Group #30638186 — subgroups

Subgroups — #30638186

Order 16256.

144 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 55 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	#22550226	C₂	1	1	yes	maximal, characteristic
2	#22550171	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#22549628	C₂	1	1	yes	maximal, characteristic
5	#22549624	C₂	1	1	yes	maximal, characteristic
6	#22549622	C₂	1	1	yes	maximal, characteristic
7	#475×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	#16861209	C₂²	1	1	yes	characteristic
9	#16861142	—	2	1	no	—
10	C₄⋊(C₁₂₇⋊C₈)	—	2	1	no	—
11	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#16861133	—	2	1	no	—
14	#16861133	—	2	1	no	—
15	#16861133	—	2	1	no	—
16	C₄×C₁₂₇⋊C₈	—	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	C₂×C₁₂₇⋊C₈	—	2	1	no	—
25	C₂×C₁₂₇⋊C₈	—	2	1	no	—
26	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₂₇	—	2	1	no	—
29	C₄⋊Dic₁₂₇	—	2	1	no	—
30	C₂×Q₈×C₁₂₇	—	2	1	no	—
31	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₂₇	—	2	1	no	—
33	C₄⋊C₄×C₁₂₇	—	2	1	no	—
34	C₄⋊C₄×C₁₂₇	—	2	1	no	—
35	C₄⋊C₄×C₁₂₇	—	2	1	no	—
36	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
37	C₄²×C₁₂₇	—	2	1	no	—
38	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
39	Dic₂₅₄	—	8	1	no	—
40	C₂×Dic₁₂₇	—	4	1	no	—
41	C₁₂₇⋊C₈	—	2	1	no	—
42	C₁₂₇⋊C₈	—	2	1	no	—
43	C₁₂₇⋊C₈	—	4	1	no	—
44	C₁₂₇⋊C₈	—	4	1	no	—
45	Q₈×C₁₂₇	—	2	1	no	—
46	Q₈×C₁₂₇	—	4	1	no	—
47	Q₈×C₁₂₇	—	4	1	no	—
48	Q₈×C₁₂₇	Q₁₆	1	2	yes	—
49	C₂×C₄×C₁₂₇	—	2	1	no	—
50	C₂×C₄×C₁₂₇	—	2	1	no	—
51	C₂×C₄×C₁₂₇	—	4	1	no	—
52	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
55	C₈×C₁₂₇	—	4	1	no	—
56	C₂²×C₁₂₇	#121	1	1	yes	characteristic
57	C₄×C₁₂₇	—	2	2	no	—
58	C₄×C₁₂₇	—	2	1	no	—
59	C₄×C₁₂₇	—	4	1	no	—
60	C₄×C₁₂₇	—	8	1	no	—
61	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₁₂₇	#135	1	1	yes	characteristic
63	C₄×C₁₂₇	#135	1	1	yes	characteristic
64	C₄×C₁₂₇	#137	1	1	yes	characteristic
65	Dic₁₂₇	—	8	1	no	—
66	C₂×C₁₂₇	#464	1	1	yes	characteristic
67	C₂×C₁₂₇	#467	1	1	yes	characteristic
68	C₂×C₁₂₇	#471	1	1	yes	characteristic
69	#1684	—	127	1	no	maximal, Sylow
70	C₁₂₇	#1684	1	1	yes	characteristic, complemented, Sylow
71	#487	—	127	1	no	—
72	#477	—	127	1	no	—
73	#475	D₁₂₇	1	1	yes	characteristic
74	C₈×Q₈	—	127	1	no	—
75	#335	—	127	1	no	—
76	#333	—	127	1	no	—
77	#332	—	127	1	no	—
78	C₄⋊Q₈	—	127	1	no	—
79	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
80	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
81	#107	—	254	1	no	—
82	#106	—	2	1	no	—
83	C₄⋊C₈	—	127	1	no	—
84	C₄⋊C₈	—	127	1	no	—
85	C₄⋊C₈	—	254	1	no	—
86	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
87	#103	—	2	1	no	—
88	#103	—	254	1	no	—
89	#103	—	254	1	no	—
90	#103	—	254	1	no	—
91	C₄×C₈	—	127	1	no	—
92	C₄×C₈	—	254	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	254	1	no	—
95	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	127	1	no	—
98	C₂×C₈	—	127	1	no	—
99	C₂×C₈	—	254	1	no	—
100	C₂×C₈	—	254	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	254	1	no	—
105	C₄⋊C₄	—	254	1	no	—
106	C₄⋊C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	Q₈	—	4	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	1016	1	no	—
113	Q₈	#14523238	1	2	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	508	1	no	—
118	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
120	C₂×C₄	#14523247	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	508	1	no	—
123	C₈	—	508	1	no	—
124	C₈	—	254	1	no	—
125	C₈	—	254	1	no	—
126	C₂²	#16861181	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	1016	1	no	—
132	C₄	C₂×#14523238	1	1	yes	characteristic
133	C₄	#16861229	1	1	yes	characteristic
134	C₄	#16861230	1	1	yes	characteristic
135	C₄	#16861238	1	1	yes	characteristic
136	C₂	#22550096	1	1	yes	characteristic
137	C₂	#22550115	1	1	yes	characteristic
138	C₂	#22550141	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550226

