Group #30640269 — subgroups

Subgroups — #30640269

Order 16256.

254 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 91 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×#16861142	C₂	1	2	yes	maximal
2	C₂×#16861132	C₂	1	2	yes	maximal
3	#22549782	C₂	1	1	yes	maximal, characteristic
4	#22549631	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
7	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
9	#16861142	—	2	4	no	—
10	#16861132	—	2	4	no	—
11	C₁₂₇⋊#95	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₅₄	—	2	2	no	—
16	C₂×Dic₂₅₄	—	2	1	no	—
17	C₂×Dic₂₅₄	—	2	2	no	—
18	C₂×Dic₂₅₄	Q₈	1	2	yes	—
19	C₂²×Dic₁₂₇	—	2	1	no	—
20	C₂²×Dic₁₂₇	—	2	1	no	—
21	C₂×C₁₂₇⋊C₈	—	2	2	no	—
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
24	C₄⋊Dic₁₂₇	—	2	2	no	—
25	C₂²×C₄×C₁₂₇	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₂₇	—	2	2	no	—
29	C₄⋊C₄×C₁₂₇	—	2	2	no	—
30	C₄⋊C₄×C₁₂₇	—	4	4	no	—
31	C₄²×C₁₂₇	—	2	2	no	—
32	Dic₂₅₄	—	2	4	no	—
33	Dic₂₅₄	—	4	2	no	—
34	C₂×Dic₁₂₇	—	2	2	no	—
35	C₂×Dic₁₂₇	—	4	1	no	—
36	C₂×Dic₁₂₇	—	4	1	no	—
37	C₂×Dic₁₂₇	—	4	1	no	—
38	C₂×Dic₁₂₇	—	4	2	no	—
39	C₁₂₇⋊C₈	—	2	4	no	—
40	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	—	2	2	no	—
43	C₂×C₄×C₁₂₇	—	4	2	no	—
44	C₂×C₄×C₁₂₇	—	4	4	no	—
45	C₂×C₄×C₁₂₇	Q₁₆	1	4	yes	—
46	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	#41	1	2	yes	—
49	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
50	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₂₇	#123	1	1	yes	characteristic
52	C₂²×C₁₂₇	#127	1	1	yes	characteristic
53	C₂²×C₁₂₇	C₂×Q₁₆	1	2	yes	—
54	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₂₇	—	2	4	no	—
56	C₄×C₁₂₇	—	8	2	no	—
57	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₂₇	#123	1	2	yes	—
59	Dic₁₂₇	—	4	2	no	—
60	Dic₁₂₇	—	8	1	no	—
61	C₂×C₁₂₇	#399	1	1	yes	characteristic
62	C₂×C₁₂₇	#477	1	2	yes	—
63	C₂×C₁₂₇	#484	1	2	yes	—
64	C₂×C₁₂₇	#494	1	1	yes	characteristic
65	C₂×C₁₂₇	#496	1	1	yes	characteristic
66	#2095	—	127	1	no	maximal, Sylow
67	C₁₂₇	#2095	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	127	2	no	—
69	C₂×#103	—	127	2	no	—
70	#391	—	127	1	no	—
71	#384	D₁₂₇	1	1	yes	characteristic
72	#336	—	127	1	no	—
73	C₂²×Q₈	—	127	1	no	—
74	C₂²×C₈	—	127	2	no	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	127	1	no	—
77	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
78	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
79	#107	—	254	4	no	—
80	#103	—	254	4	no	—
81	#95	—	254	2	no	—
82	C₂×Q₈	—	127	2	no	—
83	C₂×Q₈	—	254	2	no	—
84	C₂×Q₈	—	254	2	no	—
85	C₂×Q₈	—	254	1	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	254	1	no	—
88	C₂²×C₄	—	254	1	no	—
89	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
91	C₂×C₈	—	127	4	no	—
92	C₂×C₈	—	254	2	no	—
93	C₂×C₈	—	254	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	4	4	no	—
97	C₄⋊C₄	—	254	2	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
100	Q₈	—	508	2	no	—
101	Q₈	—	254	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	508	1	no	—
107	C₂×C₄	—	508	2	no	—
108	C₂×C₄	—	508	1	no	—
109	C₂×C₄	—	508	1	no	—
110	C₂×C₄	—	254	2	no	—
111	C₂×C₄	#14523238	1	4	yes	—
112	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
113	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
114	C₂×C₄	#14523248	1	2	yes	—
115	C₂×C₄	#14523249	1	1	yes	characteristic
116	C₈	—	254	4	no	—
117	C₂²	#16861185	1	1	yes	characteristic
118	C₂²	#16861192	1	1	yes	characteristic
119	C₂²	#16861205	1	1	yes	characteristic
120	C₂²	C₂×#14523238	1	2	yes	—
121	C₂²	#16861235	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	8	2	no	—
124	C₄	—	1016	1	no	—
125	C₄	—	508	2	no	—
126	C₄	#16861187	1	2	yes	—
127	C₄	#16861194	1	2	yes	—
128	C₂	#22549811	1	1	yes	characteristic
129	C₂	#22550168	1	2	yes	—
130	C₂	#22550211	1	2	yes	—
131	C₂	#22550255	1	1	yes	characteristic
132	C₂	#22550263	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861142

