Group #30640237 — subgroups

Subgroups — #30640237

Order 16256.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22549918	C₂	1	2	yes	maximal
2	C₂×#16861143	C₂	1	1	yes	maximal, characteristic
3	#22549902	C₂	1	1	yes	maximal, characteristic
4	#22549644	C₂	1	2	yes	maximal
5	#341×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	C₂²	1	1	yes	characteristic
7	#16861168	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	#16861143	—	2	2	no	—
10	#16861143	—	2	2	no	—
11	#16861141	—	2	2	no	—
12	Dic₁₂₇⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₂₇	—	2	2	no	—
16	#14523232	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	—	2	1	no	—
18	C₄⋊Dic₁₂₇	—	2	2	no	—
19	C₄⋊Dic₁₂₇	—	2	2	no	—
20	C₄⋊Dic₁₂₇	D₄	1	1	yes	characteristic
21	C₄⋊Dic₁₂₇	D₄	1	1	yes	characteristic
22	C₂²⋊Dic₁₂₇	—	2	1	no	—
23	C₄×Dic₁₂₇	—	2	1	no	—
24	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₂₇	—	2	2	no	—
26	M₁₆×C₁₂₇	Q₈	1	4	yes	—
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
30	C₂×Dic₁₂₇	—	2	1	no	—
31	C₂×Dic₁₂₇	—	2	1	no	—
32	C₂×Dic₁₂₇	—	4	2	no	—
33	C₂×Dic₁₂₇	—	8	2	no	—
34	C₁₂₇⋊C₈	—	4	1	no	—
35	C₂³×C₁₂₇	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₁₂₇	#41	1	2	yes	—
40	C₈×C₁₂₇	—	2	4	no	—
41	C₈×C₁₂₇	—	2	2	no	—
42	C₂²×C₁₂₇	—	2	1	no	—
43	C₂²×C₁₂₇	—	2	1	no	—
44	C₂²×C₁₂₇	#121	1	1	yes	characteristic
45	C₂²×C₁₂₇	#125	1	2	yes	—
46	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₂₇	#123	1	1	yes	characteristic
48	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
49	Dic₁₂₇	—	4	1	no	—
50	Dic₁₂₇	—	4	1	no	—
51	Dic₁₂₇	—	8	2	no	—
52	C₂×C₁₂₇	—	2	2	no	—
53	C₂×C₁₂₇	#398	1	1	yes	characteristic
54	C₂×C₁₂₇	#472	1	1	yes	characteristic
55	C₂×C₁₂₇	#473	1	1	yes	characteristic
56	#2089	—	127	1	no	maximal, Sylow
57	C₁₂₇	#2089	1	1	yes	characteristic, complemented, Sylow
58	#428	—	127	2	no	—
59	C₂×#107	—	127	1	no	—
60	#424	—	127	1	no	—
61	#343	—	127	2	no	—
62	#341	D₁₂₇	1	1	yes	characteristic
63	C₂×M₁₆	—	127	1	no	—
64	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
65	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
66	#117	—	127	1	no	—
67	C₂×C₄⋊C₄	—	127	2	no	—
68	#107	—	254	2	no	—
69	#107	—	254	2	no	—
70	#106	—	254	2	no	—
71	C₄⋊C₈	—	254	2	no	—
72	C₂²×C₄	—	254	2	no	—
73	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	254	2	no	—
76	M₁₆	Dic₂₅₄	1	4	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	254	1	no	—
80	C₂×C₈	D₅₀₈	1	2	yes	—
81	C₄⋊C₄	—	127	1	no	—
82	C₄⋊C₄	—	127	1	no	—
83	C₄⋊C₄	—	254	2	no	—
84	C₄⋊C₄	—	254	2	no	—
85	C₂²⋊C₄	—	254	1	no	—
86	C₄²	—	254	1	no	—
87	C₂³	#14523249	1	1	yes	characteristic
88	C₂×C₄	—	1016	2	no	—
89	C₂×C₄	—	508	2	no	—
90	C₂×C₄	—	254	1	no	—
91	C₂×C₄	—	254	1	no	—
92	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
93	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
95	C₂×C₄	#14523247	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	2	no	—
98	C₈	—	508	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#16861184	1	1	yes	characteristic
102	C₂²	#16861200	1	2	yes	—
103	C₄	—	1016	2	no	—
104	C₄	—	508	1	no	—
105	C₄	—	508	1	no	—
106	C₄	#16861186	1	2	yes	—
107	C₄	#16861193	1	1	yes	characteristic
108	C₄	#16861209	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#22549808	1	1	yes	characteristic
111	C₂	#22550147	1	1	yes	characteristic
112	C₂	#22550152	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549918

C₂

yes

maximal

C₂×#16861143

C₂

yes

maximal, characteristic

#22549902

C₂

yes

maximal, characteristic

#22549644

C₂

yes

maximal

#341×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

characteristic

#16861168

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861143

—

#16861143

—

#16861141

—

Dic₁₂₇⋊C₈

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

D₄

yes

characteristic

C₄⋊Dic₁₂₇

D₄

yes

characteristic

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

Q₈

yes

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

characteristic

C₂²×C₁₂₇

#125

yes

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

#123

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2089

yes

characteristic, complemented, Sylow

#428

—

127

—

C₂×#107

—

127

—

#424

—

127

—

#343

—

127

—

#341

D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

—

127

—

C₂×C₄⋊C₄

—

127

—

#107

—

254

—

#107

—

254

—

#106

—

254

—

C₄⋊C₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

254

—

M₁₆

Dic₂₅₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

D₅₀₈

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

254

—

C₂³

#14523249

yes

characteristic

C₂×C₄

—

1016

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

D₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₂×C₄

C₂×Dic₂₅₄

yes

—

C₂×C₄

#14523247

yes

—

C₈

—

C₈

—

C₈

—

508

—

C₂²

—

100

C₂²

—

101

C₂²

#16861184

yes

characteristic

102

C₂²

#16861200

yes

—

103

C₄

—

1016

—

104

C₄

—

508

—

105

C₄

—

508

—

106

C₄

#16861186

yes

—

107

C₄

#16861193

yes

characteristic

108

C₄

#16861209

yes

characteristic

109

C₂

—

110

C₂

#22549808

yes

characteristic

111

C₂

#22550147

yes

characteristic

112

C₂

#22550152

yes

characteristic

	subgroup	G/H
center	C₂²	#16861184
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#341×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
prime radical	C₂³×C₁₂₇	#41
hypercenter	#341	D₁₂₇
hypocenter	C₁₂₇	#2089
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#16861184

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#341×C₁₂₇

C₂

C₂²×C₁₂₇

C₂³×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
57. C₁₂₇	#2089	semidirect

H ≅ G/N

type

57. C₁₂₇

#2089

semidirect

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

subgroup	index
#22549918	2
C₂×#16861143	2
#22549902	2
#22549644	2
#341×C₁₂₇	2
#2089	127

subgroup

index

#341×C₁₂₇

127

Subgroups — #30640237

Conjugacy classes of subgroups

Complements

Maximal subgroups