C₂

yes

maximal, characteristic

#22550171

C₂

yes

maximal, characteristic

Q₈×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

#22549628

C₂

yes

maximal, characteristic

#22549624

C₂

yes

maximal, characteristic

#22549622

C₂

yes

maximal, characteristic

#475×C₁₂₇

C₂

yes

maximal, characteristic

#16861209

C₂²

yes

characteristic

#16861142

—

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

—

—

C₄×C₁₂₇⋊C₈

—

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

—

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

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

Q₁₆

yes

—

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×Q₁₆

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1684

yes

characteristic, complemented, Sylow

#487

—

127

—

#477

—

127

—

#475

D₁₂₇

yes

characteristic

C₈×Q₈

—

127

—

#335

—

127

—

#333

—

127

—

#332

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

C₂×D₁₂₇

yes

characteristic

C₄×Q₈

C₂×D₁₂₇

yes

characteristic

#107

—

254

—

#106

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

C₄⋊C₈

C₂×D₁₂₇

yes

characteristic

#103

—

#103

—

254

—

#103

—

254

—

#103

—

254

—

C₄×C₈

—

127

—

C₄×C₈

—

254

—

C₂×Q₈

—

C₂×Q₈

—

254

—

C₂×Q₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

100

C₂×C₈

—

254

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

254

—

105

C₄⋊C₄

—

254

—

106

C₄⋊C₄

C₂²⋊D₁₂₇

yes

characteristic

107

C₄²

—

108

C₄²

C₂²×D₁₂₇

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

1016

—

113

Q₈

#14523238

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

508

—

118

C₂×C₄

D₄×D₁₂₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

120

C₂×C₄

#14523247

yes

characteristic

121

C₈

—

122

C₈

—

508

—

123

C₈

—

508

—

124

C₈

—

254

—

125

C₈

—

254

—

126

C₂²

#16861181

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

1016

—

132

C₄

C₂×#14523238

yes

characteristic

133

C₄

#16861229

yes

characteristic

134

C₄

#16861230

yes

characteristic

135

C₄

#16861238

yes

characteristic

136

C₂

#22550096

yes

characteristic

137

C₂

#22550115

yes

characteristic

138

C₂

#22550141

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	#1684
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
70. C₁₂₇	#1684	semidirect

H ≅ G/N

type

70. C₁₂₇

#1684

semidirect

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

subgroup	index
#22550226	2
#22550171	2
Q₈×C₁₂₇⋊C₈	2
#22549628	2
#22549624	2
#22549622	2
#475×C₁₂₇	2
#1684	127

subgroup

index

Q₈×C₁₂₇⋊C₈

#475×C₁₂₇

127

Subgroups — #30638186

Conjugacy classes of subgroups

Complements

Maximal subgroups