Group #41013231 — subgroups

Subgroups — #41013231

Order 27008.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₄₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269900	C₂	1	2	yes	maximal
3	C₂×#20269888	C₂	1	1	yes	maximal, characteristic
4	#27212908	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₂	C₂²	1	1	yes	characteristic
7	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
8	C₄×Dic₄₂₂	—	2	2	no	—
9	C₄×Dic₄₂₂	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₁₁⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₂₁₁	—	2	1	no	—
13	#20269900	C₄	1	8	yes	complemented
14	#20269888	—	2	2	no	—
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₁₁	—	2	1	no	—
17	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
18	#95×C₂₁₁	—	2	1	no	—
19	C₂×Dic₄₂₂	—	2	1	no	—
20	C₂×Dic₄₂₂	—	2	2	no	—
21	C₂×Dic₄₂₂	C₂×C₄	1	4	yes	—
22	C₂×Dic₄₂₂	C₂×C₄	1	2	yes	—
23	C₂²×Dic₂₁₁	—	2	1	no	—
24	C₂²×Dic₂₁₁	—	2	1	no	—
25	C₂×C₂₁₁⋊C₈	—	2	2	no	—
26	C₂×C₂₁₁⋊C₈	—	2	2	no	—
27	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
28	C₄⋊Dic₂₁₁	—	2	1	no	—
29	C₄⋊Dic₂₁₁	C₂×C₄	1	2	yes	—
30	Dic₂₁₁⋊C₄	—	2	4	no	—
31	C₄×Dic₂₁₁	—	2	4	no	—
32	C₂²×C₄×C₂₁₁	—	2	2	no	—
33	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₁₁	—	2	2	no	—
37	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
38	C₄²×C₂₁₁	—	4	2	no	—
39	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
40	C₄²×C₂₁₁	D₄	1	2	yes	—
41	Dic₄₂₂	—	2	2	no	—
42	Dic₄₂₂	—	2	4	no	—
43	Dic₄₂₂	C₄²	1	4	yes	complemented
44	C₂×Dic₂₁₁	—	2	2	no	—
45	C₂×Dic₂₁₁	—	2	4	no	—
46	C₂×Dic₂₁₁	—	2	2	no	—
47	C₂×Dic₂₁₁	—	4	1	no	—
48	C₂×Dic₂₁₁	—	4	1	no	—
49	C₂₁₁⋊C₈	—	2	4	no	—
50	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₁₁	—	2	1	no	—
52	C₂×C₄×C₂₁₁	—	2	2	no	—
53	C₂×C₄×C₂₁₁	—	2	1	no	—
54	C₂×C₄×C₂₁₁	—	2	4	no	—
55	C₂×C₄×C₂₁₁	—	4	2	no	—
56	C₂×C₄×C₂₁₁	—	4	2	no	—
57	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₁₁	#41	1	2	yes	—
63	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₂₁₁	C₄×D₄	1	2	yes	—
65	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₂₁₁	#137	1	2	yes	—
67	C₂²×C₂₁₁	#137	1	1	yes	characteristic
68	C₄×C₂₁₁	—	2	2	no	—
69	C₄×C₂₁₁	—	2	2	no	—
70	C₄×C₂₁₁	—	4	4	no	—
71	C₄×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₂₁₁	#117	1	1	yes	characteristic
73	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
74	Dic₂₁₁	—	2	4	no	—
75	Dic₂₁₁	—	4	2	no	—
