557×C₆₇	—	2	2	no	—
21	#557×C₆₇	C₂²	1	2	yes	—
22	#557×C₆₇	C₂²	1	4	yes	—
23	#554×C₆₇	C₂²	1	2	yes	—
24	#554×C₆₇	C₂²	1	2	yes	—
25	#550×C₆₇	C₂²	1	2	yes	—
26	#550×C₆₇	C₂²	1	2	yes	—
27	#547×C₆₇	—	2	1	no	—
28	#547×C₆₇	—	2	1	no	—
29	#544×C₆₇	—	2	2	no	—
30	#544×C₆₇	C₂²	1	2	yes	—
31	#540×C₆₇	C₂²	1	4	yes	—
32	#537×C₆₇	—	2	2	no	—
33	#537×C₆₇	—	2	2	no	—
34	#533×C₆₇	—	2	2	no	—
35	#533×C₆₇	C₂²	1	1	yes	characteristic
36	#533×C₆₇	C₂²	1	1	yes	characteristic
37	#532×C₆₇	C₂²	1	1	yes	characteristic
38	#532×C₆₇	C₂²	1	1	yes	characteristic
39	#519×C₆₇	C₂²	1	2	yes	—
40	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
41	#501×C₆₇	C₂²	1	4	yes	—
42	#501×C₆₇	C₂²	1	2	yes	—
43	#500×C₆₇	C₂²	1	1	yes	characteristic
44	#499×C₆₇	C₂²	1	4	yes	—
45	#498×C₆₇	C₂²	1	1	yes	characteristic
46	#490×C₆₇	C₂²	1	2	yes	—
47	#490×C₆₇	C₂²	1	4	yes	—
48	#486×C₆₇	C₂²	1	1	yes	characteristic
49	#486×C₆₇	C₂²	1	4	yes	—
50	#486×C₆₇	C₂²	1	1	yes	characteristic
51	#482×C₆₇	C₂²	1	4	yes	—
52	#482×C₆₇	C₂²	1	4	yes	—
53	#481×C₆₇	C₂²	1	2	yes	—
54	#480×C₆₇	C₂²	1	2	yes	—
55	#479×C₆₇	C₂²	1	4	yes	—
56	#479×C₆₇	C₂²	1	4	yes	—
57	#478×C₆₇	C₂²	1	4	yes	—
58	#478×C₆₇	C₂²	1	4	yes	—
59	#478×C₆₇	C₂²	1	4	yes	—
60	#477×C₆₇	C₂²	1	2	yes	—
61	#476×C₆₇	C₂²	1	2	yes	—
62	#476×C₆₇	C₂²	1	4	yes	—
63	#475×C₆₇	C₂²	1	2	yes	—
64	#474×C₆₇	C₂²	1	2	yes	—
65	#474×C₆₇	C₂²	1	4	yes	—
66	#460×C₆₇	C₂²	1	2	yes	—
67	#460×C₆₇	C₂²	1	2	yes	—
68	#460×C₆₇	C₂²	1	4	yes	—
69	#460×C₆₇	C₂²	1	4	yes	—
70	#448×C₆₇	C₂²	1	2	yes	—
71	#448×C₆₇	C₂²	1	2	yes	—
72	#446×C₆₇	C₂²	1	2	yes	—
73	#446×C₆₇	C₂²	1	4	yes	—
74	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
75	#440×C₆₇	C₂²	1	4	yes	—
76	#440×C₆₇	C₂²	1	4	yes	—
77	#440×C₆₇	C₂²	1	4	yes	—
78	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
79	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
80	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
81	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
82	#433×C₆₇	C₂²	1	4	yes	—
83	#432×C₆₇	C₂²	1	1	yes	characteristic
84	#431×C₆₇	C₂²	1	1	yes	characteristic
85	C₂×#41×C₆₇	—	2	2	no	—
86	C₂×#41×C₆₇	—	2	2	no	—
87	C₂×#41×C₆₇	—	2	1	no	—
88	C₂×#41×C₆₇	—	2	1	no	—
89	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
91	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
92	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
93	C₄⋊Q₈×C₆₇	—	2	2	no	—
94	C₄⋊Q₈×C₆₇	—	2	4	no	—
95	C₄⋊Q₈×C₆₇	—	2	2	no	—
96	C₄⋊Q₈×C₆₇	—	2	4	no	—
97	C₄⋊Q₈×C₆₇	—	2	2	no	—
98	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
99	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
100	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
102	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
103	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
104	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
105	#127×C₆₇	C₂³	1	1	yes	characteristic
106	#127×C₆₇	C₂³	1	4	yes	—
107	#127×C₆₇	C₂³	1	1	yes	characteristic
108	#125×C₆₇	—	2	4	no	—
109	#125×C₆₇	—	2	4	no	—
110	#125×C₆₇	—	2	4	no	—
111	#125×C₆₇	—	2	2	no	—
112	#125×C₆₇	D₄	1	2	yes	—
