Group #30640282 — subgroups

Subgroups — #30640282

Order 16256.

210 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 63 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	#22549904	C₂	1	2	yes	maximal, complemented
2	#22549892	C₂	1	2	yes	maximal
3	#22549844	C₂	1	1	yes	maximal, characteristic
4	#22549633	C₂	1	1	yes	maximal, characteristic, complemented
5	#430×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523248	C₂²	1	1	yes	characteristic
7	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
8	#16861168	C₂²	1	2	yes	—
9	Dic₁₂₇⋊C₈	—	2	4	no	—
10	#16861135	—	2	4	no	—
11	(C₂×D₁₂₇)⋊C₈	—	2	2	no	—
12	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	#108×C₁₂₇	—	2	2	no	—
15	#14523248	—	2	2	no	—
16	#14523248	—	2	2	no	—
17	C₂×Dic₂₅₄	Q₈	1	1	yes	characteristic
18	C₂×D₅₀₈	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
20	C₂×C₄×D₁₂₇	—	2	1	no	—
21	C₂×C₁₂₇⋊C₈	—	2	1	no	—
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₂²⋊Dic₁₂₇	—	2	2	no	—
25	C₄×Dic₁₂₇	D₄	1	4	yes	—
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₂₇	—	2	2	no	—
28	M₁₆×C₁₂₇	—	2	2	no	—
29	M₁₆×C₁₂₇	—	2	2	no	—
30	C₂×C₈×C₁₂₇	—	2	2	no	—
31	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
32	C₂²×D₁₂₇	—	2	1	no	—
33	Dic₂₅₄	—	2	2	no	—
34	D₅₀₈	—	2	2	no	—
35	C₂²⋊D₁₂₇	—	4	2	no	—
36	C₂×Dic₁₂₇	—	2	1	no	—
37	C₂×Dic₁₂₇	—	2	4	no	—
38	C₄×D₁₂₇	—	4	1	no	—
39	C₄×D₁₂₇	—	4	1	no	—
40	C₁₂₇⋊C₈	—	4	1	no	—
41	C₂³×C₁₂₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	#41	1	2	yes	—
46	C₈×C₁₂₇	—	2	2	no	—
47	C₈×C₁₂₇	—	4	2	no	—
48	C₂²×C₁₂₇	—	4	1	no	—
49	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₂₇	#123	1	2	yes	—
51	C₂×D₁₂₇	—	4	2	no	—
52	C₂×D₁₂₇	—	4	1	no	—
53	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
54	C₄×C₁₂₇	#123	1	1	yes	characteristic
55	C₄×C₁₂₇	#127	1	1	yes	characteristic
56	Dic₁₂₇	—	4	4	no	—
57	Dic₁₂₇	—	4	1	no	—
58	C₂×C₁₂₇	—	2	2	no	—
59	C₂×C₁₂₇	—	2	1	no	—
60	C₂×C₁₂₇	#399	1	1	yes	characteristic
61	D₁₂₇	—	8	1	no	—
62	#2098	—	127	1	no	maximal, Sylow
63	C₁₂₇	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	D₁₂₇	1	1	yes	characteristic, complemented
65	#424	—	127	2	no	—
66	#421	—	127	2	no	—
67	#408	—	127	1	no	—
68	H(3,ℤ/4)	—	127	1	no	—
69	C₂×#41	—	127	1	no	—
70	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
71	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
72	C₂²×C₈	—	127	1	no	—
73	#117	—	127	2	no	—
74	#108	—	2	2	no	—
75	C₄⋊C₈	—	254	4	no	—
76	#104	—	254	4	no	—
77	C₂²⋊C₈	—	254	2	no	—
78	#41	—	254	2	no	—
79	#41	—	254	2	no	—
80	C₂×Q₈	—	127	1	no	—
81	C₂×D₄	—	127	1	no	—
82	C₂×D₄	—	254	1	no	—
83	C₂²×C₄	—	254	1	no	—
84	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	254	2	no	—
90	C₂×C₈	—	254	1	no	—
91	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
92	C₄⋊C₄	—	254	2	no	—
93	C₂²⋊C₄	—	254	2	no	—
94	C₄²	—	127	4	no	—
95	C₂³	—	254	1	no	—
96	C₂³	#14523249	1	1	yes	characteristic
97	Q₈	—	254	2	no	—
98	D₄	—	508	2	no	—
99	D₄	—	254	2	no	—
100	C₂×C₄	—	508	1	no	—
101	C₂×C₄	—	508	1	no	—
102	C₂×C₄	—	254	1	no	—
103	C₂×C₄	—	254	4	no	—
104	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
107	C₂×C₄	#14523248	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	508	1	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	508	2	no	—
113	C₂²	—	508	1	no	—
114	C₂²	#16861185	1	1	yes	characteristic
115	C₂²	#16861194	1	2	yes	—
116	C₄	—	508	4	no	—
117	C₄	—	508	1	no	—
118	C₄	#16861187	1	2	yes	—
119	C₄	#16861192	1	1	yes	characteristic
120	C₄	#16861205	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1016	1	no	—
124	C₂	#22549811	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549904

