Subgroups — #72269543

Order 3520.

536 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#67042⋊C₅	C₂	1	2	yes	maximal, complemented
2	#135×C₁₁⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	#67002⋊C₅	C₂	1	2	yes	maximal, complemented
4	#66972⋊C₅	C₂	1	2	yes	maximal, complemented
5	#66967⋊C₅	C₂	1	4	yes	maximal, complemented
6	D₈×AGL(1,𝔽₁₁)	C₂	1	2	yes	maximal, complemented
7	C₂×D₈₈⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#66954⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented
9	#41×C₁₁⋊C₅	C₂²	1	2	yes	complemented
10	#4343⋊C₅	C₂²	1	4	yes	complemented
11	D₄×AGL(1,𝔽₁₁)	—	2	4	no	—
12	D₄×AGL(1,𝔽₁₁)	C₂²	1	4	yes	complemented
13	#4339⋊C₅	—	2	2	no	—
14	#4339⋊C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊AGL(1,𝔽₁₁)	—	2	2	no	—
16	C₂×C₄⋊AGL(1,𝔽₁₁)	C₂²	1	2	yes	complemented
17	Q₁₆×C₁₁⋊C₅	C₂²	1	1	yes	characteristic, complemented
18	SD₁₆×C₁₁⋊C₅	C₂²	1	2	yes	complemented
19	D₈×C₁₁⋊C₅	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₈×C₁₁⋊C₅	C₂²	1	1	yes	characteristic, complemented
21	#4325⋊C₅	C₂²	1	4	yes	complemented
22	#4323⋊C₅	C₂²	1	4	yes	complemented
23	#4318⋊C₅	C₂²	1	1	yes	characteristic, complemented
24	D₈₈⋊C₅	C₂²	1	2	yes	complemented
25	D₈₈⋊C₅	C₂²	1	2	yes	complemented
26	#4313⋊C₅	C₂²	1	2	yes	complemented
27	C₈×AGL(1,𝔽₁₁)	C₂²	1	2	yes	complemented
28	#10623300	C₅	1	1	yes	maximal, characteristic, complemented, Hall
29	C₂²×AGL(1,𝔽₁₁)	—	2	2	no	—
30	C₂²×AGL(1,𝔽₁₁)	—	2	4	no	—
31	Q₈×C₁₁⋊C₅	C₂³	1	2	yes	—
32	D₄×C₁₁⋊C₅	—	2	2	no	—
33	D₄×C₁₁⋊C₅	C₂³	1	2	yes	—
34	C₂×C₄×C₁₁⋊C₅	—	2	2	no	—
35	C₂×C₄×C₁₁⋊C₅	C₂³	1	1	yes	characteristic
36	C₂²⋊AGL(1,𝔽₁₁)	—	2	4	no	—
37	C₂²⋊AGL(1,𝔽₁₁)	D₄	1	2	yes	—
38	C₄⋊AGL(1,𝔽₁₁)	—	2	4	no	—
39	C₄⋊AGL(1,𝔽₁₁)	—	2	2	no	—
40	C₄⋊AGL(1,𝔽₁₁)	D₄	1	1	yes	characteristic
41	C₄⋊AGL(1,𝔽₁₁)	C₂³	1	4	yes	—
42	C₄×AGL(1,𝔽₁₁)	—	2	4	no	—
43	C₄×AGL(1,𝔽₁₁)	C₂³	1	2	yes	—
44	(C₁₁⋊C₅)⋊Q₈	D₄	1	1	yes	characteristic
45	C₈×C₁₁⋊C₅	C₂³	1	1	yes	characteristic
46	C₈×C₁₁⋊C₅	C₂³	1	1	yes	characteristic
47	(C₁₁⋊C₅)⋊C₈	C₂³	1	2	yes	—
