Subgroups — #22600930

Order 9088.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986713	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986649	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986396	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986321	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986320	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986316	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986315	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986287	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986277	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986249	C₂	1	1	yes	maximal, characteristic, complemented
11	D₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986075	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986041	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986038	C₂	1	1	yes	maximal, characteristic, complemented
15	#469×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765864	—	2	1	no	—
17	C₂×#11765848	C₂²	1	1	yes	characteristic
18	#14556127	C₂²	1	1	yes	characteristic
19	#14556127	C₂²	1	1	yes	characteristic
20	#14556115	—	2	1	no	—
21	#14556114	—	2	1	no	—
22	#14556110	—	2	1	no	—
23	#14556109	—	2	1	no	—
24	#14556105	—	2	1	no	—
25	#14556100	—	2	1	no	—
26	#14556099	C₂²	1	1	yes	characteristic
27	#14556099	C₂²	1	1	yes	characteristic
28	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
29	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
31	#14556087	C₂²	1	1	yes	characteristic
32	#14556086	C₂²	1	1	yes	characteristic
33	#14556084	—	2	1	no	—
34	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
36	#14556062	C₂²	1	1	yes	characteristic
37	#14556062	C₂²	1	1	yes	characteristic
38	#14556061	C₂²	1	1	yes	characteristic
39	#14556060	C₂²	1	1	yes	characteristic
40	#14556060	C₂²	1	1	yes	characteristic
41	#14556059	C₂²	1	1	yes	characteristic
42	#14556059	C₂²	1	1	yes	characteristic
43	#14556059	C₂²	1	1	yes	characteristic
44	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
45	#14556049	C₂²	1	1	yes	characteristic
46	#14556049	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	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₇₁	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
54	#121×C₇₁	C₂²	1	1	yes	characteristic
55	#121×C₇₁	C₂²	1	1	yes	characteristic
56	#106×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	#102×C₇₁	C₂²	1	1	yes	characteristic
59	#11765864	—	4	4	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	—	2	1	no	—
62	C₂×C₂²⋊D₇₁	—	2	1	no	—
63	C₂×C₂²⋊D₇₁	—	2	1	no	—
64	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
65	C₂²×Dic₇₁	—	2	1	no	—
66	C₂²×Dic₇₁	—	2	1	no	—
67	C₂²×Dic₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	—	2	1	no	—
69	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
70	#11765848	—	2	2	no	—
71	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₇₁	—	2	1	no	—
74	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
78	Dic₇₁⋊C₄	—	2	1	no	—
79	Dic₇₁⋊C₄	—	2	1	no	—
80	Dic₇₁⋊C₄	—	2	1	no	—
81	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₇₁	—	2	1	no	—
83	C₂²⋊Dic₇₁	—	2	1	no	—
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	—	2	1	no	—
86	C₂²⋊Dic₇₁	—	2	1	no	—
87	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
88	(C₂×D₇₁)⋊C₄	—	2	1	no	—
89	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
