Group #41014808 — subgroups

Subgroups — #41014808

Order 27008.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20269902	C₂	1	2	yes	maximal
2	#27213052	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	C₂²×Dic₄₂₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
8	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
9	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
11	#20269902	—	2	4	no	—
12	C₂₁₁⋊#95	—	2	1	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	C₂×Dic₄₂₂	—	2	1	no	—
17	C₂×Dic₄₂₂	—	2	2	no	—
18	C₂×Dic₄₂₂	—	2	2	no	—
19	C₂×Dic₄₂₂	D₄	1	2	yes	—
20	C₂²×Dic₂₁₁	—	2	1	no	—
21	C₂²×Dic₂₁₁	—	2	1	no	—
22	C₂²×Dic₂₁₁	—	2	2	no	—
23	C₄⋊Dic₂₁₁	—	2	2	no	—
24	C₄⋊Dic₂₁₁	—	2	2	no	—
25	C₄⋊Dic₂₁₁	—	2	2	no	—
26	C₄⋊Dic₂₁₁	—	4	2	no	—
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	2	no	—
33	C₂×C₈×C₂₁₁	—	2	2	no	—
34	C₄²×C₂₁₁	—	2	2	no	—
35	Dic₄₂₂	—	2	4	no	—
36	Dic₄₂₂	—	4	2	no	—
37	C₂×Dic₂₁₁	—	2	2	no	—
38	C₂×Dic₂₁₁	—	4	1	no	—
39	C₂×Dic₂₁₁	—	4	1	no	—
40	C₂×Dic₂₁₁	—	4	2	no	—
41	C₂×Dic₂₁₁	—	4	1	no	—
42	C₂×Dic₂₁₁	—	4	2	no	—
43	C₂×Dic₂₁₁	—	4	2	no	—
44	C₂×Dic₂₁₁	—	4	1	no	—
45	C₂×Dic₂₁₁	—	4	1	no	—
46	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	—	2	1	no	—
48	C₂×C₄×C₂₁₁	—	2	1	no	—
49	C₂×C₄×C₂₁₁	—	2	1	no	—
50	C₂×C₄×C₂₁₁	—	2	1	no	—
51	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
52	C₂×C₄×C₂₁₁	Q₁₆	1	2	yes	—
53	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₁₁	#41	1	2	yes	—
55	C₂×C₄×C₂₁₁	#41	1	2	yes	—
56	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
57	C₈×C₂₁₁	—	4	2	no	—
58	C₂²×C₂₁₁	#121	1	1	yes	characteristic
59	C₂²×C₂₁₁	#123	1	1	yes	characteristic
60	C₂²×C₂₁₁	#124	1	1	yes	characteristic
61	C₂²×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₂₁₁	#137	1	1	yes	characteristic
65	C₄×C₂₁₁	—	2	2	no	—
66	C₄×C₂₁₁	—	2	2	no	—
67	C₄×C₂₁₁	#123	1	2	yes	—
68	C₄×C₂₁₁	#126	1	2	yes	—
69	Dic₂₁₁	—	4	2	no	—
70	Dic₂₁₁	—	8	2	no	—
71	Dic₂₁₁	—	8	1	no	—
72	C₂×C₂₁₁	#397	1	1	yes	characteristic
73	C₂×C₂₁₁	#460	1	1	yes	characteristic
74	C₂×C₂₁₁	#462	1	1	yes	characteristic
75	C₂×C₂₁₁	#483	1	1	yes	characteristic
76	C₂×C₂₁₁	#484	1	1	yes	characteristic
77	C₂×C₂₁₁	#487	1	1	yes	characteristic
78	C₂×C₂₁₁	#489	1	1	yes	characteristic
79	C₂₁₁	#2072	1	1	yes	characteristic, complemented, Sylow
80	#2072	—	211	1	no	maximal, Sylow
81	C₂×#103	—	211	2	no	—
82	#391	—	211	1	no	—
83	#384	—	211	1	no	—
84	#340	—	211	2	no	—
85	#336	D₂₁₁	1	1	yes	characteristic
86	C₂²×Q₈	—	211	1	no	—
87	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
88	C₂×C₄⋊C₄	—	422	1	no	—
89	C₂×C₄⋊C₄	—	422	1	no	—
90	C₂×C₄⋊C₄	—	211	1	no	—
91	C₂×C₄⋊C₄	—	211	2	no	—
92	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
93	#103	—	422	4	no	—
94	#95	—	422	1	no	—
95	#95	—	422	1	no	—
96	C₂×Q₈	—	422	2	no	—
97	C₂×Q₈	—	422	2	no	—
98	C₂×Q₈	—	422	1	no	—
99	C₂×Q₈	—	211	2	no	—
100	C₂²×C₄	—	422	1	no	—
101	C₂²×C₄	—	422	2	no	—
102	C₂²×C₄	—	422	1	no	—
103	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
104	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₄⋊C₄	—	844	2	no	—
110	C₄⋊C₄	—	844	2	no	—
111	C₄⋊C₄	—	422	2	no	—
112	C₄⋊C₄	—	422	2	no	—
113	C₄⋊C₄	—	422	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
116	Q₈	—	844	2	no	—
117	Q₈	—	422	4	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	844	1	no	—
123	C₂×C₄	—	844	1	no	—
124	C₂×C₄	—	844	2	no	—
125	C₂×C₄	—	844	1	no	—
126	C₂×C₄	—	844	2	no	—
127	C₂×C₄	—	844	2	no	—
128	C₂×C₄	—	844	1	no	—
129	C₂×C₄	—	844	1	no	—
130	C₂×C₄	—	422	2	no	—
131	C₂×C₄	#16809951	1	2	yes	—
132	C₂×C₄	Dic₈₄₄	1	2	yes	—
133	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
134	C₂×C₄	#16809961	1	2	yes	—
135	C₂×C₄	#16809962	1	2	yes	—
136	C₂×C₄	#16809963	1	1	yes	characteristic
137	C₈	—	4	2	no	—
138	C₂²	#20269952	1	1	yes	characteristic
139	C₂²	#20269961	1	1	yes	characteristic
140	C₂²	#20269965	1	1	yes	characteristic
141	C₂²	C₂×#16809951	1	1	yes	characteristic
142	C₂²	C₂×Dic₈₄₄	1	1	yes	characteristic
143	C₂²	#20270005	1	1	yes	characteristic
144	C₂²	#20270010	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	844	2	no	—
148	C₄	—	1688	2	no	—
149	C₄	—	1688	1	no	—
150	C₄	#20269959	1	2	yes	—
151	C₄	#20269972	1	2	yes	—
152	C₂	#27213073	1	1	yes	characteristic
153	C₂	#27213347	1	1	yes	characteristic
154	C₂	#27213357	1	1	yes	characteristic
155	C₂	#27213477	1	1	yes	characteristic
156	C₂	#27213482	1	1	yes	characteristic
157	C₂	#27213497	1	1	yes	characteristic
158	C₂	#27213506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269902

