Subgroups — #22600796

Order 9088.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986704	C₂	1	1	yes	maximal, characteristic
2	#17986688	C₂	1	1	yes	maximal, characteristic
3	#17986407	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986340	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986336	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986331	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986329	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986305	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986283	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986257	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986241	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986105	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986068	C₂	1	1	yes	maximal, characteristic
14	#106×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#428×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
18	#14556127	C₂²	1	1	yes	characteristic
19	#14556121	—	2	1	no	—
20	#14556120	—	2	1	no	—
21	#14556119	C₂²	1	1	yes	characteristic
22	#14556118	—	2	1	no	—
23	#14556110	—	2	1	no	—
24	#14556108	C₂²	1	1	yes	characteristic
25	#14556108	C₂²	1	1	yes	characteristic
26	#14556107	—	2	1	no	—
27	#14556107	C₂²	1	1	yes	characteristic
28	#14556107	C₂²	1	1	yes	characteristic
29	#14556106	—	2	1	no	—
30	#14556093	C₂²	1	1	yes	characteristic
31	#14556093	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₇₁	—	2	1	no	—
33	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
34	#14556085	—	2	1	no	—
35	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
37	#14556061	C₂²	1	1	yes	characteristic
38	#14556060	C₂²	1	1	yes	characteristic
39	#14556059	C₂²	1	1	yes	characteristic
40	#14556059	C₂²	1	1	yes	characteristic
41	#14556059	C₂²	1	1	yes	characteristic
42	Dic₇₁⋊C₈	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	#14556051	C₂²	1	1	yes	characteristic
48	#14556051	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×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	#117×C₇₁	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	#107×C₇₁	C₂²	1	1	yes	characteristic
57	#106×C₇₁	C₂²	1	1	yes	characteristic
58	#106×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
63	C₂×C₄×D₇₁	—	2	1	no	—
64	C₂×C₄×D₇₁	—	2	1	no	—
65	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
66	Dic₂₈₄	—	2	2	no	—
67	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₈×D₇₁	—	2	2	no	—
70	C₄⋊Dic₇₁	—	2	1	no	—
71	C₄⋊Dic₇₁	—	2	1	no	—
72	C₄⋊Dic₇₁	—	2	1	no	—
73	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
75	Dic₇₁⋊C₄	—	2	1	no	—
76	Dic₇₁⋊C₄	—	2	1	no	—
77	Dic₇₁⋊C₄	—	2	1	no	—
78	Dic₇₁⋊C₄	—	2	1	no	—
79	Dic₇₁⋊C₄	—	2	1	no	—
80	Dic₇₁⋊C₄	—	2	1	no	—
81	Dic₇₁⋊C₄	—	2	1	no	—
82	Dic₇₁⋊C₄	—	2	1	no	—
83	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	—	2	1	no	—
86	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
87	(C₂×D₇₁)⋊C₄	—	2	1	no	—
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	C₂²×C₄×C₇₁	—	2	1	no	—
