Subgroups — #22598287

Order 9088.

410 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊Q₈×D₇₁	C₂	1	1	yes	maximal, characteristic
2	#117×D₇₁	C₂	1	1	yes	maximal, characteristic
3	#17986341	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986340	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986337	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986336	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986333	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986331	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986328	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986040	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986038	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986006	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986006	C₂	1	1	yes	maximal, characteristic, complemented
14	#17985961	C₂	1	1	yes	maximal, characteristic
15	#485×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₇₁	C₂²	1	1	yes	characteristic
18	C₂×#11765848	C₂²	1	1	yes	characteristic
19	C₂×#11765848	C₂²	1	1	yes	characteristic
20	#14556108	C₂²	1	1	yes	characteristic
21	#14556107	C₂²	1	1	yes	characteristic
22	#14556106	—	2	1	no	—
23	#14556105	—	2	1	no	—
24	#14556104	C₂²	1	1	yes	characteristic
25	#14556087	C₂²	1	1	yes	characteristic
26	#14556086	C₂²	1	1	yes	characteristic
27	#14556086	C₂²	1	1	yes	characteristic
28	#14556085	—	2	1	no	—
29	#14556084	—	2	1	no	—
30	C₄⋊C₄×D₇₁	—	2	1	no	—
31	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×D₇₁	—	2	1	no	—
35	C₂²⋊C₄×D₇₁	—	2	1	no	—
36	C₄²×D₇₁	—	2	1	no	—
37	#14556062	C₂²	1	1	yes	characteristic, complemented
38	#14556061	C₂²	1	1	yes	characteristic, complemented
39	#14556061	C₂²	1	1	yes	characteristic, complemented
40	#14556060	C₂²	1	1	yes	characteristic, complemented
41	#14556059	C₂²	1	1	yes	characteristic, complemented
42	#14556059	C₂²	1	1	yes	characteristic, complemented
43	#14556053	C₂²	1	1	yes	characteristic, complemented
44	#14556053	C₂²	1	1	yes	characteristic, complemented
45	#14556052	C₂²	1	1	yes	characteristic, complemented
46	#14556052	C₂²	1	1	yes	characteristic, complemented
47	#14556051	C₂²	1	1	yes	characteristic, complemented
48	#14556051	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
53	#117×C₇₁	C₂²	1	1	yes	characteristic
54	#107×C₇₁	C₂²	1	1	yes	characteristic, complemented
55	#106×C₇₁	C₂²	1	1	yes	characteristic, complemented
56	#103×C₇₁	C₂²	1	1	yes	characteristic, complemented
57	#103×C₇₁	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
59	C₂³×D₇₁	D₄	1	1	yes	characteristic
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	Q₈×D₇₁	—	2	2	no	—
63	Q₈×D₇₁	—	2	2	no	—
64	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
65	C₂×C₄×D₇₁	—	2	2	no	—