48	#67042	C₂×C₅	1	2	yes	complemented
49	C₁₁×#135	C₂×C₅	1	1	yes	characteristic, complemented
50	#67002	C₂×C₅	1	2	yes	complemented
51	#66972	C₂×C₅	1	2	yes	complemented
52	#66967	C₂×C₅	1	4	yes	complemented
53	D₈×D₁₁	C₂×C₅	1	2	yes	complemented
54	C₂×D₈₈	C₂×C₅	1	1	yes	characteristic, complemented
55	#66954	C₂×C₅	1	1	yes	characteristic, complemented
56	C₅×#576	—	11	1	no	maximal, Hall
57	C₂²×C₁₁⋊C₅	—	2	2	no	—
58	C₂²×C₁₁⋊C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×AGL(1,𝔽₁₁)	—	2	4	no	—
60	C₂×AGL(1,𝔽₁₁)	—	8	4	no	—
61	C₂×AGL(1,𝔽₁₁)	—	8	2	no	—
62	C₂×AGL(1,𝔽₁₁)	C₂×D₄	1	2	yes	—
63	C₄×C₁₁⋊C₅	—	2	2	no	—
64	C₄×C₁₁⋊C₅	C₂×D₄	1	1	yes	characteristic
65	C₄×C₁₁⋊C₅	C₂⁴	1	1	yes	characteristic
66	Dic₁₁⋊C₅	C₂×D₄	1	2	yes	—
67	C₁₁×#41	C₂²×C₅	1	2	yes	complemented
68	#4343	C₂²×C₅	1	4	yes	complemented
69	D₄×D₁₁	—	2	4	no	—
70	D₄×D₁₁	C₂²×C₅	1	4	yes	complemented
71	#4339	—	2	2	no	—
72	#4339	C₂²×C₅	1	1	yes	characteristic
73	C₂×D₄₄	—	2	2	no	—
74	C₂×D₄₄	C₂²×C₅	1	2	yes	complemented
75	C₁₁×Q₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
76	C₁₁×SD₁₆	C₂²×C₅	1	2	yes	complemented
77	C₁₁×D₈	C₂²×C₅	1	1	yes	characteristic, complemented
78	C₂×C₈×C₁₁	C₂²×C₅	1	1	yes	characteristic, complemented
79	#4325	C₂²×C₅	1	4	yes	complemented
80	#4323	C₂²×C₅	1	4	yes	complemented
81	#4318	C₂²×C₅	1	1	yes	characteristic, complemented
82	D₈₈	C₂²×C₅	1	2	yes	complemented
83	D₈₈	C₂²×C₅	1	2	yes	complemented
84	#4313	C₂²×C₅	1	2	yes	complemented
85	C₈×D₁₁	C₂²×C₅	1	2	yes	complemented
86	C₅×H(5,𝔽₂)	—	11	2	no	—
87	C₅×AGL(1,ℤ/8)	—	11	2	no	—
88	C₅×AGL(1,ℤ/8)	—	11	4	no	—
89	C₅×#135	—	11	2	no	—
90	C₅×#135	—	22	1	no	—
91	C₂×C₅×D₈	—	11	2	no	—
92	C₂×C₅×D₈	—	11	1	no	—
93	C₅×#131	—	11	1	no	—
94	C₂×C₁₁⋊C₅	—	2	1	no	—
95	C₂×C₁₁⋊C₅	—	4	2	no	—
96	C₂×C₁₁⋊C₅	C₂²×D₄	1	1	yes	characteristic
97	AGL(1,𝔽₁₁)	—	2	2	no	—
98	AGL(1,𝔽₁₁)	—	4	4	no	—
99	C₂²×D₁₁	—	2	2	no	—
100	C₂²×D₁₁	—	2	4	no	—
101	Q₈×C₁₁	C₂³×C₅	1	2	yes	—
102	D₄×C₁₁	—	2	2	no	—
103	D₄×C₁₁	C₂³×C₅	1	2	yes	—
104	C₂×C₄×C₁₁	—	2	2	no	—
105	C₂×C₄×C₁₁	C₂³×C₅	1	1	yes	characteristic
