Group #21422385 — subgroups

Subgroups — #21422385

Order 7552.

208 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16837455	C₂	1	1	yes	maximal, characteristic
2	C₄×#11745664	C₂	1	1	yes	maximal, characteristic
3	#16837124	C₂	1	2	yes	maximal, complemented
4	#16837123	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₄	1	2	yes	—
7	C₄⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
9	#14273638	C₄	1	2	yes	—
10	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
11	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₉	C₄	1	2	yes	—
13	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₉	C₄	1	2	yes	—
15	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	—	2	4	no	—
21	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₅₉	C₂³	1	1	yes	characteristic
27	#11745664	—	2	4	no	—
28	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
30	C₂³×C₅₉	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₅₉	—	2	2	no	—
32	C₂×C₄×C₅₉	—	2	2	no	—
33	C₂×C₄×C₅₉	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₅₉	M₁₆	1	4	yes	—
37	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₅₉	—	2	2	no	—
41	C₈×C₅₉	M₁₆	1	4	yes	complemented
42	C₅₉⋊C₈	—	2	4	no	—
43	C₅₉⋊C₈	—	2	4	no	—
44	C₂²×C₅₉	—	2	1	no	—
45	C₂²×C₅₉	—	2	1	no	—
46	C₂²×C₅₉	#97	1	2	yes	—
47	C₂²×C₅₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₅₉	—	2	1	no	—
49	C₄×C₅₉	—	2	1	no	—
50	C₄×C₅₉	#97	1	2	yes	—
51	C₄×C₅₉	C₂×M₁₆	1	2	yes	—
52	C₄×C₅₉	C₂×M₁₆	1	2	yes	—
53	C₄×C₅₉	#131	1	2	yes	—
54	#1451	—	59	1	no	maximal, Sylow
55	C₂×C₅₉	—	2	2	no	—
56	C₂×C₅₉	C₂×#97	1	1	yes	characteristic
57	C₂×C₅₉	C₄×M₁₆	1	1	yes	characteristic
58	C₂×C₅₉	#405	1	1	yes	characteristic
59	#431	—	59	1	no	—
60	C₄×M₁₆	—	59	1	no	—
61	C₂×C₄×C₈	D₅₉	1	1	yes	characteristic
62	#322	—	59	2	no	—
63	#322	—	59	2	no	—
64	C₅₉	#1451	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₁₆	—	59	2	no	—
66	C₂²×C₈	Dic₅₉	1	2	yes	—
67	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
68	C₄⋊C₈	—	59	2	no	—
69	C₂²⋊C₈	—	59	2	no	—
70	#97	—	59	2	no	—
71	C₄×C₈	—	59	2	no	—
72	C₄×C₈	—	59	2	no	—
73	C₄×C₈	Dic₅₉	1	2	yes	—
74	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
75	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
79	M₁₆	—	118	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	59	4	no	—
82	C₂×C₈	—	59	4	no	—
83	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
84	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
85	C₄²	C₄×D₅₉	1	1	yes	characteristic
86	C₄²	C₄×D₅₉	1	1	yes	characteristic
87	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
88	C₄²	C₂²×D₅₉	1	1	yes	characteristic
89	C₂³	C₄×Dic₅₉	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	#11745659	1	4	yes	—
93	C₂×C₄	C₄×Dic₅₉	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₅₉	1	1	yes	characteristic
95	C₂×C₄	C₄×Dic₅₉	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	118	4	no	—
101	C₈	—	118	4	no	—
102	C₈	#11745664	1	4	yes	complemented
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#14273637	1	2	yes	—
106	C₂²	C₂×C₄×Dic₅₉	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	#14273637	1	2	yes	—
110	C₄	C₂×#11745659	1	2	yes	—
111	C₄	C₂×#11745664	1	2	yes	—
112	C₄	#14273733	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×#14273637	1	1	yes	characteristic
115	C₂	C₄×#11745664	1	1	yes	characteristic
116	C₂	#16837357	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837455

C₂

yes

maximal, characteristic

C₄×#11745664

C₂

yes

maximal, characteristic

#16837124

C₂

yes

maximal, complemented

#16837123

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₄

yes

—

C₄⋊(C₅₉⋊C₈)

C₄

yes

—

C₂²⋊(C₅₉⋊C₈)

C₄

yes

—

#14273638

C₄

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₂²×C₈×C₅₉

C₄

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₄

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₄²

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄²

yes

characteristic

C₂×C₄×C₅₉

C₄²

yes

characteristic

C₂×C₄×C₅₉

C₄²

yes

characteristic

C₂×C₄×C₅₉

M₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

M₁₆

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#97

yes

—

C₂²×C₅₉

C₂×C₄²

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#97

yes

—

C₄×C₅₉

C₂×M₁₆

yes

—

C₄×C₅₉

C₂×M₁₆

yes

—

C₄×C₅₉

#131

yes

—

#1451

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×#97

yes

characteristic

C₂×C₅₉

C₄×M₁₆

yes

C₂×C₅₉

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

D₅₉

yes

characteristic

#322

—

#322

—

C₅₉

#1451

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

Dic₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₅₉

yes

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₄²

C₄×D₅₉

yes

characteristic

C₄²

C₄×D₅₉

yes

characteristic

C₄²

C₂×Dic₅₉

yes

characteristic

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

C₄×Dic₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11745659

yes

—

C₂×C₄

C₄×Dic₅₉

yes

characteristic

C₂×C₄

C₄×Dic₅₉

yes

characteristic

C₂×C₄

C₄×Dic₅₉

yes

characteristic

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

C₈

—

100

C₈

—

118

—

101

C₈

—

118

—

102

C₈

#11745664

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

#14273637

yes

—

106

C₂²

C₂×C₄×Dic₅₉

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#14273637

yes

—

110

C₄

C₂×#11745659

yes

—

111

C₄

C₂×#11745664

yes

—

112

C₄

#14273733

yes

—

113

C₂

—

114

C₂

C₂×#14273637

yes

characteristic

115

C₂

C₄×#11745664

yes

characteristic

116

C₂

#16837357

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂×C₄²
Fitting	C₂×C₄×C₈×C₅₉	C₂
hypercenter	C₂×C₄×C₈	D₅₉
hypocenter	C₅₉	#1451
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₄²

C₂²×D₅₉

commutator, socle

C₂²×C₅₉

C₂×C₄²

Fitting

C₂×C₄×C₈×C₅₉

C₂

hypercenter

C₂×C₄×C₈

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #16837124	C₂	semidirect
4. #16837123	C₂	semidirect
41. C₈×C₅₉	M₁₆	semidirect
64. C₅₉	#1451	semidirect
102. C₈	#11745664	semidirect

H ≅ G/N

type

3. #16837124

C₂

semidirect

4. #16837123

C₂

semidirect

41. C₈×C₅₉

M₁₆

semidirect

64. C₅₉

#1451

semidirect

102. C₈

#11745664

semidirect

Subgroups — #21422385

Conjugacy classes of subgroups

Complements