Group #21425884 — subgroups

Subgroups — #21425884

Order 7552.

156 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16837440	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273661	C₂	1	1	yes	maximal, characteristic
3	#16837230	C₂	1	2	yes	maximal, complemented
4	#16837228	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#14273686	—	2	1	no	—
7	C₂×C₄⋊Dic₅₉	—	2	1	no	—
8	#14273661	—	2	1	no	—
9	#14273661	C₂²	1	2	yes	—
10	#14273661	C₂²	1	1	yes	characteristic
11	#14273661	C₂²	1	1	yes	characteristic
12	#14273659	—	2	1	no	—
13	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	M₃₂×C₅₉	C₄	1	4	yes	complemented
15	C₂×C₁₆×C₅₉	C₂²	1	2	yes	—
16	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₉	—	4	1	no	—
18	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	Q₈	1	2	yes	—
21	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²⋊Dic₅₉	—	4	1	no	—
23	C₄⋊Dic₅₉	—	2	1	no	—
24	C₄⋊Dic₅₉	—	2	1	no	—
25	C₄⋊Dic₅₉	—	2	2	no	—
26	C₄⋊Dic₅₉	—	4	1	no	—
27	C₄×Dic₅₉	—	4	1	no	—
28	C₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
29	C₂³×C₅₉	D₈	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
32	C₂×C₄×C₅₉	Q₁₆	1	2	yes	—
33	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₅₉	—	4	1	no	—
35	C₂×Dic₅₉	—	4	1	no	—
36	C₂×Dic₅₉	—	4	2	no	—
37	C₂×Dic₅₉	—	8	2	no	—
38	C₈×C₅₉	C₄⋊C₄	1	2	yes	—
39	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₅₉	—	2	1	no	—
42	C₂²×C₅₉	—	2	1	no	—
43	C₂²×C₅₉	#107	1	2	yes	—
44	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
45	C₄×C₅₉	#107	1	2	yes	—
46	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₅₉	—	8	1	no	—
49	Dic₅₉	—	8	2	no	—
50	Dic₅₉	—	8	1	no	—
51	#2159	—	59	1	no	maximal, Sylow
52	C₂×C₅₉	—	2	2	no	—
53	C₂×C₅₉	C₂×#107	1	1	yes	characteristic
54	C₂×C₅₉	#509	1	1	yes	characteristic
55	C₂×C₅₉	#510	1	1	yes	characteristic
56	C₂×M₃₂	D₅₉	1	1	yes	characteristic
57	#427	—	59	1	no	—
58	C₂×#107	—	59	1	no	—
59	#367	—	59	2	no	—
60	#366	—	59	2	no	—
61	C₅₉	#2159	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
63	#117	—	118	1	no	—
64	C₂×C₄⋊C₄	—	118	1	no	—
65	M₃₂	Dic₅₉	1	4	yes	complemented
66	C₂×C₁₆	C₂×D₅₉	1	2	yes	—
67	#107	—	59	1	no	—
68	#107	—	59	2	no	—
69	#107	—	59	1	no	—
70	#107	—	118	1	no	—
71	#106	—	118	1	no	—
72	C₂²×C₄	—	236	1	no	—
73	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
74	C₂×C₈	Dic₁₁₈	1	2	yes	—
75	C₂×C₈	D₂₃₆	1	1	yes	characteristic
76	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
77	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
78	C₄⋊C₄	—	118	1	no	—
79	C₄⋊C₄	—	118	1	no	—
80	C₄⋊C₄	—	118	2	no	—
81	C₄⋊C₄	—	236	1	no	—
82	C₂²⋊C₄	—	236	1	no	—
83	C₄²	—	236	1	no	—
84	C₁₆	C₂×Dic₅₉	1	4	yes	complemented
85	C₂³	D₄₇₂	1	1	yes	characteristic
86	C₂×C₄	—	472	2	no	—
87	C₂×C₄	—	236	1	no	—
88	C₂×C₄	—	236	1	no	—
89	C₂×C₄	—	236	2	no	—
90	C₂×C₄	D₄₇₂	1	1	yes	characteristic
91	C₂×C₄	Dic₂₃₆	1	2	yes	—
92	C₂×C₄	C₄⋊Dic₅₉	1	2	yes	—
93	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
94	C₈	C₄⋊Dic₅₉	1	2	yes	—
95	C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
96	C₈	C₂²×Dic₅₉	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#14273661	1	2	yes	—
100	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
101	C₄	—	472	2	no	—
102	C₄	—	472	1	no	—
103	C₄	—	472	1	no	—
104	C₄	#14273661	1	2	yes	—
105	C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
106	C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
107	C₂	—	2	2	no	—
108	C₂	C₂×#14273661	1	1	yes	characteristic
109	C₂	#16837840	1	1	yes	characteristic
110	C₂	#16837845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837440

