Group #30638110 — subgroups

Subgroups — #30638110

Order 16256.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22550272	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	#22549626	C₂	1	1	yes	maximal, characteristic
5	#22549624	C₂	1	1	yes	maximal, characteristic
6	#22549622	C₂	1	1	yes	maximal, characteristic
7	#462×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	#16861209	C₂²	1	1	yes	characteristic
9	#16861142	—	2	1	no	—
10	#16861140	—	2	1	no	—
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	C₄×C₁₂₇⋊C₈	—	2	1	no	—
17	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
19	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
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	Q₁₆×C₁₂₇	—	4	2	no	—
33	C₂×C₈×C₁₂₇	—	2	1	no	—
34	C₄⋊C₄×C₁₂₇	—	2	1	no	—
35	C₄⋊C₄×C₁₂₇	—	2	1	no	—
36	C₄⋊C₄×C₁₂₇	—	2	1	no	—
37	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₂₇	—	2	1	no	—
39	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
40	Dic₂₅₄	—	8	1	no	—
41	C₂×Dic₁₂₇	—	4	1	no	—
42	C₁₂₇⋊C₈	—	2	1	no	—
43	C₁₂₇⋊C₈	—	4	1	no	—
44	C₁₂₇⋊C₈	—	4	1	no	—
45	C₁₂₇⋊C₈	Q₁₆	1	2	yes	complemented
46	Q₈×C₁₂₇	—	2	1	no	—
47	Q₈×C₁₂₇	—	4	1	no	—
48	Q₈×C₁₂₇	—	4	1	no	—
49	Q₈×C₁₂₇	SD₁₆	1	2	yes	—
50	C₂×C₄×C₁₂₇	—	2	1	no	—
51	C₂×C₄×C₁₂₇	—	2	1	no	—
52	C₂×C₄×C₁₂₇	—	4	1	no	—
53	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
56	C₈×C₁₂₇	—	4	1	no	—
57	C₂²×C₁₂₇	#121	1	1	yes	characteristic
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	—	2	1	no	—
60	C₄×C₁₂₇	—	4	1	no	—
61	C₄×C₁₂₇	—	8	1	no	—
62	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₁₂₇	#135	1	1	yes	characteristic
65	C₄×C₁₂₇	#137	1	1	yes	characteristic
66	Dic₁₂₇	—	8	1	no	—
67	C₂×C₁₂₇	#462	1	1	yes	characteristic
68	C₂×C₁₂₇	#465	1	1	yes	characteristic
69	C₂×C₁₂₇	#471	1	1	yes	characteristic
70	#1672	—	127	1	no	maximal, Sylow
71	C₁₂₇	#1672	1	1	yes	characteristic, complemented, Sylow
72	#498	—	127	1	no	—
73	#477	—	127	1	no	—
74	#462	D₁₂₇	1	1	yes	characteristic
75	C₈×Q₈	—	127	1	no	—
76	#334	—	127	1	no	—
77	#333	—	127	1	no	—
78	#332	—	127	1	no	—
79	C₂×Q₁₆	—	2	1	no	—
80	C₄⋊Q₈	—	127	1	no	—
81	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
83	#107	—	254	1	no	—
84	#106	—	254	1	no	—
85	#106	—	254	1	no	—
86	C₄⋊C₈	—	127	1	no	—
87	C₄⋊C₈	—	127	1	no	—
88	C₄⋊C₈	—	254	1	no	—
89	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
90	#103	—	2	1	no	—
91	#103	—	254	1	no	—
92	C₄×C₈	—	127	1	no	—
93	C₄×C₈	—	254	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	254	1	no	—
96	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
97	Q₁₆	—	4	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	127	1	no	—
100	C₂×C₈	—	127	1	no	—
101	C₂×C₈	—	254	1	no	—
102	C₂×C₈	—	254	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	254	1	no	—
107	C₄⋊C₄	—	254	1	no	—
108	C₄⋊C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	1016	1	no	—
115	Q₈	#14523236	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	508	1	no	—
120	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
122	C₂×C₄	#14523247	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	127	2	no	—
125	C₈	—	508	1	no	—
126	C₈	—	508	1	no	—
127	C₈	—	254	1	no	—
128	C₂²	#16861181	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	1016	1	no	—
134	C₄	C₂×#14523236	1	1	yes	characteristic
135	C₄	Q₁₆×D₁₂₇	1	1	yes	characteristic
136	C₄	#16861229	1	1	yes	characteristic
137	C₄	#16861241	1	1	yes	characteristic
138	C₂	#22550084	1	1	yes	characteristic
139	C₂	#22550103	1	1	yes	characteristic
140	C₂	#22550141	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550272

