Subgroups — C₆₇×#17930 #30697181

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3101	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1590	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1583	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1570	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1486	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1481	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₆₇	—	2	1	no	—
11	#533×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	#483×C₆₇	C₄	1	2	yes	—
14	#476×C₆₇	C₄	1	2	yes	—
15	#461×C₆₇	C₄	1	2	yes	—
16	#450×C₆₇	C₄	1	2	yes	—
17	#446×C₆₇	—	2	1	no	—
18	#446×C₆₇	C₂²	1	2	yes	complemented
19	C₈×Q₈×C₆₇	—	2	1	no	—
20	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
21	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
22	#436×C₆₇	—	2	1	no	—
23	#436×C₆₇	—	2	1	no	—
24	#436×C₆₇	C₂²	1	2	yes	—
25	#436×C₆₇	C₂²	1	2	yes	—
26	#431×C₆₇	C₂²	1	1	yes	characteristic
27	#431×C₆₇	C₂²	1	1	yes	characteristic
28	#424×C₆₇	—	2	1	no	—
29	#423×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
32	#405×C₆₇	—	2	1	no	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	#334×C₆₇	C₂²	1	4	yes	complemented
35	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	complemented
36	D₄⋊C₈×C₆₇	C₂²	1	4	yes	—
37	#322×C₆₇	C₂²	1	2	yes	complemented
38	C₈²×C₆₇	C₂²	1	2	yes	complemented
39	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	—	2	1	no	—
41	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
42	C₂×M₁₆×C₆₇	—	2	1	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	—	4	4	no	—
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	#125×C₆₇	—	2	1	no	—
52	C₂²⋊Q₈×C₆₇	—	2	1	no	—
53	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
54	C₄×Q₈×C₆₇	—	2	1	no	—
55	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
56	C₄×D₄×C₆₇	—	2	1	no	—
57	C₄×D₄×C₆₇	—	2	1	no	—
58	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
59	C₄×D₄×C₆₇	C₂³	1	2	yes	—
60	#117×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#106×C₆₇	C₂×C₄	1	4	yes	—
65	C₄⋊C₈×C₆₇	—	2	2	no	—
66	C₄⋊C₈×C₆₇	—	2	2	no	—
67	C₄⋊C₈×C₆₇	—	2	1	no	—
68	C₄⋊C₈×C₆₇	—	2	1	no	—
69	C₄⋊C₈×C₆₇	—	2	1	no	—
70	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
71	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
72	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
73	#103×C₆₇	C₂×C₄	1	4	yes	—
74	#102×C₆₇	C₂×C₄	1	4	yes	—
75	C₂²⋊C₈×C₆₇	—	2	1	no	—
76	C₂²⋊C₈×C₆₇	—	2	2	no	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
79	#97×C₆₇	—	2	2	no	—
80	C₄×C₈×C₆₇	—	2	2	no	—
81	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
82	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	C₂³	1	2	yes	—
84	C₄×C₈×C₆₇	C₂³	1	2	yes	—
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₂⁴×C₆₇	—	2	1	no	—
88	C₂×Q₈×C₆₇	—	2	1	no	—
89	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₆₇	—	2	1	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	—	2	1	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	4	2	no	—
97	C₂²×C₄×C₆₇	—	4	2	no	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
101	SD₁₆×C₆₇	—	4	4	no	—
102	M₁₆×C₆₇	—	4	2	no	—
103	M₁₆×C₆₇	—	4	4	no	—
104	M₁₆×C₆₇	—	4	4	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	4	2	no	—
110	C₂×C₈×C₆₇	SD₁₆	1	4	yes	complemented
111	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄⋊C₄×C₆₇	—	2	1	no	—
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂²⋊C₄×C₆₇	—	2	2	no	—
125	C₂²⋊C₄×C₆₇	—	2	2	no	—
126	C₂²⋊C₄×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	#41	1	1	yes	characteristic
131	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	4	2	no	—
134	C₂³×C₆₇	—	4	2	no	—
135	C₂³×C₆₇	—	4	2	no	—
