Group #30638222 — subgroups

Subgroups — #30638222

Order 16256.

152 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22550255	C₂	1	1	yes	maximal, characteristic
2	#22550161	C₂	1	1	yes	maximal, characteristic
3	#22550007	C₂	1	1	yes	maximal, characteristic
4	#22549626	C₂	1	1	yes	maximal, characteristic
5	#22549622	C₂	1	1	yes	maximal, characteristic
6	#22549621	C₂	1	1	yes	maximal, characteristic
7	#475×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523238	—	2	1	no	—
9	C₂×#14523238	—	2	1	no	—
10	#16861209	C₂²	1	1	yes	characteristic
11	#16861140	—	2	1	no	—
12	C₄⋊(C₁₂₇⋊C₈)	—	2	1	no	—
13	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
15	#16861133	—	2	1	no	—
16	#16861120	—	2	1	no	—
17	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	#106×C₁₂₇	—	2	1	no	—
21	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	#103×C₁₂₇	—	2	1	no	—
23	C₂×Dic₂₅₄	—	2	1	no	—
24	#14523238	—	4	2	no	—
25	#14523238	—	4	2	no	—
26	C₂×C₁₂₇⋊C₈	—	2	1	no	—
27	C₂×C₁₂₇⋊C₈	—	2	1	no	—
28	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
30	C₄⋊Dic₁₂₇	—	2	1	no	—
31	C₄⋊Dic₁₂₇	—	2	1	no	—
32	C₂×Q₈×C₁₂₇	—	2	1	no	—
33	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₂₇	—	2	1	no	—
35	C₄⋊C₄×C₁₂₇	—	2	1	no	—
36	C₄⋊C₄×C₁₂₇	—	2	1	no	—
37	C₄⋊C₄×C₁₂₇	—	2	1	no	—
38	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
39	C₄²×C₁₂₇	—	2	1	no	—
40	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
41	Dic₂₅₄	—	4	1	no	—
42	Dic₂₅₄	—	4	1	no	—
43	C₂×Dic₁₂₇	—	4	1	no	—
44	C₁₂₇⋊C₈	—	2	1	no	—
45	C₁₂₇⋊C₈	—	4	1	no	—
46	C₁₂₇⋊C₈	—	4	1	no	—
47	C₁₂₇⋊C₈	SD₁₆	1	2	yes	—
48	Q₈×C₁₂₇	—	2	1	no	—
49	Q₈×C₁₂₇	—	2	1	no	—
50	Q₈×C₁₂₇	—	4	1	no	—
51	Q₈×C₁₂₇	—	4	1	no	—
52	C₂×C₄×C₁₂₇	—	2	1	no	—
53	C₂×C₄×C₁₂₇	—	2	1	no	—
54	C₂×C₄×C₁₂₇	—	4	1	no	—
55	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
58	C₈×C₁₂₇	—	4	1	no	—
59	C₂²×C₁₂₇	#121	1	1	yes	characteristic
60	C₄×C₁₂₇	—	2	1	no	—
61	C₄×C₁₂₇	—	4	1	no	—
62	C₄×C₁₂₇	—	4	1	no	—
63	C₄×C₁₂₇	—	8	1	no	—
64	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₄×C₁₂₇	#137	1	1	yes	characteristic
67	C₄×C₁₂₇	#137	1	1	yes	characteristic
68	Dic₁₂₇	—	8	1	no	—
69	C₂×C₁₂₇	#460	1	1	yes	characteristic
70	C₂×C₁₂₇	#470	1	1	yes	characteristic
71	C₂×C₁₂₇	#473	1	1	yes	characteristic
72	#1690	—	127	1	no	maximal, Sylow
73	C₁₂₇	#1690	1	1	yes	characteristic, complemented, Sylow
74	#494	—	127	1	no	—
75	#475	—	127	1	no	—
76	#475	D₁₂₇	1	1	yes	characteristic
77	#446	—	127	1	no	—
78	#334	—	127	1	no	—
79	#332	—	127	1	no	—
80	#332	—	127	1	no	—
81	C₂×Q₁₆	—	254	1	no	—
82	C₂×Q₁₆	—	254	1	no	—
83	C₄⋊Q₈	—	127	1	no	—
84	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
85	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
86	#106	—	2	1	no	—
87	#106	—	254	1	no	—
88	C₄⋊C₈	—	127	1	no	—
89	C₄⋊C₈	—	127	1	no	—
90	C₄⋊C₈	—	254	1	no	—
91	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
92	#103	—	2	1	no	—
93	#103	—	254	1	no	—
94	#97	—	254	1	no	—
95	C₄×C₈	—	127	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	254	1	no	—
98	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
99	Q₁₆	—	508	2	no	—
100	Q₁₆	—	508	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	127	1	no	—
103	C₂×C₈	—	127	1	no	—
104	C₂×C₈	—	254	1	no	—
105	C₂×C₈	—	254	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	254	1	no	—
110	C₄⋊C₄	—	254	1	no	—
111	C₄⋊C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	508	1	no	—
119	Q₈	—	508	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	508	1	no	—
124	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
126	C₂×C₄	#14523247	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	127	2	no	—
129	C₈	—	508	1	no	—
130	C₈	—	508	1	no	—
131	C₈	—	254	1	no	—
132	C₂²	#16861181	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	8	1	no	—
137	C₄	—	1016	1	no	—
138	C₄	SD₁₆×D₁₂₇	1	1	yes	characteristic
139	C₄	#16861236	1	1	yes	characteristic
140	C₄	#16861239	1	1	yes	characteristic
141	C₄	#16861240	1	1	yes	characteristic
142	C₂	#22550074	1	1	yes	characteristic
143	C₂	#22550132	1	1	yes	characteristic
144	C₂	#22550149	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550255