C₂

yes

maximal

#27213052

C₂

yes

maximal, characteristic

#27213016

C₂

yes

maximal, characteristic

#27212909

C₂

yes

maximal

#336×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

characteristic

#20269902

—

C₂₁₁⋊#95

—

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

D₄

yes

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

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

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

—

Dic₄₂₂

—

Dic₄₂₂

—

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

SD₁₆

yes

—

C₂×C₄×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×Q₁₆

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#123

yes

—

C₄×C₂₁₁

#126

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

#2072

—

211

maximal, Sylow

C₂×#103

—

211

—

#391

—

211

—

#384

—

211

—

#340

—

211

—

#336

D₂₁₁

yes

characteristic

C₂²×Q₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#103

—

422

—

#95

—

422

—

#95

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

211

—

100

C₂²×C₄

—

422

—

101

C₂²×C₄

—

422

—

102

C₂²×C₄

—

422

—

103

C₂²×C₄

D₈₄₄

yes

characteristic

104

C₂²×C₄

D₈₄₄

yes

characteristic

105

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

844

—

110

C₄⋊C₄

—

844

—

111

C₄⋊C₄

—

422

—

112

C₄⋊C₄

—

422

—

113

C₄⋊C₄

—

422

—

114

C₄²

—

115

C₂³

C₂×D₈₄₄

yes

characteristic

116

Q₈

—

844

—

117

Q₈

—

422

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

844

—

123

C₂×C₄

—

844

—

124

C₂×C₄

—

844

—

125

C₂×C₄

—

844

—

126

C₂×C₄

—

844

—

127

C₂×C₄

—

844

—

128

C₂×C₄

—

844

—

129

C₂×C₄

—

844

—

130

C₂×C₄

—

422

—

131

C₂×C₄

#16809951

yes

—

132

C₂×C₄

Dic₈₄₄

yes

—

133

C₂×C₄

D₄×D₂₁₁

yes

characteristic

134

C₂×C₄

#16809961

yes

—

135

C₂×C₄

#16809962

yes

—

136

C₂×C₄

#16809963

yes

characteristic

137

C₈

—

138

C₂²

#20269952

yes

characteristic

139

C₂²

#20269961

yes

characteristic

140

C₂²

#20269965

yes

characteristic

141

C₂²

C₂×#16809951

yes

characteristic

142

C₂²

C₂×Dic₈₄₄

yes

characteristic

143

C₂²

#20270005

yes

characteristic

144

C₂²

#20270010

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

844

—

148

C₄

—

1688

—

149

C₄

—

1688

—

150

C₄

#20269959

yes

—

151

C₄

#20269972

yes

—

152

C₂

#27213073

yes

characteristic

153

C₂

#27213347

yes

characteristic

154

C₂

#27213357

yes

characteristic

155

C₂

#27213477

yes

characteristic

156

C₂

#27213482

yes

characteristic

157

C₂

#27213497

yes

characteristic

158

C₂

#27213506

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	#2072
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
79. C₂₁₁	#2072	semidirect

H ≅ G/N

type

79. C₂₁₁

#2072

semidirect

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

subgroup	index
C₂×#20269902	2
#27213052	2
#27213016	2
#27212909	2
#336×C₂₁₁	2
#2072	211

subgroup

index

#336×C₂₁₁

211

Subgroups — #41014808

Conjugacy classes of subgroups

Complements

Maximal subgroups