Group #44485522 — subgroups

Subgroups — #44485522

Order 30592.

276 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 113 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×#21439871	C₂	1	1	yes	maximal, characteristic
2	#29504638	C₂	1	1	yes	maximal, characteristic
3	#29504534	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₃₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
7	#21439871	—	2	2	no	—
8	C₂₃₉⋊#95	—	2	1	no	—
9	C₂₃₉⋊#95	—	2	1	no	—
10	C₂²×Q₈×C₂₃₉	—	2	1	no	—
11	C₄⋊Q₈×C₂₃₉	—	2	2	no	—
12	C₄⋊Q₈×C₂₃₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₂₃₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₂₃₉	—	2	2	no	—
17	C₂×C₂₃₉⋊C₈	—	2	2	no	—
18	C₂×C₂₃₉⋊C₈	—	2	2	no	—
19	C₂×C₂₃₉⋊C₈	Q₈	1	4	yes	complemented
20	C₄⋊Dic₂₃₉	—	2	4	no	—
21	C₂×Q₈×C₂₃₉	—	2	4	no	—
22	C₂×Q₈×C₂₃₉	—	4	4	no	—
23	C₂²×C₄×C₂₃₉	—	2	2	no	—
24	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₃₉	—	2	4	no	—
28	C₄⋊C₄×C₂₃₉	—	2	2	no	—
29	C₄⋊C₄×C₂₃₉	C₂×C₄	1	4	yes	—
30	C₄²×C₂₃₉	C₂×C₄	1	2	yes	—
31	C₄²×C₂₃₉	D₄	1	2	yes	—
32	C₂×Dic₂₃₉	—	4	2	no	—
33	C₂×Dic₂₃₉	—	4	2	no	—
34	C₂×Dic₂₃₉	—	4	2	no	—
35	C₂₃₉⋊C₈	—	2	4	no	—
36	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₂₃₉	—	4	8	no	—
38	C₂×C₄×C₂₃₉	—	2	2	no	—
39	C₂×C₄×C₂₃₉	—	2	2	no	—
40	C₂×C₄×C₂₃₉	—	2	4	no	—
41	C₂×C₄×C₂₃₉	—	4	4	no	—
42	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₂₃₉	#41	1	2	yes	—
48	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₃₉	#124	1	1	yes	characteristic
50	C₂²×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₂₃₉	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₂₃₉	#137	1	2	yes	—
53	C₄×C₂₃₉	—	2	2	no	—
54	C₄×C₂₃₉	—	2	2	no	—
55	C₄×C₂₃₉	—	4	4	no	—
56	C₄×C₂₃₉	C₄×Q₈	1	2	yes	—
57	C₄×C₂₃₉	C₂²⋊Q₈	1	2	yes	—
58	Dic₂₃₉	—	8	2	no	—
59	C₂×C₂₃₉	#391	1	1	yes	characteristic
60	C₂×C₂₃₉	#418	1	1	yes	characteristic
61	C₂×C₂₃₉	#419	1	1	yes	characteristic
62	C₂×C₂₃₉	#489	1	2	yes	—
63	C₂×C₂₃₉	#501	1	2	yes	—
64	C₂₃₉	#1993	1	1	yes	characteristic, complemented, Sylow
65	#1993	—	239	1	no	maximal, Sylow
66	C₂×C₄⋊Q₈	D₂₃₉	1	1	yes	characteristic
67	C₂×#97	—	239	1	no	—
68	#382	—	239	1	no	—
69	#340	—	239	4	no	—
70	C₂²×Q₈	—	2	1	no	—
71	C₂²×C₈	—	239	2	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₄⋊Q₈	Dic₂₃₉	1	4	yes	complemented
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	239	2	no	—
76	C₂×C₄⋊C₄	C₂×D₂₃₉	1	2	yes	—
77	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
78	#97	—	478	2	no	—
79	#95	—	478	1	no	—
80	#95	—	478	1	no	—
81	C₂×Q₈	—	2	4	no	—
82	C₂×Q₈	—	4	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	478	2	no	—
85	C₂²×C₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
88	C₂×C₈	—	478	2	no	—
89	C₂×C₈	—	478	2	no	—
90	C₂×C₈	—	239	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	478	4	no	—
94	C₄⋊C₄	C₂×Dic₂₃₉	1	4	yes	—
95	C₄²	C₂×Dic₂₃₉	1	2	yes	—
96	C₄²	C₂²⋊D₂₃₉	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
98	Q₈	—	4	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	956	2	no	—
104	C₂×C₄	—	956	2	no	—
105	C₂×C₄	—	956	2	no	—
106	C₂×C₄	C₂²⋊Dic₂₃₉	1	2	yes	—
107	C₂×C₄	C₂²⋊Dic₂₃₉	1	2	yes	—
108	C₂×C₄	C₂²×Dic₂₃₉	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₂₃₉	1	2	yes	—
111	C₂×C₄	#16838549	1	2	yes	—
112	C₈	—	478	4	no	—
113	C₂²	C₂×C₂²⋊Dic₂₃₉	1	1	yes	characteristic
114	C₂²	#21439949	1	1	yes	characteristic
115	C₂²	#21439958	1	1	yes	characteristic
116	C₂²	#21439986	1	2	yes	—
117	C₂²	#21439991	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	4	4	no	—
121	C₄	—	1912	2	no	—
122	C₄	Q₈×Dic₂₃₉	1	2	yes	—
123	C₄	#21439936	1	2	yes	—
124	C₂	#29504679	1	1	yes	characteristic
125	C₂	#29504777	1	1	yes	characteristic
126	C₂	#29504781	1	1	yes	characteristic
127	C₂	#29505134	1	2	yes	—
128	C₂	#29505178	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439871

