Subgroups — C₆₇×#32916 #30712167

Order 17152.

1406 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#462×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#452×C₆₇	C₂	1	2	yes	maximal
5	C₆₇×#2621	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1878	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1872	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1805	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1804	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
17	#520×C₆₇	—	2	4	no	—
18	#520×C₆₇	—	2	4	no	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	—	2	1	no	—
22	C₂×C₄×D₄×C₆₇	—	2	1	no	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₆₇	—	2	1	no	—
26	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#462×C₆₇	—	2	4	no	—
30	#461×C₆₇	—	2	4	no	—
31	#452×C₆₇	—	2	8	no	—
32	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#103×C₆₇	C₂²	1	2	yes	—
35	C₂×#103×C₆₇	C₂²	1	2	yes	—
36	C₂×#103×C₆₇	C₂²	1	2	yes	—
37	C₂×#102×C₆₇	C₂²	1	2	yes	—
38	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
39	#398×C₆₇	—	2	1	no	—
40	#398×C₆₇	—	2	1	no	—
41	#397×C₆₇	—	2	1	no	—
42	#397×C₆₇	—	2	1	no	—
43	#391×C₆₇	C₂²	1	2	yes	—
44	#382×C₆₇	C₂²	1	1	yes	characteristic
45	#340×C₆₇	C₂²	1	2	yes	—
46	#340×C₆₇	C₂²	1	2	yes	—
47	#336×C₆₇	C₂²	1	2	yes	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	—	2	2	no	—
50	C₂×#41×C₆₇	—	2	2	no	—
51	C₂×#41×C₆₇	—	2	2	no	—
52	C₂×#41×C₆₇	—	2	2	no	—
53	C₂×#41×C₆₇	D₄	1	4	yes	—
54	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₆₇	—	2	1	no	—
57	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	—	2	1	no	—
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂³×C₄×C₆₇	—	2	1	no	—
61	C₂³×C₄×C₆₇	—	2	1	no	—
62	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×Q₁₆×C₆₇	—	2	4	no	—
64	C₂×Q₁₆×C₆₇	—	2	4	no	—
65	C₂×Q₁₆×C₆₇	—	2	4	no	—
66	C₂×SD₁₆×C₆₇	—	2	4	no	—
67	C₂×SD₁₆×C₆₇	—	2	4	no	—
68	C₂×SD₁₆×C₆₇	—	2	4	no	—
69	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
70	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
71	C₄⋊Q₈×C₆₇	—	2	2	no	—
72	C₄⋊Q₈×C₆₇	—	2	2	no	—
73	C₂²⋊Q₈×C₆₇	—	2	4	no	—
74	C₂²⋊Q₈×C₆₇	—	2	4	no	—
75	C₄×Q₈×C₆₇	—	2	2	no	—
76	C₄×Q₈×C₆₇	—	2	2	no	—
77	C₄×Q₈×C₆₇	—	2	2	no	—
78	C₄×Q₈×C₆₇	D₄	1	4	yes	—
79	C₄×D₄×C₆₇	—	2	8	no	—
80	C₄×D₄×C₆₇	—	2	8	no	—
81	C₄×D₄×C₆₇	—	2	2	no	—
82	C₄×D₄×C₆₇	—	2	2	no	—
83	C₄×D₄×C₆₇	—	2	2	no	—
84	C₄×D₄×C₆₇	D₄	1	4	yes	—
85	#117×C₆₇	—	2	8	no	—
86	#117×C₆₇	—	2	2	no	—
87	#117×C₆₇	—	2	2	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
90	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
96	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
97	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₆₇	—	2	1	no	—
108	C₂×C₄²×C₆₇	—	2	1	no	—
109	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
111	C₄⋊C₈×C₆₇	—	2	2	no	—
112	C₄⋊C₈×C₆₇	—	2	2	no	—
113	#103×C₆₇	—	2	4	no	—
114	#103×C₆₇	—	2	4	no	—
115	#103×C₆₇	—	2	4	no	—
116	#102×C₆₇	—	2	4	no	—
117	C₂²⋊C₈×C₆₇	—	2	4	no	—
118	#95×C₆₇	—	2	1	no	—
119	#95×C₆₇	—	2	1	no	—
120	#95×C₆₇	—	2	2	no	—
121	#95×C₆₇	—	2	2	no	—
122	C₂⁴×C₆₇	—	2	1	no	—
123	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
124	#41×C₆₇	—	2	8	no	—
125	#41×C₆₇	—	2	8	no	—
126	#41×C₆₇	—	2	8	no	—
