Group #41014320 — subgroups

Subgroups — #41014320

Order 27008.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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×#20269977	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269902	C₂	1	4	yes	maximal
3	C₄×C₄⋊Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₂₂	C₂²	1	2	yes	—
6	#20269977	—	2	2	no	—
7	#20269977	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
9	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
10	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₁₁	—	2	1	no	—
12	#20269902	—	2	8	no	—
13	C₂₁₁⋊#95	—	2	1	no	—
14	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₁₁	—	2	2	no	—
17	C₂×Dic₄₂₂	—	2	2	no	—
18	C₂×Dic₄₂₂	—	2	4	no	—
19	C₂×Dic₄₂₂	—	2	4	no	—
20	C₂×Dic₄₂₂	C₂×C₄	1	4	yes	—
21	C₂²×Dic₂₁₁	—	2	2	no	—
22	C₂²×Dic₂₁₁	—	2	2	no	—
23	C₄⋊Dic₂₁₁	—	2	4	no	—
24	C₄⋊Dic₂₁₁	—	2	4	no	—
25	C₄⋊Dic₂₁₁	—	2	2	no	—
26	C₄⋊Dic₂₁₁	D₄	1	4	yes	—
27	C₄×Dic₂₁₁	—	4	2	no	—
28	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₂₁₁	—	2	2	no	—
32	C₂×C₈×C₂₁₁	—	2	4	no	—
33	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
34	C₄²×C₂₁₁	D₄	1	2	yes	—
35	Dic₄₂₂	—	2	8	no	—
36	Dic₄₂₂	—	4	2	no	—
37	Dic₄₂₂	—	4	2	no	—
38	C₂×Dic₂₁₁	—	2	4	no	—
39	C₂×Dic₂₁₁	—	2	4	no	—
40	C₂×Dic₂₁₁	—	4	4	no	—
41	C₂×Dic₂₁₁	—	4	2	no	—
42	C₂×Dic₂₁₁	—	4	2	no	—
43	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	—	2	2	no	—
45	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₂₁₁	Q₁₆	1	2	yes	—
49	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₁₁	#41	1	2	yes	—
53	C₈×C₂₁₁	—	4	2	no	—
54	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₁₁	#121	1	2	yes	—
56	C₂²×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₂₁₁	#137	1	2	yes	—
59	C₄×C₂₁₁	—	2	2	no	—
60	C₄×C₂₁₁	#103	1	4	yes	complemented
61	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
62	C₄×C₂₁₁	#124	1	1	yes	characteristic
63	C₄×C₂₁₁	#127	1	1	yes	characteristic
64	Dic₂₁₁	—	4	4	no	—
65	Dic₂₁₁	—	4	4	no	—
66	C₂×C₂₁₁	#390	1	1	yes	characteristic
67	C₂×C₂₁₁	C₂×#103	1	1	yes	characteristic
68	C₂×C₂₁₁	#419	1	1	yes	characteristic
69	C₂×C₂₁₁	#461	1	2	yes	—
70	C₂×C₂₁₁	#462	1	2	yes	—
71	C₂₁₁	#1981	1	1	yes	characteristic, complemented, Sylow
72	#1981	—	211	1	no	maximal, Sylow
73	C₂×C₄⋊Q₈	—	211	1	no	—
74	C₂×C₄⋊C₈	D₂₁₁	1	1	yes	characteristic
75	C₂×#103	—	211	4	no	—
76	C₄×C₄⋊C₄	—	211	1	no	—
77	C₂²×Q₈	—	211	2	no	—
78	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
79	C₄⋊Q₈	—	422	2	no	—
80	C₄⋊Q₈	—	211	4	no	—
81	C₂×C₄⋊C₄	—	422	1	no	—
82	C₂×C₄⋊C₄	—	422	1	no	—
83	C₂×C₄⋊C₄	—	211	2	no	—
84	C₂×C₄²	—	422	1	no	—
85	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
86	C₄⋊C₈	—	2	2	no	—
87	#103	—	422	8	no	—
88	#95	—	422	1	no	—
89	C₂×Q₈	—	422	4	no	—
90	C₂×Q₈	—	422	4	no	—
91	C₂×Q₈	—	422	2	no	—
92	C₂×Q₈	—	211	4	no	—
93	C₂²×C₄	—	422	2	no	—
94	C₂²×C₄	—	422	2	no	—
95	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
96	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₄⋊C₄	—	422	4	no	—
101	C₄⋊C₄	—	422	4	no	—
102	C₄⋊C₄	—	422	2	no	—
103	C₄⋊C₄	—	211	4	no	—
104	C₄²	—	844	2	no	—
105	C₄²	C₄×D₂₁₁	1	2	yes	—
106	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
107	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
108	Q₈	—	844	2	no	—
109	Q₈	—	844	2	no	—
110	Q₈	—	422	8	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	844	4	no	—
113	C₂×C₄	—	844	2	no	—
114	C₂×C₄	—	844	2	no	—
115	C₂×C₄	—	422	4	no	—
116	C₂×C₄	—	422	4	no	—
117	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
118	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
119	C₂×C₄	#16809951	1	2	yes	—
120	C₂×C₄	Dic₈₄₄	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
122	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
123	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
124	C₂×C₄	#16809963	1	2	yes	—
125	C₈	—	4	2	no	—
126	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
127	C₂²	#20269952	1	2	yes	—
128	C₂²	C₂×#16809951	1	1	yes	characteristic
129	C₂²	C₂×Dic₈₄₄	1	1	yes	characteristic
130	C₂²	#20270010	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	844	4	no	—
133	C₄	—	844	4	no	—
134	C₄	#20269902	1	4	yes	complemented
135	C₄	#20269939	1	2	yes	—
136	C₄	#20269966	1	1	yes	characteristic
137	C₄	#20269973	1	1	yes	characteristic
138	C₂	#27213047	1	1	yes	characteristic
139	C₂	C₂×#20269902	1	1	yes	characteristic
140	C₂	#27213156	1	1	yes	characteristic
141	C₂	#27213352	1	2	yes	—
142	C₂	#27213357	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269977

