Group #30639553 — subgroups

Subgroups — #30639553

Order 16256.

412 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16861206	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861129	C₂	1	4	yes	maximal, complemented
3	C₂×#16861120	C₂	1	1	yes	maximal, characteristic, complemented
4	#382×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₀₈	—	2	2	no	—
6	C₂²×D₅₀₈	C₂²	1	2	yes	—
7	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
8	#16861206	—	2	2	no	—
9	#16861206	C₄	1	4	yes	complemented
10	#16861129	—	2	8	no	—
11	#16861120	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	#95×C₁₂₇	—	2	2	no	—
15	C₂³×D₁₂₇	—	2	2	no	—
16	C₂×D₅₀₈	—	2	8	no	—
17	C₂×D₅₀₈	—	2	4	no	—
18	C₂×D₅₀₈	—	2	2	no	—
19	C₂×D₅₀₈	—	2	4	no	—
20	C₂×D₅₀₈	—	4	8	no	—
21	C₂×D₅₀₈	C₂×C₄	1	4	yes	—
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	—	2	2	no	—
24	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	complemented
25	C₂²×C₄×C₁₂₇	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₂₇	—	2	4	no	—
29	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₂₇	D₄	1	2	yes	—
31	C₂²×D₁₂₇	—	2	4	no	—
32	C₂²×D₁₂₇	—	4	2	no	—
33	C₂²×D₁₂₇	—	4	2	no	—
34	C₂²×D₁₂₇	—	4	8	no	—
35	D₅₀₈	—	2	8	no	—
36	D₅₀₈	—	4	16	no	—
37	D₅₀₈	—	4	4	no	—
38	C₁₂₇⋊C₈	—	2	4	no	—
39	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	—	2	4	no	—
41	C₂×C₄×C₁₂₇	—	4	2	no	—
42	C₂×C₄×C₁₂₇	—	4	4	no	—
43	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
44	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	#41	1	2	yes	—
48	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₂₇	#124	1	1	yes	characteristic
50	C₂²×C₁₂₇	#127	1	1	yes	characteristic
51	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
52	C₂×D₁₂₇	—	4	4	no	—
53	C₂×D₁₂₇	—	4	8	no	—
54	C₂×D₁₂₇	—	8	4	no	—
55	C₂×D₁₂₇	—	8	4	no	—
56	C₄×C₁₂₇	—	2	4	no	—
57	C₄×C₁₂₇	—	8	2	no	—
58	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
59	C₄×C₁₂₇	#121	1	2	yes	—
60	C₂×C₁₂₇	#390	1	1	yes	characteristic
61	C₂×C₁₂₇	#418	1	2	yes	—
62	C₂×C₁₂₇	#490	1	2	yes	—
63	C₂×C₁₂₇	#496	1	2	yes	—
64	D₁₂₇	—	8	4	no	—
65	#1961	—	127	1	no	maximal, Sylow
66	C₁₂₇	#1961	1	1	yes	characteristic, complemented, Sylow
67	C₂×#127	—	127	1	no	—
68	C₂×#102	—	127	4	no	—
69	C₂×#97	—	127	1	no	—
70	#382	D₁₂₇	1	1	yes	characteristic, complemented
71	C₂²×D₄	—	127	2	no	—
72	C₂²×D₄	—	254	2	no	—
73	C₂²×C₈	—	127	2	no	—
74	#127	—	127	4	no	—
75	#127	—	254	2	no	—
76	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
77	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
78	#102	—	254	8	no	—
79	#97	—	254	2	no	—
80	#95	—	2	2	no	—
81	C₂⁴	—	254	2	no	—
82	C₂×D₄	—	127	4	no	—
83	C₂×D₄	—	508	8	no	—
84	C₂×D₄	—	254	8	no	—
85	C₂×D₄	—	254	4	no	—
86	C₂×D₄	—	254	2	no	—
87	C₂×D₄	—	254	4	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
91	C₂×C₈	—	127	4	no	—
92	C₂×C₈	—	254	2	no	—
93	C₂×C₈	—	254	2	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₄²	C₄×D₁₂₇	1	2	yes	—
96	C₄²	D₅₀₈	1	2	yes	—
97	C₂³	—	508	2	no	—
98	C₂³	—	508	2	no	—
99	C₂³	—	508	8	no	—
100	C₂³	—	254	4	no	—
101	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
102	D₄	—	508	16	no	—
103	D₄	—	508	4	no	—
104	D₄	—	254	8	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
110	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
111	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
112	C₂×C₄	#14523249	1	2	yes	—
113	C₈	—	254	4	no	—
114	C₂²	—	1016	4	no	—
115	C₂²	—	1016	4	no	—
116	C₂²	—	508	4	no	—
117	C₂²	—	508	8	no	—
118	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
119	C₂²	#16861198	1	1	yes	characteristic
120	C₂²	#16861205	1	1	yes	characteristic
121	C₂²	#16861234	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	8	2	no	—
124	C₄	#16861171	1	2	yes	—
125	C₄	#16861184	1	2	yes	—
126	C₂	—	1016	4	no	—
127	C₂	#22549777	1	1	yes	characteristic
128	C₂	#22549880	1	2	yes	—
129	C₂	#22550242	1	2	yes	—
130	C₂	#22550266	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861206

