Group #44486091 — subgroups

Subgroups — #44486091

Order 30592.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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×#21439891	C₂	1	2	yes	maximal
2	C₂×#21439884	C₂	1	2	yes	maximal
3	#29504642	C₂	1	1	yes	maximal, characteristic
4	#29504527	C₂	1	1	yes	maximal, characteristic
5	#391×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₃₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₃₉	—	2	2	no	—
8	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
9	#21439891	—	2	4	no	—
10	#21439884	—	2	4	no	—
11	C₂²×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
14	#95×C₂₃₉	—	2	2	no	—
15	C₂²×Dic₂₃₉	—	2	2	no	—
16	C₂×C₂₃₉⋊C₈	—	2	2	no	—
17	C₂×C₂₃₉⋊C₈	—	2	2	no	—
18	C₂×C₂₃₉⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₂₃₉	—	2	2	no	—
20	C₄⋊Dic₂₃₉	—	2	2	no	—
21	C₄⋊Dic₂₃₉	—	4	4	no	—
22	C₂×Q₈×C₂₃₉	—	2	1	no	—
23	C₂×Q₈×C₂₃₉	—	2	2	no	—
24	C₂×Q₈×C₂₃₉	—	2	2	no	—
25	C₂×Q₈×C₂₃₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₂₃₉	—	2	1	no	—
27	C₂²×C₄×C₂₃₉	—	2	1	no	—
28	C₂²×C₄×C₂₃₉	D₄	1	2	yes	—
29	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₂₃₉	—	2	2	no	—
31	C₄²×C₂₃₉	—	2	2	no	—
32	C₂×Dic₂₃₉	—	4	2	no	—
33	C₂×Dic₂₃₉	—	4	4	no	—
34	C₂₃₉⋊C₈	—	2	4	no	—
35	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₂₃₉	—	2	4	no	—
37	Q₈×C₂₃₉	—	4	2	no	—
38	C₂×C₄×C₂₃₉	—	2	2	no	—
39	C₂×C₄×C₂₃₉	—	2	2	no	—
40	C₂×C₄×C₂₃₉	—	2	2	no	—
41	C₂×C₄×C₂₃₉	—	4	1	no	—
42	C₂×C₄×C₂₃₉	—	4	1	no	—
43	C₂×C₄×C₂₃₉	—	4	1	no	—
44	C₂×C₄×C₂₃₉	—	4	2	no	—
45	C₂×C₄×C₂₃₉	SD₁₆	1	4	yes	—
46	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₃₉	#41	1	2	yes	—
49	C₂×C₄×C₂₃₉	#41	1	1	yes	characteristic
50	C₂²×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₂₃₉	#123	1	1	yes	characteristic
52	C₂²×C₂₃₉	#127	1	1	yes	characteristic
53	C₂²×C₂₃₉	C₂×SD₁₆	1	2	yes	—
54	C₂²×C₂₃₉	#137	1	2	yes	—
55	C₄×C₂₃₉	—	2	4	no	—
56	C₄×C₂₃₉	—	4	2	no	—
57	C₄×C₂₃₉	—	8	1	no	—
58	C₄×C₂₃₉	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₂₃₉	#123	1	2	yes	—
60	Dic₂₃₉	—	8	2	no	—
61	C₂×C₂₃₉	#399	1	1	yes	characteristic
62	C₂×C₂₃₉	#475	1	2	yes	—
63	C₂×C₂₃₉	#483	1	2	yes	—
64	C₂×C₂₃₉	#492	1	1	yes	characteristic
65	C₂×C₂₃₉	#497	1	1	yes	characteristic
66	C₂₃₉	#2096	1	1	yes	characteristic, complemented, Sylow
67	#2096	—	239	1	no	maximal, Sylow
68	C₂×#106	—	239	2	no	—
69	C₂×#103	—	239	2	no	—
70	#391	D₂₃₉	1	1	yes	characteristic
71	#384	—	239	1	no	—
72	#336	—	239	1	no	—
73	C₂²×Q₈	C₂×D₂₃₉	1	1	yes	characteristic
74	C₂²×C₈	—	239	2	no	—
75	C₂×C₄⋊C₄	—	478	2	no	—
76	C₂×C₄⋊C₄	—	239	2	no	—
77	C₂×C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
79	#106	—	478	4	no	—
80	#103	—	478	4	no	—
81	#95	—	2	2	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	Dic₄₇₈	1	2	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	478	2	no	—
89	C₂²×C₄	C₂²⋊D₂₃₉	1	2	yes	—
90	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
91	C₂×C₈	—	478	2	no	—
92	C₂×C₈	—	478	2	no	—
93	C₂×C₈	—	239	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	956	4	no	—
96	C₄⋊C₄	—	478	2	no	—
97	C₄⋊C₄	—	478	2	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	956	2	no	—
110	C₂×C₄	—	956	4	no	—
111	C₂×C₄	#16838535	1	4	yes	—
112	C₂×C₄	D₄×D₂₃₉	1	2	yes	—
113	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
114	C₂×C₄	#16838547	1	2	yes	—
115	C₂×C₄	#16838548	1	1	yes	characteristic
116	C₈	—	478	4	no	—
117	C₂²	#21439939	1	1	yes	characteristic
118	C₂²	#21439941	1	1	yes	characteristic
119	C₂²	#21439956	1	1	yes	characteristic
120	C₂²	C₂×#16838535	1	2	yes	—
121	C₂²	#21439990	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	1	no	—
125	C₄	—	1912	2	no	—
126	C₄	#21439937	1	2	yes	—
127	C₄	#21439942	1	2	yes	—
128	C₂	#29504708	1	1	yes	characteristic
129	C₂	#29505057	1	2	yes	—
130	C₂	#29505100	1	2	yes	—
131	C₂	#29505146	1	1	yes	characteristic
132	C₂	#29505167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439891

