Group #21433116 — subgroups

Subgroups — #21433116

Order 7552.

402 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 221 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	Q₈×C₅₉⋊C₈	C₂	1	2	yes	maximal, complemented
2	#16837480	C₂	1	6	yes	maximal, complemented
3	#16837459	C₂	1	3	yes	maximal, complemented
4	C₄×#11745664	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	6	yes	—
7	C₄⋊(C₅₉⋊C₈)	C₂²	1	6	yes	—
8	C₂²⋊(C₅₉⋊C₈)	C₂²	1	6	yes	—
9	#14273638	C₂²	1	3	yes	—
10	#14273638	C₂²	1	1	yes	characteristic
11	C₄×C₅₉⋊C₈	C₂²	1	6	yes	—
12	C₂×#41×C₅₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₅₉	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₅₉	C₄	1	3	yes	—
16	C₄×D₄×C₅₉	C₂²	1	3	yes	—
17	#117×C₅₉	C₄	1	3	yes	—
18	C₂×C₄²×C₅₉	C₂²	1	3	yes	—
19	#41×C₅₉	—	2	4	no	—
20	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₅₉	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₅₉	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₅₉	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
25	C₄⋊C₄×C₅₉	C₂×C₄	1	3	yes	—
26	C₄⋊C₄×C₅₉	C₂×C₄	1	3	yes	—
27	C₂²⋊C₄×C₅₉	C₂×C₄	1	6	yes	—
28	C₄²×C₅₉	C₂×C₄	1	3	yes	—
29	C₄²×C₅₉	C₂×C₄	1	3	yes	—
30	C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄²×C₅₉	C₂³	1	3	yes	—
32	#11745664	—	2	12	no	—
33	C₂×C₅₉⋊C₈	C₂³	1	6	yes	—
34	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
35	C₂³×C₅₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₅₉	M₁₆	1	4	yes	complemented
37	D₄×C₅₉	—	2	6	no	—
38	C₂×C₄×C₅₉	—	2	6	no	—
39	C₂×C₄×C₅₉	—	2	6	no	—
40	C₂×C₄×C₅₉	—	4	3	no	—
41	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
47	C₅₉⋊C₈	—	2	6	no	—
48	C₅₉⋊C₈	#41	1	4	yes	complemented
49	C₂²×C₅₉	—	2	3	no	—
50	C₂²×C₅₉	—	2	3	no	—
51	C₂²×C₅₉	—	2	3	no	—
52	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₅₉	—	2	3	no	—
54	C₄×C₅₉	—	2	1	no	—
55	C₄×C₅₉	—	2	1	no	—
56	C₄×C₅₉	—	2	3	no	—
57	C₄×C₅₉	C₂×M₁₆	1	6	yes	—
58	C₄×C₅₉	C₂×#41	1	2	yes	—
59	#2971	—	59	1	no	maximal, Sylow
60	C₂×C₅₉	—	4	3	no	—
61	C₂×C₅₉	C₄×#41	1	1	yes	characteristic
62	C₂×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
63	C₂×C₅₉	#568	1	1	yes	characteristic
64	C₄×#41	D₅₉	1	1	yes	characteristic
65	C₈×Q₈	—	59	2	no	—
66	#436	—	59	6	no	—
67	#432	—	59	3	no	—
68	C₄×M₁₆	—	59	3	no	—
69	C₅₉	#2971	1	1	yes	characteristic, complemented, Sylow
70	C₂×#41	Dic₅₉	1	1	yes	characteristic
71	C₂×M₁₆	—	59	6	no	—
72	C₄×Q₈	Dic₅₉	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
74	C₄×D₄	Dic₅₉	1	3	yes	—
75	C₄×D₄	C₂×D₅₉	1	3	yes	—
76	#117	Dic₅₉	1	3	yes	—
77	C₂×C₄²	C₂×D₅₉	1	3	yes	—
78	C₄⋊C₈	—	59	6	no	—
79	C₂²⋊C₈	—	59	6	no	—
80	#97	—	59	1	no	—
81	#97	—	59	3	no	—
82	C₄×C₈	—	59	6	no	—
83	#41	—	2	4	no	—
84	C₂×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
85	C₂×D₄	C₂×Dic₅₉	1	3	yes	—
86	C₂²×C₄	C₂×Dic₅₉	1	3	yes	—
87	C₂²×C₄	C₂×Dic₅₉	1	3	yes	—
88	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
89	M₁₆	—	118	12	no	—
90	C₂×C₈	—	59	2	no	—
91	C₂×C₈	—	59	6	no	—
92	C₄⋊C₄	C₂×Dic₅₉	1	3	yes	—
93	C₄⋊C₄	C₂×Dic₅₉	1	3	yes	—
94	C₂²⋊C₄	C₂×Dic₅₉	1	6	yes	—
95	C₄²	C₂×Dic₅₉	1	3	yes	—
96	C₄²	C₂×Dic₅₉	1	3	yes	—
97	C₄²	C₂²×D₅₉	1	3	yes	—
98	C₄²	C₂²×D₅₉	1	1	yes	characteristic
99	C₂³	C₂²×Dic₅₉	1	3	yes	—
100	Q₈	#11745664	1	4	yes	complemented
101	D₄	—	2	6	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	—	4	3	no	—
105	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₅₉	1	3	yes	—
108	C₂×C₄	C₂²×Dic₅₉	1	3	yes	—
109	C₂×C₄	C₂²×Dic₅₉	1	3	yes	—
110	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
111	C₈	—	59	4	no	—
112	C₈	—	118	6	no	—
113	C₂²	—	2	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	—	2	3	no	—
116	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	3	no	—
120	C₄	—	2	3	no	—
121	C₄	C₂×#11745664	1	6	yes	—
122	C₄	#41×D₅₉	1	2	yes	—
123	C₂	—	4	3	no	—
124	C₂	#41×Dic₅₉	1	1	yes	characteristic
125	C₂	C₂²×#11745664	1	1	yes	characteristic
126	C₂	#16838169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₅₉⋊C₈

