Subgroups — #15660396

Order 2816.

10930 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×Dic₁₁	C₂	1	32	yes	maximal, complemented
2	#13081819	C₂	1	20	yes	maximal, complemented
3	C₂×#10623343	C₂	1	10	yes	maximal, complemented
4	C₂²×C₁₁×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×2₋¹⁺⁴	C₄	1	32	yes	complemented
6	C₂×C₁₁×2₋¹⁺⁴	C₂²	1	16	yes	—
7	C₂²×C₁₁×#41	C₂²	1	10	yes	—
8	C₂³×Q₈×C₁₁	C₂²	1	5	yes	—
9	#10623358	C₂²	1	160	yes	complemented
10	#41×Dic₁₁	C₂²	1	160	yes	complemented
11	#10623343	C₂²	1	40	yes	—
12	#10623343	C₂²	1	80	yes	complemented
13	C₂×Q₈×Dic₁₁	C₂²	1	80	yes	complemented
14	#10623329	C₂²	1	40	yes	—
15	C₂×#66979	C₂²	1	60	yes	—
16	C₁₁×2₋¹⁺⁴	C₂×C₄	1	256	yes	complemented
17	C₂×C₁₁×#41	C₂×C₄	1	160	yes	complemented
18	C₂×C₁₁×#41	C₂³	1	80	yes	—
19	C₂²×Q₈×C₁₁	C₂×C₄	1	80	yes	complemented
20	C₂²×Q₈×C₁₁	C₂³	1	40	yes	—
21	C₂²×Q₈×C₁₁	C₂³	1	10	yes	—
22	C₂²×D₄×C₁₁	C₂³	1	10	yes	—
23	C₂³×C₄×C₁₁	C₂³	1	15	yes	—
24	C₂×C₂²⋊Dic₁₁	C₂³	1	40	yes	—
25	Q₈×Dic₁₁	C₂³	1	320	yes	—
26	D₄×Dic₁₁	C₂³	1	320	yes	—
27	#66979	C₂³	1	80	yes	—
28	#66979	C₂³	1	240	yes	—
29	C₂×C₄⋊Dic₁₁	C₂³	1	120	yes	—
30	C₂×C₄×Dic₁₁	C₂³	1	120	yes	—
31	#62653	—	11	1	no	maximal, Sylow
32	C₂⁴×C₁₁	C₂⁴	1	5	yes	—
33	C₁₁×#41	C₂²×C₄	1	640	yes	—
34	C₂×Q₈×C₁₁	C₂²×C₄	1	80	yes	—
35	C₂×Q₈×C₁₁	C₂²×C₄	1	320	yes	—
36	C₂×Q₈×C₁₁	C₂⁴	1	40	yes	—
37	C₂×D₄×C₁₁	C₂²×C₄	1	80	yes	—
38	C₂×D₄×C₁₁	C₂⁴	1	40	yes	—
39	C₂²×C₄×C₁₁	C₂²×C₄	1	120	yes	—
40	C₂²×C₄×C₁₁	C₂⁴	1	60	yes	—
41	C₂²×C₄×C₁₁	C₂⁴	1	10	yes	—
42	C₂²×Dic₁₁	C₂⁴	1	80	yes	—
43	C₂²⋊Dic₁₁	C₂⁴	1	16	yes	—
44	C₂²⋊Dic₁₁	C₂⁴	1	80	yes	—
45	C₄⋊Dic₁₁	C₂⁴	1	160	yes	—
46	C₄×Dic₁₁	C₂⁴	1	160	yes	—
47	C₂²×2₋¹⁺⁴	D₁₁	1	1	yes	characteristic
48	C₄×2₋¹⁺⁴	—	11	32	no	—
49	#3428	—	11	20	no	—
50	C₂×#519	—	11	10	no	—
51	C₂³×C₁₁	—	4	10	no	—
52	C₂³×C₁₁	C₂³×C₄	1	20	yes	—
53	C₂³×C₁₁	C₂⁵	1	1	yes	characteristic
54	C₂³×C₁₁	C₂⁵	1	10	yes	—
55	Q₈×C₁₁	C₂³×C₄	1	160	yes	—
56	D₄×C₁₁	C₂³×C₄	1	160	yes	—
57	C₂×C₄×C₁₁	C₂³×C₄	1	40	yes	—
58	C₂×C₄×C₁₁	C₂³×C₄	1	240	yes	—
59	C₂×C₄×C₁₁	C₂⁵	1	20	yes	—
60	C₂×Dic₁₁	—	4	80	no	—
61	C₂×Dic₁₁	C₂⁵	1	32	yes	—
62	C₂×2₋¹⁺⁴	Dic₁₁	1	32	yes	complemented
63	C₂×2₋¹⁺⁴	C₂×D₁₁	1	16	yes	—
64	C₂²×#41	C₂×D₁₁	1	10	yes	—
65	C₂³×Q₈	C₂×D₁₁	1	5	yes	—
66	#520	—	11	160	no	—
67	#519	—	11	40	no	—
68	#519	—	11	80	no	—
69	#518	—	11	40	no	—
70	C₄×#41	—	11	160	no	—
71	C₂×C₄×Q₈	—	11	80	no	—
72	C₂×#117	—	11	60	no	—
73	C₂²×C₁₁	—	2	2	no	—
74	C₂²×C₁₁	—	2	20	no	—
75	C₂²×C₁₁	—	4	20	no	—
76	C₂²×C₁₁	C₂⁴×C₄	1	2	yes	—
77	C₂²×C₁₁	C₂⁴×C₄	1	20	yes	—
78	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
79	C₄×C₁₁	C₂⁴×C₄	1	40	yes	—
80	Dic₁₁	—	2	32	no	—
