Group #21434610 — subgroups

Subgroups — #21434610

Order 7552.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	#16837959	C₂	1	2	yes	maximal
2	#16837807	C₂	1	2	yes	maximal, complemented
3	#16837804	C₂	1	4	yes	maximal, complemented
4	#16837800	C₂	1	2	yes	maximal, complemented
5	#16837440	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
7	#14273727	C₂²	1	4	yes	—
8	#14273719	C₂²	1	4	yes	—
9	#14273706	—	2	4	no	—
10	C₄×Dic₁₁₈	—	2	4	no	—
11	#14273686	C₂²	1	4	yes	—
12	#14273661	C₂²	1	8	yes	—
13	#14273659	C₂²	1	8	yes	—
14	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₅₉	C₂²	1	2	yes	—
18	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	D₄	1	2	yes	—
20	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₁₈	—	2	4	no	—
22	C₂×C₈×C₅₉	Q₈	1	8	yes	complemented
23	C₂×C₈×C₅₉	C₂³	1	2	yes	—
24	C₂×C₈×C₅₉	C₂³	1	2	yes	—
25	C₄²×C₅₉	D₄	1	2	yes	—
26	C₄²×C₅₉	C₂³	1	1	yes	characteristic
27	C₄²×C₅₉	C₂³	1	1	yes	characteristic
28	C₂²⋊Dic₅₉	—	2	4	no	—
29	C₄⋊Dic₅₉	—	2	4	no	—
30	C₄⋊Dic₅₉	C₂³	1	8	yes	—
31	Dic₅₉⋊C₄	—	2	8	no	—
32	C₄×Dic₅₉	—	2	4	no	—
33	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	—	2	4	no	—
35	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₅₉	—	2	8	no	—
41	Dic₁₁₈	—	4	8	no	—
42	C₈×C₅₉	C₂×Q₈	1	4	yes	—
43	C₈×C₅₉	C₂×Q₈	1	4	yes	—
44	C₂²×C₅₉	—	2	2	no	—
45	C₂²×C₅₉	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₅₉	—	2	2	no	—
48	C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
49	C₄×C₅₉	#135	1	4	yes	—
50	C₄×C₅₉	C₂²×Q₈	1	2	yes	—
51	Dic₅₉	—	4	8	no	—
52	#3160	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	—	2	2	no	—
54	C₂×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₅₉	C₂×#135	1	2	yes	—
56	#533	—	59	2	no	—
57	#500	—	59	2	no	—
58	#499	—	59	4	no	—
59	#498	—	59	2	no	—
60	#427	—	59	4	no	—
61	C₂×C₄×C₈	D₅₉	1	1	yes	characteristic
62	C₅₉	#3160	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	C₂×D₅₉	1	2	yes	—
64	C₄⋊Q₈	—	59	4	no	—
65	#125	—	59	4	no	—
66	C₂²⋊Q₈	—	118	4	no	—
67	C₄×Q₈	—	118	4	no	—
68	#117	—	59	4	no	—
69	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
70	#107	—	59	8	no	—
71	#106	—	59	8	no	—
72	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
73	C₄×C₈	C₂×D₅₉	1	2	yes	—
74	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
75	C₂×Q₈	—	118	4	no	—
76	C₂²×C₄	D₂₃₆	1	2	yes	—
77	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
78	C₂×C₈	Dic₁₁₈	1	8	yes	complemented
79	C₂×C₈	C₂²×D₅₉	1	2	yes	—
80	C₂×C₈	C₂²×D₅₉	1	2	yes	—
81	C₄⋊C₄	—	59	8	no	—
82	C₄⋊C₄	—	118	8	no	—
83	C₄⋊C₄	—	118	4	no	—
84	C₂²⋊C₄	—	118	4	no	—
85	C₄²	—	118	4	no	—
86	C₄²	D₂₃₆	1	2	yes	—
87	C₄²	C₂²×D₅₉	1	1	yes	characteristic
88	C₄²	C₂²×D₅₉	1	1	yes	characteristic
89	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
90	Q₈	—	236	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	118	8	no	—
93	C₂×C₄	C₂×Dic₁₁₈	1	4	yes	—
94	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
95	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
96	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
97	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
98	C₈	C₂×Dic₁₁₈	1	4	yes	—
99	C₈	C₂×Dic₁₁₈	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#14273727	1	2	yes	—
102	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	236	8	no	—
105	C₄	#14273727	1	2	yes	—
106	C₄	#14273749	1	4	yes	—
107	C₄	C₂²×Dic₁₁₈	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#14273727	1	1	yes	characteristic
110	C₂	C₂×#14273749	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal

#16837807

C₂

yes

maximal, complemented

#16837804

C₂

yes

maximal, complemented

#16837800

C₂

yes

maximal, complemented

#16837440

C₂

yes

maximal

C₂×C₄×C₈×C₅₉

C₂

yes

maximal, characteristic

#14273727

C₂²

yes

—

#14273719

C₂²

yes

—

#14273706

—

C₄×Dic₁₁₈

—

#14273686

C₂²

yes

—

#14273661

C₂²

yes

—

#14273659

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

Q₈

yes

complemented

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

C₂×Q₈

yes

—

C₈×C₅₉

C₂×Q₈

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄⋊Q₈

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₄⋊Q₈

yes

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

C₂²×Q₈

yes

—

Dic₅₉

—

#3160

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₉

C₂×#135

yes

—

—

—

—

—

—

C₂×C₄×C₈

D₅₉

yes

characteristic

C₅₉

#3160

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₉

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

118

—

#117

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

Dic₁₁₈

yes

complemented

C₂×C₈

C₂²×D₅₉

yes

—

C₂×C₈

C₂²×D₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₄²

—

118

—

C₄²

D₂₃₆

yes

—

C₄²

C₂²×D₅₉

yes

characteristic

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

C₂×D₂₃₆

yes

characteristic

Q₈

—

236

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

C₂×Dic₁₁₈

yes

—

C₂×C₄

C₂×D₂₃₆

yes

—

C₂×C₄

C₂×D₂₃₆

yes

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

C₂³×D₅₉

yes

characteristic

C₈

C₂×Dic₁₁₈

yes

—

C₈

C₂×Dic₁₁₈

yes

—

100

C₂²

—

101

C₂²

#14273727

yes

—

102

C₂²

C₂²×D₂₃₆

yes

characteristic

103

C₄

—

104

C₄

—

236

—

105

C₄

#14273727

yes

—

106

C₄

#14273749

yes

—

107

C₄

C₂²×Dic₁₁₈

yes

—

108

C₂

—

109

C₂

C₂×#14273727

yes

characteristic

110

C₂

C₂×#14273749

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×C₄×C₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₂×C₄×C₈	D₅₉
hypocenter	C₅₉	#3160
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂×C₄×C₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

hypercenter

C₂×C₄×C₈

D₅₉

hypocenter

C₅₉

#3160

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16837807	C₂	semidirect
3. #16837804	C₂	semidirect
4. #16837800	C₂	semidirect
22. C₂×C₈×C₅₉	Q₈	semidirect
62. C₅₉	#3160	semidirect
78. C₂×C₈	Dic₁₁₈	semidirect

H ≅ G/N

type

2. #16837807

C₂

semidirect

3. #16837804

C₂

semidirect

4. #16837800

C₂

semidirect

22. C₂×C₈×C₅₉

Q₈

semidirect

62. C₅₉

#3160

semidirect

78. C₂×C₈

Dic₁₁₈

semidirect

Subgroups — #21434610

Conjugacy classes of subgroups

Complements