Subgroups — C₆₇×#60612 #30739863

Order 17152.

2050 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3449	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3137	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3136	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3135	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3131	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2896	C₂	1	2	yes	maximal, complemented
8	C₂×#416×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	C₂²	1	2	yes	—
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#544×C₆₇	—	2	2	no	—
14	#544×C₆₇	C₂²	1	8	yes	complemented
15	#543×C₆₇	—	2	2	no	—
16	#537×C₆₇	—	2	2	no	—
17	#536×C₆₇	C₂²	1	4	yes	—
18	#535×C₆₇	C₂²	1	4	yes	—
19	#534×C₆₇	—	2	2	no	—
20	#533×C₆₇	—	2	4	no	—
21	#533×C₆₇	C₂²	1	4	yes	complemented
22	#532×C₆₇	C₂²	1	4	yes	—
23	#525×C₆₇	—	2	4	no	—
24	C₂×#117×C₆₇	C₂²	1	2	yes	—
25	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
26	#491×C₆₇	C₂²	1	8	yes	complemented
27	#490×C₆₇	C₂²	1	8	yes	complemented
28	#489×C₆₇	C₂²	1	16	yes	complemented
29	#488×C₆₇	C₂²	1	16	yes	complemented
30	#487×C₆₇	C₂²	1	8	yes	complemented
31	#486×C₆₇	C₂²	1	8	yes	complemented
32	#419×C₆₇	C₂²	1	4	yes	complemented
33	#418×C₆₇	C₂²	1	4	yes	complemented
34	#417×C₆₇	C₂²	1	8	yes	—
35	#416×C₆₇	C₂²	1	8	yes	—
36	#416×C₆₇	C₂²	1	4	yes	complemented
37	#416×C₆₇	C₂²	1	8	yes	complemented
38	#416×C₆₇	C₂²	1	4	yes	complemented
39	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
40	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
41	#405×C₆₇	C₂²	1	8	yes	complemented
42	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
43	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
44	C₂×#41×C₆₇	—	2	2	no	—
45	C₂×#41×C₆₇	—	2	2	no	—
46	C₂×#41×C₆₇	—	2	8	no	—
47	C₂×#41×C₆₇	—	2	8	no	—
48	C₂×#41×C₆₇	C₂³	1	4	yes	—
49	C₂×#41×C₆₇	C₂³	1	4	yes	—
50	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×D₄×C₆₇	—	2	2	no	—
52	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	—	2	2	no	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₆₇	—	2	4	no	—
56	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
57	C₂²×C₈×C₆₇	—	2	4	no	—
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₆₇	—	2	8	no	—
63	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
64	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
65	#127×C₆₇	C₂³	1	4	yes	—
66	#125×C₆₇	—	2	8	no	—
67	#125×C₆₇	C₂³	1	8	yes	—
68	#124×C₆₇	C₂³	1	8	yes	—
69	#123×C₆₇	—	2	8	no	—
70	C₂²⋊Q₈×C₆₇	—	2	8	no	—
71	C₂²⋊Q₈×C₆₇	—	2	8	no	—
72	C₂²⋊Q₈×C₆₇	—	2	16	no	—
73	#121×C₆₇	—	2	8	no	—
74	#121×C₆₇	—	2	8	no	—
75	#120×C₆₇	—	2	8	no	—
76	C₄×Q₈×C₆₇	—	2	8	no	—
77	C₄×D₄×C₆₇	—	2	8	no	—
78	#117×C₆₇	—	2	2	no	—
79	#117×C₆₇	—	2	2	no	—
80	#117×C₆₇	D₄	1	4	yes	—
81	#117×C₆₇	C₂³	1	4	yes	—
82	#117×C₆₇	C₂³	1	4	yes	—
83	#117×C₆₇	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
88	C₂×C₄²×C₆₇	—	2	2	no	—
89	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
90	#103×C₆₇	C₂³	1	8	yes	—
91	#103×C₆₇	C₂³	1	16	yes	—
92	#103×C₆₇	C₂³	1	8	yes	—