C₂

yes

maximal

C₂×#21439884

C₂

yes

maximal

#29504642

C₂

yes

maximal, characteristic

#29504527

C₂

yes

maximal, characteristic

#391×C₂₃₉

C₂

yes

maximal, characteristic

C₂²×C₂₃₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₃₉

—

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

#21439891

—

#21439884

—

C₂²×Q₈×C₂₃₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃₉

C₂²

yes

characteristic

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

#95×C₂₃₉

—

C₂²×Dic₂₃₉

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

D₄

yes

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

Q₈

yes

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

D₄

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

C₂×D₄

yes

characteristic

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

SD₁₆

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

#41

yes

—

C₂×C₄×C₂₃₉

#41

yes

characteristic

C₂²×C₂₃₉

C₂²⋊Q₈

yes

C₂²×C₂₃₉

yes

C₂²×C₂₃₉

yes

C₂²×C₂₃₉

C₂×SD₁₆

yes

—

C₂²×C₂₃₉

#137

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂²⋊Q₈

yes

—

C₄×C₂₃₉

#123

yes

—

Dic₂₃₉

—

C₂×C₂₃₉

#399

yes

characteristic

C₂×C₂₃₉

#475

yes

—

C₂×C₂₃₉

#483

yes

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂₃₉

yes

characteristic, complemented, Sylow

#2096

—

239

maximal, Sylow

C₂×#106

—

239

—

C₂×#103

—

239

—

#391

D₂₃₉

yes

characteristic

#384

—

239

—

#336

—

239

—

C₂²×Q₈

C₂×D₂₃₉

yes

characteristic

C₂²×C₈

—

239

—

C₂×C₄⋊C₄

—

478

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄⋊C₄

C₂×D₂₃₉

yes

characteristic

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

#106

—

478

—

#103

—

478

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₄₇₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

478

—

C₂²×C₄

C₂²⋊D₂₃₉

yes

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

—

239

—

C₄⋊C₄

—

C₄⋊C₄

—

956

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄²

—

C₂³

C₂×C₂²⋊D₂₃₉

yes

characteristic

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

956

—

110

C₂×C₄

—

956

—

111

C₂×C₄

#16838535

yes

—

112

C₂×C₄

D₄×D₂₃₉

yes

—

113

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

114

C₂×C₄

#16838547

yes

—

115

C₂×C₄

#16838548

yes

characteristic

116

C₈

—

478

—

117

C₂²

#21439939

yes

characteristic

118

C₂²

#21439941

yes

characteristic

119

C₂²

#21439956

yes

characteristic

120

C₂²

C₂×#16838535

yes

—

121

C₂²

#21439990

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

1912

—

126

C₄

#21439937

yes

—

127

C₄

#21439942

yes

—

128

C₂

#29504708

yes

characteristic

129

C₂

#29505057

yes

—

130

C₂

#29505100

yes

—

131

C₂

#29505146

yes

characteristic

132

C₂

#29505167

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃₉
commutator, cosocle	C₂²×C₄×C₂₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	#391×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	C₂×D₄
hypercenter	#391	D₂₃₉
hypocenter	C₂₃₉	#2096
LCS	G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃₉

commutator, cosocle

C₂²×C₄×C₂₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

#391×C₂₃₉

C₂

socle, prime radical

C₂³×C₂₃₉

C₂×D₄

D₂₃₉

C₂₃₉

G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
66. C₂₃₉	#2096	semidirect

H ≅ G/N

type

66. C₂₃₉

#2096

semidirect

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

subgroup	index
C₂×#21439891	2
C₂×#21439884	2
#29504642	2
#29504527	2
#391×C₂₃₉	2
#2096	239

subgroup

index

#391×C₂₃₉

239

Subgroups — #44486091

Conjugacy classes of subgroups

Complements

Maximal subgroups