Group #30639678 — subgroups

Subgroups — #30639678

Order 16256.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16861209	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	—	2	1	no	—
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	#16861209	—	2	2	no	—
8	#16861209	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₂₇	—	2	1	no	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	#95×C₁₂₇	—	2	1	no	—
16	C₂×Dic₂₅₄	—	2	4	no	—
17	C₂×Dic₂₅₄	—	4	4	no	—
18	C₂²×Dic₁₂₇	—	2	2	no	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₂×C₁₂₇⋊C₈	—	2	4	no	—
21	C₄⋊Dic₁₂₇	—	2	4	no	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₂₇	—	2	2	no	—
25	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₂₇	—	2	2	no	—
29	C₄⋊C₄×C₁₂₇	Q₈	1	4	yes	complemented
30	C₄²×C₁₂₇	—	4	2	no	—
31	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₄²×C₁₂₇	D₄	1	2	yes	—
33	Dic₂₅₄	—	4	8	no	—
34	C₂×Dic₁₂₇	—	2	4	no	—
35	C₂×Dic₁₂₇	—	4	4	no	—
36	C₁₂₇⋊C₈	—	4	2	no	—
37	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	—	2	2	no	—
39	C₂×C₄×C₁₂₇	—	2	4	no	—
40	C₂×C₄×C₁₂₇	—	4	2	no	—
41	C₂×C₄×C₁₂₇	—	4	2	no	—
42	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
45	C₂×C₄×C₁₂₇	Q₁₆	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₂₇	#41	1	2	yes	—
50	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₂₇	—	2	2	no	—
56	C₄×C₁₂₇	—	4	4	no	—
57	C₄×C₁₂₇	#103	1	4	yes	—
58	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
59	C₄×C₁₂₇	#124	1	1	yes	characteristic
60	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₁₂₇	—	4	4	no	—
62	C₂×C₁₂₇	#391	1	1	yes	characteristic
63	C₂×C₁₂₇	C₂×#103	1	1	yes	characteristic
64	C₂×C₁₂₇	#418	1	1	yes	characteristic
65	C₂×C₁₂₇	#476	1	2	yes	—
66	C₂×C₁₂₇	#477	1	2	yes	—
67	#1987	—	127	1	no	maximal, Sylow
68	C₁₂₇	#1987	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₄⋊Q₈	—	127	1	no	—
70	C₂×C₄⋊C₈	—	127	1	no	—
71	C₄×C₄⋊C₄	D₁₂₇	1	1	yes	characteristic
72	#340	—	127	4	no	—
73	C₂²×Q₈	—	254	1	no	—
74	C₂²×C₈	—	127	2	no	—
75	C₄⋊Q₈	—	127	4	no	—
76	C₄⋊Q₈	—	254	2	no	—
77	C₂×C₄⋊C₄	—	127	2	no	—
78	C₂×C₄⋊C₄	—	254	1	no	—
79	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
80	C₂×C₄²	—	2	1	no	—
81	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
82	C₄⋊C₈	—	254	2	no	—
83	#95	—	2	1	no	—
84	C₂×Q₈	—	508	4	no	—
85	C₂×Q₈	—	254	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	254	2	no	—
88	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
91	C₂×C₈	—	254	2	no	—
92	C₂×C₈	—	254	4	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	127	4	no	—
95	C₄⋊C₄	—	254	4	no	—
96	C₄⋊C₄	—	254	2	no	—
97	C₄⋊C₄	Dic₂₅₄	1	4	yes	complemented
98	C₄²	—	4	2	no	—
99	C₄²	C₄×D₁₂₇	1	2	yes	—
100	C₄²	D₅₀₈	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
102	Q₈	—	508	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	508	4	no	—
108	C₂×C₄	—	254	4	no	—
109	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
110	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
111	C₂×C₄	#14523236	1	2	yes	—
112	C₂×C₄	#14523238	1	2	yes	—
113	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
114	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
116	C₂×C₄	#14523248	1	2	yes	—
117	C₈	—	508	2	no	—
118	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
119	C₂²	#16861188	1	2	yes	—
120	C₂²	C₂×#14523236	1	1	yes	characteristic
121	C₂²	C₂×#14523238	1	1	yes	characteristic
122	C₂²	#16861234	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	508	4	no	—
126	C₄	#16861132	1	4	yes	—
127	C₄	C₄×Dic₂₅₄	1	2	yes	—
128	C₄	#16861197	1	1	yes	characteristic
129	C₄	#16861209	1	1	yes	characteristic
130	C₂	#22549782	1	1	yes	characteristic
131	C₂	C₂×#16861132	1	1	yes	characteristic
132	C₂	#22549880	1	1	yes	characteristic
133	C₂	#22550166	1	2	yes	—
134	C₂	#22550171	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861209

