Group #44402454 — subgroups

Subgroups — #44402454

Order 29056.

516 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q₈×Dic₂₂₇	C₂	1	2	yes	maximal
2	#28408939	C₂	1	2	yes	maximal
3	#28408918	C₂	1	8	yes	maximal
4	#28408898	C₂	1	1	yes	maximal, characteristic
5	#28408878	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
7	Q₈×Dic₂₂₇	—	2	8	no	—
8	C₂×C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₂₇	C₂²	1	4	yes	—
10	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₂₇	C₂²	1	4	yes	—
12	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
13	C₂₂₇⋊#95	C₂²	1	4	yes	—
14	C₂₂₇⋊#95	C₂²	1	4	yes	—
15	C₂₂₇⋊#95	C₂²	1	8	yes	—
16	C₂²×Q₈×C₂₂₇	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₂₂₇	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	4	yes	—
19	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
21	C₂²×Dic₂₂₇	C₂³	1	4	yes	—
22	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
23	C₄⋊Dic₂₂₇	—	2	8	no	—
24	C₄⋊Dic₂₂₇	—	2	4	no	—
25	C₄×Dic₂₂₇	—	2	8	no	—
26	C₄×Dic₂₂₇	—	2	2	no	—
27	C₄×Dic₂₂₇	D₄	1	4	yes	—
28	C₂×Q₈×C₂₂₇	—	2	4	no	—
29	C₂×Q₈×C₂₂₇	—	2	4	no	—
30	C₂×Q₈×C₂₂₇	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₂₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₂₂₇	C₂×C₄	1	16	yes	—
35	C₄²×C₂₂₇	C₂×C₄	1	4	yes	—
36	C₂×Dic₂₂₇	—	2	4	no	—
37	C₂×Dic₂₂₇	—	2	4	no	—
38	C₂×Dic₂₂₇	—	2	2	no	—
39	C₂×Dic₂₂₇	—	2	8	no	—
40	C₂×Dic₂₂₇	—	2	4	no	—
41	C₂×Dic₂₂₇	C₂×D₄	1	4	yes	—
42	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
43	Q₈×C₂₂₇	—	2	16	no	—
44	C₂×C₄×C₂₂₇	—	2	4	no	—
45	C₂×C₄×C₂₂₇	—	2	4	no	—
46	C₂×C₄×C₂₂₇	—	2	4	no	—
47	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₂₂₇	C₂²×C₄	1	8	yes	—
49	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₂₇	#41	1	2	yes	—
52	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
55	C₂²×C₂₂₇	2₋¹⁺⁴	1	4	yes	—
56	C₄×C₂₂₇	—	2	8	no	—
57	C₄×C₂₂₇	—	2	4	no	—
58	C₄×C₂₂₇	C₄×D₄	1	4	yes	—
59	Dic₂₂₇	—	2	4	no	—
60	Dic₂₂₇	—	4	4	no	—
61	Dic₂₂₇	—	4	2	no	—
62	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
63	C₂×C₂₂₇	#519	1	2	yes	—
