Subgroups — #26033101

Order 11392.

332 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152323	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152320	C₂	1	1	yes	maximal, characteristic
3	#19151965	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151960	C₂	1	1	yes	maximal, characteristic
5	#19151954	C₂	1	1	yes	maximal, characteristic
6	#19151904	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151901	C₂	1	1	yes	maximal, characteristic
8	#19151823	C₂	1	1	yes	maximal, characteristic
9	#19151821	C₂	1	1	yes	maximal, characteristic
10	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151727	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151703	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151697	C₂	1	1	yes	maximal, characteristic
14	#19151692	C₂	1	1	yes	maximal, characteristic, complemented
15	#477×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
17	C₂×#13083564	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	#15668010	—	2	1	no	—
20	#15668010	C₂²	1	1	yes	characteristic
21	#15668007	—	2	1	no	—
22	#15668006	—	2	1	no	—
23	#15668003	C₂²	1	1	yes	characteristic
24	#15668003	C₂²	1	1	yes	characteristic
25	#15668002	—	2	1	no	—
26	#15668002	—	2	1	no	—
27	#15668001	C₂²	1	1	yes	characteristic
28	#15668000	C₂²	1	1	yes	characteristic
29	#15667990	—	2	1	no	—
30	Q₈×Dic₈₉	—	2	1	no	—
31	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
33	#15667977	C₂²	1	1	yes	characteristic
34	#15667977	C₂²	1	1	yes	characteristic
35	#15667977	C₂²	1	1	yes	characteristic
36	#15667974	C₂²	1	1	yes	characteristic
37	#15667969	—	2	1	no	—
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	#15667936	C₂²	1	1	yes	characteristic
43	#15667935	C₂²	1	1	yes	characteristic
44	#15667935	C₂²	1	1	yes	characteristic
45	#15667935	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	#15667923	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₈₉	—	2	1	no	—
64	Dic₃₅₆	—	2	2	no	—
65	#13083564	—	2	2	no	—
66	#13083563	—	2	2	no	—
67	C₂×#10623494	C₂³	1	1	yes	characteristic
68	C₂×#10623494	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₈₉	—	2	1	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	D₄	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	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	D₄	1	1	yes	characteristic
82	#13083553	D₄	1	1	yes	characteristic
83	#13083551	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₄×Dic₈₉	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
93	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₈₉	—	2	1	no	—
95	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₄²×C₈₉	—	2	1	no	—
104	C₄²×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²×D₈₉	—	2	1	no	—
106	Q₈×C₈₉	—	2	1	no	—
107	Q₈×C₈₉	—	2	1	no	—
108	Q₈×C₈₉	—	4	2	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₈₉	—	2	1	no	—
117	C₂×Dic₈₉	—	2	1	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
123	D₃₅₆	—	2	1	no	—
124	D₃₅₆	—	2	1	no	—
125	C₄×D₈₉	—	4	2	no	—
126	Dic₁₇₈	—	2	1	no	—
127	Dic₁₇₈	—	2	1	no	—
128	Dic₁₇₈	—	2	1	no	—
129	Dic₁₇₈	—	4	1	no	—
130	Dic₁₇₈	#41	1	2	yes	—
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	—	2	1	no	—
133	#10623494	—	2	1	no	—
134	#10623494	—	2	1	no	—
135	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₈₉	—	4	1	no	—
137	C₂×D₈₉	—	4	1	no	—
138	C₂×D₈₉	—	4	1	no	—
139	C₄×C₈₉	—	2	1	no	—
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	—	4	1	no	—
142	C₄×C₈₉	—	4	1	no	—
143	C₄×C₈₉	—	4	1	no	—
144	C₄×C₈₉	—	4	1	no	—
145	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
146	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₈₉	—	2	2	no	—
148	Dic₈₉	—	2	1	no	—
149	Dic₈₉	—	2	1	no	—
150	Dic₈₉	—	2	1	no	—
151	Dic₈₉	—	2	1	no	—
152	Dic₈₉	—	4	1	no	—
153	Dic₈₉	—	4	1	no	—
154	C₂×C₈₉	#548	1	1	yes	characteristic
155	C₂×C₈₉	#575	1	1	yes	characteristic
156	C₂×C₈₉	#578	1	1	yes	characteristic
157	D₈₉	—	8	1	no	—
158	#3377	—	89	1	no	maximal, Sylow
159	C₈₉	#3377	1	1	yes	characteristic, complemented, Sylow
160	#557	—	89	1	no	—
161	#556	—	89	1	no	—
162	#491	—	89	1	no	—
163	#489	—	89	1	no	—
164	#488	—	89	1	no	—
