Subgroups — C₆₇×#61015 #30740266

Order 17152.

1768 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3511	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3263	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3229	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3227	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3210	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3207	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3204	C₂	1	4	yes	maximal, complemented
8	C₆₇×#3185	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3184	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3167	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3125	C₂	1	8	yes	maximal, complemented
12	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
13	#556×C₆₇	—	2	2	no	—
14	#555×C₆₇	—	2	2	no	—
15	#554×C₆₇	—	2	2	no	—
16	#553×C₆₇	—	2	2	no	—
17	#553×C₆₇	—	2	2	no	—
18	#548×C₆₇	—	2	2	no	—
19	#548×C₆₇	—	2	2	no	—
20	#547×C₆₇	—	2	2	no	—
21	#542×C₆₇	C₂²	1	4	yes	—
22	#542×C₆₇	C₂²	1	4	yes	—
23	#537×C₆₇	C₂²	1	4	yes	—
24	#533×C₆₇	C₂²	1	2	yes	—
25	#533×C₆₇	C₂²	1	2	yes	—
26	#532×C₆₇	C₂²	1	2	yes	—
27	#532×C₆₇	C₂²	1	2	yes	—
28	#532×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
30	#520×C₆₇	C₂²	1	4	yes	—
31	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
32	#498×C₆₇	C₂²	1	2	yes	—
33	#498×C₆₇	C₂²	1	1	yes	characteristic
34	#498×C₆₇	C₂²	1	1	yes	characteristic
35	#494×C₆₇	C₂²	1	1	yes	characteristic
36	#493×C₆₇	C₂²	1	1	yes	characteristic
37	#492×C₆₇	C₂²	1	2	yes	—
38	#487×C₆₇	C₂²	1	2	yes	—
39	#487×C₆₇	C₂²	1	2	yes	—
40	#486×C₆₇	C₂²	1	2	yes	—
41	#486×C₆₇	C₂²	1	2	yes	—
42	#483×C₆₇	C₂²	1	8	yes	—
43	#481×C₆₇	C₂²	1	8	yes	—
44	#476×C₆₇	C₂²	1	8	yes	—
45	#475×C₆₇	C₂²	1	8	yes	—
46	#470×C₆₇	C₂²	1	4	yes	—
47	#469×C₆₇	C₂²	1	4	yes	—
48	#465×C₆₇	C₂²	1	8	yes	—
49	#464×C₆₇	C₂²	1	8	yes	—
50	#463×C₆₇	C₂²	1	8	yes	—
51	#452×C₆₇	C₂²	1	8	yes	—
52	#449×C₆₇	C₂²	1	8	yes	—
53	#442×C₆₇	C₂²	1	4	yes	—
54	#441×C₆₇	C₂²	1	4	yes	—
55	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	—
56	#433×C₆₇	C₂²	1	4	yes	—
57	#431×C₆₇	C₂²	1	2	yes	—
58	#431×C₆₇	C₂²	1	2	yes	—
59	#424×C₆₇	C₂²	1	2	yes	—
60	#408×C₆₇	C₂²	1	2	yes	—
61	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
62	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
63	C₂×#41×C₆₇	—	2	2	no	—
64	C₂×#41×C₆₇	—	2	2	no	—
65	C₂×#41×C₆₇	C₂³	1	4	yes	—
66	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×Q₈×C₆₇	—	2	2	no	—
68	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
69	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
70	C₂×D₈×C₆₇	C₂³	1	4	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₆₇	—	2	2	no	—
74	C₄⋊Q₈×C₆₇	—	2	2	no	—
75	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
76	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
77	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
79	#127×C₆₇	C₂³	1	2	yes	—
80	#126×C₆₇	—	2	4	no	—
81	#126×C₆₇	—	2	4	no	—
82	#125×C₆₇	—	2	2	no	—
83	#125×C₆₇	—	2	2	no	—
84	#124×C₆₇	—	2	4	no	—
85	#124×C₆₇	—	2	4	no	—
86	#124×C₆₇	—	2	2	no	—
87	#124×C₆₇	—	2	2	no	—
88	#124×C₆₇	C₂³	1	4	yes	—
89	#123×C₆₇	—	2	4	no	—
90	#123×C₆₇	—	2	4	no	—
91	C₂²⋊Q₈×C₆₇	—	2	4	no	—
92	C₂²⋊Q₈×C₆₇	—	2	2	no	—
93	C₂²⋊Q₈×C₆₇	—	2	2	no	—
94	C₂²⋊Q₈×C₆₇	—	2	4	no	—
95	C₂²⋊Q₈×C₆₇	—	2	4	no	—
96	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
97	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
98	#121×C₆₇	—	2	4	no	—
99	#121×C₆₇	—	2	4	no	—
