Subgroups — #26031513

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	#19152225	C₂	1	1	yes	maximal, characteristic
2	#19152209	C₂	1	1	yes	maximal, characteristic
3	#19152004	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152000	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151966	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151938	C₂	1	1	yes	maximal, characteristic
7	#19151937	C₂	1	1	yes	maximal, characteristic
8	#19151932	C₂	1	1	yes	maximal, characteristic
9	#19151927	C₂	1	1	yes	maximal, characteristic
10	#19151907	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151903	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151891	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151881	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151657	C₂	1	1	yes	maximal, characteristic
15	#487×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	#15668011	C₂²	1	1	yes	characteristic
17	#15668010	—	2	1	no	—
18	#15668007	—	2	1	no	—
19	#15668007	—	2	1	no	—
20	#15668004	C₂²	1	1	yes	characteristic
21	#15668004	C₂²	1	1	yes	characteristic
22	#15668003	C₂²	1	1	yes	characteristic
23	#15668002	—	2	1	no	—
24	#15668002	—	2	1	no	—
25	#15668002	—	2	1	no	—
26	#15667990	—	2	1	no	—
27	#15667988	—	2	1	no	—
28	#15667988	C₂²	1	1	yes	characteristic
29	#15667988	C₂²	1	1	yes	characteristic
30	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
31	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
32	#15667970	C₂²	1	1	yes	characteristic
33	#15667969	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
35	#15667946	C₂²	1	1	yes	characteristic
36	#15667946	C₂²	1	1	yes	characteristic
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667944	C₂²	1	1	yes	characteristic
40	#15667944	C₂²	1	1	yes	characteristic
41	#15667943	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667936	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667923	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	#103×C₈₉	C₂²	1	1	yes	characteristic
55	#103×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×C₄×D₈₉	—	2	1	no	—
61	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
62	C₂×#10623494	C₂³	1	1	yes	characteristic
63	C₂×#10623494	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₈₉	—	2	1	no	—
65	C₄⋊Dic₈₉	—	2	1	no	—
66	C₄⋊Dic₈₉	—	2	1	no	—
67	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
71	#13083553	—	2	1	no	—
72	#13083553	—	2	1	no	—
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	D₄	1	1	yes	characteristic
81	#13083553	D₄	1	1	yes	characteristic
82	#13083551	—	2	1	no	—
83	#13083551	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	D₄	1	1	yes	characteristic
86	#13083551	D₄	1	1	yes	characteristic
87	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₄×Dic₈₉	—	2	1	no	—
90	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₈₉	—	2	1	no	—
92	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	—	2	1	no	—
97	C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₄⋊C₄×C₈₉	—	2	1	no	—
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	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	Q₈×C₈₉	—	8	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	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₈₉	—	2	1	no	—
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	C₄×D₈₉	—	2	1	no	—
123	C₄×D₈₉	—	2	1	no	—
124	C₄×D₈₉	—	8	1	no	—
125	Dic₁₇₈	—	8	1	no	—
126	C₈×C₈₉	—	2	1	no	—
127	C₈×C₈₉	—	2	1	no	—
128	#10623494	—	2	1	no	—
129	#10623494	—	2	1	no	—
130	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₈₉	—	2	1	no	—
132	C₂×D₈₉	—	2	1	no	—
133	C₂×D₈₉	—	2	1	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	4	1	no	—
137	C₄×C₈₉	—	4	1	no	—
138	C₄×C₈₉	—	4	1	no	—
139	C₄×C₈₉	—	4	1	no	—
140	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
141	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
142	Dic₈₉	—	2	1	no	—
143	Dic₈₉	—	2	1	no	—
144	Dic₈₉	—	2	1	no	—
