Subgroups — #22600401

Order 9088.

380 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	#17986720	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986655	C₂	1	1	yes	maximal, characteristic
3	#17986357	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986342	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986320	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986293	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986239	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986219	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986158	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986154	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986103	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986068	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986009	C₂	1	1	yes	maximal, characteristic, complemented
14	#17985992	C₂	1	1	yes	maximal, characteristic, complemented
15	#468×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765864	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₄₂	C₂²	1	1	yes	characteristic
18	C₂×#11765855	C₂²	1	1	yes	characteristic
19	C₂×#11765855	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
21	#14556127	C₂²	1	1	yes	characteristic
22	#14556123	—	2	1	no	—
23	#14556121	—	2	1	no	—
24	#14556115	C₂²	1	1	yes	characteristic
25	#14556114	—	2	1	no	—
26	#14556114	—	2	1	no	—
27	#14556108	C₂²	1	1	yes	characteristic
28	#14556105	—	2	1	no	—
29	#14556104	C₂²	1	1	yes	characteristic
30	#14556100	—	2	1	no	—
31	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
32	#14556093	—	2	1	no	—
33	#14556093	C₂²	1	1	yes	characteristic
34	D₄×Dic₇₁	—	2	1	no	—
35	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
36	#14556087	C₂²	1	1	yes	characteristic
37	#14556086	C₂²	1	1	yes	characteristic
38	#14556061	C₂²	1	1	yes	characteristic
39	#14556059	C₂²	1	1	yes	characteristic
40	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
41	#14556053	C₂²	1	1	yes	characteristic
42	#14556053	C₂²	1	1	yes	characteristic
43	#14556053	C₂²	1	1	yes	characteristic
44	#14556053	C₂²	1	1	yes	characteristic
45	#14556051	C₂²	1	1	yes	characteristic
46	#14556051	C₂²	1	1	yes	characteristic
47	#14556049	C₂²	1	1	yes	characteristic
48	#14556049	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	#14556038	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
55	#121×C₇₁	C₂²	1	1	yes	characteristic
56	#107×C₇₁	C₂²	1	1	yes	characteristic
57	#103×C₇₁	C₂²	1	1	yes	characteristic
58	#102×C₇₁	C₂²	1	1	yes	characteristic
59	#11765864	—	2	2	no	—
60	#11765864	—	4	1	no	—
61	C₂×Dic₁₄₂	—	2	2	no	—
62	C₂×Dic₁₄₂	—	2	1	no	—
63	C₂×Dic₁₄₂	—	2	1	no	—
64	C₂×Dic₁₄₂	—	4	1	no	—
65	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₇₁	—	2	1	no	—
68	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
69	C₂²×Dic₇₁	—	2	1	no	—
70	C₂²×Dic₇₁	—	2	1	no	—
71	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
72	#11765855	—	2	2	no	—
73	#11765855	—	4	1	no	—
74	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×D₇₁	—	4	1	no	—
77	C₄⋊Dic₇₁	—	2	1	no	—
78	C₄⋊Dic₇₁	—	2	1	no	—
79	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
81	Dic₇₁⋊C₄	—	2	1	no	—
82	Dic₇₁⋊C₄	—	2	1	no	—
83	Dic₇₁⋊C₄	—	2	1	no	—
84	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₇₁	—	2	1	no	—
86	C₂²⋊Dic₇₁	—	2	1	no	—