94	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
95	M₁₆×C₇₁	—	2	2	no	—
96	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₇₁	—	2	1	no	—
99	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₇₁	—	2	1	no	—
104	C₄²×C₇₁	—	2	1	no	—
105	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	8	1	no	—
114	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₇₁	—	2	2	no	—
117	C₂×Dic₇₁	—	2	1	no	—
118	C₂×Dic₇₁	—	2	1	no	—
119	C₂×Dic₇₁	—	2	1	no	—
120	C₂×Dic₇₁	—	2	1	no	—
121	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
124	C₄×D₇₁	—	2	1	no	—
125	C₄×D₇₁	—	2	1	no	—
126	C₄×D₇₁	—	4	2	no	—
127	C₄×D₇₁	—	8	1	no	—
128	Dic₁₄₂	—	2	1	no	—
129	Dic₁₄₂	—	2	1	no	—
130	Dic₁₄₂	—	2	1	no	—
131	Dic₁₄₂	—	2	1	no	—
132	Dic₁₄₂	—	8	1	no	—
133	C₈×C₇₁	—	2	1	no	—
134	C₈×C₇₁	#41	1	2	yes	—
135	C₇₁⋊C₈	—	2	1	no	—
136	C₇₁⋊C₈	—	2	1	no	—
137	C₂²×C₇₁	—	2	1	no	—
138	C₂²×C₇₁	—	2	1	no	—
139	C₂²×C₇₁	—	2	1	no	—
140	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₇₁	—	2	1	no	—
142	C₂×D₇₁	—	2	1	no	—
143	C₂×D₇₁	#137	1	2	yes	—
144	C₄×C₇₁	—	2	1	no	—
145	C₄×C₇₁	—	4	1	no	—
146	C₄×C₇₁	—	4	1	no	—
147	C₄×C₇₁	—	4	1	no	—
148	C₄×C₇₁	—	4	1	no	—
149	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
150	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₇₁	—	2	1	no	—
152	Dic₇₁	—	2	1	no	—
153	Dic₇₁	—	2	1	no	—
154	Dic₇₁	—	4	1	no	—
155	Dic₇₁	—	4	1	no	—
156	Dic₇₁	—	4	1	no	—
157	Dic₇₁	—	4	1	no	—
158	C₂×C₇₁	—	4	1	no	—
159	C₂×C₇₁	#547	1	1	yes	characteristic
160	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
161	C₂×C₇₁	#577	1	1	yes	characteristic
162	D₇₁	—	2	2	no	—
163	#3344	—	71	1	no	maximal, Sylow
164	C₇₁	#3344	1	1	yes	characteristic, complemented, Sylow
165	#554	—	71	1	no	—
166	#552	—	71	1	no	—
167	#501	—	71	1	no	—
168	#485	—	71	1	no	—
169	#484	—	71	1	no	—
170	#483	—	71	1	no	—
171	#483	—	71	1	no	—
172	#479	—	71	1	no	—
173	#475	—	71	1	no	—
174	#470	—	71	1	no	—
175	#467	—	71	1	no	—
176	#441	—	71	1	no	—
177	#435	—	71	1	no	—
178	#428	D₇₁	1	1	yes	characteristic, complemented
179	C₂×#106	—	71	1	no	—
180	C₂×Q₁₆	—	71	1	no	—
181	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
182	C₂²×C₈	—	71	1	no	—
183	C₄⋊Q₈	—	71	1	no	—
184	#126	—	142	1	no	—
185	#125	—	71	1	no	—
186	#125	—	142	1	no	—
187	#125	—	142	1	no	—
188	#123	—	142	1	no	—
189	C₂²⋊Q₈	—	71	1	no	—
190	C₂²⋊Q₈	—	71	1	no	—
191	C₂²⋊Q₈	—	71	1	no	—
192	C₂²⋊Q₈	—	71	1	no	—
193	C₂²⋊Q₈	—	142	1	no	—
194	C₂²⋊Q₈	—	142	1	no	—
195	C₄×Q₈	—	71	1	no	—
196	C₄×Q₈	—	71	1	no	—
197	C₄×D₄	—	71	1	no	—
198	C₄×D₄	—	142	1	no	—
199	#117	—	142	1	no	—
200	#117	C₂×D₇₁	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	71	1	no	—
202	C₂×C₄⋊C₄	—	71	1	no	—
203	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
204	#107	—	71	1	no	—
205	#107	C₂×D₇₁	1	1	yes	characteristic
206	#107	C₂×D₇₁	1	1	yes	characteristic
207	#106	—	71	1	no	—
208	#106	—	71	1	no	—
209	#106	—	71	1	no	—
210	#106	—	71	1	no	—
211	#106	C₂×D₇₁	1	1	yes	characteristic
212	#106	C₂×D₇₁	1	1	yes	characteristic
213	C₄⋊C₈	—	71	1	no	—
