Subgroups — #26031432

Order 11392.

332 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152154	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152133	C₂	1	1	yes	maximal, characteristic
3	#19151997	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151955	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151953	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151942	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151941	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151925	C₂	1	1	yes	maximal, characteristic
9	#19151924	C₂	1	1	yes	maximal, characteristic
10	#19151892	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151887	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151882	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151877	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151649	C₂	1	1	yes	maximal, characteristic, complemented
15	#499×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	#15668011	C₂²	1	1	yes	characteristic
17	#15668011	C₂²	1	1	yes	characteristic
18	#15668007	—	2	1	no	—
19	#15668004	C₂²	1	1	yes	characteristic
20	#15668000	C₂²	1	1	yes	characteristic
21	#15667997	—	2	1	no	—
22	#15667992	C₂²	1	1	yes	characteristic
23	#15667990	—	2	1	no	—
24	#15667990	—	2	1	no	—
25	#15667987	C₂²	1	1	yes	characteristic
26	#15667977	—	2	1	no	—
27	#15667977	—	2	1	no	—
28	#15667977	C₂²	1	1	yes	characteristic
29	#15667974	—	2	1	no	—
30	#15667974	C₂²	1	1	yes	characteristic
31	#15667970	—	2	1	no	—
32	#15667970	C₂²	1	1	yes	characteristic
33	#15667970	C₂²	1	1	yes	characteristic
34	C₄²×D₈₉	C₂²	1	1	yes	characteristic
35	#15667945	C₂²	1	1	yes	characteristic
36	#15667945	C₂²	1	1	yes	characteristic
37	#15667943	C₂²	1	1	yes	characteristic
38	#15667943	C₂²	1	1	yes	characteristic
39	#15667941	C₂²	1	1	yes	characteristic
40	#15667941	C₂²	1	1	yes	characteristic
41	#15667937	C₂²	1	1	yes	characteristic
42	#15667937	C₂²	1	1	yes	characteristic
43	#15667935	C₂²	1	1	yes	characteristic
44	#15667935	C₂²	1	1	yes	characteristic
45	#15667934	C₂²	1	1	yes	characteristic
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	C₄×#10623494	C₂²	1	1	yes	characteristic
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	#125×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#106×C₈₉	C₂²	1	1	yes	characteristic
57	#106×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	—	2	1	no	—
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
67	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
68	C₂×#10623494	C₂³	1	1	yes	characteristic
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
73	#13083553	—	2	1	no	—
74	#13083553	—	2	1	no	—
75	#13083553	—	2	1	no	—
76	#13083553	—	2	1	no	—
77	#13083553	—	2	1	no	—
78	#13083551	—	2	1	no	—
79	#13083551	—	2	1	no	—
80	#13083551	—	2	1	no	—
81	#13083551	—	2	1	no	—
82	#13083551	—	2	1	no	—
83	#13083551	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	C₄×Dic₈₉	—	2	1	no	—
86	C₄×Dic₈₉	—	2	1	no	—
87	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	D₄	1	1	yes	characteristic