106	C₂²⋊D₁₁	—	2	4	no	—
107	C₂²⋊D₁₁	C₅×D₄	1	2	yes	—
108	D₄₄	—	2	4	no	—
109	D₄₄	—	2	2	no	—
110	D₄₄	C₅×D₄	1	1	yes	characteristic
111	D₄₄	C₂³×C₅	1	4	yes	—
112	C₄×D₁₁	—	2	4	no	—
113	C₄×D₁₁	C₂³×C₅	1	2	yes	—
114	Dic₂₂	C₅×D₄	1	1	yes	characteristic
115	C₈×C₁₁	C₂³×C₅	1	1	yes	characteristic
116	C₈×C₁₁	C₂³×C₅	1	1	yes	characteristic
117	C₁₁⋊C₈	C₂³×C₅	1	2	yes	—
118	C₅×#41	—	11	1	no	—
119	C₅×#41	—	11	4	no	—
120	C₅×#41	—	22	2	no	—
121	C₅×#41	—	22	2	no	—
122	C₂×C₅×D₄	—	11	4	no	—
123	C₂×C₅×D₄	—	11	2	no	—
124	C₂×C₅×D₄	—	22	4	no	—
125	C₂×C₅×D₄	—	22	2	no	—
126	C₅×Q₁₆	—	22	1	no	—
127	C₅×SD₁₆	—	11	4	no	—
128	C₅×SD₁₆	—	22	2	no	—
129	C₅×D₈	—	11	4	no	—
130	C₅×D₈	—	11	2	no	—
131	C₅×D₈	—	11	2	no	—
132	C₅×D₈	—	22	1	no	—
133	C₅×M₁₆	—	11	1	no	—
134	C₅×M₁₆	—	11	2	no	—
135	C₂×C₅×C₈	—	11	2	no	—
136	C₂×C₅×C₈	—	22	1	no	—
137	#576	—	11	1	no	Sylow
138	C₁₁⋊C₅	#576	1	1	yes	characteristic, complemented, Hall
139	C₂²×C₁₁	—	2	2	no	—
140	C₂²×C₁₁	C₂×C₅×D₄	1	1	yes	characteristic
141	C₂×D₁₁	—	2	4	no	—
142	C₂×D₁₁	—	8	4	no	—
143	C₂×D₁₁	—	8	2	no	—
144	C₂×D₁₁	C₂×C₅×D₄	1	2	yes	—
145	C₄×C₁₁	—	2	2	no	—
146	C₄×C₁₁	C₂×C₅×D₄	1	1	yes	characteristic
147	C₄×C₁₁	C₂⁴×C₅	1	1	yes	characteristic
148	Dic₁₁	C₂×C₅×D₄	1	2	yes	—
149	C₂³×C₅	—	22	4	no	—
150	C₂³×C₅	—	22	2	no	—
151	C₅×Q₈	—	11	1	no	—
152	C₅×Q₈	—	22	2	no	—
153	C₅×D₄	—	11	2	no	—
154	C₅×D₄	—	11	1	no	—
155	C₅×D₄	—	11	4	no	—
156	C₅×D₄	—	22	4	no	—
157	C₅×D₄	—	22	2	no	—
158	C₅×D₄	—	22	4	no	—
159	C₅×D₄	—	22	2	no	—
160	C₅×D₄	—	44	2	no	—
161	C₂×C₄×C₅	—	11	2	no	—
162	C₂×C₄×C₅	—	22	4	no	—
163	C₂×C₄×C₅	—	22	1	no	—
164	C₂×C₄×C₅	—	44	2	no	—
165	C₅×C₈	—	11	2	no	—
166	C₅×C₈	—	22	1	no	—
167	C₅×C₈	—	22	1	no	—
168	H(5,𝔽₂)	—	11	2	no	—
169	AGL(1,ℤ/8)	—	11	2	no	—
170	AGL(1,ℤ/8)	—	11	4	no	—
171	#135	—	11	2	no	—
172	#135	AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented
173	C₂×D₈	—	11	2	no	—
174	C₂×D₈	—	11	1	no	—
175	#131	—	11	1	no	—
176	C₂×C₁₁	—	2	1	no	—
177	C₂×C₁₁	—	4	2	no	—
178	C₂×C₁₁	C₂²×C₅×D₄	1	1	yes	characteristic
179	D₁₁	—	2	2	no	—
180	D₁₁	—	4	4	no	—
181	C₂²×C₅	—	11	2	no	—
