Group #44408451 — subgroups

Subgroups — #44408451

Order 29056.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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×#20300296	C₂	1	2	yes	maximal, complemented
2	#28409582	C₂	1	1	yes	maximal, characteristic
3	#28409102	C₂	1	8	yes	maximal, complemented
4	#28409000	C₂	1	2	yes	maximal, complemented
5	C₂×#20300199	C₂	1	1	yes	maximal, characteristic, complemented
6	#423×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16813884	C₂²	1	2	yes	—
8	#20300296	—	2	8	no	—
9	C₂×#16813867	C₂²	1	4	yes	—
10	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₂₂₇	C₂²	1	4	yes	—
13	#20300288	C₄	1	1	yes	characteristic
14	#20300285	C₄	1	1	yes	characteristic
15	#20300276	C₄	1	2	yes	—
16	#20300262	C₄	1	4	yes	—
17	C₄×D₉₀₈	C₂²	1	4	yes	—
18	(C₂×D₂₂₇)⋊C₈	C₂²	1	8	yes	—
19	#20300199	C₂²	1	4	yes	—
20	C₂×M₁₆×C₂₂₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₂₂₇	C₂²	1	4	yes	—
23	#16813884	—	2	4	no	—
24	#16813884	—	2	4	no	—
25	C₂×Dic₄₅₄	C₂×C₄	1	2	yes	—
26	C₂×D₉₀₈	C₂×C₄	1	2	yes	—
27	C₂×D₉₀₈	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₂₂₇	C₂³	1	4	yes	—
30	#16813867	—	2	8	no	—
31	C₂×C₂₂₇⋊C₈	—	2	2	no	—
32	C₂×C₂₂₇⋊C₈	D₄	1	4	yes	complemented
33	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
34	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
35	C₈×D₂₂₇	—	2	8	no	—
36	C₄⋊Dic₂₂₇	C₂×C₄	1	4	yes	—
37	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	8	yes	—
38	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
40	M₁₆×C₂₂₇	—	2	4	no	—
41	C₂×C₈×C₂₂₇	C₂³	1	4	yes	—
42	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₂₂₇	C₂³	1	2	yes	—
45	C₂²×D₂₂₇	C₂²×C₄	1	4	yes	—
46	Dic₄₅₄	—	2	4	no	—
47	D₉₀₈	—	2	4	no	—
48	D₉₀₈	—	4	4	no	—
49	C₂²⋊D₂₂₇	—	2	8	no	—
50	C₂×Dic₂₂₇	C₂²×C₄	1	4	yes	—
51	C₄×D₂₂₇	—	2	4	no	—
52	C₄×D₂₂₇	—	2	4	no	—
53	C₂₂₇⋊C₈	—	2	2	no	—
54	C₂₂₇⋊C₈	C₂×D₄	1	4	yes	—
55	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₂₇	—	4	2	no	—
57	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₂₇	#41	1	2	yes	—
62	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
63	C₈×C₂₂₇	—	2	4	no	—
64	C₂²×C₂₂₇	—	2	2	no	—
65	C₂²×C₂₂₇	C₄×D₄	1	2	yes	—
66	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₂₂₇	—	2	4	no	—
68	C₂×D₂₂₇	—	2	8	no	—
69	C₄×C₂₂₇	—	2	2	no	—
70	C₄×C₂₂₇	—	2	2	no	—
71	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
72	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
74	Dic₂₂₇	—	2	4	no	—
75	C₂×C₂₂₇	—	2	2	no	—
76	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
77	C₂×C₂₂₇	#568	1	2	yes	—
78	D₂₂₇	—	4	4	no	—
79	C₂₂₇	#2932	1	1	yes	characteristic, complemented, Sylow
80	#2932	—	227	1	no	maximal, Sylow
81	C₂×#131	—	227	2	no	—
82	#532	—	227	1	no	—
83	#435	—	227	8	no	—
84	#423	D₂₂₇	1	1	yes	characteristic, complemented
85	#408	—	227	2	no	—
86	C₂×#97	—	227	1	no	—
87	C₂×#41	—	227	2	no	—
88	#131	—	454	8	no	—
89	C₂×M₁₆	—	227	4	no	—
90	C₂×M₁₆	C₂×D₂₂₇	1	2	yes	—
91	C₂²×C₈	—	227	1	no	—
92	C₂²×C₈	—	227	4	no	—
93	C₂²×C₈	—	227	1	no	—
94	C₄⋊Q₈	—	227	1	no	—
95	#127	—	227	1	no	—
96	#124	—	227	2	no	—
97	#121	—	227	4	no	—
98	C₄×D₄	—	227	4	no	—
99	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
100	C₄⋊C₈	C₂×D₂₂₇	1	4	yes	—
101	C₂²⋊C₈	—	227	8	no	—
102	#97	—	227	4	no	—
103	#41	—	454	4	no	—
104	#41	—	454	4	no	—
105	C₂×Q₈	—	227	2	no	—
106	C₂×D₄	—	227	4	no	—
107	C₂×D₄	—	227	4	no	—
108	C₂×D₄	—	227	2	no	—
109	C₂²×C₄	—	227	4	no	—
110	C₂²×C₄	C₄×D₂₂₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	454	8	no	—
114	C₂×C₈	—	454	8	no	—
115	C₂×C₈	—	454	2	no	—
116	C₂×C₈	—	227	2	no	—
117	C₂×C₈	—	227	4	no	—
118	C₂×C₈	—	227	2	no	—
119	C₂×C₈	C₂²×D₂₂₇	1	4	yes	—
120	C₄⋊C₄	—	227	4	no	—
121	C₂²⋊C₄	—	227	8	no	—
122	C₄²	C₄×D₂₂₇	1	1	yes	characteristic
123	C₄²	C₄×D₂₂₇	1	1	yes	characteristic
124	C₄²	C₂²×D₂₂₇	1	2	yes	—
125	C₂³	—	227	4	no	—
126	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
127	Q₈	—	454	4	no	—
128	D₄	—	908	4	no	—
129	D₄	—	454	8	no	—
130	D₄	—	454	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	454	4	no	—
133	C₂×C₄	—	454	4	no	—
134	C₂×C₄	—	227	4	no	—
135	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
138	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
139	C₂×C₄	#16813885	1	2	yes	—
140	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	454	2	no	—
143	C₈	—	227	4	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	454	4	no	—
146	C₂²	—	454	8	no	—
147	C₂²	#20300250	1	2	yes	—
148	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	454	4	no	—
152	C₄	#20300250	1	2	yes	—
153	C₄	C₂×D₄×D₂₂₇	1	1	yes	characteristic
154	C₄	C₂×#16813885	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	908	4	no	—
157	C₂	C₂×#20300250	1	1	yes	characteristic
158	C₂	#28409798	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300296

