Subgroups — #26033094

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	#19152321	C₂	1	1	yes	maximal, characteristic
2	#19151968	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151960	C₂	1	1	yes	maximal, characteristic
4	#19151955	C₂	1	1	yes	maximal, characteristic
5	#19151902	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151895	C₂	1	1	yes	maximal, characteristic
7	#19151890	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151820	C₂	1	1	yes	maximal, characteristic
9	#19151820	C₂	1	1	yes	maximal, characteristic
10	#19151808	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151729	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151703	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151696	C₂	1	1	yes	maximal, characteristic
14	C₄×#13083564	C₂	1	1	yes	maximal, characteristic, complemented
15	#557×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂×#13083564	C₂²	1	1	yes	characteristic
18	C₂×#13083561	C₂²	1	1	yes	characteristic
19	#15668012	C₂²	1	1	yes	characteristic
20	#15668007	—	2	1	no	—
21	#15668007	—	2	1	no	—
22	#15668000	C₂²	1	1	yes	characteristic
23	#15668000	C₂²	1	1	yes	characteristic
24	#15667988	—	2	1	no	—
25	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
27	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
28	#15667970	—	2	1	no	—
29	#15667945	C₂²	1	1	yes	characteristic
30	#15667945	C₂²	1	1	yes	characteristic
31	#15667943	C₂²	1	1	yes	characteristic
32	C₄⋊#10623494	C₂²	1	1	yes	characteristic
33	C₄⋊#10623494	C₂²	1	1	yes	characteristic
34	C₄⋊#10623494	C₂²	1	1	yes	characteristic
35	#15667936	C₂²	1	1	yes	characteristic
36	#15667935	C₂²	1	1	yes	characteristic
37	#15667935	C₂²	1	1	yes	characteristic
38	#15667935	C₂²	1	1	yes	characteristic
39	#15667935	C₂²	1	1	yes	characteristic
40	#15667935	C₂²	1	1	yes	characteristic
41	#15667935	C₂²	1	1	yes	characteristic
42	#15667932	C₂²	1	1	yes	characteristic
43	#15667932	C₂²	1	1	yes	characteristic
44	#15667932	C₂²	1	1	yes	characteristic
45	#15667923	C₂²	1	1	yes	characteristic
46	#15667923	C₂²	1	1	yes	characteristic
47	C₄×#10623494	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₉	—	2	1	no	—
49	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
50	#125×C₈₉	—	2	1	no	—
51	#125×C₈₉	—	2	1	no	—
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	#125×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	—	2	1	no	—
55	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₈₉	—	2	1	no	—
63	#13083564	—	2	2	no	—
64	#13083564	—	2	2	no	—
65	#13083561	—	2	2	no	—
66	C₂×#10623494	C₂³	1	1	yes	characteristic
67	C₂×#10623494	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	—	2	1	no	—
72	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
73	#13083553	—	2	1	no	—
74	#13083553	—	2	1	no	—
75	#13083551	—	2	1	no	—
76	#13083551	—	2	1	no	—
77	#13083551	—	2	1	no	—
78	#13083551	—	2	1	no	—
79	#13083551	—	2	1	no	—
80	C₄×Dic₈₉	—	2	1	no	—
81	C₄×Dic₈₉	—	2	1	no	—
82	C₂×Q₈×C₈₉	—	2	1	no	—
83	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
84	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₈₉	—	2	1	no	—
86	C₄⋊C₄×C₈₉	—	2	1	no	—
87	C₄⋊C₄×C₈₉	—	2	1	no	—
88	C₄⋊C₄×C₈₉	—	2	1	no	—
89	C₄⋊C₄×C₈₉	—	2	1	no	—
90	C₄⋊C₄×C₈₉	—	2	1	no	—
91	C₄⋊C₄×C₈₉	—	2	1	no	—
92	C₄⋊C₄×C₈₉	—	2	1	no	—
93	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₄²×C₈₉	—	2	1	no	—
100	C₄²×C₈₉	—	2	1	no	—
101	C₄²×C₈₉	—	2	1	no	—
102	C₄²×C₈₉	C₂³	1	1	yes	characteristic
103	C₄²×C₈₉	C₂³	1	1	yes	characteristic
104	C₄²×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²×D₈₉	—	2	1	no	—
106	Q₈×C₈₉	—	2	1	no	—
107	Q₈×C₈₉	—	4	1	no	—
108	Q₈×C₈₉	—	4	2	no	—
109	Q₈×C₈₉	#41	1	2	yes	—
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	—	2	1	no	—
114	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	D₃₅₆	—	2	1	no	—
125	D₃₅₆	—	2	1	no	—