136	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
137	Q₈×C₆₇	—	2	4	no	—
138	Q₈×C₆₇	—	4	2	no	—
139	D₄×C₆₇	—	2	2	no	—
140	D₄×C₆₇	—	2	2	no	—
141	D₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	4	no	—
151	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
155	C₂×C₄×C₆₇	#135	1	2	yes	—
156	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
159	C₈×C₆₇	—	2	4	no	—
160	C₈×C₆₇	—	2	1	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	4	2	no	—
163	C₈×C₆₇	—	4	2	no	—
164	C₈×C₆₇	—	4	2	no	—
165	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
166	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	—	4	2	no	—
170	C₂²×C₆₇	—	4	2	no	—
171	C₂²×C₆₇	—	4	2	no	—
172	C₂²×C₆₇	—	8	4	no	—
173	C₂²×C₆₇	C₄×SD₁₆	1	2	yes	—
174	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	—	4	2	no	—
180	C₄×C₆₇	—	4	2	no	—
181	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
182	C₄×C₆₇	#568	1	2	yes	—
183	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
184	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
185	#17930	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂×C₆₇	—	2	2	no	—
187	C₂×C₆₇	—	8	2	no	—
188	C₂×C₆₇	#2930	1	1	yes	characteristic
189	C₂×C₆₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
190	C₂×C₆₇	#2955	1	1	yes	characteristic
191	#3101	C₂×C₆₇	1	1	yes	characteristic
192	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
193	C₂×#436	C₂×C₆₇	1	1	yes	characteristic
194	#1590	C₂×C₆₇	1	4	yes	complemented
195	#1583	C₂×C₆₇	1	4	yes	complemented
196	#1570	C₂×C₆₇	1	1	yes	characteristic, complemented
197	#1486	C₂×C₆₇	1	1	yes	characteristic
198	#1481	C₂×C₆₇	1	1	yes	characteristic, complemented
199	#1450	C₂×C₆₇	1	1	yes	characteristic, complemented
200	C₆₇	#17930	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²×M₁₆	—	2	1	no	—
202	#533	C₂²×C₆₇	1	1	yes	characteristic
203	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
204	#483	C₄×C₆₇	1	2	yes	—
205	#476	C₄×C₆₇	1	2	yes	—
206	#461	C₄×C₆₇	1	2	yes	—
207	#450	C₄×C₆₇	1	2	yes	—
208	#446	—	2	1	no	—
209	#446	C₂²×C₆₇	1	2	yes	complemented
210	C₈×Q₈	—	2	1	no	—
211	C₈×Q₈	C₂²×C₆₇	1	2	yes	complemented
212	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	—
213	#436	—	2	1	no	—
214	#436	—	2	1	no	—
215	#436	C₂²×C₆₇	1	2	yes	—
216	#436	C₂²×C₆₇	1	2	yes	—
217	#431	C₂²×C₆₇	1	1	yes	characteristic
218	#431	C₂²×C₆₇	1	1	yes	characteristic
219	#424	—	2	1	no	—
220	#423	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×C₂²⋊C₈	—	2	1	no	—
223	#405	—	2	1	no	—
224	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
225	#334	C₂²×C₆₇	1	4	yes	complemented
226	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	complemented
227	D₄⋊C₈	C₂²×C₆₇	1	4	yes	—
228	#322	C₂²×C₆₇	1	2	yes	complemented
229	C₈²	C₂²×C₆₇	1	2	yes	complemented
230	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
231	C₂³×C₄	—	2	1	no	—
232	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
233	C₂×M₁₆	—	2	1	no	—
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	—	4	4	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
239	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
240	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
241	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
242	#125	—	2	1	no	—
243	C₂²⋊Q₈	—	2	1	no	—
244	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
245	C₄×Q₈	—	2	1	no	—
246	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
247	C₄×D₄	—	2	1	no	—
248	C₄×D₄	—	2	1	no	—
249	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
250	C₄×D₄	C₂³×C₆₇	1	2	yes	—
251	#117	—	2	1	no	—
252	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
253	C₂×C₂²⋊C₄	—	2	1	no	—
254	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