64	C₂×C₂₂₇	#536	1	2	yes	—
65	C₂×C₂₂₇	#541	1	2	yes	—
66	C₂₂₇	#2327	1	1	yes	characteristic, complemented, Sylow
67	#2327	—	227	1	no	maximal, Sylow
68	C₂×C₄⋊Q₈	D₂₂₇	1	1	yes	characteristic
69	C₂×C₄×Q₈	—	227	2	no	—
70	#391	—	227	2	no	—
71	#387	—	227	8	no	—
72	#382	—	227	1	no	—
73	#376	—	227	1	no	—
74	C₂²×Q₈	C₂×D₂₂₇	1	2	yes	—
75	C₄⋊Q₈	Dic₂₂₇	1	8	yes	complemented
76	C₄×Q₈	—	454	8	no	—
77	C₂×C₄⋊C₄	—	227	2	no	—
78	C₂×C₄⋊C₄	—	227	4	no	—
79	C₂×C₄⋊C₄	C₂×D₂₂₇	1	4	yes	—
80	C₂×C₄²	—	227	4	no	—
81	C₂×C₄²	—	227	1	no	—
82	C₂×C₄²	—	227	1	no	—
83	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
84	#95	—	227	4	no	—
85	#95	—	227	4	no	—
86	#95	—	227	8	no	—
87	C₂×Q₈	—	2	4	no	—
88	C₂×Q₈	—	2	4	no	—
89	C₂×Q₈	C₂×Dic₂₂₇	1	8	yes	—
90	C₂²×C₄	—	227	2	no	—
91	C₂²×C₄	—	227	4	no	—
92	C₂²×C₄	—	227	2	no	—
93	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
94	C₂²×C₄	C₂²×D₂₂₇	1	4	yes	—
95	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
96	C₄⋊C₄	—	454	8	no	—
97	C₄⋊C₄	—	454	4	no	—
98	C₄⋊C₄	C₂×Dic₂₂₇	1	16	yes	—
99	C₄²	—	454	8	no	—
100	C₄²	—	454	2	no	—
101	C₄²	—	227	4	no	—
102	C₄²	C₂×Dic₂₂₇	1	4	yes	—
103	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
104	Q₈	—	2	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	454	4	no	—
109	C₂×C₄	—	454	2	no	—
110	C₂×C₄	—	454	8	no	—
111	C₂×C₄	—	454	4	no	—
112	C₂×C₄	—	454	4	no	—
113	C₂×C₄	—	227	4	no	—
114	C₂×C₄	C₂²×Dic₂₂₇	1	4	yes	—
115	C₂×C₄	C₂²×Dic₂₂₇	1	8	yes	—
116	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
117	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
118	C₂×C₄	#16813883	1	2	yes	—
119	C₂²	C₂³×Dic₂₂₇	1	1	yes	characteristic
120	C₂²	C₂×D₄×D₂₂₇	1	1	yes	characteristic
121	C₂²	C₂×#16813883	1	1	yes	characteristic
122	C₂²	#20300336	1	4	yes	—
123	C₄	—	2	8	no	—
124	C₄	—	2	4	no	—
125	C₄	—	908	4	no	—
126	C₄	—	908	2	no	—
127	C₄	—	454	4	no	—
128	C₄	D₄×Dic₂₂₇	1	4	yes	—
129	C₂	C₂×D₄×Dic₂₂₇	1	1	yes	characteristic
130	C₂	#28409510	1	2	yes	—
131	C₂	#28409607	1	2	yes	—
132	C₂	#28409643	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₂₂₇