81	2₋¹⁺⁴	C₂×Dic₁₁	1	256	yes	complemented
82	C₂×#41	C₂×Dic₁₁	1	160	yes	complemented
83	C₂×#41	C₂²×D₁₁	1	80	yes	—
84	C₂²×Q₈	C₂×Dic₁₁	1	80	yes	complemented
85	C₂²×Q₈	C₂²×D₁₁	1	40	yes	—
86	C₂²×Q₈	C₂²×D₁₁	1	10	yes	—
87	C₂²×D₄	C₂²×D₁₁	1	10	yes	—
88	C₂³×C₄	C₂²×D₁₁	1	15	yes	—
89	C₄×Q₈	—	11	320	no	—
90	C₄×D₄	—	11	320	no	—
91	#117	—	11	80	no	—
92	#117	—	11	240	no	—
93	C₂×C₄⋊C₄	—	11	120	no	—
94	C₂×C₂²⋊C₄	—	11	40	no	—
95	C₂×C₄²	—	11	120	no	—
96	C₂×C₁₁	—	2	2	no	—
97	C₂×C₁₁	—	2	20	no	—
98	C₂×C₁₁	C₂⁵×C₄	1	1	yes	characteristic
99	C₂×C₁₁	2₋¹⁺⁶	1	2	yes	—
100	C₂⁴	C₂³×D₁₁	1	5	yes	—
101	#41	C₂²×Dic₁₁	1	640	yes	—
102	C₂×Q₈	C₂²×Dic₁₁	1	80	yes	—
103	C₂×Q₈	C₂²×Dic₁₁	1	320	yes	—
104	C₂×Q₈	C₂³×D₁₁	1	40	yes	—
105	C₂×D₄	C₂²×Dic₁₁	1	80	yes	—
106	C₂×D₄	C₂³×D₁₁	1	40	yes	—
107	C₂²×C₄	—	11	80	no	—
108	C₂²×C₄	C₂²×Dic₁₁	1	120	yes	—
109	C₂²×C₄	C₂³×D₁₁	1	60	yes	—
110	C₂²×C₄	C₂³×D₁₁	1	10	yes	—
111	C₄⋊C₄	—	11	160	no	—
112	C₂²⋊C₄	—	11	16	no	—
113	C₂²⋊C₄	—	11	80	no	—
114	C₄²	—	11	160	no	—
115	C₁₁	#62653	1	1	yes	characteristic, complemented, Sylow
116	C₂³	—	4	10	no	—
117	C₂³	C₂³×Dic₁₁	1	20	yes	—
118	C₂³	C₂⁴×D₁₁	1	1	yes	characteristic
119	C₂³	C₂⁴×D₁₁	1	10	yes	—
120	Q₈	C₂³×Dic₁₁	1	160	yes	—
121	D₄	C₂³×Dic₁₁	1	160	yes	—
122	C₂×C₄	—	11	32	no	—
123	C₂×C₄	—	44	80	no	—
124	C₂×C₄	C₂³×Dic₁₁	1	40	yes	—
125	C₂×C₄	C₂³×Dic₁₁	1	240	yes	—
126	C₂×C₄	C₂⁴×D₁₁	1	20	yes	—
127	C₂²	—	2	20	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	20	no	—
130	C₂²	C₂⁴×Dic₁₁	1	2	yes	—
131	C₂²	C₂⁴×Dic₁₁	1	20	yes	—
132	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
133	C₄	—	22	32	no	—
134	C₄	C₂⁴×Dic₁₁	1	40	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	2	20	no	—
137	C₂	C₂⁵×Dic₁₁	1	1	yes	characteristic
138	C₂	#13083428	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₁₁
commutator	C₂×C₁₁	C₂⁵×C₄
Fitting, prime radical	C₂²×C₁₁×2₋¹⁺⁴	C₂
socle, cosocle	C₂²×C₁₁	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₁₁
hypocenter	C₁₁	#62653
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×Dic₁₁	C₂	semidirect
2. #13081819	C₂	semidirect
3. C₂×#10623343	C₂	semidirect
5. C₂×C₁₁×2₋¹⁺⁴	C₄	semidirect
9. #10623358	C₂²	semidirect
10. #41×Dic₁₁	C₂²	semidirect
12. #10623343	C₂²	semidirect
13. C₂×Q₈×Dic₁₁	C₂²	semidirect
16. C₁₁×2₋¹⁺⁴	C₂×C₄	semidirect
17. C₂×C₁₁×#41	C₂×C₄	semidirect
19. C₂²×Q₈×C₁₁	C₂×C₄	semidirect
62. C₂×2₋¹⁺⁴	Dic₁₁	semidirect
81. 2₋¹⁺⁴	C₂×Dic₁₁	semidirect
82. C₂×#41	C₂×Dic₁₁	semidirect
84. C₂²×Q₈	C₂×Dic₁₁	semidirect
115. C₁₁	#62653	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×Dic₁₁	2
#13081819	2
C₂×#10623343	2
C₂²×C₁₁×2₋¹⁺⁴	2
#62653	11