66	C₂×C₄×D₇₁	—	2	2	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	—	2	1	no	—
69	C₂×C₄×D₇₁	—	2	1	no	—
70	C₂×C₄×D₇₁	—	2	1	no	—
71	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
72	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
73	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
75	#11765848	—	2	2	no	—
76	#11765848	—	2	2	no	—
77	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₇₁	—	2	1	no	—
80	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
83	Dic₇₁⋊C₄	—	2	1	no	—
84	Dic₇₁⋊C₄	—	2	1	no	—
85	Dic₇₁⋊C₄	—	2	1	no	—
86	Dic₇₁⋊C₄	—	2	1	no	—
87	Dic₇₁⋊C₄	—	2	1	no	—
88	C₂²⋊Dic₇₁	—	2	1	no	—
89	C₂²⋊Dic₇₁	—	2	1	no	—
90	(C₂×D₇₁)⋊C₄	—	2	1	no	—
91	(C₂×D₇₁)⋊C₄	—	2	1	no	—
92	(C₂×D₇₁)⋊C₄	—	2	1	no	—
93	(C₂×D₇₁)⋊C₄	—	2	1	no	—
94	C₄×Dic₇₁	—	2	1	no	—
95	C₄×Dic₇₁	—	2	1	no	—
96	C₄×Dic₇₁	—	2	1	no	—
97	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	—	2	1	no	—
102	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₇₁	—	2	1	no	—
106	C₂²⋊C₄×C₇₁	—	2	1	no	—
107	C₄²×C₇₁	—	2	1	no	—
108	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₇₁	—	2	2	no	—
110	C₂²×D₇₁	—	2	2	no	—
111	C₂²×D₇₁	—	4	1	no	—
112	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₇₁	—	2	1	no	—
115	Q₈×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	2	1	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	—	2	1	no	—
119	C₂×C₄×C₇₁	—	2	1	no	—
120	C₂×C₄×C₇₁	—	2	1	no	—
121	C₂×C₄×C₇₁	—	2	1	no	—
122	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₇₁	—	2	2	no	—
125	C₂×Dic₇₁	—	2	1	no	—
126	C₂×Dic₇₁	—	2	1	no	—
127	C₂×Dic₇₁	—	2	1	no	—
128	C₂×Dic₇₁	—	2	1	no	—
129	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
131	C₄×D₇₁	—	2	2	no	—
132	C₄×D₇₁	—	2	1	no	—
133	C₄×D₇₁	—	2	1	no	—
134	C₄×D₇₁	—	2	2	no	—
135	C₄×D₇₁	—	4	2	no	—
136	C₄×D₇₁	—	4	2	no	—
137	C₄×D₇₁	—	4	2	no	—
138	C₄×D₇₁	—	4	2	no	—
139	C₄×D₇₁	#41	1	2	yes	—
140	C₄×D₇₁	#41	1	2	yes	—
141	Dic₁₄₂	—	2	1	no	—
142	Dic₁₄₂	—	2	1	no	—
143	Dic₁₄₂	—	4	2	no	—
144	C₈×C₇₁	—	2	1	no	—
145	C₈×C₇₁	—	2	1	no	—
146	C₇₁⋊C₈	—	2	1	no	—
147	C₇₁⋊C₈	—	2	1	no	—
148	C₂²×C₇₁	—	2	1	no	—
149	C₂²×C₇₁	—	2	1	no	—
150	C₂²×C₇₁	—	2	1	no	—
151	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₇₁	—	2	1	no	—
153	C₂×D₇₁	—	2	1	no	—
154	C₂×D₇₁	—	2	1	no	—
155	C₂×D₇₁	—	2	1	no	—
156	C₂×D₇₁	—	4	4	no	—
157	C₂×D₇₁	#123	1	2	yes	—
158	C₂×D₇₁	#137	1	2	yes	—
159	C₄×C₇₁	—	2	1	no	—
160	C₄×C₇₁	—	4	1	no	—
161	C₄×C₇₁	—	4	1	no	—
162	C₄×C₇₁	—	4	1	no	—
163	C₄×C₇₁	—	4	1	no	—
164	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
165	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
166	Dic₇₁	—	2	1	no	—
167	Dic₇₁	—	4	1	no	—
168	Dic₇₁	—	4	1	no	—
169	Dic₇₁	—	4	1	no	—
170	Dic₇₁	—	4	1	no	—
171	Dic₇₁	#123	1	2	yes	—
172	C₂×C₇₁	—	4	1	no	—
