Group #30647664 — subgroups

Subgroups — #30647664

Order 16256.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#22550433	C₂	1	1	yes	maximal, characteristic
2	#22550006	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₂₅₄	C₂	1	4	yes	maximal, complemented
4	#22549902	C₂	1	2	yes	maximal
5	#22549900	C₂	1	1	yes	maximal, characteristic
6	#22549831	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523232	C₂²	1	2	yes	—
9	#16861209	C₄	1	2	yes	—
10	#16861201	C₄	1	2	yes	—
11	#16861188	C₄	1	4	yes	—
12	C₄×Dic₂₅₄	C₂²	1	4	yes	—
13	#16861168	C₂²	1	2	yes	—
14	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
15	Dic₁₂₇⋊C₈	C₂²	1	8	yes	—
16	#16861119	C₂²	1	4	yes	—
17	C₈×Dic₁₂₇	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
22	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
23	C₂×Dic₂₅₄	C₂×C₄	1	4	yes	—
24	#14523232	—	2	4	no	—
25	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
27	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
28	Dic₁₂₇⋊C₄	C₂×C₄	1	8	yes	—
29	C₂²⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
30	C₄×Dic₁₂₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	—	2	2	no	—
34	C₂×C₈×C₁₂₇	Q₈	1	4	yes	complemented
35	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
36	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₂₇	C₂³	1	2	yes	—
41	Dic₂₅₄	—	2	8	no	—
42	C₂×Dic₁₂₇	C₂²×C₄	1	8	yes	—
43	C₁₂₇⋊C₈	—	2	4	no	—
44	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	—	2	4	no	—
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₂₇	#41	1	2	yes	—
51	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
52	C₈×C₁₂₇	—	2	2	no	—
53	C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
54	C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
55	C₂²×C₁₂₇	—	2	2	no	—
56	C₂²×C₁₂₇	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
60	C₄×C₁₂₇	#131	1	4	yes	—
61	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
63	Dic₁₂₇	—	2	8	no	—
64	C₂×C₁₂₇	—	2	2	no	—
65	C₂×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₂₇	C₂×#131	1	2	yes	—
67	#2960	—	127	1	no	maximal, Sylow
68	C₁₂₇	#2960	1	1	yes	characteristic, complemented, Sylow
69	#533	—	127	1	no	—
70	#446	—	127	4	no	—
71	C₈×Q₈	—	127	4	no	—
72	#424	—	127	2	no	—
73	#423	—	127	1	no	—
74	#405	—	127	2	no	—
75	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
76	C₂×M₁₆	—	127	2	no	—
77	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
79	C₄⋊Q₈	—	127	2	no	—
80	#125	—	127	2	no	—
81	C₂²⋊Q₈	—	127	4	no	—
82	C₄×Q₈	—	127	4	no	—
83	#117	—	127	2	no	—
84	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
85	C₄⋊C₈	—	127	8	no	—
86	C₄⋊C₈	—	127	4	no	—
87	#97	—	127	4	no	—
88	C₄×C₈	—	127	4	no	—
89	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
90	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
91	C₂×Q₈	—	127	4	no	—
92	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
94	M₁₆	—	254	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	127	4	no	—
97	C₂×C₈	Dic₂₅₄	1	4	yes	complemented
98	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
99	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
101	C₄⋊C₄	—	127	8	no	—
102	C₄⋊C₄	—	127	4	no	—
103	C₄⋊C₄	—	127	4	no	—
104	C₂²⋊C₄	—	127	4	no	—
105	C₄²	—	127	4	no	—
106	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
107	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
108	C₄²	C₂²×D₁₂₇	1	2	yes	—
109	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
110	Q₈	—	254	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	127	8	no	—
113	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
115	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
116	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
117	C₂×C₄	#14523248	1	2	yes	—
118	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	254	4	no	—
121	C₈	C₂×Dic₂₅₄	1	2	yes	—
122	C₈	C₂×Dic₂₅₄	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₂₅₄	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	254	8	no	—
128	C₄	C₄×Dic₂₅₄	1	2	yes	—
129	C₄	#16861215	1	4	yes	—
130	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
131	C₄	C₂×#14523248	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₂₅₄	1	1	yes	characteristic
134	C₂	C₂×#16861215	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550433