87	C₂²⋊Dic₇₁	—	2	1	no	—
88	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
89	(C₂×D₇₁)⋊C₄	—	2	1	no	—
90	(C₂×D₇₁)⋊C₄	—	2	1	no	—
91	(C₂×D₇₁)⋊C₄	—	2	1	no	—
92	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
93	C₄×Dic₇₁	—	2	1	no	—
94	C₄×Dic₇₁	—	2	1	no	—
95	C₄×Dic₇₁	—	2	1	no	—
96	C₄×Dic₇₁	—	2	1	no	—
97	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₇₁	—	2	1	no	—
100	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
102	SD₁₆×C₇₁	—	4	1	no	—
103	M₁₆×C₇₁	—	4	1	no	—
104	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₇₁	—	2	1	no	—
107	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₇₁	—	2	1	no	—
110	C₂²⋊C₄×C₇₁	—	2	1	no	—
111	C₂³×C₇₁	—	2	1	no	—
112	C₂³×C₇₁	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	D₄×C₇₁	—	2	1	no	—
117	D₄×C₇₁	—	2	1	no	—
118	D₄×C₇₁	—	4	2	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	—	2	1	no	—
123	C₂×C₄×C₇₁	—	2	1	no	—
124	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₇₁	—	4	1	no	—
127	C₂²⋊D₇₁	—	4	2	no	—
128	C₂×Dic₇₁	—	2	1	no	—
129	C₂×Dic₇₁	—	2	1	no	—
130	C₂×Dic₇₁	—	2	1	no	—
131	C₂×Dic₇₁	—	2	1	no	—
132	C₂×Dic₇₁	—	4	2	no	—
133	C₂×Dic₇₁	—	4	2	no	—
134	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
137	C₄×D₇₁	—	2	1	no	—
138	C₄×D₇₁	—	2	1	no	—
139	Dic₁₄₂	—	2	1	no	—
140	Dic₁₄₂	—	2	1	no	—
141	Dic₁₄₂	—	2	1	no	—
142	Dic₁₄₂	—	4	2	no	—
143	Dic₁₄₂	—	4	2	no	—
144	Dic₁₄₂	#41	1	2	yes	—
145	C₈×C₇₁	—	2	1	no	—
146	C₈×C₇₁	—	2	1	no	—
147	C₇₁⋊C₈	—	2	1	no	—
148	C₇₁⋊C₈	—	2	1	no	—
149	C₂²×C₇₁	—	2	1	no	—
150	C₂²×C₇₁	—	2	1	no	—
151	C₂²×C₇₁	—	2	1	no	—
152	C₂²×C₇₁	—	4	1	no	—
153	C₂²×C₇₁	—	4	1	no	—
154	C₂²×C₇₁	—	4	1	no	—
155	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₇₁	—	2	1	no	—
157	C₂×D₇₁	—	2	1	no	—
158	C₂×D₇₁	—	2	1	no	—
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	C₂×#41	1	1	yes	characteristic
164	C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₇₁	—	2	2	no	—
166	Dic₇₁	—	2	1	no	—
167	Dic₇₁	—	2	1	no	—
168	Dic₇₁	—	2	1	no	—
169	Dic₇₁	—	4	1	no	—
170	Dic₇₁	—	4	1	no	—
171	Dic₇₁	—	4	1	no	—
172	C₂×C₇₁	—	4	1	no	—
173	C₂×C₇₁	—	8	1	no	—
174	C₂×C₇₁	#546	1	1	yes	characteristic
175	C₂×C₇₁	#575	1	1	yes	characteristic
176	C₂×C₇₁	#578	1	1	yes	characteristic
177	D₇₁	—	4	1	no	—
178	#3313	—	71	1	no	maximal, Sylow
179	C₇₁	#3313	1	1	yes	characteristic, complemented, Sylow
180	#556	—	71	1	no	—
181	#548	—	71	1	no	—
182	#489	—	71	1	no	—
183	#485	—	71	1	no	—
184	#482	—	71	1	no	—
185	#477	—	71	1	no	—
186	#468	D₇₁	1	1	yes	characteristic, complemented
187	#467	—	71	1	no	—
188	#464	—	71	1	no	—
189	#452	—	71	1	no	—
190	#451	—	71	1	no	—
191	#441	—	71	1	no	—
192	#435	—	71	1	no	—
193	#419	—	71	1	no	—
194	#416	—	71	1	no	—
195	C₂×#41	—	71	1	no	—
196	C₂²×Q₈	—	71	1	no	—
197	C₂×Q₁₆	—	71	1	no	—
198	C₂×Q₁₆	—	71	1	no	—
199	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
200	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
201	C₂²×C₈	—	71	1	no	—
202	C₄⋊Q₈	—	71	1	no	—
203	#126	—	142	1	no	—
204	#126	—	142	1	no	—
205	#124	—	71	1	no	—
206	#124	—	142	1	no	—
207	#124	—	142	1	no	—
208	C₂²⋊Q₈	—	71	1	no	—
209	C₂²⋊Q₈	—	71	1	no	—
210	C₂²⋊Q₈	—	142	1	no	—
211	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
212	#121	—	142	1	no	—
213	#121	C₂×D₇₁	1	1	yes	characteristic
214	C₄×Q₈	—	71	1	no	—
215	C₄×Q₈	—	71	1	no	—
216	C₄×Q₈	—	142	1	no	—
217	C₄×D₄	—	71	1	no	—