C₂

yes

maximal

#28408939

C₂

yes

maximal

#28408918

C₂

yes

maximal

#28408898

C₂

yes

maximal, characteristic

#28408878

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₂₂₇

C₂

yes

maximal, characteristic

Q₈×Dic₂₂₇

—

C₂×C₄⋊Dic₂₂₇

C₂²

yes

—

C₂×C₄⋊Dic₂₂₇

C₂²

yes

—

C₂×C₄×Dic₂₂₇

C₂²

yes

characteristic

C₂×C₄×Dic₂₂₇

C₂²

yes

—

C₂×C₄×Dic₂₂₇

C₂²

yes

characteristic

C₂₂₇⋊#95

C₂²

yes

—

C₂₂₇⋊#95

C₂²

yes

—

C₂₂₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₂₂₇

C₂²

yes

—

C₄⋊Q₈×C₂₂₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₂₇

C₂²

yes

—

C₂×C₄²×C₂₂₇

C₂²

yes

characteristic

C₂²×Dic₂₂₇

C₂³

yes

—

C₂²×Dic₂₂₇

C₂³

yes

—

C₂²×Dic₂₂₇

C₂³

yes

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

D₄

yes

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₂₇

C₂×C₄

yes

—

C₄²×C₂₂₇

C₂×C₄

yes

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

C₂×D₄

yes

—

C₂³×C₂₂₇

C₂⁴

yes

characteristic

Q₈×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

#41

yes

—

C₂²×C₂₂₇

C₂³×C₄

yes

characteristic

C₂²×C₂₂₇

C₂²×D₄

yes

characteristic

C₂²×C₂₂₇

C₂×#41

yes

characteristic

C₂²×C₂₂₇

2₋¹⁺⁴

yes

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₄×D₄

yes

—

Dic₂₂₇

—

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

C₂×C₄×D₄

yes

characteristic

C₂×C₂₂₇

#519

yes

—

C₂×C₂₂₇

#536

yes

—

C₂×C₂₂₇

#541

yes

—

C₂₂₇

#2327

yes

characteristic, complemented, Sylow

#2327

—

227

maximal, Sylow

C₂×C₄⋊Q₈

D₂₂₇

yes

characteristic

C₂×C₄×Q₈

—

227

—

#391

—

227

—

#387

—

227

—

#382

—

227

—

#376

—

227

—

C₂²×Q₈

C₂×D₂₂₇

yes

—

C₄⋊Q₈

Dic₂₂₇

yes

complemented

C₄×Q₈

—

454

—

C₂×C₄⋊C₄

—

227

—

C₂×C₄⋊C₄

—

227

—

C₂×C₄⋊C₄

C₂×D₂₂₇

yes

—

C₂×C₄²

—

227

—

C₂×C₄²

—

227

—

C₂×C₄²

—

227

—

C₂×C₄²

C₂×D₂₂₇

yes

characteristic

#95

—

227

—

#95

—

227

—

#95

—

227

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₂₇

yes

—

C₂²×C₄

—

227

—

C₂²×C₄

—

227

—

C₂²×C₄

—

227

—

C₂²×C₄

C₂²×D₂₂₇

yes

—

C₂²×C₄

C₂²×D₂₂₇

yes

—

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

C₄⋊C₄

—

454

—

C₄⋊C₄

—

454

—

C₄⋊C₄

C₂×Dic₂₂₇

yes

—

C₄²

—

454

—

100

C₄²

—

454

—

101

C₄²

—

227

—

102

C₄²

C₂×Dic₂₂₇

yes

—

103

C₂³

C₂³×D₂₂₇

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

454

—

109

C₂×C₄

—

454

—

110

C₂×C₄

—

454

—

111

C₂×C₄

—

454

—

112

C₂×C₄

—

454

—

113

C₂×C₄

—

227

—

114

C₂×C₄

C₂²×Dic₂₂₇

yes

—

115

C₂×C₄

C₂²×Dic₂₂₇

yes

—

116

C₂×C₄

C₂²×Dic₂₂₇

yes

—

117

C₂×C₄

D₄×D₂₂₇

yes

—

118

C₂×C₄

#16813883

yes

—

119

C₂²

C₂³×Dic₂₂₇

yes

characteristic

120

C₂²

C₂×D₄×D₂₂₇

yes

characteristic

121

C₂²

C₂×#16813883

yes

characteristic

122

C₂²

#20300336

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

908

—

126

C₄

—

908

—

127

C₄

—

454

—

128

C₄

D₄×Dic₂₂₇

yes

—

129

C₂

C₂×D₄×Dic₂₂₇

yes

characteristic

130

C₂

#28409510

yes

—

131

C₂

#28409607

yes

—

132

C₂

#28409643

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₂₂₇	C₂
socle, prime radical, cosocle	C₂³×C₂₂₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₂₂₇
hypocenter	C₂₂₇	#2327
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₂₇

commutator

C₂²×C₂₂₇

C₂³×C₄

Fitting

C₂×C₄⋊Q₈×C₂₂₇

C₂

socle, prime radical, cosocle

C₂³×C₂₂₇

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₂₂₇

C₂₂₇

G ▷ C₂²×C₂₂₇ ▷ C₂₂₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
17. C₄⋊Q₈×C₂₂₇	C₄	semidirect
66. C₂₂₇	#2327	semidirect
75. C₄⋊Q₈	Dic₂₂₇	semidirect

H ≅ G/N

type

17. C₄⋊Q₈×C₂₂₇

C₄

semidirect

66. C₂₂₇

#2327

semidirect

75. C₄⋊Q₈

Dic₂₂₇

semidirect

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

subgroup	index
C₂×Q₈×Dic₂₂₇	2
#28408939	2
#28408918	2
#28408898	2
#28408878	2
C₂×C₄⋊Q₈×C₂₂₇	2
#2327	227

subgroup

index

C₂×Q₈×Dic₂₂₇

C₂×C₄⋊Q₈×C₂₂₇

227

Subgroups — #44402454

Conjugacy classes of subgroups

Complements

Maximal subgroups