Group #21425988 — subgroups

Subgroups — #21425988

Order 7552.

142 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 59 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	#16837524	C₂	1	1	yes	maximal, characteristic
2	#16837509	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837225	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837218	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837212	C₂	1	1	yes	maximal, characteristic
6	#16837179	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#510	C₂	1	1	yes	maximal, characteristic
8	#14273734	—	2	1	no	—
9	Q₈×Dic₅₉	—	2	1	no	—
10	#14273667	C₂²	1	1	yes	characteristic
11	#14273663	C₂²	1	1	yes	characteristic
12	#14273659	C₂²	1	1	yes	characteristic
13	#14273654	—	2	1	no	—
14	#14273651	—	2	1	no	—
15	#14273637	C₂²	1	1	yes	characteristic
16	#135×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	Q₃₂×C₅₉	C₄	1	2	yes	complemented
19	SD₃₂×C₅₉	C₄	1	2	yes	complemented
20	M₃₂×C₅₉	C₂²	1	1	yes	characteristic
21	#41×C₅₉	—	2	1	no	—
22	C₂×Q₈×C₅₉	—	2	1	no	—
23	Q₁₆×C₅₉	—	2	1	no	—
24	Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
25	Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
26	SD₁₆×C₅₉	—	2	1	no	—
27	D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₅₉	—	2	1	no	—
30	C₄⋊Dic₅₉	—	4	1	no	—
31	C₄×Dic₅₉	—	4	1	no	—
32	C₄×Dic₅₉	D₄	1	1	yes	characteristic
33	#11745664	—	2	1	no	—
34	#11745664	—	4	1	no	—
35	C₂×C₅₉⋊C₈	—	4	1	no	—
36	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
37	C₁₆×C₅₉	C₂×C₄	1	2	yes	—
38	C₅₉⋊C₁₆	—	2	1	no	—
39	Q₈×C₅₉	—	2	1	no	—
40	Q₈×C₅₉	—	2	2	no	—
41	Q₈×C₅₉	—	4	1	no	—
42	D₄×C₅₉	—	2	1	no	—
43	D₄×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₅₉	—	2	1	no	—
48	C₂×Dic₅₉	—	4	1	no	—
49	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₈×C₅₉	#41	1	1	yes	characteristic
51	C₅₉⋊C₈	—	2	1	no	—
52	C₅₉⋊C₈	—	4	1	no	—
53	C₂²×C₅₉	—	4	1	no	—
54	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₄×C₅₉	—	4	1	no	—
56	C₄×C₅₉	—	4	2	no	—
57	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
58	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₅₉	—	4	1	no	—
60	Dic₅₉	—	8	1	no	—
61	#2180	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	—	2	1	no	—
63	C₂×C₅₉	—	8	1	no	—
64	C₂×C₅₉	#442	1	1	yes	characteristic
65	#510	D₅₉	1	1	yes	characteristic
66	#444	—	59	1	no	—
67	#441	—	59	1	no	—
68	#365	—	59	1	no	—
69	#362	—	59	1	no	—
70	#360	—	59	1	no	—
71	#347	—	59	1	no	—
72	C₅₉	#2180	1	1	yes	characteristic, complemented, Sylow
73	#135	C₂×D₅₉	1	1	yes	characteristic
74	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
75	#131	—	118	1	no	—
76	C₄×Q₈	—	118	1	no	—
77	Q₃₂	Dic₅₉	1	2	yes	complemented
78	SD₃₂	Dic₅₉	1	2	yes	complemented
79	M₃₂	—	59	1	no	—
80	M₃₂	C₂×D₅₉	1	1	yes	characteristic
81	#108	—	59	1	no	—
82	#106	—	59	1	no	—
83	#104	—	118	1	no	—
84	#103	—	118	1	no	—
85	#97	—	59	1	no	—
86	#41	—	2	1	no	—
87	C₂×Q₈	—	2	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	C₂×Dic₅₉	1	2	yes	—
90	Q₁₆	C₂×Dic₅₉	1	1	yes	characteristic
91	SD₁₆	—	2	1	no	—
92	D₈	C₂×Dic₅₉	1	1	yes	characteristic
93	M₁₆	—	118	1	no	—
94	M₁₆	—	236	1	no	—
95	C₂×C₈	—	59	1	no	—
96	C₂×C₈	—	236	1	no	—
97	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
98	C₄⋊C₄	—	118	1	no	—
99	C₄⋊C₄	—	236	1	no	—
100	C₄²	—	59	1	no	—
101	C₄²	—	236	1	no	—
102	C₁₆	—	118	1	no	—
103	C₁₆	C₂×Dic₅₉	1	2	yes	—
104	Q₈	—	2	1	no	—
105	Q₈	—	2	2	no	—
106	Q₈	—	4	1	no	—
107	D₄	—	2	1	no	—
108	D₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	118	1	no	—
112	C₂×C₄	—	236	1	no	—
113	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
114	C₈	—	118	1	no	—
115	C₈	—	236	1	no	—
116	C₈	#11745687	1	1	yes	characteristic
117	C₈	C₂²×Dic₅₉	1	1	yes	characteristic
118	C₂²	—	4	1	no	—
119	C₂²	#14273751	1	1	yes	characteristic
120	C₄	—	4	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	472	1	no	—
123	C₄	—	236	1	no	—
124	C₄	D₄×Dic₅₉	1	1	yes	characteristic
125	C₄	#14273751	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	8	1	no	—
128	C₂	#16837514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837524