93	#102×C₆₇	C₂³	1	8	yes	—
94	#102×C₆₇	C₂³	1	16	yes	—
95	#102×C₆₇	C₂³	1	8	yes	—
96	#97×C₆₇	C₂³	1	8	yes	—
97	C₄×C₈×C₆₇	C₂³	1	8	yes	—
98	C₂⁴×C₆₇	—	2	2	no	—
99	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
100	#41×C₆₇	—	2	8	no	—
101	#41×C₆₇	—	2	8	no	—
102	#41×C₆₇	—	2	8	no	—
103	#41×C₆₇	—	2	8	no	—
104	#41×C₆₇	—	4	8	no	—
105	#41×C₆₇	—	4	8	no	—
106	C₂×Q₈×C₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	2	8	no	—
108	C₂×Q₈×C₆₇	—	2	4	no	—
109	C₂×Q₈×C₆₇	—	2	4	no	—
110	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
111	C₂×D₄×C₆₇	—	2	4	no	—
112	C₂×D₄×C₆₇	—	2	4	no	—
113	C₂×D₄×C₆₇	—	2	2	no	—
114	C₂×D₄×C₆₇	—	2	8	no	—
115	C₂×D₄×C₆₇	—	2	4	no	—
116	C₂×D₄×C₆₇	—	2	4	no	—
117	C₂×D₄×C₆₇	—	4	8	no	—
118	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	4	8	no	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₆₇	#41	1	4	yes	—
128	C₂²×C₄×C₆₇	#41	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	M₁₆×C₆₇	—	2	8	no	—
132	C₂×C₈×C₆₇	—	2	8	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	4	no	—
136	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
137	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
138	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
139	C₄⋊C₄×C₆₇	—	2	4	no	—
140	C₄⋊C₄×C₆₇	—	2	8	no	—
141	C₄⋊C₄×C₆₇	—	2	8	no	—
142	C₄⋊C₄×C₆₇	—	2	16	no	—
143	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
144	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
145	C₂²⋊C₄×C₆₇	—	2	4	no	—
146	C₂²⋊C₄×C₆₇	—	2	4	no	—
147	C₂²⋊C₄×C₆₇	—	2	16	no	—
148	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
149	C₄²×C₆₇	—	2	4	no	—
150	C₄²×C₆₇	—	2	4	no	—
151	C₄²×C₆₇	C₂×D₄	1	4	yes	—
152	C₄²×C₆₇	C₂⁴	1	4	yes	—
153	C₂³×C₆₇	—	2	4	no	—
154	C₂³×C₆₇	—	4	2	no	—
155	C₂³×C₆₇	—	4	4	no	—
156	C₂³×C₆₇	—	8	4	no	—
157	C₂³×C₆₇	#124	1	4	yes	—
158	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
160	Q₈×C₆₇	—	2	4	no	—
161	Q₈×C₆₇	—	2	4	no	—
162	Q₈×C₆₇	—	4	4	no	—
163	Q₈×C₆₇	—	8	8	no	—
164	D₄×C₆₇	—	2	4	no	—
165	D₄×C₆₇	—	2	4	no	—
166	D₄×C₆₇	—	4	8	no	—
167	D₄×C₆₇	—	4	8	no	—
168	D₄×C₆₇	—	4	4	no	—
169	D₄×C₆₇	—	8	8	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	4	4	no	—
173	C₂×C₄×C₆₇	—	4	4	no	—
174	C₂×C₄×C₆₇	—	4	8	no	—
175	C₂×C₄×C₆₇	—	4	8	no	—
176	C₂×C₄×C₆₇	—	8	4	no	—
177	C₂×C₄×C₆₇	#124	1	8	yes	—
178	C₂×C₄×C₆₇	#124	1	4	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
180	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
181	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
183	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
184	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
185	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
186	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
187	C₈×C₆₇	—	2	2	no	—
188	C₈×C₆₇	—	2	2	no	—
189	C₈×C₆₇	—	2	4	no	—
190	C₂²×C₆₇	—	2	2	no	—
191	C₂²×C₆₇	—	2	4	no	—
192	C₂²×C₆₇	—	4	4	no	—
193	C₂²×C₆₇	—	4	8	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	—	8	8	no	—
196	C₂²×C₆₇	C₂×#124	1	4	yes	—
197	C₂²×C₆₇	C₂×#124	1	2	yes	—
