Group #21435900 — subgroups

Subgroups — #21435900

Order 7552.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16838215	C₂	1	4	yes	maximal, complemented
2	#16837674	C₂	1	4	yes	maximal, complemented
3	#16837584	C₂	1	4	yes	maximal, complemented
4	#16837520	C₂	1	2	yes	maximal, complemented
5	C₅₉×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	#14273775	C₂²	1	4	yes	—
7	#14273759	—	2	8	no	—
8	#14273758	C₂²	1	8	yes	—
9	#14273750	—	2	4	no	—
10	#14273749	C₂²	1	2	yes	—
11	Q₁₆×D₅₉	—	2	4	no	—
12	#14273733	C₂²	1	4	yes	—
13	#14273663	C₂²	1	2	yes	—
14	#14273655	C₂²	1	4	yes	—
15	#14273646	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₅₉	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
18	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₉	C₂³	1	4	yes	—
20	#11745689	—	2	8	no	—
21	Q₈×D₅₉	—	2	8	no	—
22	#11745685	—	4	4	no	—
23	#11745685	C₂³	1	4	yes	—
24	Q₁₆×C₅₉	—	2	4	no	—
25	Q₁₆×C₅₉	—	2	4	no	—
26	C₂×C₈×C₅₉	C₂³	1	2	yes	—
27	C₄⋊C₄×C₅₉	—	4	2	no	—
28	C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	#11745672	—	2	8	no	—
30	#11745671	—	2	8	no	—
31	#11745664	C₂³	1	4	yes	—
32	Dic₂₃₆	D₄	1	2	yes	—
33	D₄₇₂	D₄	1	2	yes	—
34	#11745660	D₄	1	4	yes	—
35	#11745659	—	2	4	no	—
36	C₈×D₅₉	—	2	4	no	—
37	Q₈×C₅₉	—	2	8	no	—
38	C₂×C₄×C₅₉	—	2	1	no	—
39	C₂×C₄×C₅₉	—	2	4	no	—
40	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₅₉	—	2	4	no	—
42	D₂₃₆	—	4	4	no	—
43	D₂₃₆	C₂×D₄	1	4	yes	—
44	C₄×D₅₉	—	2	4	no	—
45	C₄×D₅₉	—	4	8	no	—
46	Dic₁₁₈	—	4	4	no	—
47	Dic₁₁₈	C₂×D₄	1	4	yes	—
48	C₈×C₅₉	C₂×D₄	1	2	yes	—
49	C₈×C₅₉	C₂×D₄	1	2	yes	—
50	C₅₉⋊C₈	—	2	4	no	—
51	C₂²×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₅₉	—	2	4	no	—
53	C₄×C₅₉	—	2	2	no	—
54	C₄×C₅₉	—	4	4	no	—
55	C₄×C₅₉	C₂²×D₄	1	2	yes	—
56	Dic₅₉	—	2	4	no	—
57	#3293	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	—	2	1	no	—
59	C₂×C₅₉	D₄²	1	1	yes	characteristic
60	D₅₉	—	4	4	no	—
61	#578	—	59	4	no	—
62	#494	D₅₉	1	1	yes	characteristic, complemented
63	#473	—	59	4	no	—
64	#456	—	59	4	no	—
65	#443	—	59	2	no	—
66	C₅₉	#3293	1	1	yes	characteristic, complemented, Sylow
67	2₋¹⁺⁴	—	59	4	no	—
68	#137	—	59	8	no	—
69	#137	—	118	8	no	—
70	#135	—	59	2	no	—
71	#135	—	118	4	no	—
72	C₂×Q₁₆	—	118	4	no	—
73	C₂×Q₁₆	C₂×D₅₉	1	4	yes	—
74	#131	—	59	4	no	—
75	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
76	#108	—	59	2	no	—
77	#104	—	59	4	no	—
78	#101	—	59	4	no	—
79	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
80	#41	—	59	4	no	—
81	#41	—	118	8	no	—
82	#41	—	236	4	no	—
83	C₂×Q₈	—	118	8	no	—
84	C₂×Q₈	C₂²×D₅₉	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	59	2	no	—
88	Q₁₆	—	118	8	no	—
89	SD₁₆	—	59	4	no	—
90	SD₁₆	—	118	8	no	—
91	D₈	—	59	2	no	—
92	M₁₆	—	59	4	no	—
93	M₁₆	—	118	4	no	—
94	C₂×C₈	—	118	4	no	—
95	C₂×C₈	C₂²×D₅₉	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₅₉	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	59	4	no	—
100	Q₈	—	236	4	no	—
101	D₄	—	59	4	no	—
102	D₄	—	118	4	no	—
103	D₄	—	236	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	118	4	no	—
107	C₂×C₄	—	236	8	no	—
108	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
109	C₈	—	118	4	no	—
110	C₈	D₄×D₅₉	1	2	yes	—
111	C₈	D₄×D₅₉	1	2	yes	—
112	C₂²	—	118	4	no	—
113	C₂²	#14273772	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	118	4	no	—
117	C₄	C₂×D₄×D₅₉	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	236	4	no	—
120	C₂	#16838037	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838215

