Group #13068779 — subgroups

Subgroups — #13068779

Order 1408.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×#66882	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	#10622590	C₂	1	2	yes	maximal
5	C₂×#66872	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₁₁×#107	—	2	2	no	—
12	C₁₁⋊#95	—	2	4	no	—
13	#66882	—	2	2	no	—
14	C₈×Dic₁₁	C₄	1	4	yes	complemented
15	#66872	—	2	4	no	—
16	C₂²×C₄×C₁₁	—	2	2	no	—
17	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₁	—	2	2	no	—
19	C₂²×Dic₁₁	D₄	1	2	yes	—
20	C₂×C₈×C₁₁	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₁₁	D₄	1	2	yes	—
22	C₂×C₈×C₁₁	Q₈	1	2	yes	—
23	C₁₁×C₄⋊C₄	—	2	2	no	—
24	C₁₁×C₄⋊C₄	—	2	2	no	—
25	C₄⋊Dic₁₁	—	2	2	no	—
26	C₄⋊Dic₁₁	—	2	2	no	—
27	C₄×Dic₁₁	C₂×C₄	1	2	yes	—
28	C₄×Dic₁₁	Q₈	1	2	yes	—
29	C₂×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₁₁⋊C₈	D₄	1	2	yes	—
31	C₂×C₁₁⋊C₈	Q₈	1	2	yes	—
32	#1846	—	11	1	no	maximal, Sylow
33	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₁	—	4	2	no	—
35	C₂×C₄×C₁₁	—	4	4	no	—
36	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×Dic₁₁	—	2	2	no	—
43	C₂×Dic₁₁	—	2	2	no	—
44	C₂×Dic₁₁	—	4	2	no	—
45	C₂×Dic₁₁	—	4	4	no	—
46	C₂×Dic₁₁	C₄⋊C₄	1	4	yes	—
47	C₈×C₁₁	C₄⋊C₄	1	4	yes	—
48	C₁₁⋊C₈	C₄⋊C₄	1	4	yes	—
49	C₂×#107	—	11	1	no	—
50	C₂×#107	D₁₁	1	1	yes	characteristic
51	C₂×#106	—	11	2	no	—
52	C₂×C₄×C₈	—	11	1	no	—
53	#382	—	11	2	no	—
54	C₂²×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₁	#127	1	1	yes	characteristic
56	C₂²×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₁₁	#135	1	2	yes	—
58	C₂²×C₁₁	#135	1	2	yes	—
59	C₄×C₁₁	—	8	2	no	—
60	C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
63	C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
64	Dic₁₁	—	2	4	no	—
65	Dic₁₁	—	8	2	no	—
66	C₂²×C₈	—	11	1	no	—
67	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
68	C₂×C₄⋊C₄	—	11	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
70	C₂×C₄²	—	11	1	no	—
71	#107	—	2	2	no	—
72	#107	—	22	2	no	—
73	#106	—	22	4	no	—
74	C₄×C₈	—	11	4	no	—
75	#95	—	22	4	no	—
76	C₂×C₁₁	#384	1	1	yes	characteristic
77	C₂×C₁₁	#427	1	2	yes	—
78	C₂×C₁₁	#495	1	1	yes	characteristic
79	C₂×C₁₁	#495	1	1	yes	characteristic
80	C₂×C₁₁	#499	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	11	2	no	—
83	C₂²×C₄	—	22	2	no	—
84	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
85	C₂×C₈	—	11	2	no	—
86	C₂×C₈	—	11	2	no	—
87	C₂×C₈	—	11	2	no	—
88	C₂×C₈	Dic₂₂	1	2	yes	—
89	C₂×C₈	C₄×D₁₁	1	2	yes	—
90	C₂×C₈	C₂²⋊D₁₁	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	22	2	no	—
94	C₄⋊C₄	—	22	2	no	—
95	C₄²	—	11	2	no	—
96	C₄²	—	11	2	no	—
97	C₁₁	#1846	1	1	yes	characteristic, complemented, Sylow
98	C₂³	D₄×D₁₁	1	1	yes	characteristic
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	11	4	no	—
102	C₂×C₄	—	22	2	no	—
103	C₂×C₄	—	22	2	no	—
104	C₂×C₄	—	44	4	no	—
105	C₂×C₄	—	44	2	no	—
106	C₂×C₄	C₂×Dic₂₂	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
108	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₁₁	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₁₁	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
112	C₈	—	11	4	no	—
113	C₈	Dic₁₁⋊C₄	1	4	yes	—
114	C₂²	#66941	1	1	yes	characteristic
115	C₂²	C₄⋊C₄×D₁₁	1	1	yes	characteristic
116	C₂²	#66965	1	2	yes	—
117	C₂²	#66972	1	2	yes	—
118	C₂²	#66993	1	1	yes	characteristic
119	C₄	—	8	2	no	—
120	C₄	—	22	4	no	—
121	C₄	—	88	2	no	—
122	C₄	#66941	1	1	yes	characteristic
123	C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
124	C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
125	C₄	#66997	1	1	yes	characteristic
126	C₂	#10622491	1	2	yes	—
127	C₂	#10622493	1	2	yes	—
128	C₂	#10622589	1	1	yes	characteristic
129	C₂	#10622768	1	1	yes	characteristic
130	C₂	#10622804	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₁×#107

