Group #21424264 — subgroups

Subgroups — #21424264

Order 7552.

256 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14273661	C₂	1	2	yes	maximal
2	C₂×#14273659	C₂	1	2	yes	maximal
3	#16837272	C₂	1	1	yes	maximal, characteristic
4	#16837268	C₂	1	1	yes	maximal, characteristic
5	C₂×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₅₉	—	2	2	no	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
10	#14273661	—	2	4	no	—
11	#14273659	—	2	4	no	—
12	C₅₉⋊#95	—	2	2	no	—
13	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	#97×C₅₉	C₄	1	4	yes	complemented
16	C₂²×C₄×C₅₉	D₄	1	2	yes	—
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₉	—	2	1	no	—
19	C₂²×Dic₅₉	—	2	1	no	—
20	C₂²×Dic₅₉	—	2	2	no	—
21	C₂×C₈×C₅₉	C₂×C₄	1	4	yes	—
22	C₂×C₈×C₅₉	D₄	1	4	yes	—
23	C₂×C₈×C₅₉	Q₈	1	4	yes	—
24	C₄²×C₅₉	C₂×C₄	1	2	yes	—
25	C₄²×C₅₉	Q₈	1	2	yes	—
26	C₄⋊Dic₅₉	—	2	2	no	—
27	C₄⋊Dic₅₉	—	2	2	no	—
28	C₄⋊Dic₅₉	—	2	2	no	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	C₄⋊Dic₅₉	—	4	4	no	—
31	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	—	2	2	no	—
33	C₂×C₄×C₅₉	—	2	2	no	—
34	C₂×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
35	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×Dic₅₉	—	4	2	no	—
40	C₂×Dic₅₉	—	4	1	no	—
41	C₂×Dic₅₉	—	4	1	no	—
42	C₂×Dic₅₉	—	4	4	no	—
43	C₂×Dic₅₉	—	4	2	no	—
44	C₂×Dic₅₉	—	4	2	no	—
45	C₈×C₅₉	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₅₉	#127	1	1	yes	characteristic
48	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₅₉	#137	1	2	yes	—
51	C₄×C₅₉	—	2	4	no	—
52	C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
53	C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
54	Dic₅₉	—	8	2	no	—
55	Dic₅₉	—	8	1	no	—
56	Dic₅₉	—	8	1	no	—
57	#1847	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	#384	1	1	yes	characteristic
59	C₂×C₅₉	#428	1	2	yes	—
60	C₂×C₅₉	#496	1	1	yes	characteristic
61	C₂×C₅₉	#497	1	1	yes	characteristic
62	C₂×C₅₉	#501	1	2	yes	—
63	C₂×#107	—	59	2	no	—
64	C₂×#106	—	59	2	no	—
65	C₂×#97	D₅₉	1	1	yes	characteristic
66	#384	—	59	1	no	—
67	#382	—	59	1	no	—
68	C₅₉	#1847	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	C₂×D₅₉	1	2	yes	—
70	C₂×C₄⋊C₄	—	59	1	no	—
71	C₂×C₄⋊C₄	—	59	1	no	—
72	C₂×C₄⋊C₄	—	59	2	no	—
73	C₂×C₄⋊C₄	—	118	2	no	—
74	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
75	#107	—	118	4	no	—
76	#106	—	118	4	no	—
77	#97	Dic₅₉	1	4	yes	complemented
78	#95	—	118	2	no	—
79	C₂²×C₄	—	118	1	no	—
80	C₂²×C₄	—	118	1	no	—
81	C₂²×C₄	—	118	2	no	—
82	C₂²×C₄	D₂₃₆	1	2	yes	—
83	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
84	C₂×C₈	Dic₁₁₈	1	4	yes	—
85	C₂×C₈	D₂₃₆	1	4	yes	—
86	C₂×C₈	C₂×Dic₅₉	1	4	yes	—
87	C₄⋊C₄	—	118	2	no	—
88	C₄⋊C₄	—	118	2	no	—
89	C₄⋊C₄	—	118	2	no	—
90	C₄⋊C₄	—	118	2	no	—
91	C₄⋊C₄	—	236	4	no	—
92	C₄²	Dic₁₁₈	1	2	yes	—
93	C₄²	C₂×Dic₅₉	1	2	yes	—
94	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	236	2	no	—
98	C₂×C₄	—	236	1	no	—
99	C₂×C₄	—	236	1	no	—
100	C₂×C₄	—	236	4	no	—
101	C₂×C₄	—	236	2	no	—
102	C₂×C₄	—	236	2	no	—
103	C₂×C₄	C₄⋊Dic₅₉	1	4	yes	—
104	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
105	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
106	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
107	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
108	C₈	C₄⋊Dic₅₉	1	8	yes	complemented
109	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
110	C₂²	#14273724	1	1	yes	characteristic
111	C₂²	#14273727	1	1	yes	characteristic
112	C₂²	#14273755	1	2	yes	—
113	C₂²	#14273760	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	472	2	no	—
116	C₄	—	472	1	no	—
117	C₄	—	472	1	no	—
118	C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
119	C₄	#14273727	1	2	yes	—
120	C₂	#16837272	1	1	yes	characteristic
121	C₂	#16837444	1	2	yes	—
122	C₂	#16837791	1	1	yes	characteristic
123	C₂	#16837796	1	1	yes	characteristic
124	C₂	#16837811	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273661

