Group #30640807 — subgroups

Subgroups — #30640807

Order 16256.

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

normal subgroups only

This group has 170 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	#22550314	C₂	1	2	yes	maximal, complemented
2	#22550253	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549994	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549690	C₂	1	2	yes	maximal, complemented
5	#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861231	C₂²	1	1	yes	characteristic
7	C₂×D₁₀₁₆	—	2	1	no	—
8	C₂×D₁₀₁₆	C₂²	1	2	yes	—
9	#16861215	—	2	1	no	—
10	#16861206	—	2	1	no	—
11	#16861154	—	2	2	no	—
12	D₂₀₃₂	—	2	2	no	—
13	#16861145	C₂²	1	1	yes	characteristic
14	#16861137	—	2	1	no	—
15	M₃₂×C₁₂₇	C₂²	1	2	yes	—
16	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	#14523248	—	2	1	no	—
18	C₂×D₅₀₈	—	2	2	no	—
19	C₂×D₅₀₈	—	4	2	no	—
20	Dic₅₀₈	D₄	1	1	yes	characteristic, complemented
21	#14523237	—	2	1	no	—
22	D₁₀₁₆	—	2	2	no	—
23	D₁₀₁₆	—	2	2	no	—
24	D₁₀₁₆	—	4	1	no	—
25	D₁₀₁₆	—	4	1	no	—
26	D₁₀₁₆	D₄	1	1	yes	characteristic, complemented
27	#14523232	—	2	1	no	—
28	#14523231	—	4	1	no	—
29	C₈×D₁₂₇	—	4	1	no	—
30	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
33	C₁₆×C₁₂₇	—	2	2	no	—
34	C₂²×D₁₂₇	—	4	2	no	—
35	Dic₂₅₄	—	2	1	no	—
36	D₅₀₈	—	2	1	no	—
37	D₅₀₈	—	4	2	no	—
38	D₅₀₈	—	4	2	no	—
39	D₅₀₈	—	8	4	no	—
40	C₂²⋊D₁₂₇	—	4	1	no	—
41	C₄×D₁₂₇	—	4	1	no	—
42	C₁₂₇⋊C₈	—	4	1	no	—
43	C₂×C₄×C₁₂₇	—	2	1	no	—
44	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₂₇	D₈	1	1	yes	characteristic
46	C₈×C₁₂₇	D₈	1	1	yes	characteristic
47	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₈×C₁₂₇	#41	1	1	yes	characteristic
49	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₂₇	—	4	1	no	—
51	C₂×D₁₂₇	—	8	4	no	—
52	C₂×D₁₂₇	—	8	2	no	—
53	C₄×C₁₂₇	—	2	2	no	—
54	C₄×C₁₂₇	#121	1	1	yes	characteristic
55	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
56	Dic₁₂₇	—	4	1	no	—
57	C₂×C₁₂₇	—	2	1	no	—
58	C₂×C₁₂₇	#459	1	1	yes	characteristic
59	D₁₂₇	—	8	1	no	—
60	D₁₂₇	—	16	2	no	—
61	#2213	—	127	1	no	maximal, Sylow
62	C₁₂₇	#2213	1	1	yes	characteristic, complemented, Sylow
63	#509	—	127	2	no	—
64	#493	—	127	1	no	—
65	#443	—	127	1	no	—
66	#364	D₁₂₇	1	1	yes	characteristic, complemented
67	#361	—	127	2	no	—
68	#135	—	127	1	no	—
69	C₂×D₈	—	127	2	no	—
70	C₂×D₈	—	254	1	no	—
71	#131	—	254	1	no	—
72	#127	—	254	1	no	—
73	SD₃₂	—	254	2	no	—
74	D₁₆	—	254	2	no	—
75	M₃₂	C₂×D₁₂₇	1	2	yes	—
76	#108	—	127	1	no	—
77	#104	—	254	1	no	—
78	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
79	#41	—	254	1	no	—
80	C₂×D₄	—	508	2	no	—
81	C₂×D₄	—	254	2	no	—
82	Q₁₆	—	127	1	no	—
83	SD₁₆	—	254	1	no	—
84	D₈	—	127	1	no	—
85	D₈	—	508	1	no	—
86	D₈	—	508	1	no	—
87	D₈	—	254	2	no	—
88	D₈	—	254	2	no	—
89	M₁₆	—	508	1	no	—
90	M₁₆	—	254	1	no	—
91	C₂×C₈	—	508	1	no	—
92	C₂×C₈	D₅₀₈	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
94	C₄²	D₅₀₈	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	508	2	no	—
97	Q₈	—	254	1	no	—
98	D₄	—	1016	4	no	—
99	D₄	—	508	1	no	—
100	D₄	—	508	2	no	—
101	D₄	—	508	2	no	—
102	D₄	—	254	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	508	1	no	—
105	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
106	C₈	—	508	1	no	—
107	C₈	D₁₀₁₆	1	1	yes	characteristic
108	C₈	D₁₀₁₆	1	1	yes	characteristic
109	C₈	D₄×D₁₂₇	1	1	yes	characteristic
110	C₈	#14523249	1	1	yes	characteristic
111	C₂²	—	1016	4	no	—
112	C₂²	—	1016	2	no	—
113	C₂²	—	508	1	no	—
114	C₂²	#16861237	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	508	1	no	—
117	C₄	#16861184	1	1	yes	characteristic
118	C₄	C₂×D₁₀₁₆	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2032	2	no	—
121	C₂	—	1016	1	no	—
122	C₂	#22550072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550314

