Group #30650124 — subgroups

Subgroups — #30650124

Order 16256.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#22550433	C₂	1	2	yes	maximal
2	#22550162	C₂	1	4	yes	maximal, complemented
3	#22550134	C₂	1	4	yes	maximal
4	#22550087	C₂	1	4	yes	maximal, complemented
5	#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₀₈	C₂²	1	4	yes	—
7	#16861209	C₂²	1	2	yes	—
8	#16861209	C₂²	1	2	yes	—
9	#16861201	—	2	2	no	—
10	#16861188	—	2	2	no	—
11	#16861188	C₂²	1	4	yes	—
12	C₄×Dic₂₅₄	—	2	2	no	—
13	C₄×Dic₂₅₄	C₂²	1	4	yes	—
14	#16861168	—	2	2	no	—
15	#16861141	C₂²	1	4	yes	—
16	#16861134	C₂²	1	8	yes	—
17	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
20	C₂×Dic₂₅₄	—	2	2	no	—
21	C₂×Dic₂₅₄	C₂³	1	4	yes	—
22	Dic₅₀₈	—	4	4	no	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₄⋊Dic₁₂₇	—	2	2	no	—
25	C₄⋊Dic₁₂₇	—	2	2	no	—
26	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
27	Dic₁₂₇⋊C₄	—	2	4	no	—
28	Dic₁₂₇⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₁₂₇	—	2	4	no	—
30	C₄×Dic₁₂₇	—	2	4	no	—
31	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₂₇	—	4	2	no	—
34	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
35	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
36	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
37	C₄²×C₁₂₇	C₂³	1	2	yes	—
38	Dic₂₅₄	—	2	4	no	—
39	Dic₂₅₄	—	2	4	no	—
40	Dic₂₅₄	—	4	4	no	—
41	C₂×Dic₁₂₇	—	2	4	no	—
42	C₂×Dic₁₂₇	—	2	4	no	—
43	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	—	2	4	no	—
45	C₂×C₄×C₁₂₇	—	2	1	no	—
46	C₂×C₄×C₁₂₇	—	2	1	no	—
47	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₂₇	—	2	4	no	—
52	C₂²×C₁₂₇	—	2	1	no	—
53	C₂²×C₁₂₇	—	2	2	no	—
54	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₁₂₇	—	2	1	no	—
56	C₄×C₁₂₇	—	2	2	no	—
57	C₄×C₁₂₇	#137	1	4	yes	—
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	C₂×#137	1	2	yes	—
65	#3255	—	127	1	no	maximal, Sylow
66	C₁₂₇	#3255	1	1	yes	characteristic, complemented, Sylow
67	#533	—	127	2	no	—
68	#475	—	127	4	no	—
69	#470	—	127	4	no	—
70	#462	—	127	4	no	—
71	#423	D₁₂₇	1	1	yes	characteristic
72	C₂×Q₁₆	—	127	4	no	—
73	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
74	C₄⋊Q₈	—	127	2	no	—
75	C₄⋊Q₈	—	127	2	no	—
76	#125	—	254	2	no	—
77	C₂²⋊Q₈	—	127	4	no	—
78	C₂²⋊Q₈	—	254	2	no	—
79	C₄×Q₈	—	127	4	no	—
80	C₄×Q₈	—	254	2	no	—
81	#117	—	254	2	no	—
82	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
83	#106	—	127	4	no	—
84	C₄⋊C₈	C₂×D₁₂₇	1	4	yes	—
85	#103	—	127	8	no	—
86	C₂×Q₈	—	127	4	no	—
87	C₂×Q₈	—	254	2	no	—
88	C₂²×C₄	D₅₀₈	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
90	Q₁₆	—	508	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
93	C₄⋊C₄	—	127	4	no	—
94	C₄⋊C₄	—	254	4	no	—
95	C₄⋊C₄	—	254	4	no	—
96	C₄⋊C₄	—	254	2	no	—
97	C₄⋊C₄	—	254	2	no	—
98	C₄⋊C₄	—	254	2	no	—
99	C₂²⋊C₄	—	254	4	no	—
100	C₄²	—	254	4	no	—
101	C₄²	D₅₀₈	1	1	yes	characteristic
102	C₄²	D₅₀₈	1	1	yes	characteristic
103	C₄²	C₂²×D₁₂₇	1	2	yes	—
104	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
105	Q₈	—	508	4	no	—
106	Q₈	—	254	4	no	—
107	Q₈	—	254	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	254	4	no	—
112	C₂×C₄	—	254	4	no	—
113	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
114	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
115	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
116	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	508	4	no	—
124	C₄	—	508	4	no	—
125	C₄	#16861242	1	4	yes	—
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	C₂×#16861242	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550433

C₂

yes

maximal

#22550162

C₂

yes

maximal, complemented

#22550134

C₂

yes

maximal

#22550087

C₂

yes

maximal, complemented

#423×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

C₂²

yes

—

#16861209

C₂²

yes

—

#16861209

C₂²

yes

—

#16861201

—

#16861188

—

#16861188

C₂²

yes

—

C₄×Dic₂₅₄

—

C₄×Dic₂₅₄

C₂²

yes

—

#16861168

—

#16861141

C₂²

yes

—

#16861134

C₂²

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂³

yes

—

Dic₅₀₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#137

yes

—

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

C₂×#137

yes

—

#3255

—

127

maximal, Sylow

C₁₂₇

#3255

yes

characteristic, complemented, Sylow

#533

—

127

—

#475

—

127

—

#470

—

127

—

#462

—

127

—

#423

D₁₂₇

yes

characteristic

C₂×Q₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

#125

—

254

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

254

—

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

#117

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#106

—

127

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

#103

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂²×C₄

D₅₀₈

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

Q₁₆

—

508

—

M₁₆

—

C₂×C₈

C₂²×D₁₂₇

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

100

C₄²

—

254

—

101

C₄²

D₅₀₈

yes

characteristic

102

C₄²

D₅₀₈

yes

characteristic

103

C₄²

C₂²×D₁₂₇

yes

—

104

C₂³

C₂×D₅₀₈

yes

characteristic

105

Q₈

—

508

—

106

Q₈

—

254

—

107

Q₈

—

254

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

254

—

112

C₂×C₄

—

254

—

113

C₂×C₄

C₂×D₅₀₈

yes

characteristic

114

C₂×C₄

C₂×D₅₀₈

yes

—

115

C₂×C₄

C₂×D₅₀₈

yes

—

116

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₅₀₈

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

508

—

124

C₄

—

508

—

125

C₄

#16861242

yes

—

126

C₄

#16861254

yes

characteristic

127

C₄

#16861257

yes

characteristic

128

C₂

—

129

C₂

#22550457

yes

characteristic

130

C₂

C₂×#16861242

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#423×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#423	D₁₂₇
hypocenter	C₁₂₇	#3255
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₅₀₈

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#423×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 ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. #22550162	C₂	semidirect
4. #22550087	C₂	semidirect
66. C₁₂₇	#3255	semidirect

H ≅ G/N

type

2. #22550162

C₂

semidirect

4. #22550087

C₂

semidirect

66. C₁₂₇

#3255

semidirect

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

subgroup	index
#22550433	2
#22550162	2
#22550134	2
#22550087	2
#423×C₁₂₇	2
#3255	127

subgroup

index

#423×C₁₂₇

127

Subgroups — #30650124

Conjugacy classes of subgroups

Complements

Maximal subgroups