Subgroups — C₆₇×#61335 #30740586

Order 17152.

1730 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3507	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3377	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3374	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3371	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3369	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3137	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3135	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#3084	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3078	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3044	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3037	C₂	1	1	yes	maximal, characteristic
13	C₆₇×#2961	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2947	C₂	1	2	yes	maximal, complemented
15	C₂×#441×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137×C₆₇	C₂²	1	2	yes	—
17	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	#559×C₆₇	C₂²	1	2	yes	—
19	Q₈²×C₆₇	—	2	1	no	—
20	Q₈²×C₆₇	C₂²	1	2	yes	—
21	#557×C₆₇	—	2	1	no	—
22	#557×C₆₇	—	2	2	no	—
23	#557×C₆₇	C₂²	1	2	yes	—
24	#557×C₆₇	C₂²	1	4	yes	—
25	#557×C₆₇	C₂²	1	2	yes	—
26	#556×C₆₇	—	2	1	no	—
27	#556×C₆₇	C₂²	1	4	yes	—
28	#554×C₆₇	—	2	1	no	—
29	#552×C₆₇	—	2	2	no	—
30	#544×C₆₇	—	2	2	no	—
31	#544×C₆₇	C₂²	1	2	yes	—
32	#539×C₆₇	—	2	2	no	—
33	#536×C₆₇	C₂²	1	2	yes	—
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	#529×C₆₇	—	2	2	no	—
39	#520×C₆₇	C₂²	1	2	yes	—
40	#519×C₆₇	C₂²	1	2	yes	—
41	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
43	#491×C₆₇	C₂²	1	2	yes	—
44	#491×C₆₇	C₂²	1	4	yes	—
45	#490×C₆₇	C₂²	1	2	yes	—
46	#489×C₆₇	C₂²	1	2	yes	—
47	#489×C₆₇	C₂²	1	4	yes	—
48	#489×C₆₇	C₂²	1	2	yes	—
49	#488×C₆₇	C₂²	1	4	yes	—
50	#487×C₆₇	C₂²	1	4	yes	—
51	#479×C₆₇	C₂²	1	4	yes	—
52	#479×C₆₇	C₂²	1	4	yes	—
53	#478×C₆₇	C₂²	1	4	yes	—
54	#477×C₆₇	C₂²	1	2	yes	—
55	#476×C₆₇	C₂²	1	4	yes	—
56	#475×C₆₇	C₂²	1	2	yes	—
57	#475×C₆₇	C₂²	1	4	yes	—
58	#464×C₆₇	C₂²	1	4	yes	—
59	#464×C₆₇	C₂²	1	4	yes	—
60	#464×C₆₇	C₂²	1	4	yes	—
61	#462×C₆₇	C₂²	1	2	yes	—
62	#462×C₆₇	C₂²	1	4	yes	—
63	#462×C₆₇	C₂²	1	4	yes	—
64	#460×C₆₇	C₂²	1	2	yes	—
65	#446×C₆₇	C₂²	1	4	yes	—
66	#446×C₆₇	C₂²	1	4	yes	—
67	#441×C₆₇	C₂²	1	4	yes	—
68	#441×C₆₇	C₂²	1	4	yes	—
69	#441×C₆₇	C₂²	1	4	yes	—
70	#440×C₆₇	C₂²	1	4	yes	—
71	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	—
72	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
73	#428×C₆₇	C₂²	1	2	yes	—
74	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
75	#424×C₆₇	C₂²	1	2	yes	—
76	#423×C₆₇	C₂²	1	1	yes	characteristic
77	#419×C₆₇	C₂²	1	2	yes	—
78	#417×C₆₇	C₂²	1	2	yes	—
79	#416×C₆₇	C₂²	1	2	yes	—
80	#416×C₆₇	C₂²	1	2	yes	—
81	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
82	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
83	#405×C₆₇	C₂²	1	2	yes	—
84	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
85	C₂×#41×C₆₇	C₂³	1	2	yes	—
86	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
88	#137×C₆₇	—	2	8	no	—
89	C₂×Q₁₆×C₆₇	—	2	4	no	—
90	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