173	C₂×C₇₁	C₂×#123	1	1	yes	characteristic
174	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
175	C₂×C₇₁	#575	1	1	yes	characteristic
176	D₇₁	—	2	2	no	—
177	D₇₁	—	2	2	no	—
178	#3095	—	71	1	no	maximal, Sylow
179	C₇₁	#3095	1	1	yes	characteristic, complemented, Sylow
180	C₂×C₂²⋊Q₈	—	71	1	no	—
181	C₂×#117	—	71	1	no	—
182	#485	—	71	1	no	—
183	#485	—	71	1	no	—
184	#485	—	71	1	no	—
185	#485	D₇₁	1	1	yes	characteristic, complemented
186	#484	—	71	1	no	—
187	#484	—	71	1	no	—
188	#483	—	71	1	no	—
189	#483	—	71	1	no	—
190	#428	—	71	1	no	—
191	#428	—	71	1	no	—
192	#419	—	71	1	no	—
193	#419	—	71	1	no	—
194	#407	—	71	1	no	—
195	C₂²×Q₈	—	71	1	no	—
196	C₂³×C₄	—	71	1	no	—
197	C₂×M₁₆	—	71	1	no	—
198	C₂×M₁₆	—	71	1	no	—
199	C₂²⋊Q₈	—	71	1	no	—
200	C₂²⋊Q₈	—	71	1	no	—
201	C₂²⋊Q₈	—	71	1	no	—
202	C₂²⋊Q₈	—	142	1	no	—
203	C₂²⋊Q₈	—	142	1	no	—
204	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
205	#117	—	71	1	no	—
206	#117	—	71	1	no	—
207	#117	—	71	1	no	—
208	#117	—	142	1	no	—
209	#117	—	142	1	no	—
210	#117	C₂×D₇₁	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	71	1	no	—
212	C₂×C₄⋊C₄	—	71	1	no	—
213	C₂×C₄⋊C₄	—	71	1	no	—
214	C₂×C₄⋊C₄	—	142	1	no	—
215	C₂×C₂²⋊C₄	—	142	1	no	—
216	C₂×C₂²⋊C₄	—	142	1	no	—
217	C₂×C₄²	—	142	1	no	—
218	#107	—	71	1	no	—
219	#107	—	71	1	no	—
220	#107	—	71	1	no	—
221	#107	C₂×D₇₁	1	1	yes	characteristic, complemented
222	#106	—	71	1	no	—
223	#106	—	71	1	no	—
224	#106	—	71	1	no	—
225	#106	C₂×D₇₁	1	1	yes	characteristic, complemented
226	#103	—	71	1	no	—
227	#103	—	71	1	no	—
228	#103	—	71	1	no	—
229	#103	—	71	1	no	—
230	#103	—	71	1	no	—
231	#103	—	71	1	no	—
232	#103	C₂×D₇₁	1	1	yes	characteristic, complemented
233	#103	C₂×D₇₁	1	1	yes	characteristic, complemented
234	C₂²⋊C₈	—	71	1	no	—
235	C₂²⋊C₈	—	71	1	no	—
236	C₂²⋊C₈	—	71	1	no	—
237	C₂²⋊C₈	C₂×D₇₁	1	1	yes	characteristic
238	C₂⁴	—	71	1	no	—
239	C₂×Q₈	—	71	1	no	—
240	C₂×Q₈	—	142	1	no	—
241	C₂×Q₈	—	142	2	no	—
242	C₂×Q₈	—	142	2	no	—
243	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
244	C₂²×C₄	—	71	1	no	—
245	C₂²×C₄	—	71	1	no	—
246	C₂²×C₄	—	71	1	no	—
247	C₂²×C₄	—	71	1	no	—
248	C₂²×C₄	—	71	1	no	—
249	C₂²×C₄	—	142	1	no	—
250	C₂²×C₄	—	142	1	no	—
251	C₂²×C₄	—	142	1	no	—
252	C₂²×C₄	—	142	1	no	—
253	C₂²×C₄	—	142	2	no	—
254	C₂²×C₄	—	142	2	no	—
255	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
256	M₁₆	—	142	2	no	—
257	M₁₆	—	142	2	no	—
258	C₂×C₈	—	71	1	no	—
259	C₂×C₈	—	71	1	no	—
260	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	71	1	no	—
264	C₄⋊C₄	—	71	1	no	—
265	C₄⋊C₄	—	71	1	no	—
266	C₄⋊C₄	—	142	1	no	—
267	C₄⋊C₄	—	142	1	no	—
268	C₄⋊C₄	—	142	1	no	—
269	C₄⋊C₄	—	142	1	no	—
270	C₄⋊C₄	—	142	1	no	—
271	C₄⋊C₄	—	142	1	no	—
272	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	142	1	no	—
