Group #30640531 — subgroups

Subgroups — #30640531

Order 16256.

152 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	Dic₁₂₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	(C₂×D₁₂₇)⋊C₁₆	C₂	1	2	yes	maximal
4	#22549652	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
8	C₄²×D₁₂₇	C₄	1	1	yes	characteristic
9	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
10	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
11	C₈×Dic₁₂₇	C₄	1	1	yes	characteristic
12	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₂₇	C₈	1	1	yes	characteristic
16	C₂×C₄×D₁₂₇	C₈	1	1	yes	characteristic
17	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₂₇	—	2	2	no	—
21	C₈×D₁₂₇	—	2	2	no	—
22	C₄×Dic₁₂₇	C₈	1	1	yes	characteristic
23	C₄×Dic₁₂₇	C₈	1	1	yes	characteristic
24	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₁₂₇⋊C₁₆	—	2	2	no	—
26	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₁₆×C₁₂₇	—	2	2	no	—
30	C₂²×D₁₂₇	C₂×C₈	1	1	yes	characteristic
31	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
32	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
33	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
34	C₄×D₁₂₇	—	2	2	no	—
35	C₄×D₁₂₇	—	2	1	no	—
36	C₄×D₁₂₇	—	2	1	no	—
37	C₄×D₁₂₇	—	2	2	no	—
38	C₁₂₇⋊C₈	—	2	1	no	—
39	C₁₂₇⋊C₈	—	2	1	no	—
40	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₂₇	M₁₆	1	2	yes	—
44	C₈×C₁₂₇	#41	1	2	yes	—
45	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
46	C₂×D₁₂₇	—	2	1	no	—
47	C₂×D₁₂₇	—	2	1	no	—
48	C₂×D₁₂₇	M₃₂	1	2	yes	—
49	C₄×C₁₂₇	—	2	1	no	—
50	C₄×C₁₂₇	—	2	1	no	—
51	C₄×C₁₂₇	#117	1	1	yes	characteristic
52	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
53	Dic₁₂₇	—	2	1	no	—
54	Dic₁₂₇	—	2	1	no	—
55	Dic₁₂₇	M₃₂	1	2	yes	—
56	C₂×C₁₂₇	#431	1	1	yes	characteristic
57	C₂×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
59	D₁₂₇	—	2	2	no	—
60	#2163	—	127	1	no	maximal, Sylow
61	C₁₂₇	#2163	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₄×C₈	—	127	1	no	—
63	C₄⋊C₁₆	—	127	2	no	—
64	C₂²⋊C₁₆	—	127	2	no	—
65	#346	—	127	1	no	—
66	#346	D₁₂₇	1	1	yes	characteristic, complemented
67	C₂²×C₈	—	127	1	no	—
68	C₂²×C₈	—	127	1	no	—
69	C₂×C₄²	—	127	1	no	—
70	C₂×C₁₆	—	127	2	no	—
71	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
72	C₄×C₈	—	127	1	no	—
73	C₄×C₈	—	127	1	no	—
74	C₄×C₈	—	127	1	no	—
75	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
76	C₂²×C₄	—	127	1	no	—
77	C₂²×C₄	—	127	1	no	—
78	C₂²×C₄	—	127	1	no	—
79	C₂×C₈	—	127	1	no	—
80	C₂×C₈	—	127	1	no	—
81	C₂×C₈	—	254	2	no	—
82	C₂×C₈	—	254	2	no	—
83	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₄²	—	127	1	no	—
86	C₄²	—	127	1	no	—
87	C₄²	—	127	1	no	—
88	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
89	C₁₆	—	2	2	no	—
90	C₁₆	—	254	2	no	—
91	C₂³	—	127	1	no	—
92	C₂×C₄	—	127	1	no	—
93	C₂×C₄	—	127	1	no	—
94	C₂×C₄	—	127	1	no	—
95	C₂×C₄	—	254	2	no	—
96	C₂×C₄	—	254	2	no	—
97	C₂×C₄	—	254	1	no	—
98	C₂×C₄	—	254	1	no	—
99	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
100	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
102	C₈	—	254	1	no	—
103	C₈	—	254	1	no	—
104	C₈	#14523231	1	2	yes	—
105	C₈	#14523248	1	2	yes	—
106	C₂²	—	127	2	no	—
107	C₂²	—	254	1	no	—
108	C₂²	—	254	1	no	—
109	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	127	2	no	—
113	C₄	—	254	1	no	—
114	C₄	—	254	1	no	—
115	C₄	#16861166	1	1	yes	characteristic
116	C₄	C₂×#14523231	1	1	yes	characteristic
117	C₂	—	254	2	no	—
118	C₂	#22549927	1	1	yes	characteristic
119	C₂	M₃₂×D₁₂₇	1	1	yes	characteristic
120	C₂	M₃₂×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×D₁₂₇

