Group #41014871 — subgroups

Subgroups — #41014871

Order 27008.

230 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 95 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	2	yes	maximal
2	#27213044	C₂	1	1	yes	maximal, characteristic
3	#27213016	C₂	1	1	yes	maximal, characteristic
4	#27212909	C₂	1	2	yes	maximal
5	#336×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
7	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
8	C₂×C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₁₁⋊C₄	—	2	1	no	—
12	#20269911	—	2	4	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	C₂²×Dic₂₁₁	—	2	1	no	—
17	C₂²×Dic₂₁₁	—	2	1	no	—
18	C₂²×Dic₂₁₁	—	2	2	no	—
19	C₄⋊Dic₂₁₁	—	2	2	no	—
20	C₄⋊Dic₂₁₁	—	2	1	no	—
21	C₄⋊Dic₂₁₁	—	2	2	no	—
22	C₄⋊Dic₂₁₁	—	2	2	no	—
23	C₄⋊Dic₂₁₁	—	4	2	no	—
24	C₄⋊Dic₂₁₁	—	4	2	no	—
25	C₄⋊Dic₂₁₁	D₄	1	2	yes	—
26	Dic₂₁₁⋊C₄	—	4	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	C₂×C₈×C₂₁₁	—	2	2	no	—
31	C₂×C₈×C₂₁₁	—	2	2	no	—
32	C₂×C₈×C₂₁₁	Q₈	1	4	yes	—
33	C₄²×C₂₁₁	—	2	2	no	—
34	C₂×Dic₂₁₁	—	2	2	no	—
35	C₂×Dic₂₁₁	—	4	1	no	—
36	C₂×Dic₂₁₁	—	4	1	no	—
37	C₂×Dic₂₁₁	—	4	2	no	—
38	C₂×Dic₂₁₁	—	4	1	no	—
39	C₂×Dic₂₁₁	—	4	1	no	—
40	C₂×Dic₂₁₁	—	4	2	no	—
41	C₂×Dic₂₁₁	—	4	2	no	—
42	C₂×Dic₂₁₁	—	4	1	no	—
43	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	—	2	1	no	—
45	C₂×C₄×C₂₁₁	—	2	1	no	—
46	C₂×C₄×C₂₁₁	—	2	1	no	—
47	C₂×C₄×C₂₁₁	—	2	1	no	—
48	C₂×C₄×C₂₁₁	D₈	1	2	yes	—
49	C₂×C₄×C₂₁₁	Q₁₆	1	2	yes	—
50	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₂₁₁	#41	1	2	yes	—
53	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
54	C₈×C₂₁₁	—	2	4	no	—
55	C₂²×C₂₁₁	#121	1	1	yes	characteristic
56	C₂²×C₂₁₁	#123	1	1	yes	characteristic
57	C₂²×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₂₁₁	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₂₁₁	#137	1	1	yes	characteristic
62	C₄×C₂₁₁	—	2	2	no	—
63	C₄×C₂₁₁	—	2	2	no	—
64	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₂₁₁	#125	1	2	yes	—
66	Dic₂₁₁	—	4	2	no	—
67	Dic₂₁₁	—	8	2	no	—
68	Dic₂₁₁	—	8	1	no	—
69	C₂×C₂₁₁	#398	1	1	yes	characteristic
70	C₂×C₂₁₁	#459	1	1	yes	characteristic
71	C₂×C₂₁₁	#462	1	1	yes	characteristic
72	C₂×C₂₁₁	#480	1	1	yes	characteristic
73	C₂×C₂₁₁	#484	1	1	yes	characteristic
74	C₂×C₂₁₁	#500	1	1	yes	characteristic
75	C₂×C₂₁₁	#501	1	1	yes	characteristic
76	C₂₁₁	#2083	1	1	yes	characteristic, complemented, Sylow
77	#2083	—	211	1	no	maximal, Sylow
78	C₂×#107	—	211	2	no	—
79	#389	—	211	1	no	—
80	#384	—	211	1	no	—
81	#340	—	211	2	no	—
82	#336	D₂₁₁	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
84	C₂×C₄⋊C₄	—	422	1	no	—
85	C₂×C₄⋊C₄	—	422	1	no	—
86	C₂×C₄⋊C₄	—	422	1	no	—
87	C₂×C₄⋊C₄	—	211	1	no	—
88	C₂×C₄⋊C₄	—	211	2	no	—
89	C₂×C₄⋊C₄	—	211	1	no	—
90	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
91	#107	—	422	4	no	—
92	#95	—	422	1	no	—
93	C₂²×C₄	—	422	1	no	—
94	C₂²×C₄	—	422	2	no	—
95	C₂²×C₄	—	422	1	no	—
96	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
97	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	Dic₄₂₂	1	4	yes	—
102	C₄⋊C₄	—	844	2	no	—
103	C₄⋊C₄	—	844	2	no	—
104	C₄⋊C₄	—	844	2	no	—
105	C₄⋊C₄	—	422	2	no	—
106	C₄⋊C₄	—	422	2	no	—
107	C₄⋊C₄	—	422	2	no	—
108	C₄⋊C₄	—	422	1	no	—
109	C₄⋊C₄	—	211	2	no	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	844	1	no	—
117	C₂×C₄	—	844	1	no	—
118	C₂×C₄	—	844	2	no	—
119	C₂×C₄	—	844	1	no	—
120	C₂×C₄	—	844	2	no	—
121	C₂×C₄	—	844	1	no	—
122	C₂×C₄	—	844	2	no	—
123	C₂×C₄	—	844	1	no	—
124	C₂×C₄	—	422	2	no	—
125	C₂×C₄	D₁₆₈₈	1	2	yes	—
126	C₂×C₄	Dic₈₄₄	1	2	yes	—
127	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
128	C₂×C₄	C₂×Dic₄₂₂	1	2	yes	—
129	C₂×C₄	#16809961	1	2	yes	—
130	C₂×C₄	#16809963	1	1	yes	characteristic
131	C₈	—	2	4	no	—
132	C₂²	#20269952	1	1	yes	characteristic
133	C₂²	#20269961	1	1	yes	characteristic
134	C₂²	#20269977	1	1	yes	characteristic
135	C₂²	C₂×D₁₆₈₈	1	1	yes	characteristic
136	C₂²	C₂×Dic₈₄₄	1	1	yes	characteristic
137	C₂²	#20270005	1	1	yes	characteristic
138	C₂²	#20270010	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	844	2	no	—
142	C₄	—	1688	2	no	—
143	C₄	—	1688	1	no	—
144	C₄	#20269954	1	2	yes	—
145	C₄	#20269968	1	2	yes	—
146	C₂	#27213078	1	1	yes	characteristic
147	C₂	#27213342	1	1	yes	characteristic
148	C₂	#27213357	1	1	yes	characteristic
149	C₂	#27213460	1	1	yes	characteristic
150	C₂	#27213482	1	1	yes	characteristic
151	C₂	#27213551	1	1	yes	characteristic
152	C₂	#27213555	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269911