90	C₄×Dic₇₁	—	2	1	no	—
91	C₄×Dic₇₁	—	2	1	no	—
92	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₇₁	—	2	1	no	—
94	C₂×D₄×C₇₁	—	2	1	no	—
95	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
98	D₈×C₇₁	—	2	2	no	—
99	M₁₆×C₇₁	—	2	2	no	—
100	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₇₁	—	2	1	no	—
105	C₂²⋊C₄×C₇₁	—	2	1	no	—
106	C₂³×C₇₁	—	2	1	no	—
107	C₂³×C₇₁	—	2	1	no	—
108	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
110	D₄×C₇₁	—	2	1	no	—
111	D₄×C₇₁	—	2	1	no	—
112	D₄×C₇₁	—	2	1	no	—
113	D₄×C₇₁	—	2	1	no	—
114	D₄×C₇₁	—	4	2	no	—
115	D₄×C₇₁	—	8	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	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₇₁	—	2	2	no	—
123	C₂²⋊D₇₁	—	4	2	no	—
124	C₂²⋊D₇₁	—	8	1	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	—	4	2	no	—
130	C₂×Dic₇₁	—	4	2	no	—
131	C₂×Dic₇₁	—	4	2	no	—
132	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
135	C₄×D₇₁	—	2	1	no	—
136	C₄×D₇₁	—	2	1	no	—
137	C₄×D₇₁	—	4	2	no	—
138	Dic₁₄₂	—	4	2	no	—
139	Dic₁₄₂	—	8	1	no	—
140	C₈×C₇₁	—	2	1	no	—
141	C₈×C₇₁	#41	1	2	yes	complemented
142	C₇₁⋊C₈	—	2	1	no	—
143	C₇₁⋊C₈	—	2	1	no	—
144	C₂²×C₇₁	—	2	1	no	—
145	C₂²×C₇₁	—	2	1	no	—
146	C₂²×C₇₁	—	2	1	no	—
147	C₂²×C₇₁	—	4	1	no	—
148	C₂²×C₇₁	—	4	1	no	—
149	C₂²×C₇₁	—	4	1	no	—
150	C₂²×C₇₁	—	4	1	no	—
151	C₂²×C₇₁	—	4	1	no	—
152	C₂²×C₇₁	—	4	1	no	—
153	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₇₁	—	2	1	no	—
155	C₂×D₇₁	—	2	1	no	—
156	C₂×D₇₁	—	2	1	no	—
157	C₄×C₇₁	—	2	1	no	—
158	C₄×C₇₁	—	4	1	no	—
159	C₄×C₇₁	—	4	1	no	—
160	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
161	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
162	Dic₇₁	—	2	1	no	—
163	Dic₇₁	—	2	1	no	—
164	Dic₇₁	—	4	1	no	—
165	Dic₇₁	—	4	1	no	—
166	Dic₇₁	—	4	1	no	—
167	Dic₇₁	—	4	1	no	—
168	Dic₇₁	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₇₁	—	4	1	no	—
170	C₂×C₇₁	—	8	1	no	—
171	C₂×C₇₁	—	8	1	no	—
172	C₂×C₇₁	#547	1	1	yes	characteristic
173	C₂×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₇₁	#577	1	1	yes	characteristic
175	D₇₁	—	4	1	no	—
176	#3354	—	71	1	no	maximal, Sylow
177	C₇₁	#3354	1	1	yes	characteristic, complemented, Sylow
178	#555	—	71	1	no	—
179	#547	—	71	1	no	—
180	#498	—	71	1	no	—
181	#482	—	71	1	no	—
182	#482	—	71	1	no	—
183	#481	—	71	1	no	—
184	#481	—	71	1	no	—
185	#476	—	71	1	no	—
186	#474	—	71	1	no	—
187	#469	—	71	1	no	—
188	#469	D₇₁	1	1	yes	characteristic, complemented
189	C₄×D₈	—	71	1	no	—
190	#436	—	71	1	no	—
191	#428	—	71	1	no	—
192	#428	—	71	1	no	—
193	C₂×#41	—	142	1	no	—
194	C₂×D₈	C₂×D₇₁	1	1	yes	characteristic
195	C₂×M₁₆	—	71	1	no	—
196	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
197	C₄⋊Q₈	—	71	1	no	—
198	C₄⋊Q₈	—	71	1	no	—
199	#124	—	142	1	no	—
200	#124	—	142	1	no	—
201	#123	—	142	1	no	—
202	#123	—	142	1	no	—
203	C₂²⋊Q₈	—	142	1	no	—
204	#121	—	71	1	no	—
205	#121	—	71	1	no	—
206	#121	—	142	1	no	—
207	#121	C₂×D₇₁	1	1	yes	characteristic
208	#121	C₂×D₇₁	1	1	yes	characteristic
209	C₄×D₄	—	71	1	no	—
210	C₄×D₄	—	71	1	no	—
211	C₄×D₄	—	71	1	no	—
212	#117	—	71	1	no	—
213	#117	—	71	1	no	—
214	#117	—	142	1	no	—
215	C₂×C₄⋊C₄	—	71	1	no	—
216	C₂×C₄⋊C₄	—	71	1	no	—
217	#107	—	71	1	no	—
218	#107	—	71	1	no	—
219	#107	—	71	1	no	—
220	#106	—	71	1	no	—
221	#106	—	71	1	no	—
222	#106	—	71	1	no	—
223	#106	—	71	1	no	—
224	#106	—	71	1	no	—
225	#106	C₂×D₇₁	1	1	yes	characteristic
226	C₄⋊C₈	—	71	1	no	—
227	#102	—	71	1	no	—
