Group #44486102 — subgroups

Subgroups — #44486102

Order 30592.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#29504821	C₂	1	1	yes	maximal, characteristic, complemented
2	#29504799	C₂	1	2	yes	maximal, complemented
3	#29504790	C₂	1	2	yes	maximal
4	#29504740	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16838548	C₂²	1	1	yes	characteristic
7	C₂×#16838532	C₂²	1	2	yes	—
8	C₂²×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	#21439896	—	2	2	no	—
10	C₄⋊(C₂₃₉⋊C₈)	—	2	4	no	—
11	#21439888	—	2	4	no	—
12	(C₂×D₂₃₉)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
14	#117×C₂₃₉	C₂²	1	2	yes	—
15	#16838548	—	2	2	no	—
16	#16838548	—	2	2	no	—
17	C₂×Dic₄₇₈	Q₈	1	1	yes	characteristic
18	C₂×D₉₅₆	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₃₉	—	2	1	no	—
20	C₂×C₄×D₂₃₉	—	2	1	no	—
21	#16838532	—	2	2	no	—
22	#16838532	—	2	2	no	—
23	C₂×C₂₃₉⋊C₈	—	2	1	no	—
24	C₂×C₂₃₉⋊C₈	—	2	2	no	—
25	C₂×C₂₃₉⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₃₉	—	2	2	no	—
28	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₂₃₉	—	2	2	no	—
30	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
31	C₄²×C₂₃₉	D₄	1	4	yes	—
32	C₂²×D₂₃₉	—	2	1	no	—
33	Dic₄₇₈	—	2	2	no	—
34	D₉₅₆	—	2	2	no	—
35	C₂²⋊D₂₃₉	—	4	2	no	—
36	C₂×Dic₂₃₉	—	2	1	no	—
37	C₄×D₂₃₉	—	4	1	no	—
38	C₄×D₂₃₉	—	4	1	no	—
39	C₂₃₉⋊C₈	—	4	1	no	—
40	C₂₃₉⋊C₈	—	4	2	no	—
41	C₂³×C₂₃₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₂₃₉	—	2	4	no	—
43	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₃₉	#41	1	2	yes	—
47	C₈×C₂₃₉	—	2	2	no	—
48	C₂²×C₂₃₉	—	4	1	no	—
49	C₂²×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₃₉	#123	1	2	yes	—
51	C₂×D₂₃₉	—	4	2	no	—
52	C₂×D₂₃₉	—	4	1	no	—
53	C₄×C₂₃₉	—	4	4	no	—
54	C₄×C₂₃₉	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₂₃₉	#123	1	1	yes	characteristic
56	C₄×C₂₃₉	#127	1	1	yes	characteristic
57	Dic₂₃₉	—	4	1	no	—
58	C₂×C₂₃₉	—	2	2	no	—
59	C₂×C₂₃₉	—	2	1	no	—
60	C₂×C₂₃₉	#399	1	1	yes	characteristic
61	D₂₃₉	—	8	1	no	—
62	C₂₃₉	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	239	1	no	maximal, Sylow
64	#430	—	239	1	no	—
65	#424	—	239	2	no	—
66	#421	—	239	2	no	—
67	#408	—	239	1	no	—
68	H(3,ℤ/4)	D₂₃₉	1	1	yes	characteristic, complemented
69	C₂×#41	—	239	1	no	—
70	C₂×M₁₆	—	239	2	no	—
71	C₂×M₁₆	C₂×D₂₃₉	1	1	yes	characteristic
72	C₂²×C₈	—	239	1	no	—
73	#117	C₂×D₂₃₉	1	2	yes	—
74	#108	—	478	2	no	—
75	C₄⋊C₈	—	478	4	no	—
76	#104	—	478	4	no	—
77	C₂²⋊C₈	—	478	2	no	—
78	#41	—	478	2	no	—
79	#41	—	478	2	no	—
80	C₂×Q₈	—	239	1	no	—
81	C₂×D₄	—	478	1	no	—
82	C₂×D₄	—	239	1	no	—
83	C₂²×C₄	—	478	1	no	—
84	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	478	2	no	—
87	M₁₆	—	478	2	no	—
88	C₂×C₈	—	478	1	no	—
89	C₂×C₈	—	478	2	no	—
90	C₂×C₈	—	478	2	no	—
91	C₂×C₈	D₉₅₆	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₉₅₆	1	4	yes	—
95	C₂³	—	478	1	no	—
96	C₂³	#16838547	1	1	yes	characteristic
97	Q₈	—	478	2	no	—
98	D₄	—	956	2	no	—
99	D₄	—	478	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	956	1	no	—
102	C₂×C₄	—	956	1	no	—
103	C₂×C₄	—	478	1	no	—
104	C₂×C₄	C₂×D₉₅₆	1	2	yes	—
105	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₃₉	1	1	yes	characteristic
107	C₂×C₄	#16838547	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	956	1	no	—
110	C₈	—	956	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	956	2	no	—
113	C₂²	—	956	1	no	—
114	C₂²	#21439940	1	1	yes	characteristic
115	C₂²	#21439944	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	956	1	no	—
118	C₄	#21439940	1	2	yes	—
119	C₄	#21439944	1	1	yes	characteristic
120	C₄	#21439957	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1912	1	no	—
124	C₂	#29504709	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29504821