C₂

yes

maximal, characteristic

C₂×#20269902

C₂

yes

maximal

C₄×C₄⋊Dic₂₁₁

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₂²

yes

—

#20269977

—

#20269977

C₄

yes

complemented

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄×Dic₂₁₁

—

#20269902

—

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

C₂×C₄

yes

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

D₄

yes

—

C₄×Dic₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

#121

yes

—

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×Q₁₆

yes

characteristic

C₂²×C₂₁₁

#137

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#103

yes

complemented

C₄×C₂₁₁

C₄×D₄

yes

—

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂×C₂₁₁

#462

yes

—

C₂₁₁

#1981

yes

characteristic, complemented, Sylow

#1981

—

211

maximal, Sylow

C₂×C₄⋊Q₈

—

211

—

C₂×C₄⋊C₈

D₂₁₁

yes

characteristic

C₂×#103

—

211

—

C₄×C₄⋊C₄

—

211

—

C₂²×Q₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

—

C₄⋊Q₈

—

422

—

C₄⋊Q₈

—

211

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

—

422

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

#103

—

422

—

#95

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

211

—

C₂²×C₄

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

D₈₄₄

yes

characteristic

C₂²×C₄

D₈₄₄

yes

characteristic

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

422

—

101

C₄⋊C₄

—

422

—

102

C₄⋊C₄

—

422

—

103

C₄⋊C₄

—

211

—

104

C₄²

—

844

—

105

C₄²

C₄×D₂₁₁

yes

—

106

C₄²

C₂²⋊D₂₁₁

yes

—

107

C₂³

C₂×D₈₄₄

yes

characteristic

108

Q₈

—

844

—

109

Q₈

—

844

—

110

Q₈

—

422

—

111

C₂×C₄

—

112

C₂×C₄

—

844

—

113

C₂×C₄

—

844

—

114

C₂×C₄

—

844

—

115

C₂×C₄

—

422

—

116

C₂×C₄

—

422

—

117

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

118

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

119

C₂×C₄

#16809951

yes

—

120

C₂×C₄

Dic₈₄₄

yes

—

121

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

122

C₂×C₄

D₄×D₂₁₁

yes

—

123

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

124

C₂×C₄

#16809963

yes

—

125

C₈

—

126

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

127

C₂²

#20269952

yes

—

128

C₂²

C₂×#16809951

yes

characteristic

129

C₂²

C₂×Dic₈₄₄

yes

characteristic

130

C₂²

#20270010

yes

—

131

C₄

—

132

C₄

—

844

—

133

C₄

—

844

—

134

C₄

#20269902

yes

complemented

135

C₄

#20269939

yes

—

136

C₄

#20269966

yes

characteristic

137

C₄

#20269973

yes

characteristic

138

C₂

#27213047

yes

characteristic

139

C₂

C₂×#20269902

yes

characteristic

140

C₂

#27213156

yes

characteristic

141

C₂

#27213352

yes

—

142

C₂

#27213357

yes

—

	subgroup	G/H
center	C₂³	C₂×D₈₄₄
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×C₄⋊C₈×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₁₁
hypocenter	C₂₁₁	#1981
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₈₄₄

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×C₄⋊C₈×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₁₁

hypocenter

C₂₁₁

#1981

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #20269977	C₄	semidirect
60. C₄×C₂₁₁	#103	semidirect
71. C₂₁₁	#1981	semidirect
134. C₄	#20269902	semidirect

H ≅ G/N

type

7. #20269977

C₄

semidirect

60. C₄×C₂₁₁

#103

semidirect

71. C₂₁₁

#1981

semidirect

134. C₄

#20269902

semidirect

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

subgroup	index
C₂×#20269977	2
C₂×#20269902	2
C₄×C₄⋊Dic₂₁₁	2
C₂×C₄⋊C₈×C₂₁₁	2
#1981	211

subgroup

index

C₂×#20269977

C₂×#20269902

C₄×C₄⋊Dic₂₁₁

C₂×C₄⋊C₈×C₂₁₁

#1981

211

Subgroups — #41014320

Conjugacy classes of subgroups

Complements

Maximal subgroups