113	#125×C₆₇	C₂³	1	4	yes	—
114	#125×C₆₇	C₂³	1	2	yes	—
115	#125×C₆₇	C₂³	1	4	yes	—
116	#124×C₆₇	—	2	1	no	—
117	#124×C₆₇	—	2	1	no	—
118	#123×C₆₇	—	2	4	no	—
119	#123×C₆₇	—	2	4	no	—
120	C₂²⋊Q₈×C₆₇	—	2	2	no	—
121	C₂²⋊Q₈×C₆₇	—	2	2	no	—
122	C₂²⋊Q₈×C₆₇	—	2	1	no	—
123	C₂²⋊Q₈×C₆₇	—	2	1	no	—
124	C₂²⋊Q₈×C₆₇	—	2	4	no	—
125	C₂²⋊Q₈×C₆₇	—	2	4	no	—
126	C₂²⋊Q₈×C₆₇	—	2	4	no	—
127	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
128	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
129	#121×C₆₇	—	2	1	no	—
130	#121×C₆₇	—	2	1	no	—
131	#121×C₆₇	—	2	2	no	—
132	#121×C₆₇	—	2	2	no	—
133	#121×C₆₇	—	2	4	no	—
134	#121×C₆₇	—	2	4	no	—
135	#121×C₆₇	—	2	4	no	—
136	#121×C₆₇	—	2	4	no	—
137	#121×C₆₇	C₂³	1	2	yes	—
138	#121×C₆₇	C₂³	1	2	yes	—
139	C₄×Q₈×C₆₇	—	2	2	no	—
140	C₄×Q₈×C₆₇	—	2	2	no	—
141	C₄×Q₈×C₆₇	—	2	2	no	—
142	C₄×Q₈×C₆₇	—	2	4	no	—
143	C₄×Q₈×C₆₇	—	2	2	no	—
144	C₄×Q₈×C₆₇	—	2	1	no	—
145	C₄×Q₈×C₆₇	—	2	1	no	—
146	C₄×Q₈×C₆₇	—	2	1	no	—
147	C₄×Q₈×C₆₇	—	2	1	no	—
148	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
149	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
150	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
151	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
152	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
153	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
154	C₄×D₄×C₆₇	—	2	1	no	—
155	C₄×D₄×C₆₇	—	2	1	no	—
156	C₄×D₄×C₆₇	C₂³	1	2	yes	—
157	C₄×D₄×C₆₇	C₂³	1	4	yes	—
158	C₄×D₄×C₆₇	C₂³	1	4	yes	—
159	C₄×D₄×C₆₇	C₂³	1	2	yes	—
160	#117×C₆₇	—	2	4	no	—
161	#117×C₆₇	C₂³	1	2	yes	—
162	#117×C₆₇	C₂³	1	2	yes	—
163	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
164	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
166	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
167	C₂×C₄²×C₆₇	—	2	1	no	—
168	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
169	#107×C₆₇	C₂³	1	4	yes	—
170	#107×C₆₇	C₂³	1	4	yes	—
171	#107×C₆₇	C₂³	1	4	yes	—
172	#106×C₆₇	C₂³	1	2	yes	—
173	#106×C₆₇	C₂³	1	2	yes	—
174	#106×C₆₇	C₂³	1	4	yes	—
175	#106×C₆₇	C₂³	1	4	yes	—
176	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
177	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
178	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
179	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
180	#103×C₆₇	C₂³	1	2	yes	—
181	#103×C₆₇	C₂³	1	2	yes	—
182	#103×C₆₇	C₂³	1	4	yes	—
183	#102×C₆₇	C₂³	1	2	yes	—
184	#102×C₆₇	C₂³	1	4	yes	—
185	#102×C₆₇	C₂³	1	4	yes	—
186	#102×C₆₇	C₂³	1	4	yes	—
187	#102×C₆₇	C₂³	1	4	yes	—
188	#102×C₆₇	C₂³	1	2	yes	—
189	#102×C₆₇	C₂³	1	4	yes	—
190	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
191	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
192	#97×C₆₇	C₂³	1	2	yes	—
193	#97×C₆₇	C₂³	1	4	yes	—
194	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
195	C₄×C₈×C₆₇	C₂³	1	4	yes	—
196	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
197	#41×C₆₇	—	8	2	no	—
198	#41×C₆₇	—	8	2	no	—
199	#41×C₆₇	—	8	4	no	—
200	#41×C₆₇	—	8	4	no	—
201	C₂×Q₈×C₆₇	—	2	4	no	—
202	C₂×Q₈×C₆₇	—	2	1	no	—