278	C₂²⋊C₄	—	142	1	no	—
279	C₂²⋊C₄	—	142	1	no	—
280	C₂²⋊C₄	—	142	1	no	—
281	C₂²⋊C₄	—	142	1	no	—
282	C₂²⋊C₄	—	142	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	142	1	no	—
285	C₄²	—	142	1	no	—
286	C₄²	—	142	1	no	—
287	C₂³	—	284	1	no	—
288	C₂³	—	71	1	no	—
289	C₂³	—	71	1	no	—
290	C₂³	—	142	2	no	—
291	C₂³	—	142	2	no	—
292	C₂³	D₄×D₇₁	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	284	2	no	—
296	Q₈	—	142	1	no	—
297	Q₈	—	142	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	284	2	no	—
305	C₂×C₄	—	284	2	no	—
306	C₂×C₄	—	284	2	no	—
307	C₂×C₄	—	284	2	no	—
308	C₂×C₄	—	71	1	no	—
309	C₂×C₄	—	71	2	no	—
310	C₂×C₄	—	71	2	no	—
311	C₂×C₄	—	71	1	no	—
312	C₂×C₄	—	142	2	no	—
313	C₂×C₄	—	142	2	no	—
314	C₂×C₄	—	142	1	no	—
315	C₂×C₄	—	142	1	no	—
316	C₂×C₄	—	142	1	no	—
317	C₂×C₄	—	142	1	no	—
318	C₂×C₄	—	142	2	no	—
319	C₂×C₄	—	142	1	no	—
320	C₂×C₄	—	142	1	no	—
321	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	142	1	no	—
326	C₈	—	142	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	284	4	no	—
331	C₂²	—	71	2	no	—
332	C₂²	—	71	2	no	—
333	C₂²	—	142	1	no	—
334	C₂²	—	142	1	no	—
335	C₂²	—	142	1	no	—
336	C₂²	—	142	1	no	—
337	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	284	1	no	—
344	C₄	—	284	1	no	—
345	C₄	—	284	1	no	—
346	C₄	—	284	1	no	—
347	C₄	—	71	2	no	—
348	C₄	—	142	1	no	—
349	C₄	#41×D₇₁	1	1	yes	characteristic
350	C₄	#41×D₇₁	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	142	2	no	—
353	C₂	—	142	2	no	—
354	C₂	#123×D₇₁	1	1	yes	characteristic
355	C₂	#137×D₇₁	1	1	yes	characteristic
356	C₂	#17986796	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#485×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#485	D₇₁
hypocenter	C₇₁	#3095
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #17986341	C₂	semidirect
4. #17986340	C₂	semidirect
5. #17986337	C₂	semidirect
6. #17986336	C₂	semidirect
7. #17986333	C₂	semidirect
8. #17986331	C₂	semidirect
9. #17986328	C₂	semidirect
10. #17986040	C₂	semidirect
11. #17986038	C₂	semidirect
12. #17986006	C₂	semidirect
13. #17986006	C₂	semidirect
15. #485×C₇₁	C₂	semidirect
37. #14556062	C₂²	semidirect
38. #14556061	C₂²	semidirect
39. #14556061	C₂²	semidirect
40. #14556060	C₂²	semidirect
41. #14556059	C₂²	semidirect
42. #14556059	C₂²	semidirect
43. #14556053	C₂²	semidirect
44. #14556053	C₂²	semidirect
45. #14556052	C₂²	semidirect
46. #14556052	C₂²	semidirect
47. #14556051	C₂²	semidirect
48. #14556051	C₂²	semidirect
54. #107×C₇₁	C₂²	semidirect
55. #106×C₇₁	C₂²	semidirect
56. #103×C₇₁	C₂²	semidirect
57. #103×C₇₁	C₂²	semidirect
179. C₇₁	#3095	semidirect
185. #485	D₇₁	semidirect
221. #107	C₂×D₇₁	semidirect
225. #106	C₂×D₇₁	semidirect
232. #103	C₂×D₇₁	semidirect
233. #103	C₂×D₇₁	semidirect