91	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
92	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
93	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
94	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₄⋊Q₈×C₆₇	—	2	2	no	—
97	C₄⋊Q₈×C₆₇	—	2	4	no	—
98	C₄⋊Q₈×C₆₇	—	2	2	no	—
99	C₄⋊Q₈×C₆₇	—	2	4	no	—
100	C₄⋊Q₈×C₆₇	—	2	2	no	—
101	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
102	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
103	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
104	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
105	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
106	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
107	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
108	#127×C₆₇	C₂³	1	1	yes	characteristic
109	#126×C₆₇	—	2	4	no	—
110	#126×C₆₇	—	2	4	no	—
111	#125×C₆₇	—	2	2	no	—
112	#125×C₆₇	—	2	4	no	—
113	#125×C₆₇	—	2	4	no	—
114	#125×C₆₇	—	2	4	no	—
115	#125×C₆₇	—	2	4	no	—
116	#125×C₆₇	—	2	2	no	—
117	#125×C₆₇	D₄	1	2	yes	—
118	#125×C₆₇	C₂³	1	4	yes	—
119	#125×C₆₇	C₂³	1	2	yes	—
120	#124×C₆₇	—	2	2	no	—
121	#124×C₆₇	C₂³	1	4	yes	—
122	#124×C₆₇	C₂³	1	2	yes	—
123	#123×C₆₇	—	2	4	no	—
124	#123×C₆₇	—	2	4	no	—
125	C₂²⋊Q₈×C₆₇	—	2	4	no	—
126	C₂²⋊Q₈×C₆₇	—	2	1	no	—
127	C₂²⋊Q₈×C₆₇	—	2	1	no	—
128	C₂²⋊Q₈×C₆₇	—	2	2	no	—
129	C₂²⋊Q₈×C₆₇	—	2	2	no	—
130	C₂²⋊Q₈×C₆₇	—	2	4	no	—
131	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
132	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
133	#121×C₆₇	—	2	1	no	—
134	#121×C₆₇	—	2	1	no	—
135	C₄×Q₈×C₆₇	—	2	2	no	—
136	C₄×Q₈×C₆₇	—	2	2	no	—
137	C₄×Q₈×C₆₇	—	2	2	no	—
138	C₄×Q₈×C₆₇	—	2	2	no	—
139	C₄×Q₈×C₆₇	—	2	4	no	—
140	C₄×Q₈×C₆₇	—	2	2	no	—
141	C₄×Q₈×C₆₇	—	2	1	no	—
142	C₄×Q₈×C₆₇	—	2	1	no	—
143	C₄×Q₈×C₆₇	—	2	1	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	C₂³	1	2	yes	—
148	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
149	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
150	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
151	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
152	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
153	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
154	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
155	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
156	C₄×D₄×C₆₇	—	2	2	no	—
157	C₄×D₄×C₆₇	—	2	4	no	—
158	C₄×D₄×C₆₇	—	2	1	no	—
159	C₄×D₄×C₆₇	—	2	1	no	—
160	C₄×D₄×C₆₇	C₂³	1	4	yes	—
161	#117×C₆₇	—	2	2	no	—
162	#117×C₆₇	—	2	4	no	—
163	#117×C₆₇	—	2	4	no	—
164	#117×C₆₇	C₂³	1	2	yes	—
165	#117×C₆₇	C₂³	1	2	yes	—
166	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
167	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
168	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
170	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
171	C₂×C₄²×C₆₇	—	2	1	no	—
172	C₂×C₄²×C₆₇	—	2	1	no	—
173	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
174	#107×C₆₇	C₂³	1	4	yes	—
175	#106×C₆₇	C₂³	1	4	yes	—
176	#106×C₆₇	C₂³	1	4	yes	—
177	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
