Group #21432660 — subgroups

Subgroups — #21432660

Order 7552.

484 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 293 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	C₂²×#11745664	C₂	1	2	yes	maximal
2	#16837429	C₂	1	8	yes	maximal, complemented
3	#16837426	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	—	2	8	no	—
6	C₂×#11745664	C₂²	1	4	yes	—
7	C₂×#11745664	C₂²	1	4	yes	—
8	C₂²×C₅₉⋊C₈	C₂²	1	4	yes	—
9	C₄⋊(C₅₉⋊C₈)	C₂²	1	16	yes	complemented
10	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
11	#117×C₅₉	C₄	1	4	yes	—
12	#117×C₅₉	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₅₉	C₄	1	2	yes	—
14	C₂×C₂²⋊C₄×C₅₉	C₄	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂²	1	2	yes	—
16	C₂⁴×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₉	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₉	D₄	1	4	yes	—
21	C₂²×C₄×C₅₉	Q₈	1	4	yes	—
22	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
24	C₄⋊C₄×C₅₉	C₂×C₄	1	8	yes	—
25	C₂²⋊C₄×C₅₉	C₂×C₄	1	8	yes	—
26	C₄²×C₅₉	C₂×C₄	1	4	yes	—
27	C₄²×C₅₉	C₂³	1	4	yes	—
28	#11745664	—	2	8	no	—
29	#11745664	—	2	8	no	—
30	C₂×C₅₉⋊C₈	—	2	8	no	—
31	C₂×C₅₉⋊C₈	C₂³	1	8	yes	—
32	C₂³×C₅₉	—	4	2	no	—
33	C₂³×C₅₉	C₄⋊C₄	1	4	yes	—
34	C₂³×C₅₉	C₂²×C₄	1	2	yes	—
35	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	—	4	4	no	—
37	C₂×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₅₉	C₄⋊C₄	1	8	yes	—
39	C₂×C₄×C₅₉	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₅₉	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₅₉	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
47	C₅₉⋊C₈	—	2	8	no	—
48	C₂²×C₅₉	—	2	2	no	—
49	C₂²×C₅₉	—	2	4	no	—
50	C₂²×C₅₉	—	4	4	no	—
51	C₂²×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
52	C₂²×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
53	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₅₉	—	2	8	no	—
55	C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
56	C₄×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
57	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
59	#2905	—	59	1	no	maximal, Sylow
60	C₂×C₅₉	—	2	2	no	—
61	C₂×C₅₉	—	2	4	no	—
62	C₂×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₅₉	#568	1	2	yes	—
64	C₂²×M₁₆	—	59	2	no	—
65	C₂×#117	D₅₉	1	1	yes	characteristic
66	#424	—	59	8	no	—
67	#423	—	59	4	no	—
68	C₅₉	#2905	1	1	yes	characteristic, complemented, Sylow
69	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
70	C₂×M₁₆	—	59	4	no	—
71	C₂×M₁₆	—	59	4	no	—
72	C₂×M₁₆	—	118	8	no	—
73	C₂²×C₈	—	59	4	no	—
74	#117	Dic₅₉	1	4	yes	—
75	#117	C₂×D₅₉	1	4	yes	—
76	C₂×C₄⋊C₄	Dic₅₉	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₅₉	1	2	yes	—
78	C₂×C₄²	C₂×D₅₉	1	2	yes	—
79	C₄⋊C₈	—	59	16	no	—
80	C₂⁴	C₂×Dic₅₉	1	1	yes	characteristic
81	C₂²×C₄	Dic₁₁₈	1	4	yes	—
82	C₂²×C₄	D₂₃₆	1	4	yes	—
83	C₂²×C₄	C₂×Dic₅₉	1	4	yes	—
84	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
86	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
88	M₁₆	—	118	8	no	—
89	M₁₆	—	118	8	no	—
90	C₂×C₈	—	59	8	no	—
91	C₂×C₈	—	118	8	no	—
92	C₄⋊C₄	C₂×Dic₅₉	1	8	yes	—
93	C₂²⋊C₄	C₂×Dic₅₉	1	8	yes	—
94	C₄²	C₂×Dic₅₉	1	4	yes	—
95	C₄²	C₂²×D₅₉	1	4	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₄⋊Dic₅₉	1	4	yes	—
98	C₂³	C₂²×Dic₅₉	1	2	yes	—
99	C₂³	C₂²×Dic₅₉	1	1	yes	characteristic
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄⋊Dic₅₉	1	4	yes	—
102	C₂×C₄	C₄⋊Dic₅₉	1	8	yes	—
103	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
104	C₂×C₄	C₂×Dic₁₁₈	1	4	yes	—
105	C₂×C₄	C₂×D₂₃₆	1	4	yes	—
106	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
107	C₂×C₄	C₂²×Dic₅₉	1	8	yes	—
108	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₅₉	1	2	yes	—
110	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
111	C₈	—	118	8	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂×C₄⋊Dic₅₉	1	4	yes	—
116	C₂²	C₂×C₄⋊Dic₅₉	1	2	yes	—
117	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
118	C₄	—	2	8	no	—
119	C₄	C₂×C₄⋊Dic₅₉	1	4	yes	—
120	C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
121	C₄	C₂²×D₂₃₆	1	1	yes	characteristic
122	C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	2	4	no	—
125	C₂	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
126	C₂	#16838169	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745664

