Group #30639720 — subgroups

Subgroups — #30639720

Order 16256.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	#22549900	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523232	C₂	1	1	yes	maximal, characteristic
3	#22549645	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	C₂²	1	2	yes	—
6	C₂×#14523232	C₂²	1	2	yes	—
7	C₄⋊(C₁₂₇⋊C₈)	—	2	1	no	—
8	C₄⋊(C₁₂₇⋊C₈)	—	2	1	no	—
9	#16861120	—	2	1	no	—
10	C₄×C₁₂₇⋊C₈	—	2	1	no	—
11	C₂²×Q₈×C₁₂₇	C₄	1	2	yes	—
12	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
13	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
14	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
15	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
16	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
19	#14523232	—	2	2	no	—
20	#14523232	—	2	4	no	—
21	#14523232	Q₈	1	4	yes	complemented
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	—	2	2	no	—
24	C₂×Q₈×C₁₂₇	—	2	2	no	—
25	C₂×Q₈×C₁₂₇	—	2	2	no	—
26	C₂×Q₈×C₁₂₇	—	4	4	no	—
27	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₂₇	—	2	2	no	—
32	C₄⋊C₄×C₁₂₇	—	2	2	no	—
33	C₄⋊C₄×C₁₂₇	—	2	2	no	—
34	C₄⋊C₄×C₁₂₇	—	2	2	no	—
35	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
36	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
37	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
39	C₁₂₇⋊C₈	—	2	4	no	—
40	C₁₂₇⋊C₈	—	4	2	no	—
41	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₁₂₇	—	4	4	no	—
43	Q₈×C₁₂₇	—	4	4	no	—
44	C₂×C₄×C₁₂₇	—	2	2	no	—
45	C₂×C₄×C₁₂₇	—	2	2	no	—
46	C₂×C₄×C₁₂₇	—	2	4	no	—
47	C₂×C₄×C₁₂₇	—	4	4	no	—
48	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₁₂₇	#41	1	2	yes	—
56	C₂²×C₁₂₇	—	2	1	no	—
57	C₂²×C₁₂₇	—	2	1	no	—
58	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₂₇	C₄×Q₈	1	2	yes	—
60	C₄×C₁₂₇	—	2	1	no	—
61	C₄×C₁₂₇	—	2	1	no	—
62	C₄×C₁₂₇	—	4	4	no	—
63	C₄×C₁₂₇	#100	1	2	yes	—
64	C₄×C₁₂₇	#101	1	2	yes	—
65	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₂₇	#124	1	1	yes	characteristic
67	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₁₂₇	—	2	2	no	—
69	C₂×C₁₂₇	#391	1	1	yes	characteristic
70	C₂×C₁₂₇	C₂×#100	1	1	yes	characteristic
71	C₂×C₁₂₇	C₂×#101	1	1	yes	characteristic
72	#1997	—	127	1	no	maximal, Sylow
73	C₁₂₇	#1997	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic
75	#423	—	127	1	no	—
76	C₄×M₁₆	—	127	1	no	—
77	#343	—	127	4	no	—
78	C₂²×Q₈	Dic₁₂₇	1	2	yes	—
79	C₂×M₁₆	—	127	2	no	—
80	C₂×M₁₆	—	127	2	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	—	2	1	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₂×C₄⋊C₄	Dic₁₂₇	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
87	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
88	C₄⋊C₈	—	254	1	no	—
89	C₄⋊C₈	—	254	1	no	—
90	#97	—	254	1	no	—
91	C₄×C₈	—	254	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	4	no	—
95	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
98	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
99	M₁₆	—	127	4	no	—
100	M₁₆	—	254	4	no	—
101	M₁₆	—	254	2	no	—
102	C₂×C₈	—	254	2	no	—
103	C₂×C₈	—	254	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
109	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
110	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
111	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
112	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	Q₈	—	4	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
124	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
126	C₂×C₄	#14523249	1	2	yes	—
127	C₈	—	508	2	no	—
128	C₈	—	254	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
132	C₂²	Q₈×Dic₁₂₇	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	4	no	—
136	C₄	#16861126	1	2	yes	—
137	C₄	#16861128	1	2	yes	—
138	C₄	#16861185	1	2	yes	—
139	C₄	#16861198	1	1	yes	characteristic
140	C₄	#16861207	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#22549783	1	1	yes	characteristic
143	C₂	C₂×#16861126	1	1	yes	characteristic
144	C₂	C₂×#16861128	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549900

C₂

yes

maximal, characteristic

C₄×#14523232

C₂

yes

maximal, characteristic

#22549645

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

C₂×#14523232

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

—

#16861120

—

C₄×C₁₂₇⋊C₈

—

C₂²×Q₈×C₁₂₇

C₄

yes

—

C₄⋊Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₄

yes

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

—

Q₈

yes

complemented

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#100

yes

—

C₄×C₁₂₇

#101

yes

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

#124

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1997

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₂₇

yes

characteristic

#423

—

127

—

C₄×M₁₆

—

127

—

#343

—

127

—

C₂²×Q₈

Dic₁₂₇

yes

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₁₂₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

254

—

C₄⋊C₈

—

254

—

#97

—

254

—

C₄×C₈

—

254

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

127

—

100

M₁₆

—

254

—

101

M₁₆

—

254

—

102

C₂×C₈

—

254

—

103

C₂×C₈

—

254

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₂²⋊D₁₂₇

yes

characteristic

109

C₄²

C₂²⋊D₁₂₇

yes

characteristic

110

C₄²

C₂²⋊D₁₂₇

yes

characteristic

111

C₄²

C₂²⋊D₁₂₇

yes

characteristic

112

C₂³

C₂²×Dic₁₂₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊Dic₁₂₇

yes

characteristic

120

C₂×C₄

C₂²⋊Dic₁₂₇

yes

characteristic

121

C₂×C₄

C₂²⋊Dic₁₂₇

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₁₂₇

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

124

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

125

C₂×C₄

Q₈×D₁₂₇

yes

—

126

C₂×C₄

#14523249

yes

—

127

C₈

—

508

—

128

C₈

—

254

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×C₂²⋊Dic₁₂₇

yes

characteristic

132

C₂²

Q₈×Dic₁₂₇

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#16861126

yes

—

137

C₄

#16861128

yes

—

138

C₄

#16861185

yes

—

139

C₄

#16861198

yes

characteristic

140

C₄

#16861207

yes

characteristic

141

C₂

—

142

C₂

#22549783

yes

characteristic

143

C₂

C₂×#16861126

yes

characteristic

144

C₂

C₂×#16861128

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₂₇
commutator, prime radical	C₂³×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×C₄⋊Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#1997
LCS	G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₂₇

commutator, prime radical

C₂³×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×C₄⋊Q₈×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₂₇

hypocenter

C₁₂₇

#1997

LCS

G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
21. #14523232	Q₈	semidirect
73. C₁₂₇	#1997	semidirect

H ≅ G/N

type

21. #14523232

Q₈

semidirect

73. C₁₂₇

#1997

semidirect

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

subgroup	index
#22549900	2
C₄×#14523232	2
#22549645	2
C₂×C₄⋊Q₈×C₁₂₇	2
#1997	127

subgroup

index

C₂×C₄⋊Q₈×C₁₂₇

127

Subgroups — #30639720

Conjugacy classes of subgroups

Complements

Maximal subgroups