Group #30640484 — subgroups

Subgroups — #30640484

Order 16256.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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×#16861148	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₁₂₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
8	C₄²×D₁₂₇	C₄	1	1	yes	characteristic
9	#16861148	—	2	4	no	—
10	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
11	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
12	C₈×Dic₁₂₇	C₄	1	1	yes	characteristic
13	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
15	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₂₇	C₈	1	1	yes	characteristic
17	C₂×C₄×D₁₂₇	C₈	1	1	yes	characteristic
18	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₈×D₁₂₇	—	2	2	no	—
22	C₈×D₁₂₇	D₄	1	2	yes	—
23	C₈×D₁₂₇	Q₈	1	2	yes	—
24	C₄×Dic₁₂₇	C₈	1	1	yes	characteristic
25	C₄×Dic₁₂₇	C₈	1	1	yes	characteristic
26	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₁₂₇⋊C₁₆	—	2	2	no	—
28	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₁₆×C₁₂₇	—	2	2	no	—
32	C₂²×D₁₂₇	C₂×C₈	1	1	yes	characteristic
33	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
34	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
35	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
36	C₄×D₁₂₇	—	2	2	no	—
37	C₄×D₁₂₇	—	2	2	no	—
38	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
39	C₄×D₁₂₇	M₁₆	1	2	yes	—
40	C₁₂₇⋊C₈	—	2	1	no	—
41	C₁₂₇⋊C₈	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₈×C₁₂₇	—	2	1	no	—
46	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
49	C₂×D₁₂₇	—	2	1	no	—
50	C₂×D₁₂₇	—	2	1	no	—
51	C₂×D₁₂₇	C₄⋊C₈	1	2	yes	—
52	C₄×C₁₂₇	—	2	1	no	—
53	C₄×C₁₂₇	M₃₂	1	2	yes	complemented
54	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
56	Dic₁₂₇	—	2	1	no	—
57	Dic₁₂₇	C₄⋊C₈	1	2	yes	—
58	Dic₁₂₇	M₃₂	1	2	yes	complemented
59	C₂×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
61	C₂×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
62	D₁₂₇	—	2	2	no	—
63	#2150	—	127	1	no	maximal, Sylow
64	C₁₂₇	#2150	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₃₂	—	127	2	no	—
66	C₂×C₄×C₈	—	127	1	no	—
67	C₄⋊C₁₆	—	127	2	no	—
68	C₄⋊C₁₆	—	127	1	no	—
69	C₄⋊C₁₆	D₁₂₇	1	1	yes	characteristic, complemented
70	C₂²×C₈	—	127	1	no	—
71	C₂²×C₈	—	127	1	no	—
72	C₂×C₄²	—	127	1	no	—
73	M₃₂	—	254	4	no	—
74	C₂×C₁₆	—	127	2	no	—
75	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
76	C₄×C₈	—	127	1	no	—
77	C₄×C₈	—	127	1	no	—
78	C₄×C₈	—	127	1	no	—
79	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
80	C₂²×C₄	—	127	1	no	—
81	C₂²×C₄	—	127	1	no	—
82	C₂²×C₄	—	127	1	no	—
83	C₂×C₈	—	127	1	no	—
84	C₂×C₈	—	127	2	no	—
85	C₂×C₈	—	127	2	no	—
86	C₂×C₈	—	127	1	no	—
87	C₂×C₈	—	254	2	no	—
88	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
89	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
90	C₄²	—	127	1	no	—
91	C₄²	—	127	1	no	—
92	C₄²	—	127	1	no	—
93	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
94	C₁₆	—	2	2	no	—
95	C₁₆	—	254	2	no	—
96	C₂³	—	127	1	no	—
97	C₂×C₄	—	127	1	no	—
98	C₂×C₄	—	127	1	no	—
99	C₂×C₄	—	127	2	no	—
100	C₂×C₄	—	127	2	no	—
101	C₂×C₄	—	127	1	no	—
102	C₂×C₄	—	254	2	no	—
103	C₂×C₄	—	254	2	no	—
104	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
105	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	127	2	no	—
109	C₈	—	254	1	no	—
110	C₈	D₄×D₁₂₇	1	1	yes	characteristic
111	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
112	C₂²	—	127	2	no	—
113	C₂²	—	254	1	no	—
114	C₂²	—	254	1	no	—
115	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	127	2	no	—
118	C₄	—	127	2	no	—
119	C₄	—	254	1	no	—
120	C₄	#16861148	1	2	yes	complemented
121	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
122	C₄	M₁₆×D₁₂₇	1	1	yes	characteristic
123	C₂	—	254	2	no	—
124	C₂	C₄⋊C₈×D₁₂₇	1	1	yes	characteristic
125	C₂	M₃₂×D₁₂₇	1	1	yes	characteristic
126	C₂	C₂×#16861148	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861148

