Group #30638630 — subgroups

Subgroups — #30638630

Order 16256.

272 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×#16861143	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861119	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₂₇	C₂	1	2	yes	maximal
4	#22549638	C₂	1	2	yes	maximal
5	C₂×#106×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₂₇	—	2	2	no	—
9	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
10	#16861143	C₄	1	4	yes	complemented
11	#16861119	C₄	1	4	yes	complemented
12	C₁₂₇⋊#95	—	2	2	no	—
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
15	#106×C₁₂₇	C₄	1	4	yes	complemented
16	C₂²×Dic₁₂₇	—	2	2	no	—
17	C₂²×Dic₁₂₇	D₄	1	2	yes	—
18	C₂×C₁₂₇⋊C₈	—	2	1	no	—
19	C₂×C₁₂₇⋊C₈	—	2	1	no	—
20	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₄⋊Dic₁₂₇	—	2	2	no	—
22	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
23	C₄×Dic₁₂₇	—	4	4	no	—
24	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₂₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₂₇	—	2	2	no	—
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₄⋊C₄×C₁₂₇	—	2	2	no	—
32	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
33	C₂×Dic₁₂₇	—	2	2	no	—
34	C₂×Dic₁₂₇	—	2	2	no	—
35	C₂×Dic₁₂₇	—	2	4	no	—
36	C₂×Dic₁₂₇	—	4	2	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₂×Dic₁₂₇	C₄⋊C₄	1	4	yes	—
39	C₁₂₇⋊C₈	—	2	2	no	—
40	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	—	2	4	no	—
42	C₂×C₄×C₁₂₇	—	4	2	no	—
43	C₂×C₄×C₁₂₇	—	4	2	no	—
44	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
50	C₈×C₁₂₇	C₄²	1	4	yes	complemented
51	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₁₂₇	#137	1	2	yes	—
56	C₄×C₁₂₇	—	4	4	no	—
57	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
58	C₄×C₁₂₇	#117	1	1	yes	characteristic
59	C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
61	Dic₁₂₇	—	2	4	no	—
62	Dic₁₂₇	—	4	4	no	—
63	C₂×C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₁₂₇	#428	1	2	yes	—
65	C₂×C₁₂₇	#440	1	2	yes	—
66	C₂×C₁₂₇	#441	1	1	yes	characteristic
67	C₂×C₁₂₇	#442	1	1	yes	characteristic
68	#1776	—	127	1	no	maximal, Sylow
69	C₁₂₇	#1776	1	1	yes	characteristic, complemented, Sylow
70	C₂×#107	—	127	1	no	—
71	C₂×#106	D₁₂₇	1	1	yes	characteristic
72	C₂×#97	—	127	1	no	—
73	C₄×C₄⋊C₄	—	127	2	no	—
74	#340	—	127	2	no	—
75	C₂²×C₈	—	127	1	no	—
76	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	127	2	no	—
78	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
79	C₂×C₄²	—	127	1	no	—
80	C₂×C₄²	—	254	2	no	—
81	#107	—	127	4	no	—
82	#106	Dic₁₂₇	1	4	yes	complemented
83	#97	—	127	4	no	—
84	#95	—	254	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	127	2	no	—
87	C₂²×C₄	—	254	2	no	—
88	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
89	C₂×C₈	—	127	2	no	—
90	C₂×C₈	—	254	1	no	—
91	C₂×C₈	—	254	1	no	—
92	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
93	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
94	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	127	4	no	—
97	C₄⋊C₄	—	254	2	no	—
98	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
99	C₄²	—	127	2	no	—
100	C₄²	—	127	2	no	—
101	C₄²	—	508	4	no	—
102	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	127	4	no	—
107	C₂×C₄	—	508	2	no	—
108	C₂×C₄	—	508	2	no	—
109	C₂×C₄	—	254	4	no	—
110	C₂×C₄	—	254	2	no	—
111	C₂×C₄	—	254	2	no	—
112	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
114	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
116	C₂×C₄	#14523247	1	1	yes	characteristic
117	C₂×C₄	#14523249	1	1	yes	characteristic
118	C₈	—	254	2	no	—
119	C₈	C₄×Dic₁₂₇	1	4	yes	complemented
120	C₂²	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
121	C₂²	#16861171	1	1	yes	characteristic
122	C₂²	D₄×Dic₁₂₇	1	1	yes	characteristic
123	C₂²	#16861236	1	2	yes	—
124	C₂²	#16861238	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	508	4	no	—
127	C₄	—	254	4	no	—
128	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
129	C₄	#16861167	1	1	yes	characteristic
130	C₄	#16861174	1	1	yes	characteristic
131	C₄	Q₈×Dic₁₂₇	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₁₂₇	1	1	yes	characteristic
133	C₂	#22549915	1	2	yes	—
134	C₂	#22549977	1	2	yes	—
135	C₂	#22549982	1	1	yes	characteristic
136	C₂	#22549987	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861143

