Group #44485500 — subgroups

Subgroups — #44485500

Order 30592.

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×#21439960	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#29504534	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₇₈	—	2	1	no	—
6	C₂²×C₂₃₉⋊C₈	C₂²	1	2	yes	—
7	#21439960	—	2	2	no	—
8	#21439960	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₃₉	—	2	1	no	—
10	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
11	C₄⋊(C₂₃₉⋊C₈)	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₃₉	—	2	1	no	—
14	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
15	#95×C₂₃₉	—	2	1	no	—
16	C₂×Dic₄₇₈	—	2	4	no	—
17	C₂×Dic₄₇₈	—	4	2	no	—
18	C₂×Dic₄₇₈	—	4	2	no	—
19	C₂²×Dic₂₃₉	—	2	2	no	—
20	C₂×C₂₃₉⋊C₈	—	2	2	no	—
21	C₂×C₂₃₉⋊C₈	—	2	4	no	—
22	C₄⋊Dic₂₃₉	—	2	4	no	—
23	C₄⋊Dic₂₃₉	—	2	2	no	—
24	C₄⋊Dic₂₃₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₂₃₉	—	2	2	no	—
26	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₃₉	—	2	2	no	—
30	C₄⋊C₄×C₂₃₉	Q₈	1	4	yes	complemented
31	C₄²×C₂₃₉	—	4	2	no	—
32	C₄²×C₂₃₉	C₂×C₄	1	2	yes	—
33	C₄²×C₂₃₉	D₄	1	2	yes	—
34	Dic₄₇₈	—	4	8	no	—
35	C₂×Dic₂₃₉	—	2	4	no	—
36	C₂×Dic₂₃₉	—	4	2	no	—
37	C₂×Dic₂₃₉	—	4	2	no	—
38	C₂₃₉⋊C₈	—	4	2	no	—
39	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃₉	—	2	2	no	—
41	C₂×C₄×C₂₃₉	—	2	4	no	—
42	C₂×C₄×C₂₃₉	—	4	2	no	—
43	C₂×C₄×C₂₃₉	—	4	2	no	—
44	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₃₉	D₈	1	2	yes	—
47	C₂×C₄×C₂₃₉	SD₁₆	1	2	yes	—
48	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₂₃₉	#41	1	2	yes	—
52	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₂₃₉	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₂₃₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₂₃₉	#137	1	2	yes	—
57	C₄×C₂₃₉	—	2	2	no	—
58	C₄×C₂₃₉	—	4	4	no	—
59	C₄×C₂₃₉	#102	1	4	yes	—
60	C₄×C₂₃₉	C₄×Q₈	1	2	yes	—
61	C₄×C₂₃₉	#124	1	1	yes	characteristic
62	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₂₃₉	—	4	4	no	—
64	C₂×C₂₃₉	#391	1	1	yes	characteristic
65	C₂×C₂₃₉	C₂×#102	1	1	yes	characteristic
66	C₂×C₂₃₉	#419	1	1	yes	characteristic
67	C₂×C₂₃₉	#474	1	2	yes	—
68	C₂×C₂₃₉	#475	1	2	yes	—