100	#121×C₆₇	—	2	4	no	—
101	#121×C₆₇	—	2	2	no	—
102	#121×C₆₇	—	2	2	no	—
103	#121×C₆₇	—	2	2	no	—
104	#121×C₆₇	—	2	2	no	—
105	#121×C₆₇	C₂³	1	4	yes	—
106	#121×C₆₇	C₂³	1	4	yes	—
107	C₄×Q₈×C₆₇	—	2	2	no	—
108	C₄×Q₈×C₆₇	—	2	2	no	—
109	C₄×Q₈×C₆₇	D₄	1	2	yes	—
110	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
111	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
112	C₄×D₄×C₆₇	—	2	4	no	—
113	C₄×D₄×C₆₇	—	2	4	no	—
114	C₄×D₄×C₆₇	—	2	2	no	—
115	C₄×D₄×C₆₇	—	2	2	no	—
116	C₄×D₄×C₆₇	—	2	2	no	—
117	C₄×D₄×C₆₇	—	2	2	no	—
118	C₄×D₄×C₆₇	D₄	1	2	yes	—
119	C₄×D₄×C₆₇	D₄	1	2	yes	—
120	C₄×D₄×C₆₇	D₄	1	2	yes	—
121	C₄×D₄×C₆₇	C₂³	1	4	yes	—
122	C₄×D₄×C₆₇	C₂³	1	4	yes	—
123	#117×C₆₇	—	2	2	no	—
124	#117×C₆₇	—	2	2	no	—
125	#117×C₆₇	—	2	4	no	—
126	#117×C₆₇	C₂³	1	2	yes	—
127	#117×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
133	#106×C₆₇	C₂³	1	8	yes	—
134	#106×C₆₇	C₂³	1	4	yes	—
135	#106×C₆₇	C₂³	1	4	yes	—
136	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
137	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
138	#103×C₆₇	C₂³	1	8	yes	—
139	#103×C₆₇	C₂³	1	8	yes	—
140	#102×C₆₇	C₂³	1	8	yes	—
141	#102×C₆₇	C₂³	1	8	yes	—
142	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
143	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
144	#97×C₆₇	C₂³	1	2	yes	—
145	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₆₇	C₂³	1	2	yes	—
147	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
148	C₄×C₈×C₆₇	C₂³	1	2	yes	—
149	#41×C₆₇	—	4	2	no	—
150	#41×C₆₇	—	4	8	no	—
151	#41×C₆₇	—	8	4	no	—
152	#41×C₆₇	—	8	4	no	—
153	C₂×Q₈×C₆₇	—	2	2	no	—
154	C₂×Q₈×C₆₇	—	2	2	no	—
155	C₂×Q₈×C₆₇	—	2	4	no	—
156	C₂×Q₈×C₆₇	—	8	4	no	—
157	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
159	C₂×D₄×C₆₇	—	2	4	no	—
160	C₂×D₄×C₆₇	—	2	4	no	—
161	C₂×D₄×C₆₇	—	2	4	no	—
162	C₂×D₄×C₆₇	—	2	2	no	—
163	C₂×D₄×C₆₇	—	2	2	no	—
164	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₇	—	2	4	no	—
168	C₂²×C₄×C₆₇	—	2	4	no	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	—	2	4	no	—
171	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
176	Q₁₆×C₆₇	—	4	4	no	—
177	SD₁₆×C₆₇	—	4	8	no	—
178	D₈×C₆₇	—	4	4	no	—
179	M₁₆×C₆₇	—	4	2	no	—
180	C₂×C₈×C₆₇	—	4	2	no	—
181	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
182	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
183	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
184	C₄⋊C₄×C₆₇	—	2	4	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	4	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
195	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
196	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
197	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
198	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
199	C₂²⋊C₄×C₆₇	—	2	4	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	4	no	—
202	C₂²⋊C₄×C₆₇	—	2	4	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	—	2	4	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	—	2	4	no	—
207	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
208	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
209	C₂²⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
210	C₂²⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₆₇	—	2	4	no	—
212	C₄²×C₆₇	—	2	4	no	—
213	C₄²×C₆₇	C₂×D₄	1	2	yes	—
214	C₄²×C₆₇	C₂×D₄	1	2	yes	—
215	C₄²×C₆₇	C₂×D₄	1	2	yes	—
216	C₄²×C₆₇	C₂⁴	1	2	yes	—
217	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
