Group #30650529 — subgroups

Subgroups — #30650529

Order 16256.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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	#22550677	C₂	1	4	yes	maximal, complemented
2	#22550143	C₂	1	4	yes	maximal, complemented
3	#22550043	C₂	1	4	yes	maximal, complemented
4	#22549994	C₂	1	2	yes	maximal, complemented
5	#493×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861254	C₂²	1	4	yes	complemented
7	#16861235	C₂²	1	8	yes	complemented
8	#16861233	—	2	8	no	—
9	#16861231	C₂²	1	2	yes	—
10	#16861228	—	2	4	no	—
11	D₈×D₁₂₇	—	2	4	no	—
12	#16861215	C₂²	1	4	yes	complemented
13	#16861145	C₂²	1	2	yes	complemented
14	#16861137	C₂²	1	4	yes	complemented
15	#16861126	C₂²	1	4	yes	complemented
16	C₂×D₈×C₁₂₇	C₂²	1	4	yes	complemented
17	#127×C₁₂₇	C₂²	1	2	yes	complemented
18	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
19	#14523248	C₂³	1	4	yes	—
20	#14523247	—	2	8	no	—
21	C₂×C₂²⋊D₁₂₇	—	4	4	no	—
22	D₄×D₁₂₇	—	2	8	no	—
23	Dic₅₀₈	D₄	1	2	yes	complemented
24	#14523237	D₄	1	4	yes	complemented
25	#14523236	—	2	8	no	—
26	D₁₀₁₆	D₄	1	2	yes	complemented
27	#14523233	—	2	8	no	—
28	#14523232	C₂³	1	4	yes	—
29	#14523231	—	2	4	no	—
30	C₈×D₁₂₇	—	2	4	no	—
31	C₂×D₄×C₁₂₇	—	4	2	no	—
32	C₂×D₄×C₁₂₇	C₂³	1	4	yes	—
33	D₈×C₁₂₇	—	2	4	no	—
34	D₈×C₁₂₇	—	2	4	no	—
35	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
36	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×D₁₂₇	—	4	4	no	—
38	Dic₂₅₄	C₂×D₄	1	4	yes	—
39	D₅₀₈	C₂×D₄	1	4	yes	—
40	C₂²⋊D₁₂₇	—	2	4	no	—
41	C₂²⋊D₁₂₇	—	4	8	no	—
42	C₂×Dic₁₂₇	—	4	4	no	—
43	C₄×D₁₂₇	—	2	4	no	—
44	C₁₂₇⋊C₈	—	2	4	no	—
45	C₂³×C₁₂₇	—	2	4	no	—
46	D₄×C₁₂₇	—	2	8	no	—
47	D₄×C₁₂₇	—	8	4	no	—
48	C₂×C₄×C₁₂₇	—	2	1	no	—
49	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
50	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
51	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
52	C₂²×C₁₂₇	—	4	4	no	—
53	C₂²×C₁₂₇	—	8	4	no	—
54	C₂²×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂×D₁₂₇	—	2	4	no	—
56	C₂×D₁₂₇	—	16	4	no	—
57	C₄×C₁₂₇	—	2	2	no	—
58	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
59	Dic₁₂₇	—	2	4	no	—
60	C₂×C₁₂₇	—	2	1	no	—
61	C₂×C₁₂₇	—	8	4	no	—
62	C₂×C₁₂₇	D₄²	1	1	yes	characteristic
63	D₁₂₇	—	4	4	no	—
64	#3292	—	127	1	no	maximal, Sylow
65	C₁₂₇	#3292	1	1	yes	characteristic, complemented, Sylow
66	#576	—	127	4	no	—
67	#493	D₁₂₇	1	1	yes	characteristic, complemented
68	#472	—	127	4	no	—
69	#453	—	127	4	no	—
70	#443	—	127	2	no	—
71	H(5,𝔽₂)	—	127	4	no	—
72	AGL(1,ℤ/8)	—	127	8	no	—
73	AGL(1,ℤ/8)	—	254	8	no	—
74	#135	—	127	2	no	—
75	#135	—	254	4	no	—
76	C₂×D₈	—	254	4	no	—
77	C₂×D₈	C₂×D₁₂₇	1	4	yes	complemented
78	#131	—	127	4	no	—
79	#127	C₂×D₁₂₇	1	2	yes	complemented
80	#108	—	127	2	no	—
81	#104	—	127	4	no	—
82	#100	—	127	4	no	—
83	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
84	#41	—	127	4	no	—
85	#41	—	254	8	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	508	4	no	—
88	C₂×D₄	—	254	8	no	—
89	C₂×D₄	C₂²×D₁₂₇	1	4	yes	—
90	Q₁₆	—	127	2	no	—
91	SD₁₆	—	127	4	no	—
92	SD₁₆	—	254	8	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	127	2	no	—
96	D₈	—	254	8	no	—
97	M₁₆	—	127	4	no	—
98	M₁₆	—	254	4	no	—
99	C₂×C₈	—	254	4	no	—
100	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
101	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	508	4	no	—
104	Q₈	—	127	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	127	4	no	—
108	D₄	—	508	8	no	—
109	D₄	—	254	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	508	4	no	—
112	C₂×C₄	—	254	4	no	—
113	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
114	C₈	—	254	4	no	—
115	C₈	D₄×D₁₂₇	1	2	yes	—
116	C₈	D₄×D₁₂₇	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	2032	4	no	—
120	C₂²	—	254	4	no	—
121	C₂²	#16861254	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	254	4	no	—
124	C₄	C₂×D₄×D₁₂₇	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	508	4	no	—
128	C₂	#22550511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550677

