Subgroups — #26034054

Order 11392.

1100 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152309	C₂	1	4	yes	maximal, complemented
2	#19152304	C₂	1	4	yes	maximal, complemented
3	#19152304	C₂	1	4	yes	maximal, complemented
4	#19152263	C₂	1	4	yes	maximal, complemented
5	#19152229	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152223	C₂	1	4	yes	maximal, complemented
7	#19152151	C₂	1	2	yes	maximal, complemented
8	C₂²⋊Q₈×D₈₉	C₂	1	4	yes	maximal, complemented
9	#19152072	C₂	1	2	yes	maximal
10	C₂×#15667974	C₂	1	1	yes	maximal, characteristic
11	#544×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₈₉	C₂²	1	4	yes	complemented
13	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₈₉	C₂²	1	2	yes	—
15	#15668011	C₂²	1	2	yes	complemented
16	#15668011	C₂²	1	2	yes	complemented
17	#15668011	C₂²	1	4	yes	complemented
18	#15668003	C₂²	1	2	yes	complemented
19	#15668003	C₂²	1	2	yes	complemented
20	#15668003	C₂²	1	4	yes	complemented
21	#15667992	C₂²	1	8	yes	complemented
22	#15667992	C₂²	1	4	yes	complemented
23	#15667991	C₂²	1	4	yes	—
24	#15667990	C₂²	1	8	yes	complemented
25	#15667990	C₂²	1	8	yes	complemented
26	#15667990	C₂²	1	8	yes	complemented
27	#15667989	C₂²	1	4	yes	—
28	#15667989	C₂²	1	4	yes	—
29	#15667988	C₂²	1	8	yes	complemented
30	#15667988	C₂²	1	4	yes	complemented
31	C₄×Dic₁₇₈	C₂²	1	4	yes	complemented
32	C₄×Dic₁₇₈	C₂²	1	4	yes	complemented
33	C₄×D₃₅₆	C₂²	1	4	yes	—
34	C₄×D₃₅₆	C₂²	1	4	yes	—
35	#15667974	C₂²	1	4	yes	—
36	#15667974	C₂²	1	4	yes	—
37	#15667974	C₂²	1	4	yes	—
38	#15667974	C₂²	1	4	yes	—
39	#15667971	C₂²	1	2	yes	—
40	C₄⋊C₄×D₈₉	C₂²	1	4	yes	complemented
41	C₄⋊C₄×D₈₉	C₂²	1	8	yes	complemented
42	C₂×#13083551	C₂²	1	2	yes	—
43	C₂²⋊C₄×D₈₉	C₂²	1	8	yes	—
44	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
45	C₄⋊Q₈×C₈₉	C₂²	1	4	yes	complemented
46	#125×C₈₉	C₂²	1	4	yes	complemented
47	C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
48	#117×C₈₉	C₂²	1	2	yes	—
49	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
50	C₂³×D₈₉	C₂³	1	2	yes	—
51	C₂×Dic₁₇₈	C₂³	1	4	yes	—
52	C₂×Dic₁₇₈	C₂³	1	4	yes	—
53	C₂×Dic₁₇₈	C₂³	1	4	yes	—
54	Q₈×D₈₉	—	4	8	no	—
55	C₂×D₃₅₆	Q₈	1	8	yes	complemented
56	C₂×D₃₅₆	C₂³	1	4	yes	—
57	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₈₉	—	4	4	no	—
60	C₂×C₄×D₈₉	C₂³	1	4	yes	—
61	C₂×C₄×D₈₉	C₂³	1	4	yes	—
62	C₂×C₄×D₈₉	C₂³	1	8	yes	—
63	C₂×C₄×D₈₉	C₂³	1	8	yes	—
64	C₄⋊Dic₈₉	C₂³	1	2	yes	—
65	C₄⋊Dic₈₉	C₂³	1	2	yes	—
66	C₄⋊Dic₈₉	C₂³	1	4	yes	—
67	C₄⋊Dic₈₉	C₂³	1	4	yes	—
68	C₄⋊Dic₈₉	C₂³	1	4	yes	—