C₂

yes

maximal, characteristic

#29504638

C₂

yes

maximal, characteristic

#29504534

C₂

yes

maximal

C₂×C₄⋊Q₈×C₂₃₉

C₂

yes

maximal, characteristic

C₂²×C₂₃₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

#21439871

—

C₂₃₉⋊#95

—

C₂₃₉⋊#95

—

C₂²×Q₈×C₂₃₉

—

C₄⋊Q₈×C₂₃₉

—

C₄⋊Q₈×C₂₃₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₃₉

—

C₂×C₄⋊C₄×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₂²×Dic₂₃₉

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

Q₈

yes

complemented

C₄⋊Dic₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

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

—

C₄⋊C₄×C₂₃₉

C₂×C₄

yes

—

C₄²×C₂₃₉

C₂×C₄

yes

—

C₄²×C₂₃₉

D₄

yes

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

C₂×D₄

yes

characteristic

Q₈×C₂₃₉

—

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

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

#124

yes

characteristic

C₂²×C₂₃₉

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃₉

AGL(1,ℤ/8)

yes

—

C₂²×C₂₃₉

#137

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₄×Q₈

yes

—

C₄×C₂₃₉

C₂²⋊Q₈

yes

—

Dic₂₃₉

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

—

C₂×C₂₃₉

#501

yes

—

C₂₃₉

#1993

yes

characteristic, complemented, Sylow

#1993

—

239

maximal, Sylow

C₂×C₄⋊Q₈

D₂₃₉

yes

characteristic

C₂×#97

—

239

—

#382

—

239

—

#340

—

239

—

C₂²×Q₈

—

C₂²×C₈

—

239

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₂₃₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄⋊C₄

C₂×D₂₃₉

yes

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

#97

—

478

—

#95

—

478

—

#95

—

478

—

C₂×Q₈

—

C₂×Q₈

—

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

—

C₂×C₈

—

239

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

478

—

C₄⋊C₄

C₂×Dic₂₃₉

yes

—

C₄²

C₂×Dic₂₃₉

yes

—

C₄²

C₂²⋊D₂₃₉

yes

—

C₂³

C₂×C₂²⋊D₂₃₉

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

956

—

104

C₂×C₄

—

956

—

105

C₂×C₄

—

956

—

106

C₂×C₄

C₂²⋊Dic₂₃₉

yes

—

107

C₂×C₄

C₂²⋊Dic₂₃₉

yes

—

108

C₂×C₄

C₂²×Dic₂₃₉

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

110

C₂×C₄

Q₈×D₂₃₉

yes

—

111

C₂×C₄

#16838549

yes

—

112

C₈

—

478

—

113

C₂²

C₂×C₂²⋊Dic₂₃₉

yes

characteristic

114

C₂²

#21439949

yes

characteristic

115

C₂²

#21439958

yes

characteristic

116

C₂²

#21439986

yes

—

117

C₂²

#21439991

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

1912

—

122

C₄

Q₈×Dic₂₃₉

yes

—

123

C₄

#21439936

yes

—

124

C₂

#29504679

yes

characteristic

125

C₂

#29504777

yes

characteristic

126

C₂

#29504781

yes

characteristic

127

C₂

#29505134

yes

—

128

C₂

#29505178

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃₉
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₂×C₄⋊Q₈×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	C₂×D₄
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₃₉
hypocenter	C₂₃₉	#1993
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃₉

commutator

C₂×C₄×C₂₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

C₂×C₄⋊Q₈×C₂₃₉

C₂

socle, prime radical

C₂³×C₂₃₉

C₂×D₄

cosocle

C₂²×C₄×C₂₃₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₃₉

hypocenter

C₂₃₉

#1993

LCS

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₂₃₉	C₄	semidirect
19. C₂×C₂₃₉⋊C₈	Q₈	semidirect
64. C₂₃₉	#1993	semidirect
73. C₄⋊Q₈	Dic₂₃₉	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₂₃₉

C₄

semidirect

19. C₂×C₂₃₉⋊C₈

Q₈

semidirect

64. C₂₃₉

#1993

semidirect

73. C₄⋊Q₈

Dic₂₃₉

semidirect

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

subgroup	index
C₂×#21439871	2
#29504638	2
#29504534	2
C₂×C₄⋊Q₈×C₂₃₉	2
#1993	239

subgroup

index

C₂×C₄⋊Q₈×C₂₃₉

239

Subgroups — #44485522

Conjugacy classes of subgroups

Complements

Maximal subgroups