Group #30650184 — subgroups

Subgroups — #30650184

Order 16256.

322 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 117 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	#22550488	C₂	1	2	yes	maximal
2	#22550184	C₂	1	4	yes	maximal, complemented
3	#22550134	C₂	1	2	yes	maximal
4	#22550117	C₂	1	2	yes	maximal
5	#22550087	C₂	1	4	yes	maximal, complemented
6	#424×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₅₀₈	C₂²	1	2	yes	—
8	C₂×Dic₅₀₈	C₂²	1	2	yes	—
9	#16861209	C₂²	1	2	yes	—
10	#16861202	—	2	4	no	—
11	#16861201	C₂²	1	2	yes	—
12	#16861199	—	2	4	no	—
13	#16861188	C₂²	1	4	yes	—
14	C₄×Dic₂₅₄	C₂²	1	4	yes	—
15	#16861143	C₂²	1	2	yes	—
16	#16861141	C₂²	1	2	yes	—
17	#16861134	C₂²	1	4	yes	—
18	#16861134	C₂²	1	4	yes	—
19	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	#117×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
23	C₂×Dic₂₅₄	C₂³	1	4	yes	—
24	Dic₅₀₈	—	4	2	no	—
25	Dic₅₀₈	—	4	2	no	—
26	C₄⋊Dic₁₂₇	—	2	4	no	—
27	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
28	Dic₁₂₇⋊C₄	—	2	4	no	—
29	Dic₁₂₇⋊C₄	—	2	4	no	—
30	Dic₁₂₇⋊C₄	—	2	4	no	—
31	C₂²⋊Dic₁₂₇	—	2	4	no	—
32	C₄×Dic₁₂₇	—	2	4	no	—
33	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₂₇	—	4	1	no	—
35	C₂×C₈×C₁₂₇	—	4	1	no	—
36	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
37	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₂₇	D₄	1	2	yes	—
39	C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
40	C₄²×C₁₂₇	C₂³	1	2	yes	—
41	Dic₂₅₄	—	2	4	no	—
42	Dic₂₅₄	—	2	4	no	—
43	C₂×Dic₁₂₇	—	2	4	no	—
44	C₂×Dic₁₂₇	—	2	4	no	—
45	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	—	2	1	no	—
47	C₂×C₄×C₁₂₇	—	2	1	no	—
48	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₂₇	—	2	2	no	—
52	C₈×C₁₂₇	—	2	2	no	—
53	C₂²×C₁₂₇	—	2	1	no	—
54	C₂²×C₁₂₇	—	2	2	no	—
55	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₂₇	—	2	1	no	—
57	C₄×C₁₂₇	—	2	4	no	—
58	C₄×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₁₂₇	—	4	4	no	—
61	Dic₁₂₇	—	4	4	no	—
62	C₂×C₁₂₇	—	4	1	no	—
63	C₂×C₁₂₇	#537	1	1	yes	characteristic
64	C₂×C₁₂₇	#578	1	2	yes	—
65	#3261	—	127	1	no	maximal, Sylow
66	C₁₂₇	#3261	1	1	yes	characteristic, complemented, Sylow
67	#541	—	127	2	no	—
68	#479	—	127	4	no	—
69	#470	—	127	2	no	—
70	#467	—	127	2	no	—
71	#462	—	127	4	no	—
72	#424	D₁₂₇	1	1	yes	characteristic
73	C₂×Q₁₆	—	127	2	no	—
74	C₂×Q₁₆	—	127	2	no	—
75	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
77	C₄⋊Q₈	—	127	2	no	—
78	#126	—	254	4	no	—
79	#125	—	127	2	no	—
80	#125	—	254	4	no	—
81	C₂²⋊Q₈	—	127	4	no	—
82	C₄×Q₈	—	127	4	no	—
83	#117	C₂×D₁₂₇	1	1	yes	characteristic
84	#107	—	127	2	no	—
85	#106	—	127	2	no	—
86	C₄⋊C₈	C₂×D₁₂₇	1	4	yes	—
87	#103	—	127	4	no	—
88	#103	—	127	4	no	—
89	C₂×Q₈	—	127	4	no	—
90	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
91	Q₁₆	—	508	2	no	—
92	Q₁₆	—	508	2	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
96	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
97	C₄⋊C₄	—	127	4	no	—
98	C₄⋊C₄	—	254	4	no	—
99	C₄⋊C₄	—	254	4	no	—
100	C₄⋊C₄	—	254	4	no	—
101	C₄⋊C₄	—	254	4	no	—
102	C₄⋊C₄	D₅₀₈	1	2	yes	—
103	C₂²⋊C₄	—	254	4	no	—
104	C₂²⋊C₄	D₅₀₈	1	2	yes	—
105	C₄²	—	254	4	no	—
106	C₄²	C₂²×D₁₂₇	1	2	yes	—
107	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
108	Q₈	—	254	4	no	—
109	Q₈	—	254	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	254	4	no	—
113	C₂×C₄	—	254	4	no	—
114	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
115	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	508	4	no	—
125	C₄	—	508	4	no	—
126	C₄	#16861254	1	1	yes	characteristic
127	C₄	#16861257	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#22550457	1	1	yes	characteristic
130	C₂	#22550693	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550488

