Group #41014874 — subgroups

Subgroups — #41014874

Order 27008.

230 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20269908	C₂	1	2	yes	maximal
2	#27213016	C₂	1	1	yes	maximal, characteristic
3	#27212908	C₂	1	2	yes	maximal
4	#27212901	C₂	1	1	yes	maximal, characteristic
5	#389×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
8	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
9	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
10	#20269908	—	2	4	no	—
11	C₂×C₄⋊C₄×C₂₁₁	—	2	1	no	—
12	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	#95×C₂₁₁	—	2	1	no	—
16	C₂²×Dic₂₁₁	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	—	2	2	no	—
19	C₂×C₂₁₁⋊C₈	Q₈	1	4	yes	—
20	C₄⋊Dic₂₁₁	—	2	2	no	—
21	C₄⋊Dic₂₁₁	—	2	2	no	—
22	C₄⋊Dic₂₁₁	—	4	2	no	—
23	C₄⋊Dic₂₁₁	—	4	2	no	—
24	C₂²×C₄×C₂₁₁	—	2	1	no	—
25	C₂²×C₄×C₂₁₁	—	2	1	no	—
26	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₁₁	—	2	1	no	—
30	C₄⋊C₄×C₂₁₁	—	2	2	no	—
31	C₄⋊C₄×C₂₁₁	—	4	2	no	—
32	C₄⋊C₄×C₂₁₁	D₄	1	2	yes	—
33	C₄²×C₂₁₁	—	2	2	no	—
34	C₂×Dic₂₁₁	—	4	2	no	—
35	C₂×Dic₂₁₁	—	4	2	no	—
36	C₂×Dic₂₁₁	—	4	2	no	—
37	C₂₁₁⋊C₈	—	2	4	no	—
38	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	—	2	1	no	—
40	C₂×C₄×C₂₁₁	—	2	1	no	—
41	C₂×C₄×C₂₁₁	—	2	1	no	—
42	C₂×C₄×C₂₁₁	—	2	1	no	—
43	C₂×C₄×C₂₁₁	—	2	2	no	—
44	C₂×C₄×C₂₁₁	—	4	1	no	—
45	C₂×C₄×C₂₁₁	—	4	1	no	—
46	C₂×C₄×C₂₁₁	—	4	1	no	—
47	C₂×C₄×C₂₁₁	—	4	1	no	—
48	C₂×C₄×C₂₁₁	—	4	1	no	—
49	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
50	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
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	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
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×SD₁₆	1	1	yes	characteristic
59	C₂²×C₂₁₁	C₂×SD₁₆	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	—	4	2	no	—
65	C₄×C₂₁₁	—	8	1	no	—
66	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₂₁₁	#125	1	2	yes	—
68	Dic₂₁₁	—	8	2	no	—
69	C₂×C₂₁₁	#398	1	1	yes	characteristic
70	C₂×C₂₁₁	#460	1	1	yes	characteristic
71	C₂×C₂₁₁	#461	1	1	yes	characteristic
72	C₂×C₂₁₁	#481	1	1	yes	characteristic
73	C₂×C₂₁₁	#483	1	1	yes	characteristic
74	C₂×C₂₁₁	#498	1	1	yes	characteristic
75	C₂×C₂₁₁	#501	1	1	yes	characteristic
76	C₂₁₁	#2084	1	1	yes	characteristic, complemented, Sylow
77	#2084	—	211	1	no	maximal, Sylow
78	C₂×#106	—	211	2	no	—
79	#389	D₂₁₁	1	1	yes	characteristic
80	#384	—	211	1	no	—
81	#340	—	211	2	no	—
82	#336	—	211	1	no	—
83	C₂²×C₈	—	211	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	422	1	no	—
86	C₂×C₄⋊C₄	—	422	1	no	—
87	C₂×C₄⋊C₄	—	211	2	no	—
88	C₂×C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
91	#106	—	422	4	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	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	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	844	2	no	—
106	C₄⋊C₄	—	844	2	no	—
107	C₄⋊C₄	—	422	2	no	—
108	C₄⋊C₄	—	422	2	no	—
109	C₄⋊C₄	D₈₄₄	1	2	yes	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₂₁₁	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	—	2	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	844	2	no	—
123	C₂×C₄	—	844	2	no	—
124	C₂×C₄	—	844	2	no	—
125	C₂×C₄	#16809949	1	2	yes	—
126	C₂×C₄	#16809950	1	2	yes	—
127	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₂₁₁	1	2	yes	—
129	C₂×C₄	#16809961	1	2	yes	—
130	C₂×C₄	#16809962	1	1	yes	characteristic
131	C₈	—	422	4	no	—
132	C₂²	#20269951	1	1	yes	characteristic
133	C₂²	#20269958	1	1	yes	characteristic
134	C₂²	#20269975	1	1	yes	characteristic
135	C₂²	C₂×#16809949	1	1	yes	characteristic
136	C₂²	C₂×#16809950	1	1	yes	characteristic
137	C₂²	#20270002	1	1	yes	characteristic
138	C₂²	#20270007	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	8	1	no	—
143	C₄	—	1688	2	no	—
144	C₄	#20269957	1	2	yes	—
145	C₄	#20269967	1	2	yes	—
146	C₂	#27213076	1	1	yes	characteristic
147	C₂	#27213346	1	1	yes	characteristic
148	C₂	#27213351	1	1	yes	characteristic
149	C₂	#27213462	1	1	yes	characteristic
150	C₂	#27213474	1	1	yes	characteristic
151	C₂	#27213542	1	1	yes	characteristic
152	C₂	#27213556	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269908

