Group #30649252 — subgroups

Subgroups — #30649252

Order 16256.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	1	yes	maximal, characteristic
2	#22550286	C₂	1	4	yes	maximal, complemented
3	#22550282	C₂	1	4	yes	maximal, complemented
4	#22549912	C₂	1	4	yes	maximal
5	C₂×#16861120	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
8	#16861209	C₂²	1	2	yes	—
9	#16861201	C₂²	1	2	yes	—
10	#16861188	—	2	2	no	—
11	C₄×Dic₂₅₄	—	2	2	no	—
12	#16861168	C₂²	1	2	yes	—
13	#16861142	C₂²	1	8	yes	—
14	#16861140	C₂²	1	8	yes	—
15	#16861120	C₂²	1	2	yes	—
16	#16861120	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
18	#125×C₁₂₇	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
20	C₄×Q₈×C₁₂₇	—	2	2	no	—
21	#117×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₅₄	—	2	2	no	—
24	C₂×C₁₂₇⋊C₈	Q₈	1	8	yes	complemented
25	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₁₂₇	—	2	2	no	—
27	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
28	Dic₁₂₇⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₁₂₇	—	2	2	no	—
30	C₄×Dic₁₂₇	—	2	2	no	—
31	C₂×Q₈×C₁₂₇	—	2	2	no	—
32	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
33	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₂₇	—	2	4	no	—
35	C₄⋊C₄×C₁₂₇	—	2	2	no	—
36	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
38	C₄²×C₁₂₇	—	2	2	no	—
39	C₄²×C₁₂₇	D₄	1	2	yes	—
40	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
42	Dic₂₅₄	—	4	4	no	—
43	C₂×Dic₁₂₇	—	2	4	no	—
44	C₁₂₇⋊C₈	C₂×Q₈	1	8	yes	—
45	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₁₂₇	—	4	4	no	—
47	C₂×C₄×C₁₂₇	—	2	4	no	—
48	C₂×C₄×C₁₂₇	—	4	2	no	—
49	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
54	C₂²×C₁₂₇	—	2	2	no	—
55	C₂²×C₁₂₇	C₄⋊Q₈	1	2	yes	—
56	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₁₂₇	—	2	2	no	—
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	—	4	4	no	—
60	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
61	C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
62	Dic₁₂₇	—	4	4	no	—
63	C₂×C₁₂₇	—	2	2	no	—
64	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₂₇	#575	1	2	yes	—
66	#3162	—	127	1	no	maximal, Sylow
67	C₁₂₇	#3162	1	1	yes	characteristic, complemented, Sylow
68	#533	—	127	1	no	—
69	#533	D₁₂₇	1	1	yes	characteristic
70	#501	—	127	4	no	—
71	#501	—	127	4	no	—
72	#427	—	127	4	no	—
73	C₂×#97	—	127	1	no	—
74	C₂²×C₈	—	127	2	no	—
75	C₄⋊Q₈	—	127	2	no	—
76	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	—
77	#125	—	127	2	no	—
78	#125	C₂×D₁₂₇	1	2	yes	—
79	C₂²⋊Q₈	—	2	2	no	—
80	C₂²⋊Q₈	—	254	2	no	—
81	C₄×Q₈	—	2	2	no	—
82	C₄×Q₈	—	254	2	no	—
83	#117	—	127	2	no	—
84	#117	C₂×D₁₂₇	1	2	yes	—
85	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
86	#107	—	127	8	no	—
87	#106	—	127	8	no	—
88	#97	—	127	2	no	—
89	#97	—	127	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	254	2	no	—
92	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
94	C₂×C₈	—	127	8	no	—
95	C₂×C₈	—	127	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	127	4	no	—
99	C₄⋊C₄	—	254	4	no	—
100	C₄⋊C₄	—	254	2	no	—
101	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	254	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	254	2	no	—
106	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
107	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
108	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	508	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	254	4	no	—
115	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
118	C₂×C₄	Q₈×D₁₂₇	1	4	yes	—
119	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
120	C₈	—	127	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#16861207	1	2	yes	—
123	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	508	4	no	—
128	C₄	#16861207	1	2	yes	—
129	C₄	C₂×Q₈×D₁₂₇	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₂×#16861207	1	1	yes	characteristic
132	C₂	#22550674	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550433

C₂

yes

maximal, characteristic

#22550286

C₂

yes

maximal, complemented

#22550282

C₂

yes

maximal, complemented

#22549912

C₂

yes

maximal

C₂×#16861120

C₂

yes

maximal, characteristic

#533×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

#16861209

C₂²

yes

—

#16861201

C₂²

yes

—

#16861188

—

C₄×Dic₂₅₄

—

#16861168

C₂²

yes

—

#16861142

C₂²

yes

—

#16861140

C₂²

yes

—

#16861120

C₂²

yes

—

#16861120

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

—

#125×C₁₂₇

C₂²

yes

—

C₂²⋊Q₈×C₁₂₇

—

C₄×Q₈×C₁₂₇

—

#117×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×C₁₂₇⋊C₈

Q₈

yes

complemented

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄⋊Q₈

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄⋊Q₈

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₂₇

#575

yes

—

#3162

—

127

maximal, Sylow

C₁₂₇

#3162

yes

characteristic, complemented, Sylow

#533

—

127

—

#533

D₁₂₇

yes

characteristic

#501

—

127

—

#501

—

127

—

#427

—

127

—

C₂×#97

—

127

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

C₂×D₁₂₇

yes

—

#125

—

127

—

#125

C₂×D₁₂₇

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

254

—

C₄×Q₈

—

C₄×Q₈

—

254

—

#117

—

127

—

#117

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#107

—

127

—

#106

—

127

—

#97

—

127

—

#97

—

127

—

C₂×Q₈

—

C₂×Q₈

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

100

C₄⋊C₄

—

254

—

101

C₄⋊C₄

C₂²×D₁₂₇

yes

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

254

—

104

C₄²

—

105

C₄²

—

254

—

106

C₄²

C₂²⋊D₁₂₇

yes

—

107

C₄²

C₂²×D₁₂₇

yes

characteristic

108

C₄²

C₂²×D₁₂₇

yes

characteristic

109

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

110

Q₈

—

111

Q₈

—

508

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

254

—

115

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

116

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

118

C₂×C₄

Q₈×D₁₂₇

yes

—

119

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

120

C₈

—

127

—

121

C₂²

—

122

C₂²

#16861207

yes

—

123

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

508

—

128

C₄

#16861207

yes

—

129

C₄

C₂×Q₈×D₁₂₇

yes

—

130

C₂

—

131

C₂

C₂×#16861207

yes

characteristic

132

C₂

#22550674

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#533×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#533	D₁₂₇
hypocenter	C₁₂₇	#3162
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#533×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #22550286	C₂	semidirect
3. #22550282	C₂	semidirect
24. C₂×C₁₂₇⋊C₈	Q₈	semidirect
67. C₁₂₇	#3162	semidirect

H ≅ G/N

type

2. #22550286

C₂

semidirect

3. #22550282

C₂

semidirect

24. C₂×C₁₂₇⋊C₈

Q₈

semidirect

67. C₁₂₇

#3162

semidirect

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

subgroup	index
#22550433	2
#22550286	2
#22550282	2
#22549912	2
C₂×#16861120	2
#533×C₁₂₇	2
#3162	127

subgroup

index

#533×C₁₂₇

127

Subgroups — #30649252

Conjugacy classes of subgroups

Complements

Maximal subgroups