182	C₂²×C₅	—	22	4	no	—
183	C₂²×C₅	—	22	1	no	—
184	C₂²×C₅	—	44	2	no	—
185	C₂²×C₅	—	88	4	no	—
186	C₂²×C₅	—	88	2	no	—
187	C₄×C₅	—	11	2	no	—
188	C₄×C₅	—	22	1	no	—
189	C₄×C₅	—	22	1	no	—
190	C₄×C₅	—	44	2	no	—
191	#41	—	11	1	no	—
192	#41	—	11	4	no	—
193	#41	—	22	2	no	—
194	#41	C₂×AGL(1,𝔽₁₁)	1	2	yes	complemented
195	C₂×D₄	—	11	4	no	—
196	C₂×D₄	—	11	2	no	—
197	C₂×D₄	—	22	2	no	—
198	C₂×D₄	—	22	4	no	—
199	Q₁₆	C₂×AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented
200	SD₁₆	—	11	4	no	—
201	SD₁₆	C₂×AGL(1,𝔽₁₁)	1	2	yes	complemented
202	D₈	—	11	4	no	—
203	D₈	—	11	2	no	—
204	D₈	—	11	2	no	—
205	D₈	C₂×AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented
206	M₁₆	—	11	1	no	—
207	M₁₆	—	11	2	no	—
208	C₂×C₈	—	11	2	no	—
209	C₂×C₈	C₂×AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented
210	C₁₁	C₅×#576	1	1	yes	characteristic, complemented, Sylow
211	C₂×C₅	—	22	2	no	—
212	C₂×C₅	—	22	1	no	—
213	C₂×C₅	—	44	1	no	—
214	C₂×C₅	—	44	4	no	—
215	C₂×C₅	—	88	2	no	—
216	C₂³	—	22	4	no	—
217	C₂³	—	22	2	no	—
218	Q₈	—	11	1	no	—
219	Q₈	C₂²×AGL(1,𝔽₁₁)	1	2	yes	—
220	D₄	—	2	2	no	—
221	D₄	—	11	2	no	—
222	D₄	—	11	1	no	—
223	D₄	—	11	4	no	—
224	D₄	—	22	4	no	—
225	D₄	—	22	4	no	—
226	D₄	—	22	2	no	—
227	D₄	C₂²×AGL(1,𝔽₁₁)	1	2	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	11	2	no	—
230	C₂×C₄	—	22	4	no	—
231	C₂×C₄	C₂²×AGL(1,𝔽₁₁)	1	1	yes	characteristic
232	C₈	—	11	2	no	—
233	C₈	C₂²×AGL(1,𝔽₁₁)	1	1	yes	characteristic
234	C₈	C₂²×AGL(1,𝔽₁₁)	1	1	yes	characteristic
235	C₅	—	22	1	no	Sylow
236	C₂²	—	2	2	no	—
237	C₂²	—	11	2	no	—
238	C₂²	—	22	4	no	—
239	C₂²	—	88	2	no	—
240	C₂²	—	88	4	no	—
241	C₂²	D₄×AGL(1,𝔽₁₁)	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	11	2	no	—
244	C₄	D₄×AGL(1,𝔽₁₁)	1	1	yes	characteristic
245	C₄	C₂³×AGL(1,𝔽₁₁)	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	4	2	no	—
248	C₂	—	22	2	no	—
249	C₂	—	44	4	no	—
250	C₂	C₂×D₄×AGL(1,𝔽₁₁)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×AGL(1,𝔽₁₁)
commutator, cosocle	C₄×C₁₁	C₂⁴×C₅
Frattini	C₄	C₂³×AGL(1,𝔽₁₁)
