Group #30638215 — subgroups

Subgroups — #30638215

Order 16256.

184 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 55 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	#22550253	C₂	1	1	yes	maximal, characteristic
2	#22550077	C₂	1	2	yes	maximal, complemented
3	#22550006	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549619	C₂	1	2	yes	maximal, complemented
5	#334×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523237	—	2	2	no	—
7	C₂×D₁₀₁₆	—	2	1	no	—
8	C₂×D₁₀₁₆	—	2	1	no	—
9	#16861206	C₂²	1	2	yes	—
10	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₁₂₇⋊C₈	—	2	1	no	—
13	#16861131	—	2	2	no	—
14	#16861119	—	2	1	no	—
15	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
16	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
18	C₂×D₅₀₈	—	2	2	no	—
19	C₂×D₅₀₈	—	2	2	no	—
20	C₂×D₅₀₈	—	2	2	no	—
21	#14523237	—	4	4	no	—
22	D₁₀₁₆	—	4	2	no	—
23	D₁₀₁₆	—	4	2	no	—
24	C₂×C₁₂₇⋊C₈	—	2	1	no	—
25	C₄⋊Dic₁₂₇	D₄	1	1	yes	characteristic
26	Dic₁₂₇⋊C₄	—	2	1	no	—
27	C₄×Dic₁₂₇	—	2	1	no	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
31	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²×D₁₂₇	—	4	2	no	—
33	Dic₂₅₄	—	2	2	no	—
34	D₅₀₈	—	4	2	no	—
35	D₅₀₈	—	4	2	no	—
36	D₅₀₈	—	4	2	no	—
37	D₅₀₈	—	4	2	no	—
38	D₅₀₈	—	8	2	no	—
39	C₂×Dic₁₂₇	—	2	1	no	—
40	C₂×Dic₁₂₇	—	2	1	no	—
41	C₁₂₇⋊C₈	—	4	1	no	—
42	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
45	C₈×C₁₂₇	—	2	1	no	—
46	C₈×C₁₂₇	—	4	2	no	—
47	C₈×C₁₂₇	SD₁₆	1	2	yes	complemented
48	C₂²×C₁₂₇	#121	1	1	yes	characteristic
49	C₂×D₁₂₇	—	8	2	no	—
50	C₂×D₁₂₇	—	8	2	no	—
51	C₂×D₁₂₇	—	8	2	no	—
52	C₄×C₁₂₇	—	2	1	no	—
53	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
56	Dic₁₂₇	—	4	1	no	—
57	Dic₁₂₇	—	4	1	no	—
58	C₂×C₁₂₇	#460	1	1	yes	characteristic
59	C₂×C₁₂₇	#469	1	1	yes	characteristic
60	C₂×C₁₂₇	#472	1	1	yes	characteristic
61	D₁₂₇	—	16	2	no	—
62	#1688	—	127	1	no	maximal, Sylow
63	C₁₂₇	#1688	1	1	yes	characteristic, complemented, Sylow
64	#493	—	127	1	no	—
65	#460	—	127	2	no	—
66	#446	—	127	1	no	—
67	#334	D₁₂₇	1	1	yes	characteristic, complemented
68	#331	—	127	2	no	—
69	C₂×SD₁₆	—	254	2	no	—
70	C₂×D₈	—	254	1	no	—
71	C₂×D₈	—	254	1	no	—
72	#127	—	127	2	no	—
73	C₄×Q₈	—	127	1	no	—
74	C₄⋊C₈	—	127	1	no	—
75	C₄⋊C₈	—	254	1	no	—
76	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
77	#102	—	254	2	no	—
78	#97	—	254	1	no	—
79	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
80	C₂×Q₈	—	127	1	no	—
81	C₂×D₄	—	254	2	no	—
82	C₂×D₄	—	254	2	no	—
83	C₂×D₄	—	254	2	no	—
84	SD₁₆	—	508	4	no	—
85	D₈	—	508	2	no	—
86	D₈	—	508	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	254	1	no	—
89	C₂×C₈	D₅₀₈	1	1	yes	characteristic
90	C₂×C₈	D₅₀₈	1	1	yes	characteristic
91	C₄⋊C₄	—	127	1	no	—
92	C₄⋊C₄	—	254	1	no	—
93	C₄²	—	254	1	no	—
94	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
95	C₂³	—	508	2	no	—
96	Q₈	—	254	2	no	—
97	D₄	—	1016	2	no	—
98	D₄	—	508	2	no	—
99	D₄	—	508	2	no	—
100	D₄	—	508	2	no	—
101	D₄	—	508	2	no	—
102	C₂×C₄	—	254	1	no	—
103	C₂×C₄	—	254	1	no	—
104	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
105	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
106	C₂×C₄	#14523249	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	4	2	no	—
109	C₈	—	508	1	no	—
110	C₈	#14523237	1	2	yes	complemented
111	C₂²	—	1016	2	no	—
112	C₂²	—	1016	2	no	—
113	C₂²	—	1016	2	no	—
114	C₂²	#16861184	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	508	1	no	—
117	C₄	—	508	1	no	—
118	C₄	C₂×#14523237	1	1	yes	characteristic
119	C₄	#16861236	1	2	yes	—
120	C₄	#16861237	1	1	yes	characteristic
121	C₂	—	2032	2	no	—
122	C₂	#22550077	1	1	yes	characteristic
123	C₂	#22550130	1	1	yes	characteristic
124	C₂	#22550147	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550253