165	#479	—	89	1	no	—
166	#478	—	89	1	no	—
167	#477	D₈₉	1	1	yes	characteristic, complemented
168	#464	—	89	1	no	—
169	#464	—	89	1	no	—
170	#462	—	89	1	no	—
171	#446	—	89	1	no	—
172	#441	—	89	1	no	—
173	#440	—	89	1	no	—
174	C₄×Q₁₆	—	89	1	no	—
175	C₂×Q₁₆	—	89	1	no	—
176	C₂×Q₁₆	—	89	1	no	—
177	C₂×SD₁₆	—	89	1	no	—
178	C₄⋊Q₈	—	89	1	no	—
179	C₄⋊Q₈	—	178	1	no	—
180	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
181	#126	—	178	1	no	—
182	#126	—	178	1	no	—
183	#125	—	89	1	no	—
184	#125	—	89	1	no	—
185	#125	—	178	1	no	—
186	#125	—	178	1	no	—
187	#124	—	89	1	no	—
188	#124	—	89	1	no	—
189	C₂²⋊Q₈	—	178	1	no	—
190	C₄×Q₈	—	89	1	no	—
191	C₄×Q₈	—	89	1	no	—
192	C₄×Q₈	—	89	1	no	—
193	C₄×Q₈	—	89	1	no	—
194	C₄×Q₈	—	89	1	no	—
195	C₄×Q₈	—	178	1	no	—
196	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
197	C₄×D₄	—	89	1	no	—
198	#117	—	178	1	no	—
199	#107	C₂×D₈₉	1	1	yes	characteristic
200	#107	C₂×D₈₉	1	1	yes	characteristic
201	#106	—	89	1	no	—
202	C₄⋊C₈	—	89	1	no	—
203	C₄⋊C₈	—	89	1	no	—
204	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
205	#103	—	89	1	no	—
206	#103	—	89	1	no	—
207	#103	—	89	1	no	—
208	#103	—	89	1	no	—
209	#103	—	89	1	no	—
210	#103	C₂×D₈₉	1	1	yes	characteristic
211	#103	C₂×D₈₉	1	1	yes	characteristic
212	#102	—	89	1	no	—
213	#102	—	89	1	no	—
214	#102	—	89	1	no	—
215	#97	—	89	1	no	—
216	#97	—	89	1	no	—
217	C₄×C₈	—	89	1	no	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×Q₈	—	89	1	no	—
220	C₂×Q₈	—	89	1	no	—
221	C₂×Q₈	—	89	1	no	—
222	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
223	C₂×D₄	—	89	1	no	—
224	C₂²×C₄	—	178	1	no	—
225	Q₁₆	—	178	2	no	—
226	Q₁₆	—	178	2	no	—
227	SD₁₆	—	178	2	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	—	89	1	no	—
235	C₄⋊C₄	—	89	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	—	178	1	no	—
245	C₄⋊C₄	—	178	1	no	—
246	C₄⋊C₄	—	178	1	no	—
247	C₄⋊C₄	—	178	1	no	—
248	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
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	C₂²⋊C₄	—	178	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	89	1	no	—
258	C₄²	—	89	1	no	—
259	C₄²	—	178	1	no	—
260	C₄²	—	178	1	no	—
261	C₄²	—	178	1	no	—
262	C₄²	—	178	1	no	—
263	C₄²	C₂²×D₈₉	1	1	yes	characteristic
264	C₂³	—	178	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	4	2	no	—
268	Q₈	—	89	2	no	—
269	Q₈	—	356	1	no	—
270	Q₈	—	178	1	no	—
271	Q₈	—	178	1	no	—
272	Q₈	—	178	1	no	—
273	D₄	—	178	1	no	—
274	D₄	—	178	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	89	1	no	—
280	C₂×C₄	—	89	1	no	—
281	C₂×C₄	—	89	1	no	—
282	C₂×C₄	—	89	1	no	—
283	C₂×C₄	—	356	2	no	—
284	C₂×C₄	—	178	1	no	—
285	C₂×C₄	—	178	1	no	—
286	C₂×C₄	—	178	1	no	—
287	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
288	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	178	1	no	—
293	C₈	—	178	1	no	—
294	C₂²	—	356	1	no	—
295	C₂²	—	356	1	no	—
296	C₂²	—	356	1	no	—
297	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	356	1	no	—
305	C₄	—	356	1	no	—
306	C₄	—	178	1	no	—
307	C₄	—	178	1	no	—
308	C₄	—	178	1	no	—
309	C₄	—	178	1	no	—
310	C₄	—	178	2	no	—
311	C₄	#41×D₈₉	1	1	yes	characteristic
312	C₄	#15668060	1	1	yes	characteristic
313	C₂	—	712	1	no	—
314	C₂	#19152258	1	1	yes	characteristic
315	C₂	#19152394	1	1	yes	characteristic
316	C₂	#19152411	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#477×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₃₅₆	C₂³
hypercenter	#477	D₈₉
hypocenter	C₈₉	#3377
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

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

N	H ≅ G/N	type
1. #19152323	C₂	semidirect
3. #19151965	C₂	semidirect
6. #19151904	C₂	semidirect
10. #19151807	C₂	semidirect
11. #19151727	C₂	semidirect
12. #19151703	C₂	semidirect
14. #19151692	C₂	semidirect
15. #477×C₈₉	C₂	semidirect
159. C₈₉	#3377	semidirect
167. #477	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152323	2
#19152320	2
#19151965	2
#19151960	2
#19151954	2
#19151904	2
#19151901	2
#19151823	2
#19151821	2
#19151807	2
#19151727	2
#19151703	2
#19151697	2
#19151692	2
#477×C₈₉	2
#3377	89