Fitting	C₁₁×#135	C₂×C₅
socle	C₂×C₁₁	C₂²×C₅×D₄
hypercenter	#135	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#576
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #67042⋊C₅	C₂	semidirect
2. #135×C₁₁⋊C₅	C₂	semidirect
3. #67002⋊C₅	C₂	semidirect
4. #66972⋊C₅	C₂	semidirect
5. #66967⋊C₅	C₂	semidirect
6. D₈×AGL(1,𝔽₁₁)	C₂	semidirect
7. C₂×D₈₈⋊C₅	C₂	semidirect
8. #66954⋊C₅	C₂	semidirect
9. #41×C₁₁⋊C₅	C₂²	semidirect
10. #4343⋊C₅	C₂²	semidirect
12. D₄×AGL(1,𝔽₁₁)	C₂²	semidirect
16. C₂×C₄⋊AGL(1,𝔽₁₁)	C₂²	semidirect
17. Q₁₆×C₁₁⋊C₅	C₂²	semidirect
18. SD₁₆×C₁₁⋊C₅	C₂²	semidirect
19. D₈×C₁₁⋊C₅	C₂²	semidirect
20. C₂×C₈×C₁₁⋊C₅	C₂²	semidirect
21. #4325⋊C₅	C₂²	semidirect
22. #4323⋊C₅	C₂²	semidirect
23. #4318⋊C₅	C₂²	semidirect
24. D₈₈⋊C₅	C₂²	semidirect
25. D₈₈⋊C₅	C₂²	semidirect
26. #4313⋊C₅	C₂²	semidirect
27. C₈×AGL(1,𝔽₁₁)	C₂²	semidirect
28. #10623300	C₅	semidirect
48. #67042	C₂×C₅	semidirect
49. C₁₁×#135	C₂×C₅	semidirect
50. #67002	C₂×C₅	semidirect
51. #66972	C₂×C₅	semidirect
52. #66967	C₂×C₅	semidirect
53. D₈×D₁₁	C₂×C₅	semidirect
54. C₂×D₈₈	C₂×C₅	semidirect
55. #66954	C₂×C₅	semidirect
67. C₁₁×#41	C₂²×C₅	semidirect
68. #4343	C₂²×C₅	semidirect
70. D₄×D₁₁	C₂²×C₅	semidirect
74. C₂×D₄₄	C₂²×C₅	semidirect
75. C₁₁×Q₁₆	C₂²×C₅	semidirect
76. C₁₁×SD₁₆	C₂²×C₅	semidirect
77. C₁₁×D₈	C₂²×C₅	semidirect
78. C₂×C₈×C₁₁	C₂²×C₅	semidirect
79. #4325	C₂²×C₅	semidirect
80. #4323	C₂²×C₅	semidirect
81. #4318	C₂²×C₅	semidirect
82. D₈₈	C₂²×C₅	semidirect
83. D₈₈	C₂²×C₅	semidirect
84. #4313	C₂²×C₅	semidirect
85. C₈×D₁₁	C₂²×C₅	semidirect
138. C₁₁⋊C₅	#576	semidirect
172. #135	AGL(1,𝔽₁₁)	semidirect
194. #41	C₂×AGL(1,𝔽₁₁)	semidirect
199. Q₁₆	C₂×AGL(1,𝔽₁₁)	semidirect
201. SD₁₆	C₂×AGL(1,𝔽₁₁)	semidirect
205. D₈	C₂×AGL(1,𝔽₁₁)	semidirect
209. C₂×C₈	C₂×AGL(1,𝔽₁₁)	semidirect
210. C₁₁	C₅×#576	semidirect

Maximal subgroups

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

subgroup	index
#67042⋊C₅	2
#135×C₁₁⋊C₅	2
#67002⋊C₅	2
#66972⋊C₅	2
#66967⋊C₅	2
D₈×AGL(1,𝔽₁₁)	2
C₂×D₈₈⋊C₅	2
#66954⋊C₅	2
#10623300	5
C₅×#576	11