Group #30640176 — subgroups

Subgroups — #30640176

Order 16256.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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	#22549875	C₂	1	2	yes	maximal
2	#22549842	C₂	1	1	yes	maximal, characteristic
3	#22549645	C₂	1	2	yes	maximal, complemented
4	#22549641	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#106×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523248	C₂²	1	1	yes	characteristic
7	C₂×#14523232	C₂²	1	1	yes	characteristic
8	C₂×#14523232	C₂²	1	2	yes	—
9	#16861132	—	2	2	no	—
10	#16861129	—	2	2	no	—
11	(C₂×D₁₂₇)⋊C₈	—	2	2	no	—
12	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	#106×C₁₂₇	—	2	2	no	—
15	#14523248	—	2	2	no	—
16	#14523248	—	2	2	no	—
17	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
18	C₂×D₅₀₈	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
20	C₂×C₄×D₁₂₇	—	2	1	no	—
21	#14523232	—	2	2	no	—
22	#14523232	—	2	2	no	—
23	#14523232	—	2	2	no	—
24	C₂×C₁₂₇⋊C₈	—	2	1	no	—
25	C₂×C₁₂₇⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	—	2	2	no	—
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
30	C₄⋊C₄×C₁₂₇	—	2	2	no	—
31	C₄⋊C₄×C₁₂₇	—	2	2	no	—
32	C₂²×D₁₂₇	—	2	1	no	—
33	Dic₂₅₄	—	2	2	no	—
34	D₅₀₈	—	2	2	no	—
35	C₂²⋊D₁₂₇	—	4	2	no	—
36	C₂×Dic₁₂₇	—	2	1	no	—
37	C₄×D₁₂₇	—	4	1	no	—
38	C₄×D₁₂₇	—	4	1	no	—
39	C₁₂₇⋊C₈	—	4	2	no	—
40	C₁₂₇⋊C₈	—	4	1	no	—
41	C₂³×C₁₂₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	—	4	2	no	—
43	C₂×C₄×C₁₂₇	—	8	2	no	—
44	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	#41	1	2	yes	—
47	C₂×C₄×C₁₂₇	#41	1	2	yes	—
48	C₈×C₁₂₇	—	2	2	no	—
49	C₂²×C₁₂₇	—	2	2	no	—
50	C₂²×C₁₂₇	#121	1	1	yes	characteristic
51	C₂²×C₁₂₇	#126	1	2	yes	—
52	C₂×D₁₂₇	—	4	1	no	—
53	C₂×D₁₂₇	—	4	2	no	—
54	C₄×C₁₂₇	—	8	2	no	—
55	C₄×C₁₂₇	#123	1	1	yes	characteristic
56	C₄×C₁₂₇	#123	1	2	yes	—
57	C₄×C₁₂₇	#124	1	1	yes	characteristic
58	Dic₁₂₇	—	4	1	no	—
59	C₂×C₁₂₇	—	2	2	no	—
60	C₂×C₁₂₇	#397	1	1	yes	characteristic
61	C₂×C₁₂₇	#471	1	2	yes	—
62	D₁₂₇	—	8	1	no	—
63	#2079	—	127	1	no	maximal, Sylow
64	C₁₂₇	#2079	1	1	yes	characteristic, complemented, Sylow
65	C₂×#106	D₁₂₇	1	1	yes	characteristic, complemented
66	#417	—	127	2	no	—
67	#408	—	127	1	no	—
68	#343	—	127	2	no	—
69	#341	—	127	1	no	—
70	C₂×#41	—	127	1	no	—
71	C₂×M₁₆	—	127	2	no	—
72	C₂×M₁₆	—	127	1	no	—
73	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
74	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
75	#106	—	2	2	no	—
76	#103	—	254	2	no	—
77	#102	—	254	2	no	—
78	C₂²⋊C₈	—	254	2	no	—
79	#41	—	254	2	no	—
80	#41	—	254	2	no	—
81	C₂×Q₈	—	127	1	no	—
82	C₂×D₄	—	127	1	no	—
83	C₂×D₄	—	254	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	254	1	no	—
86	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
87	M₁₆	—	254	2	no	—
88	M₁₆	—	254	2	no	—
89	M₁₆	—	254	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	254	2	no	—
92	C₂×C₈	—	254	1	no	—
93	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₂³	—	254	1	no	—
97	C₂³	#14523247	1	1	yes	characteristic
98	Q₈	—	254	2	no	—
99	D₄	—	508	2	no	—
100	D₄	—	254	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	—	508	1	no	—
104	C₂×C₄	—	508	1	no	—
105	C₂×C₄	—	254	1	no	—
106	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
108	C₂×C₄	#14523248	1	2	yes	—
109	C₂×C₄	#14523249	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	508	2	no	—
112	C₈	—	508	1	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	508	1	no	—
115	C₂²	—	508	2	no	—
116	C₂²	#16861180	1	1	yes	characteristic
117	C₂²	#16861204	1	2	yes	—
118	C₄	—	8	2	no	—
119	C₄	—	508	1	no	—
120	C₄	#16861192	1	1	yes	characteristic
121	C₄	#16861194	1	2	yes	—
122	C₄	#16861198	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	1016	1	no	—
125	C₂	#22549802	1	1	yes	characteristic
126	C₂	#22550136	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22549875

