Subgroups — #30748549

Order 17152.

472 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560326	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
4	#22558050	C₂	1	2	yes	maximal
5	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
9	#16863151	—	2	1	no	—
10	#16863151	—	2	1	no	—
11	#16863151	—	2	1	no	—
12	Q₈⋊(C₆₇⋊C₈)	—	2	4	no	—
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	#391×C₆₇	C₄	1	1	yes	characteristic
15	#391×C₆₇	C₄	1	1	yes	characteristic
16	#389×C₆₇	C₄	1	1	yes	characteristic
17	#389×C₆₇	C₄	1	1	yes	characteristic
18	#387×C₆₇	—	2	1	no	—
19	#387×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	#376×C₆₇	—	2	1	no	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄×C₆₇⋊C₈	—	2	2	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₆₇	—	2	1	no	—
33	C₄×Q₈×C₆₇	—	2	1	no	—
34	C₄×Q₈×C₆₇	D₄	1	2	yes	—
35	C₄×Q₈×C₆₇	Q₈	1	2	yes	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	—	2	1	no	—
43	C₂×C₄²×C₆₇	—	2	1	no	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	#95×C₆₇	—	2	1	no	—
52	#95×C₆₇	—	2	1	no	—
53	#95×C₆₇	—	2	1	no	—
54	#95×C₆₇	—	2	1	no	—
55	#95×C₆₇	—	2	1	no	—
56	#95×C₆₇	—	2	1	no	—
57	#95×C₆₇	—	2	1	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₂×Q₈×C₆₇	—	2	1	no	—
65	C₂×Q₈×C₆₇	—	2	2	no	—
66	C₂×Q₈×C₆₇	—	2	2	no	—
67	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	—	2	1	no	—
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₆₇	—	2	1	no	—
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₄⋊C₄×C₆₇	—	4	2	no	—
85	C₄⋊C₄×C₆₇	—	4	2	no	—
86	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
87	C₄²×C₆₇	—	2	2	no	—
88	C₄²×C₆₇	—	2	2	no	—
89	C₄²×C₆₇	—	2	2	no	—
90	C₄²×C₆₇	—	2	2	no	—
91	C₄²×C₆₇	—	2	2	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	4	1	no	—
95	C₄²×C₆₇	—	4	2	no	—
96	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₆₇	#41	1	1	yes	characteristic
99	C₄²×C₆₇	#41	1	1	yes	characteristic
100	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	Q₈×C₆₇	—	2	4	no	—
102	Q₈×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	—	4	1	no	—
120	C₂×C₄×C₆₇	—	4	1	no	—
121	C₂×C₄×C₆₇	—	4	1	no	—
122	C₂×C₄×C₆₇	—	4	1	no	—
123	C₂×C₄×C₆₇	—	4	1	no	—
124	C₂×C₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	1	no	—
128	C₂×C₄×C₆₇	#104	1	2	yes	—
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
131	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#131	1	2	yes	—
136	C₂×C₄×C₆₇	#131	1	2	yes	—
137	C₂×C₄×C₆₇	#135	1	2	yes	—
138	C₂×C₄×C₆₇	#137	1	2	yes	—
139	C₆₇⋊C₈	—	4	4	no	—
140	C₆₇⋊C₈	—	8	2	no	—
141	C₂²×C₆₇	#380	1	1	yes	characteristic
142	C₂²×C₆₇	#408	1	1	yes	characteristic
143	C₂²×C₆₇	#408	1	1	yes	characteristic
144	C₂²×C₆₇	#416	1	1	yes	characteristic
145	C₂²×C₆₇	#419	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
147	C₂²×C₆₇	#421	1	1	yes	characteristic
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	4	1	no	—
154	C₄×C₆₇	—	4	1	no	—
155	C₄×C₆₇	—	4	1	no	—
156	C₄×C₆₇	—	8	1	no	—
157	C₄×C₆₇	—	8	1	no	—
158	C₄×C₆₇	#424	1	2	yes	—
159	C₄×C₆₇	#433	1	2	yes	—
160	C₄×C₆₇	#464	1	2	yes	—
161	C₄×C₆₇	#479	1	2	yes	—
162	#8570	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	#1482	1	1	yes	characteristic
164	C₂×C₆₇	#1485	1	1	yes	characteristic
165	C₂×C₆₇	#1494	1	1	yes	characteristic
166	C₂×C₆₇	#1497	1	1	yes	characteristic
167	C₂×C₆₇	#1798	1	1	yes	characteristic
168	C₂×C₆₇	#1805	1	1	yes	characteristic
169	C₂×C₆₇	#1806	1	1	yes	characteristic
170	#2295	D₆₇	1	1	yes	characteristic
