Group #30640345 — subgroups

Subgroups — #30640345

Order 16256.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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×#16861215	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₁₂₇)⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14523248	C₄	1	1	yes	characteristic
6	#16861215	—	2	2	no	—
7	#16861215	—	2	2	no	—
8	C₂×#14523232	C₄	1	1	yes	characteristic
9	C₂×#14523231	C₄	1	2	yes	—
10	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
11	C₂×C₁₂₇⋊C₁₆	—	2	2	no	—
12	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	M₃₂×C₁₂₇	—	2	2	no	—
15	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
16	#14523248	—	2	2	no	—
17	#14523248	—	2	2	no	—
18	C₂×Dic₂₅₄	C₈	1	1	yes	characteristic
19	C₂×D₅₀₈	C₈	1	1	yes	characteristic
20	C₂×C₂²⋊D₁₂₇	C₈	1	2	yes	—
21	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
22	#14523232	—	2	2	no	—
23	#14523231	—	2	4	no	—
24	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₈×D₁₂₇	—	2	2	no	—
26	C₈×D₁₂₇	—	2	2	no	—
27	C₁₂₇⋊C₁₆	—	2	2	no	—
28	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
31	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
32	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₁₆×C₁₂₇	—	2	2	no	—
34	C₂²×D₁₂₇	C₂×C₈	1	2	yes	—
35	Dic₂₅₄	—	2	2	no	—
36	D₅₀₈	—	2	2	no	—
37	C₂²⋊D₁₂₇	—	2	4	no	—
38	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
39	C₄×D₁₂₇	—	2	2	no	—
40	C₄×D₁₂₇	—	2	2	no	—
41	C₁₂₇⋊C₈	—	2	2	no	—
42	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
48	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₁₂₇	—	2	2	no	—
51	C₂²×C₁₂₇	C₂²⋊C₈	1	2	yes	—
52	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×D₁₂₇	—	2	2	no	—
54	C₂×D₁₂₇	—	2	4	no	—
55	C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
56	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
58	Dic₁₂₇	—	2	2	no	—
59	C₂×C₁₂₇	—	2	2	no	—
60	C₂×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₁₂₇	#504	1	2	yes	—
62	D₁₂₇	—	4	2	no	—
63	#2113	—	127	1	no	maximal, Sylow
64	C₁₂₇	#2113	1	1	yes	characteristic, complemented, Sylow
65	C₂×#131	—	127	1	no	—
66	C₂×M₃₂	D₁₂₇	1	1	yes	characteristic, complemented
67	C₂²×C₁₆	—	127	1	no	—
68	C₂²⋊C₁₆	—	127	4	no	—
69	C₂×#41	—	127	1	no	—
70	#131	—	254	2	no	—
71	#131	—	254	2	no	—
72	C₂×M₁₆	—	127	1	no	—
73	C₂×M₁₆	—	127	2	no	—
74	C₂²×C₈	—	127	2	no	—
75	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
76	M₃₂	—	2	2	no	—
77	C₂×C₁₆	—	127	2	no	—
78	C₂×C₁₆	—	254	2	no	—
79	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
80	#41	—	254	2	no	—
81	#41	—	254	2	no	—
82	C₂×Q₈	—	127	1	no	—
83	C₂×D₄	—	127	2	no	—
84	C₂×D₄	—	127	1	no	—
85	C₂²×C₄	—	127	2	no	—
86	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
87	M₁₆	—	254	2	no	—
88	M₁₆	—	254	4	no	—
89	C₂×C₈	—	127	2	no	—
90	C₂×C₈	—	254	2	no	—
91	C₂×C₈	—	254	2	no	—
92	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
94	C₂×C₈	D₅₀₈	1	2	yes	—
95	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
96	C₁₆	—	2	2	no	—
97	C₁₆	—	254	2	no	—
98	C₂³	—	127	2	no	—
99	C₂³	C₈×D₁₂₇	1	1	yes	characteristic
100	Q₈	—	254	2	no	—
101	D₄	—	254	4	no	—
102	D₄	—	254	2	no	—
103	C₂×C₄	—	127	2	no	—
104	C₂×C₄	—	254	2	no	—
105	C₂×C₄	—	254	2	no	—
106	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
107	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
108	C₂×C₄	#14523231	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
110	C₈	—	254	2	no	—
111	C₈	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
112	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
113	C₈	C₂×D₅₀₈	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	254	4	no	—
116	C₂²	—	254	2	no	—
117	C₂²	(C₂×D₁₂₇)⋊C₈	1	2	yes	—
118	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
119	C₄	—	254	2	no	—
120	C₄	(C₂×D₁₂₇)⋊C₈	1	2	yes	—
121	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
122	C₄	C₂×#14523231	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	508	2	no	—
125	C₂	C₂×(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
126	C₂	#22550294	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861215

