Subgroups — C₆₇×#18071 #30697322

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3167	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2935	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1592	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1588	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1585	C₂	1	2	yes	maximal, complemented
7	C₆₇×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1481	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1480	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₆₇	—	2	1	no	—
13	#533×C₆₇	C₂²	1	1	yes	characteristic
14	#532×C₆₇	C₂²	1	1	yes	characteristic
15	#498×C₆₇	C₄	1	1	yes	characteristic
16	#492×C₆₇	C₄	1	1	yes	characteristic
17	#487×C₆₇	C₄	1	1	yes	characteristic
18	#486×C₆₇	C₄	1	1	yes	characteristic
19	#465×C₆₇	C₄	1	2	yes	—
20	#465×C₆₇	C₄	1	2	yes	—
21	#446×C₆₇	—	2	2	no	—
22	#446×C₆₇	C₂²	1	2	yes	—
23	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
25	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
26	#436×C₆₇	C₂²	1	2	yes	—
27	#435×C₆₇	—	2	2	no	—
28	C₈×D₄×C₆₇	C₂²	1	2	yes	—
29	#431×C₆₇	C₂²	1	1	yes	characteristic
30	#424×C₆₇	—	2	1	no	—
31	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#408×C₆₇	—	2	1	no	—
34	#405×C₆₇	—	2	1	no	—
35	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#335×C₆₇	C₂²	1	1	yes	characteristic
38	#335×C₆₇	C₂²	1	1	yes	characteristic
39	#335×C₆₇	C₂²	1	1	yes	characteristic
40	#335×C₆₇	C₂²	1	1	yes	characteristic
41	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
42	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
44	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
45	#322×C₆₇	C₂²	1	2	yes	—
46	#322×C₆₇	C₂²	1	2	yes	—
47	C₂×#41×C₆₇	—	2	1	no	—
48	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	#131×C₆₇	—	4	4	no	—
51	C₂×M₁₆×C₆₇	—	2	2	no	—
52	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	—	2	1	no	—
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	—	2	2	no	—
57	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
63	#125×C₆₇	—	2	1	no	—
64	#124×C₆₇	—	2	1	no	—
65	C₂²⋊Q₈×C₆₇	—	2	1	no	—
66	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
67	#121×C₆₇	—	2	1	no	—
68	#121×C₆₇	C₂×C₄	1	2	yes	—
69	C₄×Q₈×C₆₇	—	2	1	no	—
70	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
71	C₄×D₄×C₆₇	—	2	1	no	—
72	C₄×D₄×C₆₇	C₂³	1	2	yes	—
73	#117×C₆₇	—	2	1	no	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	#106×C₆₇	C₂×C₄	1	2	yes	—
76	#106×C₆₇	C₂×C₄	1	2	yes	—
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	1	no	—
80	C₄⋊C₈×C₆₇	—	2	1	no	—
81	C₄⋊C₈×C₆₇	—	2	1	no	—
82	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
86	#103×C₆₇	C₂×C₄	1	2	yes	—
87	#103×C₆₇	C₂×C₄	1	2	yes	—
88	#102×C₆₇	C₂×C₄	1	2	yes	—
89	#102×C₆₇	C₂×C₄	1	2	yes	—
90	C₂²⋊C₈×C₆₇	—	2	1	no	—
91	C₂²⋊C₈×C₆₇	—	2	2	no	—
92	C₂²⋊C₈×C₆₇	—	2	2	no	—
93	#97×C₆₇	—	2	2	no	—
94	#97×C₆₇	D₄	1	2	yes	—
95	C₄×C₈×C₆₇	—	2	2	no	—
96	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₄×C₈×C₆₇	C₂³	1	2	yes	—
99	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₆₇	C₂³	1	2	yes	—
102	#41×C₆₇	—	4	2	no	—
103	#41×C₆₇	—	4	2	no	—
104	C₂×Q₈×C₆₇	—	2	1	no	—
105	C₂×Q₈×C₆₇	—	2	1	no	—
106	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	1	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
115	SD₁₆×C₆₇	—	2	4	no	—
116	SD₁₆×C₆₇	—	4	2	no	—
117	M₁₆×C₆₇	—	2	2	no	—
118	M₁₆×C₆₇	—	2	2	no	—
119	M₁₆×C₆₇	—	4	4	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	4	1	no	—
127	C₂×C₈×C₆₇	—	4	4	no	—
128	C₂×C₈×C₆₇	—	4	2	no	—
129	C₂×C₈×C₆₇	—	4	2	no	—
130	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	2	1	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
143	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂²⋊C₄×C₆₇	—	2	2	no	—
145	C₂²⋊C₄×C₆₇	—	2	2	no	—
146	C₂²⋊C₄×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₆₇	#41	1	1	yes	characteristic
151	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
154	Q₈×C₆₇	—	2	2	no	—
155	Q₈×C₆₇	—	2	2	no	—
156	Q₈×C₆₇	—	4	2	no	—
157	Q₈×C₆₇	—	8	1	no	—
158	D₄×C₆₇	—	2	2	no	—
159	D₄×C₆₇	—	2	2	no	—
160	D₄×C₆₇	—	4	4	no	—
161	D₄×C₆₇	—	4	2	no	—
162	D₄×C₆₇	—	8	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
175	C₂×C₄×C₆₇	#137	1	2	yes	—