89	C₄×Dic₈₉	D₄	1	1	yes	characteristic
90	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₄⋊C₄×C₈₉	—	2	1	no	—
95	C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₄⋊C₄×C₈₉	—	2	1	no	—
97	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₄²×C₈₉	C₂³	1	1	yes	characteristic
102	C₂²×D₈₉	—	2	1	no	—
103	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₈₉	—	2	1	no	—
112	C₂×Dic₈₉	—	2	1	no	—
113	C₂×Dic₈₉	—	2	1	no	—
114	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
117	D₃₅₆	—	2	1	no	—
118	D₃₅₆	—	2	1	no	—
119	D₃₅₆	—	4	2	no	—
120	C₄×D₈₉	—	2	1	no	—
121	C₄×D₈₉	—	2	1	no	—
122	C₄×D₈₉	—	2	2	no	—
123	C₄×D₈₉	—	4	1	no	—
124	C₄×D₈₉	—	4	2	no	—
125	Dic₁₇₈	—	2	1	no	—
126	Dic₁₇₈	—	2	1	no	—
127	Dic₁₇₈	—	4	2	no	—
128	Dic₁₇₈	—	4	2	no	—
129	C₈×C₈₉	—	2	1	no	—
130	C₈×C₈₉	—	2	1	no	—
131	#10623494	—	2	1	no	—
132	#10623494	—	2	1	no	—
133	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×D₈₉	—	2	1	no	—
135	C₂×D₈₉	—	2	1	no	—
136	C₂×D₈₉	—	2	1	no	—
137	C₂×D₈₉	—	4	1	no	—
138	C₂×D₈₉	—	4	1	no	—
139	C₂×D₈₉	—	4	1	no	—
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	4	1	no	—
143	C₄×C₈₉	—	4	1	no	—
144	C₄×C₈₉	—	4	1	no	—
145	C₄×C₈₉	—	4	1	no	—
146	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₈₉	—	2	1	no	—
149	Dic₈₉	—	2	1	no	—
150	Dic₈₉	—	4	1	no	—
151	Dic₈₉	—	4	1	no	—
152	Dic₈₉	—	4	1	no	—
153	Dic₈₉	#135	1	2	yes	—
154	C₂×C₈₉	#536	1	1	yes	characteristic
155	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
156	C₂×C₈₉	#575	1	1	yes	characteristic
157	D₈₉	—	4	1	no	—
158	D₈₉	—	8	1	no	—
159	#3233	—	89	1	no	maximal, Sylow
160	C₈₉	#3233	1	1	yes	characteristic, complemented, Sylow
161	#533	—	89	1	no	—
162	#529	—	89	1	no	—
163	#499	—	89	1	no	—
164	#499	D₈₉	1	1	yes	characteristic, complemented
165	#488	—	89	1	no	—
166	#488	—	89	1	no	—
167	#485	—	89	1	no	—
168	#485	—	89	1	no	—
169	#482	—	89	1	no	—
170	#482	—	89	1	no	—
171	#477	—	89	1	no	—
172	#476	—	89	1	no	—
173	#475	—	89	1	no	—
174	#474	—	89	1	no	—
175	#431	—	89	1	no	—
176	C₄⋊Q₈	—	89	1	no	—
177	C₄⋊Q₈	—	89	1	no	—
178	#126	—	178	1	no	—
179	#125	—	89	1	no	—
180	#125	C₂×D₈₉	1	1	yes	characteristic
181	#125	C₂×D₈₉	1	1	yes	characteristic
182	#124	—	89	1	no	—
183	#123	—	178	1	no	—
184	C₂²⋊Q₈	—	89	1	no	—
185	C₂²⋊Q₈	—	178	1	no	—
186	C₂²⋊Q₈	—	178	1	no	—
187	#121	—	89	1	no	—
188	C₄×Q₈	—	89	1	no	—
189	C₄×Q₈	—	178	1	no	—
190	C₄×Q₈	—	178	1	no	—
191	C₄×D₄	—	89	1	no	—
192	C₄×D₄	—	178	1	no	—
193	#117	—	89	1	no	—
194	#117	—	89	1	no	—
195	#117	—	178	1	no	—
196	C₂×C₄²	—	89	1	no	—
197	#107	—	89	1	no	—
198	#107	—	89	1	no	—
199	#107	C₂×D₈₉	1	1	yes	characteristic
200	#107	C₂×D₈₉	1	1	yes	characteristic
201	#106	—	89	1	no	—
202	#106	—	89	1	no	—
203	#106	C₂×D₈₉	1	1	yes	characteristic
204	#106	C₂×D₈₉	1	1	yes	characteristic
205	C₄⋊C₈	—	89	1	no	—
206	C₄⋊C₈	—	89	1	no	—
207	#103	—	89	1	no	—
208	#103	—	89	1	no	—
209	#103	—	89	1	no	—
210	#103	—	89	1	no	—
211	#102	—	89	1	no	—
212	#102	—	89	1	no	—
213	#102	—	89	1	no	—
214	#102	—	89	1	no	—
215	C₂²⋊C₈	—	89	1	no	—
216	C₂²⋊C₈	—	89	1	no	—
217	C₄×C₈	—	89	1	no	—