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₂₇)⋊C₁₆

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₄

yes

—

—

C₄

yes

characteristic

C₂×#14523231

C₄

yes

—

C₂×C₈×D₁₂₇

C₂²

yes

—

C₂×C₁₂₇⋊C₁₆

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

M₃₂×C₁₂₇

—

C₂×C₁₆×C₁₂₇

C₂²

yes

—

#14523248

—

#14523248

—

C₂×Dic₂₅₄

C₈

yes

characteristic

C₂×D₅₀₈

C₈

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₈

yes

—

C₂×C₄×D₁₂₇

C₂×C₄

yes

—

#14523232

—

#14523231

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₈×D₁₂₇

—

C₈×D₁₂₇

—

C₁₂₇⋊C₁₆

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₁₆×C₁₂₇

—

C₂²×D₁₂₇

C₂×C₈

yes

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₂×C₈

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

C₂²⋊C₄

yes

—

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²⋊C₈

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

C₂²⋊C₈

yes

—

C₄×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₂₇

#504

yes

—

D₁₂₇

—

#2113

—

127

maximal, Sylow

C₁₂₇

#2113

yes

characteristic, complemented, Sylow

C₂×#131

—

127

—

C₂×M₃₂

D₁₂₇

yes

characteristic, complemented

C₂²×C₁₆

—

127

—

C₂²⋊C₁₆

—

127

—

C₂×#41

—

127

—

#131

—

254

—

#131

—

254

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

127

—

C₂×C₁₆

—

254

—

C₂×C₁₆

C₂×D₁₂₇

yes

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

—

C₂×C₈

D₅₀₈

yes

—

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₁₆

—

C₁₆

—

254

—

C₂³

—

127

—

C₂³

C₈×D₁₂₇

yes

characteristic

100

Q₈

—

254

—

101

D₄

—

254

—

102

D₄

—

254

—

103

C₂×C₄

—

127

—

104

C₂×C₄

—

254

—

105

C₂×C₄

—

254

—

106

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

107

C₂×C₄

C₈×D₁₂₇

yes

characteristic

108

C₂×C₄

#14523231

yes

—

109

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

110

C₈

—

254

—

111

C₈

(C₂×D₁₂₇)⋊C₄

yes

—

112

C₈

C₂×C₂²⋊D₁₂₇

yes

characteristic

113

C₈

C₂×D₅₀₈

yes

characteristic

114

C₂²

—

115

C₂²

—

254

—

116

C₂²

—

254

—

117

C₂²

(C₂×D₁₂₇)⋊C₈

yes

—

118

C₂²

C₂×C₈×D₁₂₇

yes

characteristic

119

C₄

—

254

—

120

C₄

(C₂×D₁₂₇)⋊C₈

yes

—

121

C₄

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

122

C₄

C₂×#14523231

yes

characteristic

123

C₂

—

124

C₂

—

508

—

125

C₂

C₂×(C₂×D₁₂₇)⋊C₈

yes

characteristic

126

C₂

#22550294

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₈
Fitting	C₂×M₃₂×C₁₂₇	C₂
prime radical	C₂×#14523248	C₄
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂×M₃₂	D₁₂₇
hypocenter	C₁₂₇	#2113
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂²×C₈

Fitting

C₂×M₃₂×C₁₂₇

C₂

prime radical

C₂×#14523248

C₄

cosocle

C₂×C₈×C₁₂₇

C₂³

hypercenter

C₂×M₃₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. C₂²×C₁₂₇⋊C₁₆	C₂	semidirect
3. (C₂×D₁₂₇)⋊C₁₆	C₂	semidirect
4. C₂×M₃₂×C₁₂₇	C₂	semidirect
64. C₁₂₇	#2113	semidirect
66. C₂×M₃₂	D₁₂₇	semidirect

H ≅ G/N

type

2. C₂²×C₁₂₇⋊C₁₆

C₂

semidirect

3. (C₂×D₁₂₇)⋊C₁₆

C₂

semidirect

4. C₂×M₃₂×C₁₂₇

C₂

semidirect

64. C₁₂₇

#2113

semidirect

66. C₂×M₃₂

D₁₂₇

semidirect

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

subgroup	index
C₂×#16861215	2
C₂²×C₁₂₇⋊C₁₆	2
(C₂×D₁₂₇)⋊C₁₆	2
C₂×M₃₂×C₁₂₇	2
#2113	127

subgroup

index

C₂×#16861215

C₂²×C₁₂₇⋊C₁₆

(C₂×D₁₂₇)⋊C₁₆

C₂×M₃₂×C₁₂₇

#2113

127

Subgroups — #30640345

Conjugacy classes of subgroups

Complements

Maximal subgroups