127	#41×C₆₇	—	2	8	no	—
128	C₂×Q₈×C₆₇	—	2	1	no	—
129	C₂×Q₈×C₆₇	—	2	1	no	—
130	C₂×Q₈×C₆₇	—	2	1	no	—
131	C₂×Q₈×C₆₇	—	2	2	no	—
132	C₂×Q₈×C₆₇	—	2	2	no	—
133	C₂×Q₈×C₆₇	—	2	2	no	—
134	C₂×Q₈×C₆₇	—	2	2	no	—
135	C₂×Q₈×C₆₇	—	2	2	no	—
136	C₂×Q₈×C₆₇	—	2	2	no	—
137	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
138	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×Q₈×C₆₇	#41	1	2	yes	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	—	2	4	no	—
142	C₂×D₄×C₆₇	—	2	4	no	—
143	C₂×D₄×C₆₇	—	2	1	no	—
144	C₂×D₄×C₆₇	—	2	1	no	—
145	C₂×D₄×C₆₇	—	2	1	no	—
146	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
148	C₂×D₄×C₆₇	#41	1	2	yes	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	—	2	4	no	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	—	2	1	no	—
156	C₂²×C₄×C₆₇	—	2	1	no	—
157	C₂²×C₄×C₆₇	—	2	1	no	—
158	C₂²×C₄×C₆₇	—	2	1	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	2	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	—	8	2	no	—
165	C₂²×C₄×C₆₇	—	8	2	no	—
166	C₂²×C₄×C₆₇	—	8	2	no	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₆₇	#41	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	Q₁₆×C₆₇	—	4	8	no	—
176	SD₁₆×C₆₇	—	4	8	no	—
177	C₂×C₈×C₆₇	—	2	2	no	—
178	C₂×C₈×C₆₇	—	2	2	no	—
179	C₂×C₈×C₆₇	—	2	2	no	—
180	C₂×C₈×C₆₇	—	2	2	no	—
181	C₂×C₈×C₆₇	—	2	2	no	—
182	C₂×C₈×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	2	4	no	—
184	C₄⋊C₄×C₆₇	—	2	4	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	4	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	—	2	1	no	—
189	C₄⋊C₄×C₆₇	—	2	1	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	4	2	no	—
193	C₄⋊C₄×C₆₇	—	4	2	no	—
194	C₄⋊C₄×C₆₇	—	4	4	no	—
195	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
196	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	2	4	no	—
199	C₂²⋊C₄×C₆₇	—	2	2	no	—
200	C₂²⋊C₄×C₆₇	—	2	2	no	—
201	C₂²⋊C₄×C₆₇	—	4	4	no	—
202	C₄²×C₆₇	—	2	4	no	—
203	C₄²×C₆₇	—	2	4	no	—
204	C₄²×C₆₇	—	2	1	no	—
205	C₄²×C₆₇	—	2	1	no	—
206	C₄²×C₆₇	C₂×D₄	1	2	yes	—
207	C₄²×C₆₇	C₂×D₄	1	2	yes	—
208	C₂³×C₆₇	—	2	1	no	—
209	C₂³×C₆₇	—	2	2	no	—
210	C₂³×C₆₇	—	2	2	no	—
211	C₂³×C₆₇	—	2	1	no	—
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	2	2	no	—
214	C₂³×C₆₇	—	2	2	no	—
215	C₂³×C₆₇	—	4	2	no	—
216	C₂³×C₆₇	—	4	2	no	—
217	C₂³×C₆₇	#121	1	2	yes	—
218	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₆₇	—	2	2	no	—
220	Q₈×C₆₇	—	2	2	no	—
221	Q₈×C₆₇	—	2	2	no	—
222	Q₈×C₆₇	—	4	2	no	—
223	Q₈×C₆₇	—	4	2	no	—
224	Q₈×C₆₇	—	4	2	no	—
225	Q₈×C₆₇	—	4	2	no	—
226	Q₈×C₆₇	#121	1	4	yes	—
227	D₄×C₆₇	—	2	16	no	—
228	D₄×C₆₇	—	2	2	no	—
229	D₄×C₆₇	—	2	2	no	—
230	D₄×C₆₇	—	2	2	no	—
231	D₄×C₆₇	#121	1	4	yes	—
232	C₂×C₄×C₆₇	—	2	16	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	1	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	1	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	1	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	4	2	no	—
251	C₂×C₄×C₆₇	—	4	2	no	—
252	C₂×C₄×C₆₇	—	4	2	no	—
253	C₂×C₄×C₆₇	—	4	2	no	—
254	C₂×C₄×C₆₇	—	4	2	no	—
255	C₂×C₄×C₆₇	—	4	2	no	—
256	C₂×C₄×C₆₇	—	4	2	no	—
257	C₂×C₄×C₆₇	—	4	2	no	—
258	C₂×C₄×C₆₇	—	8	4	no	—
259	C₂×C₄×C₆₇	—	8	4	no	—
260	C₂×C₄×C₆₇	—	8	4	no	—
261	C₂×C₄×C₆₇	#120	1	2	yes	—
262	C₂×C₄×C₆₇	#120	1	2	yes	—
