Group #30637476 — subgroups

Subgroups — #30637476

Order 16256.

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

normal subgroups only

This group has 180 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	#22549929	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22549628	C₂	1	2	yes	maximal, complemented
4	#22549626	C₂	1	2	yes	maximal, complemented
5	#431×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂²×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
9	C₂²⋊(C₁₂₇⋊C₈)	—	2	1	no	—
10	C₄×C₁₂₇⋊C₈	—	2	1	no	—
11	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
12	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
13	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
18	C₄×C₈×C₁₂₇	—	2	1	no	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₂×C₁₂₇⋊C₈	—	2	2	no	—
21	C₂×C₁₂₇⋊C₈	C₈	1	4	yes	complemented
22	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
32	C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
33	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₁₂₇⋊C₈	—	2	1	no	—
35	C₁₂₇⋊C₈	—	2	1	no	—
36	C₁₂₇⋊C₈	—	4	2	no	—
37	C₁₂₇⋊C₈	C₂×C₈	1	2	yes	—
38	C₁₂₇⋊C₈	C₂×C₈	1	2	yes	—
39	C₂³×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
50	C₈×C₁₂₇	—	4	2	no	—
51	C₂²×C₁₂₇	—	2	1	no	—
52	C₂²×C₁₂₇	—	2	1	no	—
53	C₂²×C₁₂₇	C₄⋊C₈	1	2	yes	—
54	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₁₂₇	—	2	1	no	—
56	C₄×C₁₂₇	—	2	1	no	—
57	C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
58	C₄×C₁₂₇	#108	1	2	yes	—
59	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
60	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
61	C₄×C₁₂₇	#135	1	2	yes	—
62	C₂×C₁₂₇	—	2	2	no	—
63	C₂×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₁₂₇	#427	1	1	yes	characteristic
65	C₂×C₁₂₇	C₂×#108	1	1	yes	characteristic
66	#1564	—	127	1	no	maximal, Sylow
67	C₁₂₇	#1564	1	1	yes	characteristic, complemented, Sylow
68	#431	—	127	1	no	—
69	#431	D₁₂₇	1	1	yes	characteristic
70	C₂×C₄×C₈	—	127	1	no	—
71	#335	—	127	2	no	—
72	#334	—	127	2	no	—
73	C₂²×C₈	—	127	1	no	—
74	C₂²×C₈	—	127	1	no	—
75	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
76	C₄⋊C₈	—	127	2	no	—
77	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
78	C₂²⋊C₈	—	2	1	no	—
79	C₂²⋊C₈	—	254	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	127	1	no	—
82	C₄×C₈	—	127	1	no	—
83	C₄×C₈	—	127	1	no	—
84	C₄×C₈	—	127	1	no	—
85	C₄×C₈	—	254	1	no	—
86	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
88	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	127	4	no	—
91	C₂×C₈	—	127	1	no	—
92	C₂×C₈	—	127	1	no	—
93	C₂×C₈	—	127	1	no	—
94	C₂×C₈	—	127	1	no	—
95	C₂×C₈	—	254	2	no	—
96	C₂×C₈	—	254	2	no	—
97	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
98	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
99	C₄²	Dic₂₅₄	1	1	yes	characteristic
100	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
101	C₂³	Dic₁₂₇⋊C₄	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
105	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
106	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
107	C₂×C₄	C₈×D₁₂₇	1	2	yes	—
108	C₂×C₄	#14523231	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	127	2	no	—
114	C₈	—	127	2	no	—
115	C₈	—	508	2	no	—
116	C₈	—	254	1	no	—
117	C₈	—	254	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	Dic₁₂₇⋊C₈	1	2	yes	—
121	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	Dic₁₂₇⋊C₈	1	2	yes	—
125	C₄	#16861144	1	2	yes	—
126	C₄	C₂×C₈×D₁₂₇	1	1	yes	characteristic
127	C₄	C₂×#14523231	1	1	yes	characteristic
128	C₄	#16861229	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	C₂×Dic₁₂₇⋊C₈	1	1	yes	characteristic
131	C₂	#22549912	1	1	yes	characteristic
132	C₂	C₂×#16861144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549929