C₂

yes

maximal, characteristic

C₂×#14273661

C₂

yes

maximal, characteristic

#16837230

C₂

yes

maximal, complemented

#16837228

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₉

C₂

yes

maximal, characteristic

#14273686

—

C₂×C₄⋊Dic₅₉

—

#14273661

—

#14273661

C₂²

yes

—

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273659

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

M₃₂×C₅₉

C₄

yes

complemented

C₂×C₁₆×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

Q₈

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₁₆×C₅₉

C₂×C₄

yes

complemented

C₂³×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

Q₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

C₄⋊C₄

yes

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#107

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

#107

yes

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2159

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×M₃₂

D₅₉

yes

—

—

—

—

C₅₉

#2159

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

118

—

C₂×C₄⋊C₄

—

118

—

M₃₂

Dic₅₉

yes

complemented

C₂×C₁₆

C₂×D₅₉

yes

—

—

—

—

—

118

—

#106

—

118

—

C₂²×C₄

—

236

—

C₂²×C₄

D₂₃₆

yes

characteristic

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₂²⋊C₄

—

236

—

C₄²

—

236

—

C₁₆

C₂×Dic₅₉

yes

complemented

C₂³

D₄₇₂

yes

characteristic

C₂×C₄

—

472

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

D₄₇₂

yes

characteristic

C₂×C₄

Dic₂₃₆

yes

—

C₂×C₄

C₄⋊Dic₅₉

yes

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₈

C₄⋊Dic₅₉

yes

—

C₈

C₂×Dic₁₁₈

yes

characteristic

C₈

C₂²×Dic₅₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

#14273661

yes

—

100

C₂²

C₂×D₄₇₂

yes

characteristic

101

C₄

—

472

—

102

C₄

—

472

—

103

C₄

—

472

—

104

C₄

#14273661

yes

—

105

C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

106

C₄

C₂×Dic₂₃₆

yes

characteristic

107

C₂

—

108

C₂

C₂×#14273661

yes

characteristic

109

C₂

#16837840

yes

characteristic

110

C₂

#16837845

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₇₂
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₂×M₃₂	D₅₉
hypocenter	C₅₉	#2159
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₄₇₂

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂×M₃₂×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₂×M₃₂

D₅₉

hypocenter

C₅₉

#2159

LCS

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16837230	C₂	semidirect
4. #16837228	C₂	semidirect
14. M₃₂×C₅₉	C₄	semidirect
28. C₁₆×C₅₉	C₂×C₄	semidirect
61. C₅₉	#2159	semidirect
65. M₃₂	Dic₅₉	semidirect
84. C₁₆	C₂×Dic₅₉	semidirect

H ≅ G/N

type

3. #16837230

C₂

semidirect

4. #16837228

C₂

semidirect

14. M₃₂×C₅₉

C₄

semidirect

28. C₁₆×C₅₉

C₂×C₄

semidirect

61. C₅₉

#2159

semidirect

65. M₃₂

Dic₅₉

semidirect

84. C₁₆

C₂×Dic₅₉

semidirect

Subgroups — #21425884

Conjugacy classes of subgroups

Complements