C₂

yes

maximal, complemented

#28409582

C₂

yes

maximal, characteristic

#28409102

C₂

yes

maximal, complemented

#28409000

C₂

yes

maximal, complemented

C₂×#20300199

C₂

yes

maximal, characteristic, complemented

#423×C₂₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#16813884

C₂²

yes

—

#20300296

—

C₂×#16813867

C₂²

yes

—

C₂²×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₂²×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₂₂₇

C₂²

yes

—

#20300288

C₄

yes

characteristic

#20300285

C₄

yes

characteristic

#20300276

C₄

yes

—

#20300262

C₄

yes

—

C₄×D₉₀₈

C₂²

yes

—

(C₂×D₂₂₇)⋊C₈

C₂²

yes

—

#20300199

C₂²

yes

—

C₂×M₁₆×C₂₂₇

C₂²

yes

—

C₂×C₄²×C₂₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₇

C₂²

yes

—

#16813884

—

#16813884

—

C₂×Dic₄₅₄

C₂×C₄

yes

—

C₂×D₉₀₈

C₂×C₄

yes

—

C₂×D₉₀₈

C₂×C₄

yes

—

C₂×C₂²⋊D₂₂₇

C₂×C₄

yes

—

C₂×C₄×D₂₂₇

C₂³

yes

—

#16813867

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

D₄

yes

complemented

C₂×C₂₂₇⋊C₈

C₂³

yes

—

C₂×C₂₂₇⋊C₈

C₂³

yes

—

C₈×D₂₂₇

—

C₄⋊Dic₂₂₇

C₂×C₄

yes

—

(C₂×D₂₂₇)⋊C₄

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

M₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

C₂³

yes

—

C₄²×C₂₂₇

C₂×C₄

yes

characteristic

C₄²×C₂₂₇

C₂×C₄

yes

characteristic

C₄²×C₂₂₇

C₂³

yes

—

C₂²×D₂₂₇

C₂²×C₄

yes

—

Dic₄₅₄

—

D₉₀₈

—

D₉₀₈

—

C₂²⋊D₂₂₇

—

C₂×Dic₂₂₇

C₂²×C₄

yes

—

C₄×D₂₂₇

—

C₄×D₂₂₇

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

C₂×D₄

yes

—

C₂³×C₂₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

#41

yes

—

C₂×C₄×C₂₂₇

C₂⁴

yes

characteristic

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₄×D₄

yes

—

C₂²×C₂₂₇

C₂³×C₄

yes

characteristic

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₄×D₄

yes

—

C₄×C₂₂₇

C₂²×D₄

yes

characteristic

C₄×C₂₂₇

C₂×#41

yes

characteristic

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

C₂×C₄×D₄

yes

characteristic

C₂×C₂₂₇

#568

yes

—

D₂₂₇

—

C₂₂₇

#2932

yes

characteristic, complemented, Sylow

#2932

—

227

maximal, Sylow

C₂×#131

—

227

—

#532

—

227

—

#435

—

227

—

#423

D₂₂₇

yes

characteristic, complemented

#408

—

227

—

C₂×#97

—

227

—

C₂×#41

—

227

—

#131

—

454

—

C₂×M₁₆

—

227

—

C₂×M₁₆

C₂×D₂₂₇

yes

—

C₂²×C₈

—

227

—

C₂²×C₈

—

227

—

C₂²×C₈

—

227

—

C₄⋊Q₈

—

227

—

#127

—

227

—

#124

—

227

—

#121

—

227

—

C₄×D₄

—

227

—

C₂×C₄²

C₂×D₂₂₇

yes

characteristic

100

C₄⋊C₈

C₂×D₂₂₇

yes

—

101

C₂²⋊C₈

—

227

—

102

#97

—

227

—

103

#41

—

454

—

104

#41