C₂

yes

maximal

C₂×#16861132

C₂

yes

maximal

#22549782

C₂

yes

maximal, characteristic

#22549631

C₂

yes

maximal, characteristic

#384×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

#16861142

—

#16861132

—

C₁₂₇⋊#95

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

Q₈

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

Q₁₆

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂×Q₁₆

yes

—

C₂²×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

#123

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#399

yes

characteristic

C₂×C₁₂₇

#477

yes

—

C₂×C₁₂₇

#484

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2095

yes

characteristic, complemented, Sylow

C₂×#107

—

127

—

C₂×#103

—

127

—

#391

—

127

—

#384

D₁₂₇

yes

characteristic

#336

—

127

—

C₂²×Q₈

—

127

—

C₂²×C₈

—

127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#107

—

254

—

#103

—

254

—

#95

—

254

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

254

—

C₄²

—

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

100

Q₈

—

508

—

101

Q₈

—

254

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

508

—

107

C₂×C₄

—

508

—

108

C₂×C₄

—

508

—

109

C₂×C₄

—

508

—

110

C₂×C₄

—

254

—

111

C₂×C₄

#14523238

yes

—

112

C₂×C₄

D₄×D₁₂₇

yes

—

113

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

114

C₂×C₄

#14523248

yes

—

115

C₂×C₄

#14523249

yes

characteristic

116

C₈

—

254

—

117

C₂²

#16861185

yes

characteristic

118

C₂²

#16861192

yes

characteristic

119

C₂²

#16861205

yes

characteristic

120

C₂²

C₂×#14523238

yes

—

121

C₂²

#16861235

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

1016

—

125

C₄

—

508

—

126

C₄

#16861187

yes

—

127

C₄

#16861194

yes

—

128

C₂

#22549811

yes

characteristic

129

C₂

#22550168

yes

—

130

C₂

#22550211

yes

—

131

C₂

#22550255

yes

characteristic

132

C₂

#22550263

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#384×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#384	D₁₂₇
hypocenter	C₁₂₇	#2095
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#384×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
67. C₁₂₇	#2095	semidirect

H ≅ G/N

type

67. C₁₂₇

#2095

semidirect

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

subgroup	index
C₂×#16861142	2
C₂×#16861132	2
#22549782	2
#22549631	2
#384×C₁₂₇	2
#2095	127

subgroup

index

#384×C₁₂₇

127

Subgroups — #30640269

Conjugacy classes of subgroups

Complements

Maximal subgroups