203	C₂×Q₈×C₆₇	—	2	1	no	—
204	C₂×Q₈×C₆₇	—	2	1	no	—
205	C₂×Q₈×C₆₇	—	2	4	no	—
206	C₂×Q₈×C₆₇	—	2	1	no	—
207	C₂×Q₈×C₆₇	—	2	1	no	—
208	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
209	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
210	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
211	C₂×D₄×C₆₇	—	2	2	no	—
212	C₂×D₄×C₆₇	—	2	2	no	—
213	C₂×D₄×C₆₇	—	2	1	no	—
214	C₂×D₄×C₆₇	—	2	1	no	—
215	C₂×D₄×C₆₇	—	2	4	no	—
216	C₂×D₄×C₆₇	—	2	4	no	—
217	C₂×D₄×C₆₇	—	2	1	no	—
218	C₂×D₄×C₆₇	—	2	1	no	—
219	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
220	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
221	C₂²×C₄×C₆₇	—	2	1	no	—
222	C₂²×C₄×C₆₇	—	2	1	no	—
223	C₂²×C₄×C₆₇	—	2	2	no	—
224	C₂²×C₄×C₆₇	—	2	2	no	—
225	C₂²×C₄×C₆₇	—	2	4	no	—
226	C₂²×C₄×C₆₇	—	2	4	no	—
227	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
228	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
229	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
230	SD₁₆×C₆₇	—	2	4	no	—
231	SD₁₆×C₆₇	—	2	4	no	—
232	SD₁₆×C₆₇	—	2	8	no	—
233	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
234	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
235	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
236	C₄⋊C₄×C₆₇	—	2	1	no	—
237	C₄⋊C₄×C₆₇	—	2	1	no	—
238	C₄⋊C₄×C₆₇	—	2	2	no	—
239	C₄⋊C₄×C₆₇	—	2	2	no	—
240	C₄⋊C₄×C₆₇	—	2	4	no	—
241	C₄⋊C₄×C₆₇	—	2	4	no	—
242	C₄⋊C₄×C₆₇	—	2	1	no	—
243	C₄⋊C₄×C₆₇	—	2	4	no	—
244	C₄⋊C₄×C₆₇	—	2	4	no	—
245	C₄⋊C₄×C₆₇	—	2	2	no	—
246	C₄⋊C₄×C₆₇	—	2	1	no	—
247	C₄⋊C₄×C₆₇	—	2	1	no	—
248	C₄⋊C₄×C₆₇	—	2	4	no	—
249	C₄⋊C₄×C₆₇	—	2	4	no	—
250	C₄⋊C₄×C₆₇	—	2	2	no	—
251	C₄⋊C₄×C₆₇	—	2	4	no	—
252	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
253	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
254	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
255	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
256	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
257	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
258	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
259	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
260	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
261	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
262	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
263	C₂²⋊C₄×C₆₇	—	2	2	no	—
264	C₂²⋊C₄×C₆₇	—	2	2	no	—
265	C₂²⋊C₄×C₆₇	—	2	2	no	—
266	C₂²⋊C₄×C₆₇	—	2	4	no	—
267	C₂²⋊C₄×C₆₇	—	2	4	no	—
268	C₂²⋊C₄×C₆₇	—	2	2	no	—
269	C₂²⋊C₄×C₆₇	—	2	2	no	—
270	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
271	C₄²×C₆₇	—	2	2	no	—
272	C₄²×C₆₇	—	2	2	no	—
273	C₄²×C₆₇	—	2	4	no	—
274	C₄²×C₆₇	—	2	4	no	—
275	C₄²×C₆₇	—	2	2	no	—
276	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
277	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
278	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
279	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
280	C₄²×C₆₇	C₂⁴	1	4	yes	—
281	C₂³×C₆₇	—	2	2	no	—
282	C₂³×C₆₇	—	2	2	no	—
283	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