228	#102	—	71	1	no	—
229	#102	—	71	1	no	—
230	#102	—	71	1	no	—
231	#102	C₂×D₇₁	1	1	yes	characteristic
232	#102	C₂×D₇₁	1	1	yes	characteristic
233	C₂²⋊C₈	—	71	1	no	—
234	C₂²⋊C₈	—	71	1	no	—
235	C₄×C₈	—	71	1	no	—
236	#41	—	284	4	no	—
237	C₂×Q₈	—	142	1	no	—
238	C₂×Q₈	—	142	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	71	1	no	—
242	C₂×D₄	—	142	1	no	—
243	C₂×D₄	—	142	1	no	—
244	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
245	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
246	C₂²×C₄	—	71	1	no	—
247	C₂²×C₄	—	142	1	no	—
248	C₂²×C₄	—	142	1	no	—
249	C₂²×C₄	—	142	1	no	—
250	C₂²×C₄	—	142	1	no	—
251	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
252	D₈	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	142	2	no	—
255	C₂×C₈	—	71	1	no	—
256	C₂×C₈	—	71	1	no	—
257	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
259	C₄⋊C₄	—	71	1	no	—
260	C₄⋊C₄	—	71	1	no	—
261	C₄⋊C₄	—	71	1	no	—
262	C₄⋊C₄	—	71	1	no	—
263	C₄⋊C₄	—	71	1	no	—
264	C₄⋊C₄	—	142	1	no	—
265	C₄⋊C₄	—	142	1	no	—
266	C₄⋊C₄	—	142	1	no	—
267	C₄⋊C₄	—	142	1	no	—
268	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	71	1	no	—
273	C₂²⋊C₄	—	71	1	no	—
274	C₂²⋊C₄	—	142	1	no	—
275	C₂²⋊C₄	—	142	1	no	—
276	C₂²⋊C₄	—	142	1	no	—
277	C₂²⋊C₄	—	142	1	no	—
278	C₂²⋊C₄	—	142	1	no	—
279	C₂²⋊C₄	—	142	1	no	—
280	C₄²	—	71	1	no	—
281	C₄²	—	142	1	no	—
282	C₄²	—	142	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	71	1	no	—
286	C₂³	D₄×D₇₁	1	1	yes	characteristic
287	Q₈	—	284	2	no	—
288	Q₈	—	568	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	4	2	no	—
294	D₄	—	8	1	no	—
295	D₄	—	284	2	no	—
296	D₄	—	568	1	no	—
297	D₄	—	142	2	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	—	284	2	no	—
303	C₂×C₄	—	284	2	no	—
304	C₂×C₄	—	284	2	no	—
305	C₂×C₄	—	284	2	no	—
306	C₂×C₄	—	71	1	no	—
307	C₂×C₄	—	71	1	no	—
308	C₂×C₄	—	71	1	no	—
309	C₂×C₄	—	142	1	no	—
310	C₂×C₄	—	142	1	no	—
311	C₂×C₄	—	142	1	no	—
312	C₂×C₄	—	142	1	no	—
313	C₂×C₄	—	142	1	no	—
314	C₂×C₄	—	142	1	no	—
315	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	142	1	no	—
319	C₈	—	142	1	no	—
320	C₈	#11765864	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	142	1	no	—
331	C₂²	—	142	1	no	—
332	C₂²	—	142	1	no	—
333	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	284	1	no	—
338	C₄	—	284	1	no	—
339	C₄	—	284	1	no	—
340	C₄	—	284	1	no	—
341	C₄	—	71	2	no	—
342	C₄	—	142	1	no	—
343	C₄	—	142	1	no	—
344	C₄	C₂×#11765864	1	1	yes	characteristic
345	C₄	#14556175	1	1	yes	characteristic
346	C₂	—	4	1	no	—
347	C₂	—	8	1	no	—
348	C₂	—	8	1	no	—
349	C₂	—	284	1	no	—
350	C₂	#17986649	1	1	yes	characteristic
351	C₂	AGL(1,ℤ/8)×D₇₁	1	1	yes	characteristic
352	C₂	#17986809	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#469×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#469	D₇₁
hypocenter	C₇₁	#3354
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #17986713	C₂	semidirect
2. #17986649	C₂	semidirect
3. #17986396	C₂	semidirect
4. #17986321	C₂	semidirect
5. #17986320	C₂	semidirect
6. #17986316	C₂	semidirect
7. #17986315	C₂	semidirect
8. #17986287	C₂	semidirect
9. #17986277	C₂	semidirect
10. #17986249	C₂	semidirect
11. D₈×Dic₇₁	C₂	semidirect
12. #17986075	C₂	semidirect
13. #17986041	C₂	semidirect
14. #17986038	C₂	semidirect
15. #469×C₇₁	C₂	semidirect
141. C₈×C₇₁	#41	semidirect
177. C₇₁	#3354	semidirect
188. #469	D₇₁	semidirect
320. C₈	#11765864	semidirect