Group #30639515 — subgroups

Subgroups — #30639515

Order 16256.

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

normal subgroups only

This group has 196 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×#16861145	C₂	1	1	yes	maximal, characteristic
2	#22549919	C₂	1	2	yes	maximal
3	C₂×#16861143	C₂	1	1	yes	maximal, characteristic
4	#22549644	C₂	1	2	yes	maximal
5	#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
8	#16861145	C₄	1	4	yes	complemented
9	#16861143	—	2	1	no	—
10	#16861143	—	2	1	no	—
11	#16861142	—	2	2	no	—
12	#16861140	—	2	2	no	—
13	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	#117×C₁₂₇	C₂²	1	1	yes	characteristic
16	#97×C₁₂₇	—	2	1	no	—
17	C₄×C₈×C₁₂₇	—	2	1	no	—
18	C₂²×Dic₁₂₇	—	2	2	no	—
19	#14523232	—	2	2	no	—
20	#14523232	C₂×C₄	1	4	yes	—
21	C₂×C₁₂₇⋊C₈	—	2	2	no	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₂₇	—	2	1	no	—
26	M₁₆×C₁₂₇	Q₈	1	2	yes	—
27	C₂×C₈×C₁₂₇	—	2	1	no	—
28	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
32	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
35	C₄²×C₁₂₇	—	2	1	no	—
36	C₂×Dic₁₂₇	—	4	2	no	—
37	C₂×Dic₁₂₇	—	8	2	no	—
38	C₁₂₇⋊C₈	—	2	4	no	—
39	C₂³×C₁₂₇	#41	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	—	2	1	no	—
41	C₂×C₄×C₁₂₇	—	2	1	no	—
42	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
48	C₈×C₁₂₇	—	2	2	no	—
49	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
50	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₁₂₇	—	2	1	no	—
52	C₂²×C₁₂₇	—	2	1	no	—
53	C₂²×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₁₂₇	#121	1	1	yes	characteristic
55	C₂²×C₁₂₇	#125	1	1	yes	characteristic
56	C₄×C₁₂₇	—	4	1	no	—
57	C₄×C₁₂₇	—	4	1	no	—
58	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
60	C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₁₂₇	—	8	2	no	—
63	C₂×C₁₂₇	—	2	1	no	—
64	C₂×C₁₂₇	—	2	1	no	—
65	C₂×C₁₂₇	#389	1	1	yes	characteristic
66	C₂×C₁₂₇	#472	1	1	yes	characteristic
67	C₂×C₁₂₇	#473	1	1	yes	characteristic
68	#1952	—	127	1	no	maximal, Sylow
69	C₁₂₇	#1952	1	1	yes	characteristic, complemented, Sylow
70	C₂×#108	—	127	1	no	—
71	#428	—	127	2	no	—
72	C₂×#107	—	127	1	no	—
73	#405	D₁₂₇	1	1	yes	characteristic
74	#343	—	127	2	no	—
75	C₂×M₁₆	—	127	2	no	—
76	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
78	#117	C₂×D₁₂₇	1	1	yes	characteristic
79	C₂×C₄⋊C₄	—	127	2	no	—
80	#108	—	127	4	no	—
81	#107	—	254	1	no	—
82	#107	—	254	1	no	—
83	#107	—	254	2	no	—
84	#106	—	254	2	no	—
85	#97	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₂²×C₄	—	254	2	no	—
88	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
89	M₁₆	—	2	1	no	—
90	M₁₆	—	127	4	no	—
91	M₁₆	—	254	2	no	—
92	M₁₆	Dic₂₅₄	1	2	yes	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	254	2	no	—
95	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
96	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
98	C₂×C₈	Dic₂₅₄	1	2	yes	—
99	C₄⋊C₄	—	254	2	no	—
100	C₄⋊C₄	—	254	2	no	—
101	C₄⋊C₄	D₅₀₈	1	1	yes	characteristic
102	C₄⋊C₄	D₅₀₈	1	1	yes	characteristic
103	C₂²⋊C₄	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₂³	#14523248	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	1016	2	no	—
109	C₂×C₄	—	508	2	no	—
110	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
112	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
115	C₂×C₄	#14523248	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	254	4	no	—
118	C₈	Dic₁₂₇⋊C₄	1	2	yes	—
119	C₈	Dic₁₂₇⋊C₄	1	2	yes	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	C₄×Dic₂₅₄	1	1	yes	characteristic
123	C₂²	#16861183	1	1	yes	characteristic
124	C₂²	#16861201	1	1	yes	characteristic
125	C₄	—	4	1	no	—
126	C₄	—	4	1	no	—
127	C₄	—	1016	2	no	—
128	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
129	C₄	C₄×D₅₀₈	1	1	yes	characteristic
130	C₄	#16861188	1	1	yes	characteristic
131	C₄	#16861209	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#22549774	1	1	yes	characteristic
135	C₂	#22550146	1	1	yes	characteristic
136	C₂	#22550151	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861145

