Group #30638101 — subgroups

Subgroups — #30638101

Order 16256.

156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22550162	C₂	1	2	yes	maximal
3	C₈×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
4	#22549621	C₂	1	2	yes	maximal
5	#335×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	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
10	#16861141	—	2	2	no	—
11	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₁₂₇⋊C₈	—	2	1	no	—
13	#16861134	—	2	2	no	—
14	C₈×Dic₁₂₇	—	2	1	no	—
15	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
16	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₅₄	—	2	2	no	—
18	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
19	Dic₅₀₈	—	4	2	no	—
20	Dic₅₀₈	—	4	2	no	—
21	C₂×C₁₂₇⋊C₈	—	2	1	no	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₄⋊Dic₁₂₇	D₄	1	1	yes	characteristic
25	Dic₁₂₇⋊C₄	—	2	1	no	—
26	C₄×Dic₁₂₇	—	2	1	no	—
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
30	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
31	Dic₂₅₄	—	2	1	no	—
32	Dic₂₅₄	—	4	2	no	—
33	Dic₂₅₄	—	4	2	no	—
34	Dic₂₅₄	Q₁₆	1	2	yes	complemented
35	C₂×Dic₁₂₇	—	2	1	no	—
36	C₂×Dic₁₂₇	—	2	1	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₁₂₇⋊C₈	—	4	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	2	no	—
44	C₈×C₁₂₇	SD₁₆	1	2	yes	—
45	C₂²×C₁₂₇	#121	1	1	yes	characteristic
46	C₄×C₁₂₇	—	2	1	no	—
47	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
48	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×C₁₂₇	#135	1	1	yes	characteristic
50	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
51	Dic₁₂₇	—	2	2	no	—
52	Dic₁₂₇	—	4	1	no	—
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	—	127	1	no	—
62	#335	D₁₂₇	1	1	yes	characteristic
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	—	127	1	no	—
68	#106	—	254	2	no	—
69	C₄⋊C₈	—	127	1	no	—
70	C₄⋊C₈	—	254	1	no	—
71	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
72	#103	—	254	2	no	—
73	C₄×C₈	—	254	1	no	—
74	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
75	C₂×Q₈	—	127	1	no	—
76	C₂×Q₈	—	254	2	no	—
77	Q₁₆	—	508	2	no	—
78	Q₁₆	—	508	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	254	1	no	—
81	C₂×C₈	D₅₀₈	1	1	yes	characteristic
82	C₂×C₈	D₅₀₈	1	1	yes	characteristic
83	C₄⋊C₄	—	127	1	no	—
84	C₄⋊C₄	—	254	2	no	—
85	C₄⋊C₄	—	254	2	no	—
86	C₄⋊C₄	—	254	1	no	—
87	C₄²	—	254	1	no	—
88	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
89	Q₈	—	127	2	no	—
90	Q₈	—	508	2	no	—
91	Q₈	—	508	2	no	—
92	Q₈	—	254	1	no	—
93	C₂×C₄	—	508	2	no	—
94	C₂×C₄	—	254	1	no	—
95	C₂×C₄	—	254	1	no	—
96	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
97	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
98	C₂×C₄	#14523249	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	2	no	—
101	C₈	—	508	1	no	—
102	C₈	#14523237	1	2	yes	—
103	C₂²	#16861184	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	1016	2	no	—
106	C₄	—	508	1	no	—
107	C₄	—	254	2	no	—
108	C₄	C₂×#14523237	1	1	yes	characteristic
109	C₄	Q₁₆×D₁₂₇	1	1	yes	characteristic
110	C₄	#16861228	1	1	yes	characteristic
111	C₄	#16861237	1	1	yes	characteristic
112	C₂	#22550077	1	1	yes	characteristic
113	C₂	#22550118	1	1	yes	characteristic
114	C₂	#22550152	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550256

C₂

yes

maximal, characteristic

#22550162

C₂

yes

maximal

C₈×Dic₂₅₄

C₂

yes

maximal, characteristic

#22549621

C₂

yes

maximal

#335×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

—

C₂×Dic₅₀₈

—

#16861209

C₂²

yes

—

C₄×Dic₂₅₄

C₂²

yes

characteristic

#16861141

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

characteristic

Dic₁₂₇⋊C₈

—

#16861134

—

C₈×Dic₁₂₇

—

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

D₄

yes

characteristic

Dic₅₀₈

—

Dic₅₀₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

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

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

Q₁₆

yes

complemented

C₂×Dic₁₂₇

—

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

—

C₂²×C₁₂₇

#121

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₂₇

C₂×Q₁₆

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

Dic₁₂₇

—

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

—

127

—

#335

D₁₂₇

yes

characteristic

#332

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₄⋊Q₈

—

127

—

C₄×Q₈

—

127

—

#106

—

254

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

#103

—

254

—

C₄×C₈

—

254

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

Q₁₆

—

508

—

Q₁₆

—

508

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

D₅₀₈

yes

characteristic

C₂×C₈

D₅₀₈

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

Q₈

—

127

—

Q₈

—

508

—

Q₈

—

508

—

Q₈

—

254

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

D₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₂×C₄

#14523249

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

508

—

102

C₈

#14523237

yes

—

103

C₂²

#16861184

yes

characteristic

104

C₄

—

105

C₄

—

1016

—

106

C₄

—

508

—

107

C₄

—

254

—

108

C₄

C₂×#14523237

yes

characteristic

109

C₄

Q₁₆×D₁₂₇

yes

characteristic

110

C₄

#16861228

yes

characteristic

111

C₄

#16861237

yes

characteristic

112

C₂

#22550077

yes

characteristic

113

C₂

#22550118

yes

characteristic

114

C₂

#22550152

yes

characteristic

	subgroup	G/H
center	C₂²	#16861184
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#335×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#121
hypercenter	#335	D₁₂₇
hypocenter	C₁₂₇	#1670
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#16861184

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#335×C₁₂₇

C₂

C₂²×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
34. Dic₂₅₄	Q₁₆	semidirect
58. C₁₂₇	#1670	semidirect

H ≅ G/N

type

34. Dic₂₅₄

Q₁₆

semidirect

58. C₁₂₇

#1670

semidirect

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

subgroup	index
#22550256	2
#22550162	2
C₈×Dic₂₅₄	2
#22549621	2
#335×C₁₂₇	2
#1670	127

subgroup

index

C₈×Dic₂₅₄

#335×C₁₂₇

127

Subgroups — #30638101

Conjugacy classes of subgroups

Complements

Maximal subgroups