C₂

yes

maximal

#16837429

C₂

yes

maximal, complemented

#16837426

C₂

yes

maximal, complemented

C₂×#117×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₂×#11745664

C₂²

yes

—

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

complemented

C₂³×C₄×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₄

yes

—

#117×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₄

yes

—

C₂×C₂²⋊C₄×C₅₉

C₄

yes

—

C₂×C₄²×C₅₉

C₂²

yes

—

C₂⁴×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₂²⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂³

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

C₄⋊C₄

yes

—

C₂³×C₅₉

C₂²×C₄

yes

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₄×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

C₂²×Q₈

yes

characteristic

#2905

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₅₉

#568

yes

—

C₂²×M₁₆

—

C₂×#117

D₅₉

yes

characteristic

#424

—

#423

—

C₅₉

#2905

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂²×C₈

—

#117

Dic₅₉

yes

—

#117

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

Dic₅₉

yes

—

C₂×C₂²⋊C₄

Dic₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

—

C₄⋊C₈

—

C₂⁴

C₂×Dic₅₉

yes

characteristic

C₂²×C₄

Dic₁₁₈

yes

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₂²⋊C₄

C₂×Dic₅₉

yes

—

C₄²

C₂×Dic₅₉

yes

—

C₄²

C₂²×D₅₉

yes

—

C₂³

—

C₂³

C₄⋊Dic₅₉

yes

—

C₂³

C₂²×Dic₅₉

yes

—

C₂³

C₂²×Dic₅₉

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₅₉

yes

—

102

C₂×C₄

C₄⋊Dic₅₉

yes

—

103

C₂×C₄

C₂×Dic₁₁₈

yes

—

104

C₂×C₄

C₂×Dic₁₁₈

yes

—

105

C₂×C₄

C₂×D₂₃₆

yes

—

106

C₂×C₄

C₂×D₂₃₆

yes

—

107

C₂×C₄

C₂²×Dic₅₉

yes

—

108

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

109

C₂×C₄

C₂²×Dic₅₉

yes

—

110

C₂×C₄

C₂³×D₅₉

yes

characteristic

111

C₈

—

118

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄⋊Dic₅₉

yes

—

116

C₂²

C₂×C₄⋊Dic₅₉

yes

—

117

C₂²

C₂³×Dic₅₉

yes

characteristic

118

C₄

—

119

C₄

C₂×C₄⋊Dic₅₉

yes

—

120

C₄

C₂×C₄⋊Dic₅₉

yes

—

121

C₄

C₂²×D₂₃₆

yes

characteristic

122

C₄

C₂²×Dic₁₁₈

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂²×C₄⋊Dic₅₉

yes

characteristic

126

C₂

#16838169

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₂×#117×C₅₉	C₂
hypercenter	C₂×#117	D₅₉
hypocenter	C₅₉	#2905
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₉

commutator, socle

C₂²×C₅₉

C₂³×C₄

Fitting

C₂×#117×C₅₉

C₂

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. #16837429	C₂	semidirect
3. #16837426	C₂	semidirect
9. C₄⋊(C₅₉⋊C₈)	C₂²	semidirect
68. C₅₉	#2905	semidirect

H ≅ G/N

type

2. #16837429

C₂

semidirect

3. #16837426

C₂

semidirect

9. C₄⋊(C₅₉⋊C₈)

C₂²

semidirect

68. C₅₉

#2905

semidirect

Subgroups — #21432660

Conjugacy classes of subgroups

Complements