Group #30639699 — subgroups

Subgroups — #30639699

Order 16256.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#16861201	C₂	1	1	yes	maximal, characteristic
2	#22549746	C₂	1	1	yes	maximal, characteristic
3	#22549639	C₂	1	4	yes	maximal
4	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#16861201	—	2	2	no	—
6	#16861201	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	C₂×Dic₁₂₇⋊C₄	—	2	2	no	—
11	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	#97×C₁₂₇	—	2	2	no	—
14	C₂²×Dic₁₂₇	—	2	2	no	—
15	C₂²×Dic₁₂₇	—	2	2	no	—
16	C₄⋊Dic₁₂₇	—	2	2	no	—
17	C₄⋊Dic₁₂₇	—	2	4	no	—
18	C₄⋊Dic₁₂₇	—	4	4	no	—
19	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
20	Dic₁₂₇⋊C₄	—	2	8	no	—
21	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
22	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	—	2	2	no	—
24	C₂×C₈×C₁₂₇	—	2	2	no	—
25	C₂×C₈×C₁₂₇	Q₈	1	4	yes	—
26	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₄²×C₁₂₇	D₄	1	2	yes	—
28	C₂×Dic₁₂₇	—	2	4	no	—
29	C₂×Dic₁₂₇	—	4	2	no	—
30	C₂×Dic₁₂₇	—	4	2	no	—
31	C₂×Dic₁₂₇	—	4	2	no	—
32	C₂×Dic₁₂₇	—	4	4	no	—
33	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	—	2	2	no	—
35	C₂×C₄×C₁₂₇	—	2	2	no	—
36	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₂₇	#41	1	2	yes	—
41	C₈×C₁₂₇	—	2	4	no	—
42	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₂₇	#124	1	1	yes	characteristic
44	C₂²×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₂₇	#137	1	2	yes	—
47	C₄×C₁₂₇	—	2	4	no	—
48	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
49	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
50	Dic₁₂₇	—	4	4	no	—
51	Dic₁₂₇	—	8	2	no	—
52	C₂×C₁₂₇	#391	1	1	yes	characteristic
53	C₂×C₁₂₇	#417	1	2	yes	—
54	C₂×C₁₂₇	#490	1	1	yes	characteristic
55	C₂×C₁₂₇	#491	1	1	yes	characteristic
56	C₂×C₁₂₇	#501	1	2	yes	—
57	#1992	—	127	1	no	maximal, Sylow
58	C₁₂₇	#1992	1	1	yes	characteristic, complemented, Sylow
59	C₂×#125	—	127	1	no	—
60	C₂×#97	D₁₂₇	1	1	yes	characteristic
61	#384	—	127	1	no	—
62	#340	—	127	4	no	—
63	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
64	#125	—	127	4	no	—
65	#125	—	254	2	no	—
66	C₂×C₄⋊C₄	—	127	2	no	—
67	C₂×C₄⋊C₄	—	127	2	no	—
68	C₂×C₄⋊C₄	—	254	2	no	—
69	C₂×C₄⋊C₄	—	254	2	no	—
70	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
71	#97	—	2	2	no	—
72	C₂²×C₄	—	254	2	no	—
73	C₂²×C₄	—	254	2	no	—
74	C₂²×C₄	D₅₀₈	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Dic₂₅₄	1	4	yes	—
79	C₄⋊C₄	—	127	4	no	—
80	C₄⋊C₄	—	508	4	no	—
81	C₄⋊C₄	—	254	4	no	—
82	C₄⋊C₄	—	254	8	no	—
83	C₄⋊C₄	—	254	2	no	—
84	C₄²	C₄×D₁₂₇	1	2	yes	—
85	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
86	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	508	2	no	—
90	C₂×C₄	—	508	4	no	—
91	C₂×C₄	—	508	2	no	—
92	C₂×C₄	—	508	2	no	—
93	C₂×C₄	—	254	4	no	—
94	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
98	C₂×C₄	#14523248	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
101	C₂²	#16861197	1	1	yes	characteristic
102	C₂²	#16861209	1	1	yes	characteristic
103	C₂²	#16861237	1	2	yes	—
104	C₂²	#16861242	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	1016	2	no	—
107	C₄	—	508	4	no	—
108	C₄	C₄×Dic₂₅₄	1	2	yes	—
109	C₄	#16861188	1	2	yes	—
110	C₂	#22549782	1	1	yes	characteristic
111	C₂	#22549878	1	2	yes	—
112	C₂	#22550241	1	1	yes	characteristic
113	C₂	#22550245	1	1	yes	characteristic
114	C₂	#22550285	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861201

C₂

yes

maximal, characteristic

#22549746

C₂

yes

maximal, characteristic

#22549639

C₂

yes

maximal

C₂×#97×C₁₂₇

C₂

yes

maximal, characteristic

#16861201

—

#16861201

C₄

yes

complemented

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#97×C₁₂₇

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

Dic₁₂₇⋊C₄

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

Q₈

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

#124

yes

characteristic

C₂²×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#391

yes

characteristic

C₂×C₁₂₇

#417

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

#1992

—

127

maximal, Sylow

C₁₂₇

#1992

yes

characteristic, complemented, Sylow

C₂×#125

—

127

—

C₂×#97

D₁₂₇

yes

characteristic

#384

—

127

—

#340

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

—

#125

—

127

—

#125

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#97

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

D₅₀₈

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₅₄

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

C₄×D₁₂₇

yes

—

C₄²

C₂²⋊D₁₂₇

yes

—

C₂³

C₂×D₅₀₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂×Dic₂₅₄

yes

—

C₂×C₄

#14523248

yes

—

C₈

—

100

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

101

C₂²

#16861197

yes

characteristic

102

C₂²

#16861209

yes

characteristic

103

C₂²

#16861237

yes

—

104

C₂²

#16861242

yes

—

105

C₄

—

106

C₄

—

1016

—

107

C₄

—

508

—

108

C₄

C₄×Dic₂₅₄

yes

—

109

C₄

#16861188

yes

—

110

C₂

#22549782

yes

characteristic

111

C₂

#22549878

yes

—

112

C₂

#22550241

yes

characteristic

113

C₂

#22550245

yes

characteristic

114

C₂

#22550285

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#97×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#97	D₁₂₇
hypocenter	C₁₂₇	#1992
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₅₀₈

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#97×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×#97

D₁₂₇

hypocenter

C₁₂₇

#1992

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
6. #16861201	C₄	semidirect
58. C₁₂₇	#1992	semidirect

H ≅ G/N

type

6. #16861201

C₄

semidirect

58. C₁₂₇

#1992

semidirect

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

subgroup	index
C₂×#16861201	2
#22549746	2
#22549639	2
C₂×#97×C₁₂₇	2
#1992	127

subgroup

index

C₂×#97×C₁₂₇

127

Subgroups — #30639699

Conjugacy classes of subgroups

Complements

Maximal subgroups