69	#13083553	C₂³	1	4	yes	—
70	#13083553	C₂³	1	4	yes	—
71	#13083553	C₂³	1	4	yes	—
72	#13083553	C₂³	1	4	yes	—
73	#13083553	C₂³	1	8	yes	—
74	C₂²⋊Dic₈₉	C₂³	1	4	yes	—
75	#13083551	C₂³	1	8	yes	—
76	#13083551	C₂³	1	8	yes	—
77	#13083551	C₂³	1	8	yes	—
78	C₄×Dic₈₉	C₂³	1	4	yes	—
79	C₄×Dic₈₉	C₂³	1	2	yes	—
80	C₄×Dic₈₉	C₂³	1	2	yes	—
81	C₂×Q₈×C₈₉	C₂³	1	4	yes	—
82	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
85	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
86	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
87	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
88	C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
89	C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
90	C₄²×C₈₉	C₂³	1	4	yes	—
91	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
92	C₂²×D₈₉	—	4	2	no	—
93	C₂²×D₈₉	—	4	2	no	—
94	C₂²×D₈₉	C₂×Q₈	1	8	yes	complemented
95	C₂²×D₈₉	C₂⁴	1	4	yes	—
96	Q₈×C₈₉	—	4	4	no	—
97	C₂×C₄×C₈₉	—	4	2	no	—
98	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	complemented
99	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
100	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
101	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
102	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
103	C₂×Dic₈₉	—	2	2	no	—
104	C₂×Dic₈₉	—	4	1	no	—
105	C₂×Dic₈₉	C₂⁴	1	2	yes	—
106	C₂×Dic₈₉	C₂⁴	1	1	yes	characteristic
107	C₂×Dic₈₉	C₂⁴	1	4	yes	—
108	C₂×Dic₈₉	C₂⁴	1	4	yes	—
109	C₂×Dic₈₉	C₂⁴	1	1	yes	characteristic
110	D₃₅₆	C₂×Q₈	1	16	yes	complemented
111	C₄×D₈₉	—	2	8	no	—
112	C₄×D₈₉	—	4	4	no	—
113	C₄×D₈₉	—	4	8	no	—
114	C₄×D₈₉	—	4	8	no	—
115	Dic₁₇₈	—	2	8	no	—
116	Dic₁₇₈	—	4	4	no	—
117	Dic₁₇₈	—	4	4	no	—
118	C₂²×C₈₉	—	2	1	no	—
119	C₂²×C₈₉	—	2	1	no	—
120	C₂²×C₈₉	C₂²×Q₈	1	2	yes	—
121	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
122	C₂×D₈₉	—	2	4	no	—
123	C₂×D₈₉	—	2	4	no	—
124	C₂×D₈₉	—	4	8	no	—
125	C₂×D₈₉	C₂²×Q₈	1	8	yes	—
126	C₄×C₈₉	—	2	4	no	—
127	C₄×C₈₉	—	2	4	no	—
128	C₄×C₈₉	—	2	4	no	—
129	C₄×C₈₉	C₂²×Q₈	1	4	yes	—
130	Dic₈₉	—	2	4	no	—
131	Dic₈₉	—	2	1	no	—
132	Dic₈₉	—	2	2	no	—
133	Dic₈₉	—	2	4	no	—
134	Dic₈₉	H(5,𝔽₂)	1	2	yes	—
135	C₂×C₈₉	—	2	2	no	—
136	C₂×C₈₉	C₂³×Q₈	1	1	yes	characteristic
137	C₂×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₈₉	#585	1	1	yes	characteristic
139	D₈₉	—	2	8	no	—
140	#3501	—	89	1	no	maximal, Sylow
141	C₈₉	#3501	1	1	yes	characteristic, complemented, Sylow
142	#554	—	89	4	no	—
143	#554	—	89	4	no	—
144	#554	—	89	4	no	—
145	D₄×Q₈	—	89	4	no	—
146	#544	—	89	1	no	—
