Group #44408801 — subgroups

Subgroups — #44408801

Order 29056.

410 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 205 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	#28409110	C₂	1	4	yes	maximal, complemented
2	D₄×C₂₂₇⋊C₈	C₂	1	4	yes	maximal, complemented
3	#28409085	C₂	1	2	yes	maximal, complemented
4	#28408999	C₂	1	4	yes	maximal, complemented
5	#532×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#16813868	C₂²	1	4	yes	—
7	C₂²×C₂₂₇⋊C₈	C₂²	1	4	yes	—
8	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
9	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
10	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	8	yes	—
11	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	4	yes	—
12	C₄×C₂₂₇⋊C₈	C₂²	1	4	yes	—
13	C₂×#41×C₂₂₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₂₂₇	C₄	1	1	yes	characteristic
15	#127×C₂₂₇	C₄	1	1	yes	characteristic
16	#124×C₂₂₇	C₄	1	2	yes	—
17	#121×C₂₂₇	C₄	1	4	yes	—
18	C₄×D₄×C₂₂₇	C₂²	1	4	yes	—
19	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
20	#16813868	—	4	4	no	—
21	C₂×C₂₂₇⋊C₈	—	4	4	no	—
22	C₂×C₂₂₇⋊C₈	C₂³	1	4	yes	—
23	C₂×C₂₂₇⋊C₈	C₂³	1	4	yes	—
24	#41×C₂₂₇	—	4	2	no	—
25	#41×C₂₂₇	—	4	2	no	—
26	C₂×Q₈×C₂₂₇	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₂₂₇	C₂×C₄	1	4	yes	—
28	C₂×D₄×C₂₂₇	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₂₂₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₂₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	—
34	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	8	yes	—
35	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₂₂₇	C₂³	1	2	yes	—
38	C₂₂₇⋊C₈	—	2	4	no	—
39	C₂₂₇⋊C₈	—	2	4	no	—
40	C₂³×C₂₂₇	C₂²×C₄	1	4	yes	—
41	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₂₂₇	—	4	2	no	—
43	D₄×C₂₂₇	—	2	8	no	—
44	D₄×C₂₂₇	—	4	2	no	—
45	D₄×C₂₂₇	—	4	4	no	—
46	C₂×C₄×C₂₂₇	—	2	4	no	—
47	C₂×C₄×C₂₂₇	—	2	1	no	—
48	C₂×C₄×C₂₂₇	—	2	1	no	—
49	C₂×C₄×C₂₂₇	—	2	4	no	—
50	C₂×C₄×C₂₂₇	—	2	4	no	—
51	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
56	C₂²×C₂₂₇	—	2	1	no	—
57	C₂²×C₂₂₇	—	2	4	no	—
58	C₂²×C₂₂₇	—	2	2	no	—
59	C₂²×C₂₂₇	—	2	8	no	—
60	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₂₂₇	—	2	1	no	—
