Group #21426308 — subgroups

Subgroups — #21426308

Order 7552.

134 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16837451	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837224	C₂	1	2	yes	maximal, complemented
3	#16837215	C₂	1	2	yes	maximal, complemented
4	#16837184	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#469	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	—	2	1	no	—
7	#14273667	C₂²	1	2	yes	—
8	#14273663	—	2	1	no	—
9	#14273663	C₂²	1	2	yes	—
10	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
12	#121×C₅₉	—	2	1	no	—
13	#106×C₅₉	C₂²	1	1	yes	characteristic
14	#102×C₅₉	—	2	1	no	—
15	C₂×D₄×C₅₉	—	2	1	no	—
16	C₂×D₄×C₅₉	—	4	1	no	—
17	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
18	D₈×C₅₉	—	2	2	no	—
19	M₁₆×C₅₉	—	2	2	no	—
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₅₉	—	2	1	no	—
23	C₂²⋊C₄×C₅₉	—	4	1	no	—
24	#11745664	—	2	2	no	—
25	#11745664	—	4	1	no	—
26	C₂×C₅₉⋊C₈	—	4	1	no	—
27	C₅₉⋊C₁₆	—	2	2	no	—
28	C₂³×C₅₉	—	4	1	no	—
29	C₂³×C₅₉	SD₁₆	1	1	yes	characteristic
30	D₄×C₅₉	—	4	1	no	—
31	D₄×C₅₉	—	4	1	no	—
32	D₄×C₅₉	—	8	2	no	—
33	C₂×C₄×C₅₉	—	2	1	no	—
34	C₂×C₄×C₅₉	—	2	1	no	—
35	C₂×C₄×C₅₉	—	4	1	no	—
36	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₅₉	—	2	1	no	—
39	C₈×C₅₉	#41	1	2	yes	—
40	C₅₉⋊C₈	—	4	2	no	—
41	C₂²×C₅₉	—	2	1	no	—
42	C₂²×C₅₉	—	4	1	no	—
43	C₂²×C₅₉	—	8	1	no	—
44	C₂²×C₅₉	—	8	1	no	—
45	C₂²×C₅₉	—	8	1	no	—
46	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
47	C₄×C₅₉	—	2	1	no	—
48	C₄×C₅₉	—	8	1	no	—
49	C₄×C₅₉	#123	1	1	yes	characteristic
50	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
51	#2238	—	59	1	no	maximal, Sylow
52	C₂×C₅₉	—	2	1	no	—
53	C₂×C₅₉	—	4	1	no	—
54	C₂×C₅₉	—	16	1	no	—
55	C₂×C₅₉	#481	1	1	yes	characteristic
56	#469	D₅₉	1	1	yes	characteristic
57	#430	—	59	1	no	—
58	#365	—	59	2	no	—
59	#361	—	59	2	no	—
60	#349	—	59	1	no	—
61	C₅₉	#2238	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
63	C₂×M₁₆	—	118	1	no	—
64	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
65	#121	—	2	1	no	—
66	M₃₂	—	59	2	no	—
67	#108	—	59	2	no	—
68	#108	—	118	1	no	—
69	#106	C₂×D₅₉	1	1	yes	characteristic
70	#102	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	C₂×D₄	—	4	1	no	—
73	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
74	D₈	—	2	2	no	—
75	M₁₆	—	2	2	no	—
76	M₁₆	—	118	2	no	—
77	M₁₆	—	236	1	no	—
78	C₂×C₈	—	236	1	no	—
79	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
80	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
81	C₄⋊C₄	—	2	1	no	—
82	C₂²⋊C₄	—	4	1	no	—
83	C₁₆	—	118	2	no	—
84	C₂³	—	4	1	no	—
85	C₂³	#11745670	1	1	yes	characteristic
86	D₄	—	4	1	no	—
87	D₄	—	4	1	no	—
88	D₄	—	8	2	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	#11745670	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	236	2	no	—
96	C₈	#11745687	1	2	yes	—
97	C₂²	—	2	1	no	—
98	C₂²	—	4	1	no	—
99	C₂²	—	8	1	no	—
100	C₂²	—	8	1	no	—
101	C₂²	—	8	1	no	—
102	C₂²	C₂×#11745670	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	8	1	no	—
105	C₄	#14273708	1	1	yes	characteristic
106	C₄	#14273752	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	4	1	no	—
109	C₂	—	16	1	no	—
110	C₂	#16837718	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837451

C₂

yes

maximal, characteristic, complemented

#16837224

C₂

yes

maximal, complemented

#16837215

C₂

yes

maximal, complemented

#16837184

C₂

yes

maximal, characteristic, complemented

C₅₉×#469

C₂

yes

maximal, characteristic

C₂×#11745664

—

#14273667

C₂²

yes

—

#14273663

—

#14273663

C₂²

yes

—

C₂×D₈×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#121×C₅₉

—

#106×C₅₉

C₂²

yes

characteristic

#102×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

D₈×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂³×C₅₉

SD₁₆

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

#41

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#481

yes

characteristic

#469

D₅₉

yes

—

—

—

—

C₅₉

#2238

yes

characteristic, complemented, Sylow

C₂×D₈

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

118

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

#121

—

M₃₂

—

#108

—

#108

—

118

—

#106

C₂×D₅₉

yes

characteristic

#102

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

D₈

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

236

—

C₂×C₈

—

236

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₁₆

—

118

—

C₂³

—

C₂³

#11745670

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11745670

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

C₈

—

236

—

C₈

#11745687

yes

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×#11745670

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

#14273708

yes

characteristic

106

C₄

#14273752

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

110

C₂

#16837718

yes

characteristic

	subgroup	G/H
center	C₂	#16837718
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#469	C₂
socle	C₂×C₅₉	#481
hypercenter	#469	D₅₉
hypocenter	C₅₉	#2238
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #469

subgroup

G/H

center

C₂

#16837718

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#469

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #469

N	H ≅ G/N	type
1. #16837451	C₂	semidirect
2. #16837224	C₂	semidirect
3. #16837215	C₂	semidirect
4. #16837184	C₂	semidirect
61. C₅₉	#2238	semidirect

H ≅ G/N

type

1. #16837451

C₂

semidirect

2. #16837224

C₂

semidirect

3. #16837215

C₂

semidirect

4. #16837184

C₂

semidirect

61. C₅₉

#2238

semidirect

Subgroups — #21426308

Conjugacy classes of subgroups

Complements