284	Q₈×C₆₇	—	2	4	no	—
285	Q₈×C₆₇	—	2	4	no	—
286	Q₈×C₆₇	—	2	2	no	—
287	Q₈×C₆₇	—	4	2	no	—
288	Q₈×C₆₇	—	4	2	no	—
289	Q₈×C₆₇	—	8	1	no	—
290	Q₈×C₆₇	—	8	4	no	—
291	Q₈×C₆₇	—	8	1	no	—
292	Q₈×C₆₇	—	8	1	no	—
293	Q₈×C₆₇	2₋¹⁺⁴	1	4	yes	—
294	D₄×C₆₇	—	2	2	no	—
295	D₄×C₆₇	—	2	2	no	—
296	D₄×C₆₇	—	2	4	no	—
297	D₄×C₆₇	—	4	2	no	—
298	D₄×C₆₇	—	4	2	no	—
299	D₄×C₆₇	—	8	1	no	—
300	D₄×C₆₇	—	8	4	no	—
301	D₄×C₆₇	—	8	1	no	—
302	D₄×C₆₇	—	8	4	no	—
303	D₄×C₆₇	—	8	2	no	—
304	D₄×C₆₇	—	8	2	no	—
305	C₂×C₄×C₆₇	—	2	4	no	—
306	C₂×C₄×C₆₇	—	2	2	no	—
307	C₂×C₄×C₆₇	—	2	2	no	—
308	C₂×C₄×C₆₇	—	2	2	no	—
309	C₂×C₄×C₆₇	—	2	2	no	—
310	C₂×C₄×C₆₇	—	2	2	no	—
311	C₂×C₄×C₆₇	—	2	2	no	—
312	C₂×C₄×C₆₇	—	2	2	no	—
313	C₂×C₄×C₆₇	—	2	2	no	—
314	C₂×C₄×C₆₇	—	4	1	no	—
315	C₂×C₄×C₆₇	—	4	4	no	—
316	C₂×C₄×C₆₇	—	4	4	no	—
317	C₂×C₄×C₆₇	—	8	4	no	—
318	C₂×C₄×C₆₇	—	8	4	no	—
319	C₂×C₄×C₆₇	—	8	1	no	—
320	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
322	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
323	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
324	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
325	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
326	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
327	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
328	C₈×C₆₇	—	2	2	no	—
329	C₈×C₆₇	—	2	2	no	—
330	C₈×C₆₇	—	2	4	no	—
331	C₂²×C₆₇	—	2	1	no	—
332	C₂²×C₆₇	—	2	1	no	—
333	C₂²×C₆₇	—	2	1	no	—
334	C₂²×C₆₇	—	4	2	no	—
335	C₂²×C₆₇	—	4	2	no	—
336	C₂²×C₆₇	—	4	2	no	—
337	C₂²×C₆₇	—	4	2	no	—
338	C₂²×C₆₇	—	4	2	no	—
339	C₂²×C₆₇	—	4	2	no	—
340	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
341	C₄×C₆₇	—	2	1	no	—
342	C₄×C₆₇	—	2	1	no	—
343	C₄×C₆₇	—	2	4	no	—
344	C₄×C₆₇	—	2	4	no	—
345	C₄×C₆₇	—	2	1	no	—
346	C₄×C₆₇	—	2	1	no	—
347	C₄×C₆₇	—	2	4	no	—
348	C₄×C₆₇	—	4	2	no	—
349	C₄×C₆₇	—	4	2	no	—
350	C₄×C₆₇	—	4	2	no	—
351	C₄×C₆₇	—	4	2	no	—
352	C₄×C₆₇	—	4	2	no	—
353	C₄×C₆₇	#577	1	2	yes	—
354	C₄×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
355	#61851	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
356	C₂×C₆₇	—	4	1	no	—
357	C₂×C₆₇	—	8	2	no	—
358	C₂×C₆₇	—	8	2	no	—
359	C₂×C₆₇	#3486	1	1	yes	characteristic
360	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
361	C₂×C₆₇	#3585	1	1	yes	characteristic
362	#3548	C₂×C₆₇	1	1	yes	characteristic
363	#3502	C₂×C₆₇	1	1	yes	characteristic, complemented
364	#3388	C₂×C₆₇	1	4	yes	complemented
365	#3385	C₂×C₆₇	1	2	yes	complemented
366	#3381	C₂×C₆₇	1	2	yes	complemented
367	#3368	C₂×C₆₇	1	2	yes	complemented
368	#3368	C₂×C₆₇	1	4	yes	complemented
369	#3362	C₂×C₆₇	1	2	yes	complemented
370	#3241	C₂×C₆₇	1	4	yes	complemented
371	#3235	C₂×C₆₇	1	1	yes	characteristic, complemented
372	#3228	C₂×C₆₇	1	1	yes	characteristic, complemented
373	#3126	C₂×C₆₇	1	4	yes	complemented
374	#3112	C₂×C₆₇	1	1	yes	characteristic, complemented
375	#3103	C₂×C₆₇	1	1	yes	characteristic, complemented
376	#2985	C₂×C₆₇	1	1	yes	characteristic, complemented
377	C₆₇	#61851	1	1	yes	characteristic, direct product factor, Sylow
378	#562	—	2	2	no	—