C₂

yes

maximal

C₂×#14273659

C₂

yes

maximal

#16837272

C₂

yes

maximal, characteristic

#16837268

C₂

yes

maximal, characteristic

C₂×#97×C₅₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

#14273661

—

#14273659

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#97×C₅₉

C₄

yes

complemented

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

Q₈

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

C₄⋊C₄

yes

complemented

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#127

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₄×C₅₉

C₄⋊Q₈

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

D₅₉

yes

characteristic

#384

—

#382

—

C₅₉

#1847

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#107

—

118

—

#106

—

118

—

#97

Dic₅₉

yes

complemented

#95

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

D₂₃₆

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄²

Dic₁₁₈

yes

—

C₄²

C₂×Dic₅₉

yes

—

C₂³

C₂×D₂₃₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

100

C₂×C₄

—

236

—

101

C₂×C₄

—

236

—

102

C₂×C₄

—

236

—

103

C₂×C₄

C₄⋊Dic₅₉

yes

—

104

C₂×C₄

C₂×Dic₁₁₈

yes

—

105

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

106

C₂×C₄

C₂×D₂₃₆

yes

—

107

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

108

C₈

C₄⋊Dic₅₉

yes

complemented

109

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

110

C₂²

#14273724

yes

characteristic

111

C₂²

#14273727

yes

characteristic

112

C₂²

#14273755

yes

—

113

C₂²

#14273760

yes

—

114

C₄

—

115

C₄

—

472

—

116

C₄

—

472

—

117

C₄

—

472

—

118

C₄

C₂×C₄⋊Dic₅₉

yes

—

119

C₄

#14273727

yes

—

120

C₂

#16837272

yes

characteristic

121

C₂

#16837444

yes

—

122

C₂

#16837791

yes

characteristic

123

C₂

#16837796

yes

characteristic

124

C₂

#16837811

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#97×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#97	D₅₉
hypocenter	C₅₉	#1847
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#97×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×#97

D₅₉

hypocenter

C₅₉

#1847

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
15. #97×C₅₉	C₄	semidirect
45. C₈×C₅₉	C₄⋊C₄	semidirect
68. C₅₉	#1847	semidirect
77. #97	Dic₅₉	semidirect
108. C₈	C₄⋊Dic₅₉	semidirect

H ≅ G/N

type

15. #97×C₅₉

C₄

semidirect

45. C₈×C₅₉

C₄⋊C₄

semidirect

68. C₅₉

#1847

semidirect

77. #97

Dic₅₉

semidirect

108. C₈

C₄⋊Dic₅₉

semidirect

Subgroups — #21424264

Conjugacy classes of subgroups

Complements