178	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
179	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
180	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
181	#103×C₆₇	C₂³	1	2	yes	—
182	#103×C₆₇	C₂³	1	2	yes	—
183	#103×C₆₇	C₂³	1	4	yes	—
184	#103×C₆₇	C₂³	1	4	yes	—
185	#103×C₆₇	C₂³	1	4	yes	—
186	#103×C₆₇	C₂³	1	2	yes	—
187	#103×C₆₇	C₂³	1	4	yes	—
188	#103×C₆₇	C₂³	1	2	yes	—
189	#103×C₆₇	C₂³	1	4	yes	—
190	#102×C₆₇	C₂³	1	4	yes	—
191	#102×C₆₇	C₂³	1	2	yes	—
192	#102×C₆₇	C₂³	1	2	yes	—
193	#102×C₆₇	C₂³	1	2	yes	—
194	#102×C₆₇	C₂³	1	2	yes	—
195	#97×C₆₇	C₂³	1	2	yes	—
196	#97×C₆₇	C₂³	1	4	yes	—
197	#97×C₆₇	C₂³	1	2	yes	—
198	C₄×C₈×C₆₇	C₂³	1	4	yes	—
199	#41×C₆₇	—	2	4	no	—
200	#41×C₆₇	—	2	4	no	—
201	C₂×Q₈×C₆₇	—	2	1	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	2	no	—
207	C₂×Q₈×C₆₇	—	2	2	no	—
208	C₂×Q₈×C₆₇	—	2	2	no	—
209	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
210	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
211	C₂×Q₈×C₆₇	#41	1	4	yes	—
212	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
213	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
214	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
215	C₂×D₄×C₆₇	—	2	2	no	—
216	C₂×D₄×C₆₇	—	2	1	no	—
217	C₂×D₄×C₆₇	—	2	1	no	—
218	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
219	C₂²×C₄×C₆₇	—	2	1	no	—
220	C₂²×C₄×C₆₇	—	2	1	no	—
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	4	no	—
225	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
226	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
227	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
228	Q₁₆×C₆₇	—	2	8	no	—
229	Q₁₆×C₆₇	—	2	8	no	—
230	SD₁₆×C₆₇	—	2	8	no	—
231	M₁₆×C₆₇	—	2	4	no	—
232	C₂×C₈×C₆₇	—	2	2	no	—
233	C₂×C₈×C₆₇	—	2	2	no	—
234	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
235	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
236	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
237	C₄⋊C₄×C₆₇	—	2	1	no	—
238	C₄⋊C₄×C₆₇	—	2	4	no	—
239	C₄⋊C₄×C₆₇	—	2	2	no	—
240	C₄⋊C₄×C₆₇	—	2	2	no	—
241	C₄⋊C₄×C₆₇	—	2	2	no	—
242	C₄⋊C₄×C₆₇	—	2	2	no	—
243	C₄⋊C₄×C₆₇	—	2	1	no	—
244	C₄⋊C₄×C₆₇	—	2	4	no	—
245	C₄⋊C₄×C₆₇	—	2	4	no	—
246	C₄⋊C₄×C₆₇	—	2	4	no	—
247	C₄⋊C₄×C₆₇	—	2	1	no	—
248	C₄⋊C₄×C₆₇	—	2	1	no	—
249	C₄⋊C₄×C₆₇	—	2	2	no	—
250	C₄⋊C₄×C₆₇	—	2	1	no	—
251	C₄⋊C₄×C₆₇	—	2	4	no	—
252	C₄⋊C₄×C₆₇	—	2	1	no	—
253	C₄⋊C₄×C₆₇	—	2	4	no	—
254	C₄⋊C₄×C₆₇	—	2	4	no	—
255	C₄⋊C₄×C₆₇	—	2	2	no	—
256	C₄⋊C₄×C₆₇	—	2	4	no	—
257	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
258	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
259	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
260	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
261	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
262	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
263	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
264	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
265	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
266	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
