Group #30638798 — subgroups

Subgroups — #30638798

Order 16256.

380 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 149 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×C₂²⋊(C₁₂₇⋊C₈)	C₂	1	2	yes	maximal
2	#22549631	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
5	C₂²⋊(C₁₂₇⋊C₈)	—	2	4	no	—
6	C₂³×C₄×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₁₂₇	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₂₇	—	2	4	no	—
10	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
11	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₁₂₇⋊C₈	—	2	4	no	—
13	C₂×C₁₂₇⋊C₈	—	2	8	no	—
14	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₂₇	—	2	2	no	—
16	C₂²×C₄×C₁₂₇	—	2	2	no	—
17	C₂²×C₄×C₁₂₇	—	2	2	no	—
18	C₂²×C₄×C₁₂₇	—	2	4	no	—
19	C₂²×C₄×C₁₂₇	—	2	4	no	—
20	C₂²×C₄×C₁₂₇	C₈	1	4	yes	complemented
21	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
25	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₄²×C₁₂₇	—	2	4	no	—
27	C₄²×C₁₂₇	—	2	4	no	—
28	C₁₂₇⋊C₈	—	4	4	no	—
29	C₂³×C₁₂₇	—	2	2	no	—
30	C₂³×C₁₂₇	—	2	2	no	—
31	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
32	C₂³×C₁₂₇	C₂×C₈	1	2	yes	—
33	C₂³×C₁₂₇	M₁₆	1	2	yes	—
34	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	—	2	8	no	—
36	C₂×C₄×C₁₂₇	—	2	8	no	—
37	C₂×C₄×C₁₂₇	—	2	2	no	—
38	C₂×C₄×C₁₂₇	—	2	2	no	—
39	C₂×C₄×C₁₂₇	—	2	2	no	—
40	C₂×C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	—	2	8	no	—
43	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₂₇	C₂×C₈	1	4	yes	—
45	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	—
46	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₂₇	#41	1	4	yes	—
48	C₂²×C₁₂₇	—	2	2	no	—
49	C₂²×C₁₂₇	—	2	2	no	—
50	C₂²×C₁₂₇	—	2	4	no	—
51	C₂²×C₁₂₇	—	2	4	no	—
52	C₂²×C₁₂₇	C₂²⋊C₈	1	4	yes	—
53	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₂₇	#117	1	2	yes	—
55	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	—	2	4	no	—
60	C₄×C₁₂₇	—	2	4	no	—
61	C₄×C₁₂₇	#123	1	4	yes	—
62	C₂×C₁₂₇	—	2	4	no	—
63	C₂×C₁₂₇	#381	1	1	yes	characteristic
64	C₂×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
65	C₂×C₁₂₇	#407	1	1	yes	characteristic
66	C₂×C₁₂₇	#431	1	2	yes	—
67	C₂×C₁₂₇	#432	1	2	yes	—
68	#1817	—	127	1	no	maximal, Sylow
69	C₁₂₇	#1817	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic
71	C₂×C₂²⋊C₈	—	127	2	no	—
72	#336	—	127	4	no	—
73	C₂³×C₄	Dic₁₂₇	1	1	yes	characteristic
74	C₂³×C₄	Dic₁₂₇	1	1	yes	characteristic
75	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
76	C₂²×C₈	—	127	4	no	—
77	C₂×C₄²	—	2	4	no	—
78	C₂×C₄²	Dic₁₂₇	1	2	yes	—
79	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
80	C₂²⋊C₈	—	254	4	no	—
81	C₂⁴	C₂×Dic₁₂₇	1	1	yes	characteristic
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	2	4	no	—
87	C₂²×C₄	C₁₂₇⋊C₈	1	4	yes	complemented
88	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
89	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
90	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
91	C₂²×C₄	C₂²⋊D₁₂₇	1	4	yes	—
92	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
93	C₂×C₈	—	254	4	no	—
94	C₂×C₈	—	254	8	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₂³	—	2	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	C₂²⋊Dic₁₂₇	1	2	yes	—
100	C₂³	C₂×C₁₂₇⋊C₈	1	2	yes	—
101	C₂³	#14523232	1	2	yes	—
102	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	8	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
112	C₂×C₄	C₂×C₁₂₇⋊C₈	1	4	yes	—
113	C₂×C₄	#14523232	1	4	yes	—
114	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
115	C₂×C₄	#14523248	1	4	yes	—
116	C₈	—	508	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	4	no	—
120	C₂²	—	2	2	no	—
121	C₂²	C₂²⋊(C₁₂₇⋊C₈)	1	4	yes	—
122	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
123	C₂²	#16861168	1	2	yes	—
124	C₂²	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
125	C₂²	C₂×#14523232	1	1	yes	characteristic
126	C₂²	C₂×#14523232	1	2	yes	—
127	C₄	—	2	4	no	—
128	C₄	—	2	4	no	—
129	C₄	—	2	2	no	—
130	C₄	#16861192	1	4	yes	—
131	C₂	—	2	4	no	—
132	C₂	#22549739	1	1	yes	characteristic
133	C₂	C₂×C₂²⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
134	C₂	#22549840	1	1	yes	characteristic
135	C₂	#22549929	1	2	yes	—
136	C₂	#22549933	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal

#22549631

C₂

yes

maximal, complemented

C₂²×C₄²×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂²⋊(C₁₂₇⋊C₈)

—

C₂³×C₄×C₁₂₇

C₄

yes

characteristic

C₂³×C₄×C₁₂₇

C₄

yes

characteristic

C₂³×C₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂⁴×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₈

yes

complemented

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂²⋊C₄

yes

—

C₂³×C₁₂₇

C₂×C₈

yes

—

C₂³×C₁₂₇

M₁₆

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²⋊C₈

yes

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

#117

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₂²×C₁₂₇

C₂×M₁₆

yes

characteristic

C₂²×C₁₂₇

C₂×M₁₆

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#123

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#381

yes

characteristic

C₂×C₁₂₇

C₂×C₂²⋊C₈

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×C₁₂₇

#432

yes

—

#1817

—

127

maximal, Sylow

C₁₂₇

#1817

yes

characteristic, complemented, Sylow

C₂²×C₄²

D₁₂₇

yes

characteristic

C₂×C₂²⋊C₈

—

127

—

#336

—

127

—

C₂³×C₄

Dic₁₂₇

yes

characteristic

C₂³×C₄

Dic₁₂₇

yes

characteristic

C₂³×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

C₂×C₄²

—

C₂×C₄²

Dic₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

—

C₂²⋊C₈

—

254

—

C₂⁴

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₁₂₇⋊C₈

yes

complemented

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄²

—

C₄²

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₁₂₇

yes

—

100

C₂³

C₂×C₁₂₇⋊C₈

yes

—

101

C₂³

#14523232

yes

—

102

C₂³

C₂²×Dic₁₂₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²⋊Dic₁₂₇

yes

—

112

C₂×C₄

C₂×C₁₂₇⋊C₈

yes

—

113

C₂×C₄

#14523232

yes

—

114

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

115

C₂×C₄

#14523248

yes

—

116

C₈

—

508

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²⋊(C₁₂₇⋊C₈)

yes

—

122

C₂²

C₂×C₂²⋊Dic₁₂₇

yes

characteristic

123

C₂²

#16861168

yes

—

124

C₂²

C₂²×C₁₂₇⋊C₈

yes

characteristic

125

C₂²

C₂×#14523232

yes

characteristic

126

C₂²

C₂×#14523232

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#16861192

yes

—

131

C₂

—

132

C₂

#22549739

yes

characteristic

133

C₂

C₂×C₂²⋊(C₁₂₇⋊C₈)

yes

characteristic

134

C₂

#22549840

yes

characteristic

135

C₂

#22549929

yes

—

136

C₂

#22549933

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₈
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	#1817
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂²×C₈

Fitting

C₂²×C₄²×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂²×C₄

prime radical

C₂⁴×C₁₂₇

C₂×C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂²×C₄²

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. #22549631	C₂	semidirect
20. C₂²×C₄×C₁₂₇	C₈	semidirect
69. C₁₂₇	#1817	semidirect
87. C₂²×C₄	C₁₂₇⋊C₈	semidirect

H ≅ G/N

type

2. #22549631

C₂

semidirect

20. C₂²×C₄×C₁₂₇

C₈

semidirect

69. C₁₂₇

#1817

semidirect

87. C₂²×C₄

C₁₂₇⋊C₈

semidirect

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

subgroup	index
C₂×C₂²⋊(C₁₂₇⋊C₈)	2
#22549631	2
C₂²×C₄²×C₁₂₇	2
#1817	127

subgroup

index

C₂×C₂²⋊(C₁₂₇⋊C₈)

#22549631

C₂²×C₄²×C₁₂₇

#1817

127

Subgroups — #30638798

Conjugacy classes of subgroups

Complements

Maximal subgroups