76	C₂×C₂₁₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₂₁₁	#417	1	1	yes	characteristic
78	C₂×C₂₁₁	#419	1	1	yes	characteristic
79	C₂×C₂₁₁	#440	1	2	yes	—
80	C₂×C₂₁₁	#441	1	2	yes	—
81	C₂₁₁	#1763	1	1	yes	characteristic, complemented, Sylow
82	#1763	—	211	1	no	maximal, Sylow
83	C₂×C₄×Q₈	—	211	1	no	—
84	C₂×#103	—	211	2	no	—
85	C₂×#97	—	211	1	no	—
86	C₄×C₄⋊C₄	D₂₁₁	1	1	yes	characteristic
87	#340	—	211	2	no	—
88	C₂²×Q₈	—	211	1	no	—
89	C₂²×C₈	—	211	2	no	—
90	C₄×Q₈	—	422	2	no	—
91	C₄×Q₈	—	211	4	no	—
92	C₂×C₄⋊C₄	—	422	1	no	—
93	C₂×C₄⋊C₄	—	211	1	no	—
94	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
95	C₂×C₄²	—	2	1	no	—
96	C₂×C₄²	—	422	1	no	—
97	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
98	#103	—	211	8	no	—
99	#97	—	422	2	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	422	1	no	—
102	C₂×Q₈	—	422	2	no	—
103	C₂×Q₈	—	211	4	no	—
104	C₂×Q₈	—	211	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	422	1	no	—
107	C₂²×C₄	—	422	1	no	—
108	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
111	C₂×C₈	—	422	2	no	—
112	C₂×C₈	—	422	2	no	—
113	C₂×C₈	—	211	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	422	4	no	—
116	C₄⋊C₄	—	422	1	no	—
117	C₄⋊C₄	—	211	2	no	—
118	C₄⋊C₄	C₄×D₂₁₁	1	4	yes	—
119	C₄²	—	4	2	no	—
120	C₄²	—	422	4	no	—
121	C₄²	C₄×D₂₁₁	1	2	yes	—
122	C₄²	D₈₄₄	1	2	yes	—
123	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
124	Q₈	—	422	2	no	—
125	Q₈	—	422	4	no	—
126	Q₈	—	211	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	844	1	no	—
134	C₂×C₄	—	844	1	no	—
135	C₂×C₄	—	422	2	no	—
136	C₂×C₄	—	422	2	no	—
137	C₂×C₄	—	422	4	no	—
138	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₂₁₁	1	2	yes	—
142	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
143	C₂×C₄	#16809962	1	2	yes	—
144	C₈	—	422	4	no	—
145	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
146	C₂²	C₄×C₂²⋊D₂₁₁	1	2	yes	—
147	C₂²	#20270003	1	1	yes	characteristic
148	C₂²	#20270007	1	2	yes	—
149	C₂²	#20270008	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	844	2	no	—
154	C₄	—	422	4	no	—
155	C₄	C₄²×D₂₁₁	1	1	yes	characteristic
156	C₄	#20269934	1	1	yes	characteristic
157	C₄	C₄×D₈₄₄	1	2	yes	—
158	C₂	C₄×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
159	C₂	#27213145	1	1	yes	characteristic
160	C₂	#27213154	1	1	yes	characteristic
161	C₂	#27213246	1	2	yes	—
162	C₂	#27213251	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₄₂₂

