Group #41014358 — subgroups

Subgroups — #41014358

Order 27008.

300 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 137 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×C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#27212908	C₂	1	4	yes	maximal
4	#384×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₂₂	—	2	1	no	—
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	#20269977	—	2	2	no	—
8	#20269977	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
10	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
11	C₄×C₂₁₁⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₁₁	—	2	1	no	—
13	C₂×C₄⋊C₄×C₂₁₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₂₂	—	2	4	no	—
17	C₂×Dic₄₂₂	—	4	2	no	—
18	C₂×Dic₄₂₂	—	4	2	no	—
19	C₂²×Dic₂₁₁	—	2	2	no	—
20	C₂×C₂₁₁⋊C₈	—	2	2	no	—
21	C₂×C₂₁₁⋊C₈	—	2	2	no	—
22	C₂×C₂₁₁⋊C₈	Q₈	1	4	yes	complemented
23	C₄⋊Dic₂₁₁	—	2	4	no	—
24	C₄⋊Dic₂₁₁	—	2	2	no	—
25	C₄⋊Dic₂₁₁	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₁₁	—	2	2	no	—
27	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₂₁₁	—	2	4	no	—
31	C₄⋊C₄×C₂₁₁	—	4	2	no	—
32	C₄⋊C₄×C₂₁₁	—	4	2	no	—
33	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
34	C₄²×C₂₁₁	D₄	1	2	yes	—
35	Dic₄₂₂	—	4	8	no	—
36	C₂×Dic₂₁₁	—	2	4	no	—
37	C₂×Dic₂₁₁	—	4	2	no	—
38	C₂×Dic₂₁₁	—	4	2	no	—
39	C₂₁₁⋊C₈	—	2	4	no	—
40	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	—	4	2	no	—
42	C₂×C₄×C₂₁₁	—	4	2	no	—
43	C₂×C₄×C₂₁₁	—	4	2	no	—
44	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₁₁	D₈	1	2	yes	—
47	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
49	C₂×C₄×C₂₁₁	Q₁₆	1	2	yes	—
50	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₂₁₁	#41	1	2	yes	—
54	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₁₁	#124	1	1	yes	characteristic
56	C₂²×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₂₁₁	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₂₁₁	—	8	2	no	—
62	C₄×C₂₁₁	#102	1	4	yes	—
63	C₄×C₂₁₁	#103	1	4	yes	—
64	C₄×C₂₁₁	C₄×Q₈	1	2	yes	—
65	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
66	Dic₂₁₁	—	4	4	no	—
67	C₂×C₂₁₁	#391	1	1	yes	characteristic
68	C₂×C₂₁₁	C₂×#102	1	1	yes	characteristic
69	C₂×C₂₁₁	C₂×#103	1	1	yes	characteristic
70	C₂×C₂₁₁	#486	1	1	yes	characteristic
71	C₂×C₂₁₁	#487	1	1	yes	characteristic
72	C₂×C₂₁₁	#498	1	1	yes	characteristic
73	C₂×C₂₁₁	#500	1	1	yes	characteristic
74	C₂₁₁	#1990	1	1	yes	characteristic, complemented, Sylow
75	#1990	—	211	1	no	maximal, Sylow
76	C₂×C₄⋊Q₈	—	211	1	no	—
77	C₂×C₄×C₈	—	211	1	no	—
78	#384	D₂₁₁	1	1	yes	characteristic
79	#340	—	211	4	no	—
80	C₂²×Q₈	—	422	1	no	—
81	C₂²×C₈	—	211	2	no	—
82	C₄⋊Q₈	—	422	2	no	—
83	C₄⋊Q₈	—	211	4	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	422	1	no	—
87	C₂×C₄⋊C₄	—	211	2	no	—
88	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
89	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
90	C₄×C₈	—	422	2	no	—
91	C₂×Q₈	—	844	2	no	—
92	C₂×Q₈	—	844	2	no	—
93	C₂×Q₈	—	422	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	422	2	no	—
96	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
99	C₂×C₈	—	422	2	no	—
100	C₂×C₈	—	422	2	no	—
101	C₂×C₈	—	211	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	422	4	no	—
106	C₄⋊C₄	—	422	2	no	—
107	C₄⋊C₄	—	211	4	no	—
108	C₄²	C₄×D₂₁₁	1	2	yes	—
109	C₄²	D₈₄₄	1	2	yes	—
110	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
111	Q₈	—	844	8	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	844	2	no	—
116	C₂×C₄	—	844	2	no	—
117	C₂×C₄	—	422	4	no	—
118	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
119	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
120	C₂×C₄	#16809947	1	2	yes	—
121	C₂×C₄	#16809949	1	2	yes	—
122	C₂×C₄	#16809950	1	2	yes	—
123	C₂×C₄	#16809952	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
125	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₂₁₁	1	2	yes	—
127	C₂×C₄	#16809961	1	2	yes	—
128	C₈	—	422	4	no	—
129	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
130	C₂²	#20269964	1	1	yes	characteristic
131	C₂²	#20269975	1	1	yes	characteristic
132	C₂²	C₂×#16809947	1	1	yes	characteristic
133	C₂²	C₂×#16809949	1	1	yes	characteristic
134	C₂²	C₂×#16809950	1	1	yes	characteristic
135	C₂²	C₂×#16809952	1	1	yes	characteristic
136	C₄	—	8	2	no	—
137	C₄	—	844	4	no	—
138	C₄	#20269897	1	4	yes	—
139	C₄	#20269900	1	4	yes	—
140	C₄	#20269942	1	2	yes	—
141	C₄	#20269957	1	2	yes	—
142	C₂	#27213051	1	1	yes	characteristic
143	C₂	C₂×#20269897	1	1	yes	characteristic
144	C₂	C₂×#20269900	1	1	yes	characteristic
145	C₂	#27213492	1	1	yes	characteristic
146	C₂	#27213496	1	1	yes	characteristic
147	C₂	#27213542	1	1	yes	characteristic
148	C₂	#27213549	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269977