C₂

yes

maximal, characteristic

#22550006

C₂

yes

maximal, complemented

C₈×Dic₂₅₄

C₂

yes

maximal, complemented

#22549902

C₂

yes

maximal

#22549900

C₂

yes

maximal, characteristic

#22549831

C₂

yes

maximal

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

#16861209

C₄

yes

—

#16861201

C₄

yes

—

#16861188

C₄

yes

—

C₄×Dic₂₅₄

C₂²

yes

—

#16861168

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

Dic₁₂₇⋊C₈

C₂²

yes

—

#16861119

C₂²

yes

—

C₈×Dic₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

C₂×C₄

yes

—

#14523232

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

Dic₁₂₇⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₁₂₇

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

Q₈

yes

complemented

C₂×C₈×C₁₂₇

C₂³

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

C₂×Dic₁₂₇

C₂²×C₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

C₂×Q₈

yes

—

C₈×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄×Q₈

yes

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

#131

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

C₄×C₁₂₇

C₂×#41

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₂₇

C₂×#131

yes

—

#2960

—

127

maximal, Sylow

C₁₂₇

#2960

yes

characteristic, complemented, Sylow

#533

—

127

—

#446

—

127

—

C₈×Q₈

—

127

—

#424

—

127

—

#423

—

127

—

#405

—

127

—

C₂×C₄×C₈

D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊Q₈

—

127

—

#125

—

127

—

C₂²⋊Q₈

—

127

—

C₄×Q₈

—

127

—

#117

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

—

C₄×C₈

C₂×D₁₂₇

yes

—

C₂×Q₈

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

Dic₂₅₄

yes

complemented

C₂×C₈

C₂²×D₁₂₇

yes

—

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

100

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

101

C₄⋊C₄

—

127

—

102

C₄⋊C₄

—

127

—

103

C₄⋊C₄

—

127

—

104

C₂²⋊C₄

—

127

—

105

C₄²

—

127

—

106

C₄²

C₄×D₁₂₇

yes

characteristic

107

C₄²

C₄×D₁₂₇

yes

characteristic

108

C₄²

C₂²×D₁₂₇

yes

—

109

C₂³

C₂×C₄×D₁₂₇

yes

characteristic

110

Q₈

—

254

—

111

C₂×C₄

—

112

C₂×C₄

—

127

—

113

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

114

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

115

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

116

C₂×C₄

C₂×Dic₂₅₄

yes

—

117

C₂×C₄

#14523248

yes

—

118

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

119

C₈

—

120

C₈

—

254

—

121

C₈

C₂×Dic₂₅₄

yes

—

122

C₈

C₂×Dic₂₅₄

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₂₅₄

yes

—

125

C₂²

C₂²×C₄×D₁₂₇

yes

characteristic

126

C₄

—

127

C₄

—

254

—

128

C₄

C₄×Dic₂₅₄

yes

—

129

C₄

#16861215

yes

—

130

C₄

C₂²×Dic₂₅₄

yes

characteristic

131

C₄

C₂×#14523248

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₂₅₄

yes

characteristic

134

C₂

C₂×#16861215

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#2960
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×C₄×C₈×C₁₂₇

C₂

prime radical

C₂³×C₁₂₇

C₂²×C₄

cosocle

C₂×C₄×C₁₂₇

C₂⁴

hypercenter

C₂×C₄×C₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #22550006	C₂	semidirect
3. C₈×Dic₂₅₄	C₂	semidirect
34. C₂×C₈×C₁₂₇	Q₈	semidirect
68. C₁₂₇	#2960	semidirect
97. C₂×C₈	Dic₂₅₄	semidirect

H ≅ G/N

type

2. #22550006

C₂

semidirect

3. C₈×Dic₂₅₄

C₂

semidirect

34. C₂×C₈×C₁₂₇

Q₈

semidirect

68. C₁₂₇

#2960

semidirect

97. C₂×C₈

Dic₂₅₄

semidirect

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

subgroup	index
#22550433	2
#22550006	2
C₈×Dic₂₅₄	2
#22549902	2
#22549900	2
#22549831	2
C₂×C₄×C₈×C₁₂₇	2
#2960	127

subgroup

index

C₈×Dic₂₅₄

C₂×C₄×C₈×C₁₂₇

127

Subgroups — #30647664

Conjugacy classes of subgroups

Complements

Maximal subgroups