Group #44400673 — subgroups

Subgroups — #44400673

Order 29056.

280 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 109 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×Dic₄₅₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#20300199	C₂	1	1	yes	maximal, characteristic
3	#28408786	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₅₄	C₄	1	1	yes	characteristic
6	C₂²×C₂₂₇⋊C₈	C₂²	1	2	yes	—
7	C₄×Dic₄₅₄	—	2	2	no	—
8	C₄×Dic₄₅₄	—	2	2	no	—
9	C₂×C₄⋊Dic₂₂₇	C₄	1	1	yes	characteristic
10	C₂×Dic₂₂₇⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
12	#20300199	—	2	2	no	—
13	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₂₂₇	—	2	2	no	—
16	C₂×Dic₄₅₄	—	2	4	no	—
17	C₂×Dic₄₅₄	—	2	2	no	—
18	C₂²×Dic₂₂₇	C₂×C₄	1	2	yes	—
19	C₂²×Dic₂₂₇	C₂×C₄	1	2	yes	—
20	C₂×C₂₂₇⋊C₈	—	2	2	no	—
21	C₂×C₂₂₇⋊C₈	—	2	2	no	—
22	C₂×C₂₂₇⋊C₈	Q₈	1	4	yes	complemented
23	C₄⋊Dic₂₂₇	—	2	2	no	—
24	Dic₂₂₇⋊C₄	—	2	4	no	—
25	C₄×Dic₂₂₇	—	2	2	no	—
26	C₄×Dic₂₂₇	—	2	2	no	—
27	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₂₂₇	—	2	2	no	—
31	C₂×C₈×C₂₂₇	—	2	4	no	—
32	C₄²×C₂₂₇	D₄	1	2	yes	—
33	C₄²×C₂₂₇	D₄	1	2	yes	—
34	Dic₄₅₄	—	2	8	no	—
35	C₂×Dic₂₂₇	—	2	2	no	—
36	C₂×Dic₂₂₇	—	2	2	no	—
37	C₂×Dic₂₂₇	—	2	2	no	—
38	C₂×Dic₂₂₇	—	2	4	no	—
39	C₂×Dic₂₂₇	M₁₆	1	4	yes	—
40	C₂₂₇⋊C₈	—	2	4	no	—
41	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₂₇	—	2	2	no	—
43	C₂×C₄×C₂₂₇	—	2	2	no	—
44	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₂₂₇	#41	1	2	yes	—
50	C₈×C₂₂₇	—	4	2	no	—
51	C₂²×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₂₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₂₂₇	C₂×M₁₆	1	2	yes	—
54	C₂²×C₂₂₇	#131	1	2	yes	—
55	C₄×C₂₂₇	—	2	2	no	—
56	C₄×C₂₂₇	—	2	2	no	—
57	C₄×C₂₂₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₂₂₇	#124	1	1	yes	characteristic
59	C₄×C₂₂₇	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₂₂₇	—	2	4	no	—
61	Dic₂₂₇	—	4	2	no	—
62	C₂×C₂₂₇	#391	1	1	yes	characteristic
63	C₂×C₂₂₇	#407	1	1	yes	characteristic
