Group #21422876 — subgroups

Subgroups — #21422876

Order 7552.

150 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 81 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	#16837459	C₂	1	1	yes	maximal, characteristic
2	#16837455	C₂	1	1	yes	maximal, characteristic
3	#16837155	C₂	1	2	yes	maximal, complemented
4	#16837153	C₂	1	2	yes	maximal, complemented
5	C₅₉×#431	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
9	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
10	#14273638	—	2	1	no	—
11	C₄×C₅₉⋊C₈	—	2	1	no	—
12	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₉	C₄	1	2	yes	—
14	C₂²⋊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	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₉	D₄	1	1	yes	characteristic
24	C₄²×C₅₉	Q₈	1	1	yes	characteristic
25	C₄²×C₅₉	C₂³	1	1	yes	characteristic
26	C₂×C₅₉⋊C₈	—	2	2	no	—
27	C₂×C₅₉⋊C₈	—	2	2	no	—
28	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	—	2	2	no	—
30	C₂×C₄×C₅₉	—	2	2	no	—
31	C₂×C₄×C₅₉	—	4	1	no	—
32	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₅₉	—	2	2	no	—
39	C₈×C₅₉	—	2	2	no	—
40	C₅₉⋊C₈	—	4	2	no	—
41	C₅₉⋊C₈	—	4	2	no	—
42	C₂²×C₅₉	—	2	1	no	—
43	C₂²×C₅₉	—	2	1	no	—
44	C₂²×C₅₉	—	2	1	no	—
45	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₅₉	—	2	1	no	—
47	C₄×C₅₉	—	2	1	no	—
48	C₄×C₅₉	—	2	1	no	—
49	C₄×C₅₉	—	2	1	no	—
50	C₄×C₅₉	#131	1	2	yes	—
51	C₄×C₅₉	#135	1	2	yes	—
52	#1572	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	—	4	1	no	—
54	C₂×C₅₉	#424	1	1	yes	characteristic
55	C₂×C₅₉	#427	1	1	yes	characteristic
56	C₂×C₅₉	#430	1	1	yes	characteristic
57	#432	—	59	1	no	—
58	#431	—	59	1	no	—
59	#431	D₅₉	1	1	yes	characteristic
60	#335	—	59	2	no	—
61	#334	—	59	2	no	—
62	C₅₉	#1572	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
64	C₄⋊C₈	—	59	2	no	—
65	C₄⋊C₈	—	59	2	no	—
66	C₄⋊C₈	Dic₅₉	1	2	yes	—
67	C₂²⋊C₈	—	118	1	no	—
68	C₂²⋊C₈	—	118	1	no	—
69	C₂²⋊C₈	Dic₅₉	1	2	yes	—
70	#97	—	118	1	no	—
71	C₄×C₈	—	118	1	no	—
72	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
73	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
74	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
75	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
77	C₂×C₈	—	118	2	no	—
78	C₂×C₈	—	118	2	no	—
79	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
80	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
81	C₄²	Dic₁₁₈	1	1	yes	characteristic
82	C₄²	D₂₃₆	1	1	yes	characteristic
83	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
84	C₄²	C₂²×D₅₉	1	1	yes	characteristic
85	C₂³	C₄⋊Dic₅₉	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
92	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
93	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
95	C₈	—	2	2	no	—
96	C₈	—	2	2	no	—
97	C₈	—	236	2	no	—
98	C₈	—	236	2	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	#14273734	1	2	yes	—
108	C₄	#14273749	1	2	yes	—
109	C₂	—	4	1	no	—
110	C₂	#16837429	1	1	yes	characteristic
111	C₂	#16837440	1	1	yes	characteristic
112	C₂	#16837451	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837459

C₂

yes

maximal, characteristic

#16837455

C₂

yes

maximal, characteristic

#16837155

C₂

yes

maximal, complemented

#16837153

C₂

yes

maximal, complemented

C₅₉×#431

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₄

yes

—

C₂²⋊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

D₄

yes

characteristic

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

Q₈

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

#135

yes

—

#1572

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

D₅₉

yes

characteristic

#335

—

#334

—

C₅₉

#1572

yes

characteristic, complemented, Sylow

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₅₉

yes

—

C₂²⋊C₈

—

118

—

C₂²⋊C₈

—

118

—

C₂²⋊C₈

Dic₅₉

yes

—

#97

—

118

—

C₄×C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₄²

Dic₁₁₈

yes

characteristic

C₄²

D₂₃₆

yes

characteristic

C₄²

C₂×Dic₅₉

yes

characteristic

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

C₈

—

C₈

—

C₈

—

236

—

C₈

—

236

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#14273734

yes

—

108

C₄

#14273749

yes

—

109

C₂

—

110

C₂

#16837429

yes

characteristic

111

C₂

#16837440

yes

characteristic

112

C₂

#16837451

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#431	D₅₉
hypocenter	C₅₉	#1572
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#431

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16837155	C₂	semidirect
4. #16837153	C₂	semidirect
62. C₅₉	#1572	semidirect

H ≅ G/N

type

3. #16837155

C₂

semidirect

4. #16837153

C₂

semidirect

62. C₅₉

#1572

semidirect

Subgroups — #21422876

Conjugacy classes of subgroups

Complements