Subgroups — #26031514

Order 11392.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152226	C₂	1	1	yes	maximal, characteristic
2	#19152008	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151997	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151966	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151948	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151936	C₂	1	1	yes	maximal, characteristic
7	#19151936	C₂	1	1	yes	maximal, characteristic
8	#19151931	C₂	1	1	yes	maximal, characteristic
9	#19151926	C₂	1	1	yes	maximal, characteristic
10	#19151905	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151905	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151893	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151883	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151659	C₂	1	1	yes	maximal, characteristic
15	#541×C₈₉	C₂	1	1	yes	maximal, characteristic
16	#15668012	C₂²	1	1	yes	characteristic
17	#15668012	C₂²	1	1	yes	characteristic
18	#15668011	—	2	1	no	—
19	#15668004	C₂²	1	1	yes	characteristic
20	#15668004	C₂²	1	1	yes	characteristic
21	#15668003	—	2	1	no	—
22	#15667989	—	2	1	no	—
23	#15667989	—	2	1	no	—
24	#15667971	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
26	#15667945	C₂²	1	1	yes	characteristic
27	#15667945	C₂²	1	1	yes	characteristic
28	#15667945	C₂²	1	1	yes	characteristic
29	#15667945	C₂²	1	1	yes	characteristic
30	#15667943	C₂²	1	1	yes	characteristic
31	#15667943	C₂²	1	1	yes	characteristic
32	#15667943	C₂²	1	1	yes	characteristic
33	#15667943	C₂²	1	1	yes	characteristic
34	C₄⋊#10623494	C₂²	1	1	yes	characteristic
35	C₄⋊#10623494	C₂²	1	1	yes	characteristic
36	#15667936	C₂²	1	1	yes	characteristic
37	#15667936	C₂²	1	1	yes	characteristic
38	#15667936	C₂²	1	1	yes	characteristic
39	#15667936	C₂²	1	1	yes	characteristic
40	#15667936	C₂²	1	1	yes	characteristic
41	#15667936	C₂²	1	1	yes	characteristic
42	#15667936	C₂²	1	1	yes	characteristic
43	#15667936	C₂²	1	1	yes	characteristic
44	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
45	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
46	#15667923	C₂²	1	1	yes	characteristic
47	C₄×#10623494	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
49	#126×C₈₉	—	2	1	no	—
50	#126×C₈₉	—	2	1	no	—
51	#125×C₈₉	—	2	1	no	—
52	#125×C₈₉	—	2	1	no	—
53	#125×C₈₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²⋊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	#117×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂²×Dic₈₉	—	2	1	no	—
62	C₂×#10623494	C₂³	1	1	yes	characteristic
63	C₂×#10623494	C₂³	1	1	yes	characteristic
64	C₂×#10623494	C₂³	1	1	yes	characteristic
65	C₂×#10623494	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₈₉	—	2	1	no	—
67	C₄⋊Dic₈₉	—	2	1	no	—
68	C₄⋊Dic₈₉	—	2	1	no	—
69	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₈₉	—	2	1	no	—
78	C₄×Dic₈₉	—	2	1	no	—
79	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₈₉	—	2	1	no	—
83	C₄⋊C₄×C₈₉	—	2	1	no	—
84	C₄⋊C₄×C₈₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
91	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₂²⋊C₄×C₈₉	—	2	1	no	—
95	C₂²⋊C₄×C₈₉	—	2	1	no	—
96	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
97	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
98	C₄²×C₈₉	—	2	1	no	—
99	C₄²×C₈₉	—	2	1	no	—
100	C₄²×C₈₉	C₂³	1	1	yes	characteristic
101	C₄²×C₈₉	C₂³	1	1	yes	characteristic
102	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₈₉	—	2	1	no	—
104	Q₈×C₈₉	—	2	1	no	—
105	Q₈×C₈₉	—	2	1	no	—
106	Q₈×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	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	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	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	8	1	no	—
124	Dic₁₇₈	—	8	1	no	—
125	Dic₁₇₈	—	8	1	no	—
126	#10623494	—	2	1	no	—
127	#10623494	—	2	1	no	—
128	#10623494	—	2	1	no	—
129	#10623494	—	2	1	no	—
130	C₂²×C₈₉	—	2	1	no	—
131	C₂²×C₈₉	—	2	1	no	—
132	C₂²×C₈₉	—	2	1	no	—
133	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	2	1	no	—
139	C₄×C₈₉	—	4	1	no	—
140	C₄×C₈₉	—	4	1	no	—
141	C₄×C₈₉	—	4	1	no	—
142	C₄×C₈₉	—	4	1	no	—
143	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
145	Dic₈₉	—	4	1	no	—
146	Dic₈₉	—	4	1	no	—
147	Dic₈₉	—	4	1	no	—
148	Dic₈₉	—	4	1	no	—
149	C₂×C₈₉	—	4	1	no	—
