Group #41014270 — subgroups

Subgroups — #41014270

Order 27008.

304 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20269909	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269901	C₂	1	2	yes	maximal
3	C₂×#20269887	C₂	1	1	yes	maximal, characteristic
4	#27213053	C₂	1	2	yes	maximal
5	C₂²×Q₁₆×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	#20269909	—	2	2	no	—
10	#20269901	—	2	4	no	—
11	#20269887	—	2	2	no	—
12	C₂₁₁⋊#95	—	2	4	no	—
13	C₂²×Q₈×C₂₁₁	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₂₁₁	—	2	2	no	—
15	C₂×Q₁₆×C₂₁₁	—	2	2	no	—
16	C₂×Q₁₆×C₂₁₁	C₄	1	4	yes	complemented
17	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₁₁	—	2	2	no	—
19	C₂²×Dic₂₁₁	D₄	1	2	yes	—
20	C₂×C₂₁₁⋊C₈	—	2	1	no	—
21	C₂×C₂₁₁⋊C₈	—	2	1	no	—
22	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
23	C₄⋊Dic₂₁₁	—	2	2	no	—
24	C₄⋊Dic₂₁₁	—	2	2	no	—
25	C₄×Dic₂₁₁	—	2	1	no	—
26	C₄×Dic₂₁₁	—	2	1	no	—
27	C₂×Q₈×C₂₁₁	—	2	2	no	—
28	C₂×Q₈×C₂₁₁	—	2	2	no	—
29	C₂×Q₈×C₂₁₁	—	2	2	no	—
30	C₂×Q₈×C₂₁₁	—	2	2	no	—
31	C₂×Q₈×C₂₁₁	—	2	2	no	—
32	C₂×Q₈×C₂₁₁	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₂₁₁	—	2	2	no	—
34	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
35	Q₁₆×C₂₁₁	—	2	8	no	—
36	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
38	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
39	C₂×Dic₂₁₁	—	2	2	no	—
40	C₂×Dic₂₁₁	—	2	4	no	—
41	C₂×Dic₂₁₁	—	4	2	no	—
42	C₂×Dic₂₁₁	—	4	4	no	—
43	C₂₁₁⋊C₈	—	2	2	no	—
44	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₂₁₁	—	2	4	no	—
46	Q₈×C₂₁₁	—	2	4	no	—
47	Q₈×C₂₁₁	—	4	2	no	—
48	Q₈×C₂₁₁	—	4	2	no	—
49	C₂×C₄×C₂₁₁	—	2	4	no	—
50	C₂×C₄×C₂₁₁	—	4	2	no	—
51	C₂×C₄×C₂₁₁	—	4	2	no	—
52	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
57	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
58	C₈×C₂₁₁	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
60	C₂²×C₂₁₁	#121	1	1	yes	characteristic
61	C₂²×C₂₁₁	#127	1	1	yes	characteristic
62	C₂²×C₂₁₁	#137	1	4	yes	—
63	C₄×C₂₁₁	—	4	4	no	—
64	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₂₁₁	C₄×D₄	1	1	yes	characteristic
66	C₄×C₂₁₁	#121	1	1	yes	characteristic
67	C₄×C₂₁₁	#124	1	1	yes	characteristic
68	Dic₂₁₁	—	4	2	no	—
69	Dic₂₁₁	—	8	2	no	—
70	C₂×C₂₁₁	#390	1	1	yes	characteristic
71	C₂×C₂₁₁	#441	1	2	yes	—
72	C₂×C₂₁₁	#470	1	2	yes	—
73	C₂×C₂₁₁	#497	1	2	yes	—
74	C₂₁₁	#1971	1	1	yes	characteristic, complemented, Sylow
75	#1971	—	211	1	no	maximal, Sylow
76	C₂²×Q₁₆	D₂₁₁	1	1	yes	characteristic
77	C₂×#106	—	211	1	no	—
78	C₂×#103	—	211	2	no	—
79	C₂×#97	—	211	1	no	—
80	#391	—	211	2	no	—
81	C₂²×Q₈	C₂×D₂₁₁	1	2	yes	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×Q₁₆	—	2	2	no	—
84	C₂×Q₁₆	Dic₂₁₁	1	4	yes	complemented
85	C₂²×C₈	—	211	1	no	—
86	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	211	2	no	—
88	C₂×C₄²	—	211	1	no	—
89	#106	—	422	2	no	—
90	#103	—	422	4	no	—
91	#97	—	422	2	no	—
92	#95	—	422	4	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂×Dic₂₁₁	1	4	yes	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	422	2	no	—
101	C₂²×C₄	—	211	2	no	—
102	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
103	Q₁₆	—	2	8	no	—
104	C₂×C₈	—	422	1	no	—
105	C₂×C₈	—	422	1	no	—
106	C₂×C₈	—	211	2	no	—
107	C₂×C₈	C₂×Dic₂₁₁	1	2	yes	—
108	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
109	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
110	C₄⋊C₄	—	422	2	no	—
111	C₄⋊C₄	—	422	2	no	—
112	C₄²	—	422	1	no	—
113	C₄²	—	422	1	no	—
114	C₂³	D₄×D₂₁₁	1	1	yes	characteristic
115	Q₈	—	2	4	no	—
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	844	2	no	—
123	C₂×C₄	—	844	4	no	—
124	C₂×C₄	—	422	2	no	—
125	C₂×C₄	—	422	4	no	—
126	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
127	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
130	C₂×C₄	#16809961	1	1	yes	characteristic
131	C₂×C₄	#16809961	1	1	yes	characteristic
132	C₈	—	422	2	no	—
133	C₈	C₂²⋊Dic₂₁₁	1	4	yes	—
134	C₂²	D₄×Dic₂₁₁	1	1	yes	characteristic
135	C₂²	#20269948	1	1	yes	characteristic
136	C₂²	#20269973	1	1	yes	characteristic
137	C₂²	#20270009	1	4	yes	—
138	C₄	—	4	4	no	—
139	C₄	—	844	2	no	—
140	C₄	—	1688	2	no	—
141	C₄	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
142	C₄	D₄×Dic₂₁₁	1	1	yes	characteristic
143	C₄	#20269948	1	1	yes	characteristic
144	C₄	#20269964	1	1	yes	characteristic
145	C₂	#27213049	1	1	yes	characteristic
146	C₂	#27213253	1	2	yes	—
147	C₂	#27213401	1	2	yes	—
148	C₂	#27213539	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269909