379	Q₈²	—	2	1	no	—
380	Q₈²	C₂²×C₆₇	1	2	yes	—
381	#557	—	2	1	no	—
382	#557	—	2	2	no	—
383	#557	C₂²×C₆₇	1	2	yes	—
384	#557	C₂²×C₆₇	1	4	yes	—
385	#554	C₂²×C₆₇	1	2	yes	—
386	#554	C₂²×C₆₇	1	2	yes	—
387	#550	C₂²×C₆₇	1	2	yes	—
388	#550	C₂²×C₆₇	1	2	yes	—
389	#547	—	2	1	no	—
390	#547	—	2	1	no	—
391	#544	—	2	2	no	—
392	#544	C₂²×C₆₇	1	2	yes	—
393	#540	C₂²×C₆₇	1	4	yes	—
394	#537	—	2	2	no	—
395	#537	—	2	2	no	—
396	#533	—	2	2	no	—
397	#533	C₂²×C₆₇	1	1	yes	characteristic
398	#533	C₂²×C₆₇	1	1	yes	characteristic
399	#532	C₂²×C₆₇	1	1	yes	characteristic
400	#532	C₂²×C₆₇	1	1	yes	characteristic
401	#519	C₂²×C₆₇	1	2	yes	—
402	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
403	#501	C₂²×C₆₇	1	4	yes	—
404	#501	C₂²×C₆₇	1	2	yes	—
405	#500	C₂²×C₆₇	1	1	yes	characteristic
406	#499	C₂²×C₆₇	1	4	yes	—
407	#498	C₂²×C₆₇	1	1	yes	characteristic
408	#490	C₂²×C₆₇	1	2	yes	—
409	#490	C₂²×C₆₇	1	4	yes	—
410	#486	C₂²×C₆₇	1	1	yes	characteristic
411	#486	C₂²×C₆₇	1	4	yes	—
412	#486	C₂²×C₆₇	1	1	yes	characteristic
413	#482	C₂²×C₆₇	1	4	yes	—
414	#482	C₂²×C₆₇	1	4	yes	—
415	#481	C₂²×C₆₇	1	2	yes	—
416	#480	C₂²×C₆₇	1	2	yes	—
417	#479	C₂²×C₆₇	1	4	yes	—
418	#479	C₂²×C₆₇	1	4	yes	—
419	#478	C₂²×C₆₇	1	4	yes	—
420	#478	C₂²×C₆₇	1	4	yes	—
421	#478	C₂²×C₆₇	1	4	yes	—
422	#477	C₂²×C₆₇	1	2	yes	—
423	#476	C₂²×C₆₇	1	2	yes	—
424	#476	C₂²×C₆₇	1	4	yes	—
425	#475	C₂²×C₆₇	1	2	yes	—
426	#474	C₂²×C₆₇	1	2	yes	—
427	#474	C₂²×C₆₇	1	4	yes	—
428	#460	C₂²×C₆₇	1	2	yes	—
429	#460	C₂²×C₆₇	1	2	yes	—
430	#460	C₂²×C₆₇	1	4	yes	—
431	#460	C₂²×C₆₇	1	4	yes	—
432	#448	C₂²×C₆₇	1	2	yes	—
433	#448	C₂²×C₆₇	1	2	yes	—
434	#446	C₂²×C₆₇	1	2	yes	—
435	#446	C₂²×C₆₇	1	4	yes	—
436	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
437	#440	C₂²×C₆₇	1	4	yes	—
438	#440	C₂²×C₆₇	1	4	yes	—
439	#440	C₂²×C₆₇	1	4	yes	—
440	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
441	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
442	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	—
443	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	—
444	#433	C₂²×C₆₇	1	4	yes	—
445	#432	C₂²×C₆₇	1	1	yes	characteristic
446	#431	C₂²×C₆₇	1	1	yes	characteristic
447	C₂×#41	—	2	2	no	—
448	C₂×#41	—	2	2	no	—
449	C₂×#41	—	2	1	no	—
450	C₂×#41	—	2	1	no	—
451	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
452	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
453	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
454	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
455	C₄⋊Q₈	—	2	2	no	—
456	C₄⋊Q₈	—	2	4	no	—
457	C₄⋊Q₈	—	2	2	no	—
458	C₄⋊Q₈	—	2	4	no	—
459	C₄⋊Q₈	—	2	2	no	—
460	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
461	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
462	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
463	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
464	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
465	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
466	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
467	#127	C₂³×C₆₇	1	1	yes	characteristic