C₂

yes

maximal, complemented

#22549892

C₂

yes

maximal

#22549844

C₂

yes

maximal, characteristic

#22549633

C₂

yes

maximal, characteristic, complemented

#430×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

#16861168

C₂²

yes

—

Dic₁₂₇⋊C₈

—

#16861135

—

(C₂×D₁₂₇)⋊C₈

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

#108×C₁₂₇

—

#14523248

—

#14523248

—

C₂×Dic₂₅₄

Q₈

yes

characteristic

C₂×D₅₀₈

Q₈

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₂₇

#123

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#399

yes

characteristic

D₁₂₇

—

#2098

—

127

maximal, Sylow

C₁₂₇

#2098

yes

characteristic, complemented, Sylow

#430

D₁₂₇

yes

characteristic, complemented

#424

—

127

—

#421

—

127

—

#408

—

127

—

H(3,ℤ/4)

—

127

—

C₂×#41

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

#117

—

127

—

#108

—

C₄⋊C₈

—

254

—

#104

—

254

—

C₂²⋊C₈

—

254

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₂²⋊D₁₂₇

yes

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₂³

—

254

—

C₂³

#14523249

yes

characteristic

Q₈

—

254

—

D₄

—

508

—

D₄

—

254

—

100

C₂×C₄

—

508

—

101

C₂×C₄

—

508

—

102

C₂×C₄

—

254

—

103

C₂×C₄

—

254

—

104

C₂×C₄

D₄×D₁₂₇

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

106

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

107

C₂×C₄

#14523248

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

508

—

111

C₂²

—

112

C₂²

—

508

—

113

C₂²

—

508

—

114

C₂²

#16861185

yes

characteristic

115

C₂²

#16861194

yes

—

116

C₄

—

508

—

117

C₄

—

508

—

118

C₄

#16861187

yes

—

119

C₄

#16861192

yes

characteristic

120

C₄

#16861205

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1016

—

124

C₂

#22549811

yes

characteristic

	subgroup	G/H
center	C₄	#16861205
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#430×C₁₂₇	C₂
socle	C₂×C₁₂₇	#399
prime radical	C₂×#14523248	C₂²
hypercenter	#430	D₁₂₇
hypocenter	C₁₂₇	#2098
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

#16861205

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#430×C₁₂₇

C₂

C₂×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
1. #22549904	C₂	semidirect
4. #22549633	C₂	semidirect
5. #430×C₁₂₇	C₂	semidirect
63. C₁₂₇	#2098	semidirect
64. #430	D₁₂₇	semidirect

H ≅ G/N

type

1. #22549904

C₂

semidirect

4. #22549633

C₂

semidirect

5. #430×C₁₂₇

C₂

semidirect

63. C₁₂₇

#2098

semidirect

64. #430

D₁₂₇

semidirect

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

subgroup	index
#22549904	2
#22549892	2
#22549844	2
#22549633	2
#430×C₁₂₇	2
#2098	127

subgroup

index

#430×C₁₂₇

127

Subgroups — #30640282

Conjugacy classes of subgroups

Complements

Maximal subgroups