267	C₂²⋊C₄×C₆₇	—	2	2	no	—
268	C₂²⋊C₄×C₆₇	—	2	2	no	—
269	C₂²⋊C₄×C₆₇	—	2	4	no	—
270	C₂²⋊C₄×C₆₇	—	2	4	no	—
271	C₂²⋊C₄×C₆₇	—	2	2	no	—
272	C₂²⋊C₄×C₆₇	—	2	4	no	—
273	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
274	C₄²×C₆₇	—	2	2	no	—
275	C₄²×C₆₇	—	2	2	no	—
276	C₄²×C₆₇	—	2	2	no	—
277	C₄²×C₆₇	—	2	2	no	—
278	C₄²×C₆₇	—	2	4	no	—
279	C₄²×C₆₇	—	2	4	no	—
280	C₄²×C₆₇	—	2	4	no	—
281	C₄²×C₆₇	—	2	2	no	—
282	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
283	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
284	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
285	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
286	C₄²×C₆₇	C₂⁴	1	4	yes	—
287	C₂³×C₆₇	—	2	2	no	—
288	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
289	Q₈×C₆₇	—	2	4	no	—
290	Q₈×C₆₇	—	2	2	no	—
291	Q₈×C₆₇	—	2	2	no	—
292	Q₈×C₆₇	—	2	2	no	—
293	Q₈×C₆₇	—	2	2	no	—
294	Q₈×C₆₇	—	2	2	no	—
295	Q₈×C₆₇	—	4	2	no	—
296	Q₈×C₆₇	—	4	2	no	—
297	Q₈×C₆₇	—	4	2	no	—
298	Q₈×C₆₇	—	4	2	no	—
299	Q₈×C₆₇	—	8	1	no	—
300	Q₈×C₆₇	—	8	4	no	—
301	Q₈×C₆₇	C₂×#41	1	4	yes	—
302	D₄×C₆₇	—	2	2	no	—
303	D₄×C₆₇	—	2	2	no	—
304	D₄×C₆₇	—	4	2	no	—
305	D₄×C₆₇	—	4	4	no	—
306	D₄×C₆₇	—	8	1	no	—
307	C₂×C₄×C₆₇	—	2	4	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	—	2	2	no	—
315	C₂×C₄×C₆₇	—	4	1	no	—
316	C₂×C₄×C₆₇	—	4	1	no	—
317	C₂×C₄×C₆₇	—	4	2	no	—
318	C₂×C₄×C₆₇	—	4	4	no	—
319	C₂×C₄×C₆₇	—	8	4	no	—
320	C₂×C₄×C₆₇	—	8	1	no	—
321	C₂×C₄×C₆₇	—	8	1	no	—
322	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
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	4	yes	—
327	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
328	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
329	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
330	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
331	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
332	C₈×C₆₇	—	2	2	no	—
333	C₈×C₆₇	—	2	4	no	—
334	C₈×C₆₇	—	2	2	no	—
335	C₂²×C₆₇	—	2	1	no	—
336	C₂²×C₆₇	—	2	1	no	—
337	C₂²×C₆₇	—	4	2	no	—
338	C₂²×C₆₇	—	4	2	no	—
339	C₂²×C₆₇	—	4	2	no	—
340	C₂²×C₆₇	#548	1	2	yes	—
341	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
342	C₄×C₆₇	—	2	1	no	—
343	C₄×C₆₇	—	2	1	no	—
344	C₄×C₆₇	—	2	1	no	—
345	C₄×C₆₇	—	2	1	no	—
346	C₄×C₆₇	—	2	4	no	—
347	C₄×C₆₇	—	2	4	no	—
348	C₄×C₆₇	—	2	4	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	—	4	2	no	—
354	C₄×C₆₇	—	4	2	no	—
355	C₄×C₆₇	#548	1	2	yes	—
356	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
357	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
358	#61335	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
359	C₂×C₆₇	—	2	2	no	—
360	C₂×C₆₇	—	8	2	no	—
361	C₂×C₆₇	C₂×#548	1	1	yes	characteristic
362	C₂×C₆₇	#3586	1	1	yes	characteristic
363	C₂×C₆₇	#3586	1	1	yes	characteristic
364	#3507	C₂×C₆₇	1	1	yes	characteristic
365	#3502	C₂×C₆₇	1	1	yes	characteristic, complemented
366	#3377	C₂×C₆₇	1	4	yes	complemented
367	#3374	C₂×C₆₇	1	4	yes	complemented
368	#3371	C₂×C₆₇	1	4	yes	complemented
369	#3369	C₂×C₆₇	1	4	yes	complemented
370	#3137	C₂×C₆₇	1	2	yes	complemented
371	#3135	C₂×C₆₇	1	1	yes	characteristic
