Group #21426381 — subgroups

Subgroups — #21426381

Order 7552.

120 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16837806	C₂	1	1	yes	maximal, characteristic
2	#16837803	C₂	1	1	yes	maximal, characteristic
3	#16837229	C₂	1	1	yes	maximal, characteristic
4	#16837228	C₂	1	1	yes	maximal, characteristic
5	#16837227	C₂	1	1	yes	maximal, characteristic
6	#16837177	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#367	C₂	1	1	yes	maximal, characteristic
8	#14273726	—	2	1	no	—
9	#14273718	—	2	1	no	—
10	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14273661	C₂²	1	1	yes	characteristic
12	#14273661	C₂²	1	1	yes	characteristic
13	#14273660	—	2	1	no	—
14	#14273658	—	2	1	no	—
15	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	#107×C₅₉	C₂²	1	1	yes	characteristic
18	#107×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₁₈	—	4	1	no	—
20	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₅₉	—	2	1	no	—
22	C₄⋊C₄×C₅₉	—	2	1	no	—
23	C₄⋊Dic₅₉	—	2	1	no	—
24	C₄⋊Dic₅₉	—	2	1	no	—
25	C₄⋊Dic₅₉	—	4	1	no	—
26	Dic₅₉⋊C₄	—	4	1	no	—
27	Dic₅₉⋊C₄	—	4	1	no	—
28	C₄×Dic₅₉	D₄	1	1	yes	characteristic
29	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
30	C₁₆×C₅₉	Q₈	1	2	yes	complemented
31	C₅₉⋊C₁₆	Q₈	1	2	yes	complemented
32	C₂×C₄×C₅₉	—	4	1	no	—
33	C₂×C₄×C₅₉	—	4	1	no	—
34	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₅₉	—	4	1	no	—
36	C₂×Dic₅₉	—	4	1	no	—
37	C₂×Dic₅₉	D₈	1	1	yes	characteristic
38	C₂×Dic₅₉	D₈	1	1	yes	characteristic
39	Dic₁₁₈	—	8	2	no	—
40	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₅₉⋊C₈	Q₁₆	1	2	yes	—
43	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
44	C₄×C₅₉	—	8	1	no	—
45	C₄×C₅₉	—	8	1	no	—
46	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₅₉	—	2	1	no	—
49	Dic₅₉	—	2	1	no	—
50	Dic₅₉	—	8	1	no	—
51	Dic₅₉	—	8	1	no	—
52	#2255	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	#500	1	1	yes	characteristic
54	C₂×C₅₉	#509	1	1	yes	characteristic
55	C₂×C₅₉	#510	1	1	yes	characteristic
56	#500	—	59	1	no	—
57	#499	—	59	1	no	—
58	#367	—	59	1	no	—
59	#367	D₅₉	1	1	yes	characteristic
60	#366	—	59	1	no	—
61	#366	—	59	1	no	—
62	#346	—	59	1	no	—
63	C₅₉	#2255	1	1	yes	characteristic, complemented, Sylow
64	C₄⋊Q₈	—	118	1	no	—
65	#125	—	118	1	no	—
66	C₂×C₁₆	—	59	1	no	—
67	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
68	#107	—	59	1	no	—
69	#107	—	59	1	no	—
70	#107	—	118	1	no	—
71	#107	C₂×D₅₉	1	1	yes	characteristic
72	#107	C₂×D₅₉	1	1	yes	characteristic
73	#106	—	118	1	no	—
74	C₄×C₈	—	59	1	no	—
75	C₂×Q₈	—	236	1	no	—
76	C₂×C₈	—	59	1	no	—
77	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
78	C₄⋊C₄	—	2	1	no	—
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	118	1	no	—
81	C₄⋊C₄	—	118	1	no	—
82	C₄⋊C₄	—	236	1	no	—
83	C₄⋊C₄	—	236	1	no	—
84	C₄⋊C₄	—	236	1	no	—
85	C₄²	—	59	1	no	—
86	C₁₆	—	59	2	no	—
87	C₁₆	Dic₁₁₈	1	2	yes	complemented
88	Q₈	—	472	2	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	59	1	no	—
92	C₂×C₄	—	59	1	no	—
93	C₂×C₄	—	236	1	no	—
94	C₂×C₄	—	236	1	no	—
95	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
96	C₈	—	59	2	no	—
97	C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
98	C₈	Q₈×D₅₉	1	1	yes	characteristic
99	C₂²	D₈×D₅₉	1	1	yes	characteristic
100	C₄	—	8	1	no	—
101	C₄	—	8	1	no	—
102	C₄	—	118	1	no	—
103	C₄	—	118	1	no	—
104	C₄	—	472	1	no	—
105	C₄	—	472	1	no	—
106	C₄	#14273726	1	1	yes	characteristic
107	C₄	Q₁₆×D₅₉	1	1	yes	characteristic
108	C₂	#16837806	1	1	yes	characteristic
109	C₂	#16837839	1	1	yes	characteristic
110	C₂	#16837841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837806

C₂

yes

maximal, characteristic

#16837803

C₂

yes

maximal, characteristic

#16837229

C₂

yes

maximal, characteristic

#16837228

C₂

yes

maximal, characteristic

#16837227

C₂

yes

maximal, characteristic

#16837177

C₂

yes

maximal, characteristic

C₅₉×#367

C₂

yes

maximal, characteristic

#14273726

—

#14273718

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273660

—

#14273658

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₁₆×C₅₉

Q₈

yes

complemented

C₅₉⋊C₁₆

Q₈

yes

complemented

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

Dic₁₁₈

—

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₅₉⋊C₈

Q₁₆

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

D₅₉

yes

—

—

—

C₅₉

#2255

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

118

—

#125

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

—

—

—

118

—

#107

C₂×D₅₉

yes

characteristic

#107

C₂×D₅₉

yes

characteristic

#106

—

118

—

C₄×C₈

—

C₂×Q₈

—

236

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄²

—

C₁₆

—

C₁₆

Dic₁₁₈

yes

complemented

Q₈

—

472

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

D₄×D₅₉

yes

characteristic

C₈

—

C₈

C₂×Dic₁₁₈

yes

characteristic

C₈

Q₈×D₅₉

yes

characteristic

C₂²

D₈×D₅₉

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

118

—

103

C₄

—

118

—

104

C₄

—

472

—

105

C₄

—

472

—

106

C₄

#14273726

yes

characteristic

107

C₄

Q₁₆×D₅₉

yes

characteristic

108

C₂

#16837806

yes

characteristic

109

C₂

#16837839

yes

characteristic

110

C₂

#16837841

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#367	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#367	D₅₉
hypocenter	C₅₉	#2255
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#367

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
30. C₁₆×C₅₉	Q₈	semidirect
31. C₅₉⋊C₁₆	Q₈	semidirect
63. C₅₉	#2255	semidirect
87. C₁₆	Dic₁₁₈	semidirect

H ≅ G/N

type

30. C₁₆×C₅₉

Q₈

semidirect

31. C₅₉⋊C₁₆

Q₈

semidirect

63. C₅₉

#2255

semidirect

87. C₁₆

Dic₁₁₈

semidirect

Subgroups — #21426381

Conjugacy classes of subgroups

Complements