C₂

yes

maximal, characteristic

C₂×#20269901

C₂

yes

maximal

C₂×#20269887

C₂

yes

maximal, characteristic

#27213053

C₂

yes

maximal

C₂²×Q₁₆×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄×Dic₂₁₁

C₂²

yes

—

—

—

C₂₁₁⋊#95

—

C₂²×Q₈×C₂₁₁

C₂²

yes

—

C₂×Q₁₆×C₂₁₁

—

C₂×Q₁₆×C₂₁₁

—

C₂×Q₁₆×C₂₁₁

C₄

yes

complemented

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

D₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

Q₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

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×D₄

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

C₈×C₂₁₁

C₂²⋊C₄

yes

—

C₂²×C₂₁₁

C₄×D₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₁₁

C₄×D₄

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

#390

yes

characteristic

C₂×C₂₁₁

#441

yes

—

C₂×C₂₁₁

#470

yes

—

C₂×C₂₁₁

#497

yes

—

C₂₁₁

#1971

yes

characteristic, complemented, Sylow

#1971

—

211

maximal, Sylow

C₂²×Q₁₆

D₂₁₁

yes

characteristic

C₂×#106

—

211

—

C₂×#103

—

211

—

C₂×#97

—

211

—

#391

—

211

—

C₂²×Q₈

C₂×D₂₁₁

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₂₁₁

yes

complemented

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

—

211

—

#106

—

422

—

#103

—

422

—

#97

—

422

—

#95

—

422

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₁₁

yes

—

C₂²×C₄

—

100

C₂²×C₄

—

422

—

101

C₂²×C₄

—

211

—

102

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

103

Q₁₆

—

104

C₂×C₈

—

422

—

105

C₂×C₈

—

422

—

106

C₂×C₈

—

211

—

107

C₂×C₈

C₂×Dic₂₁₁

yes

—

108

C₂×C₈

C₂²⋊D₂₁₁

yes

—

109

C₂×C₈

C₂²⋊D₂₁₁

yes

—

110

C₄⋊C₄

—

422

—

111

C₄⋊C₄

—

422

—

112

C₄²

—

422

—

113

C₄²

—

422

—

114

C₂³

D₄×D₂₁₁

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

844

—

123

C₂×C₄

—

844

—

124

C₂×C₄

—

422

—

125

C₂×C₄

—

422

—

126

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

127

C₂×C₄

D₄×D₂₁₁

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

130

C₂×C₄

#16809961

yes

characteristic

131

C₂×C₄

#16809961

yes

characteristic

132

C₈

—

422

—

133

C₈

C₂²⋊Dic₂₁₁

yes

—

134

C₂²

D₄×Dic₂₁₁

yes

characteristic

135

C₂²

#20269948

yes

characteristic

136

C₂²

#20269973

yes

characteristic

137

C₂²

#20270009

yes

—

138

C₄

—

139

C₄

—

844

—

140

C₄

—

1688

—

141

C₄

C₂×C₂²⋊Dic₂₁₁

yes

characteristic

142

C₄

D₄×Dic₂₁₁

yes

characteristic

143

C₄

#20269948

yes

characteristic

144

C₄

#20269964

yes

characteristic

145

C₂

#27213049

yes

characteristic

146

C₂

#27213253

yes

—

147

C₂

#27213401

yes

—

148

C₂

#27213539

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂²×Q₁₆×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂²×Q₁₆	D₂₁₁
hypocenter	C₂₁₁	#1971
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂²×Q₁₆×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂²×Q₁₆

D₂₁₁

hypocenter

C₂₁₁

#1971

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
16. C₂×Q₁₆×C₂₁₁	C₄	semidirect
74. C₂₁₁	#1971	semidirect
84. C₂×Q₁₆	Dic₂₁₁	semidirect

H ≅ G/N

type

16. C₂×Q₁₆×C₂₁₁

C₄

semidirect

74. C₂₁₁

#1971

semidirect

84. C₂×Q₁₆

Dic₂₁₁

semidirect

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

subgroup	index
C₂×#20269909	2
C₂×#20269901	2
C₂×#20269887	2
#27213053	2
C₂²×Q₁₆×C₂₁₁	2
#1971	211

subgroup

index

C₂²×Q₁₆×C₂₁₁

211

Subgroups — #41014270

Conjugacy classes of subgroups

Complements

Maximal subgroups