468	#127	C₂³×C₆₇	1	1	yes	characteristic
469	#127	C₂³×C₆₇	1	4	yes	—
470	#125	—	2	4	no	—
471	#125	—	2	4	no	—
472	#125	—	2	4	no	—
473	#125	—	2	2	no	—
474	#125	D₄×C₆₇	1	2	yes	—
475	#125	C₂³×C₆₇	1	2	yes	—
476	#125	C₂³×C₆₇	1	4	yes	—
477	#125	C₂³×C₆₇	1	4	yes	—
478	#124	—	2	1	no	—
479	#124	—	2	1	no	—
480	#123	—	2	4	no	—
481	#123	—	2	4	no	—
482	C₂²⋊Q₈	—	2	2	no	—
483	C₂²⋊Q₈	—	2	2	no	—
484	C₂²⋊Q₈	—	2	1	no	—
485	C₂²⋊Q₈	—	2	1	no	—
486	C₂²⋊Q₈	—	2	4	no	—
487	C₂²⋊Q₈	—	2	4	no	—
488	C₂²⋊Q₈	—	2	4	no	—
489	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
490	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
491	#121	—	2	1	no	—
492	#121	—	2	1	no	—
493	#121	—	2	2	no	—
494	#121	—	2	2	no	—
495	#121	—	2	4	no	—
496	#121	—	2	4	no	—
497	#121	—	2	4	no	—
498	#121	—	2	4	no	—
499	#121	C₂³×C₆₇	1	2	yes	—
500	#121	C₂³×C₆₇	1	2	yes	—
501	C₄×Q₈	—	2	2	no	—
502	C₄×Q₈	—	2	2	no	—
503	C₄×Q₈	—	2	2	no	—
504	C₄×Q₈	—	2	4	no	—
505	C₄×Q₈	—	2	2	no	—
506	C₄×Q₈	—	2	1	no	—
507	C₄×Q₈	—	2	1	no	—
508	C₄×Q₈	—	2	1	no	—
509	C₄×Q₈	—	2	1	no	—
510	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
511	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
512	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
513	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
514	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
515	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
516	C₄×D₄	—	2	1	no	—
517	C₄×D₄	—	2	1	no	—
518	C₄×D₄	C₂³×C₆₇	1	2	yes	—
519	C₄×D₄	C₂³×C₆₇	1	2	yes	—
520	C₄×D₄	C₂³×C₆₇	1	4	yes	—
521	C₄×D₄	C₂³×C₆₇	1	4	yes	—
522	#117	—	2	4	no	—
523	#117	C₂³×C₆₇	1	2	yes	—
524	#117	C₂³×C₆₇	1	2	yes	—
525	C₂×C₄⋊C₄	—	2	1	no	—
526	C₂×C₄⋊C₄	—	2	2	no	—
527	C₂×C₄⋊C₄	—	2	2	no	—
528	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
529	C₂×C₄²	—	2	1	no	—
530	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
531	#107	C₂³×C₆₇	1	4	yes	—
532	#107	C₂³×C₆₇	1	4	yes	—
533	#107	C₂³×C₆₇	1	4	yes	—
534	#106	C₂³×C₆₇	1	2	yes	—
535	#106	C₂³×C₆₇	1	2	yes	—
536	#106	C₂³×C₆₇	1	4	yes	—
537	#106	C₂³×C₆₇	1	4	yes	—
538	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
539	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
540	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
541	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
542	#103	C₂³×C₆₇	1	2	yes	—
543	#103	C₂³×C₆₇	1	2	yes	—
544	#103	C₂³×C₆₇	1	4	yes	—
545	#102	C₂³×C₆₇	1	2	yes	—
546	#102	C₂³×C₆₇	1	4	yes	—
547	#102	C₂³×C₆₇	1	4	yes	—
548	#102	C₂³×C₆₇	1	2	yes	—
549	#102	C₂³×C₆₇	1	4	yes	—
550	#102	C₂³×C₆₇	1	4	yes	—
551	#102	C₂³×C₆₇	1	4	yes	—
552	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
553	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
554	#97	C₂³×C₆₇	1	2	yes	—
555	#97	C₂³×C₆₇	1	4	yes	—
556	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
557	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
558	C₄×C₈	C₂³×C₆₇	1	4	yes	—
559	#41	—	8	2	no	—
560	#41	—	8	2	no	—
561	#41	—	8	4	no	—
562	#41	—	8	4	no	—
563	C₂×Q₈	—	2	4	no	—
564	C₂×Q₈	—	2	1	no	—
565	C₂×Q₈	—	2	1	no	—