64	C₂×C₂₂₇	#408	1	1	yes	characteristic
65	C₂×C₂₂₇	#446	1	4	yes	—
66	C₂₂₇	#1985	1	1	yes	characteristic, complemented, Sylow
67	#1985	—	227	1	no	maximal, Sylow
68	C₂×C₄×Q₈	—	227	1	no	—
69	C₂×C₄⋊C₈	D₂₂₇	1	1	yes	characteristic
70	C₂×#97	—	227	1	no	—
71	#336	—	227	4	no	—
72	C₂²×Q₈	—	227	1	no	—
73	C₂²×C₈	—	227	2	no	—
74	C₂²×C₈	C₂×D₂₂₇	1	2	yes	—
75	C₄×Q₈	—	454	2	no	—
76	C₄×Q₈	—	454	2	no	—
77	C₂×C₄⋊C₄	—	227	2	no	—
78	C₂×C₄⋊C₄	—	227	1	no	—
79	C₂×C₄²	—	227	2	no	—
80	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#97	—	454	2	no	—
83	C₂×Q₈	—	454	4	no	—
84	C₂×Q₈	—	454	2	no	—
85	C₂²×C₄	—	227	2	no	—
86	C₂²×C₄	—	227	2	no	—
87	C₂²×C₄	C₄×D₂₂₇	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₂₂₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	454	2	no	—
93	C₂×C₈	—	454	2	no	—
94	C₂×C₈	—	227	4	no	—
95	C₄⋊C₄	—	454	4	no	—
96	C₄⋊C₄	—	454	2	no	—
97	C₄²	—	454	2	no	—
98	C₄²	—	454	2	no	—
99	C₄²	C₂²⋊D₂₂₇	1	2	yes	—
100	C₄²	D₉₀₈	1	2	yes	—
101	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
102	Q₈	—	454	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	454	4	no	—
106	C₂×C₄	—	454	2	no	—
107	C₂×C₄	—	454	2	no	—
108	C₂×C₄	—	454	2	no	—
109	C₂×C₄	—	227	4	no	—
110	C₂×C₄	(C₂×D₂₂₇)⋊C₄	1	2	yes	—
111	C₂×C₄	(C₂×D₂₂₇)⋊C₄	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
113	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₂₂₇	1	2	yes	—
115	C₂×C₄	#16813883	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	454	4	no	—
118	C₂²	C₂×(C₂×D₂₂₇)⋊C₄	1	1	yes	characteristic
119	C₂²	#20300253	1	2	yes	—
120	C₂²	M₁₆×D₂₂₇	1	2	yes	—
121	C₂²	#20300294	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	908	2	no	—
125	C₄	—	454	4	no	—
126	C₄	#20300268	1	2	yes	—
127	C₄	#20300275	1	1	yes	characteristic
128	C₄	#20300286	1	1	yes	characteristic
129	C₂	#28408937	1	1	yes	characteristic
130	C₂	#28408995	1	1	yes	characteristic
131	C₂	#28408998	1	1	yes	characteristic
132	C₂	#28409161	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₄₅₄