C₂

yes

maximal, complemented

#16837674

C₂

yes

maximal, complemented

#16837584

C₂

yes

maximal, complemented

#16837520

C₂

yes

maximal, complemented

C₅₉×#494

C₂

yes

maximal, characteristic, complemented

#14273775

C₂²

yes

—

#14273759

—

#14273758

C₂²

yes

—

#14273750

—

#14273749

C₂²

yes

—

Q₁₆×D₅₉

—

#14273733

C₂²

yes

—

#14273663

C₂²

yes

—

#14273655

C₂²

yes

—

#14273646

C₂²

yes

—

C₂×Q₁₆×C₅₉

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

C₂³

yes

—

#11745689

—

Q₈×D₅₉

—

#11745685

—

#11745685

C₂³

yes

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

—

—

C₂³

yes

—

Dic₂₃₆

D₄

yes

—

D₄₇₂

D₄

yes

—

#11745660

D₄

yes

—

#11745659

—

C₈×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂²⋊D₅₉

—

D₂₃₆

—

D₂₃₆

C₂×D₄

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

C₂×D₄

yes

—

C₈×C₅₉

C₂×D₄

yes

—

C₈×C₅₉

C₂×D₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

H(5,𝔽₂)

yes

characteristic

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²×D₄

yes

—

Dic₅₉

—

#3293

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

D₄²

yes

characteristic

D₅₉

—

#578

—

#494

D₅₉

yes

characteristic, complemented

#473

—

#456

—

#443

—

C₅₉

#3293

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

#137

—

#137

—

118

—

#135

—

#135

—

118

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

C₂×D₅₉

yes

—

#131

—

C₄⋊Q₈

C₂×D₅₉

yes

—

#108

—

#104

—

#101

—

C₄×C₈

C₂×D₅₉

yes

characteristic

#41

—

#41

—

118

—

#41

—

236

—

C₂×Q₈

—

118

—

C₂×Q₈

C₂²×D₅₉

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

118

—

SD₁₆

—

SD₁₆

—

118

—

D₈

—

M₁₆

—

M₁₆

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₂²×D₅₉

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₅₉

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

236

—

101

D₄

—

102

D₄

—

118

—

103

D₄

—

236

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

118

—

107

C₂×C₄

—

236

—

108

C₂×C₄

C₂³×D₅₉

yes

characteristic

109

C₈

—

118

—

110

C₈

D₄×D₅₉

yes

—

111

C₈

D₄×D₅₉

yes

—

112

C₂²

—

118

—

113

C₂²

#14273772

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

118

—

117

C₄

C₂×D₄×D₅₉

yes

—

118

C₂

—

119

C₂

—

236

—

120

C₂

#16838037

yes

characteristic

	subgroup	G/H
center	C₂	#16838037
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#494	C₂
socle	C₂×C₅₉	D₄²
hypercenter	#494	D₅₉
hypocenter	C₅₉	#3293
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#16838037

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#494

C₂

socle

C₂×C₅₉

D₄²

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #16838215	C₂	semidirect
2. #16837674	C₂	semidirect
3. #16837584	C₂	semidirect
4. #16837520	C₂	semidirect
5. C₅₉×#494	C₂	semidirect
62. #494	D₅₉	semidirect
66. C₅₉	#3293	semidirect

H ≅ G/N

type

1. #16838215

C₂

semidirect

2. #16837674

C₂

semidirect

3. #16837584

C₂

semidirect

4. #16837520

C₂

semidirect

5. C₅₉×#494

C₂

semidirect

62. #494

D₅₉

semidirect

66. C₅₉

#3293

semidirect

Subgroups — #21435900

Conjugacy classes of subgroups

Complements