Group #30640974 — subgroups

Subgroups — #30640974

Order 16256.

132 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 47 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	#22550277	C₂	1	1	yes	maximal, characteristic
2	#22550254	C₂	1	1	yes	maximal, characteristic
3	#22549681	C₂	1	1	yes	maximal, characteristic
4	#22549680	C₂	1	1	yes	maximal, characteristic
5	#22549679	C₂	1	1	yes	maximal, characteristic
6	#22549651	C₂	1	1	yes	maximal, characteristic
7	#358×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
9	C₂×#14523238	—	2	1	no	—
10	#16861207	—	2	1	no	—
11	#16861200	—	2	1	no	—
12	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16861143	C₂²	1	1	yes	characteristic
14	#16861140	—	2	1	no	—
15	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
18	#107×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₅₄	—	2	1	no	—
20	Dic₅₀₈	—	2	1	no	—
21	Dic₅₀₈	—	2	1	no	—
22	#14523238	—	4	2	no	—
23	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₂₇	—	2	1	no	—
25	C₄⋊Dic₁₂₇	—	4	1	no	—
26	Dic₁₂₇⋊C₄	—	4	1	no	—
27	Dic₁₂₇⋊C₄	—	4	1	no	—
28	Dic₁₂₇⋊C₄	—	4	1	no	—
29	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
30	C₁₂₇⋊C₁₆	—	2	1	no	—
31	C₂×Q₈×C₁₂₇	—	2	1	no	—
32	Q₁₆×C₁₂₇	—	2	1	no	—
33	Q₁₆×C₁₂₇	—	2	1	no	—
34	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₂₇	—	2	1	no	—
36	C₁₆×C₁₂₇	—	2	1	no	—
37	Dic₂₅₄	—	4	1	no	—
38	Dic₂₅₄	—	4	1	no	—
39	C₂×Dic₁₂₇	—	4	1	no	—
40	C₂×Dic₁₂₇	—	4	1	no	—
41	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
42	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
43	C₁₂₇⋊C₈	SD₁₆	1	2	yes	—
44	Q₈×C₁₂₇	—	4	1	no	—
45	Q₈×C₁₂₇	—	4	1	no	—
46	C₂×C₄×C₁₂₇	—	4	1	no	—
47	C₂×C₄×C₁₂₇	—	4	1	no	—
48	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₂₇	#41	1	1	yes	characteristic
50	C₈×C₁₂₇	#41	1	1	yes	characteristic
51	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
52	C₄×C₁₂₇	—	8	1	no	—
53	C₄×C₁₂₇	—	8	1	no	—
54	C₄×C₁₂₇	#124	1	1	yes	characteristic
55	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₁₂₇	—	2	1	no	—
57	Dic₁₂₇	—	2	1	no	—
58	Dic₁₂₇	—	8	1	no	—
59	Dic₁₂₇	—	8	1	no	—
60	C₂×C₁₂₇	#486	1	1	yes	characteristic
61	C₂×C₁₂₇	#510	1	1	yes	characteristic
62	C₂×C₁₂₇	#510	1	1	yes	characteristic
63	#2245	—	127	1	no	maximal, Sylow
64	C₁₂₇	#2245	1	1	yes	characteristic, complemented, Sylow
65	#499	—	127	1	no	—
66	#494	—	127	1	no	—
67	#358	—	127	1	no	—
68	#358	—	127	1	no	—
69	#358	—	127	1	no	—
70	#358	D₁₂₇	1	1	yes	characteristic
71	#346	—	127	1	no	—
72	C₂×Q₁₆	—	127	1	no	—
73	C₂×Q₁₆	—	254	1	no	—
74	C₂×Q₁₆	C₂×D₁₂₇	1	1	yes	characteristic
75	C₄⋊Q₈	—	254	1	no	—
76	#125	—	254	1	no	—
77	C₂×C₁₆	—	127	1	no	—
78	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
79	#107	—	127	1	no	—
80	#107	C₂×D₁₂₇	1	1	yes	characteristic
81	#106	—	254	1	no	—
82	C₄×C₈	—	127	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	254	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	508	2	no	—
88	Q₁₆	—	254	1	no	—
89	Q₁₆	—	254	1	no	—
90	C₂×C₈	—	127	1	no	—
91	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	508	1	no	—
94	C₄⋊C₄	—	508	1	no	—
95	C₄⋊C₄	—	508	1	no	—
96	C₄⋊C₄	—	508	1	no	—
97	C₄⋊C₄	—	254	1	no	—
98	C₄²	—	127	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	254	1	no	—
101	Q₈	—	4	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	508	1	no	—
104	Q₈	—	508	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	127	1	no	—
108	C₂×C₄	—	127	1	no	—
109	C₂×C₄	—	508	1	no	—
110	C₂×C₄	—	508	1	no	—
111	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
112	C₈	—	127	2	no	—
113	C₈	#14523247	1	1	yes	characteristic
114	C₈	#14523248	1	1	yes	characteristic
115	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	1016	1	no	—
119	C₄	—	1016	1	no	—
120	C₄	—	254	1	no	—
121	C₄	—	254	1	no	—
122	C₄	#16861195	1	1	yes	characteristic
123	C₄	SD₁₆×D₁₂₇	1	1	yes	characteristic
124	C₂	#22550221	1	1	yes	characteristic
125	C₂	#22550315	1	1	yes	characteristic
126	C₂	#22550318	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550277