C₂

yes

maximal, characteristic

C₂×#66882

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₁

C₂

yes

maximal, characteristic

#10622590

C₂

yes

maximal

C₂×#66872

C₂

yes

maximal

C₂²×C₈×C₁₁

C₂²

yes

characteristic

C₂×C₁₁×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₁₁

C₂²

yes

—

C₂×C₄×Dic₁₁

C₂²

yes

characteristic

C₂²×C₁₁⋊C₈

C₂²

yes

characteristic

C₁₁×#107

—

C₁₁⋊#95

—

#66882

—

C₈×Dic₁₁

C₄

yes

complemented

#66872

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

—

C₂²×Dic₁₁

D₄

yes

—

C₂×C₈×C₁₁

C₂×C₄

yes

—

C₂×C₈×C₁₁

D₄

yes

—

C₂×C₈×C₁₁

Q₈

yes

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄×Dic₁₁

C₂×C₄

yes

—

C₄×Dic₁₁

Q₈

yes

—

C₂×C₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₁⋊C₈

D₄

yes

—

C₂×C₁₁⋊C₈

Q₈

yes

—

#1846

—

maximal, Sylow

C₂³×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₈

yes

characteristic

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

C₄⋊C₄

yes

—

C₈×C₁₁

C₄⋊C₄

yes

—

C₁₁⋊C₈

C₄⋊C₄

yes

—

C₂×#107

—

C₂×#107

D₁₁

yes

characteristic

C₂×#106

—

C₂×C₄×C₈

—

#382

—

C₂²×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁

#127

yes

characteristic

C₂²×C₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₁

#135

yes

—

C₂²×C₁₁

#135

yes

—

C₄×C₁₁

—

C₄×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₁

C₄⋊Q₈

yes

characteristic

C₄×C₁₁

C₄⋊Q₈

yes

characteristic

Dic₁₁

—

Dic₁₁

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

—

C₂×C₄²

—

#107

—

#107

—

#106

—

C₄×C₈

—

#95

—

C₂×C₁₁

#384

yes

characteristic

C₂×C₁₁

#427

yes

—

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₂

yes

—

C₂×C₈

C₄×D₁₁

yes

—

C₂×C₈

C₂²⋊D₁₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₁

#1846

yes

characteristic, complemented, Sylow

C₂³

D₄×D₁₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₂₂

yes

characteristic

107

C₂×C₄

C₂×C₄×D₁₁

yes

characteristic

108

C₂×C₄

D₄×D₁₁

yes

characteristic

109

C₂×C₄

Q₈×D₁₁

yes

characteristic

110

C₂×C₄

Q₈×D₁₁

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

112

C₈

—

113

C₈

Dic₁₁⋊C₄

yes

—

114

C₂²

#66941

yes

characteristic

115

C₂²

C₄⋊C₄×D₁₁

yes

characteristic

116

C₂²

#66965

yes

—

117

C₂²

#66972

yes

—

118

C₂²

#66993

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#66941

yes

characteristic

123

C₄

C₄⋊C₄×D₁₁

yes

characteristic

124

C₄

C₂×Dic₁₁⋊C₄

yes

characteristic

125

C₄

#66997

yes

characteristic

126

C₂

#10622491

yes

—

127

C₂

#10622493

yes

—

128

C₂

#10622589

yes

characteristic

129

C₂

#10622768

yes

characteristic

130

C₂

#10622804

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₂×C₁₁×#107	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₄
cosocle	C₂²×C₄×C₁₁	C₂³
hypercenter	C₂×#107	D₁₁
hypocenter	C₁₁	#1846
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₁

commutator

C₂×C₄×C₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁

Fitting

C₂×C₁₁×#107

C₂

socle, prime radical

C₂³×C₁₁

C₂×D₄

cosocle

C₂²×C₄×C₁₁

C₂³

D₁₁

C₁₁

G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
14. C₈×Dic₁₁	C₄	semidirect
97. C₁₁	#1846	semidirect

H ≅ G/N

type

14. C₈×Dic₁₁

C₄

semidirect

97. C₁₁

#1846

semidirect

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

subgroup	index
C₂×C₁₁×#107	2
C₂×#66882	2
C₂×C₈×Dic₁₁	2
#10622590	2
C₂×#66872	2
#1846	11

subgroup

index

C₂×C₁₁×#107

C₂×C₈×Dic₁₁

Subgroups — #13068779

Conjugacy classes of subgroups

Complements

Maximal subgroups