126	C₄×D₈₉	—	4	2	no	—
127	Dic₁₇₈	—	2	1	no	—
128	Dic₁₇₈	—	2	1	no	—
129	Dic₁₇₈	—	2	1	no	—
130	Dic₁₇₈	—	2	1	no	—
131	#10623494	—	2	1	no	—
132	#10623494	—	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	2	no	—
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	—	4	1	no	—
147	C₄×C₈₉	—	4	1	no	—
148	C₄×C₈₉	—	4	1	no	—
149	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
150	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₈₉	—	4	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	D₈₉	1	1	yes	characteristic, complemented
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	—	89	1	no	—
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	—	2	1	no	—
179	C₄⋊Q₈	—	89	1	no	—
180	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
181	#126	—	178	1	no	—
182	#126	—	178	1	no	—
183	#125	—	2	1	no	—
184	#125	—	2	1	no	—
185	#125	C₂×D₈₉	1	1	yes	characteristic
186	#125	C₂×D₈₉	1	1	yes	characteristic
187	#124	—	89	1	no	—
188	#124	—	89	1	no	—
189	C₂²⋊Q₈	—	178	1	no	—
190	C₄×Q₈	—	2	1	no	—
191	C₄×Q₈	—	89	1	no	—
192	C₄×Q₈	—	89	1	no	—
193	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
194	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
195	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
196	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
197	C₄×D₄	—	89	1	no	—
198	#117	—	178	1	no	—
199	#107	—	89	1	no	—
200	#107	—	89	1	no	—
201	#106	—	89	1	no	—
202	C₄⋊C₈	—	89	1	no	—
203	C₄⋊C₈	—	89	1	no	—
204	C₄⋊C₈	—	89	1	no	—
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	—	89	1	no	—
211	#103	—	89	1	no	—
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	C₂²⋊D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
231	C₂×C₈	—	89	1	no	—
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	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	89	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	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	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	—	2	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	178	1	no	—
260	C₄²	—	178	1	no	—
261	C₄²	C₂²×D₈₉	1	1	yes	characteristic
262	C₄²	C₂²×D₈₉	1	1	yes	characteristic
263	C₄²	C₂²×D₈₉	1	1	yes	characteristic
264	C₂³	—	178	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	4	1	no	—
267	Q₈	—	4	2	no	—
268	Q₈	—	178	1	no	—
269	Q₈	—	178	1	no	—
270	Q₈	—	178	1	no	—
271	Q₈	—	178	1	no	—
272	Q₈	#13083575	1	2	yes	—
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	—	356	2	no	—
280	C₂×C₄	—	178	1	no	—
281	C₂×C₄	—	178	1	no	—
282	C₂×C₄	—	178	1	no	—
283	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
290	C₈	—	178	1	no	—
291	C₈	—	178	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	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	356	1	no	—
309	C₄	—	356	1	no	—
310	C₄	—	356	1	no	—
311	C₄	C₂×#13083575	1	1	yes	characteristic
312	C₄	#15668060	1	1	yes	characteristic
313	C₂	—	712	1	no	—
314	C₂	#19152259	1	1	yes	characteristic
315	C₂	#19152396	1	1	yes	characteristic
316	C₂	#19152413	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#557×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₃₅₆	C₂³
hypercenter	#557	D₈₉
hypocenter	C₈₉	#3377
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

Complements

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

N	H ≅ G/N	type
2. #19151968	C₂	semidirect
5. #19151902	C₂	semidirect
7. #19151890	C₂	semidirect
10. #19151808	C₂	semidirect
11. #19151729	C₂	semidirect
12. #19151703	C₂	semidirect
14. C₄×#13083564	C₂	semidirect
15. #557×C₈₉	C₂	semidirect
159. C₈₉	#3377	semidirect
160. #557	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152321	2
#19151968	2
#19151960	2
#19151955	2
#19151902	2
#19151895	2
#19151890	2
#19151820	2
#19151808	2
#19151729	2
#19151703	2
#19151696	2
C₄×#13083564	2
#557×C₈₉	2
#3377	89