218	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
219	C₂×Q₈	—	89	1	no	—
220	C₂×Q₈	—	178	1	no	—
221	C₂×Q₈	—	178	1	no	—
222	C₂×D₄	—	89	1	no	—
223	C₂×D₄	—	178	1	no	—
224	C₂²×C₄	—	89	1	no	—
225	C₂²×C₄	—	89	1	no	—
226	C₂²×C₄	—	89	1	no	—
227	C₂²×C₄	—	178	1	no	—
228	C₂×C₈	—	89	1	no	—
229	C₂×C₈	—	89	1	no	—
230	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	89	1	no	—
237	C₄⋊C₄	—	89	1	no	—
238	C₄⋊C₄	—	178	1	no	—
239	C₄⋊C₄	—	178	1	no	—
240	C₄⋊C₄	—	178	1	no	—
241	C₄⋊C₄	—	178	1	no	—
242	C₄⋊C₄	—	178	1	no	—
243	C₄⋊C₄	—	178	1	no	—
244	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
246	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
247	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
248	C₂²⋊C₄	—	178	1	no	—
249	C₂²⋊C₄	—	178	1	no	—
250	C₂²⋊C₄	—	178	1	no	—
251	C₂²⋊C₄	—	178	1	no	—
252	C₂²⋊C₄	—	178	1	no	—
253	C₂²⋊C₄	—	178	1	no	—
254	C₂²⋊C₄	—	178	1	no	—
255	C₄²	—	89	1	no	—
256	C₄²	—	89	1	no	—
257	C₄²	—	89	1	no	—
258	C₄²	—	178	1	no	—
259	C₄²	—	178	1	no	—
260	C₄²	—	178	1	no	—
261	C₄²	C₂²×D₈₉	1	1	yes	characteristic
262	C₂³	—	89	1	no	—
263	C₂³	—	178	1	no	—
264	Q₈	—	356	2	no	—
265	Q₈	—	356	2	no	—
266	Q₈	—	178	1	no	—
267	Q₈	—	178	1	no	—
268	D₄	—	356	2	no	—
269	D₄	—	178	1	no	—
270	D₄	—	178	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	89	1	no	—
276	C₂×C₄	—	89	1	no	—
277	C₂×C₄	—	89	1	no	—
278	C₂×C₄	—	356	1	no	—
279	C₂×C₄	—	356	2	no	—
280	C₂×C₄	—	178	2	no	—
281	C₂×C₄	—	178	1	no	—
282	C₂×C₄	—	178	1	no	—
283	C₂×C₄	—	178	1	no	—
284	C₂×C₄	—	178	1	no	—
285	C₂×C₄	—	178	1	no	—
286	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
287	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	178	1	no	—
292	C₈	—	178	1	no	—
293	C₂²	—	356	1	no	—
294	C₂²	—	356	1	no	—
295	C₂²	—	356	1	no	—
296	C₂²	—	178	1	no	—
297	C₂²	—	178	1	no	—
298	C₂²	—	178	1	no	—
299	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	89	2	no	—
307	C₄	—	356	1	no	—
308	C₄	—	356	1	no	—
309	C₄	—	356	1	no	—
310	C₄	—	178	1	no	—
311	C₄	—	178	1	no	—
312	C₄	#15668060	1	1	yes	characteristic
313	C₄	#15668060	1	1	yes	characteristic
314	C₂	—	356	1	no	—
315	C₂	—	712	1	no	—
316	C₂	#19152173	1	1	yes	characteristic
317	C₂	#135×D₈₉	1	1	yes	characteristic
318	C₂	#19152395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#499×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#15667987	C₂²
hypercenter	#499	D₈₉
hypocenter	C₈₉	#3233
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
1. #19152154	C₂	semidirect
3. #19151997	C₂	semidirect
4. #19151955	C₂	semidirect
5. #19151953	C₂	semidirect
6. #19151942	C₂	semidirect
7. #19151941	C₂	semidirect
10. #19151892	C₂	semidirect
11. #19151887	C₂	semidirect
12. #19151882	C₂	semidirect
13. #19151877	C₂	semidirect
14. #19151649	C₂	semidirect
15. #499×C₈₉	C₂	semidirect
160. C₈₉	#3233	semidirect
164. #499	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152154	2
#19152133	2
#19151997	2
#19151955	2
#19151953	2
#19151942	2
#19151941	2
#19151925	2
#19151924	2
#19151892	2
#19151887	2
#19151882	2
#19151877	2
#19151649	2
#499×C₈₉	2
#3233	89