C₂

yes

maximal

#27213016

C₂

yes

maximal, characteristic

#27212908

C₂

yes

maximal

#27212901

C₂

yes

maximal, characteristic

#389×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

#20269908

—

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

characteristic

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#95×C₂₁₁

—

C₂²×Dic₂₁₁

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

Q₈

yes

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂²×C₄×C₂₁₁

—

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

—

C₄⋊C₄×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

—

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

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×SD₁₆

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

#125

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

#2084

—

211

maximal, Sylow

C₂×#106

—

211

—

#389

D₂₁₁

yes

characteristic

#384

—

211

—

#340

—

211

—

#336

—

211

—

C₂²×C₈

—

211

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

characteristic

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#106

—

422

—

#95

—

C₂²×C₄

—

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

—

844

—

106

C₄⋊C₄

—

844

—

107

C₄⋊C₄

—

422

—

108

C₄⋊C₄

—

422

—

109

C₄⋊C₄

D₈₄₄

yes

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

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

#16809949

yes

—

126

C₂×C₄

#16809950

yes

—

127

C₂×C₄

C₂×D₈₄₄

yes

characteristic

128

C₂×C₄

Q₈×D₂₁₁

yes

—

129

C₂×C₄

#16809961

yes

—

130

C₂×C₄

#16809962

yes

characteristic

131

C₈

—

422

—

132

C₂²

#20269951

yes

characteristic

133

C₂²

#20269958

yes

characteristic

134

C₂²

#20269975

yes

characteristic

135

C₂²

C₂×#16809949

yes

characteristic

136

C₂²

C₂×#16809950

yes

characteristic

137

C₂²

#20270002

yes

characteristic

138

C₂²

#20270007

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

1688

—

144

C₄

#20269957

yes

—

145

C₄

#20269967

yes

—

146

C₂

#27213076

yes

characteristic

147

C₂

#27213346

yes

characteristic

148

C₂

#27213351

yes

characteristic

149

C₂

#27213462

yes

characteristic

150

C₂

#27213474

yes

characteristic

151

C₂

#27213542

yes

characteristic

152

C₂

#27213556

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₁₁
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#389×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
hypercenter	#389	D₂₁₁
hypocenter	C₂₁₁	#2084
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₁₁

commutator, cosocle

C₂²×C₄×C₂₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#389×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

D₂₁₁

C₂₁₁

G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
76. C₂₁₁	#2084	semidirect

H ≅ G/N

type

76. C₂₁₁

#2084

semidirect

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

subgroup	index
C₂×#20269908	2
#27213016	2
#27212908	2
#27212901	2
#389×C₂₁₁	2
#2084	211

subgroup

index

#389×C₂₁₁

211

Subgroups — #41014874

Conjugacy classes of subgroups

Complements

Maximal subgroups