Group #30638766 — subgroups

Subgroups — #30638766

Order 16256.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861144	C₂	1	2	yes	maximal, complemented
3	#22549840	C₂	1	1	yes	maximal, characteristic
4	#22549640	C₂	1	2	yes	maximal, complemented
5	#407×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	C₂²	1	2	yes	—
7	C₂²×C₁₂₇⋊C₈	—	2	4	no	—
8	C₂²×C₁₂₇⋊C₈	C₄	1	4	yes	—
9	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	#16861144	—	2	4	no	—
12	C₂²⋊(C₁₂₇⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
16	#14523232	—	2	2	no	—
17	#14523232	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	—	2	8	no	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₂×C₁₂₇⋊C₈	—	2	2	no	—
21	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
24	C₂⁴×C₁₂₇	Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	—	2	2	no	—
27	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
29	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₂₇	—	2	2	no	—
32	M₁₆×C₁₂₇	—	2	2	no	—
33	C₂×C₈×C₁₂₇	—	2	2	no	—
34	C₁₂₇⋊C₈	—	2	2	no	—
35	C₁₂₇⋊C₈	—	4	2	no	—
36	C₁₂₇⋊C₈	C₂²⋊C₄	1	4	yes	—
37	C₂³×C₁₂₇	—	2	4	no	—
38	C₂³×C₁₂₇	—	2	2	no	—
39	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
40	C₂³×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	—	2	2	no	—
43	C₂×C₄×C₁₂₇	—	2	2	no	—
44	C₂×C₄×C₁₂₇	—	2	4	no	—
45	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	#41	1	2	yes	—
50	C₈×C₁₂₇	—	4	2	no	—
51	C₂²×C₁₂₇	—	2	8	no	—
52	C₂²×C₁₂₇	—	2	2	no	—
53	C₂²×C₁₂₇	—	2	2	no	—
54	C₂²×C₁₂₇	—	2	2	no	—
55	C₂²×C₁₂₇	#108	1	4	yes	complemented
56	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₂₇	#117	1	1	yes	characteristic
61	C₄×C₁₂₇	#121	1	2	yes	—
62	C₂×C₁₂₇	—	2	4	no	—
63	C₂×C₁₂₇	—	2	2	no	—
64	C₂×C₁₂₇	#380	1	1	yes	characteristic
65	C₂×C₁₂₇	C₂×#108	1	2	yes	—
66	#1810	—	127	1	no	maximal, Sylow
67	C₁₂₇	#1810	1	1	yes	characteristic, complemented, Sylow
68	C₂³×C₈	—	127	1	no	—
69	C₂×#108	—	127	2	no	—
70	#407	—	127	1	no	—
71	#407	D₁₂₇	1	1	yes	characteristic
72	#341	—	127	2	no	—
73	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
74	C₂×M₁₆	—	127	2	no	—
75	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
76	C₂²×C₈	—	127	1	no	—
77	C₂²×C₈	—	127	4	no	—
78	C₂²×C₈	—	127	1	no	—
79	C₂²×C₈	—	254	4	no	—
80	#108	—	254	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₂²⋊C₈	—	254	2	no	—
83	C₂⁴	Dic₂₅₄	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
88	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	254	2	no	—
93	M₁₆	—	254	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	127	2	no	—
96	C₂×C₈	—	127	4	no	—
97	C₂×C₈	—	127	2	no	—
98	C₂×C₈	—	254	8	no	—
99	C₂×C₈	—	254	2	no	—
100	C₂×C₈	—	254	2	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	2	4	no	—
103	C₂³	Dic₁₂₇⋊C₄	1	2	yes	—
104	C₂³	C₂×Dic₂₅₄	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
113	C₂×C₄	#14523247	1	2	yes	—
114	C₈	—	4	2	no	—
115	C₈	—	127	4	no	—
116	C₈	—	508	2	no	—
117	C₈	—	254	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#16861144	1	4	yes	complemented
123	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
124	C₂²	#16861186	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
127	C₄	#16861165	1	1	yes	characteristic
128	C₄	#16861181	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	4	no	—
131	C₂	#22549734	1	1	yes	characteristic
132	C₂	C₂×#16861144	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

C₂×#16861144

C₂

yes

maximal, complemented

#22549840

C₂

yes

maximal, characteristic

#22549640

C₂

yes

maximal, complemented

#407×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

—

C₂²×C₁₂₇⋊C₈

C₄

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

#16861144

—

C₂²⋊(C₁₂₇⋊C₈)

—

C₂³×C₄×C₁₂₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂²⋊C₈×C₁₂₇

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₂⁴×C₁₂₇

Q₈

yes

characteristic

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

Q₈

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₄⋊C₄

yes

—

C₂³×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#108

yes

complemented

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₂²⋊C₄

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#380

yes

characteristic

C₂×C₁₂₇

C₂×#108

yes

—

#1810

—

127

maximal, Sylow

C₁₂₇

#1810

yes

characteristic, complemented, Sylow

C₂³×C₈

—

127

—

C₂×#108

—

127

—

#407

—

127

—

#407

D₁₂₇

yes

characteristic

#341

—

127

—

C₂³×C₄

C₂×D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

254

—

#108

—

254

—

C₂²⋊C₈

—

C₂²⋊C₈

—

254

—

C₂⁴

Dic₂₅₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₂₇

yes

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

Dic₂₅₄

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

100

C₂×C₈

—

254

—

101

C₂³

—

102

C₂³

—

103

C₂³

Dic₁₂₇⋊C₄

yes

—

104

C₂³

C₂×Dic₂₅₄

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

111

C₂×C₄

D₄×D₁₂₇

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

113

C₂×C₄

#14523247

yes

—

114

C₈

—

115

C₈

—

127

—

116

C₈

—

508

—

117

C₈

—

254

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#16861144

yes

complemented

123

C₂²

C₂×Dic₁₂₇⋊C₄

yes

characteristic

124

C₂²

#16861186

yes

—

125

C₄

—

126

C₄

C₂²⋊C₄×D₁₂₇

yes

characteristic

127

C₄

#16861165

yes

characteristic

128

C₄

#16861181

yes

—

129

C₂

—

130

C₂

—

131

C₂

#22549734

yes

characteristic

132

C₂

C₂×#16861144

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#407×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₂₇	Q₈
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#407	D₁₂₇
hypocenter	C₁₂₇	#1810
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#407×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₄⋊C₄

prime radical

C₂⁴×C₁₂₇

Q₈

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#16861144	C₂	semidirect
4. #22549640	C₂	semidirect
55. C₂²×C₁₂₇	#108	semidirect
67. C₁₂₇	#1810	semidirect
122. C₂²	#16861144	semidirect

H ≅ G/N

type

2. C₂×#16861144

C₂

semidirect

4. #22549640

C₂

semidirect

55. C₂²×C₁₂₇

#108

semidirect

67. C₁₂₇

#1810

semidirect

122. C₂²

#16861144

semidirect

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

subgroup	index
C₂³×C₁₂₇⋊C₈	2
C₂×#16861144	2
#22549840	2
#22549640	2
#407×C₁₂₇	2
#1810	127

subgroup

index

C₂³×C₁₂₇⋊C₈

#407×C₁₂₇

127

Subgroups — #30638766

Conjugacy classes of subgroups

Complements

Maximal subgroups