C₂

yes

maximal, characteristic

C₂×#20269900

C₂

yes

maximal

C₂×#20269888

C₂

yes

maximal, characteristic

#27212908

C₂

yes

maximal

C₄×C₄⋊C₄×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₂²

yes

characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₄×Dic₄₂₂

—

C₄×Dic₄₂₂

C₄

yes

complemented

C₂×C₄⋊Dic₂₁₁

C₂²

yes

characteristic

C₂×Dic₂₁₁⋊C₄

—

C₂×C₄×Dic₂₁₁

—

#20269900

C₄

yes

complemented

#20269888

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#95×C₂₁₁

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

C₂×C₄

yes

—

C₂×Dic₄₂₂

C₂×C₄

yes

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

C₂×C₄

yes

—

Dic₂₁₁⋊C₄

—

C₄×Dic₂₁₁

—

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

C₂×C₄

yes

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

C₄²

yes

complemented

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

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

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₄×D₄

yes

—

C₂²×C₂₁₁

AGL(1,ℤ/8)

yes

characteristic

C₂²×C₂₁₁

#137

yes

—

C₂²×C₂₁₁

#137

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₄²

yes

characteristic

C₄×C₂₁₁

#117

yes

characteristic

C₄×C₂₁₁

C₄×D₄

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

C₄×C₂²⋊C₄

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂×C₂₁₁

#441

yes

—

C₂₁₁

#1763

yes

characteristic, complemented, Sylow

#1763

—

211

maximal, Sylow

C₂×C₄×Q₈

—

211

—

C₂×#103

—

211

—

C₂×#97

—

211

—

C₄×C₄⋊C₄

D₂₁₁

yes

characteristic

#340

—

211

—

C₂²×Q₈

—

211

—

C₂²×C₈

—

211

—

C₄×Q₈

—

422

—

C₄×Q₈

—

211

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

422

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#103

—

211

—

#97

—

422

—

100

#95

—

101

C₂×Q₈

—

422

—

102

C₂×Q₈

—

422

—

103

C₂×Q₈

—

211

—

104

C₂×Q₈

—

211

—

105

C₂²×C₄

—

106

C₂²×C₄

—

422

—

107

C₂²×C₄

—

422

—

108

C₂²×C₄

C₂²⋊D₂₁₁

yes

characteristic

109

C₂²×C₄

C₂²⋊D₂₁₁

yes

characteristic

110

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

111

C₂×C₈

—

422

—

112

C₂×C₈

—

422

—

113

C₂×C₈

—

211

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

422

—

116

C₄⋊C₄

—

422

—

117

C₄⋊C₄

—

211

—

118

C₄⋊C₄

C₄×D₂₁₁

yes

—

119

C₄²

—

120

C₄²

—

422

—

121

C₄²

C₄×D₂₁₁

yes

—

122

C₄²

D₈₄₄

yes

—

123

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

124

Q₈

—

422

—

125

Q₈

—

422

—

126

Q₈

—

211

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

844

—

134

C₂×C₄

—

844

—

135

C₂×C₄

—

422

—

136

C₂×C₄

—

422

—

137

C₂×C₄

—

422

—

138

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

139

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

140

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

141

C₂×C₄

C₂×C₄×D₂₁₁

yes

—

142

C₂×C₄

C₂×D₈₄₄

yes

characteristic

143

C₂×C₄

#16809962

yes

—

144

C₈

—

422

—

145

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

146

C₂²

C₄×C₂²⋊D₂₁₁

yes

—

147

C₂²

#20270003

yes

characteristic

148

C₂²

#20270007

yes

—

149

C₂²

#20270008

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

844

—

154

C₄

—

422

—

155

C₄

C₄²×D₂₁₁

yes

characteristic

156

C₄

#20269934

yes

characteristic

157

C₄

C₄×D₈₄₄

yes

—

158

C₂

C₄×(C₂×D₂₁₁)⋊C₄

yes

characteristic

159

C₂

#27213145

yes

characteristic

160

C₂

#27213154

yes

characteristic

161

C₂

#27213246

yes

—

162

C₂

#27213251

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₁₁
commutator	C₄×C₂₁₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₄×C₄⋊C₄×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₁₁
hypocenter	C₂₁₁	#1763
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₁₁

commutator

C₄×C₂₁₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₄×C₄⋊C₄×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₄×C₄⋊C₄

D₂₁₁

hypocenter

C₂₁₁

#1763

LCS

G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
9. C₄×Dic₄₂₂	C₄	semidirect
13. #20269900	C₄	semidirect
43. Dic₄₂₂	C₄²	semidirect
81. C₂₁₁	#1763	semidirect

H ≅ G/N

type

9. C₄×Dic₄₂₂

C₄

semidirect

13. #20269900

C₄

semidirect

43. Dic₄₂₂

C₄²

semidirect

81. C₂₁₁

#1763

semidirect

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

subgroup	index
C₂×C₄×Dic₄₂₂	2
C₂×#20269900	2
C₂×#20269888	2
#27212908	2
C₄×C₄⋊C₄×C₂₁₁	2
#1763	211

subgroup

index

C₂×C₄×Dic₄₂₂

C₄×C₄⋊C₄×C₂₁₁

211

Subgroups — #41013231

Conjugacy classes of subgroups

Complements

Maximal subgroups