Group #27171259 — subgroups

Subgroups — #27171259

Order 12928.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×#11717579	C₂	1	1	yes	maximal, characteristic
2	#20265047	C₂	1	1	yes	maximal, characteristic
3	#20264939	C₂	1	4	yes	maximal
4	#384×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717579	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
7	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
8	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
9	C₄×#11717579	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₀₁	—	2	2	no	—
15	C₂×#11717579	—	2	2	no	—
16	C₂×#11717579	—	2	2	no	—
17	C₂×#11717579	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₀₁	—	2	4	no	—
19	C₄⋊Dic₁₀₁	—	4	2	no	—
20	C₄⋊Dic₁₀₁	—	4	2	no	—
21	C₂²×C₄×C₁₀₁	—	2	2	no	—
22	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₀₁	—	2	4	no	—
26	C₄⋊C₄×C₁₀₁	—	4	2	no	—
27	C₄⋊C₄×C₁₀₁	—	4	2	no	—
28	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
29	C₄²×C₁₀₁	Q₈	1	2	yes	—
30	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₁	—	4	2	no	—
32	C₂×C₄×C₁₀₁	—	4	2	no	—
33	C₂×C₄×C₁₀₁	—	4	2	no	—
34	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₁	D₈	1	2	yes	—
37	C₂×C₄×C₁₀₁	SD₁₆	1	2	yes	—
38	C₂×C₄×C₁₀₁	SD₁₆	1	2	yes	—
39	C₂×C₄×C₁₀₁	Q₁₆	1	2	yes	—
40	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₀₁	#41	1	2	yes	—
43	C₂×C₄×C₁₀₁	#41	1	2	yes	—
44	C₂×Dic₁₀₁	—	4	2	no	—
45	C₂×Dic₁₀₁	—	4	2	no	—
46	C₂×Dic₁₀₁	—	4	2	no	—
47	#11717579	—	2	4	no	—
48	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₀₁	#124	1	1	yes	characteristic
50	C₂²×C₁₀₁	#124	1	1	yes	characteristic
51	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₁₀₁	—	8	2	no	—
56	C₄×C₁₀₁	#106	1	4	yes	—
57	C₄×C₁₀₁	#107	1	4	yes	—
58	C₄×C₁₀₁	#117	1	2	yes	—
59	C₄×C₁₀₁	#125	1	2	yes	—
60	Dic₁₀₁	—	8	2	no	—
61	C₂×C₁₀₁	#382	1	1	yes	characteristic
62	C₂×C₁₀₁	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₀₁	C₂×#107	1	1	yes	characteristic
64	C₂×C₁₀₁	#486	1	1	yes	characteristic
65	C₂×C₁₀₁	#486	1	1	yes	characteristic
66	C₂×C₁₀₁	#487	1	1	yes	characteristic
67	C₂×C₁₀₁	#487	1	1	yes	characteristic
68	#1834	—	101	1	no	maximal, Sylow
69	C₁₀₁	#1834	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄×C₈	—	101	1	no	—
71	#384	—	101	1	no	—
72	#384	D₁₀₁	1	1	yes	characteristic
73	#340	—	101	4	no	—
74	C₂²×C₈	—	101	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	101	2	no	—
78	C₂×C₄⋊C₄	—	202	1	no	—
79	C₂×C₄⋊C₄	—	202	1	no	—
80	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
81	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
82	C₄×C₈	—	101	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	202	2	no	—
85	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
88	C₂×C₈	—	101	4	no	—
89	C₂×C₈	—	202	2	no	—
90	C₂×C₈	—	202	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	404	2	no	—
95	C₄⋊C₄	—	404	2	no	—
96	C₄⋊C₄	—	202	4	no	—
97	C₄²	Dic₂₀₂	1	2	yes	—
98	C₄²	C₄×D₁₀₁	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	404	2	no	—
104	C₂×C₄	—	404	2	no	—
105	C₂×C₄	—	404	2	no	—
106	C₂×C₄	#13108053	1	2	yes	—
107	C₂×C₄	#13108053	1	2	yes	—
108	C₂×C₄	#13108059	1	2	yes	—
109	C₂×C₄	#13108061	1	2	yes	—
110	C₂×C₄	#13108062	1	2	yes	—
111	C₂×C₄	#13108064	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
114	C₂×C₄	#13108073	1	2	yes	—
115	C₂×C₄	#13108075	1	2	yes	—
116	C₈	—	202	4	no	—
117	C₂²	C₂×#13108053	1	1	yes	characteristic
118	C₂²	#15692242	1	1	yes	characteristic
119	C₂²	#15692244	1	1	yes	characteristic
120	C₂²	C₂×#13108059	1	1	yes	characteristic
121	C₂²	C₂×#13108061	1	1	yes	characteristic
122	C₂²	C₂×#13108062	1	1	yes	characteristic
123	C₂²	C₂×#13108064	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	808	2	no	—
126	C₄	#15692186	1	4	yes	—
127	C₄	#15692188	1	4	yes	—
128	C₄	#15692213	1	2	yes	—
129	C₄	#15692246	1	2	yes	—
130	C₂	#20265042	1	1	yes	characteristic
131	C₂	C₂×#15692186	1	1	yes	characteristic
132	C₂	C₂×#15692188	1	1	yes	characteristic
133	C₂	#20265523	1	1	yes	characteristic
134	C₂	#20265525	1	1	yes	characteristic
135	C₂	#20265527	1	1	yes	characteristic
136	C₂	#20265529	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11717579