C₂

yes

maximal, characteristic

#22550254

C₂

yes

maximal, characteristic

#22549681

C₂

yes

maximal, characteristic

#22549680

C₂

yes

maximal, characteristic

#22549679

C₂

yes

maximal, characteristic

#22549651

C₂

yes

maximal, characteristic

#358×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

C₂²

yes

—

—

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

characteristic

#16861143

C₂²

yes

characteristic

#16861140

—

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

characteristic

#107×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

Dic₅₀₈

—

Dic₅₀₈

—

#14523238

—

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₁₆

—

C₂×Q₈×C₁₂₇

—

Q₁₆×C₁₂₇

—

Q₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₁₆×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₈

yes

characteristic

C₂×Dic₁₂₇

D₈

yes

characteristic

C₁₂₇⋊C₈

SD₁₆

yes

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

C₈×C₁₂₇

yes

C₈×C₁₂₇

yes

C₂²×C₁₂₇

C₂×D₈

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#124

yes

characteristic

C₄×C₁₂₇

C₂×SD₁₆

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2245

yes

characteristic, complemented, Sylow

#499

—

127

—

#494

—

127

—

#358

—

127

—

#358

—

127

—

#358

—

127

—

#358

D₁₂₇

yes

characteristic

#346

—

127

—

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

C₂×D₁₂₇

yes

characteristic

C₄⋊Q₈

—

254

—

#125

—

254

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

characteristic

#107

—

127

—

#107

C₂×D₁₂₇

yes

characteristic

#106

—

254

—

C₄×C₈

—

127

—

C₂×Q₈

—

C₂×Q₈

—

254

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

508

—

Q₁₆

—

254

—

Q₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

254

—

C₄²

—

127

—

C₁₆

—

100

C₁₆

—

254

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

508

—

104

Q₈

—

508

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

127

—

108

C₂×C₄

—

127

—

109

C₂×C₄

—

508

—

110

C₂×C₄

—

508

—

111

C₂×C₄

D₄×D₁₂₇

yes

characteristic

112

C₈

—

127

—

113

C₈

#14523247

yes

characteristic

114

C₈

#14523248

yes

characteristic

115

C₂²

D₈×D₁₂₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

1016

—

119

C₄

—

1016

—

120

C₄

—

254

—

121

C₄

—

254

—

122

C₄

#16861195

yes

characteristic

123

C₄

SD₁₆×D₁₂₇

yes

characteristic

124

C₂

#22550221

yes

characteristic

125

C₂

#22550315

yes

characteristic

126

C₂

#22550318

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#358×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	#358	D₁₂₇
hypocenter	C₁₂₇	#2245
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₂₇

commutator, cosocle

C₂×C₈×C₁₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#358×C₁₂₇

C₂

socle, prime radical

C₂²×C₁₂₇

C₂×D₈

D₁₂₇

C₁₂₇

G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
64. C₁₂₇	#2245	semidirect

H ≅ G/N

type

64. C₁₂₇

#2245

semidirect

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

subgroup	index
#22550277	2
#22550254	2
#22549681	2
#22549680	2
#22549679	2
#22549651	2
#358×C₁₂₇	2
#2245	127

subgroup

index

#358×C₁₂₇

127

Subgroups — #30640974

Conjugacy classes of subgroups

Complements

Maximal subgroups