C₂

yes

maximal, characteristic, complemented

#29504799

C₂

yes

maximal, complemented

#29504790

C₂

yes

maximal

#29504740

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₂₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#16838548

C₂²

yes

characteristic

C₂×#16838532

C₂²

yes

—

C₂²×C₂₃₉⋊C₈

C₂²

yes

characteristic

#21439896

—

C₄⋊(C₂₃₉⋊C₈)

—

#21439888

—

(C₂×D₂₃₉)⋊C₈

—

C₂×M₁₆×C₂₃₉

C₂²

yes

characteristic

#117×C₂₃₉

C₂²

yes

—

#16838548

—

#16838548

—

C₂×Dic₄₇₈

Q₈

yes

characteristic

C₂×D₉₅₆

Q₈

yes

characteristic

C₂×C₂²⋊D₂₃₉

—

C₂×C₄×D₂₃₉

—

#16838532

—

#16838532

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

M₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₄⋊C₄×C₂₃₉

—

C₂²⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

D₄

yes

—

C₂²×D₂₃₉

—

Dic₄₇₈

—

D₉₅₆

—

C₂²⋊D₂₃₉

—

C₂×Dic₂₃₉

—

C₄×D₂₃₉

—

C₄×D₂₃₉

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

#41

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

#41

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃₉

#123

yes

—

C₂×D₂₃₉

—

C₂×D₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂²⋊Q₈

yes

—

C₄×C₂₃₉

yes

C₄×C₂₃₉

yes

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#399

yes

characteristic

D₂₃₉

—

C₂₃₉

#2098

yes

characteristic, complemented, Sylow

#2098

—

239

maximal, Sylow

#430

—

239

—

#424

—

239

—

#421

—

239

—

#408

—

239

—

H(3,ℤ/4)

D₂₃₉

yes

characteristic, complemented

C₂×#41

—

239

—

C₂×M₁₆

—

239

—

C₂×M₁₆

C₂×D₂₃₉

yes

characteristic

C₂²×C₈

—

239

—

#117

C₂×D₂₃₉

yes

—

#108

—

478

—

C₄⋊C₈

—

478

—

#104

—

478

—

C₂²⋊C₈

—

478

—

#41

—

478

—

#41

—

478

—

C₂×Q₈

—

239

—

C₂×D₄

—

478

—

C₂×D₄

—

239

—

C₂²×C₄

—

478

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

M₁₆

—

M₁₆

—

478

—

M₁₆

—

478

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

D₉₅₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₉₅₆

yes

—

C₂³

—

478

—

C₂³

#16838547

yes

characteristic

Q₈

—

478

—

D₄

—

956

—

D₄

—

478

—

100

C₂×C₄

—

101

C₂×C₄

—

956

—

102

C₂×C₄

—

956

—

103

C₂×C₄

—

478

—

104

C₂×C₄

C₂×D₉₅₆

yes

—

105

C₂×C₄

C₂×D₉₅₆

yes

characteristic

106

C₂×C₄

Q₈×D₂₃₉

yes

characteristic

107

C₂×C₄

#16838547

yes

—

108

C₈

—

109

C₈

—

956

—

110

C₈

—

956

—

111

C₂²

—

112

C₂²

—

956

—

113

C₂²

—

956

—

114

C₂²

#21439940

yes

characteristic

115

C₂²

#21439944

yes

—

116

C₄

—

117

C₄

—

956

—

118

C₄

#21439940

yes

—

119

C₄

#21439944

yes

characteristic

120

C₄

#21439957

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1912

—

124

C₂

#29504709

yes

characteristic

	subgroup	G/H
center	C₄	#21439957
commutator, cosocle	C₂²×C₄×C₂₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	H(3,ℤ/4)×C₂₃₉	C₂
socle	C₂×C₂₃₉	#399
prime radical	C₂×#16838548	C₂²
hypercenter	H(3,ℤ/4)	D₂₃₉
hypocenter	C₂₃₉	#2098
LCS	G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#21439957

commutator, cosocle

C₂²×C₄×C₂₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

H(3,ℤ/4)×C₂₃₉

C₂

C₂×C₂₃₉

C₂²

D₂₃₉

C₂₃₉

G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #29504821	C₂	semidirect
2. #29504799	C₂	semidirect
5. H(3,ℤ/4)×C₂₃₉	C₂	semidirect
62. C₂₃₉	#2098	semidirect
68. H(3,ℤ/4)	D₂₃₉	semidirect

H ≅ G/N

type

1. #29504821

C₂

semidirect

2. #29504799

C₂

semidirect

5. H(3,ℤ/4)×C₂₃₉

C₂

semidirect

62. C₂₃₉

#2098

semidirect

68. H(3,ℤ/4)

D₂₃₉

semidirect

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

subgroup	index
#29504821	2
#29504799	2
#29504790	2
#29504740	2
H(3,ℤ/4)×C₂₃₉	2
#2098	239

subgroup

index

H(3,ℤ/4)×C₂₃₉

239

Subgroups — #44486102

Conjugacy classes of subgroups

Complements

Maximal subgroups