372	#3084	C₂×C₆₇	1	2	yes	complemented
373	#3078	C₂×C₆₇	1	1	yes	characteristic, complemented
374	#3044	C₂×C₆₇	1	2	yes	complemented
375	#3037	C₂×C₆₇	1	1	yes	characteristic
376	#2961	C₂×C₆₇	1	1	yes	characteristic, complemented
377	#2947	C₂×C₆₇	1	2	yes	complemented
378	C₂×#441	C₂×C₆₇	1	1	yes	characteristic, complemented
379	C₆₇	#61335	1	1	yes	characteristic, direct product factor, Sylow
380	C₂×#137	C₂²×C₆₇	1	2	yes	—
381	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
382	#559	C₂²×C₆₇	1	2	yes	—
383	Q₈²	—	2	1	no	—
384	Q₈²	C₂²×C₆₇	1	2	yes	—
385	#557	—	2	1	no	—
386	#557	—	2	2	no	—
387	#557	C₂²×C₆₇	1	2	yes	—
388	#557	C₂²×C₆₇	1	4	yes	—
389	#557	C₂²×C₆₇	1	2	yes	—
390	#556	—	2	1	no	—
391	#556	C₂²×C₆₇	1	4	yes	—
392	#554	—	2	1	no	—
393	#552	—	2	2	no	—
394	#544	—	2	2	no	—
395	#544	C₂²×C₆₇	1	2	yes	—
396	#539	—	2	2	no	—
397	#536	C₂²×C₆₇	1	2	yes	—
398	#533	—	2	2	no	—
399	#533	C₂²×C₆₇	1	1	yes	characteristic
400	#533	C₂²×C₆₇	1	1	yes	characteristic
401	#532	C₂²×C₆₇	1	1	yes	characteristic
402	#529	—	2	2	no	—
403	#520	C₂²×C₆₇	1	2	yes	—
404	#519	C₂²×C₆₇	1	2	yes	—
405	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
406	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
407	#491	C₂²×C₆₇	1	2	yes	—
408	#491	C₂²×C₆₇	1	4	yes	—
409	#490	C₂²×C₆₇	1	2	yes	—
410	#489	C₂²×C₆₇	1	2	yes	—
411	#489	C₂²×C₆₇	1	4	yes	—
412	#489	C₂²×C₆₇	1	2	yes	—
413	#488	C₂²×C₆₇	1	4	yes	—
414	#487	C₂²×C₆₇	1	4	yes	—
415	#479	C₂²×C₆₇	1	4	yes	—
416	#479	C₂²×C₆₇	1	4	yes	—
417	#478	C₂²×C₆₇	1	4	yes	—
418	#477	C₂²×C₆₇	1	2	yes	—
419	#476	C₂²×C₆₇	1	4	yes	—
420	#475	C₂²×C₆₇	1	2	yes	—
421	#475	C₂²×C₆₇	1	4	yes	—
422	#464	C₂²×C₆₇	1	4	yes	—
423	#464	C₂²×C₆₇	1	4	yes	—
424	#464	C₂²×C₆₇	1	4	yes	—
425	#462	C₂²×C₆₇	1	4	yes	—
426	#462	C₂²×C₆₇	1	2	yes	—
427	#462	C₂²×C₆₇	1	4	yes	—
428	#460	C₂²×C₆₇	1	2	yes	—
429	#446	C₂²×C₆₇	1	4	yes	—
430	#446	C₂²×C₆₇	1	4	yes	—
431	#441	C₂²×C₆₇	1	4	yes	—
432	#441	C₂²×C₆₇	1	4	yes	—
433	#441	C₂²×C₆₇	1	4	yes	—
434	#440	C₂²×C₆₇	1	4	yes	—
435	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	—
436	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	—
437	#428	C₂²×C₆₇	1	2	yes	—
438	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
439	#424	C₂²×C₆₇	1	2	yes	—
440	#423	C₂²×C₆₇	1	1	yes	characteristic
441	#419	C₂²×C₆₇	1	2	yes	—
442	#417	C₂²×C₆₇	1	2	yes	—
443	#416	C₂²×C₆₇	1	2	yes	—
444	#416	C₂²×C₆₇	1	2	yes	—
445	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
446	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
447	#405	C₂²×C₆₇	1	2	yes	—
448	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
449	C₂×#41	C₂³×C₆₇	1	2	yes	—
450	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
451	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
452	#137	—	2	8	no	—
453	C₂×Q₁₆	—	2	4	no	—
454	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
455	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
456	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
457	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
458	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