C₂

yes

maximal, characteristic

#22550161

C₂

yes

maximal, characteristic

#22550007

C₂

yes

maximal, characteristic

#22549626

C₂

yes

maximal, characteristic

#22549622

C₂

yes

maximal, characteristic

#22549621

C₂

yes

maximal, characteristic

#475×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523238

—

C₂×#14523238

—

#16861209

C₂²

yes

characteristic

#16861140

—

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

characteristic

#16861133

—

#16861120

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₂₇

C₂²

yes

characteristic

C₄×Q₈×C₁₂₇

C₂²

yes

characteristic

#106×C₁₂₇

—

C₄⋊C₈×C₁₂₇

C₂²

yes

characteristic

#103×C₁₂₇

—

C₂×Dic₂₅₄

—

#14523238

—

#14523238

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

SD₁₆

yes

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×SD₁₆

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1690

yes

characteristic, complemented, Sylow

#494

—

127

—

#475

—

127

—

#475

D₁₂₇

yes

characteristic

#446

—

127

—

#334

—

127

—

#332

—

127

—

#332

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

C₂×D₁₂₇

yes

characteristic

C₄×Q₈

C₂×D₁₂₇

yes

characteristic

#106

—

#106

—

254

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

C₄⋊C₈

C₂×D₁₂₇

yes

characteristic

#103

—

#103

—

254

—

#97

—

254

—

C₄×C₈

—

127

—

C₂×Q₈

—

C₂×Q₈

—

254

—

C₂×Q₈

C₂²⋊D₁₂₇

yes

characteristic

Q₁₆

—

508

—

100

Q₁₆

—

508

—

101

C₂×C₈

—

102

C₂×C₈

—

127

—

103

C₂×C₈

—

127

—

104

C₂×C₈

—

254

—

105

C₂×C₈

—

254

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

254

—

110

C₄⋊C₄

—

254

—

111

C₄⋊C₄

C₂²⋊D₁₂₇

yes

characteristic

112

C₄²

—

113

C₄²

C₂²×D₁₂₇

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

508

—

119

Q₈

—

508

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

508

—

124

C₂×C₄

D₄×D₁₂₇

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

126

C₂×C₄

#14523247

yes

characteristic

127

C₈

—

128

C₈

—

127

—

129

C₈

—

508

—

130

C₈

—

508

—

131

C₈

—

254

—

132

C₂²

#16861181

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

1016

—

138

C₄

SD₁₆×D₁₂₇

yes

characteristic

139

C₄

#16861236

yes

characteristic

140

C₄

#16861239

yes

characteristic

141

C₄

#16861240

yes

characteristic

142

C₂

#22550074

yes

characteristic

143

C₂

#22550132

yes

characteristic

144

C₂

#22550149

yes

characteristic

	subgroup	G/H
center	C₂²	#16861181
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#475×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#121
hypercenter	#475	D₁₂₇
hypocenter	C₁₂₇	#1690
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#16861181

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#475×C₁₂₇

C₂

C₂²×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
73. C₁₂₇	#1690	semidirect

H ≅ G/N

type

73. C₁₂₇

#1690

semidirect

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

subgroup	index
#22550255	2
#22550161	2
#22550007	2
#22549626	2
#22549622	2
#22549621	2
#475×C₁₂₇	2
#1690	127

subgroup

index

#475×C₁₂₇

127

Subgroups — #30638222

Conjugacy classes of subgroups

Complements

Maximal subgroups