Group #30638100 — subgroups

Subgroups — #30638100

Order 16256.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#22550256	C₂	1	1	yes	maximal, characteristic
2	#22550161	C₂	1	2	yes	maximal
3	#22549629	C₂	1	1	yes	maximal, characteristic
4	#22549623	C₂	1	2	yes	maximal
5	C₈×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₀₈	—	2	1	no	—
7	C₂×Dic₅₀₈	—	2	1	no	—
8	#16861209	C₂²	1	2	yes	—
9	#16861140	—	2	2	no	—
10	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
11	#16861133	—	2	2	no	—
12	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₂₇	—	2	1	no	—
14	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₂₇	—	2	1	no	—
16	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₅₄	—	2	2	no	—
18	Dic₅₀₈	—	4	2	no	—
19	Dic₅₀₈	—	4	2	no	—
20	C₂×C₁₂₇⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₂₇	—	2	2	no	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	—	2	1	no	—
25	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₁₂₇	—	2	1	no	—
28	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
29	C₄²×C₁₂₇	—	2	1	no	—
30	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
31	Dic₂₅₄	—	4	2	no	—
32	Dic₂₅₄	—	4	2	no	—
33	C₂×Dic₁₂₇	—	4	2	no	—
34	C₁₂₇⋊C₈	—	4	2	no	—
35	Q₈×C₁₂₇	—	2	1	no	—
36	Q₈×C₁₂₇	Q₁₆	1	2	yes	complemented
37	C₂×C₄×C₁₂₇	—	2	1	no	—
38	C₂×C₄×C₁₂₇	—	2	1	no	—
39	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
42	C₈×C₁₂₇	—	2	1	no	—
43	C₈×C₁₂₇	—	4	1	no	—
44	C₈×C₁₂₇	SD₁₆	1	2	yes	—
45	C₂²×C₁₂₇	#121	1	1	yes	characteristic
46	C₄×C₁₂₇	—	2	2	no	—
47	C₄×C₁₂₇	—	2	1	no	—
48	C₄×C₁₂₇	—	4	1	no	—
49	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₁₂₇	#135	1	1	yes	characteristic
52	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
53	Dic₁₂₇	—	8	2	no	—
54	C₂×C₁₂₇	#460	1	1	yes	characteristic
55	C₂×C₁₂₇	#467	1	1	yes	characteristic
56	C₂×C₁₂₇	#473	1	1	yes	characteristic
57	#1670	—	127	1	no	maximal, Sylow
58	C₁₂₇	#1670	1	1	yes	characteristic, complemented, Sylow
59	#494	—	127	1	no	—
60	#475	—	127	2	no	—
61	C₈×Q₈	D₁₂₇	1	1	yes	characteristic
62	#335	—	127	1	no	—
63	#332	—	127	2	no	—
64	C₂×Q₁₆	—	254	1	no	—
65	C₂×Q₁₆	—	254	1	no	—
66	C₄⋊Q₈	—	127	2	no	—
67	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
68	#106	—	254	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	127	2	no	—
71	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
72	#103	—	254	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
75	C₂×Q₈	—	254	2	no	—
76	C₂×Q₈	D₅₀₈	1	1	yes	characteristic
77	Q₁₆	—	508	2	no	—
78	Q₁₆	—	508	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	254	2	no	—
81	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	254	2	no	—
85	C₄⋊C₄	—	254	2	no	—
86	C₄⋊C₄	D₅₀₈	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	508	2	no	—
91	Q₈	—	508	2	no	—
92	Q₈	Dic₅₀₈	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	508	2	no	—
96	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
97	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
98	C₂×C₄	#14523248	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	508	2	no	—
102	C₈	#14523235	1	2	yes	—
103	C₂²	#16861183	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	1016	2	no	—
108	C₄	C₂×#14523235	1	1	yes	characteristic
109	C₄	C₂×Dic₅₀₈	1	1	yes	characteristic
110	C₄	#16861231	1	1	yes	characteristic
111	C₄	#16861234	1	1	yes	characteristic
112	C₂	#22550076	1	1	yes	characteristic
113	C₂	#22550117	1	1	yes	characteristic
114	C₂	#22550151	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550256

C₂

yes

maximal, characteristic

#22550161

C₂

yes

maximal

#22549629

C₂

yes

maximal, characteristic

#22549623

C₂

yes

maximal

C₈×Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

—

C₂×Dic₅₀₈

—

#16861209

C₂²

yes

—

#16861140

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

#16861133

—

C₄×Q₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

—

C₄⋊C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

Dic₅₀₈

—

Dic₅₀₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

Q₁₆

yes

complemented

C₂×C₄×C₁₂₇

—

C₂×C₄×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

—

C₈×C₁₂₇

SD₁₆

yes

—

C₂²×C₁₂₇

#121

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₂₇

C₂×Q₁₆

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1670

yes

characteristic, complemented, Sylow

#494

—

127

—

#475

—

127

—

C₈×Q₈

D₁₂₇

yes

characteristic

#335

—

127

—

#332

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₄⋊Q₈

—

127

—

C₄×Q₈

C₂×D₁₂₇

yes

characteristic

#106

—

254

—

C₄⋊C₈

—

C₄⋊C₈

—

127

—

C₄⋊C₈

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

C₄×C₈

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

254

—

C₂×Q₈

D₅₀₈

yes

characteristic

Q₁₆

—

508

—

Q₁₆

—

508

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

D₅₀₈

yes

characteristic

C₄²

—

C₄²

C₂²×D₁₂₇

yes

characteristic

Q₈

—

Q₈

—

508

—

Q₈

—

508

—

Q₈

Dic₅₀₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₂×C₄

#14523248

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

508

—

102

C₈

#14523235

yes

—

103

C₂²

#16861183

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

1016

—

108

C₄

C₂×#14523235

yes

characteristic

109

C₄

C₂×Dic₅₀₈

yes

characteristic

110

C₄

#16861231

yes

characteristic

111

C₄

#16861234

yes

characteristic

112

C₂

#22550076

yes

characteristic

113

C₂

#22550117

yes

characteristic

114

C₂

#22550151

yes

characteristic

	subgroup	G/H
center	C₂²	#16861183
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	C₈×Q₈×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#121
hypercenter	C₈×Q₈	D₁₂₇
hypocenter	C₁₂₇	#1670
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#16861183

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

C₈×Q₈×C₁₂₇

C₂

socle, prime radical

C₂²×C₁₂₇

#121

hypercenter

C₈×Q₈

D₁₂₇

hypocenter

C₁₂₇

#1670

LCS

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
36. Q₈×C₁₂₇	Q₁₆	semidirect
58. C₁₂₇	#1670	semidirect
92. Q₈	Dic₅₀₈	semidirect

H ≅ G/N

type

36. Q₈×C₁₂₇

Q₁₆

semidirect

58. C₁₂₇

#1670

semidirect

92. Q₈

Dic₅₀₈

semidirect

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

subgroup	index
#22550256	2
#22550161	2
#22549629	2
#22549623	2
C₈×Q₈×C₁₂₇	2
#1670	127

subgroup

index

C₈×Q₈×C₁₂₇

127

Subgroups — #30638100

Conjugacy classes of subgroups

Complements

Maximal subgroups