C₂

yes

maximal, characteristic

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, complemented

(C₂×D₁₂₇)⋊C₁₆

C₂

yes

maximal

#22549652

C₂

yes

maximal, characteristic, complemented

#346×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₂₇

C₄

yes

characteristic

C₂×C₈×D₁₂₇

C₂²

yes

characteristic

C₄²×D₁₂₇

C₄

yes

characteristic

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₄

yes

characteristic

C₈×Dic₁₂₇

C₄

yes

characteristic

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₈

yes

characteristic

C₂×C₄×D₁₂₇

C₈

yes

characteristic

C₂×C₄×D₁₂₇

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₁₂₇

—

C₈×D₁₂₇

—

C₄×Dic₁₂₇

C₈

yes

characteristic

C₄×Dic₁₂₇

C₈

yes

characteristic

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₁₂₇⋊C₁₆

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₁₆×C₁₂₇

—

C₂²×D₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

M₁₆

yes

—

C₈×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

M₃₂

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#117

yes

characteristic

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

M₃₂

yes

—

C₂×C₁₂₇

#431

yes

characteristic

C₂×C₁₂₇

C₂×M₃₂

yes

characteristic

C₂×C₁₂₇

C₂×M₃₂

yes

characteristic

D₁₂₇

—

#2163

—

127

maximal, Sylow

C₁₂₇

#2163

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

127

—

C₄⋊C₁₆

—

127

—

C₂²⋊C₁₆

—

127

—

#346

—

127

—

#346

D₁₂₇

yes

characteristic, complemented

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂×C₄²

—

127

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

C₄×D₁₂₇

yes

characteristic

C₁₆

—

C₁₆

—

254

—

C₂³

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₈×D₁₂₇

yes

characteristic

100

C₂×C₄

C₈×D₁₂₇

yes

characteristic

101

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

102

C₈

—

254

—

103

C₈

—

254

—

104

C₈

#14523231

yes

—

105

C₈

#14523248

yes

—

106

C₂²

—

127

—

107

C₂²

—

254

—

108

C₂²

—

254

—

109

C₂²

C₂×C₈×D₁₂₇

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

127

—

113

C₄

—

254

—

114

C₄

—

254

—

115

C₄

#16861166

yes

characteristic

116

C₄

C₂×#14523231

yes

characteristic

117

C₂

—

254

—

118

C₂

#22549927

yes

characteristic

119

C₂

M₃₂×D₁₂₇

yes

characteristic

120

C₂

M₃₂×D₁₂₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₈
Fitting	#346×C₁₂₇	C₂
prime radical	C₂²×D₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#346	D₁₂₇
hypocenter	C₁₂₇	#2163
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂²×C₈

Fitting

#346×C₁₂₇

C₂

prime radical

C₂²×D₁₂₇

C₂×C₈

cosocle

C₂×C₈×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
2. Dic₁₂₇⋊C₁₆	C₂	semidirect
4. #22549652	C₂	semidirect
5. #346×C₁₂₇	C₂	semidirect
61. C₁₂₇	#2163	semidirect
66. #346	D₁₂₇	semidirect

H ≅ G/N

type

2. Dic₁₂₇⋊C₁₆

C₂

semidirect

4. #22549652

C₂

semidirect

5. #346×C₁₂₇

C₂

semidirect

61. C₁₂₇

#2163

semidirect

66. #346

D₁₂₇

semidirect

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

subgroup	index
C₄×C₈×D₁₂₇	2
Dic₁₂₇⋊C₁₆	2
(C₂×D₁₂₇)⋊C₁₆	2
#22549652	2
#346×C₁₂₇	2
#2163	127

subgroup

index

C₄×C₈×D₁₂₇

Dic₁₂₇⋊C₁₆

(C₂×D₁₂₇)⋊C₁₆

#22549652

#346×C₁₂₇

#2163

127

Subgroups — #30640531

Conjugacy classes of subgroups

Complements

Maximal subgroups