Group #30639682 — subgroups

Subgroups — #30639682

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	2	no	—
18	C₂×Dic₂₅₄	—	4	2	no	—
19	C₂²×Dic₁₂₇	—	2	2	no	—
20	C₂×C₁₂₇⋊C₈	—	2	2	no	—
21	C₂×C₁₂₇⋊C₈	—	2	4	no	—
22	C₄⋊Dic₁₂₇	—	2	4	no	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₂₇	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₂₇	—	2	2	no	—
30	C₄⋊C₄×C₁₂₇	Q₈	1	4	yes	complemented
31	C₄²×C₁₂₇	—	4	2	no	—
32	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₄²×C₁₂₇	D₄	1	2	yes	—
34	Dic₂₅₄	—	4	8	no	—
35	C₂×Dic₁₂₇	—	2	4	no	—
36	C₂×Dic₁₂₇	—	4	2	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₁₂₇⋊C₈	—	4	2	no	—
39	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄×C₁₂₇	—	2	4	no	—
42	C₂×C₄×C₁₂₇	—	4	2	no	—
43	C₂×C₄×C₁₂₇	—	4	2	no	—
44	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
47	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₁₂₇	#41	1	2	yes	—
52	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₁₂₇	#137	1	2	yes	—
57	C₄×C₁₂₇	—	2	2	no	—
58	C₄×C₁₂₇	—	4	4	no	—
59	C₄×C₁₂₇	#102	1	4	yes	—
60	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
61	C₄×C₁₂₇	#124	1	1	yes	characteristic
62	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₁₂₇	—	4	4	no	—
64	C₂×C₁₂₇	#391	1	1	yes	characteristic
65	C₂×C₁₂₇	C₂×#102	1	1	yes	characteristic
66	C₂×C₁₂₇	#419	1	1	yes	characteristic
67	C₂×C₁₂₇	#474	1	2	yes	—
68	C₂×C₁₂₇	#475	1	2	yes	—
69	#1988	—	127	1	no	maximal, Sylow
70	C₁₂₇	#1988	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	127	1	no	—
72	C₂×C₄⋊C₈	—	127	1	no	—
73	C₄×C₄⋊C₄	D₁₂₇	1	1	yes	characteristic
74	#340	—	127	4	no	—
75	C₂²×Q₈	—	254	1	no	—
76	C₂²×C₈	—	127	2	no	—
77	C₄⋊Q₈	—	127	4	no	—
78	C₄⋊Q₈	—	254	2	no	—
79	C₂×C₄⋊C₄	—	127	2	no	—
80	C₂×C₄⋊C₄	—	254	1	no	—
81	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
82	C₂×C₄²	—	2	1	no	—
83	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
84	C₄⋊C₈	—	254	2	no	—
85	#95	—	2	1	no	—
86	C₂×Q₈	—	508	2	no	—
87	C₂×Q₈	—	508	2	no	—
88	C₂×Q₈	—	254	4	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	254	2	no	—
91	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
94	C₂×C₈	—	254	2	no	—
95	C₂×C₈	—	254	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	127	4	no	—
98	C₄⋊C₄	—	254	4	no	—
99	C₄⋊C₄	—	254	2	no	—
100	C₄⋊C₄	Dic₂₅₄	1	4	yes	complemented
101	C₄²	—	4	2	no	—
102	C₄²	C₄×D₁₂₇	1	2	yes	—
103	C₄²	D₅₀₈	1	2	yes	—
104	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
105	Q₈	—	508	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	508	2	no	—
111	C₂×C₄	—	508	2	no	—
112	C₂×C₄	—	254	4	no	—
113	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
114	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
115	C₂×C₄	#14523233	1	2	yes	—
116	C₂×C₄	#14523235	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
118	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
120	C₂×C₄	#14523248	1	2	yes	—
121	C₈	—	508	2	no	—
122	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
123	C₂²	#16861188	1	2	yes	—
124	C₂²	C₂×#14523233	1	1	yes	characteristic
125	C₂²	C₂×#14523235	1	1	yes	characteristic
126	C₂²	#16861239	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	4	4	no	—
129	C₄	—	508	4	no	—
130	C₄	#16861129	1	4	yes	—
131	C₄	C₄×Dic₂₅₄	1	2	yes	—
132	C₄	#16861197	1	1	yes	characteristic
133	C₄	#16861209	1	1	yes	characteristic
134	C₂	#22549782	1	1	yes	characteristic
135	C₂	C₂×#16861129	1	1	yes	characteristic
136	C₂	#22549884	1	1	yes	characteristic
137	C₂	#22550156	1	2	yes	—
138	C₂	#22550161	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

—

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

—

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

D₈

yes

—

C₂×C₄×C₁₂₇

SD₁₆

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×D₈

yes

characteristic

C₂²×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#102

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

#475

yes

—

#1988

—

127

maximal, Sylow

C₁₂₇

#1988

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

—

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

—

100

C₄⋊C₄

Dic₂₅₄

yes

complemented

101

C₄²

—

102

C₄²

C₄×D₁₂₇

yes

—

103

C₄²

D₅₀₈

yes

—

104

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

105

Q₈

—

508

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

508

—

111

C₂×C₄

—

508

—

112

C₂×C₄

—

254

—

113

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

114

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

115

C₂×C₄

#14523233

yes

—

116

C₂×C₄

#14523235

yes

—

117

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

118

C₂×C₄

C₂×D₅₀₈

yes

characteristic

119

C₂×C₄

C₂×Dic₂₅₄

yes

—

120

C₂×C₄

#14523248

yes

—

121

C₈

—

508

—

122

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

123

C₂²

#16861188

yes

—

124

C₂²

C₂×#14523233

yes

characteristic

125

C₂²

C₂×#14523235

yes

characteristic

126

C₂²

#16861239

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

508

—

130

C₄

#16861129

yes

—

131

C₄

C₄×Dic₂₅₄

yes

—

132

C₄

#16861197

yes

characteristic

133

C₄

#16861209

yes

characteristic

134

C₂

#22549782

yes

characteristic

135

C₂

C₂×#16861129

yes

characteristic

136

C₂

#22549884

yes

characteristic

137

C₂

#22550156

yes

—

138

C₂

#22550161

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	#1988
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

#1988

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
30. C₄⋊C₄×C₁₂₇	Q₈	semidirect
70. C₁₂₇	#1988	semidirect
100. C₄⋊C₄	Dic₂₅₄	semidirect

H ≅ G/N

type

8. #16861209

C₄

semidirect

30. C₄⋊C₄×C₁₂₇

Q₈

semidirect

70. C₁₂₇

#1988

semidirect

100. 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
#1988	127

subgroup

index

C₂×#16861209

C₂×C₄⋊(C₁₂₇⋊C₈)

#22549638

C₄×C₄⋊C₄×C₁₂₇

#1988

127

Subgroups — #30639682

Conjugacy classes of subgroups

Complements

Maximal subgroups