C₂

yes

maximal, characteristic

C₂×C₄×C₂₁₁⋊C₈

C₂

yes

maximal, characteristic

#27212908

C₂

yes

maximal

#384×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

#20269977

—

#20269977

C₄

yes

complemented

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₄×C₂₁₁⋊C₈

—

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂²×Dic₂₁₁

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

Q₈

yes

complemented

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

D₈

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

#124

yes

characteristic

C₂²×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₁₁

C₂×D₈

yes

characteristic

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×Q₁₆

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

#102

yes

—

C₄×C₂₁₁

#103

yes

—

C₄×C₂₁₁

C₄×Q₈

yes

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1990

—

211

maximal, Sylow

C₂×C₄⋊Q₈

—

211

—

C₂×C₄×C₈

—

211

—

#384

D₂₁₁

yes

characteristic

#340

—

211

—

C₂²×Q₈

—

422

—

C₂²×C₈

—

211

—

C₄⋊Q₈

—

422

—

C₄⋊Q₈

—

211

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄×C₈

—

422

—

C₂×Q₈

—

844

—

C₂×Q₈

—

844

—

C₂×Q₈

—

422

—

C₂²×C₄

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²⋊D₂₁₁

yes

characteristic

C₂²×C₄

C₂²⋊D₂₁₁

yes

characteristic

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

422

—

100

C₂×C₈

—

422

—

101

C₂×C₈

—

211

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

422

—

106

C₄⋊C₄

—

422

—

107

C₄⋊C₄

—

211

—

108

C₄²

C₄×D₂₁₁

yes

—

109

C₄²

D₈₄₄

yes

—

110

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

111

Q₈

—

844

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

844

—

116

C₂×C₄

—

844

—

117

C₂×C₄

—

422

—

118

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

119

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

120

C₂×C₄

#16809947

yes

—

121

C₂×C₄

#16809949

yes

—

122

C₂×C₄

#16809950

yes

—

123

C₂×C₄

#16809952

yes

—

124

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

125

C₂×C₄

C₂×D₈₄₄

yes

characteristic

126

C₂×C₄

Q₈×D₂₁₁

yes

—

127

C₂×C₄

#16809961

yes

—

128

C₈

—

422

—

129

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

130

C₂²

#20269964

yes

characteristic

131

C₂²

#20269975

yes

characteristic

132

C₂²

C₂×#16809947

yes

characteristic

133

C₂²

C₂×#16809949

yes

characteristic

134

C₂²

C₂×#16809950

yes

characteristic

135

C₂²

C₂×#16809952

yes

characteristic

136

C₄

—

137

C₄

—

844

—

138

C₄

#20269897

yes

—

139

C₄

#20269900

yes

—

140

C₄

#20269942

yes

—

141

C₄

#20269957

yes

—

142

C₂

#27213051

yes

characteristic

143

C₂

C₂×#20269897

yes

characteristic

144

C₂

C₂×#20269900

yes

characteristic

145

C₂

#27213492

yes

characteristic

146

C₂

#27213496

yes

characteristic

147

C₂

#27213542

yes

characteristic

148

C₂

#27213549

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#384×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#384	D₂₁₁
hypocenter	C₂₁₁	#1990
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#384×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 ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #20269977	C₄	semidirect
22. C₂×C₂₁₁⋊C₈	Q₈	semidirect
74. C₂₁₁	#1990	semidirect

H ≅ G/N

type

8. #20269977

C₄

semidirect

22. C₂×C₂₁₁⋊C₈

Q₈

semidirect

74. C₂₁₁

#1990

semidirect

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

subgroup	index
C₂×#20269977	2
C₂×C₄×C₂₁₁⋊C₈	2
#27212908	2
#384×C₂₁₁	2
#1990	211

subgroup

index

C₂×#20269977

C₂×C₄×C₂₁₁⋊C₈

#27212908

#384×C₂₁₁

#1990

211

Subgroups — #41014358

Conjugacy classes of subgroups

Complements

Maximal subgroups