C₂

yes

maximal

C₄×C₈×D₁₂₇

C₂

yes

maximal, characteristic

C₄⋊(C₁₂₇⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, complemented

C₄⋊C₁₆×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₂₇

C₄

yes

characteristic

C₂×C₈×D₁₂₇

C₂²

yes

characteristic

C₄²×D₁₂₇

C₄

yes

characteristic

#16861148

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₄

yes

characteristic

C₈×Dic₁₂₇

C₄

yes

characteristic

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₈

yes

characteristic

C₂×C₄×D₁₂₇

C₈

yes

characteristic

C₂×C₄×D₁₂₇

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₁₂₇

—

C₈×D₁₂₇

D₄

yes

—

C₈×D₁₂₇

Q₈

yes

—

C₄×Dic₁₂₇

C₈

yes

characteristic

C₄×Dic₁₂₇

C₈

yes

characteristic

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₁₂₇⋊C₁₆

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₁₆×C₁₂₇

—

C₂²×D₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

C₄⋊C₄

yes

—

C₄×D₁₂₇

M₁₆

yes

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

characteristic

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₄⋊C₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

M₃₂

yes

complemented

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₄⋊C₈

yes

—

Dic₁₂₇

M₃₂

yes

complemented

C₂×C₁₂₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₂₇

C₂×M₃₂

yes

characteristic

C₂×C₁₂₇

C₂×M₃₂

yes

characteristic

D₁₂₇

—

#2150

—

127

maximal, Sylow

C₁₂₇

#2150

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

127

—

C₂×C₄×C₈

—

127

—

C₄⋊C₁₆

—

127

—

C₄⋊C₁₆

—

127

—

C₄⋊C₁₆

D₁₂₇

yes

characteristic, complemented

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂×C₄²

—

127

—

M₃₂

—

254

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

C₄×D₁₂₇

yes

characteristic

C₁₆

—

C₁₆

—

254

—

C₂³

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

100

C₂×C₄

—

127

—

101

C₂×C₄

—

127

—

102

C₂×C₄

—

254

—

103

C₂×C₄

—

254

—

104

C₂×C₄

C₈×D₁₂₇

yes

characteristic

105

C₂×C₄

C₈×D₁₂₇

yes

characteristic

106

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

107

C₈

—

108

C₈

—

127

—

109

C₈

—

254

—

110

C₈

D₄×D₁₂₇

yes

characteristic

111

C₈

Q₈×D₁₂₇

yes

characteristic

112

C₂²

—

127

—

113

C₂²

—

254

—

114

C₂²

—

254

—

115

C₂²

C₂×C₈×D₁₂₇

yes

characteristic

116

C₄

—

117

C₄

—

127

—

118

C₄

—

127

—

119

C₄

—

254

—

120

C₄

#16861148

yes

complemented

121

C₄

C₄⋊C₄×D₁₂₇

yes

characteristic

122

C₄

M₁₆×D₁₂₇

yes

characteristic

123

C₂

—

254

—

124

C₂

C₄⋊C₈×D₁₂₇

yes

characteristic

125

C₂

M₃₂×D₁₂₇

yes

characteristic

126

C₂

C₂×#16861148

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₈
Fitting	C₄⋊C₁₆×C₁₂₇	C₂
prime radical	C₂²×D₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₄⋊C₁₆	D₁₂₇
hypocenter	C₁₂₇	#2150
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂²×C₈

Fitting

C₄⋊C₁₆×C₁₂₇

C₂

prime radical

C₂²×D₁₂₇

C₂×C₈

cosocle

C₂×C₈×C₁₂₇

C₂³

hypercenter

C₄⋊C₁₆

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
3. C₄⋊(C₁₂₇⋊C₁₆)	C₂	semidirect
4. Dic₁₂₇⋊C₁₆	C₂	semidirect
5. C₄⋊C₁₆×C₁₂₇	C₂	semidirect
53. C₄×C₁₂₇	M₃₂	semidirect
58. Dic₁₂₇	M₃₂	semidirect
64. C₁₂₇	#2150	semidirect
69. C₄⋊C₁₆	D₁₂₇	semidirect
120. C₄	#16861148	semidirect

H ≅ G/N

type

3. C₄⋊(C₁₂₇⋊C₁₆)

C₂

semidirect

4. Dic₁₂₇⋊C₁₆

C₂

semidirect

5. C₄⋊C₁₆×C₁₂₇

C₂

semidirect

53. C₄×C₁₂₇

M₃₂

semidirect

58. Dic₁₂₇

M₃₂

semidirect

64. C₁₂₇

#2150

semidirect

69. C₄⋊C₁₆

D₁₂₇

semidirect

120. C₄

#16861148

semidirect

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

subgroup	index
C₂×#16861148	2
C₄×C₈×D₁₂₇	2
C₄⋊(C₁₂₇⋊C₁₆)	2
Dic₁₂₇⋊C₁₆	2
C₄⋊C₁₆×C₁₂₇	2
#2150	127

subgroup

index

C₂×#16861148

C₄×C₈×D₁₂₇

C₄⋊(C₁₂₇⋊C₁₆)

Dic₁₂₇⋊C₁₆

C₄⋊C₁₆×C₁₂₇

#2150

127

Subgroups — #30640484

Conjugacy classes of subgroups

Complements

Maximal subgroups