62	C₄×C₂₂₇	—	2	4	no	—
63	C₄×C₂₂₇	—	2	2	no	—
64	C₄×C₂₂₇	#131	1	4	yes	—
65	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
67	C₂×C₂₂₇	—	4	1	no	—
68	C₂×C₂₂₇	—	4	4	no	—
69	C₂×C₂₂₇	#518	1	1	yes	characteristic
70	C₂×C₂₂₇	C₂×#131	1	2	yes	—
71	C₂₂₇	#2979	1	1	yes	characteristic, complemented, Sylow
72	#2979	—	227	1	no	maximal, Sylow
73	#532	D₂₂₇	1	1	yes	characteristic
74	#436	—	227	4	no	—
75	C₈×D₄	—	227	4	no	—
76	#431	—	227	2	no	—
77	#408	—	227	4	no	—
78	C₂×#41	C₂×D₂₂₇	1	2	yes	—
79	C₂×M₁₆	—	227	4	no	—
80	C₂²×C₈	—	227	4	no	—
81	C₄⋊Q₈	Dic₂₂₇	1	1	yes	characteristic
82	#127	Dic₂₂₇	1	1	yes	characteristic
83	#124	Dic₂₂₇	1	2	yes	—
84	#121	Dic₂₂₇	1	4	yes	—
85	C₄×D₄	C₂×D₂₂₇	1	4	yes	—
86	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
87	C₄⋊C₈	—	227	2	no	—
88	C₄⋊C₈	—	227	2	no	—
89	C₂²⋊C₈	—	227	4	no	—
90	C₂²⋊C₈	—	227	8	no	—
91	C₄×C₈	—	227	4	no	—
92	#41	—	4	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	C₂×Dic₂₂₇	1	2	yes	—
95	C₂×D₄	C₂×Dic₂₂₇	1	4	yes	—
96	C₂×D₄	C₂×Dic₂₂₇	1	4	yes	—
97	C₂×D₄	C₂×Dic₂₂₇	1	2	yes	—
98	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₂₂₇	1	4	yes	—
101	M₁₆	—	908	4	no	—
102	C₂×C₈	—	908	4	no	—
103	C₂×C₈	—	227	4	no	—
104	C₂×C₈	—	227	4	no	—
105	C₄⋊C₄	C₂×Dic₂₂₇	1	4	yes	—
106	C₂²⋊C₄	C₂×Dic₂₂₇	1	8	yes	—
107	C₄²	C₂×Dic₂₂₇	1	1	yes	characteristic
108	C₄²	C₂×Dic₂₂₇	1	1	yes	characteristic
109	C₄²	C₂²×D₂₂₇	1	2	yes	—
110	C₂³	C₂²×Dic₂₂₇	1	4	yes	—
111	C₂³	C₂²×Dic₂₂₇	1	1	yes	characteristic
112	Q₈	—	4	2	no	—
113	D₄	—	2	8	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
123	C₂×C₄	C₂²×Dic₂₂₇	1	4	yes	—
124	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
125	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
126	C₈	—	454	4	no	—
127	C₈	—	454	4	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	8	no	—
132	C₂²	C₂³×Dic₂₂₇	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	#20300295	1	4	yes	—
137	C₄	#20300333	1	1	yes	characteristic
138	C₄	#20300333	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	4	4	no	—
141	C₂	#28409507	1	1	yes	characteristic
142	C₂	C₂×#20300295	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#28409110