147	#544	D₈₉	1	1	yes	characteristic, complemented
148	#543	—	89	4	no	—
149	#533	—	89	2	no	—
150	C₂×C₂²⋊Q₈	—	89	4	no	—
151	#518	—	89	2	no	—
152	C₂×C₄×D₄	—	89	1	no	—
153	C₂²×Q₈	—	89	4	no	—
154	C₂²×D₄	—	89	1	no	—
155	C₂³×C₄	—	89	2	no	—
156	C₄⋊Q₈	—	89	2	no	—
157	C₄⋊Q₈	—	89	2	no	—
158	C₄⋊Q₈	—	89	4	no	—
159	C₄⋊Q₈	C₂×D₈₉	1	4	yes	complemented
160	#125	—	89	2	no	—
161	#125	—	89	2	no	—
162	#125	—	89	4	no	—
163	#125	C₂×D₈₉	1	4	yes	complemented
164	C₂²⋊Q₈	—	89	4	no	—
165	C₂²⋊Q₈	—	89	8	no	—
166	C₂²⋊Q₈	—	89	4	no	—
167	C₂²⋊Q₈	—	89	8	no	—
168	C₂²⋊Q₈	—	89	4	no	—
169	C₂²⋊Q₈	—	89	8	no	—
170	C₂²⋊Q₈	—	89	8	no	—
171	C₂²⋊Q₈	—	89	8	no	—
172	C₂²⋊Q₈	—	89	4	no	—
173	C₂²⋊Q₈	—	89	4	no	—
174	C₂²⋊Q₈	C₂×D₈₉	1	4	yes	—
175	C₄×Q₈	—	89	4	no	—
176	C₄×Q₈	—	89	4	no	—
177	C₄×D₄	—	89	4	no	—
178	C₄×D₄	—	89	4	no	—
179	C₄×D₄	—	89	4	no	—
180	C₄×D₄	—	89	4	no	—
181	C₄×D₄	—	89	4	no	—
182	C₄×D₄	—	89	4	no	—
183	#117	—	89	2	no	—
184	#117	C₂×D₈₉	1	2	yes	—
185	C₂×C₄⋊C₄	—	89	8	no	—
186	C₂×C₄⋊C₄	—	89	4	no	—
187	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
188	C₂×C₂²⋊C₄	—	89	2	no	—
189	C₂×C₂²⋊C₄	—	89	8	no	—
190	C₂×C₄²	—	89	1	no	—
191	C₂⁴	—	89	2	no	—
192	C₂×Q₈	—	89	4	no	—
193	C₂×Q₈	—	89	4	no	—
194	C₂×Q₈	—	89	4	no	—
195	C₂×Q₈	—	356	8	no	—
196	C₂×Q₈	C₂²×D₈₉	1	4	yes	—
197	C₂×D₄	—	89	8	no	—
198	C₂×D₄	—	89	4	no	—
199	C₂²×C₄	—	89	1	no	—
200	C₂²×C₄	—	89	1	no	—
201	C₂²×C₄	—	89	8	no	—
202	C₂²×C₄	—	89	8	no	—
203	C₂²×C₄	—	89	4	no	—
204	C₂²×C₄	—	89	4	no	—
205	C₂²×C₄	—	356	4	no	—
206	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
207	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
208	C₄⋊C₄	—	89	4	no	—
209	C₄⋊C₄	—	89	4	no	—
210	C₄⋊C₄	—	89	4	no	—
211	C₄⋊C₄	—	89	4	no	—
212	C₄⋊C₄	—	89	8	no	—
213	C₄⋊C₄	—	89	2	no	—
214	C₄⋊C₄	—	89	2	no	—
215	C₄⋊C₄	—	89	4	no	—
216	C₄⋊C₄	—	89	4	no	—
217	C₄⋊C₄	—	89	4	no	—
218	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
219	C₄⋊C₄	C₂²×D₈₉	1	8	yes	—
220	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
221	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
222	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
223	C₂²⋊C₄	—	89	4	no	—
224	C₂²⋊C₄	—	89	8	no	—
225	C₂²⋊C₄	—	89	8	no	—
226	C₂²⋊C₄	—	89	8	no	—
227	C₂²⋊C₄	C₂²×D₈₉	1	4	yes	—
228	C₄²	—	89	2	no	—
229	C₄²	—	89	2	no	—
230	C₄²	—	89	4	no	—
231	C₄²	C₂²×D₈₉	1	4	yes	—
232	C₂³	—	89	8	no	—
233	C₂³	—	89	4	no	—
234	C₂³	—	356	2	no	—
235	C₂³	—	356	2	no	—
236	C₂³	C₂³×D₈₉	1	1	yes	characteristic
237	Q₈	—	4	4	no	—
238	Q₈	—	356	4	no	—
239	Q₈	—	356	4	no	—
240	Q₈	—	178	8	no	—