C₂

yes

maximal, characteristic

#20265047

C₂

yes

maximal, characteristic

#20264939

C₂

yes

maximal

#384×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₄×#11717579

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

C₂×C₄

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

Q₈

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

D₈

yes

—

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

Q₁₆

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

#11717579

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

C₂×D₈

yes

characteristic

C₂²×C₁₀₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₁

C₂×Q₁₆

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

#106

yes

—

C₄×C₁₀₁

#107

yes

—

C₄×C₁₀₁

#117

yes

—

C₄×C₁₀₁

#125

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1834

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

101

—

#384

—

101

—

#384

D₁₀₁

yes

characteristic

#340

—

101

—

C₂²×C₈

—

101

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄×C₈

—

101

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

202

—

C₄²

Dic₂₀₂

yes

—

C₄²

C₄×D₁₀₁

yes

—

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

404

—

104

C₂×C₄

—

404

—

105

C₂×C₄

—

404

—

106

C₂×C₄

#13108053

yes

—

107

C₂×C₄

#13108053

yes

—

108

C₂×C₄

#13108059

yes

—

109

C₂×C₄

#13108061

yes

—

110

C₂×C₄

#13108062

yes

—

111

C₂×C₄

#13108064

yes

—

112

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

113

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

114

C₂×C₄

#13108073

yes

—

115

C₂×C₄

#13108075

yes

—

116

C₈

—

202

—

117

C₂²

C₂×#13108053

yes

characteristic

118

C₂²

#15692242

yes

characteristic

119

C₂²

#15692244

yes

characteristic

120

C₂²

C₂×#13108059

yes

characteristic

121

C₂²

C₂×#13108061

yes

characteristic

122

C₂²

C₂×#13108062

yes

characteristic

123

C₂²

C₂×#13108064

yes

characteristic

124

C₄

—

125

C₄

—

808

—

126

C₄

#15692186

yes

—

127

C₄

#15692188

yes

—

128

C₄

#15692213

yes

—

129

C₄

#15692246

yes

—

130

C₂

#20265042

yes

characteristic

131

C₂

C₂×#15692186

yes

characteristic

132

C₂

C₂×#15692188

yes

characteristic

133

C₂

#20265523

yes

characteristic

134

C₂

#20265525

yes

characteristic

135

C₂

#20265527

yes

characteristic

136

C₂

#20265529

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#384×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#384	D₁₀₁
hypocenter	C₁₀₁	#1834
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#384×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#11717579	C₄	semidirect
69. C₁₀₁	#1834	semidirect

H ≅ G/N

type

9. C₄×#11717579

C₄

semidirect

69. C₁₀₁

#1834

semidirect

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

subgroup	index
C₂×C₄×#11717579	2
#20265047	2
#20264939	2
#384×C₁₀₁	2
#1834	101

subgroup

index

#384×C₁₀₁

101

Subgroups — #27171259

Conjugacy classes of subgroups

Complements

Maximal subgroups