C₂

yes

maximal, characteristic

C₂×C₄×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

#22549628

C₂

yes

maximal, complemented

#22549626

C₂

yes

maximal, complemented

#431×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₄

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₄

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₂₇⋊C₈)

—

C₄×C₁₂₇⋊C₈

—

C₄×C₁₂₇⋊C₈

C₄

yes

characteristic

C₄×C₁₂₇⋊C₈

C₄

yes

characteristic

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₂²⋊C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₈

yes

complemented

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

Q₈

yes

characteristic

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

Q₈

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₂×C₈

yes

—

C₁₂₇⋊C₈

C₂×C₈

yes

—

C₂³×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄⋊C₈

yes

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄⋊C₈

yes

—

C₄×C₁₂₇

#108

yes

—

C₄×C₁₂₇

C₂²×C₈

yes

characteristic

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

C₄×C₁₂₇

#135

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄⋊C₈

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1564

yes

characteristic, complemented, Sylow

#431

—

127

—

#431

D₁₂₇

yes

characteristic

C₂×C₄×C₈

—

127

—

#335

—

127

—

#334

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

254

—

C₄×C₈

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

254

—

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂²×C₄

Dic₂₅₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄²

C₄×D₁₂₇

yes

characteristic

C₄²

C₂²⋊D₁₂₇

yes

characteristic

C₄²

Dic₂₅₄

yes

characteristic

100

C₄²

C₂²×D₁₂₇

yes

characteristic

101

C₂³

Dic₁₂₇⋊C₄

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

Dic₁₂₇⋊C₄

yes

characteristic

105

C₂×C₄

Dic₁₂₇⋊C₄

yes

characteristic

106

C₂×C₄

Dic₁₂₇⋊C₄

yes

characteristic

107

C₂×C₄

C₈×D₁₂₇

yes

—

108

C₂×C₄

#14523231

yes

—

109

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

111

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

112

C₈

—

113

C₈

—

127

—

114

C₈

—

127

—

115

C₈

—

508

—

116

C₈

—

254

—

117

C₈

—

254

—

118

C₂²

—

119

C₂²

—

120

C₂²

Dic₁₂₇⋊C₈

yes

—

121

C₂²

C₂×Dic₁₂₇⋊C₄

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

Dic₁₂₇⋊C₈

yes

—

125

C₄

#16861144

yes

—

126

C₄

C₂×C₈×D₁₂₇

yes

characteristic

127

C₄

C₂×#14523231

yes

characteristic

128

C₄

#16861229

yes

—

129

C₂

—

130

C₂

C₂×Dic₁₂₇⋊C₈

yes

characteristic

131

C₂

#22549912

yes

characteristic

132

C₂

C₂×#16861144

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₈
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#431×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₂₇	C₄⋊C₄
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	#431	D₁₂₇
hypocenter	C₁₂₇	#1564
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₈

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#431×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₂₇

C₄⋊C₄

cosocle

C₄²×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #22549628	C₂	semidirect
4. #22549626	C₂	semidirect
21. C₂×C₁₂₇⋊C₈	C₈	semidirect
67. C₁₂₇	#1564	semidirect

H ≅ G/N

type

3. #22549628

C₂

semidirect

4. #22549626

C₂

semidirect

21. C₂×C₁₂₇⋊C₈

C₈

semidirect

67. C₁₂₇

#1564

semidirect

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

subgroup	index
#22549929	2
C₂×C₄×C₁₂₇⋊C₈	2
#22549628	2
#22549626	2
#431×C₁₂₇	2
#1564	127

subgroup

index

C₂×C₄×C₁₂₇⋊C₈

#431×C₁₂₇

127

Subgroups — #30637476

Conjugacy classes of subgroups

Complements

Maximal subgroups