C₂

yes

maximal

#22549842

C₂

yes

maximal, characteristic

#22549645

C₂

yes

maximal, complemented

#22549641

C₂

yes

maximal, characteristic, complemented

C₂×#106×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₂²

yes

characteristic

C₂×#14523232

C₂²

yes

characteristic

C₂×#14523232

C₂²

yes

—

#16861132

—

#16861129

—

(C₂×D₁₂₇)⋊C₈

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

#106×C₁₂₇

—

#14523248

—

#14523248

—

C₂×Dic₂₅₄

D₄

yes

characteristic

C₂×D₅₀₈

D₄

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

#14523232

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

characteristic

C₂²×C₁₂₇

#126

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#123

yes

characteristic

C₄×C₁₂₇

#123

yes

—

C₄×C₁₂₇

#124

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#397

yes

characteristic

C₂×C₁₂₇

#471

yes

—

D₁₂₇

—

#2079

—

127

maximal, Sylow

C₁₂₇

#2079

yes

characteristic, complemented, Sylow

C₂×#106

D₁₂₇

yes

characteristic, complemented

#417

—

127

—

#408

—

127

—

#343

—

127

—

#341

—

127

—

C₂×#41

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

#106

—

#103

—

254

—

#102

—

254

—

C₂²⋊C₈

—

254

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

254

—

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₂²⋊D₁₂₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

254

—

C₂³

#14523247

yes

characteristic

Q₈

—

254

—

D₄

—

508

—

100

D₄

—

254

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

508

—

104

C₂×C₄

—

508

—

105

C₂×C₄

—

254

—

106

C₂×C₄

D₄×D₁₂₇

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

108

C₂×C₄

#14523248

yes

—

109

C₂×C₄

#14523249

yes

—

110

C₈

—

111

C₈

—

508

—

112

C₈

—

508

—

113

C₂²

—

114

C₂²

—

508

—

115

C₂²

—

508

—

116

C₂²

#16861180

yes

characteristic

117

C₂²

#16861204

yes

—

118

C₄

—

119

C₄

—

508

—

120

C₄

#16861192

yes

characteristic

121

C₄

#16861194

yes

—

122

C₄

#16861198

yes

characteristic

123

C₂

—

124

C₂

—

1016

—

125

C₂

#22549802

yes

characteristic

126

C₂

#22550136

yes

—

	subgroup	G/H
center	C₂²	#16861180
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#106×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
prime radical	C₂×#14523248	C₂²
hypercenter	C₂×#106	D₁₂₇
hypocenter	C₁₂₇	#2079
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#16861180

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#106×C₁₂₇

C₂

C₂²×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #22549645	C₂	semidirect
4. #22549641	C₂	semidirect
5. C₂×#106×C₁₂₇	C₂	semidirect
64. C₁₂₇	#2079	semidirect
65. C₂×#106	D₁₂₇	semidirect

H ≅ G/N

type

3. #22549645

C₂

semidirect

4. #22549641

C₂

semidirect

5. C₂×#106×C₁₂₇

C₂

semidirect

64. C₁₂₇

#2079

semidirect

65. C₂×#106

D₁₂₇

semidirect

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

subgroup	index
#22549875	2
#22549842	2
#22549645	2
#22549641	2
C₂×#106×C₁₂₇	2
#2079	127

subgroup

index

C₂×#106×C₁₂₇

127

Subgroups — #30640176

Conjugacy classes of subgroups

Complements

Maximal subgroups