C₂

yes

maximal

#22550184

C₂

yes

maximal, complemented

#22550134

C₂

yes

maximal

#22550117

C₂

yes

maximal

#22550087

C₂

yes

maximal, complemented

#424×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

C₂²

yes

—

C₂×Dic₅₀₈

C₂²

yes

—

#16861209

C₂²

yes

—

#16861202

—

#16861201

C₂²

yes

—

#16861199

—

#16861188

C₂²

yes

—

C₄×Dic₂₅₄

C₂²

yes

—

#16861143

C₂²

yes

—

#16861141

C₂²

yes

—

#16861134

C₂²

yes

—

#16861134

C₂²

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

C₂³

yes

—

Dic₅₀₈

—

Dic₅₀₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

D₄

yes

—

C₂²⋊C₄×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₄×C₁₂₇

2₋¹⁺⁴

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#537

yes

characteristic

C₂×C₁₂₇

#578

yes

—

#3261

—

127

maximal, Sylow

C₁₂₇

#3261

yes

characteristic, complemented, Sylow

#541

—

127

—

#479

—

127

—

#470

—

127

—

#467

—

127

—

#462

—

127

—

#424

D₁₂₇

yes

characteristic

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊Q₈

—

127

—

#126

—

254

—

#125

—

127

—

#125

—

254

—

C₂²⋊Q₈

—

127

—

C₄×Q₈

—

127

—

#117

C₂×D₁₂₇

yes

characteristic

#107

—

127

—

#106

—

127

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

#103

—

127

—

#103

—

127

—

C₂×Q₈

—

127

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

Q₁₆

—

508

—

Q₁₆

—

508

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₂₇

yes

—

C₂×C₈

C₂²×D₁₂₇

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

100

C₄⋊C₄

—

254

—

101

C₄⋊C₄

—

254

—

102

C₄⋊C₄

D₅₀₈

yes

—

103

C₂²⋊C₄

—

254

—

104

C₂²⋊C₄

D₅₀₈

yes

—

105

C₄²

—

254

—

106

C₄²

C₂²×D₁₂₇

yes

—

107

C₂³

C₂×D₅₀₈

yes

characteristic

108

Q₈

—

254

—

109

Q₈

—

254

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

254

—

113

C₂×C₄

—

254

—

114

C₂×C₄

C₂×D₅₀₈

yes

—

115

C₂×C₄

C₂×D₅₀₈

yes

characteristic

116

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×D₅₀₈

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

508

—

125

C₄

—

508

—

126

C₄

#16861254

yes

characteristic

127

C₄

#16861257

yes

characteristic

128

C₂

—

129

C₂

#22550457

yes

characteristic

130

C₂

#22550693

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#424×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#424	D₁₂₇
hypocenter	C₁₂₇	#3261
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₅₀₈

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#424×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
2. #22550184	C₂	semidirect
5. #22550087	C₂	semidirect
66. C₁₂₇	#3261	semidirect

H ≅ G/N

type

2. #22550184

C₂

semidirect

5. #22550087

C₂

semidirect

66. C₁₂₇

#3261

semidirect

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

subgroup	index
#22550488	2
#22550184	2
#22550134	2
#22550117	2
#22550087	2
#424×C₁₂₇	2
#3261	127

subgroup

index

#424×C₁₂₇

127

Subgroups — #30650184

Conjugacy classes of subgroups

Complements

Maximal subgroups