Group #21425811 — subgroups

Subgroups — #21425811

Order 7552.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×D₄₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273667	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837202	C₂	1	4	yes	maximal, complemented
4	C₅₉×#427	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₃₆	—	2	1	no	—
6	C₂×D₄₇₂	—	2	1	no	—
7	C₂×D₄₇₂	—	2	2	no	—
8	C₂×D₄₇₂	C₄	1	4	yes	complemented
9	C₂×D₄₇₂	C₂²	1	2	yes	—
10	#14273667	—	2	2	no	—
11	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	complemented
12	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	#117×C₅₉	—	2	1	no	—
14	#107×C₅₉	C₂²	1	2	yes	complemented
15	#106×C₅₉	—	2	1	no	—
16	C₂³×D₅₉	—	4	1	no	—
17	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
18	C₂×D₂₃₆	—	2	4	no	—
19	C₂×D₂₃₆	—	2	2	no	—
20	C₂×D₂₃₆	—	4	1	no	—
21	C₂×D₂₃₆	—	4	2	no	—
22	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₉	D₄	1	2	yes	—
25	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₉	—	2	2	no	—
27	C₂²⋊C₄×C₅₉	—	4	1	no	—
28	C₄²×C₅₉	—	4	1	no	—
29	D₄₇₂	—	2	4	no	—
30	D₄₇₂	—	2	2	no	—
31	D₄₇₂	C₂×C₄	1	4	yes	—
32	C₅₉⋊C₁₆	—	2	2	no	—
33	C₂³×C₅₉	D₈	1	1	yes	characteristic
34	C₂²×D₅₉	—	4	4	no	—
35	C₂²×D₅₉	—	4	2	no	—
36	C₂²×D₅₉	—	8	4	no	—
37	C₂×C₄×C₅₉	—	4	2	no	—
38	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
40	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	D₂₃₆	—	2	4	no	—
43	D₂₃₆	—	4	4	no	—
44	D₂₃₆	—	4	2	no	—
45	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₅₉	—	2	2	no	—
49	C₂²×C₅₉	#102	1	2	yes	—
50	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
51	C₂×D₅₉	—	4	4	no	—
52	C₂×D₅₉	—	8	8	no	—
53	C₂×D₅₉	—	8	4	no	—
54	C₄×C₅₉	—	8	2	no	—
55	C₄×C₅₉	#102	1	2	yes	—
56	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
58	#2140	—	59	1	no	maximal, Sylow
59	C₂×C₅₉	—	2	2	no	—
60	C₂×C₅₉	C₂×#102	1	1	yes	characteristic
61	C₂×C₅₉	#509	1	2	yes	—
62	D₅₉	—	8	4	no	—
63	C₂²×D₈	—	59	1	no	—
64	C₂×M₃₂	—	59	1	no	—
65	#427	D₅₉	1	1	yes	characteristic, complemented
66	#357	—	59	4	no	—
67	C₅₉	#2140	1	1	yes	characteristic, complemented, Sylow
68	C₂²×D₄	—	118	1	no	—
69	C₂×D₈	—	59	4	no	—
70	C₂×D₈	—	59	2	no	—
71	C₂×D₈	—	118	2	no	—
72	C₂×D₈	—	118	1	no	—
73	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
74	#117	—	2	1	no	—
75	M₃₂	—	118	2	no	—
76	C₂×C₁₆	—	59	2	no	—
77	#107	C₂×D₅₉	1	2	yes	complemented
78	#106	—	2	1	no	—
79	C₂⁴	—	236	1	no	—
80	C₂×D₄	—	118	4	no	—
81	C₂×D₄	—	118	2	no	—
82	C₂×D₄	—	236	1	no	—
83	C₂×D₄	—	236	2	no	—
84	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
85	D₈	—	59	4	no	—
86	D₈	—	118	4	no	—
87	D₈	—	118	2	no	—
88	C₂×C₈	C₄×D₅₉	1	2	yes	—
89	C₂×C₈	D₂₃₆	1	2	yes	—
90	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
91	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	4	1	no	—
94	C₄²	—	4	1	no	—
95	C₁₆	—	118	2	no	—
96	C₂³	—	472	4	no	—
97	C₂³	—	236	4	no	—
98	C₂³	—	236	2	no	—
99	C₂³	#11745669	1	1	yes	characteristic
100	D₄	—	118	4	no	—
101	D₄	—	236	4	no	—
102	D₄	—	236	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
105	C₂×C₄	#11745669	1	1	yes	characteristic
106	C₂×C₄	#11745671	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
108	C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
109	C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
110	C₈	C₂×D₂₃₆	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	472	8	no	—
113	C₂²	—	472	4	no	—
114	C₂²	—	236	4	no	—
115	C₂²	#14273647	1	2	yes	—
116	C₂²	C₂×#11745669	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	#14273647	1	2	yes	—
119	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
120	C₄	C₂×#11745671	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	472	4	no	—
123	C₂	C₂×#14273647	1	1	yes	characteristic
124	C₂	#16837837	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₇₂