176	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
179	C₈×C₆₇	—	2	1	no	—
180	C₈×C₆₇	—	2	1	no	—
181	C₈×C₆₇	—	2	1	no	—
182	C₈×C₆₇	—	2	2	no	—
183	C₈×C₆₇	—	2	2	no	—
184	C₈×C₆₇	—	4	2	no	—
185	C₈×C₆₇	—	4	1	no	—
186	C₈×C₆₇	—	4	1	no	—
187	C₈×C₆₇	—	4	1	no	—
188	C₈×C₆₇	—	4	1	no	—
189	C₈×C₆₇	#131	1	2	yes	complemented
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	#440	1	2	yes	—
196	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
197	C₄×C₆₇	—	2	1	no	—
198	C₄×C₆₇	—	2	1	no	—
199	C₄×C₆₇	—	2	1	no	—
200	C₄×C₆₇	—	4	2	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	—	4	2	no	—
203	C₄×C₆₇	#440	1	2	yes	—
204	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
205	C₄×C₆₇	#568	1	1	yes	characteristic
206	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
208	#18071	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂×C₆₇	—	2	2	no	—
210	C₂×C₆₇	—	8	2	no	—
211	C₂×C₆₇	#2935	1	1	yes	characteristic
212	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
213	C₂×C₆₇	#2957	1	1	yes	characteristic
214	#3167	C₂×C₆₇	1	1	yes	characteristic
215	#2962	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#2935	C₂×C₆₇	1	1	yes	characteristic
217	#1592	C₂×C₆₇	1	2	yes	complemented
218	#1588	C₂×C₆₇	1	2	yes	complemented
219	#1585	C₂×C₆₇	1	2	yes	complemented
220	SD₁₆⋊C₈	C₂×C₆₇	1	2	yes	complemented
221	C₂×#335	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#1481	C₂×C₆₇	1	1	yes	characteristic, complemented
223	#1480	C₂×C₆₇	1	1	yes	characteristic
224	#1451	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₆₇	#18071	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×#131	—	2	1	no	—
227	#533	C₂²×C₆₇	1	1	yes	characteristic
228	#532	C₂²×C₆₇	1	1	yes	characteristic
229	#498	C₄×C₆₇	1	1	yes	characteristic
230	#492	C₄×C₆₇	1	1	yes	characteristic
231	#487	C₄×C₆₇	1	1	yes	characteristic
232	#486	C₄×C₆₇	1	1	yes	characteristic
233	#465	C₄×C₆₇	1	2	yes	—
234	#465	C₄×C₆₇	1	2	yes	—
235	#446	—	2	2	no	—
236	#446	C₂²×C₆₇	1	2	yes	—
237	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
238	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
239	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
240	#436	C₂²×C₆₇	1	2	yes	—
241	#435	—	2	2	no	—
242	C₈×D₄	C₂²×C₆₇	1	2	yes	—
243	#431	C₂²×C₆₇	1	1	yes	characteristic
244	#424	—	2	1	no	—
245	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
247	#408	—	2	1	no	—
248	#405	—	2	1	no	—
249	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
251	#335	C₂²×C₆₇	1	1	yes	characteristic
252	#335	C₂²×C₆₇	1	1	yes	characteristic
253	#335	C₂²×C₆₇	1	1	yes	characteristic
254	#335	C₂²×C₆₇	1	1	yes	characteristic
255	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
256	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
257	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
258	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
259	#322	C₂²×C₆₇	1	2	yes	—
260	#322	C₂²×C₆₇	1	2	yes	—
261	C₂×#41	—	2	1	no	—
262	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
263	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
264	#131	—	4	4	no	—
265	C₂×M₁₆	—	2	2	no	—
266	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
267	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
268	C₂²×C₈	—	2	1	no	—
269	C₂²×C₈	—	2	1	no	—
270	C₂²×C₈	—	2	2	no	—
271	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
272	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
273	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
274	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
275	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
276	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
277	#125	—	2	1	no	—
278	#124	—	2	1	no	—
279	C₂²⋊Q₈	—	2	1	no	—
280	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
281	#121	—	2	1	no	—
282	#121	C₂×C₄×C₆₇	1	2	yes	—
283	C₄×Q₈	—	2	1	no	—
284	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
285	C₄×D₄	—	2	1	no	—
286	C₄×D₄	C₂³×C₆₇	1	2	yes	—
287	#117	—	2	1	no	—
288	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
289	#106	C₂×C₄×C₆₇	1	2	yes	—
290	#106	C₂×C₄×C₆₇	1	2	yes	—
291	C₄⋊C₈	—	2	2	no	—
292	C₄⋊C₈	—	2	2	no	—
293	C₄⋊C₈	—	2	1	no	—
294	C₄⋊C₈	—	2	1	no	—
