Group #30647665 — subgroups

Subgroups — #30647665

Order 16256.

364 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 229 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	#22550007	C₂	1	4	yes	maximal, complemented
2	Q₈×C₁₂₇⋊C₈	C₂	1	4	yes	maximal, complemented
3	#22549903	C₂	1	2	yes	maximal
4	#22549900	C₂	1	1	yes	maximal, characteristic
5	#22549832	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523232	C₂²	1	2	yes	—
9	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
12	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	8	yes	—
13	#16861120	C₂²	1	4	yes	—
14	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
15	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
16	C₄×C₁₂₇⋊C₈	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₂₇	C₄	1	2	yes	—
18	#125×C₁₂₇	C₄	1	2	yes	—
19	C₂²⋊Q₈×C₁₂₇	C₄	1	4	yes	—
20	C₄×Q₈×C₁₂₇	C₂²	1	4	yes	—
21	#117×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
23	#14523232	—	2	4	no	—
24	C₂×C₁₂₇⋊C₈	—	2	2	no	—
25	C₂×C₁₂₇⋊C₈	Q₈	1	4	yes	complemented
26	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
27	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
28	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
34	C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
35	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
36	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
37	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₂₇	C₂³	1	2	yes	—
40	C₄²×C₁₂₇	C₂³	1	4	yes	—
41	C₁₂₇⋊C₈	—	2	2	no	—
42	C₁₂₇⋊C₈	—	2	4	no	—
43	C₁₂₇⋊C₈	C₂×Q₈	1	2	yes	—
44	C₁₂₇⋊C₈	C₂×Q₈	1	2	yes	—
45	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	Q₈×C₁₂₇	—	2	8	no	—
47	C₂×C₄×C₁₂₇	—	2	4	no	—
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
50	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₂₇	#41	1	2	yes	—
54	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
55	C₂²×C₁₂₇	—	2	2	no	—
56	C₂²×C₁₂₇	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	—	2	8	no	—
60	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
61	C₄×C₁₂₇	#131	1	4	yes	—
62	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
64	C₂×C₁₂₇	—	2	2	no	—
65	C₂×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₂₇	C₂×#131	1	2	yes	—
67	#2960	—	127	1	no	maximal, Sylow
68	C₁₂₇	#2960	1	1	yes	characteristic, complemented, Sylow
69	#533	D₁₂₇	1	1	yes	characteristic
70	#446	—	127	4	no	—
71	C₈×Q₈	—	127	4	no	—
72	#424	—	127	2	no	—
73	#423	—	127	1	no	—
74	#405	—	127	2	no	—
75	C₂×C₄×C₈	—	127	1	no	—
76	C₂×M₁₆	—	127	2	no	—
77	C₂²×C₈	—	127	1	no	—
78	C₂²×C₈	—	127	1	no	—
79	C₄⋊Q₈	Dic₁₂₇	1	2	yes	—
80	#125	Dic₁₂₇	1	2	yes	—
81	C₂²⋊Q₈	Dic₁₂₇	1	4	yes	—
82	C₄×Q₈	C₂×D₁₂₇	1	4	yes	—
83	#117	C₂×D₁₂₇	1	2	yes	—
84	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
85	C₄⋊C₈	—	127	4	no	—
86	C₄⋊C₈	—	127	8	no	—
87	#97	—	127	4	no	—
88	C₄×C₈	—	127	2	no	—
89	C₄×C₈	—	127	2	no	—
90	C₄×C₈	—	127	4	no	—
91	C₂×Q₈	C₂×Dic₁₂₇	1	4	yes	—
92	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
94	M₁₆	—	254	4	no	—
95	C₂×C₈	—	127	4	no	—
96	C₂×C₈	—	127	1	no	—
97	C₂×C₈	—	127	1	no	—
98	C₂×C₈	—	127	2	no	—
99	C₂×C₈	—	127	4	no	—
100	C₂×C₈	—	254	2	no	—
101	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
102	C₄⋊C₄	C₂×Dic₁₂₇	1	8	yes	—
103	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
104	C₂²⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
105	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
106	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
107	C₄²	C₂²×D₁₂₇	1	2	yes	—
108	C₄²	C₂²×D₁₂₇	1	4	yes	—
109	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
113	C₂×C₄	C₂²×Dic₁₂₇	1	8	yes	—
114	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
115	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
117	C₂×C₄	#14523249	1	2	yes	—
118	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
119	C₈	—	127	2	no	—
120	C₈	—	127	2	no	—
121	C₈	—	254	2	no	—
122	C₈	—	254	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	Q₈×Dic₁₂₇	1	2	yes	—
125	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	Q₈×Dic₁₂₇	1	2	yes	—
129	C₄	#16861216	1	4	yes	—
130	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
131	C₄	C₂×#14523249	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×Q₈×Dic₁₂₇	1	1	yes	characteristic
134	C₂	C₂×#16861216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550007