241	D₄	—	89	16	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	89	1	no	—
244	C₂×C₄	—	89	2	no	—
245	C₂×C₄	—	89	1	no	—
246	C₂×C₄	—	89	4	no	—
247	C₂×C₄	—	89	4	no	—
248	C₂×C₄	—	356	8	no	—
249	C₂×C₄	—	356	4	no	—
250	C₂×C₄	—	356	8	no	—
251	C₂×C₄	—	356	1	no	—
252	C₂×C₄	—	178	8	no	—
253	C₂×C₄	—	178	2	no	—
254	C₂×C₄	Q₈×D₈₉	1	4	yes	complemented
255	C₂×C₄	C₂³×D₈₉	1	4	yes	—
256	C₂×C₄	C₂³×D₈₉	1	4	yes	—
257	C₂×C₄	C₂³×D₈₉	1	4	yes	—
258	C₂×C₄	C₂³×D₈₉	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	89	8	no	—
262	C₂²	—	356	8	no	—
263	C₂²	—	178	4	no	—
264	C₂²	—	178	4	no	—
265	C₂²	C₂×Q₈×D₈₉	1	2	yes	—
266	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	2	4	no	—
269	C₄	—	2	4	no	—
270	C₄	—	89	2	no	—
271	C₄	—	178	1	no	—
272	C₄	—	178	4	no	—
273	C₄	—	178	4	no	—
274	C₄	—	178	2	no	—
275	C₄	C₂×Q₈×D₈₉	1	4	yes	—
276	C₂	—	2	2	no	—
277	C₂	—	178	8	no	—
278	C₂	C₂²×Q₈×D₈₉	1	1	yes	characteristic
279	C₂	H(5,𝔽₂)×D₈₉	1	1	yes	characteristic
280	C₂	#19152435	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	#544×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	#544	D₈₉
hypocenter	C₈₉	#3501
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #544

Complements

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

N	H ≅ G/N	type
1. #19152309	C₂	semidirect
2. #19152304	C₂	semidirect
3. #19152304	C₂	semidirect
4. #19152263	C₂	semidirect
5. #19152229	C₂	semidirect
6. #19152223	C₂	semidirect
7. #19152151	C₂	semidirect
8. C₂²⋊Q₈×D₈₉	C₂	semidirect
11. #544×C₈₉	C₂	semidirect
12. C₂×Q₈×D₈₉	C₂²	semidirect
15. #15668011	C₂²	semidirect
16. #15668011	C₂²	semidirect
17. #15668011	C₂²	semidirect
18. #15668003	C₂²	semidirect
19. #15668003	C₂²	semidirect
20. #15668003	C₂²	semidirect
21. #15667992	C₂²	semidirect
22. #15667992	C₂²	semidirect
24. #15667990	C₂²	semidirect
25. #15667990	C₂²	semidirect
26. #15667990	C₂²	semidirect
29. #15667988	C₂²	semidirect
30. #15667988	C₂²	semidirect
31. C₄×Dic₁₇₈	C₂²	semidirect
32. C₄×Dic₁₇₈	C₂²	semidirect
40. C₄⋊C₄×D₈₉	C₂²	semidirect
41. C₄⋊C₄×D₈₉	C₂²	semidirect
45. C₄⋊Q₈×C₈₉	C₂²	semidirect
46. #125×C₈₉	C₂²	semidirect
55. C₂×D₃₅₆	Q₈	semidirect
94. C₂²×D₈₉	C₂×Q₈	semidirect
98. C₂×C₄×C₈₉	C₂×Q₈	semidirect
110. D₃₅₆	C₂×Q₈	semidirect
141. C₈₉	#3501	semidirect
147. #544	D₈₉	semidirect
159. C₄⋊Q₈	C₂×D₈₉	semidirect
163. #125	C₂×D₈₉	semidirect
254. C₂×C₄	Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152309	2
#19152304	2
#19152263	2
#19152229	2
#19152223	2
#19152151	2
C₂²⋊Q₈×D₈₉	2
#19152072	2
C₂×#15667974	2
#544×C₈₉	2
#3501	89