198	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
199	C₄×C₆₇	—	2	8	no	—
200	C₄×C₆₇	—	4	8	no	—
201	C₄×C₆₇	—	4	4	no	—
202	C₄×C₆₇	C₂×#124	1	2	yes	—
203	C₄×C₆₇	C₂×#124	1	4	yes	—
204	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
205	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
206	#60612	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂×C₆₇	—	2	4	no	—
208	C₂×C₆₇	—	2	2	no	—
209	C₂×C₆₇	—	8	4	no	—
210	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
211	C₂×C₆₇	#3584	1	2	yes	—
212	#3458	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#3449	C₂×C₆₇	1	1	yes	characteristic
214	#3137	C₂×C₆₇	1	8	yes	complemented
215	#3136	C₂×C₆₇	1	8	yes	complemented
216	#3135	C₂×C₆₇	1	4	yes	complemented
217	#3131	C₂×C₆₇	1	4	yes	complemented
218	#2896	C₂×C₆₇	1	2	yes	complemented
219	C₂×#416	C₂×C₆₇	1	2	yes	complemented
220	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
221	C₆₇	#60612	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²×#41	C₂²×C₆₇	1	2	yes	—
223	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
224	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
225	#544	—	2	2	no	—
226	#544	C₂²×C₆₇	1	8	yes	complemented
227	#543	—	2	2	no	—
228	#537	—	2	2	no	—
229	#536	C₂²×C₆₇	1	4	yes	—
230	#535	C₂²×C₆₇	1	4	yes	—
231	#534	—	2	2	no	—
232	#533	—	2	4	no	—
233	#533	C₂²×C₆₇	1	4	yes	complemented
234	#532	C₂²×C₆₇	1	4	yes	—
235	#525	—	2	4	no	—
236	C₂×#117	C₂²×C₆₇	1	2	yes	—
237	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
238	#491	C₂²×C₆₇	1	8	yes	complemented
239	#490	C₂²×C₆₇	1	8	yes	complemented
240	#489	C₂²×C₆₇	1	16	yes	complemented
241	#488	C₂²×C₆₇	1	16	yes	complemented
242	#487	C₂²×C₆₇	1	8	yes	complemented
243	#486	C₂²×C₆₇	1	8	yes	complemented
244	#419	C₂²×C₆₇	1	4	yes	complemented
245	#418	C₂²×C₆₇	1	4	yes	complemented
246	#417	C₂²×C₆₇	1	8	yes	—
247	#416	C₂²×C₆₇	1	8	yes	—
248	#416	C₂²×C₆₇	1	4	yes	complemented
249	#416	C₂²×C₆₇	1	8	yes	complemented
250	#416	C₂²×C₆₇	1	4	yes	complemented
251	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
252	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
253	#405	C₂²×C₆₇	1	8	yes	complemented
254	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
255	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
256	C₂×#41	—	2	2	no	—
257	C₂×#41	—	2	2	no	—
258	C₂×#41	—	2	8	no	—
259	C₂×#41	—	2	8	no	—
260	C₂×#41	C₂³×C₆₇	1	4	yes	—
261	C₂×#41	C₂³×C₆₇	1	4	yes	—
262	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
263	C₂²×D₄	—	2	2	no	—
264	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
265	C₂³×C₄	—	2	2	no	—
266	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
267	C₂×M₁₆	—	2	4	no	—
268	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
269	C₂²×C₈	—	2	4	no	—
270	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
272	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
273	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
274	C₄⋊Q₈	—	2	8	no	—
275	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
276	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