295	C₄⋊C₈	—	2	1	no	—
296	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
298	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
299	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
300	#103	C₂×C₄×C₆₇	1	2	yes	—
301	#103	C₂×C₄×C₆₇	1	2	yes	—
302	#102	C₂×C₄×C₆₇	1	2	yes	—
303	#102	C₂×C₄×C₆₇	1	2	yes	—
304	C₂²⋊C₈	—	2	1	no	—
305	C₂²⋊C₈	—	2	2	no	—
306	C₂²⋊C₈	—	2	2	no	—
307	#97	—	2	2	no	—
308	#97	D₄×C₆₇	1	2	yes	—
309	C₄×C₈	—	2	2	no	—
310	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
311	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
312	C₄×C₈	C₂³×C₆₇	1	2	yes	—
313	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
314	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
315	C₄×C₈	C₂³×C₆₇	1	2	yes	—
316	#41	—	4	2	no	—
317	#41	—	4	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
327	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
329	SD₁₆	—	2	4	no	—
330	SD₁₆	—	4	2	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	2	2	no	—
333	M₁₆	—	4	4	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	4	1	no	—
341	C₂×C₈	—	4	4	no	—
342	C₂×C₈	—	4	2	no	—
343	C₂×C₈	—	4	2	no	—
344	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
345	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
346	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
347	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
348	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
349	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
357	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	2	2	no	—
360	C₂²⋊C₄	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
363	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
364	C₄²	#41×C₆₇	1	1	yes	characteristic
365	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
366	C₂³	—	2	2	no	—
367	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
368	Q₈	—	2	2	no	—
369	Q₈	—	2	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	8	1	no	—
372	D₄	—	2	2	no	—
373	D₄	—	2	2	no	—
374	D₄	—	4	4	no	—
375	D₄	—	4	2	no	—
376	D₄	—	8	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
386	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
387	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
388	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
389	C₂×C₄	#137×C₆₇	1	2	yes	—
390	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
392	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
393	C₈	—	2	1	no	—
394	C₈	—	2	1	no	—
395	C₈	—	2	1	no	—
396	C₈	—	2	2	no	—
397	C₈	—	2	2	no	—
398	C₈	—	4	2	no	—
399	C₈	—	4	1	no	—
400	C₈	—	4	1	no	—
401	C₈	—	4	1	no	—
402	C₈	—	4	1	no	—
403	C₈	#131×C₆₇	1	2	yes	complemented
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	#440×C₆₇	1	2	yes	—
410	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	2	no	—
417	C₄	#440×C₆₇	1	2	yes	—
418	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
419	C₄	#568×C₆₇	1	1	yes	characteristic
420	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
421	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
422	C₂	—	2	2	no	—
423	C₂	—	8	2	no	—
424	C₂	C₆₇×#2935	1	1	yes	characteristic
425	C₂	C₂×#440×C₆₇	1	1	yes	characteristic
426	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#532×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₄×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2962	C₂	semidirect
4. C₆₇×#1592	C₂	semidirect
5. C₆₇×#1588	C₂	semidirect
6. C₆₇×#1585	C₂	semidirect
7. C₆₇×SD₁₆⋊C₈	C₂	semidirect
8. C₂×#335×C₆₇	C₂	semidirect
9. C₆₇×#1481	C₂	semidirect
11. C₆₇×#1451	C₂	semidirect
189. C₈×C₆₇	#131	semidirect
208. #18071	C₆₇	direct (N × H)
215. #2962	C₂×C₆₇	semidirect
217. #1592	C₂×C₆₇	semidirect
218. #1588	C₂×C₆₇	semidirect
219. #1585	C₂×C₆₇	semidirect
220. SD₁₆⋊C₈	C₂×C₆₇	semidirect
221. C₂×#335	C₂×C₆₇	semidirect
222. #1481	C₂×C₆₇	semidirect
224. #1451	C₂×C₆₇	semidirect
225. C₆₇	#18071	direct (N × H)
403. C₈	#131×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3167	2
C₆₇×#2962	2
C₆₇×#2935	2
C₆₇×#1592	2
C₆₇×#1588	2
C₆₇×#1585	2
C₆₇×SD₁₆⋊C₈	2
C₂×#335×C₆₇	2
C₆₇×#1481	2
C₆₇×#1480	2
C₆₇×#1451	2
#18071	67

Subgroups — C67×#18071 #30697322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18071 #30697322