C₂

yes

maximal, characteristic

C₂×#16861129

C₂

yes

maximal, complemented

C₂×#16861120

C₂

yes

maximal, characteristic, complemented

#382×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₀₈

—

C₂²×D₅₀₈

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

#16861206

—

#16861206

C₄

yes

complemented

#16861129

—

#16861120

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

—

C₂³×D₁₂₇

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

complemented

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×D₄

yes

—

C₄×C₁₂₇

#121

yes

—

C₂×C₁₂₇

#390

yes

characteristic

C₂×C₁₂₇

#418

yes

—

C₂×C₁₂₇

#490

yes

—

C₂×C₁₂₇

#496

yes

—

D₁₂₇

—

#1961

—

127

maximal, Sylow

C₁₂₇

#1961

yes

characteristic, complemented, Sylow

C₂×#127

—

127

—

C₂×#102

—

127

—

C₂×#97

—

127

—

#382

D₁₂₇

yes

characteristic, complemented

C₂²×D₄

—

127

—

C₂²×D₄

—

254

—

C₂²×C₈

—

127

—

#127

—

127

—

#127

—

254

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#102

—

254

—

#97

—

254

—

#95

—

C₂⁴

—

254

—

C₂×D₄

—

127

—

C₂×D₄

—

508

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄²

C₄×D₁₂₇

yes

—

C₄²

D₅₀₈

yes

—

C₂³

—

508

—

C₂³

—

508

—

C₂³

—

508

—

100

C₂³

—

254

—

101

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

102

D₄

—

508

—

103

D₄

—

508

—

104

D₄

—

254

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

109

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

110

C₂×C₄

D₄×D₁₂₇

yes

—

111

C₂×C₄

C₂×D₅₀₈

yes

characteristic

112

C₂×C₄

#14523249

yes

—

113

C₈

—

254

—

114

C₂²

—

1016

—

115

C₂²

—

1016

—

116

C₂²

—

508

—

117

C₂²

—

508

—

118

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

119

C₂²

#16861198

yes

characteristic

120

C₂²

#16861205

yes

characteristic

121

C₂²

#16861234

yes

—

122

C₄

—

123

C₄

—

124

C₄

#16861171

yes

—

125

C₄

#16861184

yes

—

126

C₂

—

1016

—

127

C₂

#22549777

yes

characteristic

128

C₂

#22549880

yes

—

129

C₂

#22550242

yes

—

130

C₂

#22550266

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#382×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×D₄
prime radical	C₂×#16861206	C₂
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#382	D₁₂₇
hypocenter	C₁₂₇	#1961
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#382×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂×D₄

prime radical

C₂×#16861206

C₂

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
2. C₂×#16861129	C₂	semidirect
3. C₂×#16861120	C₂	semidirect
4. #382×C₁₂₇	C₂	semidirect
9. #16861206	C₄	semidirect
24. C₂×C₁₂₇⋊C₈	D₄	semidirect
66. C₁₂₇	#1961	semidirect
70. #382	D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#16861129

C₂

semidirect

3. C₂×#16861120

C₂

semidirect

4. #382×C₁₂₇

C₂

semidirect

9. #16861206

C₄

semidirect

24. C₂×C₁₂₇⋊C₈

D₄

semidirect

66. C₁₂₇

#1961

semidirect

70. #382

D₁₂₇

semidirect

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

subgroup	index
C₂×#16861206	2
C₂×#16861129	2
C₂×#16861120	2
#382×C₁₂₇	2
#1961	127

subgroup

index

#382×C₁₂₇

127

Subgroups — #30639553

Conjugacy classes of subgroups

Complements

Maximal subgroups