Group #30639718 — subgroups

Subgroups — #30639718

Order 16256.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×#16861209	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523232	C₂	1	1	yes	maximal, characteristic
3	#22549644	C₂	1	4	yes	maximal
4	#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	C₄	1	2	yes	—
6	C₂×#14523232	C₂²	1	2	yes	—
7	#16861209	—	2	1	no	—
8	#16861209	—	2	1	no	—
9	#16861209	—	2	1	no	—
10	#16861209	—	2	1	no	—
11	C₂×C₄⋊Dic₁₂₇	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
13	#16861120	—	2	1	no	—
14	C₄×C₁₂₇⋊C₈	—	2	1	no	—
15	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₂₇	—	2	1	no	—
18	C₄⋊C₈×C₁₂₇	—	2	1	no	—
19	C₂×Dic₂₅₄	—	2	2	no	—
20	C₂×Dic₂₅₄	—	2	2	no	—
21	C₂×Dic₂₅₄	—	4	4	no	—
22	C₂²×Dic₁₂₇	C₂×C₄	1	4	yes	—
23	#14523232	—	2	2	no	—
24	#14523232	Q₈	1	4	yes	complemented
25	C₂×C₁₂₇⋊C₈	—	2	2	no	—
26	C₄⋊Dic₁₂₇	—	2	2	no	—
27	C₄⋊Dic₁₂₇	—	2	2	no	—
28	C₄⋊Dic₁₂₇	—	2	2	no	—
29	C₄⋊Dic₁₂₇	—	2	2	no	—
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₂₇	—	2	4	no	—
34	C₂×C₈×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	Dic₂₅₄	—	4	4	no	—
40	Dic₂₅₄	—	4	4	no	—
41	C₂×Dic₁₂₇	—	2	4	no	—
42	C₂×Dic₁₂₇	—	4	4	no	—
43	C₁₂₇⋊C₈	—	2	4	no	—
44	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	—	2	2	no	—
46	C₂×C₄×C₁₂₇	—	2	2	no	—
47	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₂₇	#41	1	2	yes	—
55	C₈×C₁₂₇	—	4	2	no	—
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	#100	1	2	yes	—
63	C₄×C₁₂₇	#101	1	2	yes	—
64	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₂₇	#124	1	1	yes	characteristic
66	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₁₂₇	—	4	4	no	—
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	—	127	1	no	—
75	#423	D₁₂₇	1	1	yes	characteristic
76	C₄×M₁₆	—	127	1	no	—
77	#343	—	127	4	no	—
78	C₂²×Q₈	—	127	2	no	—
79	C₂×M₁₆	—	127	2	no	—
80	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
81	C₄⋊Q₈	—	254	1	no	—
82	C₄⋊Q₈	—	254	1	no	—
83	C₄⋊Q₈	—	254	1	no	—
84	C₄⋊Q₈	—	254	1	no	—
85	C₂×C₄⋊C₄	—	127	2	no	—
86	C₂×C₄⋊C₄	—	127	2	no	—
87	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	#97	—	254	1	no	—
91	C₄×C₈	—	254	1	no	—
92	C₂×Q₈	—	508	4	no	—
93	C₂×Q₈	—	254	2	no	—
94	C₂×Q₈	—	254	2	no	—
95	C₂²×C₄	—	127	4	no	—
96	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	M₁₆	—	127	4	no	—
101	M₁₆	—	254	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	254	2	no	—
104	C₄⋊C₄	—	254	2	no	—
105	C₄⋊C₄	—	254	2	no	—
106	C₄⋊C₄	—	254	2	no	—
107	C₄⋊C₄	—	254	2	no	—
108	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
109	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
110	C₄²	D₅₀₈	1	1	yes	characteristic
111	C₄²	D₅₀₈	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
113	Q₈	—	508	4	no	—
114	Q₈	—	508	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	508	4	no	—
118	C₂×C₄	—	254	4	no	—
119	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
124	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
126	C₂×C₄	#14523247	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₈	—	254	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
132	C₂²	#16861174	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	508	4	no	—
136	C₄	#16861125	1	2	yes	—
137	C₄	#16861127	1	2	yes	—
138	C₄	#16861189	1	2	yes	—
139	C₄	#16861196	1	1	yes	characteristic
140	C₄	#16861207	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#22549781	1	1	yes	characteristic
143	C₂	C₂×#16861125	1	1	yes	characteristic
144	C₂	C₂×#16861127	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861209

C₂

yes

maximal, characteristic

C₄×#14523232

C₂

yes

maximal, characteristic

#22549644

C₂

yes

maximal

#423×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₄

yes

—

C₂×#14523232

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₁₂₇

C₄

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861120

—

C₄×C₁₂₇⋊C₈

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

—

C₄⋊C₈×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

C₂×C₄

yes

—

#14523232

—

#14523232

Q₈

yes

complemented

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

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

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₄×Q₈

yes

—

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

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1997

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

127

—

#423

D₁₂₇

yes

characteristic

C₄×M₁₆

—

127

—

#343

—

127

—

C₂²×Q₈

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₄⋊Q₈

—

254

—

C₄⋊Q₈

—

254

—

C₄⋊Q₈

—

254

—

C₄⋊Q₈

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

254

—

C₄×C₈

—

254

—

C₂×Q₈

—

508

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

100

M₁₆

—

127

—

101

M₁₆

—

254

—

102

C₂×C₈

—

103

C₂×C₈

—

254

—

104

C₄⋊C₄

—

254

—

105

C₄⋊C₄

—

254

—

106

C₄⋊C₄

—

254

—

107

C₄⋊C₄

—

254

—

108

C₄²

C₂²⋊D₁₂₇

yes

characteristic

109

C₄²

C₂²⋊D₁₂₇

yes

characteristic

110

C₄²

D₅₀₈

yes

characteristic

111

C₄²

D₅₀₈

yes

characteristic

112

C₂³

C₂×C₄×D₁₂₇

yes

characteristic

113

Q₈

—

508

—

114

Q₈

—

508

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

508

—

118

C₂×C₄

—

254

—

119

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

124

C₂×C₄

C₂×D₅₀₈

yes

characteristic

125

C₂×C₄

Q₈×D₁₂₇

yes

—

126

C₂×C₄

#14523247

yes

—

127

C₈

—

128

C₈

—

254

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

132

C₂²

#16861174

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

508

—

136

C₄

#16861125

yes

—

137

C₄

#16861127

yes

—

138

C₄

#16861189

yes

—

139

C₄

#16861196

yes

characteristic

140

C₄

#16861207

yes

characteristic

141

C₂

—

142

C₂

#22549781

yes

characteristic

143

C₂

C₂×#16861125

yes

characteristic

144

C₂

C₂×#16861127

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₂₇)⋊C₄
commutator, prime radical	C₂³×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#423×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#423	D₁₂₇
hypocenter	C₁₂₇	#1997
LCS	G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

commutator, prime radical

C₂³×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#423×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
24. #14523232	Q₈	semidirect
73. C₁₂₇	#1997	semidirect

H ≅ G/N

type

24. #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
C₂×#16861209	2
C₄×#14523232	2
#22549644	2
#423×C₁₂₇	2
#1997	127

subgroup

index

#423×C₁₂₇

127

Subgroups — #30639718

Conjugacy classes of subgroups

Complements

Maximal subgroups