C₂

yes

maximal, characteristic

#22549919

C₂

yes

maximal

C₂×#16861143

C₂

yes

maximal, characteristic

#22549644

C₂

yes

maximal

#405×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861145

C₄

yes

—

—

—

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

characteristic

#97×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₂²×Dic₁₂₇

—

#14523232

—

#14523232

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

Q₈

yes

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

—

C₄⋊C₄×C₁₂₇

D₄

yes

characteristic

C₄⋊C₄×C₁₂₇

D₄

yes

characteristic

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

C₄⋊C₄

yes

—

C₈×C₁₂₇

C₄⋊C₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄×Q₈

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₄×D₄

yes

characteristic

C₄×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1952

yes

characteristic, complemented, Sylow

C₂×#108

—

127

—

#428

—

127

—

C₂×#107

—

127

—

#405

D₁₂₇

yes

characteristic

#343

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

C₂×D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

—

127

—

#108

—

127

—

#107

—

254

—

#107

—

254

—

#107

—

254

—

#106

—

254

—

#97

—

C₄×C₈

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

127

—

M₁₆

—

254

—

M₁₆

Dic₂₅₄

yes

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

—

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

Dic₂₅₄

yes

—

C₄⋊C₄

—

254

—

100

C₄⋊C₄

—

254

—

101

C₄⋊C₄

D₅₀₈

yes

characteristic

102

C₄⋊C₄

D₅₀₈

yes

characteristic

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂³

#14523248

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

1016

—

109

C₂×C₄

—

508

—

110

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

112

C₂×C₄

C₂×D₅₀₈

yes

characteristic

113

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

114

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

115

C₂×C₄

#14523248

yes

characteristic

116

C₈

—

117

C₈

—

254

—

118

C₈

Dic₁₂₇⋊C₄

yes

—

119

C₈

Dic₁₂₇⋊C₄

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₄×Dic₂₅₄

yes

characteristic

123

C₂²

#16861183

yes

characteristic

124

C₂²

#16861201

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

1016

—

128

C₄

C₂×Dic₁₂₇⋊C₄

yes

characteristic

129

C₄

C₄×D₅₀₈

yes

characteristic

130

C₄

#16861188

yes

characteristic

131

C₄

#16861209

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#22549774

yes

characteristic

135

C₂

#22550146

yes

characteristic

136

C₂

#22550151

yes

characteristic

	subgroup	G/H
center	C₂²	#16861183
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#405×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
prime radical	C₂³×C₁₂₇	#41
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#405	D₁₂₇
hypocenter	C₁₂₇	#1952
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#16861183

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#405×C₁₂₇

C₂

C₂²×C₁₂₇

C₂³×C₁₂₇

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
8. #16861145	C₄	semidirect
69. C₁₂₇	#1952	semidirect

H ≅ G/N

type

8. #16861145

C₄

semidirect

69. C₁₂₇

#1952

semidirect

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

subgroup	index
C₂×#16861145	2
#22549919	2
C₂×#16861143	2
#22549644	2
#405×C₁₂₇	2
#1952	127

subgroup

index

#405×C₁₂₇

127

Subgroups — #30639515

Conjugacy classes of subgroups

Complements

Maximal subgroups