459	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
460	C₄⋊Q₈	—	2	2	no	—
461	C₄⋊Q₈	—	2	4	no	—
462	C₄⋊Q₈	—	2	2	no	—
463	C₄⋊Q₈	—	2	4	no	—
464	C₄⋊Q₈	—	2	2	no	—
465	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
466	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
467	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
468	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
469	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
470	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
471	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
472	#127	C₂³×C₆₇	1	1	yes	characteristic
473	#126	—	2	4	no	—
474	#126	—	2	4	no	—
475	#125	—	2	2	no	—
476	#125	—	2	4	no	—
477	#125	—	2	4	no	—
478	#125	—	2	4	no	—
479	#125	—	2	4	no	—
480	#125	—	2	2	no	—
481	#125	D₄×C₆₇	1	2	yes	—
482	#125	C₂³×C₆₇	1	2	yes	—
483	#125	C₂³×C₆₇	1	4	yes	—
484	#124	—	2	2	no	—
485	#124	C₂³×C₆₇	1	2	yes	—
486	#124	C₂³×C₆₇	1	4	yes	—
487	#123	—	2	4	no	—
488	#123	—	2	4	no	—
489	C₂²⋊Q₈	—	2	4	no	—
490	C₂²⋊Q₈	—	2	1	no	—
491	C₂²⋊Q₈	—	2	1	no	—
492	C₂²⋊Q₈	—	2	2	no	—
493	C₂²⋊Q₈	—	2	2	no	—
494	C₂²⋊Q₈	—	2	4	no	—
495	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
496	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
497	#121	—	2	1	no	—
498	#121	—	2	1	no	—
499	C₄×Q₈	—	2	2	no	—
500	C₄×Q₈	—	2	2	no	—
501	C₄×Q₈	—	2	2	no	—
502	C₄×Q₈	—	2	2	no	—
503	C₄×Q₈	—	2	4	no	—
504	C₄×Q₈	—	2	2	no	—
505	C₄×Q₈	—	2	1	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	—	2	1	no	—
511	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
512	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
513	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
514	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
515	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
516	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
517	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
518	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
519	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
520	C₄×D₄	—	2	2	no	—
521	C₄×D₄	—	2	4	no	—
522	C₄×D₄	—	2	1	no	—
523	C₄×D₄	—	2	1	no	—
524	C₄×D₄	C₂³×C₆₇	1	4	yes	—
525	#117	—	2	2	no	—
526	#117	—	2	4	no	—
527	#117	—	2	4	no	—
528	#117	C₂³×C₆₇	1	2	yes	—
529	#117	C₂³×C₆₇	1	2	yes	—
530	C₂×C₄⋊C₄	—	2	1	no	—
531	C₂×C₄⋊C₄	—	2	1	no	—
532	C₂×C₄⋊C₄	—	2	2	no	—
533	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
534	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
535	C₂×C₄²	—	2	1	no	—
536	C₂×C₄²	—	2	1	no	—
537	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
538	#107	C₂³×C₆₇	1	4	yes	—
539	#106	C₂³×C₆₇	1	4	yes	—
540	#106	C₂³×C₆₇	1	4	yes	—
541	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
542	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
543	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
544	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
545	#103	C₂³×C₆₇	1	4	yes	—
546	#103	C₂³×C₆₇	1	4	yes	—
547	#103	C₂³×C₆₇	1	2	yes	—
548	#103	C₂³×C₆₇	1	2	yes	—
549	#103	C₂³×C₆₇	1	2	yes	—
550	#103	C₂³×C₆₇	1	2	yes	—
551	#103	C₂³×C₆₇	1	4	yes	—