C₂

yes

maximal

#27213044

C₂

yes

maximal, characteristic

#27213016

C₂

yes

maximal, characteristic

#27212909

C₂

yes

maximal

#336×C₂₁₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

characteristic

C₂×Dic₂₁₁⋊C₄

—

#20269911

—

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

D₄

yes

—

Dic₂₁₁⋊C₄

—

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

Q₈

yes

—

C₄²×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

D₈

yes

—

C₂×C₄×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₁₁

C₂×D₈

yes

characteristic

C₂²×C₂₁₁

C₂×Q₁₆

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

#125

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

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

#2083

—

211

maximal, Sylow

C₂×#107

—

211

—

#389

—

211

—

#384

—

211

—

#340

—

211

—

#336

D₂₁₁

yes

characteristic

C₂²×C₈

C₂×D₂₁₁

yes

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#107

—

422

—

#95

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

D₈₄₄

yes

characteristic

C₂²×C₄

D₈₄₄

yes

characteristic

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

Dic₄₂₂

yes

—

102

C₄⋊C₄

—

844

—

103

C₄⋊C₄

—

844

—

104

C₄⋊C₄

—

844

—

105

C₄⋊C₄

—

422

—

106

C₄⋊C₄

—

422

—

107

C₄⋊C₄

—

422

—

108

C₄⋊C₄

—

422

—

109

C₄⋊C₄

—

211

—

110

C₄²

—

111

C₂³

C₂×D₈₄₄

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

844

—

117

C₂×C₄

—

844

—

118

C₂×C₄

—

844

—

119

C₂×C₄

—

844

—

120

C₂×C₄

—

844

—

121

C₂×C₄

—

844

—

122

C₂×C₄

—

844

—

123

C₂×C₄

—

844

—

124

C₂×C₄

—

422

—

125

C₂×C₄

D₁₆₈₈

yes

—

126

C₂×C₄

Dic₈₄₄

yes

—

127

C₂×C₄

D₄×D₂₁₁

yes

characteristic

128

C₂×C₄

C₂×Dic₄₂₂

yes

—

129

C₂×C₄

#16809961

yes

—

130

C₂×C₄

#16809963

yes

characteristic

131

C₈

—

132

C₂²

#20269952

yes

characteristic

133

C₂²

#20269961

yes

characteristic

134

C₂²

#20269977

yes

characteristic

135

C₂²

C₂×D₁₆₈₈

yes

characteristic

136

C₂²

C₂×Dic₈₄₄

yes

characteristic

137

C₂²

#20270005

yes

characteristic

138

C₂²

#20270010

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

844

—

142

C₄

—

1688

—

143

C₄

—

1688

—

144

C₄

#20269954

yes

—

145

C₄

#20269968

yes

—

146

C₂

#27213078

yes

characteristic

147

C₂

#27213342

yes

characteristic

148

C₂

#27213357

yes

characteristic

149

C₂

#27213460

yes

characteristic

150

C₂

#27213482

yes

characteristic

151

C₂

#27213551

yes

characteristic

152

C₂

#27213555

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₈₄₄
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#336×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
hypercenter	#336	D₂₁₁
hypocenter	C₂₁₁	#2083
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₈₄₄

commutator, cosocle

C₂²×C₄×C₂₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#336×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

D₂₁₁

C₂₁₁

G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
76. C₂₁₁	#2083	semidirect

H ≅ G/N

type

76. C₂₁₁

#2083

semidirect

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

subgroup	index
C₂×#20269911	2
#27213044	2
#27213016	2
#27212909	2
#336×C₂₁₁	2
#2083	211

subgroup

index

#336×C₂₁₁

211

Subgroups — #41014871

Conjugacy classes of subgroups

Complements

Maximal subgroups