C₂

yes

maximal, characteristic

C₂×#20300199

C₂

yes

maximal, characteristic

#28408786

C₂

yes

maximal

C₂×C₄⋊C₈×C₂₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₄₅₄

C₄

yes

characteristic

C₂²×C₂₂₇⋊C₈

C₂²

yes

—

C₄×Dic₄₅₄

—

C₄×Dic₄₅₄

—

C₂×C₄⋊Dic₂₂₇

C₄

yes

characteristic

C₂×Dic₂₂₇⋊C₄

C₄

yes

—

C₂×C₄×Dic₂₂₇

C₂²

yes

—

#20300199

—

C₂²×C₈×C₂₂₇

C₂²

yes

—

C₂×C₄²×C₂₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₇

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂²×Dic₂₂₇

C₂×C₄

yes

—

C₂²×Dic₂₂₇

C₂×C₄

yes

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

Q₈

yes

complemented

C₄⋊Dic₂₂₇

—

Dic₂₂₇⋊C₄

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₂²×C₄×C₂₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₄²×C₂₂₇

D₄

yes

—

C₄²×C₂₂₇

D₄

yes

—

Dic₄₅₄

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

M₁₆

yes

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×Q₈

yes

—

C₂×C₄×C₂₂₇

#41

yes

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₂₇

C₄×Q₈

yes

—

C₂²×C₂₂₇

C₂×M₁₆

yes

—

C₂²×C₂₂₇

#131

yes

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂²⋊Q₈

yes

—

C₄×C₂₂₇

#124

yes

characteristic

C₄×C₂₂₇

C₄⋊Q₈

yes

characteristic

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

—

C₂₂₇

#1985

yes

characteristic, complemented, Sylow

#1985

—

227

maximal, Sylow

C₂×C₄×Q₈

—

227

—

C₂×C₄⋊C₈

D₂₂₇

yes

characteristic

C₂×#97

—

227

—

#336

—

227

—

C₂²×Q₈

—

227

—

C₂²×C₈

—

227

—

C₂²×C₈

C₂×D₂₂₇

yes

—

C₄×Q₈

—

454

—

C₄×Q₈

—

454

—

C₂×C₄⋊C₄

—

227

—

C₂×C₄⋊C₄

—

227

—

C₂×C₄²

—

227

—

C₂×C₄²

C₂×D₂₂₇

yes

characteristic

C₄⋊C₈

—

#97

—

454

—

C₂×Q₈

—

454

—

C₂×Q₈

—

454

—

C₂²×C₄

—

227

—

C₂²×C₄

—

227

—

C₂²×C₄

C₄×D₂₂₇

yes

characteristic

C₂²×C₄

C₄×D₂₂₇

yes

characteristic

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

454

—

C₂×C₈

—

454

—

C₂×C₈

—

227

—

C₄⋊C₄

—

454

—

C₄⋊C₄

—

454

—

C₄²

—

454

—

C₄²

—

454

—

C₄²

C₂²⋊D₂₂₇

yes

—

100

C₄²

D₉₀₈

yes

—

101

C₂³

C₂×C₄×D₂₂₇

yes

characteristic

102

Q₈

—

454

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

454

—

106

C₂×C₄

—

454

—

107

C₂×C₄

—

454

—

108

C₂×C₄

—

454

—

109

C₂×C₄

—

227

—

110

C₂×C₄

(C₂×D₂₂₇)⋊C₄

yes

—

111

C₂×C₄

(C₂×D₂₂₇)⋊C₄

yes

—

112

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

113

C₂×C₄

C₂×D₉₀₈

yes

characteristic

114

C₂×C₄

Q₈×D₂₂₇

yes

—

115

C₂×C₄

#16813883

yes

—

116

C₈

—

117

C₈

—

454

—

118

C₂²

C₂×(C₂×D₂₂₇)⋊C₄

yes

characteristic

119

C₂²

#20300253

yes

—

120

C₂²

M₁₆×D₂₂₇

yes

—

121

C₂²

#20300294

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

908

—

125

C₄

—

454

—

126

C₄

#20300268

yes

—

127

C₄

#20300275

yes

characteristic

128

C₄

#20300286

yes

characteristic

129

C₂

#28408937

yes

characteristic

130

C₂

#28408995

yes

characteristic

131

C₂

#28408998

yes

characteristic

132

C₂

#28409161

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₂₇
commutator, socle, prime radical	C₂³×C₂₂₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₂₂₇	C₂
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₂₇
hypocenter	C₂₂₇	#1985
LCS	G ▷ C₂³×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₂₇

commutator, socle, prime radical

C₂³×C₂₂₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₂₂₇

C₂

cosocle

C₂²×C₄×C₂₂₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₂₇

C₂₂₇

G ▷ C₂³×C₂₂₇ ▷ C₂₂₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
22. C₂×C₂₂₇⋊C₈	Q₈	semidirect
66. C₂₂₇	#1985	semidirect

H ≅ G/N

type

22. C₂×C₂₂₇⋊C₈

Q₈

semidirect

66. C₂₂₇

#1985

semidirect

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

subgroup	index
C₂×C₄×Dic₄₅₄	2
C₂×#20300199	2
#28408786	2
C₂×C₄⋊C₈×C₂₂₇	2
#1985	227

subgroup

index

C₂×C₄×Dic₄₅₄

C₂×#20300199

#28408786

C₂×C₄⋊C₈×C₂₂₇

#1985

227

Subgroups — #44400673

Conjugacy classes of subgroups

Complements

Maximal subgroups