C₂

yes

maximal, complemented

#22550143

C₂

yes

maximal, complemented

#22550043

C₂

yes

maximal, complemented

#22549994

C₂

yes

maximal, complemented

#493×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861254

C₂²

yes

complemented

#16861235

C₂²

yes

complemented

#16861233

—

#16861231

C₂²

yes

—

#16861228

—

D₈×D₁₂₇

—

#16861215

C₂²

yes

complemented

#16861145

C₂²

yes

complemented

#16861137

C₂²

yes

complemented

#16861126

C₂²

yes

complemented

C₂×D₈×C₁₂₇

C₂²

yes

complemented

#127×C₁₂₇

C₂²

yes

complemented

C₄×C₈×C₁₂₇

C₂²

yes

characteristic, complemented

#14523248

C₂³

yes

—

#14523247

—

C₂×C₂²⋊D₁₂₇

—

D₄×D₁₂₇

—

Dic₅₀₈

D₄

yes

complemented

#14523237

D₄

yes

complemented

#14523236

—

D₁₀₁₆

D₄

yes

complemented

#14523233

—

#14523232

C₂³

yes

—

#14523231

—

C₈×D₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂³

yes

—

D₈×C₁₂₇

—

D₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

—

Dic₂₅₄

C₂×D₄

yes

—

D₅₀₈

C₂×D₄

yes

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

—

C₈×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×D₄

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

D₄²

yes

characteristic

D₁₂₇

—

#3292

—

127

maximal, Sylow

C₁₂₇

#3292

yes

characteristic, complemented, Sylow

#576

—

127

—

#493

D₁₂₇

yes

characteristic, complemented

#472

—

127

—

#453

—

127

—

#443

—

127

—

H(5,𝔽₂)

—

127

—

AGL(1,ℤ/8)

—

127

—

AGL(1,ℤ/8)