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic

#22549638

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

#16861209

—

#16861209

C₄

yes

complemented

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

Q₈

yes

complemented

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

Q₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

—

C₂²×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₂₇

C₂×Q₁₆

yes

characteristic

C₂²×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#103

yes

—

C₄×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

#124

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×C₁₂₇

#477

yes

—

#1987

—

127

maximal, Sylow

C₁₂₇

#1987

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

127

—

C₂×C₄⋊C₈

—

127

—

C₄×C₄⋊C₄

D₁₂₇

yes

characteristic

#340

—

127

—

C₂²×Q₈

—

254

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

254

—

#95

—

C₂×Q₈

—

508

—

C₂×Q₈

—

254

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

Dic₂₅₄

yes

complemented

C₄²

—

C₄²

C₄×D₁₂₇

yes

—

100

C₄²

D₅₀₈

yes

—

101

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

102

Q₈

—

508

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

508

—

108

C₂×C₄

—

254

—

109

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

110

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

111

C₂×C₄

#14523236

yes

—

112

C₂×C₄

#14523238

yes

—

113

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

114

C₂×C₄

C₂×D₅₀₈

yes

characteristic

115

C₂×C₄

C₂×Dic₂₅₄

yes

—

116

C₂×C₄

#14523248

yes

—

117

C₈

—

508

—

118

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

119

C₂²

#16861188

yes

—

120

C₂²

C₂×#14523236

yes

characteristic

121

C₂²

C₂×#14523238

yes

characteristic

122

C₂²

#16861234

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

508

—

126

C₄

#16861132

yes

—

127

C₄

C₄×Dic₂₅₄

yes

—

128

C₄

#16861197

yes

characteristic

129

C₄

#16861209

yes

characteristic

130

C₂

#22549782

yes

characteristic

131

C₂

C₂×#16861132

yes

characteristic

132

C₂

#22549880

yes

characteristic

133

C₂

#22550166

yes

—

134

C₂

#22550171

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₄×C₄⋊C₄×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₂₇
hypocenter	C₁₂₇	#1987
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₄×C₄⋊C₄×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₂₇

hypocenter

C₁₂₇

#1987

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #16861209	C₄	semidirect
29. C₄⋊C₄×C₁₂₇	Q₈	semidirect
68. C₁₂₇	#1987	semidirect
97. C₄⋊C₄	Dic₂₅₄	semidirect

H ≅ G/N

type

8. #16861209

C₄

semidirect

29. C₄⋊C₄×C₁₂₇

Q₈

semidirect

68. C₁₂₇

#1987

semidirect

97. C₄⋊C₄

Dic₂₅₄

semidirect

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

subgroup	index
C₂×#16861209	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
#22549638	2
C₄×C₄⋊C₄×C₁₂₇	2
#1987	127

subgroup

index

C₂×#16861209

C₂×C₄⋊(C₁₂₇⋊C₈)

#22549638

C₄×C₄⋊C₄×C₁₂₇

#1987

127

Subgroups — #30639678

Conjugacy classes of subgroups

Complements

Maximal subgroups