263	C₂×C₄×C₆₇	#121	1	2	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
268	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
269	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
270	C₈×C₆₇	—	4	2	no	—
271	C₈×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	2	2	no	—
273	C₂²×C₆₇	—	2	4	no	—
274	C₂²×C₆₇	—	2	4	no	—
275	C₂²×C₆₇	—	4	2	no	—
276	C₂²×C₆₇	—	4	4	no	—
277	C₂²×C₆₇	—	4	1	no	—
278	C₂²×C₆₇	—	4	2	no	—
279	C₂²×C₆₇	—	4	1	no	—
280	C₂²×C₆₇	—	4	2	no	—
281	C₂²×C₆₇	—	4	4	no	—
282	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
283	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
284	C₂²×C₆₇	#525	1	1	yes	characteristic
285	C₂²×C₆₇	#577	1	2	yes	—
286	C₂²×C₆₇	#578	1	2	yes	—
287	C₄×C₆₇	—	2	2	no	—
288	C₄×C₆₇	—	2	4	no	—
289	C₄×C₆₇	—	2	2	no	—
290	C₄×C₆₇	—	2	2	no	—
291	C₄×C₆₇	—	4	1	no	—
292	C₄×C₆₇	—	4	2	no	—
293	C₄×C₆₇	—	4	2	no	—
294	C₄×C₆₇	—	4	1	no	—
295	C₄×C₆₇	—	8	2	no	—
296	C₄×C₆₇	—	8	2	no	—
297	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
298	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
299	C₄×C₆₇	#525	1	1	yes	characteristic
300	C₄×C₆₇	#525	1	1	yes	characteristic
301	#32916	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
302	C₂×C₆₇	—	4	2	no	—
303	C₂×C₆₇	—	4	4	no	—
304	C₂×C₆₇	#2847	1	1	yes	characteristic
305	C₂×C₆₇	#3009	1	1	yes	characteristic
306	C₂×C₆₇	#3011	1	1	yes	characteristic
307	C₂×C₆₇	#3046	1	2	yes	—
308	C₂×C₆₇	#3120	1	2	yes	—
309	C₂×#520	C₂×C₆₇	1	1	yes	characteristic
310	C₂×#462	C₂×C₆₇	1	1	yes	characteristic, complemented
311	C₂×#461	C₂×C₆₇	1	1	yes	characteristic, complemented
312	C₂×#452	C₂×C₆₇	1	2	yes	—
313	#2621	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#1878	C₂×C₆₇	1	2	yes	complemented
315	#1877	C₂×C₆₇	1	2	yes	complemented
316	#1872	C₂×C₆₇	1	2	yes	complemented
317	#1805	C₂×C₆₇	1	1	yes	characteristic, complemented
318	#1804	C₂×C₆₇	1	1	yes	characteristic, complemented
319	#1798	C₂×C₆₇	1	1	yes	characteristic, complemented
320	C₆₇	#32916	1	1	yes	characteristic, direct product factor, Sylow
321	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
323	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
324	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
326	#520	—	2	4	no	—
327	#520	—	2	4	no	—
328	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
330	C₂×C₄×D₄	—	2	1	no	—
331	C₂×C₄×D₄	—	2	1	no	—
332	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
334	C₂×#117	—	2	1	no	—
335	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
336	C₂²×C₄⋊C₄	—	2	1	no	—
337	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
338	#462	—	2	4	no	—
339	#461	—	2	4	no	—
340	#452	—	2	8	no	—
341	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
342	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
343	C₂×#103	C₂²×C₆₇	1	2	yes	—
344	C₂×#103	C₂²×C₆₇	1	2	yes	—
345	C₂×#103	C₂²×C₆₇	1	2	yes	—
346	C₂×#102	C₂²×C₆₇	1	2	yes	—
347	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
348	#398	—	2	1	no	—
349	#398	—	2	1	no	—
350	#397	—	2	1	no	—
351	#397	—	2	1	no	—
352	#391	C₂²×C₆₇	1	2	yes	—
353	#382	C₂²×C₆₇	1	1	yes	characteristic
354	#340	C₂²×C₆₇	1	2	yes	—
355	#340	C₂²×C₆₇	1	2	yes	—
356	#336	C₂²×C₆₇	1	2	yes	—
357	C₂×#41	—	2	2	no	—
358	C₂×#41	—	2	2	no	—
