Group #30651404 — subgroups

Subgroups — #30651404

Order 16256.

336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	#22550226	C₂	1	3	yes	maximal
3	#22550171	C₂	1	3	yes	maximal
4	#22550083	C₂	1	3	yes	maximal
5	Q₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×#14523238	C₂	1	3	yes	maximal
7	#557×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523238	C₂²	1	3	yes	—
9	#16861209	C₂²	1	3	yes	—
10	#16861208	—	2	3	no	—
11	Q₈×Dic₁₂₇	—	2	1	no	—
12	C₄×Dic₂₅₄	C₂²	1	3	yes	—
13	#16861142	C₂²	1	3	yes	—
14	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	3	yes	—
15	#16861133	C₂²	1	6	yes	—
16	#16861133	C₂²	1	3	yes	—
17	#16861132	C₂²	1	1	yes	characteristic
18	C₄×C₁₂₇⋊C₈	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₁₂₇	C₂²	1	3	yes	—
20	#125×C₁₂₇	—	2	3	no	—
21	C₄×Q₈×C₁₂₇	—	2	1	no	—
22	C₄×Q₈×C₁₂₇	C₂²	1	3	yes	—
23	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₅₄	—	2	3	no	—
25	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
26	#14523238	—	2	6	no	—
27	C₂×C₁₂₇⋊C₈	C₂³	1	3	yes	—
28	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₂₇	—	2	3	no	—
30	C₄⋊Dic₁₂₇	C₂³	1	3	yes	—
31	Dic₁₂₇⋊C₄	—	2	3	no	—
32	C₄×Dic₁₂₇	—	2	3	no	—
33	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂×Q₈×C₁₂₇	C₂³	1	3	yes	—
35	C₄⋊C₄×C₁₂₇	—	2	3	no	—
36	C₄⋊C₄×C₁₂₇	—	2	3	no	—
37	C₄⋊C₄×C₁₂₇	—	2	3	no	—
38	C₄⋊C₄×C₁₂₇	D₄	1	3	yes	—
39	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₄²×C₁₂₇	—	2	3	no	—
41	C₄²×C₁₂₇	C₂³	1	3	yes	—
42	Dic₂₅₄	—	2	1	no	—
43	Dic₂₅₄	—	8	3	no	—
44	Dic₂₅₄	#41	1	2	yes	—
45	C₂×Dic₁₂₇	—	2	1	no	—
46	C₂×Dic₁₂₇	—	2	3	no	—
47	C₁₂₇⋊C₈	—	2	3	no	—
48	C₁₂₇⋊C₈	—	2	1	no	—
49	Q₈×C₁₂₇	—	2	3	no	—
50	Q₈×C₁₂₇	—	2	3	no	—
51	Q₈×C₁₂₇	Q₁₆	1	4	yes	complemented
52	C₂×C₄×C₁₂₇	—	2	1	no	—
53	C₂×C₄×C₁₂₇	—	2	3	no	—
54	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
57	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₂₇	—	2	3	no	—
59	C₄×C₁₂₇	—	4	3	no	—
60	C₄×C₁₂₇	—	4	1	no	—
61	C₄×C₁₂₇	C₂×Q₁₆	1	6	yes	—
62	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
63	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₁₂₇	—	2	2	no	—
65	Dic₁₂₇	—	4	3	no	—
66	C₂×C₁₂₇	#548	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₁₂₇	#577	1	1	yes	characteristic
69	#3363	—	127	1	no	maximal, Sylow
70	C₁₂₇	#3363	1	1	yes	characteristic, complemented, Sylow
71	Q₈²	—	127	1	no	—
72	#557	D₁₂₇	1	1	yes	characteristic
73	#487	—	127	3	no	—
74	#477	—	127	3	no	—
75	#462	—	127	3	no	—
76	C₈×Q₈	—	127	1	no	—
77	C₄×Q₁₆	—	127	3	no	—
78	C₂×Q₁₆	—	127	3	no	—
79	C₄⋊Q₈	—	127	3	no	—
80	C₄⋊Q₈	—	254	3	no	—
81	C₄⋊Q₈	C₂×D₁₂₇	1	3	yes	—
82	#125	—	2	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	127	3	no	—
85	C₄×Q₈	—	254	1	no	—
86	C₄×Q₈	C₂×D₁₂₇	1	3	yes	—
87	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
88	#107	—	127	3	no	—
89	C₄⋊C₈	—	127	3	no	—
90	#103	—	127	3	no	—
91	#103	—	127	6	no	—
92	#103	—	127	1	no	—
93	C₄×C₈	—	127	3	no	—
94	C₂×Q₈	—	127	1	no	—
95	C₂×Q₈	—	254	3	no	—
96	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₁₂₇	1	3	yes	—
98	Q₁₆	—	254	6	no	—
99	C₂×C₈	—	127	1	no	—
100	C₂×C₈	—	127	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	127	3	no	—
105	C₄⋊C₄	—	254	3	no	—
106	C₄⋊C₄	—	254	3	no	—
107	C₄⋊C₄	C₂²⋊D₁₂₇	1	3	yes	—
108	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
109	C₄²	—	2	3	no	—
110	C₄²	—	254	3	no	—
111	C₄²	C₂²×D₁₂₇	1	3	yes	—
112	Q₈	—	2	3	no	—
113	Q₈	—	2	3	no	—
114	Q₈	—	127	2	no	—
115	Q₈	—	1016	3	no	—
116	Q₈	—	254	1	no	—
117	Q₈	#14523238	1	4	yes	complemented
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	254	3	no	—
121	C₂×C₄	—	254	1	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
124	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
125	C₈	—	254	3	no	—
126	C₈	—	254	1	no	—
127	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	3	no	—
130	C₄	—	4	1	no	—
131	C₄	—	508	3	no	—
132	C₄	—	254	2	no	—
133	C₄	C₂×#14523238	1	6	yes	—
134	C₄	#41×D₁₂₇	1	1	yes	characteristic
135	C₄	#16861256	1	1	yes	characteristic
136	C₂	#22550533	1	1	yes	characteristic
137	C₂	C₂²×#14523238	1	1	yes	characteristic
138	C₂	#22550684	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₅₄