—

454

—

105

C₂×Q₈

—

227

—

106

C₂×D₄

—

227

—

107

C₂×D₄

—

227

—

108

C₂×D₄

—

227

—

109

C₂²×C₄

—

227

—

110

C₂²×C₄

C₄×D₂₂₇

yes

—

111

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

454

—

114

C₂×C₈

—

454

—

115

C₂×C₈

—

454

—

116

C₂×C₈

—

227

—

117

C₂×C₈

—

227

—

118

C₂×C₈

—

227

—

119

C₂×C₈

C₂²×D₂₂₇

yes

—

120

C₄⋊C₄

—

227

—

121

C₂²⋊C₄

—

227

—

122

C₄²

C₄×D₂₂₇

yes

characteristic

123

C₄²

C₄×D₂₂₇

yes

characteristic

124

C₄²

C₂²×D₂₂₇

yes

—

125

C₂³

—

227

—

126

C₂³

C₂×C₄×D₂₂₇

yes

characteristic

127

Q₈

—

454

—

128

D₄

—

908

—

129

D₄

—

454

—

130

D₄

—

454

—

131

C₂×C₄

—

132

C₂×C₄

—

454

—

133

C₂×C₄

—

454

—

134

C₂×C₄

—

227

—

135

C₂×C₄

C₂×C₄×D₂₂₇

yes

—

136

C₂×C₄

C₂×C₄×D₂₂₇

yes

—

137

C₂×C₄

C₂×C₄×D₂₂₇

yes

characteristic

138

C₂×C₄

D₄×D₂₂₇

yes

—

139

C₂×C₄

#16813885

yes

—

140

C₂×C₄

C₂³×D₂₂₇

yes

characteristic

141

C₈

—

142

C₈

—

454

—

143

C₈

—

227

—

144

C₂²

—

145

C₂²

—

454

—

146

C₂²

—

454

—

147

C₂²

#20300250

yes

—

148

C₂²

C₂²×C₄×D₂₂₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

454

—

152

C₄

#20300250

yes

—

153

C₄

C₂×D₄×D₂₂₇

yes

characteristic

154

C₄

C₂×#16813885

yes

characteristic

155

C₂

—

156

C₂

—

908

—

157

C₂

C₂×#20300250

yes

characteristic

158

C₂

#28409798

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂³×C₄
Fitting	#423×C₂₂₇	C₂
prime radical	#28409582	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#423	D₂₂₇
hypocenter	C₂₂₇	#2932
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₂₇

commutator, socle

C₂²×C₂₂₇

C₂³×C₄

Fitting

#423×C₂₂₇

C₂

prime radical

#28409582

C₂

cosocle

C₂×C₄×C₂₂₇

C₂⁴

D₂₂₇

C₂₂₇

G ▷ C₂²×C₂₂₇ ▷ C₂₂₇

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. C₂×#20300296	C₂	semidirect
3. #28409102	C₂	semidirect
4. #28409000	C₂	semidirect
5. C₂×#20300199	C₂	semidirect
6. #423×C₂₂₇	C₂	semidirect
32. C₂×C₂₂₇⋊C₈	D₄	semidirect
79. C₂₂₇	#2932	semidirect
84. #423	D₂₂₇	semidirect

H ≅ G/N

type

1. C₂×#20300296

C₂

semidirect

3. #28409102

C₂

semidirect

4. #28409000

C₂

semidirect

5. C₂×#20300199

C₂

semidirect

6. #423×C₂₂₇

C₂

semidirect

32. C₂×C₂₂₇⋊C₈

D₄

semidirect

79. C₂₂₇

#2932

semidirect

84. #423

D₂₂₇

semidirect

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

subgroup	index
C₂×#20300296	2
#28409582	2
#28409102	2
#28409000	2
C₂×#20300199	2
#423×C₂₂₇	2
#2932	227

subgroup

index

#423×C₂₂₇

227

Subgroups — #44408451

Conjugacy classes of subgroups

Complements

Maximal subgroups