359	C₂×#41	—	2	2	no	—
360	C₂×#41	—	2	2	no	—
361	C₂×#41	—	2	2	no	—
362	C₂×#41	D₄×C₆₇	1	4	yes	—
363	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
364	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
365	C₂²×D₄	—	2	1	no	—
366	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
367	C₂³×C₄	—	2	1	no	—
368	C₂³×C₄	—	2	1	no	—
369	C₂³×C₄	—	2	1	no	—
370	C₂³×C₄	—	2	1	no	—
371	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
372	C₂×Q₁₆	—	2	4	no	—
373	C₂×Q₁₆	—	2	4	no	—
374	C₂×Q₁₆	—	2	4	no	—
375	C₂×SD₁₆	—	2	4	no	—
376	C₂×SD₁₆	—	2	4	no	—
377	C₂×SD₁₆	—	2	4	no	—
378	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
379	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
380	C₄⋊Q₈	—	2	2	no	—
381	C₄⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	—	2	4	no	—
383	C₂²⋊Q₈	—	2	4	no	—
384	C₄×Q₈	—	2	2	no	—
385	C₄×Q₈	—	2	2	no	—
386	C₄×Q₈	—	2	2	no	—
387	C₄×Q₈	D₄×C₆₇	1	4	yes	—
388	C₄×D₄	—	2	8	no	—
389	C₄×D₄	—	2	8	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	D₄×C₆₇	1	4	yes	—
394	#117	—	2	8	no	—
395	#117	—	2	2	no	—
396	#117	—	2	2	no	—
397	C₂×C₄⋊C₄	—	2	8	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	—	2	2	no	—
402	C₂×C₄⋊C₄	—	2	2	no	—
403	C₂×C₄⋊C₄	—	2	2	no	—
404	C₂×C₄⋊C₄	—	2	1	no	—
405	C₂×C₄⋊C₄	—	2	1	no	—
406	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
407	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	—	2	2	no	—
414	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
415	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
416	C₂×C₄²	—	2	1	no	—
417	C₂×C₄²	—	2	1	no	—
418	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
420	C₄⋊C₈	—	2	2	no	—
421	C₄⋊C₈	—	2	2	no	—
422	#103	—	2	4	no	—
423	#103	—	2	4	no	—
424	#103	—	2	4	no	—
425	#102	—	2	4	no	—
426	C₂²⋊C₈	—	2	4	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	#95	—	2	2	no	—
430	#95	—	2	2	no	—
431	C₂⁴	—	2	1	no	—
432	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
433	#41	—	2	8	no	—
434	#41	—	2	8	no	—
435	#41	—	2	8	no	—
436	#41	—	2	8	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	2	no	—
441	C₂×Q₈	—	2	2	no	—
442	C₂×Q₈	—	2	2	no	—
443	C₂×Q₈	—	2	2	no	—
444	C₂×Q₈	—	2	2	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
447	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
448	C₂×Q₈	#41×C₆₇	1	2	yes	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
456	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
457	C₂×D₄	#41×C₆₇	1	2	yes	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	8	2	no	—
474	C₂²×C₄	—	8	2	no	—
475	C₂²×C₄	—	8	2	no	—
476	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
482	C₂²×C₄	#41×C₆₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
484	Q₁₆	—	4	8	no	—
485	SD₁₆	—	4	8	no	—
486	C₂×C₈	—	2	2	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	2	2	no	—
489	C₂×C₈	—	2	2	no	—
490	C₂×C₈	—	2	2	no	—
491	C₂×C₈	—	2	2	no	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	4	2	no	—
502	C₄⋊C₄	—	4	2	no	—
503	C₄⋊C₄	—	4	4	no	—
504	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
505	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	4	4	no	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	4	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	C₂×D₄×C₆₇	1	2	yes	—