C₂

yes

maximal, characteristic

#22550226

C₂

yes

maximal

#22550171

C₂

yes

maximal

#22550083

C₂

yes

maximal

Q₈×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

C₄×#14523238

C₂

yes

maximal

#557×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523238

C₂²

yes

—

#16861209

C₂²

yes

—

#16861208

—

Q₈×Dic₁₂₇

—

C₄×Dic₂₅₄

C₂²

yes

—

#16861142

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

#16861133

C₂²

yes

—

#16861133

C₂²

yes

—

#16861132

C₂²

yes

characteristic

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

—

#125×C₁₂₇

—

C₄×Q₈×C₁₂₇

—

C₄×Q₈×C₁₂₇

C₂²

yes

—

C₄×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂³

yes

characteristic

#14523238

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

characteristic

C₂×Q₈×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

#41

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

Q₁₆

yes

complemented

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×Q₁₆

yes

—

C₄×C₁₂₇

C₂×#41

yes

characteristic

C₄×C₁₂₇

2₋¹⁺⁴

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#548

yes

characteristic

C₂×C₁₂₇

C₂²×Q₁₆

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#3363

yes

characteristic, complemented, Sylow

Q₈²

—

127

—

#557

D₁₂₇

yes

characteristic

#487

—

127

—

#477

—

127

—

#462

—

127

—

C₈×Q₈

—

127

—

C₄×Q₁₆

—

127

—

C₂×Q₁₆

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

C₄⋊Q₈

C₂×D₁₂₇

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

C₄×Q₈

C₂×D₁₂₇

yes

—

C₄×Q₈

C₂×D₁₂₇

yes

characteristic

#107

—

127

—

C₄⋊C₈

—

127

—

#103

—

127

—

#103

—

127

—

#103

—

127

—

C₄×C₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×Q₈

C₂²×D₁₂₇

yes

—

Q₁₆

—

254

—

C₂×C₈

—

127

—

100

C₂×C₈

—

127

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

127

—

105

C₄⋊C₄

—

254

—

106

C₄⋊C₄

—

254

—

107

C₄⋊C₄

C₂²⋊D₁₂₇

yes

—

108

C₄⋊C₄

C₂²×D₁₂₇

yes

characteristic

109

C₄²

—

110

C₄²

—

254

—

111

C₄²

C₂²×D₁₂₇

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

127

—

115

Q₈

—

1016

—

116

Q₈

—

254

—

117

Q₈

#14523238

yes

complemented

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

254

—

121

C₂×C₄

—

254

—

122

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

124

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

125

C₈

—

254

—

126

C₈

—

254

—

127

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

508

—

132

C₄

—

254

—

133

C₄

C₂×#14523238

yes

—

134

C₄

#41×D₁₂₇

yes

characteristic

135

C₄

#16861256

yes

characteristic

136

C₂

#22550533

yes

characteristic

137

C₂

C₂²×#14523238

yes

characteristic

138

C₂

#22550684

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#557×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂²×D₄
hypercenter	#557	D₁₂₇
hypocenter	C₁₂₇	#3363
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#557×C₁₂₇

C₂

socle, prime radical

C₂²×C₁₂₇

C₂²×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
51. Q₈×C₁₂₇	Q₁₆	semidirect
70. C₁₂₇	#3363	semidirect
117. Q₈	#14523238	semidirect

H ≅ G/N

type

51. Q₈×C₁₂₇

Q₁₆

semidirect

70. C₁₂₇

#3363

semidirect

117. Q₈

#14523238

semidirect

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

subgroup	index
Q₈×Dic₂₅₄	2
#22550226	2
#22550171	2
#22550083	2
Q₈×C₁₂₇⋊C₈	2
C₄×#14523238	2
#557×C₁₂₇	2
#3363	127

subgroup

index

Q₈×Dic₂₅₄

Q₈×C₁₂₇⋊C₈

#557×C₁₂₇

127

Subgroups — #30651404

Conjugacy classes of subgroups

Complements

Maximal subgroups