277	#127	C₂³×C₆₇	1	4	yes	—
278	#125	—	2	8	no	—
279	#125	C₂³×C₆₇	1	8	yes	—
280	#124	C₂³×C₆₇	1	8	yes	—
281	#123	—	2	8	no	—
282	C₂²⋊Q₈	—	2	8	no	—
283	C₂²⋊Q₈	—	2	8	no	—
284	C₂²⋊Q₈	—	2	16	no	—
285	#121	—	2	8	no	—
286	#121	—	2	8	no	—
287	#120	—	2	8	no	—
288	C₄×Q₈	—	2	8	no	—
289	C₄×D₄	—	2	8	no	—
290	#117	—	2	2	no	—
291	#117	—	2	2	no	—
292	#117	D₄×C₆₇	1	4	yes	—
293	#117	C₂³×C₆₇	1	4	yes	—
294	#117	C₂³×C₆₇	1	4	yes	—
295	#117	C₂³×C₆₇	1	4	yes	—
296	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
300	C₂×C₄²	—	2	2	no	—
301	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
302	#103	C₂³×C₆₇	1	8	yes	—
303	#103	C₂³×C₆₇	1	8	yes	—
304	#103	C₂³×C₆₇	1	16	yes	—
305	#102	C₂³×C₆₇	1	8	yes	—
306	#102	C₂³×C₆₇	1	8	yes	—
307	#102	C₂³×C₆₇	1	16	yes	—
308	#97	C₂³×C₆₇	1	8	yes	—
309	C₄×C₈	C₂³×C₆₇	1	8	yes	—
310	C₂⁴	—	2	2	no	—
311	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
312	#41	—	2	8	no	—
313	#41	—	2	8	no	—
314	#41	—	2	8	no	—
315	#41	—	2	8	no	—
316	#41	—	4	8	no	—
317	#41	—	4	8	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	2	8	no	—
320	C₂×Q₈	—	2	4	no	—
321	C₂×Q₈	—	2	4	no	—
322	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
323	C₂×D₄	—	2	4	no	—
324	C₂×D₄	—	2	4	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	8	no	—
327	C₂×D₄	—	2	4	no	—
328	C₂×D₄	—	2	4	no	—
329	C₂×D₄	—	4	8	no	—
330	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	4	8	no	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
338	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
339	C₂²×C₄	#41×C₆₇	1	4	yes	—
340	C₂²×C₄	#41×C₆₇	1	4	yes	—
341	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
342	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
343	M₁₆	—	2	8	no	—
344	C₂×C₈	—	2	8	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	4	no	—
348	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
349	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
350	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	2	8	no	—
353	C₄⋊C₄	—	2	8	no	—
354	C₄⋊C₄	—	2	16	no	—
355	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
356	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₂²⋊C₄	—	2	4	no	—
359	C₂²⋊C₄	—	2	16	no	—
360	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
361	C₄²	—	2	4	no	—
362	C₄²	—	2	4	no	—
363	C₄²	C₂×D₄×C₆₇	1	4	yes	—
364	C₄²	C₂⁴×C₆₇	1	4	yes	—
365	C₂³	—	2	4	no	—
366	C₂³	—	4	2	no	—
367	C₂³	—	4	4	no	—
368	C₂³	—	8	4	no	—
369	C₂³	#124×C₆₇	1	4	yes	—
370	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
371	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
372	Q₈	—	2	4	no	—
373	Q₈	—	2	4	no	—
374	Q₈	—	4	4	no	—
375	Q₈	—	8	8	no	—
376	D₄	—	2	4	no	—
377	D₄	—	2	4	no	—
378	D₄	—	4	8	no	—
379	D₄	—	4	8	no	—
380	D₄	—	4	4	no	—
381	D₄	—	8	8	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	4	4	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	—	4	8	no	—
387	C₂×C₄	—	4	8	no	—
388	C₂×C₄	—	8	4	no	—
389	C₂×C₄	#124×C₆₇	1	8	yes	—
390	C₂×C₄	#124×C₆₇	1	4	yes	—
391	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