552	#103	C₂³×C₆₇	1	4	yes	—
553	#103	C₂³×C₆₇	1	4	yes	—
554	#102	C₂³×C₆₇	1	2	yes	—
555	#102	C₂³×C₆₇	1	2	yes	—
556	#102	C₂³×C₆₇	1	2	yes	—
557	#102	C₂³×C₆₇	1	2	yes	—
558	#102	C₂³×C₆₇	1	4	yes	—
559	#97	C₂³×C₆₇	1	2	yes	—
560	#97	C₂³×C₆₇	1	2	yes	—
561	#97	C₂³×C₆₇	1	4	yes	—
562	C₄×C₈	C₂³×C₆₇	1	4	yes	—
563	#41	—	2	4	no	—
564	#41	—	2	4	no	—
565	C₂×Q₈	—	2	1	no	—
566	C₂×Q₈	—	2	1	no	—
567	C₂×Q₈	—	2	1	no	—
568	C₂×Q₈	—	2	1	no	—
569	C₂×Q₈	—	2	4	no	—
570	C₂×Q₈	—	2	2	no	—
571	C₂×Q₈	—	2	2	no	—
572	C₂×Q₈	—	2	2	no	—
573	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
574	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
575	C₂×Q₈	#41×C₆₇	1	4	yes	—
576	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
577	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
578	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
579	C₂×D₄	—	2	2	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	—	2	1	no	—
582	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	—	2	1	no	—
585	C₂²×C₄	—	2	1	no	—
586	C₂²×C₄	—	2	1	no	—
587	C₂²×C₄	—	2	2	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	Q₁₆	—	2	8	no	—
593	Q₁₆	—	2	8	no	—
594	SD₁₆	—	2	8	no	—
595	M₁₆	—	2	4	no	—
596	C₂×C₈	—	2	2	no	—
597	C₂×C₈	—	2	2	no	—
598	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
599	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
600	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
601	C₄⋊C₄	—	2	1	no	—
602	C₄⋊C₄	—	2	4	no	—
603	C₄⋊C₄	—	2	2	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	2	no	—
607	C₄⋊C₄	—	2	1	no	—
608	C₄⋊C₄	—	2	4	no	—
609	C₄⋊C₄	—	2	4	no	—
610	C₄⋊C₄	—	2	4	no	—
611	C₄⋊C₄	—	2	1	no	—
612	C₄⋊C₄	—	2	1	no	—
613	C₄⋊C₄	—	2	2	no	—
614	C₄⋊C₄	—	2	1	no	—
615	C₄⋊C₄	—	2	4	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	4	no	—
618	C₄⋊C₄	—	2	4	no	—
619	C₄⋊C₄	—	2	2	no	—
620	C₄⋊C₄	—	2	4	no	—
621	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
622	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
623	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
624	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
625	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
626	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
627	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
628	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
629	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
630	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
631	C₂²⋊C₄	—	2	2	no	—
632	C₂²⋊C₄	—	2	2	no	—
633	C₂²⋊C₄	—	2	4	no	—
634	C₂²⋊C₄	—	2	4	no	—
635	C₂²⋊C₄	—	2	2	no	—
636	C₂²⋊C₄	—	2	4	no	—
637	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
638	C₄²	—	2	2	no	—
639	C₄²	—	2	2	no	—
640	C₄²	—	2	2	no	—
641	C₄²	—	2	2	no	—
642	C₄²	—	2	4	no	—
643	C₄²	—	2	4	no	—
644	C₄²	—	2	4	no	—
645	C₄²	—	2	2	no	—
646	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
647	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
648	C₄²	C₂⁴×C₆₇	1	4	yes	—
649	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
650	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
651	C₂³	—	2	2	no	—