218	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
219	C₂³×C₆₇	—	2	4	no	—
220	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
222	Q₈×C₆₇	—	2	8	no	—
223	Q₈×C₆₇	—	4	1	no	—
224	Q₈×C₆₇	—	4	4	no	—
225	Q₈×C₆₇	—	4	4	no	—
226	Q₈×C₆₇	—	8	4	no	—
227	D₄×C₆₇	—	2	8	no	—
228	D₄×C₆₇	—	4	2	no	—
229	D₄×C₆₇	—	4	1	no	—
230	D₄×C₆₇	—	4	8	no	—
231	D₄×C₆₇	—	4	4	no	—
232	D₄×C₆₇	—	4	4	no	—
233	D₄×C₆₇	—	8	4	no	—
234	D₄×C₆₇	—	8	4	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	4	8	no	—
244	C₂×C₄×C₆₇	—	8	4	no	—
245	C₂×C₄×C₆₇	—	8	4	no	—
246	C₂×C₄×C₆₇	—	8	4	no	—
247	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
254	C₈×C₆₇	—	2	4	no	—
255	C₈×C₆₇	—	2	2	no	—
256	C₈×C₆₇	—	2	2	no	—
257	C₂²×C₆₇	—	2	2	no	—
258	C₂²×C₆₇	—	2	1	no	—
259	C₂²×C₆₇	—	2	2	no	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	4	4	no	—
263	C₂²×C₆₇	—	4	4	no	—
264	C₂²×C₆₇	—	4	4	no	—
265	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
266	C₄×C₆₇	—	2	2	no	—
267	C₄×C₆₇	—	2	1	no	—
268	C₄×C₆₇	—	2	2	no	—
269	C₄×C₆₇	—	2	2	no	—
270	C₄×C₆₇	—	2	2	no	—
271	C₄×C₆₇	—	4	4	no	—
272	C₄×C₆₇	—	4	4	no	—
273	C₄×C₆₇	—	4	4	no	—
274	C₄×C₆₇	#577	1	4	yes	—
275	C₄×C₆₇	#585	1	2	yes	—
276	#61015	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
277	C₂×C₆₇	—	4	2	no	—
278	C₂×C₆₇	—	4	1	no	—
279	C₂×C₆₇	—	8	4	no	—
280	C₂×C₆₇	#3449	1	1	yes	characteristic
281	C₂×C₆₇	C₂×#577	1	2	yes	—
282	#3511	C₂×C₆₇	1	2	yes	complemented
283	#3263	C₂×C₆₇	1	4	yes	complemented
284	#3229	C₂×C₆₇	1	2	yes	complemented
285	#3227	C₂×C₆₇	1	2	yes	complemented
286	#3210	C₂×C₆₇	1	2	yes	complemented
287	#3207	C₂×C₆₇	1	2	yes	complemented
288	#3204	C₂×C₆₇	1	4	yes	complemented
289	#3185	C₂×C₆₇	1	1	yes	characteristic, complemented
290	#3184	C₂×C₆₇	1	1	yes	characteristic, complemented
291	#3167	C₂×C₆₇	1	2	yes	complemented
292	#3125	C₂×C₆₇	1	8	yes	complemented
293	#2922	C₂×C₆₇	1	1	yes	characteristic, complemented
294	C₆₇	#61015	1	1	yes	characteristic, direct product factor, Sylow
295	#556	—	2	2	no	—
296	#555	—	2	2	no	—
297	#554	—	2	2	no	—
298	#553	—	2	2	no	—
299	#553	—	2	2	no	—
300	#548	—	2	2	no	—
301	#548	—	2	2	no	—
302	#547	—	2	2	no	—
303	#542	C₂²×C₆₇	1	4	yes	—
304	#542	C₂²×C₆₇	1	4	yes	—
305	#537	C₂²×C₆₇	1	4	yes	—
306	#533	C₂²×C₆₇	1	2	yes	—
307	#533	C₂²×C₆₇	1	2	yes	—
308	#532	C₂²×C₆₇	1	2	yes	—
309	#532	C₂²×C₆₇	1	2	yes	—
310	#532	C₂²×C₆₇	1	2	yes	—
311	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
312	#520	C₂²×C₆₇	1	4	yes	—
313	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
314	#498	C₂²×C₆₇	1	2	yes	—
315	#498	C₂²×C₆₇	1	1	yes	characteristic
316	#498	C₂²×C₆₇	1	1	yes	characteristic
317	#494	C₂²×C₆₇	1	1	yes	characteristic
318	#493	C₂²×C₆₇	1	1	yes	characteristic
319	#492	C₂²×C₆₇	1	2	yes	—
320	#487	C₂²×C₆₇	1	2	yes	—
321	#487	C₂²×C₆₇	1	2	yes	—
322	#486	C₂²×C₆₇	1	2	yes	—
323	#486	C₂²×C₆₇	1	2	yes	—
324	#483	C₂²×C₆₇	1	8	yes	—
325	#481	C₂²×C₆₇	1	8	yes	—
326	#476	C₂²×C₆₇	1	8	yes	—
327	#475	C₂²×C₆₇	1	8	yes	—
328	#470	C₂²×C₆₇	1	4	yes	—
329	#469	C₂²×C₆₇	1	4	yes	—
330	#465	C₂²×C₆₇	1	8	yes	—
331	#464	C₂²×C₆₇	1	8	yes	—
332	#463	C₂²×C₆₇	1	8	yes	—
333	#452	C₂²×C₆₇	1	8	yes	—
334	#449	C₂²×C₆₇	1	8	yes	—
335	#442	C₂²×C₆₇	1	4	yes	—
336	#441	C₂²×C₆₇	1	4	yes	—
337	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	—
338	#433	C₂²×C₆₇	1	4	yes	—
339	#431	C₂²×C₆₇	1	2	yes	—
340	#431	C₂²×C₆₇	1	2	yes	—
341	#424	C₂²×C₆₇	1	2	yes	—
342	#408	C₂²×C₆₇	1	2	yes	—
343	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