392	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
394	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
395	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
396	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
397	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
398	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
399	C₈	—	2	2	no	—
400	C₈	—	2	2	no	—
401	C₈	—	2	4	no	—
402	C₂²	—	2	2	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	4	4	no	—
405	C₂²	—	4	8	no	—
406	C₂²	—	4	4	no	—
407	C₂²	—	8	8	no	—
408	C₂²	C₂×#124×C₆₇	1	4	yes	—
409	C₂²	C₂×#124×C₆₇	1	2	yes	—
410	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
411	C₄	—	2	8	no	—
412	C₄	—	4	8	no	—
413	C₄	—	4	4	no	—
414	C₄	C₂×#124×C₆₇	1	2	yes	—
415	C₄	C₂×#124×C₆₇	1	4	yes	—
416	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
417	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
418	C₂	—	2	4	no	—
419	C₂	—	2	2	no	—
420	C₂	—	8	4	no	—
421	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
422	C₂	C₆₇×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂³×D₄
prime radical	C₆₇×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3458	C₂	semidirect
3. C₆₇×#3137	C₂	semidirect
4. C₆₇×#3136	C₂	semidirect
5. C₆₇×#3135	C₂	semidirect
6. C₆₇×#3131	C₂	semidirect
7. C₆₇×#2896	C₂	semidirect
8. C₂×#416×C₆₇	C₂	semidirect
9. C₂×#405×C₆₇	C₂	semidirect
14. #544×C₆₇	C₂²	semidirect
21. #533×C₆₇	C₂²	semidirect
26. #491×C₆₇	C₂²	semidirect
27. #490×C₆₇	C₂²	semidirect
28. #489×C₆₇	C₂²	semidirect
29. #488×C₆₇	C₂²	semidirect
30. #487×C₆₇	C₂²	semidirect
31. #486×C₆₇	C₂²	semidirect
32. #419×C₆₇	C₂²	semidirect
33. #418×C₆₇	C₂²	semidirect
36. #416×C₆₇	C₂²	semidirect
37. #416×C₆₇	C₂²	semidirect
38. #416×C₆₇	C₂²	semidirect
39. C₂×#103×C₆₇	C₂²	semidirect
40. C₂×#102×C₆₇	C₂²	semidirect
41. #405×C₆₇	C₂²	semidirect
42. C₂×#97×C₆₇	C₂²	semidirect
43. C₂×C₄×C₈×C₆₇	C₂²	semidirect
206. #60612	C₆₇	direct (N × H)
212. #3458	C₂×C₆₇	semidirect
214. #3137	C₂×C₆₇	semidirect
215. #3136	C₂×C₆₇	semidirect
216. #3135	C₂×C₆₇	semidirect
217. #3131	C₂×C₆₇	semidirect
218. #2896	C₂×C₆₇	semidirect
219. C₂×#416	C₂×C₆₇	semidirect
220. C₂×#405	C₂×C₆₇	semidirect
221. C₆₇	#60612	direct (N × H)
226. #544	C₂²×C₆₇	semidirect
233. #533	C₂²×C₆₇	semidirect
238. #491	C₂²×C₆₇	semidirect
239. #490	C₂²×C₆₇	semidirect
240. #489	C₂²×C₆₇	semidirect
241. #488	C₂²×C₆₇	semidirect
242. #487	C₂²×C₆₇	semidirect
243. #486	C₂²×C₆₇	semidirect
244. #419	C₂²×C₆₇	semidirect
245. #418	C₂²×C₆₇	semidirect
248. #416	C₂²×C₆₇	semidirect
249. #416	C₂²×C₆₇	semidirect
250. #416	C₂²×C₆₇	semidirect
251. C₂×#103	C₂²×C₆₇	semidirect
252. C₂×#102	C₂²×C₆₇	semidirect
253. #405	C₂²×C₆₇	semidirect
254. C₂×#97	C₂²×C₆₇	semidirect
255. C₂×C₄×C₈	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×#3458	2
C₆₇×#3449	2
C₆₇×#3137	2
C₆₇×#3136	2
C₆₇×#3135	2
C₆₇×#3131	2
C₆₇×#2896	2
C₂×#416×C₆₇	2
C₂×#405×C₆₇	2
#60612	67

Subgroups — C67×#60612 #30739863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60612 #30739863