C₂

yes

maximal, characteristic

#22550077

C₂

yes

maximal, complemented

#22550006

C₂

yes

maximal, characteristic, complemented

#22549619

C₂

yes

maximal, complemented

#334×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523237

—

C₂×D₁₀₁₆

—

C₂×D₁₀₁₆

—

#16861206

C₂²

yes

—

C₄×Dic₂₅₄

C₂²

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

characteristic

Dic₁₂₇⋊C₈

—

#16861131

—

#16861119

—

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

D₄

yes

characteristic

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

#14523237

—

D₁₀₁₆

—

D₁₀₁₆

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

D₄

yes

characteristic

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

SD₁₆

yes

complemented

C₂²×C₁₂₇

#121

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₂₇

AGL(1,ℤ/8)

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

#1688

—

127

maximal, Sylow

C₁₂₇

#1688

yes

characteristic, complemented, Sylow

#493

—

127

—

#460

—

127

—

#446

—

127

—

#334

D₁₂₇

yes

characteristic, complemented

#331

—

127

—

C₂×SD₁₆

—

254

—

C₂×D₈

—

254

—

C₂×D₈

—

254

—

#127

—

127

—

C₄×Q₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

#102

—

254

—

#97

—

254

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

SD₁₆

—

508

—

D₈

—

508

—

D₈

—

508

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

D₅₀₈

yes

characteristic

C₂×C₈

D₅₀₈

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄²

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

—

508

—

Q₈

—

254

—

D₄

—

1016

—

D₄

—

508

—

D₄

—

508

—

100

D₄

—

508

—

101

D₄

—

508

—

102

C₂×C₄

—

254

—

103

C₂×C₄

—

254

—

104

C₂×C₄

D₄×D₁₂₇

yes

characteristic

105

C₂×C₄

C₂×D₅₀₈

yes

characteristic

106

C₂×C₄

#14523249

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

508

—

110

C₈

#14523237

yes

complemented

111

C₂²

—

1016

—

112

C₂²

—

1016

—

113

C₂²

—

1016

—

114

C₂²

#16861184

yes

characteristic

115

C₄

—

116

C₄

—

508

—

117

C₄

—

508

—

118

C₄

C₂×#14523237

yes

characteristic

119

C₄

#16861236

yes

—

120

C₄

#16861237

yes

characteristic

121

C₂

—

2032

—

122

C₂

#22550077

yes

characteristic

123

C₂

#22550130

yes

characteristic

124

C₂

#22550147

yes

characteristic

	subgroup	G/H
center	C₂²	#16861184
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#334×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
prime radical	#22550253	C₂
hypercenter	#334	D₁₂₇
hypocenter	C₁₂₇	#1688
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#16861184

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#334×C₁₂₇

C₂

C₂²×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #22550077	C₂	semidirect
3. #22550006	C₂	semidirect
4. #22549619	C₂	semidirect
5. #334×C₁₂₇	C₂	semidirect
47. C₈×C₁₂₇	SD₁₆	semidirect
63. C₁₂₇	#1688	semidirect
67. #334	D₁₂₇	semidirect
110. C₈	#14523237	semidirect

H ≅ G/N

type

2. #22550077

C₂

semidirect

3. #22550006

C₂

semidirect

4. #22549619

C₂

semidirect

5. #334×C₁₂₇

C₂

semidirect

47. C₈×C₁₂₇

SD₁₆

semidirect

63. C₁₂₇

#1688

semidirect

67. #334

D₁₂₇

semidirect

110. C₈

#14523237

semidirect

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

subgroup	index
#22550253	2
#22550077	2
#22550006	2
#22549619	2
#334×C₁₂₇	2
#1688	127

subgroup

index

#334×C₁₂₇

127

Subgroups — #30638215

Conjugacy classes of subgroups

Complements

Maximal subgroups