214	#103	—	71	1	no	—
215	#103	—	71	1	no	—
216	#103	—	71	1	no	—
217	#103	—	71	1	no	—
218	#103	—	71	1	no	—
219	#103	—	71	1	no	—
220	C₂²⋊C₈	—	71	1	no	—
221	C₂²⋊C₈	—	71	1	no	—
222	#97	—	71	1	no	—
223	C₂×Q₈	—	71	1	no	—
224	C₂×Q₈	—	71	1	no	—
225	C₂×Q₈	—	142	1	no	—
226	C₂×D₄	—	71	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	71	1	no	—
229	C₂²×C₄	—	142	1	no	—
230	C₂²×C₄	—	142	1	no	—
231	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
232	Q₁₆	—	142	2	no	—
233	M₁₆	—	2	2	no	—
234	C₂×C₈	—	71	1	no	—
235	C₂×C₈	—	71	1	no	—
236	C₂×C₈	—	142	2	no	—
237	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	71	1	no	—
241	C₄⋊C₄	—	71	1	no	—
242	C₄⋊C₄	—	71	1	no	—
243	C₄⋊C₄	—	142	1	no	—
244	C₄⋊C₄	—	142	1	no	—
245	C₄⋊C₄	—	142	1	no	—
246	C₄⋊C₄	—	142	1	no	—
247	C₄⋊C₄	—	142	1	no	—
248	C₄⋊C₄	—	142	1	no	—
249	C₄⋊C₄	—	142	1	no	—
250	C₄⋊C₄	—	142	1	no	—
251	C₄⋊C₄	—	142	1	no	—
252	C₄⋊C₄	—	142	1	no	—
253	C₄⋊C₄	—	142	1	no	—
254	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	71	1	no	—
260	C₂²⋊C₄	—	71	1	no	—
261	C₂²⋊C₄	—	142	1	no	—
262	C₂²⋊C₄	—	142	1	no	—
263	C₂²⋊C₄	—	142	1	no	—
264	C₂²⋊C₄	—	142	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	71	1	no	—
267	C₄²	—	142	1	no	—
268	C₄²	—	142	1	no	—
269	C₂³	—	71	1	no	—
270	C₂³	D₄×D₇₁	1	1	yes	characteristic
271	Q₈	—	568	1	no	—
272	Q₈	—	142	1	no	—
273	Q₈	—	142	1	no	—
274	Q₈	—	142	1	no	—
275	Q₈	—	142	1	no	—
276	D₄	—	142	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	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	—	8	1	no	—
284	C₂×C₄	—	284	2	no	—
285	C₂×C₄	—	568	1	no	—
286	C₂×C₄	—	71	1	no	—
287	C₂×C₄	—	71	1	no	—
288	C₂×C₄	—	71	1	no	—
289	C₂×C₄	—	142	1	no	—
290	C₂×C₄	—	142	1	no	—
291	C₂×C₄	—	142	1	no	—
292	C₂×C₄	—	142	1	no	—
293	C₂×C₄	—	142	1	no	—
294	C₂×C₄	—	142	1	no	—
295	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	142	1	no	—
299	C₈	—	142	1	no	—
300	C₈	#11765865	1	2	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	71	2	no	—
305	C₂²	—	142	1	no	—
306	C₂²	—	142	1	no	—
307	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	284	1	no	—
314	C₄	—	284	1	no	—
315	C₄	—	284	1	no	—
316	C₄	—	284	1	no	—
317	C₄	—	142	1	no	—
318	C₄	—	142	1	no	—
319	C₄	—	142	1	no	—
320	C₄	C₂×#11765865	1	1	yes	characteristic
321	C₄	#14556176	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	142	2	no	—
324	C₂	#17986650	1	1	yes	characteristic
325	C₂	#137×D₇₁	1	1	yes	characteristic
326	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	#428×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#428	D₇₁
hypocenter	C₇₁	#3344
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
3. #17986407	C₂	semidirect
4. #17986340	C₂	semidirect
5. #17986336	C₂	semidirect
6. #17986331	C₂	semidirect
7. #17986329	C₂	semidirect
8. #17986305	C₂	semidirect
9. #17986283	C₂	semidirect
10. #17986257	C₂	semidirect
11. #17986241	C₂	semidirect
12. #17986105	C₂	semidirect
14. #106×D₇₁	C₂	semidirect
15. #428×C₇₁	C₂	semidirect
164. C₇₁	#3344	semidirect
178. #428	D₇₁	semidirect