Group #30639986 — subgroups

Subgroups — #30639986

Order 16256.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×D₁₀₁₆	C₂	1	1	yes	maximal, characteristic
2	#22549880	C₂	1	2	yes	maximal, complemented
3	C₂×#16861125	C₂	1	2	yes	maximal, complemented
4	#22549641	C₂	1	1	yes	maximal, characteristic, complemented
5	#424×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₀₈	C₂²	1	2	yes	—
7	C₂×D₁₀₁₆	—	2	4	no	—
8	C₂×D₁₀₁₆	—	2	2	no	—
9	C₂×#14523232	C₂²	1	2	yes	—
10	#16861129	—	2	4	no	—
11	#16861125	—	2	4	no	—
12	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	#117×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₂₇	—	2	2	no	—
16	C₂³×D₁₂₇	—	2	2	no	—
17	C₂×D₅₀₈	—	2	2	no	—
18	C₂×D₅₀₈	—	2	2	no	—
19	C₂×D₅₀₈	—	2	2	no	—
20	C₂×D₅₀₈	—	2	2	no	—
21	C₂×D₅₀₈	—	2	2	no	—
22	C₂×D₅₀₈	D₄	1	4	yes	—
23	D₁₀₁₆	—	2	8	no	—
24	#14523232	—	2	2	no	—
25	#14523232	—	2	2	no	—
26	C₂×C₁₂₇⋊C₈	—	2	2	no	—
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₂×C₈×C₁₂₇	—	2	1	no	—
31	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
32	C₄⋊C₄×C₁₂₇	D₄	1	2	yes	—
33	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
34	C₄²×C₁₂₇	—	2	1	no	—
35	C₂²×D₁₂₇	—	2	4	no	—
36	C₂²×D₁₂₇	—	4	2	no	—
37	C₂²×D₁₂₇	—	4	2	no	—
38	C₂²×D₁₂₇	—	8	4	no	—
39	D₅₀₈	—	2	4	no	—
40	D₅₀₈	—	2	4	no	—
41	D₅₀₈	—	4	2	no	—
42	D₅₀₈	—	4	2	no	—
43	C₁₂₇⋊C₈	—	4	2	no	—
44	C₂³×C₁₂₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	—	2	2	no	—
46	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	#41	1	2	yes	—
50	C₈×C₁₂₇	—	2	2	no	—
51	C₈×C₁₂₇	—	4	1	no	—
52	C₂²×C₁₂₇	—	2	2	no	—
53	C₂²×C₁₂₇	#121	1	1	yes	characteristic
54	C₂²×C₁₂₇	#123	1	2	yes	—
55	C₂×D₁₂₇	—	4	4	no	—
56	C₂×D₁₂₇	—	8	4	no	—
57	C₂×D₁₂₇	—	8	4	no	—
58	C₂×D₁₂₇	—	8	4	no	—
59	C₄×C₁₂₇	—	4	2	no	—
60	C₄×C₁₂₇	#120	1	1	yes	characteristic
61	C₄×C₁₂₇	#121	1	2	yes	—
62	C₄×C₁₂₇	#124	1	1	yes	characteristic
63	C₂×C₁₂₇	—	2	2	no	—
64	C₂×C₁₂₇	#394	1	1	yes	characteristic
65	C₂×C₁₂₇	#472	1	2	yes	—
66	D₁₂₇	—	8	4	no	—
67	#2045	—	127	1	no	maximal, Sylow
68	C₁₂₇	#2045	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₈	—	127	1	no	—
70	#424	D₁₂₇	1	1	yes	characteristic, complemented
71	#418	—	127	2	no	—
72	C₂×#100	—	127	2	no	—
73	#341	—	127	1	no	—
74	C₂²×D₄	—	127	2	no	—
75	C₂×D₈	—	254	4	no	—
76	C₂×D₈	—	254	2	no	—
77	C₂×M₁₆	—	127	2	no	—
78	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
80	#117	C₂×D₁₂₇	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#102	—	254	4	no	—
83	#100	—	254	4	no	—
84	C₂⁴	—	254	2	no	—
85	C₂×D₄	—	127	4	no	—
86	C₂×D₄	—	254	2	no	—
87	C₂×D₄	—	254	2	no	—
88	C₂×D₄	—	254	2	no	—
89	C₂×D₄	—	254	2	no	—
90	C₂×D₄	—	254	2	no	—
91	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
92	D₈	—	254	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	254	2	no	—
95	M₁₆	—	254	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	254	2	no	—
99	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
100	C₄⋊C₄	D₅₀₈	1	2	yes	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₂³	—	1016	4	no	—
104	C₂³	—	508	2	no	—
105	C₂³	—	508	2	no	—
106	C₂³	—	254	4	no	—
107	C₂³	#14523248	1	1	yes	characteristic
108	D₄	—	508	2	no	—
109	D₄	—	508	2	no	—
110	D₄	—	254	4	no	—
111	D₄	—	254	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
115	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
116	C₂×C₄	#14523249	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	1	no	—
119	C₈	—	508	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	1016	4	no	—
122	C₂²	—	1016	4	no	—
123	C₂²	—	1016	4	no	—
124	C₂²	—	508	4	no	—
125	C₂²	#16861183	1	1	yes	characteristic
126	C₂²	#16861194	1	2	yes	—
127	C₄	—	4	2	no	—
128	C₄	#16861177	1	1	yes	characteristic
129	C₄	#16861184	1	2	yes	—
130	C₄	#16861198	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	1016	4	no	—
133	C₂	#22549793	1	1	yes	characteristic
134	C₂	#22550146	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₀₁₆