652	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
653	Q₈	—	2	4	no	—
654	Q₈	—	2	2	no	—
655	Q₈	—	2	2	no	—
656	Q₈	—	2	2	no	—
657	Q₈	—	2	2	no	—
658	Q₈	—	2	2	no	—
659	Q₈	—	4	2	no	—
660	Q₈	—	4	2	no	—
661	Q₈	—	4	2	no	—
662	Q₈	—	4	2	no	—
663	Q₈	—	8	1	no	—
664	Q₈	—	8	4	no	—
665	Q₈	C₂×#41×C₆₇	1	4	yes	—
666	D₄	—	2	2	no	—
667	D₄	—	2	2	no	—
668	D₄	—	4	2	no	—
669	D₄	—	4	4	no	—
670	D₄	—	8	1	no	—
671	C₂×C₄	—	2	4	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	—	2	2	no	—
677	C₂×C₄	—	2	2	no	—
678	C₂×C₄	—	2	2	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	4	2	no	—
682	C₂×C₄	—	4	4	no	—
683	C₂×C₄	—	8	4	no	—
684	C₂×C₄	—	8	1	no	—
685	C₂×C₄	—	8	1	no	—
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	C₂²×D₄×C₆₇	1	1	yes	characteristic
690	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
691	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
692	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
693	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
694	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
695	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
696	C₈	—	2	2	no	—
697	C₈	—	2	4	no	—
698	C₈	—	2	2	no	—
699	C₂²	—	2	1	no	—
700	C₂²	—	2	1	no	—
701	C₂²	—	4	2	no	—
702	C₂²	—	4	2	no	—
703	C₂²	—	4	2	no	—
704	C₂²	#548×C₆₇	1	2	yes	—
705	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
706	C₄	—	2	1	no	—
707	C₄	—	2	1	no	—
708	C₄	—	2	1	no	—
709	C₄	—	2	1	no	—
710	C₄	—	2	4	no	—
711	C₄	—	2	4	no	—
712	C₄	—	2	4	no	—
713	C₄	—	4	2	no	—
714	C₄	—	4	2	no	—
715	C₄	—	4	2	no	—
716	C₄	—	4	2	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	2	no	—
719	C₄	#548×C₆₇	1	2	yes	—
720	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
721	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
722	C₂	—	2	2	no	—
723	C₂	—	8	2	no	—
724	C₂	C₂×#548×C₆₇	1	1	yes	characteristic
725	C₂	C₆₇×#3586	1	1	yes	characteristic
726	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	#532×C₆₇	C₂²
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×#3377	C₂	semidirect
4. C₆₇×#3374	C₂	semidirect
5. C₆₇×#3371	C₂	semidirect
6. C₆₇×#3369	C₂	semidirect
7. C₆₇×#3137	C₂	semidirect
9. C₆₇×#3084	C₂	semidirect
10. C₆₇×#3078	C₂	semidirect
11. C₆₇×#3044	C₂	semidirect
13. C₆₇×#2961	C₂	semidirect
14. C₆₇×#2947	C₂	semidirect
15. C₂×#441×C₆₇	C₂	semidirect
358. #61335	C₆₇	direct (N × H)
365. #3502	C₂×C₆₇	semidirect
366. #3377	C₂×C₆₇	semidirect
367. #3374	C₂×C₆₇	semidirect
368. #3371	C₂×C₆₇	semidirect
369. #3369	C₂×C₆₇	semidirect
370. #3137	C₂×C₆₇	semidirect
372. #3084	C₂×C₆₇	semidirect
373. #3078	C₂×C₆₇	semidirect
374. #3044	C₂×C₆₇	semidirect
376. #2961	C₂×C₆₇	semidirect
377. #2947	C₂×C₆₇	semidirect
378. C₂×#441	C₂×C₆₇	semidirect
379. C₆₇	#61335	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×#3507	2
C₆₇×#3502	2
C₆₇×#3377	2
C₆₇×#3374	2
C₆₇×#3371	2
C₆₇×#3369	2
C₆₇×#3137	2
C₆₇×#3135	2
C₆₇×#3084	2
C₆₇×#3078	2
C₆₇×#3044	2
C₆₇×#3037	2
C₆₇×#2961	2
C₆₇×#2947	2
C₂×#441×C₆₇	2
#61335	67

Subgroups — C67×#61335 #30740586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61335 #30740586