145	Dic₈₉	—	4	1	no	—
146	Dic₈₉	—	4	1	no	—
147	Dic₈₉	—	4	1	no	—
148	Dic₈₉	—	4	1	no	—
149	C₂×C₈₉	#536	1	1	yes	characteristic
150	C₂×C₈₉	#577	1	1	yes	characteristic
151	C₂×C₈₉	#578	1	1	yes	characteristic
152	D₈₉	—	4	1	no	—
153	#3242	—	89	1	no	maximal, Sylow
154	C₈₉	#3242	1	1	yes	characteristic, complemented, Sylow
155	#544	—	89	1	no	—
156	#541	—	89	1	no	—
157	#501	—	89	1	no	—
158	#499	—	89	1	no	—
159	#491	—	89	1	no	—
160	#487	D₈₉	1	1	yes	characteristic, complemented
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	—	178	1	no	—
172	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
173	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
174	#126	—	178	1	no	—
175	#126	—	178	1	no	—
176	#125	—	89	1	no	—
177	#125	—	89	1	no	—
178	#125	—	89	1	no	—
179	#125	—	178	1	no	—
180	#125	—	178	1	no	—
181	#125	—	178	1	no	—
182	C₂²⋊Q₈	—	89	1	no	—
183	C₂²⋊Q₈	—	89	1	no	—
184	C₂²⋊Q₈	—	178	1	no	—
185	C₂²⋊Q₈	—	178	1	no	—
186	C₄×Q₈	—	89	1	no	—
187	C₄×Q₈	—	89	1	no	—
188	#117	—	89	1	no	—
189	#117	—	89	1	no	—
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	C₂×D₈₉	1	1	yes	characteristic
206	#103	C₂×D₈₉	1	1	yes	characteristic
207	#103	C₂×D₈₉	1	1	yes	characteristic
208	#103	C₂×D₈₉	1	1	yes	characteristic
209	C₂²⋊C₈	—	89	1	no	—
210	C₂²⋊C₈	—	89	1	no	—
211	#97	—	89	1	no	—
212	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
213	C₂×Q₈	—	2	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	—	89	1	no	—
218	C₂²×C₄	—	178	1	no	—
219	C₂×C₈	—	89	1	no	—
220	C₂×C₈	—	89	1	no	—
221	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	89	1	no	—
227	C₄⋊C₄	—	89	1	no	—
228	C₄⋊C₄	—	89	1	no	—
229	C₄⋊C₄	—	89	1	no	—
230	C₄⋊C₄	—	89	1	no	—
231	C₄⋊C₄	—	89	1	no	—
232	C₄⋊C₄	—	178	1	no	—
233	C₄⋊C₄	—	178	1	no	—
234	C₄⋊C₄	—	178	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	—	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	C₂²⋊C₄	—	89	1	no	—
247	C₂²⋊C₄	—	89	1	no	—
248	C₂²⋊C₄	—	178	1	no	—
249	C₂²⋊C₄	—	178	1	no	—
250	C₂²⋊C₄	—	178	1	no	—
251	C₄²	—	89	1	no	—
252	C₄²	—	178	1	no	—
253	C₄²	—	178	1	no	—
254	C₄²	—	178	1	no	—
255	C₄²	C₂²×D₈₉	1	1	yes	characteristic
256	C₂³	—	89	1	no	—
257	Q₈	—	2	1	no	—
258	Q₈	—	2	1	no	—
259	Q₈	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	8	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	—	89	1	no	—
268	C₂×C₄	—	89	1	no	—
269	C₂×C₄	—	89	1	no	—
270	C₂×C₄	—	178	1	no	—
271	C₂×C₄	—	178	1	no	—
272	C₂×C₄	—	178	1	no	—
273	C₂×C₄	—	178	1	no	—
274	C₂×C₄	—	178	1	no	—
275	C₂×C₄	—	178	1	no	—
276	C₂×C₄	—	712	1	no	—
277	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
278	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	178	1	no	—
283	C₈	—	178	1	no	—
284	C₂²	—	178	1	no	—
285	C₂²	—	178	1	no	—
286	C₂²	—	178	1	no	—
287	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	356	1	no	—
295	C₄	—	356	1	no	—
296	C₄	—	356	1	no	—
297	C₄	—	356	1	no	—
298	C₄	—	178	1	no	—
299	C₄	—	178	1	no	—
300	C₄	—	178	1	no	—
301	C₄	#15668059	1	1	yes	characteristic
302	C₄	#15668059	1	1	yes	characteristic
303	C₂	—	356	1	no	—
304	C₂	#19152168	1	1	yes	characteristic
305	C₂	#19152404	1	1	yes	characteristic
306	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	#487×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂²×D₈₉	C₂×D₄
hypercenter	#487	D₈₉
hypocenter	C₈₉	#3242
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

Complements

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

N	H ≅ G/N	type
3. #19152004	C₂	semidirect
4. #19152000	C₂	semidirect
5. #19151966	C₂	semidirect
10. #19151907	C₂	semidirect
11. #19151903	C₂	semidirect
12. #19151891	C₂	semidirect
13. #19151881	C₂	semidirect
15. #487×C₈₉	C₂	semidirect
154. C₈₉	#3242	semidirect
160. #487	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152225	2
#19152209	2
#19152004	2
#19152000	2
#19151966	2
#19151938	2
#19151937	2
#19151932	2
#19151927	2
#19151907	2
#19151903	2
#19151891	2
#19151881	2
#19151657	2
#487×C₈₉	2
#3242	89