C₂

yes

maximal, characteristic

#22550171

C₂

yes

maximal, characteristic

Q₈×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

#22549626

C₂

yes

maximal, characteristic

#22549624

C₂

yes

maximal, characteristic

#22549622

C₂

yes

maximal, characteristic

#462×C₁₂₇

C₂

yes

maximal, characteristic

#16861209

C₂²

yes

—

—

—

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

characteristic

#16861133

—

C₄×C₁₂₇⋊C₈

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

characteristic

C₄×Q₈×C₁₂₇

C₂²

yes

characteristic

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

Q₁₆×C₁₂₇

—

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

Q₁₆

yes

complemented

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

SD₁₆

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×SD₁₆

yes

characteristic

C₄×C₁₂₇

C₂×Q₁₆

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1672

yes

characteristic, complemented, Sylow

#498

—

127

—

#477

—

127

—

#462

D₁₂₇

yes

characteristic

C₈×Q₈

—

127

—

#334

—

127

—

#333

—

127

—

#332

—

127

—

C₂×Q₁₆

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

C₂×D₁₂₇

yes

characteristic

C₄×Q₈

C₂×D₁₂₇

yes

characteristic

#107

—

254

—

#106

—

254

—

#106

—

254

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

C₄⋊C₈

C₂×D₁₂₇

yes

characteristic

#103

—

#103

—

254

—

C₄×C₈

—

127

—

C₄×C₈

—

254

—

C₂×Q₈

—

C₂×Q₈

—

254

—

C₂×Q₈

C₂²⋊D₁₂₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

127

—

100

C₂×C₈

—

127

—

101

C₂×C₈

—

254

—

102

C₂×C₈

—

254

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

254

—

107

C₄⋊C₄

—

254

—

108

C₄⋊C₄

C₂²⋊D₁₂₇

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₁₂₇

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

1016

—

115

Q₈

#14523236

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

508

—

120

C₂×C₄

D₄×D₁₂₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

122

C₂×C₄

#14523247

yes

characteristic

123

C₈

—

124

C₈

—

127

—

125

C₈

—

508

—

126

C₈

—

508

—

127

C₈

—

254

—

128

C₂²

#16861181

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

1016

—

134

C₄

C₂×#14523236

yes

characteristic

135

C₄

Q₁₆×D₁₂₇

yes

characteristic

136

C₄

#16861229

yes

characteristic

137

C₄

#16861241

yes

characteristic

138

C₂

#22550084

yes

characteristic

139

C₂

#22550103

yes

characteristic

140

C₂

#22550141

yes

characteristic

	subgroup	G/H
center	C₂²	#16861181
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#462×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#121
hypercenter	#462	D₁₂₇
hypocenter	C₁₂₇	#1672
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#16861181

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#462×C₁₂₇

C₂

C₂²×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
45. C₁₂₇⋊C₈	Q₁₆	semidirect
71. C₁₂₇	#1672	semidirect

H ≅ G/N

type

45. C₁₂₇⋊C₈

Q₁₆

semidirect

71. C₁₂₇

#1672

semidirect

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

subgroup	index
#22550272	2
#22550171	2
Q₈×C₁₂₇⋊C₈	2
#22549626	2
#22549624	2
#22549622	2
#462×C₁₂₇	2
#1672	127

subgroup

index

Q₈×C₁₂₇⋊C₈

#462×C₁₂₇

127

Subgroups — #30638110

Conjugacy classes of subgroups

Complements

Maximal subgroups