Group #30649240 — subgroups

Subgroups — #30649240

Order 16256.

356 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 157 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	#22550279	C₂	1	2	yes	maximal, complemented
3	#22550275	C₂	1	4	yes	maximal, complemented
4	#22550272	C₂	1	2	yes	maximal, complemented
5	#22549912	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	#16861209	C₂²	1	2	yes	—
10	#16861201	C₂²	1	2	yes	—
11	#16861188	—	2	2	no	—
12	C₄×Dic₂₅₄	—	2	2	no	—
13	#16861168	C₂²	1	2	yes	—
14	#16861142	C₂²	1	8	yes	—
15	#16861140	C₂²	1	8	yes	—
16	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
18	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
20	#125×C₁₂₇	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
22	C₄×Q₈×C₁₂₇	—	2	2	no	—
23	#117×C₁₂₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×Dic₂₅₄	—	2	2	no	—
26	C₂×C₁₂₇⋊C₈	Q₈	1	8	yes	complemented
27	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
28	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₁₂₇	—	2	2	no	—
30	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
31	Dic₁₂₇⋊C₄	—	2	4	no	—
32	C₂²⋊Dic₁₂₇	—	2	2	no	—
33	C₄×Dic₁₂₇	—	2	2	no	—
34	C₂×Q₈×C₁₂₇	—	2	2	no	—
35	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₂₇	—	2	4	no	—
38	C₄⋊C₄×C₁₂₇	—	2	2	no	—
39	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
40	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
41	C₄²×C₁₂₇	—	2	2	no	—
42	C₄²×C₁₂₇	D₄	1	2	yes	—
43	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
45	Dic₂₅₄	—	4	4	no	—
46	C₂×Dic₁₂₇	—	2	4	no	—
47	C₁₂₇⋊C₈	C₂×Q₈	1	4	yes	—
48	C₁₂₇⋊C₈	C₂×Q₈	1	4	yes	—
49	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₁₂₇	—	4	4	no	—
51	C₂×C₄×C₁₂₇	—	2	4	no	—
52	C₂×C₄×C₁₂₇	—	2	4	no	—
53	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
58	C₂²×C₁₂₇	—	2	2	no	—
59	C₂²×C₁₂₇	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₁₂₇	—	2	2	no	—
62	C₄×C₁₂₇	—	4	4	no	—
63	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
64	C₄×C₁₂₇	#135	1	4	yes	—
65	C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
66	Dic₁₂₇	—	4	4	no	—
67	C₂×C₁₂₇	—	2	2	no	—
68	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₁₂₇	C₂×#135	1	2	yes	—
70	#3160	—	127	1	no	maximal, Sylow
71	C₁₂₇	#3160	1	1	yes	characteristic, complemented, Sylow
72	#533	—	127	1	no	—
73	#533	D₁₂₇	1	1	yes	characteristic
74	#500	—	127	2	no	—
75	#499	—	127	4	no	—
76	#498	—	127	2	no	—
77	#427	—	127	4	no	—
78	C₂×C₄×C₈	—	127	1	no	—
79	C₂²×C₈	—	127	2	no	—
80	C₄⋊Q₈	—	127	2	no	—
81	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	—
82	#125	—	127	2	no	—
83	#125	C₂×D₁₂₇	1	2	yes	—
84	C₂²⋊Q₈	—	2	2	no	—
85	C₂²⋊Q₈	—	254	2	no	—
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	—	254	2	no	—
88	#117	—	127	2	no	—
89	#117	C₂×D₁₂₇	1	2	yes	—
90	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
91	#107	—	127	8	no	—
92	#106	—	127	8	no	—
93	C₄×C₈	—	127	1	no	—
94	C₄×C₈	—	127	2	no	—
95	C₄×C₈	—	127	1	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	254	2	no	—
98	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
100	C₂×C₈	—	127	8	no	—
101	C₂×C₈	—	127	2	no	—
102	C₂×C₈	—	127	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	127	4	no	—
106	C₄⋊C₄	—	254	4	no	—
107	C₄⋊C₄	—	254	2	no	—
108	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	254	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	254	2	no	—
113	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
114	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
115	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
116	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	508	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	254	4	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₁₂₇	1	4	yes	—
126	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
127	C₈	—	127	4	no	—
128	C₈	—	127	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	#16861207	1	2	yes	—
131	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	4	no	—
134	C₄	—	508	4	no	—
135	C₄	#16861207	1	2	yes	—
136	C₄	#16861229	1	4	yes	—
137	C₄	C₂×Q₈×D₁₂₇	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	C₂×#16861207	1	1	yes	characteristic
140	C₂	C₂×#16861229	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550433

C₂

yes

maximal, characteristic

#22550279

C₂

yes

maximal, complemented

#22550275

C₂

yes

maximal, complemented

#22550272

C₂

yes

maximal, complemented

#22549912

C₂

yes

maximal

C₂×C₄×C₁₂₇⋊C₈

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

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

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

—

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

—

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⋊Q₈

yes

—

C₄×C₁₂₇

#135

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₂₇

C₂×#135

yes

—

#3160

—

127

maximal, Sylow

C₁₂₇

#3160

yes

characteristic, complemented, Sylow

#533

—

127

—

#533

D₁₂₇

yes

characteristic

#500

—

127

—

#499

—

127

—

#498

—

127

—

#427

—

127

—

C₂×C₄×C₈

—

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

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₂×Q₈

—

C₂×Q₈

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

100

C₂×C₈

—

127

—

101

C₂×C₈

—

127

—

102

C₂×C₈

—

127

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

127

—

106

C₄⋊C₄

—

254

—

107

C₄⋊C₄

—

254

—

108

C₄⋊C₄

C₂²×D₁₂₇

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

254

—

111

C₄²

—

112

C₄²

—

254

—

113

C₄²

C₂²⋊D₁₂₇

yes

—

114

C₄²

C₂²×D₁₂₇

yes

characteristic

115

C₄²

C₂²×D₁₂₇

yes

characteristic

116

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

117

Q₈

—

118

Q₈

—

508

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

254

—

122

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

124

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

125

C₂×C₄

Q₈×D₁₂₇

yes

—

126

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

127

C₈

—

127

—

128

C₈

—

127

—

129

C₂²

—

130

C₂²

#16861207

yes

—

131

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

508

—

135

C₄

#16861207

yes

—

136

C₄

#16861229

yes

—

137

C₄

C₂×Q₈×D₁₂₇

yes

—

138

C₂

—

139

C₂

C₂×#16861207

yes

characteristic

140

C₂

C₂×#16861229

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	#3160
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. #22550279	C₂	semidirect
3. #22550275	C₂	semidirect
4. #22550272	C₂	semidirect
26. C₂×C₁₂₇⋊C₈	Q₈	semidirect
71. C₁₂₇	#3160	semidirect

H ≅ G/N

type

2. #22550279

C₂

semidirect

3. #22550275

C₂

semidirect

4. #22550272

C₂

semidirect

26. C₂×C₁₂₇⋊C₈

Q₈

semidirect

71. C₁₂₇

#3160

semidirect

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

subgroup	index
#22550433	2
#22550279	2
#22550275	2
#22550272	2
#22549912	2
C₂×C₄×C₁₂₇⋊C₈	2
#533×C₁₂₇	2
#3160	127

subgroup

index

C₂×C₄×C₁₂₇⋊C₈

#533×C₁₂₇

#3160

127

Subgroups — #30649240

Conjugacy classes of subgroups

Complements

Maximal subgroups