C₂

yes

maximal, complemented

Q₈×C₁₂₇⋊C₈

C₂

yes

maximal, complemented

#22549903

C₂

yes

maximal

#22549900

C₂

yes

maximal, characteristic

#22549832

C₂

yes

maximal

C₂×C₄×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

#533×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

#16861120

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

C₄

yes

—

#125×C₁₂₇

C₄

yes

—

C₂²⋊Q₈×C₁₂₇

C₄

yes

—

C₄×Q₈×C₁₂₇

C₂²

yes

—

#117×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

Q₈

yes

complemented

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂³

yes

characteristic

C₂×Q₈×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

C₂³

yes

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₂×Q₈

yes

—

C₁₂₇⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄×Q₈

yes

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

#131

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

C₄×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₂₇

C₂×#131

yes

—

#2960

—

127

maximal, Sylow

C₁₂₇

#2960

yes

characteristic, complemented, Sylow

#533

D₁₂₇

yes

characteristic

#446

—

127

—

C₈×Q₈

—

127

—

#424

—

127

—

#423

—

127

—

#405

—

127

—

C₂×C₄×C₈

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₄⋊Q₈

Dic₁₂₇

yes

—

#125

Dic₁₂₇

yes

—

C₂²⋊Q₈

Dic₁₂₇

yes

—

C₄×Q₈

C₂×D₁₂₇

yes

—

#117

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₂×Q₈

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

100

C₂×C₈

—

254

—

101

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

102

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

103

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

104

C₂²⋊C₄

C₂×Dic₁₂₇

yes

—

105

C₄²

C₂×Dic₁₂₇

yes

characteristic

106

C₄²

C₂×Dic₁₂₇

yes

characteristic

107

C₄²

C₂²×D₁₂₇

yes

—

108

C₄²

C₂²×D₁₂₇

yes

—

109

C₂³

C₂²×Dic₁₂₇

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×Dic₁₂₇

yes

—

113

C₂×C₄

C₂²×Dic₁₂₇

yes

—

114

C₂×C₄

C₂²×Dic₁₂₇

yes

—

115

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

116

C₂×C₄

Q₈×D₁₂₇

yes

—

117

C₂×C₄

#14523249

yes

—

118

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

119

C₈

—

127

—

120

C₈

—

127

—

121

C₈

—

254

—

122

C₈

—

254

—

123

C₂²

—

124

C₂²

Q₈×Dic₁₂₇

yes

—

125

C₂²

C₂³×Dic₁₂₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

Q₈×Dic₁₂₇

yes

—

129

C₄

#16861216

yes

—

130

C₄

C₂×Q₈×D₁₂₇

yes

characteristic

131

C₄

C₂×#14523249

yes

characteristic

132

C₂

—

133

C₂

C₂×Q₈×Dic₁₂₇

yes

characteristic

134

C₂

C₂×#16861216

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#533×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#533	D₁₂₇
hypocenter	C₁₂₇	#2960
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂³×C₄

Fitting

#533×C₁₂₇

C₂

prime radical

C₂³×C₁₂₇

C₂²×C₄

cosocle

C₂×C₄×C₁₂₇

C₂⁴

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #22550007	C₂	semidirect
2. Q₈×C₁₂₇⋊C₈	C₂	semidirect
25. C₂×C₁₂₇⋊C₈	Q₈	semidirect
68. C₁₂₇	#2960	semidirect

H ≅ G/N

type

1. #22550007

C₂

semidirect

2. Q₈×C₁₂₇⋊C₈

C₂

semidirect

25. C₂×C₁₂₇⋊C₈

Q₈

semidirect

68. C₁₂₇

#2960

semidirect

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

subgroup	index
#22550007	2
Q₈×C₁₂₇⋊C₈	2
#22549903	2
#22549900	2
#22549832	2
C₂×C₄×C₁₂₇⋊C₈	2
#533×C₁₂₇	2
#2960	127

subgroup

index

Q₈×C₁₂₇⋊C₈

C₂×C₄×C₁₂₇⋊C₈

#533×C₁₂₇

#2960

127

Subgroups — #30647665

Conjugacy classes of subgroups

Complements

Maximal subgroups