C₂

yes

maximal, characteristic

#16837509

C₂

yes

maximal, characteristic, complemented

#16837225

C₂

yes

maximal, characteristic, complemented

#16837218

C₂

yes

maximal, characteristic, complemented

#16837212

C₂

yes

maximal, characteristic

#16837179

C₂

yes

maximal, characteristic, complemented

C₅₉×#510

C₂

yes

maximal, characteristic

#14273734

—

Q₈×Dic₅₉

—

#14273667

C₂²

yes

characteristic

#14273663

C₂²

yes

characteristic

#14273659

C₂²

yes

—

—

C₂²

yes

characteristic

#135×C₅₉

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

Q₃₂×C₅₉

C₄

yes

complemented

SD₃₂×C₅₉

C₄

yes

complemented

M₃₂×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

C₂×Q₈×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

C₂×C₄

yes

—

Q₁₆×C₅₉

C₂×C₄

yes

characteristic

SD₁₆×C₅₉

—

D₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

characteristic

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₁₆×C₅₉

C₂×C₄

yes

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

AGL(1,ℤ/8)

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2180

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#442

yes

characteristic

#510

D₅₉

yes

—

—

—

—

—

—

C₅₉

#2180

yes

characteristic, complemented, Sylow

#135

C₂×D₅₉

yes

characteristic

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

#131

—

118

—

C₄×Q₈

—

118

—

Q₃₂

Dic₅₉

yes

complemented

SD₃₂

Dic₅₉

yes

complemented

M₃₂

—

M₃₂

C₂×D₅₉

yes

—

—

—

118

—

#103

—

118

—

#97

—

#41

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×Dic₅₉

yes

—

Q₁₆

C₂×Dic₅₉

yes

characteristic

SD₁₆

—

D₈

C₂×Dic₅₉

yes

characteristic

M₁₆

—

118

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

100

C₄²

—

101

C₄²

—

236

—

102

C₁₆

—

118

—

103

C₁₆

C₂×Dic₅₉

yes

—

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

118

—

112

C₂×C₄

—

236

—

113

C₂×C₄

D₄×D₅₉

yes

characteristic

114

C₈

—

118

—

115

C₈

—

236

—

116

C₈

#11745687

yes

characteristic

117

C₈

C₂²×Dic₅₉

yes

characteristic

118

C₂²

—

119

C₂²

#14273751

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

472

—

123

C₄

—

236

—

124

C₄

D₄×Dic₅₉

yes

characteristic

125

C₄

#14273751

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#16837514

yes

characteristic

	subgroup	G/H
center	C₂	#16837514
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#510	C₂
socle	C₂×C₅₉	#442
hypercenter	#510	D₅₉
hypocenter	C₅₉	#2180
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

subgroup

G/H

center

C₂

#16837514

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#510

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

N	H ≅ G/N	type
2. #16837509	C₂	semidirect
3. #16837225	C₂	semidirect
4. #16837218	C₂	semidirect
6. #16837179	C₂	semidirect
18. Q₃₂×C₅₉	C₄	semidirect
19. SD₃₂×C₅₉	C₄	semidirect
72. C₅₉	#2180	semidirect
77. Q₃₂	Dic₅₉	semidirect
78. SD₃₂	Dic₅₉	semidirect

H ≅ G/N

type

2. #16837509

C₂

semidirect

3. #16837225

C₂

semidirect

4. #16837218

C₂

semidirect

6. #16837179

C₂

semidirect

18. Q₃₂×C₅₉

C₄

semidirect

19. SD₃₂×C₅₉

C₄

semidirect

72. C₅₉

#2180

semidirect

77. Q₃₂

Dic₅₉

semidirect

78. SD₃₂

Dic₅₉

semidirect

Subgroups — #21425988

Conjugacy classes of subgroups

Complements