516	C₄²	C₂×D₄×C₆₇	1	2	yes	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	1	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	4	2	no	—
526	C₂³	#121×C₆₇	1	2	yes	—
527	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
528	Q₈	—	2	2	no	—
529	Q₈	—	2	2	no	—
530	Q₈	—	2	2	no	—
531	Q₈	—	4	2	no	—
532	Q₈	—	4	2	no	—
533	Q₈	—	4	2	no	—
534	Q₈	—	4	2	no	—
535	Q₈	#121×C₆₇	1	4	yes	—
536	D₄	—	2	16	no	—
537	D₄	—	2	2	no	—
538	D₄	—	2	2	no	—
539	D₄	—	2	2	no	—
540	D₄	#121×C₆₇	1	4	yes	—
541	C₂×C₄	—	2	16	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	4	2	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	4	2	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	4	2	no	—
567	C₂×C₄	—	8	4	no	—
568	C₂×C₄	—	8	4	no	—
569	C₂×C₄	—	8	4	no	—
570	C₂×C₄	#120×C₆₇	1	2	yes	—
571	C₂×C₄	#120×C₆₇	1	2	yes	—
572	C₂×C₄	#121×C₆₇	1	2	yes	—
573	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
574	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
575	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
576	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
577	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
578	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
579	C₈	—	4	2	no	—
580	C₈	—	4	2	no	—
581	C₂²	—	2	4	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	2	4	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	2	no	—
586	C₂²	—	4	1	no	—
587	C₂²	—	4	2	no	—
588	C₂²	—	4	4	no	—
589	C₂²	—	4	2	no	—
590	C₂²	—	4	4	no	—
591	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
592	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
593	C₂²	#525×C₆₇	1	1	yes	characteristic
594	C₂²	#577×C₆₇	1	2	yes	—
595	C₂²	#578×C₆₇	1	2	yes	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	4	no	—
600	C₄	—	4	1	no	—
601	C₄	—	4	2	no	—
602	C₄	—	4	2	no	—
603	C₄	—	4	1	no	—
604	C₄	—	8	2	no	—
605	C₄	—	8	2	no	—
606	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
607	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
608	C₄	#525×C₆₇	1	1	yes	characteristic
609	C₄	#525×C₆₇	1	1	yes	characteristic
610	C₂	—	4	2	no	—
611	C₂	—	4	4	no	—
612	C₂	C₆₇×#2847	1	1	yes	characteristic
613	C₂	C₆₇×#3009	1	1	yes	characteristic
614	C₂	C₆₇×#3011	1	1	yes	characteristic
615	C₂	C₆₇×#3046	1	2	yes	—
616	C₂	C₆₇×#3120	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#462×C₆₇	C₂	semidirect
3. C₂×#461×C₆₇	C₂	semidirect
5. C₆₇×#2621	C₂	semidirect
6. C₆₇×#1878	C₂	semidirect
7. C₆₇×#1877	C₂	semidirect
8. C₆₇×#1872	C₂	semidirect
9. C₆₇×#1805	C₂	semidirect
10. C₆₇×#1804	C₂	semidirect
11. C₆₇×#1798	C₂	semidirect
301. #32916	C₆₇	direct (N × H)
310. C₂×#462	C₂×C₆₇	semidirect
311. C₂×#461	C₂×C₆₇	semidirect
313. #2621	C₂×C₆₇	semidirect
314. #1878	C₂×C₆₇	semidirect
315. #1877	C₂×C₆₇	semidirect
316. #1872	C₂×C₆₇	semidirect
317. #1805	C₂×C₆₇	semidirect
318. #1804	C₂×C₆₇	semidirect
319. #1798	C₂×C₆₇	semidirect
320. C₆₇	#32916	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#520×C₆₇	2
C₂×#462×C₆₇	2
C₂×#461×C₆₇	2
C₂×#452×C₆₇	2
C₆₇×#2621	2
C₆₇×#1878	2
C₆₇×#1877	2
C₆₇×#1872	2
C₆₇×#1805	2
C₆₇×#1804	2
C₆₇×#1798	2
#32916	67

Subgroups — C67×#32916 #30712167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32916 #30712167