—

254

—

#135

—

127

—

#135

—

254

—

C₂×D₈

—

254

—

C₂×D₈

C₂×D₁₂₇

yes

complemented

#131

—

127

—

#127

C₂×D₁₂₇

yes

complemented

#108

—

127

—

#104

—

127

—

#100

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic, complemented

#41

—

127

—

#41

—

254

—

C₂×D₄

—

C₂×D₄

—

508

—

C₂×D₄

—

254

—

C₂×D₄

C₂²×D₁₂₇

yes

—

Q₁₆

—

127

—

SD₁₆

—

127

—

SD₁₆

—

254

—

D₈

—

D₈

—

D₈

—

127

—

D₈

—

254

—

M₁₆

—

127

—

M₁₆

—

254

—

C₂×C₈

—

254

—

100

C₂×C₈

C₂²×D₁₂₇

yes

—

101

C₄²

C₂²×D₁₂₇

yes

characteristic

102

C₂³

—

103

C₂³

—

508

—

104

Q₈

—

127

—

105

D₄

—

106

D₄

—

107

D₄

—

127

—

108

D₄

—

508

—

109

D₄

—

254

—

110

C₂×C₄

—

111

C₂×C₄

—

508

—

112

C₂×C₄

—

254

—

113

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

114

C₈

—

254

—

115

C₈

D₄×D₁₂₇

yes

—

116

C₈

D₄×D₁₂₇

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

2032

—

120

C₂²

—

254

—

121

C₂²

#16861254

yes

characteristic

122

C₄

—

123

C₄

—

254

—

124

C₄

C₂×D₄×D₁₂₇

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

508

—

128

C₂

#22550511

yes

characteristic

	subgroup	G/H
center	C₂	#22550511
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#493×C₁₂₇	C₂
socle	C₂×C₁₂₇	D₄²
hypercenter	#493	D₁₂₇
hypocenter	C₁₂₇	#3292
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#22550511

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#493×C₁₂₇

C₂

socle

C₂×C₁₂₇

D₄²

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #22550677	C₂	semidirect
2. #22550143	C₂	semidirect
3. #22550043	C₂	semidirect
4. #22549994	C₂	semidirect
5. #493×C₁₂₇	C₂	semidirect
6. #16861254	C₂²	semidirect
7. #16861235	C₂²	semidirect
12. #16861215	C₂²	semidirect
13. #16861145	C₂²	semidirect
14. #16861137	C₂²	semidirect
15. #16861126	C₂²	semidirect
16. C₂×D₈×C₁₂₇	C₂²	semidirect
17. #127×C₁₂₇	C₂²	semidirect
18. C₄×C₈×C₁₂₇	C₂²	semidirect
23. Dic₅₀₈	D₄	semidirect
24. #14523237	D₄	semidirect
26. D₁₀₁₆	D₄	semidirect
65. C₁₂₇	#3292	semidirect
67. #493	D₁₂₇	semidirect
77. C₂×D₈	C₂×D₁₂₇	semidirect
79. #127	C₂×D₁₂₇	semidirect
83. C₄×C₈	C₂×D₁₂₇	semidirect

H ≅ G/N

type

1. #22550677

C₂

semidirect

2. #22550143

C₂

semidirect

3. #22550043

C₂

semidirect

4. #22549994

C₂

semidirect

5. #493×C₁₂₇

C₂

semidirect

6. #16861254

C₂²

semidirect

7. #16861235

C₂²

semidirect

12. #16861215

C₂²

semidirect

13. #16861145

C₂²

semidirect

14. #16861137

C₂²

semidirect

15. #16861126

C₂²

semidirect

16. C₂×D₈×C₁₂₇

C₂²

semidirect

17. #127×C₁₂₇

C₂²

semidirect

18. C₄×C₈×C₁₂₇

C₂²

semidirect

23. Dic₅₀₈

D₄

semidirect

24. #14523237

D₄

semidirect

26. D₁₀₁₆

D₄

semidirect

65. C₁₂₇

#3292

semidirect

67. #493

D₁₂₇

semidirect

77. C₂×D₈

C₂×D₁₂₇

semidirect

79. #127

C₂×D₁₂₇

semidirect

83. C₄×C₈

C₂×D₁₂₇

semidirect

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

subgroup	index
#22550677	2
#22550143	2
#22550043	2
#22549994	2
#493×C₁₂₇	2
#3292	127

subgroup

index

#493×C₁₂₇

127

Subgroups — #30650529

Conjugacy classes of subgroups

Complements

Maximal subgroups