344	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
345	C₂×#41	—	2	2	no	—
346	C₂×#41	—	2	2	no	—
347	C₂×#41	C₂³×C₆₇	1	4	yes	—
348	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
349	C₂²×Q₈	—	2	2	no	—
350	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
351	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
352	C₂×D₈	C₂³×C₆₇	1	4	yes	—
353	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
354	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
355	C₄⋊Q₈	—	2	2	no	—
356	C₄⋊Q₈	—	2	2	no	—
357	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
358	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
359	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
360	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
361	#127	C₂³×C₆₇	1	2	yes	—
362	#126	—	2	4	no	—
363	#126	—	2	4	no	—
364	#125	—	2	2	no	—
365	#125	—	2	2	no	—
366	#124	—	2	4	no	—
367	#124	—	2	4	no	—
368	#124	—	2	2	no	—
369	#124	—	2	2	no	—
370	#124	C₂³×C₆₇	1	4	yes	—
371	#123	—	2	4	no	—
372	#123	—	2	4	no	—
373	C₂²⋊Q₈	—	2	4	no	—
374	C₂²⋊Q₈	—	2	2	no	—
375	C₂²⋊Q₈	—	2	2	no	—
376	C₂²⋊Q₈	—	2	4	no	—
377	C₂²⋊Q₈	—	2	4	no	—
378	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
379	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
380	#121	—	2	4	no	—
381	#121	—	2	4	no	—
382	#121	—	2	4	no	—
383	#121	—	2	2	no	—
384	#121	—	2	2	no	—
385	#121	—	2	2	no	—
386	#121	—	2	2	no	—
387	#121	C₂³×C₆₇	1	4	yes	—
388	#121	C₂³×C₆₇	1	4	yes	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	D₄×C₆₇	1	2	yes	—
392	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
393	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
394	C₄×D₄	—	2	4	no	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	D₄×C₆₇	1	2	yes	—
401	C₄×D₄	D₄×C₆₇	1	2	yes	—
402	C₄×D₄	D₄×C₆₇	1	2	yes	—
403	C₄×D₄	C₂³×C₆₇	1	4	yes	—
404	C₄×D₄	C₂³×C₆₇	1	4	yes	—
405	#117	—	2	2	no	—
406	#117	—	2	2	no	—
407	#117	—	2	4	no	—
408	#117	C₂³×C₆₇	1	2	yes	—
409	#117	C₂³×C₆₇	1	1	yes	characteristic
410	C₂×C₄⋊C₄	—	2	2	no	—
411	C₂×C₄⋊C₄	—	2	4	no	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
413	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
414	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
415	#106	C₂³×C₆₇	1	8	yes	—
416	#106	C₂³×C₆₇	1	4	yes	—
417	#106	C₂³×C₆₇	1	4	yes	—
418	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
419	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
420	#103	C₂³×C₆₇	1	8	yes	—
421	#103	C₂³×C₆₇	1	8	yes	—
422	#102	C₂³×C₆₇	1	8	yes	—
423	#102	C₂³×C₆₇	1	8	yes	—
424	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
425	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
426	#97	C₂³×C₆₇	1	2	yes	—
427	C₄×C₈	C₂³×C₆₇	1	2	yes	—
428	C₄×C₈	C₂³×C₆₇	1	2	yes	—
429	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
430	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
431	#41	—	4	2	no	—
432	#41	—	4	8	no	—
433	#41	—	8	4	no	—
434	#41	—	8	4	no	—
435	C₂×Q₈	—	2	2	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	—	8	4	no	—
439	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
440	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
447	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
448	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
455	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
456	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
458	Q₁₆	—	4	4	no	—
459	SD₁₆	—	4	8	no	—
460	D₈	—	4	4	no	—
461	M₁₆	—	4	2	no	—
