Group #41013561 — subgroups

Subgroups — #41013561

Order 27008.

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×#20269911	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269908	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#27213011	C₂	1	2	yes	maximal
5	C₂×#107×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
9	#20269911	—	2	2	no	—
10	#20269908	—	2	4	no	—
11	C₈×Dic₂₁₁	C₄	1	4	yes	complemented
12	C₂₁₁⋊#95	—	2	4	no	—
13	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	#107×C₂₁₁	—	2	2	no	—
16	C₂²×Dic₂₁₁	—	2	2	no	—
17	C₂²×Dic₂₁₁	D₄	1	2	yes	—
18	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
20	C₂×C₂₁₁⋊C₈	Q₈	1	2	yes	—
21	C₄⋊Dic₂₁₁	—	2	2	no	—
22	C₄⋊Dic₂₁₁	—	2	2	no	—
23	C₄×Dic₂₁₁	C₂×C₄	1	2	yes	—
24	C₄×Dic₂₁₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₁₁	—	2	2	no	—
26	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
29	C₂×C₈×C₂₁₁	Q₈	1	2	yes	—
30	C₄⋊C₄×C₂₁₁	—	2	2	no	—
31	C₄⋊C₄×C₂₁₁	—	2	2	no	—
32	C₂×Dic₂₁₁	—	2	2	no	—
33	C₂×Dic₂₁₁	—	2	2	no	—
34	C₂×Dic₂₁₁	—	4	2	no	—
35	C₂×Dic₂₁₁	—	4	4	no	—
36	C₂×Dic₂₁₁	C₄⋊C₄	1	4	yes	—
37	C₂₁₁⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	—	4	2	no	—
40	C₂×C₄×C₂₁₁	—	4	4	no	—
41	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
47	C₈×C₂₁₁	C₄⋊C₄	1	4	yes	—
48	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₁₁	#127	1	1	yes	characteristic
50	C₂²×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₂₁₁	#135	1	2	yes	—
52	C₂²×C₂₁₁	#135	1	2	yes	—
53	C₄×C₂₁₁	—	8	2	no	—
54	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₂₁₁	—	2	4	no	—
59	Dic₂₁₁	—	8	2	no	—
60	C₂×C₂₁₁	#384	1	1	yes	characteristic
61	C₂×C₂₁₁	#427	1	2	yes	—
62	C₂×C₂₁₁	#495	1	1	yes	characteristic
63	C₂×C₂₁₁	#495	1	1	yes	characteristic
64	C₂×C₂₁₁	#499	1	2	yes	—
65	C₂₁₁	#1846	1	1	yes	characteristic, complemented, Sylow
66	#1846	—	211	1	no	maximal, Sylow
67	C₂×#107	—	211	1	no	—
68	C₂×#107	D₂₁₁	1	1	yes	characteristic
69	C₂×#106	—	211	2	no	—
70	C₂×C₄×C₈	—	211	1	no	—
71	#382	—	211	2	no	—
72	C₂²×C₈	—	211	1	no	—
73	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	211	2	no	—
75	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
76	C₂×C₄²	—	211	1	no	—
77	#107	—	2	2	no	—
78	#107	—	422	2	no	—
79	#106	—	422	4	no	—
80	C₄×C₈	—	211	4	no	—
81	#95	—	422	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	422	2	no	—
84	C₂²×C₄	—	211	2	no	—
85	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
86	C₂×C₈	—	211	2	no	—
87	C₂×C₈	—	211	2	no	—
88	C₂×C₈	—	211	2	no	—
89	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
90	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
91	C₂×C₈	Dic₄₂₂	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	422	2	no	—
95	C₄⋊C₄	—	422	2	no	—
96	C₄²	—	211	2	no	—
97	C₄²	—	211	2	no	—
98	C₂³	D₄×D₂₁₁	1	1	yes	characteristic
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	844	2	no	—
102	C₂×C₄	—	844	4	no	—
103	C₂×C₄	—	422	2	no	—
104	C₂×C₄	—	422	2	no	—
105	C₂×C₄	—	211	4	no	—
106	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
107	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₂₁₁	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₂₁₁	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
112	C₈	—	211	4	no	—
113	C₈	Dic₂₁₁⋊C₄	1	4	yes	—
114	C₂²	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
115	C₂²	#20269973	1	1	yes	characteristic
116	C₂²	#20269976	1	1	yes	characteristic
117	C₂²	#20269996	1	2	yes	—
118	C₂²	#20270000	1	2	yes	—
119	C₄	—	8	2	no	—
120	C₄	—	1688	2	no	—
121	C₄	—	422	4	no	—
122	C₄	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
123	C₄	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
124	C₄	#20269975	1	1	yes	characteristic
125	C₄	#20269976	1	1	yes	characteristic
126	C₂	#27213015	1	1	yes	characteristic
127	C₂	#27213183	1	2	yes	—
128	C₂	#27213529	1	1	yes	characteristic
129	C₂	#27213531	1	1	yes	characteristic
130	C₂	#27213547	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269911