C₂

yes

maximal, complemented

D₄×C₂₂₇⋊C₈

C₂

yes

maximal, complemented

#28409085

C₂

yes

maximal, complemented

#28408999

C₂

yes

maximal, complemented

#532×C₂₂₇

C₂

yes

maximal, characteristic

C₂×#16813868

C₂²

yes

—

C₂²×C₂₂₇⋊C₈

C₂²

yes

—

C₄⋊(C₂₂₇⋊C₈)

C₂²

yes

—

C₄⋊(C₂₂₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₂₂₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₂₂₇⋊C₈)

C₂²

yes

—

C₄×C₂₂₇⋊C₈

C₂²

yes

—

C₂×#41×C₂₂₇

C₂²

yes

—

C₄⋊Q₈×C₂₂₇

C₄

yes

characteristic

#127×C₂₂₇

C₄

yes

characteristic

#124×C₂₂₇

C₄

yes

—

#121×C₂₂₇

C₄

yes

—

C₄×D₄×C₂₂₇

C₂²

yes

—

C₂×C₄²×C₂₂₇

C₂²

yes

characteristic

#16813868

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

C₂³

yes

—

C₂×C₂₂₇⋊C₈

C₂³

yes

—

#41×C₂₂₇

—

#41×C₂₂₇

—

C₂×Q₈×C₂₂₇

C₂×C₄

yes

—

C₂×D₄×C₂₂₇

C₂×C₄

yes

—

C₂×D₄×C₂₂₇

C₂×C₄

yes

—

C₂×D₄×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₂₇

C₂×C₄

yes

—

C₂²⋊C₄×C₂₂₇

C₂×C₄

yes

—

C₄²×C₂₂₇

C₂×C₄

yes

characteristic

C₄²×C₂₂₇

C₂×C₄

yes

characteristic

C₄²×C₂₂₇

C₂³

yes

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

C₂²×C₄

yes

—

C₂³×C₂₂₇

C₂²×C₄

yes

characteristic

Q₈×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂⁴

yes

characteristic

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂³×C₄

yes

characteristic

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

#131

yes

—

C₄×C₂₂₇

yes

C₄×C₂₂₇

yes

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#518

yes

characteristic

C₂×C₂₂₇

C₂×#131

yes

—

C₂₂₇

#2979

yes

characteristic, complemented, Sylow

#2979

—

227

maximal, Sylow

#532

D₂₂₇

yes

characteristic

#436

—

227

—

C₈×D₄

—

227

—

#431

—

227

—

#408

—

227

—

C₂×#41

C₂×D₂₂₇

yes

—

C₂×M₁₆

—

227

—

C₂²×C₈

—

227

—

C₄⋊Q₈

Dic₂₂₇

yes

characteristic

#127

Dic₂₂₇

yes

characteristic

#124

Dic₂₂₇

yes

—

#121

Dic₂₂₇

yes

—

C₄×D₄

C₂×D₂₂₇

yes

—

C₂×C₄²

C₂×D₂₂₇

yes

characteristic

C₄⋊C₈

—

227

—

C₄⋊C₈

—

227

—

C₂²⋊C₈

—

227

—

C₂²⋊C₈

—

227

—

C₄×C₈

—

227

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₂₂₇

yes

—

C₂×D₄

C₂×Dic₂₂₇

yes

—

C₂×D₄

C₂×Dic₂₂₇

yes

—

C₂×D₄

C₂×Dic₂₂₇

yes

—

C₂²×C₄

C₂×Dic₂₂₇

yes

—

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

100

C₂²×C₄

C₂²×D₂₂₇

yes

—

101

M₁₆

—

908

—

102

C₂×C₈

—

908

—

103

C₂×C₈

—

227

—

104

C₂×C₈

—

227

—

105

C₄⋊C₄

C₂×Dic₂₂₇

yes

—

106

C₂²⋊C₄

C₂×Dic₂₂₇

yes

—

107

C₄²

C₂×Dic₂₂₇

yes

characteristic

108

C₄²

C₂×Dic₂₂₇

yes

characteristic

109

C₄²

C₂²×D₂₂₇

yes

—

110

C₂³

C₂²×Dic₂₂₇

yes

—

111

C₂³

C₂²×Dic₂₂₇

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×Dic₂₂₇

yes

characteristic

122

C₂×C₄

C₂²×Dic₂₂₇

yes

—

123

C₂×C₄

C₂²×Dic₂₂₇

yes

—

124

C₂×C₄

C₂²×Dic₂₂₇

yes

—

125

C₂×C₄

C₂³×D₂₂₇

yes

characteristic

126

C₈

—

454

—

127

C₈

—

454

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×Dic₂₂₇

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#20300295

yes

—

137

C₄

#20300333

yes

characteristic

138

C₄

#20300333

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#28409507

yes

characteristic

142

C₂

C₂×#20300295

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂³×C₄
Fitting, prime radical	#532×C₂₂₇	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#532	D₂₂₇
hypocenter	C₂₂₇	#2979
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₂₇

commutator, socle

C₂²×C₂₂₇

C₂³×C₄

Fitting, prime radical

#532×C₂₂₇

C₂

cosocle

C₂×C₄×C₂₂₇

C₂⁴

D₂₂₇

C₂₂₇

G ▷ C₂²×C₂₂₇ ▷ C₂₂₇

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #28409110	C₂	semidirect
2. D₄×C₂₂₇⋊C₈	C₂	semidirect
3. #28409085	C₂	semidirect
4. #28408999	C₂	semidirect
71. C₂₂₇	#2979	semidirect

H ≅ G/N

type

1. #28409110

C₂

semidirect

2. D₄×C₂₂₇⋊C₈

C₂

semidirect

3. #28409085

C₂

semidirect

4. #28408999

C₂

semidirect

71. C₂₂₇

#2979

semidirect

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

subgroup	index
#28409110	2
D₄×C₂₂₇⋊C₈	2
#28409085	2
#28408999	2
#532×C₂₂₇	2
#2979	227

subgroup

index

D₄×C₂₂₇⋊C₈

#532×C₂₂₇

227

Subgroups — #44408801

Conjugacy classes of subgroups

Complements

Maximal subgroups