Group #30638211 — subgroups

Subgroups — #30638211

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	#22550073	C₂	1	2	yes	maximal, complemented
2	#22550006	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549627	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549620	C₂	1	2	yes	maximal, complemented
5	#493×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523236	—	2	2	no	—
7	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
8	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
9	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
10	Dic₁₂₇⋊C₈	—	2	1	no	—
11	#16861130	—	2	2	no	—
12	#16861119	—	2	1	no	—
13	C₂×D₈×C₁₂₇	—	2	1	no	—
14	C₂×D₈×C₁₂₇	—	2	1	no	—
15	#127×C₁₂₇	C₂²	1	2	yes	—
16	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
18	#14523236	—	4	4	no	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₂×C₁₂₇⋊C₈	—	2	1	no	—
21	C₄⋊Dic₁₂₇	D₄	1	1	yes	characteristic
22	Dic₁₂₇⋊C₄	—	2	1	no	—
23	C₄×Dic₁₂₇	—	2	1	no	—
24	C₂×D₄×C₁₂₇	—	2	2	no	—
25	C₂×D₄×C₁₂₇	—	2	2	no	—
26	C₂×D₄×C₁₂₇	—	2	2	no	—
27	D₈×C₁₂₇	—	4	2	no	—
28	D₈×C₁₂₇	—	4	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	Dic₂₅₄	—	2	2	no	—
33	C₂×Dic₁₂₇	—	2	1	no	—
34	C₂×Dic₁₂₇	—	2	1	no	—
35	C₁₂₇⋊C₈	—	4	1	no	—
36	C₁₂₇⋊C₈	—	4	2	no	—
37	C₂³×C₁₂₇	—	4	2	no	—
38	D₄×C₁₂₇	—	4	2	no	—
39	D₄×C₁₂₇	—	4	2	no	—
40	D₄×C₁₂₇	—	4	2	no	—
41	D₄×C₁₂₇	—	4	2	no	—
42	D₄×C₁₂₇	—	8	2	no	—
43	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
46	C₈×C₁₂₇	—	2	1	no	—
47	C₈×C₁₂₇	SD₁₆	1	2	yes	complemented
48	C₂²×C₁₂₇	—	8	2	no	—
49	C₂²×C₁₂₇	—	8	2	no	—
50	C₂²×C₁₂₇	—	8	2	no	—
51	C₂²×C₁₂₇	#121	1	1	yes	characteristic
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	—	16	2	no	—
59	C₂×C₁₂₇	#460	1	1	yes	characteristic
60	C₂×C₁₂₇	#469	1	1	yes	characteristic
61	C₂×C₁₂₇	#472	1	1	yes	characteristic
62	#1688	—	127	1	no	maximal, Sylow
63	C₁₂₇	#1688	1	1	yes	characteristic, complemented, Sylow
64	#493	D₁₂₇	1	1	yes	characteristic
65	#460	—	127	2	no	—
66	#446	—	127	1	no	—
67	#334	—	127	1	no	—
68	#331	—	127	2	no	—
69	C₂×SD₁₆	—	254	2	no	—
70	C₂×D₈	—	2	1	no	—
71	C₂×D₈	—	2	1	no	—
72	#127	C₂×D₁₂₇	1	2	yes	—
73	C₄×Q₈	—	127	1	no	—
74	C₄⋊C₈	—	127	1	no	—
75	C₄⋊C₈	—	127	2	no	—
76	C₄⋊C₈	—	254	1	no	—
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	—	2	2	no	—
82	C₂×D₄	—	2	2	no	—
83	C₂×D₄	—	2	2	no	—
84	SD₁₆	—	508	4	no	—
85	D₈	—	4	2	no	—
86	D₈	—	4	2	no	—
87	C₂×C₈	—	254	2	no	—
88	C₂×C₈	—	254	1	no	—
89	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₁₂₇	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	—	4	2	no	—
96	Q₈	—	254	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	8	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×C₂²⋊D₁₂₇	1	1	yes	characteristic
106	C₂×C₄	#14523247	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	508	2	no	—
109	C₈	—	508	1	no	—
110	C₈	#14523236	1	2	yes	complemented
111	C₂²	—	8	2	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	8	2	no	—
114	C₂²	#16861180	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	508	1	no	—
117	C₄	—	508	1	no	—
118	C₄	C₂×#14523236	1	1	yes	characteristic
119	C₄	#16861233	1	2	yes	—
120	C₄	#16861235	1	1	yes	characteristic
121	C₂	—	16	2	no	—
122	C₂	#22550073	1	1	yes	characteristic
123	C₂	#22550126	1	1	yes	characteristic
124	C₂	#22550143	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550073

C₂

yes

maximal, complemented

#22550006

C₂

yes

maximal, characteristic, complemented

#22549627

C₂

yes

maximal, characteristic, complemented

#22549620

C₂

yes

maximal, complemented

#493×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523236

—

C₄×Dic₂₅₄

C₂²

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

characteristic

Dic₁₂₇⋊C₈

—

#16861130

—

#16861119

—

C₂×D₈×C₁₂₇

—

C₂×D₈×C₁₂₇

—

#127×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

D₄

yes

characteristic

#14523236

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

D₄

yes

characteristic

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

D₈×C₁₂₇

—

D₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

SD₁₆

yes

complemented

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

characteristic

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

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1688

yes

characteristic, complemented, Sylow

#493

D₁₂₇

yes

characteristic

#460

—

127

—

#446

—

127

—

#334

—

127

—

#331

—

127

—

C₂×SD₁₆

—

254

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₁₂₇

yes

—

C₄×Q₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

#102

—

254

—

#97

—

254

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

508

—

D₈

—

D₈

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄²

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

—

Q₈

—

254

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

254

—

103

C₂×C₄

—

254

—

104

C₂×C₄

D₄×D₁₂₇

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

106

C₂×C₄

#14523247

yes

characteristic

107

C₈

—

108

C₈

—

508

—

109

C₈

—

508

—

110

C₈

#14523236

yes

complemented

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#16861180

yes

characteristic

115

C₄

—

116

C₄

—

508

—

117

C₄

—

508

—

118

C₄

C₂×#14523236

yes

characteristic

119

C₄

#16861233

yes

—

120

C₄

#16861235

yes

characteristic

121

C₂

—

122

C₂

#22550073

yes

characteristic

123

C₂

#22550126

yes

characteristic

124

C₂

#22550143

yes

characteristic

	subgroup	G/H
center	C₂²	#16861180
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting, prime radical	#493×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
hypercenter	#493	D₁₂₇
hypocenter	C₁₂₇	#1688
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#16861180

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting, prime radical

#493×C₁₂₇

C₂

C₂²×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #22550073	C₂	semidirect
2. #22550006	C₂	semidirect
3. #22549627	C₂	semidirect
4. #22549620	C₂	semidirect
47. C₈×C₁₂₇	SD₁₆	semidirect
63. C₁₂₇	#1688	semidirect
110. C₈	#14523236	semidirect

H ≅ G/N

type

1. #22550073

C₂

semidirect

2. #22550006

C₂

semidirect

3. #22549627

C₂

semidirect

4. #22549620

C₂

semidirect

47. C₈×C₁₂₇

SD₁₆

semidirect

63. C₁₂₇

#1688

semidirect

110. C₈

#14523236

semidirect

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

subgroup	index
#22550073	2
#22550006	2
#22549627	2
#22549620	2
#493×C₁₂₇	2
#1688	127

subgroup

index

#493×C₁₂₇

127

Subgroups — #30638211

Conjugacy classes of subgroups

Complements

Maximal subgroups