462	C₂×C₈	—	4	2	no	—
463	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
464	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
465	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
477	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
478	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
479	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
480	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
490	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
491	C₂²⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
492	C₂²⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
493	C₄²	—	2	4	no	—
494	C₄²	—	2	4	no	—
495	C₄²	C₂×D₄×C₆₇	1	2	yes	—
496	C₄²	C₂×D₄×C₆₇	1	2	yes	—
497	C₄²	C₂×D₄×C₆₇	1	2	yes	—
498	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
499	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
500	C₄²	C₂⁴×C₆₇	1	2	yes	—
501	C₂³	—	2	4	no	—
502	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
503	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
504	Q₈	—	2	8	no	—
505	Q₈	—	4	1	no	—
506	Q₈	—	4	4	no	—
507	Q₈	—	4	4	no	—
508	Q₈	—	8	4	no	—
509	D₄	—	2	8	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	1	no	—
512	D₄	—	4	8	no	—
513	D₄	—	4	4	no	—
514	D₄	—	4	4	no	—
515	D₄	—	8	4	no	—
516	D₄	—	8	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	8	4	no	—
527	C₂×C₄	—	8	4	no	—
528	C₂×C₄	—	8	4	no	—
529	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
530	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
532	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
534	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
535	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
536	C₈	—	2	4	no	—
537	C₈	—	2	2	no	—
538	C₈	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	1	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	4	no	—
547	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
548	C₄	—	2	2	no	—
549	C₄	—	2	2	no	—
550	C₄	—	2	2	no	—
551	C₄	—	2	2	no	—
552	C₄	—	2	1	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	4	no	—
555	C₄	—	4	4	no	—
556	C₄	#577×C₆₇	1	4	yes	—
557	C₄	#585×C₆₇	1	2	yes	—
558	C₂	—	4	2	no	—
559	C₂	—	4	1	no	—
560	C₂	—	8	4	no	—
561	C₂	C₆₇×#3449	1	1	yes	characteristic
562	C₂	C₂×#577×C₆₇	1	2	yes	—

	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 ◁ C₄×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3511	C₂	semidirect
2. C₆₇×#3263	C₂	semidirect
3. C₆₇×#3229	C₂	semidirect
4. C₆₇×#3227	C₂	semidirect
5. C₆₇×#3210	C₂	semidirect
6. C₆₇×#3207	C₂	semidirect
7. C₆₇×#3204	C₂	semidirect
8. C₆₇×#3185	C₂	semidirect
9. C₆₇×#3184	C₂	semidirect
10. C₆₇×#3167	C₂	semidirect
11. C₆₇×#3125	C₂	semidirect
12. C₆₇×#2922	C₂	semidirect
276. #61015	C₆₇	direct (N × H)
282. #3511	C₂×C₆₇	semidirect
283. #3263	C₂×C₆₇	semidirect
284. #3229	C₂×C₆₇	semidirect
285. #3227	C₂×C₆₇	semidirect
286. #3210	C₂×C₆₇	semidirect
287. #3207	C₂×C₆₇	semidirect
288. #3204	C₂×C₆₇	semidirect
289. #3185	C₂×C₆₇	semidirect
290. #3184	C₂×C₆₇	semidirect
291. #3167	C₂×C₆₇	semidirect
292. #3125	C₂×C₆₇	semidirect
293. #2922	C₂×C₆₇	semidirect
294. C₆₇	#61015	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×#3511	2
C₆₇×#3263	2
C₆₇×#3229	2
C₆₇×#3227	2
C₆₇×#3210	2
C₆₇×#3207	2
C₆₇×#3204	2
C₆₇×#3185	2
C₆₇×#3184	2
C₆₇×#3167	2
C₆₇×#3125	2
C₆₇×#2922	2
#61015	67

Subgroups — C67×#61015 #30740266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61015 #30740266