C₂

yes

maximal, characteristic

#22549880

C₂

yes

maximal, complemented

C₂×#16861125

C₂

yes

maximal, complemented

#22549641

C₂

yes

maximal, characteristic, complemented

#424×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₀₈

C₂²

yes

—

C₂×D₁₀₁₆

—

C₂×D₁₀₁₆

—

C₂×#14523232

C₂²

yes

—

#16861129

—

#16861125

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

—

C₂³×D₁₂₇

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

D₄

yes

—

D₁₀₁₆

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

D₄

yes

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

characteristic

C₂²×C₁₂₇

#123

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#120

yes

characteristic

C₄×C₁₂₇

#121

yes

—

C₄×C₁₂₇

#124

yes

characteristic

C₂×C₁₂₇

—

C₂×C₁₂₇

#394

yes

characteristic

C₂×C₁₂₇

#472

yes

—

D₁₂₇

—

#2045

—

127

maximal, Sylow

C₁₂₇

#2045

yes

characteristic, complemented, Sylow

C₂²×D₈

—

127

—

#424

D₁₂₇

yes

characteristic, complemented

#418

—

127

—

C₂×#100

—

127

—

#341

—

127

—

C₂²×D₄

—

127

—

C₂×D₈

—

254

—

C₂×D₈

—

254

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

#102

—

254

—

#100

—

254

—

C₂⁴

—

254

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

D₈

—

254

—

M₁₆

—

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

C₂²⋊D₁₂₇

yes

—

100

C₄⋊C₄

D₅₀₈

yes

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₂³

—

1016

—

104

C₂³

—

508

—

105

C₂³

—

508

—

106

C₂³

—

254

—

107

C₂³

#14523248

yes

characteristic

108

D₄

—

508

—

109

D₄

—

508

—

110

D₄

—

254

—

111

D₄

—

254

—

112

C₂×C₄

—

113

C₂×C₄

D₄×D₁₂₇

yes

—

114

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

115

C₂×C₄

C₂×D₅₀₈

yes

characteristic

116

C₂×C₄

#14523249

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

508

—

120

C₂²

—

121

C₂²

—

1016

—

122

C₂²

—

1016

—

123

C₂²

—

1016

—

124

C₂²

—

508

—

125

C₂²

#16861183

yes

characteristic

126

C₂²

#16861194

yes

—

127

C₄

—

128

C₄

#16861177

yes

characteristic

129

C₄

#16861184

yes

—

130

C₄

#16861198

yes

characteristic

131

C₂

—

132

C₂

—

1016

—

133

C₂

#22549793

yes

characteristic

134

C₂

#22550146

yes

—

	subgroup	G/H
center	C₂²	#16861183
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#424×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
prime radical	C₂²×D₁₀₁₆	C₂
hypercenter	#424	D₁₂₇
hypocenter	C₁₂₇	#2045
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#16861183

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#424×C₁₂₇

C₂

socle

C₂²×C₁₂₇

#121

prime radical

C₂²×D₁₀₁₆

C₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. #22549880	C₂	semidirect
3. C₂×#16861125	C₂	semidirect
4. #22549641	C₂	semidirect
5. #424×C₁₂₇	C₂	semidirect
68. C₁₂₇	#2045	semidirect
70. #424	D₁₂₇	semidirect

H ≅ G/N

type

2. #22549880

C₂

semidirect

3. C₂×#16861125

C₂

semidirect

4. #22549641

C₂

semidirect

5. #424×C₁₂₇

C₂

semidirect

68. C₁₂₇

#2045

semidirect

70. #424

D₁₂₇

semidirect

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

subgroup	index
C₂²×D₁₀₁₆	2
#22549880	2
C₂×#16861125	2
#22549641	2
#424×C₁₂₇	2
#2045	127

subgroup

index

C₂²×D₁₀₁₆

#424×C₁₂₇

127

Subgroups — #30639986

Conjugacy classes of subgroups

Complements

Maximal subgroups