C₂

yes

maximal, complemented

#16837480

C₂

yes

maximal, complemented

#16837459

C₂

yes

maximal, complemented

C₄×#11745664

C₂

yes

maximal, complemented

C₄×#41×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273638

C₂²

yes

—

#14273638

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×#41×C₅₉

C₄

yes

characteristic

C₄×Q₈×C₅₉

C₄

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄×D₄×C₅₉

C₄

yes

—

C₄×D₄×C₅₉

C₂²

yes

—

#117×C₅₉

C₄

yes

—

C₂×C₄²×C₅₉

C₂²

yes

—

#41×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂×D₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₂²⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

—

#11745664

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂²×C₄

yes

—

Q₈×C₅₉

M₁₆

yes

complemented

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

#41

yes

complemented

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

—

C₄×C₅₉

C₂×#41

yes

—

#2971

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₄×#41

yes

characteristic

C₂×C₅₉

C₂²×M₁₆

yes

characteristic

C₂×C₅₉

#568

yes

characteristic

C₄×#41

D₅₉

yes

characteristic

C₈×Q₈

—

#436

—

#432

—

C₄×M₁₆

—

C₅₉

#2971

yes

characteristic, complemented, Sylow

C₂×#41

Dic₅₉

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

Dic₅₉

yes

characteristic

C₄×Q₈

C₂×D₅₉

yes

characteristic

C₄×D₄

Dic₅₉

yes

—

C₄×D₄

C₂×D₅₉

yes

—

#117

Dic₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×Dic₅₉

yes

characteristic

C₂×D₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₂²⋊C₄

C₂×Dic₅₉

yes

—

C₄²

C₂×Dic₅₉

yes

—

C₄²

C₂×Dic₅₉

yes

—

C₄²

C₂²×D₅₉

yes

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

C₂²×Dic₅₉

yes

—

100

Q₈

#11745664

yes

complemented

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

106

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

107

C₂×C₄

C₂²×Dic₅₉

yes

—

108

C₂×C₄

C₂²×Dic₅₉

yes

—

109

C₂×C₄

C₂²×Dic₅₉

yes

—

110

C₂×C₄

C₂³×D₅₉

yes

characteristic

111

C₈

—

112

C₈

—

118

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂³×Dic₅₉

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×#11745664

yes

—

122

C₄

#41×D₅₉

yes

—

123

C₂

—

124

C₂

#41×Dic₅₉

yes

characteristic

125

C₂

C₂²×#11745664

yes

characteristic

126

C₂

#16838169

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₄×#41×C₅₉	C₂
hypercenter	C₄×#41	D₅₉
hypocenter	C₅₉	#2971
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₉

commutator, socle

C₂²×C₅₉

C₂³×C₄

Fitting

C₄×#41×C₅₉

C₂

hypercenter

C₄×#41

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. Q₈×C₅₉⋊C₈	C₂	semidirect
2. #16837480	C₂	semidirect
3. #16837459	C₂	semidirect
4. C₄×#11745664	C₂	semidirect
36. Q₈×C₅₉	M₁₆	semidirect
48. C₅₉⋊C₈	#41	semidirect
69. C₅₉	#2971	semidirect
100. Q₈	#11745664	semidirect

H ≅ G/N

type

1. Q₈×C₅₉⋊C₈

C₂

semidirect

2. #16837480

C₂

semidirect

3. #16837459

C₂

semidirect

4. C₄×#11745664

C₂

semidirect

36. Q₈×C₅₉

M₁₆

semidirect

48. C₅₉⋊C₈

#41

semidirect

69. C₅₉

#2971

semidirect

100. Q₈

#11745664

semidirect

Subgroups — #21433116

Conjugacy classes of subgroups

Complements