171	#1831	—	67	1	no	—
172	C₄×C₄⋊C₈	—	67	1	no	—
173	C₂×Q₈⋊C₈	—	67	2	no	—
174	#1279	—	67	2	no	—
175	C₆₇	#8570	1	1	yes	characteristic, complemented, Sylow
176	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
177	C₂×C₄⋊C₈	—	67	2	no	—
178	C₂×C₄⋊C₈	—	134	1	no	—
179	C₂×C₄×C₈	—	67	2	no	—
180	#391	Dic₆₇	1	1	yes	characteristic
181	#391	Dic₆₇	1	1	yes	characteristic
182	#389	Dic₆₇	1	1	yes	characteristic
183	#389	Dic₆₇	1	1	yes	characteristic
184	#387	—	2	1	no	—
185	#387	—	2	1	no	—
186	C₄×C₄⋊C₄	—	2	1	no	—
187	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
188	#376	—	2	1	no	—
189	C₄³	C₂×D₆₇	1	1	yes	characteristic
190	#336	—	134	1	no	—
191	#336	—	134	1	no	—
192	#336	—	134	1	no	—
193	Q₈⋊C₈	—	134	4	no	—
194	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
195	C₂²×C₈	—	134	2	no	—
196	C₂²×C₈	—	134	2	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	—	2	1	no	—
199	C₄×Q₈	Dic₁₃₄	1	2	yes	—
200	C₄×Q₈	D₂₆₈	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
216	C₄⋊C₈	—	134	4	no	—
217	C₄⋊C₈	—	134	2	no	—
218	C₄⋊C₈	—	134	2	no	—
219	C₄×C₈	—	134	2	no	—
220	C₄×C₈	—	134	2	no	—
221	#95	—	2	1	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	#95	—	2	1	no	—
226	#95	—	2	1	no	—
227	#95	—	2	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
243	C₂×C₈	—	268	2	no	—
244	C₂×C₈	—	268	2	no	—
245	C₂×C₈	—	268	2	no	—
246	C₂×C₈	—	268	2	no	—
247	C₂×C₈	—	268	2	no	—
248	C₂×C₈	—	134	4	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	2	no	—
266	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
267	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
268	C₄²	#11761628	1	1	yes	characteristic
269	C₄²	#11761628	1	1	yes	characteristic
270	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
271	Q₈	—	2	4	no	—
272	Q₈	—	4	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	#14523970	1	2	yes	—
299	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
300	C₂×C₄	#14524002	1	1	yes	characteristic
301	C₂×C₄	#14524017	1	1	yes	characteristic
302	C₂×C₄	#14524017	1	1	yes	characteristic
303	C₂×C₄	#14524022	1	1	yes	characteristic
304	C₂×C₄	#14524022	1	1	yes	characteristic
305	C₂×C₄	#14524050	1	2	yes	—
306	C₂×C₄	#14524050	1	2	yes	—
307	C₂×C₄	#14524063	1	2	yes	—
308	C₂×C₄	#14524074	1	2	yes	—
309	C₈	—	268	4	no	—
310	C₈	—	536	2	no	—
311	C₂²	#16863256	1	1	yes	characteristic
312	C₂²	#16863363	1	1	yes	characteristic
313	C₂²	#16863363	1	1	yes	characteristic
314	C₂²	#16863392	1	1	yes	characteristic
315	C₂²	#16863405	1	1	yes	characteristic
316	C₂²	C₂×#14523970	1	1	yes	characteristic
317	C₂²	#16863413	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	#16863423	1	2	yes	—
329	C₄	#16863457	1	2	yes	—
330	C₄	#16863618	1	2	yes	—
331	C₄	#16863703	1	2	yes	—
332	C₂	#22558617	1	1	yes	characteristic
333	C₂	#22558630	1	1	yes	characteristic
334	C₂	#22558669	1	1	yes	characteristic
335	C₂	#22558682	1	1	yes	characteristic
336	C₂	#22560193	1	1	yes	characteristic
337	C₂	#22560224	1	1	yes	characteristic
338	C₂	#22560229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2295	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2295	D₆₇
hypocenter	C₆₇	#8570
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
175. C₆₇	#8570	semidirect

Maximal subgroups

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

subgroup	index
#22560326	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×Q₈⋊(C₆₇⋊C₈)	2
#22558050	2
C₆₇×#2295	2
#8570	67