69	C₂₃₉	#1988	1	1	yes	characteristic, complemented, Sylow
70	#1988	—	239	1	no	maximal, Sylow
71	C₂×C₄⋊Q₈	—	239	1	no	—
72	C₂×C₄⋊C₈	—	239	1	no	—
73	C₄×C₄⋊C₄	D₂₃₉	1	1	yes	characteristic
74	#340	—	239	4	no	—
75	C₂²×Q₈	—	478	1	no	—
76	C₂²×C₈	—	239	2	no	—
77	C₄⋊Q₈	—	478	2	no	—
78	C₄⋊Q₈	—	239	4	no	—
79	C₂×C₄⋊C₄	—	478	1	no	—
80	C₂×C₄⋊C₄	—	239	2	no	—
81	C₂×C₄⋊C₄	C₂×D₂₃₉	1	2	yes	—
82	C₂×C₄²	—	2	1	no	—
83	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
84	C₄⋊C₈	—	478	2	no	—
85	#95	—	2	1	no	—
86	C₂×Q₈	—	956	2	no	—
87	C₂×Q₈	—	956	2	no	—
88	C₂×Q₈	—	478	4	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	478	2	no	—
91	C₂²×C₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
94	C₂×C₈	—	478	2	no	—
95	C₂×C₈	—	478	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	478	4	no	—
98	C₄⋊C₄	—	478	2	no	—
99	C₄⋊C₄	—	239	4	no	—
100	C₄⋊C₄	Dic₄₇₈	1	4	yes	complemented
101	C₄²	—	4	2	no	—
102	C₄²	C₄×D₂₃₉	1	2	yes	—
103	C₄²	D₉₅₆	1	2	yes	—
104	C₂³	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
105	Q₈	—	956	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	—	956	2	no	—
111	C₂×C₄	—	956	2	no	—
112	C₂×C₄	—	478	4	no	—
113	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	2	yes	—
114	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	2	yes	—
115	C₂×C₄	#16838533	1	2	yes	—
116	C₂×C₄	#16838535	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
118	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₄₇₈	1	2	yes	—
120	C₂×C₄	#16838548	1	2	yes	—
121	C₈	—	956	2	no	—
122	C₂²	C₂×(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
123	C₂²	#21439939	1	2	yes	—
124	C₂²	C₂×#16838533	1	1	yes	characteristic
125	C₂²	C₂×#16838535	1	1	yes	characteristic
126	C₂²	#21439990	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	4	4	no	—
129	C₄	—	956	4	no	—
130	C₄	#21439880	1	4	yes	—
131	C₄	C₄×Dic₄₇₈	1	2	yes	—
132	C₄	#21439948	1	1	yes	characteristic
133	C₄	#21439960	1	1	yes	characteristic
134	C₂	#29504678	1	1	yes	characteristic
135	C₂	C₂×#21439880	1	1	yes	characteristic
136	C₂	#29504780	1	1	yes	characteristic
137	C₂	#29505052	1	2	yes	—
138	C₂	#29505057	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439960