150	C₂×C₈₉	#536	1	1	yes	characteristic
151	C₂×C₈₉	#577	1	1	yes	characteristic
152	C₂×C₈₉	#578	1	1	yes	characteristic
153	#3242	—	89	1	no	maximal, Sylow
154	C₈₉	#3242	1	1	yes	characteristic, complemented, Sylow
155	#544	—	89	1	no	—
156	#541	D₈₉	1	1	yes	characteristic
157	#501	—	89	1	no	—
158	#499	—	89	1	no	—
159	#491	—	89	1	no	—
160	#487	—	89	1	no	—
161	#485	—	89	1	no	—
162	#485	—	89	1	no	—
163	#484	—	89	1	no	—
164	#483	—	89	1	no	—
165	#479	—	89	1	no	—
166	#479	—	89	1	no	—
167	#477	—	89	1	no	—
168	#475	—	89	1	no	—
169	#433	—	89	1	no	—
170	C₄⋊Q₈	—	89	1	no	—
171	C₄⋊Q₈	—	89	1	no	—
172	C₄⋊Q₈	—	178	1	no	—
173	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
174	#126	—	2	1	no	—
175	#126	—	2	1	no	—
176	#125	—	2	1	no	—
177	#125	—	2	1	no	—
178	#125	—	89	1	no	—
179	#125	—	89	1	no	—
180	#125	—	178	1	no	—
181	#125	C₂×D₈₉	1	1	yes	characteristic
182	C₂²⋊Q₈	—	178	1	no	—
183	C₂²⋊Q₈	—	178	1	no	—
184	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
185	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
186	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
187	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
188	#117	—	89	1	no	—
189	#117	C₂×D₈₉	1	1	yes	characteristic
190	C₂×C₄⋊C₄	—	89	1	no	—
191	#107	—	89	1	no	—
192	#107	—	89	1	no	—
193	#107	—	89	1	no	—
194	#107	—	89	1	no	—
195	#106	—	89	1	no	—
196	#106	—	89	1	no	—
197	#106	—	89	1	no	—
198	#106	—	89	1	no	—
199	C₄⋊C₈	—	89	1	no	—
200	C₄⋊C₈	—	89	1	no	—
201	#103	—	89	1	no	—
202	#103	—	89	1	no	—
203	#103	—	89	1	no	—
204	#103	—	89	1	no	—
205	#103	—	89	1	no	—
206	#103	—	89	1	no	—
207	#103	—	89	1	no	—
208	#103	—	89	1	no	—
209	C₂²⋊C₈	—	89	1	no	—
210	C₂²⋊C₈	—	89	1	no	—
211	#97	—	89	1	no	—
212	C₄×C₈	—	89	1	no	—
213	C₂×Q₈	—	178	1	no	—
214	C₂×Q₈	—	178	1	no	—
215	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
216	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	—	178	1	no	—
218	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
219	C₂×C₈	—	89	1	no	—
220	C₂×C₈	—	89	1	no	—
221	C₂×C₈	—	89	1	no	—
222	C₂×C₈	—	89	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	89	1	no	—
232	C₄⋊C₄	—	89	1	no	—
233	C₄⋊C₄	—	89	1	no	—
234	C₄⋊C₄	—	89	1	no	—
235	C₄⋊C₄	—	178	1	no	—
236	C₄⋊C₄	—	178	1	no	—
237	C₄⋊C₄	—	178	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	C₂²⋊D₈₉	1	1	yes	characteristic
243	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
244	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	178	1	no	—
249	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
250	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	178	1	no	—
254	C₄²	C₂²×D₈₉	1	1	yes	characteristic
255	C₄²	C₂²×D₈₉	1	1	yes	characteristic
256	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
257	Q₈	—	2	1	no	—
258	Q₈	—	2	1	no	—
259	Q₈	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	712	1	no	—
262	Q₈	—	712	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	178	1	no	—
270	C₂×C₄	—	178	1	no	—
271	C₂×C₄	—	178	1	no	—
272	C₂×C₄	—	178	1	no	—
273	C₂×C₄	—	712	1	no	—
274	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
280	C₈	—	178	1	no	—
281	C₈	—	178	1	no	—
282	C₈	—	178	1	no	—
283	C₈	—	178	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	356	1	no	—
298	C₄	—	356	1	no	—
299	C₄	—	356	1	no	—
300	C₄	—	356	1	no	—
301	C₄	#15668059	1	1	yes	characteristic
302	C₄	#15668059	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	#19152169	1	1	yes	characteristic
305	C₂	#19152406	1	1	yes	characteristic
306	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	#541×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#541	D₈₉
hypocenter	C₈₉	#3242
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #541

Complements

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

N	H ≅ G/N	type
2. #19152008	C₂	semidirect
3. #19151997	C₂	semidirect
4. #19151966	C₂	semidirect
5. #19151948	C₂	semidirect
10. #19151905	C₂	semidirect
11. #19151905	C₂	semidirect
12. #19151893	C₂	semidirect
13. #19151883	C₂	semidirect
154. C₈₉	#3242	semidirect

Maximal subgroups

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

subgroup	index
#19152226	2
#19152008	2
#19151997	2
#19151966	2
#19151948	2
#19151936	2
#19151931	2
#19151926	2
#19151905	2
#19151893	2
#19151883	2
#19151659	2
#541×C₈₉	2
#3242	89