C₂

yes

maximal, characteristic

C₂×#20269908

C₂

yes

maximal

C₂×C₈×Dic₂₁₁

C₂

yes

maximal, characteristic

#27213011

C₂

yes

maximal

C₂×#107×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

#20269911

—

#20269908

—

C₈×Dic₂₁₁

C₄

yes

complemented

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

#107×C₂₁₁

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

D₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₂×C₂₁₁⋊C₈

Q₈

yes

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄×Dic₂₁₁

C₂×C₄

yes

—

C₄×Dic₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

Q₈

yes

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₂₁₁⋊C₈

C₄⋊C₄

yes

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₈×C₂₁₁

C₄⋊C₄

yes

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

#127

yes

characteristic

C₂²×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₁₁

#135

yes

—

C₂²×C₂₁₁

#135

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

#384

yes

characteristic

C₂×C₂₁₁

#427

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂₁₁

#1846

yes

characteristic, complemented, Sylow

#1846

—

211

maximal, Sylow

C₂×#107

—

211

—

C₂×#107

D₂₁₁

yes

characteristic

C₂×#106

—

211

—

C₂×C₄×C₈

—

211

—

#382

—

211

—

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

—

C₂×C₄²

—

211

—

#107

—

#107

—

422

—

#106

—

422

—

C₄×C₈

—

211

—

#95

—

422

—

C₂²×C₄

—

C₂²×C₄

—

422

—

C₂²×C₄

—

211

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

C₄×D₂₁₁

yes

—

C₂×C₈

C₂²⋊D₂₁₁

yes

—

C₂×C₈

Dic₄₂₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄²

—

211

—

C₄²

—

211

—

C₂³

D₄×D₂₁₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

844

—

102

C₂×C₄

—

844

—

103

C₂×C₄

—

422

—

104

C₂×C₄

—

422

—

105

C₂×C₄

—

211

—

106

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

107

C₂×C₄

D₄×D₂₁₁

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

109

C₂×C₄

Q₈×D₂₁₁

yes

characteristic

110

C₂×C₄

Q₈×D₂₁₁

yes

characteristic

111

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

112

C₈

—

211

—

113

C₈

Dic₂₁₁⋊C₄

yes

—

114

C₂²

C₄⋊C₄×D₂₁₁

yes

characteristic

115

C₂²

#20269973

yes

characteristic

116

C₂²

#20269976

yes

characteristic

117

C₂²

#20269996

yes

—

118

C₂²

#20270000

yes

—

119

C₄

—

120

C₄

—

1688

—

121

C₄

—

422

—

122

C₄

C₄⋊C₄×D₂₁₁

yes

characteristic

123

C₄

C₂×Dic₂₁₁⋊C₄

yes

characteristic

124

C₄

#20269975

yes

characteristic

125

C₄

#20269976

yes

characteristic

126

C₂

#27213015

yes

characteristic

127

C₂

#27213183

yes

—

128

C₂

#27213529

yes

characteristic

129

C₂

#27213531

yes

characteristic

130

C₂

#27213547

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×#107×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×#107	D₂₁₁
hypocenter	C₂₁₁	#1846
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×#107×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
11. C₈×Dic₂₁₁	C₄	semidirect
65. C₂₁₁	#1846	semidirect

H ≅ G/N

type

11. C₈×Dic₂₁₁

C₄

semidirect

65. C₂₁₁

#1846

semidirect

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

subgroup	index
C₂×#20269911	2
C₂×#20269908	2
C₂×C₈×Dic₂₁₁	2
#27213011	2
C₂×#107×C₂₁₁	2
#1846	211

subgroup

index

C₂×C₈×Dic₂₁₁

C₂×#107×C₂₁₁

211

Subgroups — #41013561

Conjugacy classes of subgroups

Complements

Maximal subgroups