C₂

yes

maximal, complemented

#22550253

C₂

yes

maximal, characteristic, complemented

#22549994

C₂

yes

maximal, characteristic, complemented

#22549690

C₂

yes

maximal, complemented

#364×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861231

C₂²

yes

characteristic

C₂×D₁₀₁₆

—

C₂×D₁₀₁₆

C₂²

yes

—

#16861215

—

#16861206

—

#16861154

—

D₂₀₃₂

—

#16861145

C₂²

yes

characteristic

#16861137

—

M₃₂×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

#14523248

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

Dic₅₀₈

D₄

yes

characteristic, complemented

#14523237

—

D₁₀₁₆

—

D₁₀₁₆

—

D₁₀₁₆

—

D₁₀₁₆

—

D₁₀₁₆

D₄

yes

characteristic, complemented

#14523232

—

#14523231

—

C₈×D₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₁₆×C₁₂₇

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

D₈

yes

characteristic

C₈×C₁₂₇

D₈

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

C₈×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#121

yes

characteristic

C₄×C₁₂₇

C₂×D₈

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#459

yes

characteristic

D₁₂₇

—

D₁₂₇

—

#2213

—

127

maximal, Sylow

C₁₂₇

#2213

yes

characteristic, complemented, Sylow

#509

—

127

—

#493

—

127

—

#443

—

127

—

#364

D₁₂₇

yes

characteristic, complemented

#361

—

127

—

#135

—

127

—

C₂×D₈

—

127

—

C₂×D₈

—

254

—

#131

—

254

—

#127

—

254

—

SD₃₂

—

254

—

D₁₆

—

254

—

M₃₂

C₂×D₁₂₇

yes

—

#108

—

127

—

#104

—

254

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

#41

—

254

—

C₂×D₄

—

508

—

C₂×D₄

—

254

—

Q₁₆

—

127

—

SD₁₆

—

254

—

D₈

—

127

—

D₈

—

508

—

D₈

—

508

—

D₈

—

254

—

D₈

—

254

—

M₁₆

—

508

—

M₁₆

—

254

—

C₂×C₈

—

508

—

C₂×C₈

D₅₀₈

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄²

D₅₀₈

yes

characteristic

C₁₆

—

C₂³

—

508

—

Q₈

—

254

—

D₄

—

1016

—

D₄

—

508

—

100

D₄

—

508

—

101

D₄

—

508

—

102

D₄

—

254

—

103

C₂×C₄

—

104

C₂×C₄

—

508

—

105

C₂×C₄

C₂×D₅₀₈

yes

characteristic

106

C₈

—

508

—

107

C₈

D₁₀₁₆

yes

characteristic

108

C₈

D₁₀₁₆

yes

characteristic

109

C₈

D₄×D₁₂₇

yes

characteristic

110

C₈

#14523249

yes

characteristic

111

C₂²

—

1016

—

112

C₂²

—

1016

—

113

C₂²

—

508

—

114

C₂²

#16861237

yes

characteristic

115

C₄

—

116

C₄

—

508

—

117

C₄

#16861184

yes

characteristic

118

C₄

C₂×D₁₀₁₆

yes

characteristic

119

C₂

—

120

C₂

—

2032

—

121

C₂

—

1016

—

122

C₂

#22550072

yes

characteristic

	subgroup	G/H
center	C₂	#22550072
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#364×C₁₂₇	C₂
socle	C₂×C₁₂₇	#459
hypercenter	#364	D₁₂₇
hypocenter	C₁₂₇	#2213
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#22550072

commutator, cosocle

C₂×C₈×C₁₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#364×C₁₂₇

C₂

C₂×C₁₂₇

D₁₂₇

C₁₂₇

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #22550314	C₂	semidirect
2. #22550253	C₂	semidirect
3. #22549994	C₂	semidirect
4. #22549690	C₂	semidirect
5. #364×C₁₂₇	C₂	semidirect
20. Dic₅₀₈	D₄	semidirect
26. D₁₀₁₆	D₄	semidirect
62. C₁₂₇	#2213	semidirect
66. #364	D₁₂₇	semidirect

H ≅ G/N

type

1. #22550314

C₂

semidirect

2. #22550253

C₂

semidirect

3. #22549994

C₂

semidirect

4. #22549690

C₂

semidirect

5. #364×C₁₂₇

C₂

semidirect

20. Dic₅₀₈

D₄

semidirect

26. D₁₀₁₆

D₄

semidirect

62. C₁₂₇

#2213

semidirect

66. #364

D₁₂₇

semidirect

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

subgroup	index
#22550314	2
#22550253	2
#22549994	2
#22549690	2
#364×C₁₂₇	2
#2213	127

subgroup

index

#364×C₁₂₇

127

Subgroups — #30640807

Conjugacy classes of subgroups

Complements

Maximal subgroups