218	C₄×D₄	—	142	1	no	—
219	#117	—	71	1	no	—
220	#117	—	71	1	no	—
221	#107	—	71	1	no	—
222	#107	C₂×D₇₁	1	1	yes	characteristic
223	#106	—	71	1	no	—
224	C₄⋊C₈	—	71	1	no	—
225	#103	—	71	1	no	—
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	C₂×D₇₁	1	1	yes	characteristic
232	#102	—	71	1	no	—
233	#102	—	71	1	no	—
234	#102	C₂×D₇₁	1	1	yes	characteristic
235	C₂²⋊C₈	—	71	1	no	—
236	C₂²⋊C₈	—	71	1	no	—
237	#97	—	71	1	no	—
238	#41	—	284	1	no	—
239	#41	—	142	2	no	—
240	C₂×Q₈	—	284	1	no	—
241	C₂×Q₈	—	71	1	no	—
242	C₂×Q₈	—	71	1	no	—
243	C₂×Q₈	—	142	2	no	—
244	C₂×Q₈	—	142	1	no	—
245	C₂×Q₈	—	142	1	no	—
246	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	71	1	no	—
249	C₂×D₄	—	142	1	no	—
250	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
251	C₂²×C₄	—	71	1	no	—
252	C₂²×C₄	—	142	1	no	—
253	C₂²×C₄	—	142	1	no	—
254	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
255	Q₁₆	—	284	1	no	—
256	Q₁₆	—	142	2	no	—
257	SD₁₆	—	4	1	no	—
258	M₁₆	—	4	1	no	—
259	C₂×C₈	—	284	1	no	—
260	C₂×C₈	—	71	1	no	—
261	C₂×C₈	—	71	1	no	—
262	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	71	1	no	—
266	C₄⋊C₄	—	71	1	no	—
267	C₄⋊C₄	—	71	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	—	142	1	no	—
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	—	71	1	no	—
278	C₂²⋊C₄	—	71	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	C₂²⋊C₄	—	142	1	no	—
284	C₂²⋊C₄	—	142	1	no	—
285	C₄²	—	71	1	no	—
286	C₄²	—	142	1	no	—
287	C₄²	—	142	1	no	—
288	C₄²	—	142	1	no	—
289	C₄²	—	142	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	71	1	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	—	284	2	no	—
297	Q₈	—	71	2	no	—
298	Q₈	—	142	1	no	—
299	Q₈	—	142	1	no	—
300	Q₈	—	142	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	4	2	no	—
304	D₄	—	284	1	no	—
305	D₄	—	284	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	284	2	no	—
312	C₂×C₄	—	284	2	no	—
313	C₂×C₄	—	71	1	no	—
314	C₂×C₄	—	71	1	no	—
315	C₂×C₄	—	71	1	no	—
316	C₂×C₄	—	142	1	no	—
317	C₂×C₄	—	142	1	no	—
318	C₂×C₄	—	142	1	no	—
319	C₂×C₄	—	142	1	no	—
320	C₂×C₄	—	142	1	no	—
321	C₂×C₄	—	142	1	no	—
322	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	142	1	no	—
327	C₈	—	142	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	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	—	284	1	no	—
343	C₄	—	284	1	no	—
344	C₄	—	284	1	no	—
345	C₄	—	142	1	no	—
346	C₄	—	142	2	no	—
347	C₄	—	142	1	no	—
348	C₄	—	142	1	no	—
349	C₄	#41×D₇₁	1	1	yes	characteristic
350	C₄	#14556173	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	8	1	no	—
353	C₂	—	284	1	no	—
354	C₂	#17986638	1	1	yes	characteristic
355	C₂	#17986795	1	1	yes	characteristic
356	C₂	#17986815	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#468×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#468	D₇₁
hypocenter	C₇₁	#3313
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #17986720	C₂	semidirect
3. #17986357	C₂	semidirect
4. #17986342	C₂	semidirect
5. #17986320	C₂	semidirect
6. #17986293	C₂	semidirect
7. #17986239	C₂	semidirect
8. #17986219	C₂	semidirect
9. #17986158	C₂	semidirect
10. #17986154	C₂	semidirect
11. #17986103	C₂	semidirect
12. #17986068	C₂	semidirect
13. #17986009	C₂	semidirect
14. #17985992	C₂	semidirect
15. #468×C₇₁	C₂	semidirect
179. C₇₁	#3313	semidirect
186. #468	D₇₁	semidirect