Group #30639681 — subgroups

Subgroups — #30639681

Order 16256.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₄×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549639	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	—	2	1	no	—
6	#16861209	—	2	2	no	—
7	#16861209	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₂₇	—	2	1	no	—
12	C₁₂₇⋊#95	—	2	1	no	—
13	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₂₇	—	2	2	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	C₂²×Dic₁₂₇	—	2	2	no	—
21	C₄⋊Dic₁₂₇	—	2	4	no	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₂₇	Q₈	1	4	yes	complemented
26	C₄×Dic₁₂₇	—	4	2	no	—
27	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₂₇	—	2	2	no	—
31	C₂×C₈×C₁₂₇	—	2	4	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	—	2	4	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₂×Dic₁₂₇	—	4	2	no	—
39	C₂×Dic₁₂₇	—	4	2	no	—
40	C₂×Dic₁₂₇	—	4	2	no	—
41	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	—	2	2	no	—
43	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
46	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₂₇	#41	1	2	yes	—
51	C₈×C₁₂₇	—	4	2	no	—
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	#102	1	4	yes	—
59	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
60	C₄×C₁₂₇	#124	1	1	yes	characteristic
61	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₁₂₇	—	4	4	no	—
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	D₁₂₇	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	127	1	no	—
74	#340	—	127	4	no	—
75	C₂²×Q₈	—	254	1	no	—
76	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
77	C₄⋊Q₈	—	127	4	no	—
78	C₄⋊Q₈	—	254	2	no	—
79	C₂×C₄⋊C₄	—	127	2	no	—
80	C₂×C₄⋊C₄	—	127	2	no	—
81	C₂×C₄⋊C₄	—	254	1	no	—
82	C₂×C₄²	—	254	1	no	—
83	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
84	C₄⋊C₈	—	2	2	no	—
85	#95	—	254	1	no	—
86	C₂×Q₈	—	508	2	no	—
87	C₂×Q₈	—	508	2	no	—
88	C₂×Q₈	—	254	4	no	—
89	C₂²×C₄	—	254	2	no	—
90	C₂²×C₄	—	254	2	no	—
91	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
92	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₄⋊C₄	—	127	4	no	—
97	C₄⋊C₄	—	127	4	no	—
98	C₄⋊C₄	—	254	4	no	—
99	C₄⋊C₄	—	254	2	no	—
100	C₄⋊C₄	—	254	2	no	—
101	C₄²	—	508	2	no	—
102	C₄²	C₄×D₁₂₇	1	2	yes	—
103	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
104	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
105	Q₈	—	508	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	508	2	no	—
108	C₂×C₄	—	508	2	no	—
109	C₂×C₄	—	508	2	no	—
110	C₂×C₄	—	508	2	no	—
111	C₂×C₄	—	254	4	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	D₁₀₁₆	1	2	yes	—
116	C₂×C₄	#14523237	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
120	C₂×C₄	#14523247	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
123	C₂²	#16861189	1	2	yes	—
124	C₂²	C₂×D₁₀₁₆	1	1	yes	characteristic
125	C₂²	C₂×#14523237	1	1	yes	characteristic
126	C₂²	#16861242	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	508	4	no	—
129	C₄	—	508	4	no	—
130	C₄	#16861131	1	4	yes	—
131	C₄	#16861174	1	2	yes	—
132	C₄	#16861196	1	1	yes	characteristic
133	C₄	#16861207	1	1	yes	characteristic
134	C₂	#22549781	1	1	yes	characteristic
135	C₂	C₂×#16861131	1	1	yes	characteristic
136	C₂	#22549886	1	1	yes	characteristic
137	C₂	#22550157	1	2	yes	—
138	C₂	#22550162	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861209

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic

#22549639

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

—

#16861209

—

#16861209

C₄

yes

complemented

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄×Dic₁₂₇

—

C₁₂₇⋊#95

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

Q₈

yes

complemented

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

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

—

C₈×C₁₂₇

—

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

#102

yes

—

C₄×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

#124

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

Dic₁₂₇

—

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

D₁₂₇

yes

characteristic

C₄×C₄⋊C₄

—

127

—

#340

—

127

—

C₂²×Q₈

—

254

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

#95

—

254

—

C₂×Q₈

—

508

—

C₂×Q₈

—

508

—

C₂×Q₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

100

C₄⋊C₄

—

254

—

101

C₄²

—

508

—

102

C₄²

C₄×D₁₂₇

yes

—

103

C₄²

C₂²⋊D₁₂₇

yes

—

104

C₂³

C₂×D₅₀₈

yes

characteristic

105

Q₈

—

508

—

106

C₂×C₄

—

107

C₂×C₄

—

508

—

108

C₂×C₄

—

508

—

109

C₂×C₄

—

508

—

110

C₂×C₄

—

508

—

111

C₂×C₄

—

254

—

112

C₂×C₄

—

254

—

113

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

114

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

115

C₂×C₄

D₁₀₁₆

yes

—

116

C₂×C₄

#14523237

yes

—

117

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

119

C₂×C₄

Q₈×D₁₂₇

yes

—

120

C₂×C₄

#14523247

yes

—

121

C₈

—

122

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

123

C₂²

#16861189

yes

—

124

C₂²

C₂×D₁₀₁₆

yes

characteristic

125

C₂²

C₂×#14523237

yes

characteristic

126

C₂²

#16861242

yes

—

127

C₄

—

128

C₄

—

508

—

129

C₄

—

508

—

130

C₄

#16861131

yes

—

131

C₄

#16861174

yes

—

132

C₄

#16861196

yes

characteristic

133

C₄

#16861207

yes

characteristic

134

C₂

#22549781

yes

characteristic

135

C₂

C₂×#16861131

yes

characteristic

136

C₂

#22549886

yes

characteristic

137

C₂

#22550157

yes

—

138

C₂

#22550162

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×C₄⋊C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#1988
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₅₀₈

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×C₄⋊C₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₂₇

hypocenter

C₁₂₇

#1988

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #16861209	C₄	semidirect
25. C₄⋊Dic₁₂₇	Q₈	semidirect
70. C₁₂₇	#1988	semidirect

H ≅ G/N

type

7. #16861209

C₄

semidirect

25. C₄⋊Dic₁₂₇

Q₈

semidirect

70. C₁₂₇

#1988

semidirect

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

subgroup	index
C₂×#16861209	2
C₄×C₄⋊Dic₁₂₇	2
#22549639	2
C₂×C₄⋊C₈×C₁₂₇	2
#1988	127

subgroup

index

C₂×#16861209

C₄×C₄⋊Dic₁₂₇

#22549639

C₂×C₄⋊C₈×C₁₂₇

#1988

127

Subgroups — #30639681

Conjugacy classes of subgroups

Complements

Maximal subgroups