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₃₉⋊C₈)

C₂

yes

maximal, characteristic

#29504534

C₂

yes

maximal

C₄×C₄⋊C₄×C₂₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₄₇₈

—

C₂²×C₂₃₉⋊C₈

C₂²

yes

—

#21439960

—

#21439960

C₄

yes

complemented

C₂×C₄⋊Dic₂₃₉

—

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

C₄⋊(C₂₃₉⋊C₈)

—

C₂×C₄⋊C₄×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

#95×C₂₃₉

—

C₂×Dic₄₇₈

—

C₂×Dic₄₇₈

—

C₂×Dic₄₇₈

—

C₂²×Dic₂₃₉

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

D₄

yes

characteristic

C₂²×C₄×C₂₃₉

D₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

Q₈

yes

complemented

C₄²×C₂₃₉

—

C₄²×C₂₃₉

C₂×C₄

yes

—

C₄²×C₂₃₉

D₄

yes

—

Dic₄₇₈

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

D₈

yes

—

C₂×C₄×C₂₃₉

SD₁₆

yes

—

C₂×C₄×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

—

C₂×C₄×C₂₃₉

#41

yes

—

C₂²×C₂₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃₉

C₂²⋊Q₈

yes

—

C₂²×C₂₃₉

C₂×D₈

yes

characteristic

C₂²×C₂₃₉

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃₉

#137

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#102

yes

—

C₄×C₂₃₉

C₄×Q₈

yes

—

C₄×C₂₃₉

#124

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

Dic₂₃₉

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

—

C₂×C₂₃₉

#475

yes

—

C₂₃₉

#1988

yes

characteristic, complemented, Sylow

#1988

—

239

maximal, Sylow

C₂×C₄⋊Q₈

—

239

—

C₂×C₄⋊C₈

—

239

—

C₄×C₄⋊C₄

D₂₃₉

yes

characteristic

#340

—

239

—

C₂²×Q₈

—

478

—

C₂²×C₈

—

239

—

C₄⋊Q₈

—

478

—

C₄⋊Q₈

—

239

—

C₂×C₄⋊C₄

—

478

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄⋊C₄

C₂×D₂₃₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

C₄⋊C₈

—

478

—

#95

—

C₂×Q₈

—

956

—

C₂×Q₈

—

956

—

C₂×Q₈

—

478

—

C₂²×C₄

—

C₂²×C₄

—

478

—

C₂²×C₄

C₂²⋊D₂₃₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃₉

yes

characteristic

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₄⋊C₄

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

239

—

100

C₄⋊C₄

Dic₄₇₈

yes

complemented

101

C₄²

—

102

C₄²

C₄×D₂₃₉

yes

—

103

C₄²

D₉₅₆

yes

—

104

C₂³

C₂×C₂²⋊D₂₃₉

yes

characteristic

105

Q₈

—

956

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

956

—

111

C₂×C₄

—

956

—

112

C₂×C₄

—

478

—

113

C₂×C₄

(C₂×D₂₃₉)⋊C₄

yes

—

114

C₂×C₄

(C₂×D₂₃₉)⋊C₄

yes

—

115

C₂×C₄

#16838533

yes

—

116

C₂×C₄

#16838535

yes

—

117

C₂×C₄

C₂×C₄×D₂₃₉

yes

characteristic

118

C₂×C₄

C₂×D₉₅₆

yes

characteristic

119

C₂×C₄

C₂×Dic₄₇₈

yes

—

120

C₂×C₄

#16838548

yes

—

121

C₈

—

956

—

122

C₂²

C₂×(C₂×D₂₃₉)⋊C₄

yes

characteristic

123

C₂²

#21439939

yes

—

124

C₂²

C₂×#16838533

yes

characteristic

125

C₂²

C₂×#16838535

yes

characteristic

126

C₂²

#21439990

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

956

—

130

C₄

#21439880

yes

—

131

C₄

C₄×Dic₄₇₈

yes

—

132

C₄

#21439948

yes

characteristic

133

C₄

#21439960

yes

characteristic

134

C₂

#29504678

yes

characteristic

135

C₂

C₂×#21439880

yes

characteristic

136

C₂

#29504780

yes

characteristic

137

C₂

#29505052

yes

—

138

C₂

#29505057

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃₉
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₄×C₄⋊C₄×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	C₂×D₄
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₃₉
hypocenter	C₂₃₉	#1988
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃₉

commutator

C₂×C₄×C₂₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

C₄×C₄⋊C₄×C₂₃₉

C₂

socle, prime radical

C₂³×C₂₃₉

C₂×D₄

cosocle

C₂²×C₄×C₂₃₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₂₃₉

hypocenter

C₂₃₉

#1988

LCS

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #21439960	C₄	semidirect
30. C₄⋊C₄×C₂₃₉	Q₈	semidirect
69. C₂₃₉	#1988	semidirect
100. C₄⋊C₄	Dic₄₇₈	semidirect

H ≅ G/N

type

8. #21439960

C₄

semidirect

30. C₄⋊C₄×C₂₃₉

Q₈

semidirect

69. C₂₃₉

#1988

semidirect

100. C₄⋊C₄

Dic₄₇₈

semidirect

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

subgroup	index
C₂×#21439960	2
C₂×C₄⋊(C₂₃₉⋊C₈)	2
#29504534	2
C₄×C₄⋊C₄×C₂₃₉	2
#1988	239

subgroup

index

C₂×#21439960

C₂×C₄⋊(C₂₃₉⋊C₈)

#29504534

C₄×C₄⋊C₄×C₂₃₉

#1988

239

Subgroups — #44485500

Conjugacy classes of subgroups

Complements

Maximal subgroups