C₂

yes

maximal, characteristic

C₂×#16861119

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₁₂₇

C₂

yes

maximal

#22549638

C₂

yes

maximal

C₂×#106×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄×Dic₁₂₇

—

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

#16861143

C₄

yes

complemented

#16861119

C₄

yes

complemented

C₁₂₇⋊#95

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

#106×C₁₂₇

C₄

yes

complemented

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

D₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₄×Dic₁₂₇

Q₈

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₄⋊C₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

C₈×C₁₂₇

C₄²

yes

complemented

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₄×D₄

yes

characteristic

C₂²×C₁₂₇

C₄×D₄

yes

characteristic

C₂²×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄²

yes

characteristic

C₄×C₁₂₇

#117

yes

characteristic

C₄×C₁₂₇

C₄×Q₈

yes

characteristic

C₄×C₁₂₇

C₄×Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₂₇

#428

yes

—

C₂×C₁₂₇

#440

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1776

yes

characteristic, complemented, Sylow

C₂×#107

—

127

—

C₂×#106

D₁₂₇

yes

characteristic

C₂×#97

—

127

—

C₄×C₄⋊C₄

—

127

—

#340

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

—

127

—

C₂×C₄²

—

254

—

#107

—

127

—

#106

Dic₁₂₇

yes

complemented

#97

—

127

—

#95

—

254

—

C₂²×C₄

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

—

C₂×C₈

C₄×D₁₂₇

yes

—

C₂×C₈

C₂×Dic₁₂₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

C₄²

—

127

—

100

C₄²

—

127

—

101

C₄²

—

508

—

102

C₂³

D₄×D₁₂₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

127

—

107

C₂×C₄

—

508

—

108

C₂×C₄

—

508

—

109

C₂×C₄

—

254

—

110

C₂×C₄

—

254

—

111

C₂×C₄

—

254

—

112

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

113

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

114

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

115

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

116

C₂×C₄

#14523247

yes

characteristic

117

C₂×C₄

#14523249

yes

characteristic

118

C₈

—

254

—

119

C₈

C₄×Dic₁₂₇

yes

complemented

120

C₂²

C₄⋊C₄×D₁₂₇

yes

characteristic

121

C₂²

#16861171

yes

characteristic

122

C₂²

D₄×Dic₁₂₇

yes

characteristic

123

C₂²

#16861236

yes

—

124

C₂²

#16861238

yes

—

125

C₄

—

126

C₄

—

508

—

127

C₄

—

254

—

128

C₄

C₂×C₄×Dic₁₂₇

yes

characteristic

129

C₄

#16861167

yes

characteristic

130

C₄

#16861174

yes

characteristic

131

C₄

Q₈×Dic₁₂₇

yes

characteristic

132

C₂

C₄⋊C₄×Dic₁₂₇

yes

characteristic

133

C₂

#22549915

yes

—

134

C₂

#22549977

yes

—

135

C₂

#22549982

yes

characteristic

136

C₂

#22549987

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#106×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#106	D₁₂₇
hypocenter	C₁₂₇	#1776
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#106×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
10. #16861143	C₄	semidirect
11. #16861119	C₄	semidirect
15. #106×C₁₂₇	C₄	semidirect
50. C₈×C₁₂₇	C₄²	semidirect
69. C₁₂₇	#1776	semidirect
82. #106	Dic₁₂₇	semidirect
119. C₈	C₄×Dic₁₂₇	semidirect

H ≅ G/N

type

10. #16861143

C₄

semidirect

11. #16861119

C₄

semidirect

15. #106×C₁₂₇

C₄

semidirect

50. C₈×C₁₂₇

C₄²

semidirect

69. C₁₂₇

#1776

semidirect

82. #106

Dic₁₂₇

semidirect

119. C₈

C₄×Dic₁₂₇

semidirect

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

subgroup	index
C₂×#16861143	2
C₂×#16861119	2
C₄⋊C₄×Dic₁₂₇	2
#22549638	2
C₂×#106×C₁₂₇	2
#1776	127

subgroup

index

C₄⋊C₄×Dic₁₂₇

C₂×#106×C₁₂₇

127

Subgroups — #30638630

Conjugacy classes of subgroups

Complements

Maximal subgroups