566	C₂×Q₈	—	2	1	no	—
567	C₂×Q₈	—	2	4	no	—
568	C₂×Q₈	—	2	1	no	—
569	C₂×Q₈	—	2	1	no	—
570	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
571	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
572	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
573	C₂×D₄	—	2	2	no	—
574	C₂×D₄	—	2	2	no	—
575	C₂×D₄	—	2	1	no	—
576	C₂×D₄	—	2	1	no	—
577	C₂×D₄	—	2	4	no	—
578	C₂×D₄	—	2	4	no	—
579	C₂×D₄	—	2	1	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
582	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	—	2	1	no	—
585	C₂²×C₄	—	2	2	no	—
586	C₂²×C₄	—	2	2	no	—
587	C₂²×C₄	—	2	4	no	—
588	C₂²×C₄	—	2	4	no	—
589	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
590	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
591	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
592	SD₁₆	—	2	4	no	—
593	SD₁₆	—	2	4	no	—
594	SD₁₆	—	2	8	no	—
595	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
596	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
597	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
598	C₄⋊C₄	—	2	1	no	—
599	C₄⋊C₄	—	2	1	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	4	no	—
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	1	no	—
605	C₄⋊C₄	—	2	4	no	—
606	C₄⋊C₄	—	2	4	no	—
607	C₄⋊C₄	—	2	2	no	—
608	C₄⋊C₄	—	2	1	no	—
609	C₄⋊C₄	—	2	1	no	—
610	C₄⋊C₄	—	2	4	no	—
611	C₄⋊C₄	—	2	4	no	—
612	C₄⋊C₄	—	2	2	no	—
613	C₄⋊C₄	—	2	4	no	—
614	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
615	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
616	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
617	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
618	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
619	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
620	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
621	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
622	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
623	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
624	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
625	C₂²⋊C₄	—	2	2	no	—
626	C₂²⋊C₄	—	2	2	no	—
627	C₂²⋊C₄	—	2	2	no	—
628	C₂²⋊C₄	—	2	4	no	—
629	C₂²⋊C₄	—	2	4	no	—
630	C₂²⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	2	2	no	—
632	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
633	C₄²	—	2	2	no	—
634	C₄²	—	2	2	no	—
635	C₄²	—	2	4	no	—
636	C₄²	—	2	4	no	—
637	C₄²	—	2	2	no	—
638	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
639	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
640	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
641	C₄²	C₂⁴×C₆₇	1	4	yes	—
642	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
643	C₂³	—	2	2	no	—
644	C₂³	—	2	2	no	—
645	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
646	Q₈	—	2	4	no	—
647	Q₈	—	2	4	no	—
648	Q₈	—	2	2	no	—
649	Q₈	—	4	2	no	—
650	Q₈	—	4	2	no	—
651	Q₈	—	8	1	no	—
652	Q₈	—	8	1	no	—
653	Q₈	—	8	1	no	—
654	Q₈	—	8	4	no	—
655	Q₈	2₋¹⁺⁴×C₆₇	1	4	yes	—
656	D₄	—	2	2	no	—
657	D₄	—	2	2	no	—
658	D₄	—	2	4	no	—