255	#106	C₂×C₄×C₆₇	1	4	yes	—
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
262	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
263	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
264	#103	C₂×C₄×C₆₇	1	4	yes	—
265	#102	C₂×C₄×C₆₇	1	4	yes	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	2	2	no	—
268	C₂²⋊C₈	—	2	2	no	—
269	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
270	#97	—	2	2	no	—
271	C₄×C₈	—	2	2	no	—
272	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
273	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
274	C₄×C₈	C₂³×C₆₇	1	2	yes	—
275	C₄×C₈	C₂³×C₆₇	1	2	yes	—
276	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
277	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
278	C₂⁴	—	2	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
291	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
292	SD₁₆	—	4	4	no	—
293	M₁₆	—	4	2	no	—
294	M₁₆	—	4	4	no	—
295	M₁₆	—	4	4	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	SD₁₆×C₆₇	1	4	yes	complemented
302	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
303	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
304	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
314	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
320	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
321	C₄²	#41×C₆₇	1	1	yes	characteristic
322	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
323	C₂³	—	2	2	no	—
324	C₂³	—	4	2	no	—
325	C₂³	—	4	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
328	Q₈	—	2	4	no	—
329	Q₈	—	4	2	no	—
330	D₄	—	2	2	no	—
331	D₄	—	2	2	no	—
332	D₄	—	4	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
343	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
344	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
346	C₂×C₄	#135×C₆₇	1	2	yes	—
347	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
349	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₈	—	4	2	no	—
356	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
357	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	8	4	no	—
364	C₂²	C₄×SD₁₆×C₆₇	1	2	yes	—
365	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	C₄×SD₁₆×C₆₇	1	2	yes	—
373	C₄	#568×C₆₇	1	2	yes	—
374	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
375	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
376	C₂	—	2	2	no	—
377	C₂	—	8	2	no	—
378	C₂	C₆₇×#2930	1	1	yes	characteristic
379	C₂	C₂×C₄×SD₁₆×C₆₇	1	1	yes	characteristic
380	C₂	C₆₇×#2955	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	C₂²×D₄×C₆₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2960	C₂	semidirect
4. C₆₇×#1590	C₂	semidirect
5. C₆₇×#1583	C₂	semidirect
6. C₆₇×#1570	C₂	semidirect
8. C₆₇×#1481	C₂	semidirect
9. C₆₇×#1450	C₂	semidirect
18. #446×C₆₇	C₂²	semidirect
20. C₈×Q₈×C₆₇	C₂²	semidirect
34. #334×C₆₇	C₂²	semidirect
35. Q₈⋊C₈×C₆₇	C₂²	semidirect
37. #322×C₆₇	C₂²	semidirect
38. C₈²×C₆₇	C₂²	semidirect
110. C₂×C₈×C₆₇	SD₁₆	semidirect
185. #17930	C₆₇	direct (N × H)
192. #2960	C₂×C₆₇	semidirect
194. #1590	C₂×C₆₇	semidirect
195. #1583	C₂×C₆₇	semidirect
196. #1570	C₂×C₆₇	semidirect
198. #1481	C₂×C₆₇	semidirect
199. #1450	C₂×C₆₇	semidirect
200. C₆₇	#17930	direct (N × H)
209. #446	C₂²×C₆₇	semidirect
211. C₈×Q₈	C₂²×C₆₇	semidirect
225. #334	C₂²×C₆₇	semidirect
226. Q₈⋊C₈	C₂²×C₆₇	semidirect
228. #322	C₂²×C₆₇	semidirect
229. C₈²	C₂²×C₆₇	semidirect
301. C₂×C₈	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3101	2
C₆₇×#2960	2
C₂×#436×C₆₇	2
C₆₇×#1590	2
C₆₇×#1583	2
C₆₇×#1570	2
C₆₇×#1486	2
C₆₇×#1481	2
C₆₇×#1450	2
#17930	67

Subgroups — C67×#17930 #30697181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17930 #30697181