C₂

yes

maximal, characteristic

C₂×#14273667

C₂

yes

maximal, characteristic, complemented

#16837202

C₂

yes

maximal, complemented

C₅₉×#427

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

C₄

yes

complemented

C₂×D₄₇₂

C₂²

yes

—

#14273667

—

C₂×C₅₉⋊C₁₆

C₂²

yes

complemented

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#117×C₅₉

—

#107×C₅₉

C₂²

yes

complemented

#106×C₅₉

—

C₂³×D₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

D₄₇₂

—

D₄₇₂

—

D₄₇₂

C₂×C₄

yes

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

D₈

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

#102

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#102

yes

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

#2140

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×#102

yes

characteristic

C₂×C₅₉

#509

yes

—

D₅₉

—

C₂²×D₈

—

C₂×M₃₂

—

#427

D₅₉

yes

characteristic, complemented

#357

—

C₅₉

#2140

yes

characteristic, complemented, Sylow

C₂²×D₄

—

118

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

118

—

C₂×D₈

—

118

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

M₃₂

—

118

—

C₂×C₁₆

—

#107

C₂×D₅₉

yes

complemented

#106

—

C₂⁴

—

236

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

D₈

—

D₈

—

118

—

D₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

D₂₃₆

yes

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

118

—

C₂³

—

472

—

C₂³

—

236

—

C₂³

—

236

—

C₂³

#11745669

yes

characteristic

100

D₄

—

118

—

101

D₄

—

236

—

102

D₄

—

236

—

103

C₂×C₄

—

104

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

105

C₂×C₄

#11745669

yes

characteristic

106

C₂×C₄

#11745671

yes

—

107

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

108

C₈

(C₂×D₅₉)⋊C₄

yes

—

109

C₈

C₂×C₄×D₅₉

yes

characteristic

110

C₈

C₂×D₂₃₆

yes

characteristic

111

C₂²

—

112

C₂²

—

472

—

113

C₂²

—

472

—

114

C₂²

—

236

—

115

C₂²

#14273647

yes

—

116

C₂²

C₂×#11745669

yes

characteristic

117

C₄

—

118

C₄

#14273647

yes

—

119

C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

120

C₄

C₂×#11745671

yes

characteristic

121

C₂

—

122

C₂

—

472

—

123

C₂

C₂×#14273647

yes

characteristic

124

C₂

#16837837

yes

—

	subgroup	G/H
center	C₂²	C₂×#11745669
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#427	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#427	D₅₉
hypocenter	C₅₉	#2140
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#11745669

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#427

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
2. C₂×#14273667	C₂	semidirect
3. #16837202	C₂	semidirect
4. C₅₉×#427	C₂	semidirect
8. C₂×D₄₇₂	C₄	semidirect
11. C₂×C₅₉⋊C₁₆	C₂²	semidirect
14. #107×C₅₉	C₂²	semidirect
65. #427	D₅₉	semidirect
67. C₅₉	#2140	semidirect
77. #107	C₂×D₅₉	semidirect

H ≅ G/N

type

2. C₂×#14273667

C₂

semidirect

3. #16837202

C₂

semidirect

4. C₅₉×#427

C₂

semidirect

8. C₂×D₄₇₂

C₄

semidirect

11. C₂×C₅₉⋊C₁₆

C₂²

semidirect

14. #107×C₅₉

C₂²

semidirect

65. #427

D₅₉

semidirect

67. C₅₉

#2140

semidirect

77. #107

C₂×D₅₉

semidirect

Subgroups — #21425811

Conjugacy classes of subgroups

Complements