659	D₄	—	4	2	no	—
660	D₄	—	4	2	no	—
661	D₄	—	8	1	no	—
662	D₄	—	8	4	no	—
663	D₄	—	8	1	no	—
664	D₄	—	8	4	no	—
665	D₄	—	8	2	no	—
666	D₄	—	8	2	no	—
667	C₂×C₄	—	2	4	no	—
668	C₂×C₄	—	2	2	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	2	2	no	—
674	C₂×C₄	—	2	2	no	—
675	C₂×C₄	—	2	2	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	4	no	—
678	C₂×C₄	—	4	4	no	—
679	C₂×C₄	—	8	4	no	—
680	C₂×C₄	—	8	4	no	—
681	C₂×C₄	—	8	1	no	—
682	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
683	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
684	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
685	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
686	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
687	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
688	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
689	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
690	C₈	—	2	2	no	—
691	C₈	—	2	2	no	—
692	C₈	—	2	4	no	—
693	C₂²	—	2	1	no	—
694	C₂²	—	2	1	no	—
695	C₂²	—	2	1	no	—
696	C₂²	—	4	2	no	—
697	C₂²	—	4	2	no	—
698	C₂²	—	4	2	no	—
699	C₂²	—	4	2	no	—
700	C₂²	—	4	2	no	—
701	C₂²	—	4	2	no	—
702	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
703	C₄	—	2	1	no	—
704	C₄	—	2	4	no	—
705	C₄	—	2	1	no	—
706	C₄	—	2	1	no	—
707	C₄	—	2	4	no	—
708	C₄	—	2	4	no	—
709	C₄	—	2	1	no	—
710	C₄	—	4	2	no	—
711	C₄	—	4	2	no	—
712	C₄	—	4	2	no	—
713	C₄	—	4	2	no	—
714	C₄	—	4	2	no	—
715	C₄	#577×C₆₇	1	2	yes	—
716	C₄	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
717	C₂	—	4	1	no	—
718	C₂	—	8	2	no	—
719	C₂	—	8	2	no	—
720	C₂	C₆₇×#3486	1	1	yes	characteristic
721	C₂	C₂×#577×C₆₇	1	1	yes	characteristic
722	C₂	C₆₇×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3548	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #533×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3502	C₂	semidirect
3. C₆₇×#3388	C₂	semidirect
4. C₆₇×#3385	C₂	semidirect
5. C₆₇×#3381	C₂	semidirect
6. C₆₇×#3368	C₂	semidirect
7. C₆₇×#3368	C₂	semidirect
8. C₆₇×#3362	C₂	semidirect
9. C₆₇×#3241	C₂	semidirect
10. C₆₇×#3235	C₂	semidirect
11. C₆₇×#3228	C₂	semidirect
12. C₆₇×#3126	C₂	semidirect
13. C₆₇×#3112	C₂	semidirect
14. C₆₇×#3103	C₂	semidirect
15. C₆₇×#2985	C₂	semidirect
355. #61851	C₆₇	direct (N × H)
363. #3502	C₂×C₆₇	semidirect
364. #3388	C₂×C₆₇	semidirect
365. #3385	C₂×C₆₇	semidirect
366. #3381	C₂×C₆₇	semidirect
367. #3368	C₂×C₆₇	semidirect
368. #3368	C₂×C₆₇	semidirect
369. #3362	C₂×C₆₇	semidirect
370. #3241	C₂×C₆₇	semidirect
371. #3235	C₂×C₆₇	semidirect
372. #3228	C₂×C₆₇	semidirect
373. #3126	C₂×C₆₇	semidirect
374. #3112	C₂×C₆₇	semidirect
375. #3103	C₂×C₆₇	semidirect
376. #2985	C₂×C₆₇	semidirect
377. C₆₇	#61851	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3548	2
C₆₇×#3502	2
C₆₇×#3388	2
C₆₇×#3385	2
C₆₇×#3381	2
C₆₇×#3368	2
C₆₇×#3362	2
C₆₇×#3241	2
C₆₇×#3235	2
C₆₇×#3228	2
C₆₇×#3126	2
C₆₇×#3112	2
C₆₇×#3103	2
C₆₇×#2985	2
#61851	67

Subgroups — C67×#61851 #30741102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61851 #30741102