Group #21423486 — subgroups

Subgroups — #21423486

Order 7552.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16837807	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837608	C₂	1	2	yes	maximal
3	C₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic
4	#16837149	C₂	1	2	yes	maximal, complemented
5	C₅₉×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745660	—	2	2	no	—
7	C₂×C₈×D₅₉	—	2	1	no	—
8	#14273727	C₂²	1	2	yes	—
9	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
10	#14273661	—	2	1	no	—
11	#14273661	—	2	1	no	—
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	#14273652	—	2	2	no	—
14	(C₂×D₅₉)⋊C₈	—	2	1	no	—
15	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×D₂₃₆	D₄	1	1	yes	characteristic
18	C₂×C₄×D₅₉	—	2	1	no	—
19	C₂×Dic₁₁₈	—	2	2	no	—
20	C₂×C₈×C₅₉	—	2	2	no	—
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₄²×C₅₉	C₂³	1	1	yes	characteristic
24	(C₂×D₅₉)⋊C₄	—	2	1	no	—
25	C₄⋊Dic₅₉	—	2	2	no	—
26	C₄⋊Dic₅₉	—	2	2	no	—
27	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
28	C₂×C₅₉⋊C₈	—	2	1	no	—
29	#11745660	—	4	4	no	—
30	C₈×D₅₉	—	4	2	no	—
31	C₂²×D₅₉	—	2	1	no	—
32	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
35	C₂×Dic₅₉	—	2	1	no	—
36	C₂×Dic₅₉	—	4	2	no	—
37	D₂₃₆	—	2	1	no	—
38	D₂₃₆	D₈	1	2	yes	—
39	C₄×D₅₉	—	4	1	no	—
40	C₄×D₅₉	—	4	1	no	—
41	Dic₁₁₈	—	4	2	no	—
42	Dic₁₁₈	—	4	2	no	—
43	C₈×C₅₉	—	2	1	no	—
44	C₈×C₅₉	—	4	2	no	—
45	C₈×C₅₉	SD₁₆	1	2	yes	complemented
46	C₅₉⋊C₈	—	4	1	no	—
47	C₂²×C₅₉	#121	1	1	yes	characteristic
48	C₂×D₅₉	—	2	2	no	—
49	C₂×D₅₉	—	4	1	no	—
50	C₂×D₅₉	—	4	1	no	—
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
53	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₅₉	#135	1	1	yes	characteristic
55	C₄×C₅₉	#137	1	1	yes	characteristic
56	Dic₅₉	—	4	1	no	—
57	Dic₅₉	—	8	2	no	—
58	#1673	—	59	1	no	maximal, Sylow
59	C₂×C₅₉	#461	1	1	yes	characteristic
60	C₂×C₅₉	#466	1	1	yes	characteristic
61	C₂×C₅₉	#473	1	1	yes	characteristic
62	D₅₉	—	4	2	no	—
63	#500	—	59	1	no	—
64	#461	—	59	2	no	—
65	C₈×D₄	—	59	1	no	—
66	#335	D₅₉	1	1	yes	characteristic, complemented
67	#332	—	59	2	no	—
68	C₅₉	#1673	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₁₆	—	118	2	no	—
70	C₂²×C₈	—	118	1	no	—
71	C₄⋊Q₈	—	59	2	no	—
72	C₄×D₄	—	59	1	no	—
73	#107	—	118	1	no	—
74	#107	—	118	1	no	—
75	C₄⋊C₈	—	59	1	no	—
76	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
77	#103	—	118	2	no	—
78	C₂²⋊C₈	—	118	1	no	—
79	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
80	C₂×Q₈	—	118	2	no	—
81	C₂×D₄	—	59	1	no	—
82	C₂²×C₄	—	118	1	no	—
83	SD₁₆	—	236	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	118	1	no	—
86	C₂×C₈	—	236	2	no	—
87	C₂×C₈	D₂₃₆	1	1	yes	characteristic
88	C₂×C₈	D₂₃₆	1	1	yes	characteristic
89	C₄⋊C₄	—	59	1	no	—
90	C₄⋊C₄	—	118	2	no	—
91	C₄⋊C₄	—	118	2	no	—
92	C₂²⋊C₄	—	118	1	no	—
93	C₄²	C₂²×D₅₉	1	1	yes	characteristic
94	C₂³	—	118	1	no	—
95	Q₈	—	236	2	no	—
96	Q₈	—	236	2	no	—
97	D₄	—	59	2	no	—
98	D₄	—	118	1	no	—
99	C₂×C₄	—	118	1	no	—
100	C₂×C₄	—	236	2	no	—
101	C₂×C₄	—	236	1	no	—
102	C₂×C₄	—	236	1	no	—
103	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
104	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
105	C₂×C₄	#11745689	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	4	2	no	—
108	C₈	—	236	1	no	—
109	C₈	#11745660	1	2	yes	complemented
110	C₂²	—	118	2	no	—
111	C₂²	—	236	1	no	—
112	C₂²	—	236	1	no	—
113	C₂²	#14273702	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	472	2	no	—
116	C₄	—	236	1	no	—
117	C₄	D₈×D₅₉	1	1	yes	characteristic
118	C₄	C₂×#11745660	1	1	yes	characteristic
119	C₄	#14273750	1	1	yes	characteristic
120	C₄	#14273760	1	1	yes	characteristic
121	C₂	—	236	2	no	—
122	C₂	#16837608	1	1	yes	characteristic
123	C₂	#16837639	1	1	yes	characteristic
124	C₂	#16837678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837807

C₂

yes

maximal, characteristic, complemented

#16837608

C₂

yes

maximal

C₈×D₂₃₆

C₂

yes

maximal, characteristic

#16837149

C₂

yes

maximal, complemented

C₅₉×#335

C₂

yes

maximal, characteristic, complemented

C₂×#11745660

—

C₂×C₈×D₅₉

—

#14273727

C₂²

yes

—

C₄×D₂₃₆

C₂²

yes

characteristic

#14273661

—

#14273661

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273652

—

(C₂×D₅₉)⋊C₈

—

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

D₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

#11745660

—

C₈×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

D₈

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

SD₁₆

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#500

—

#461

—

C₈×D₄

—

#335

D₅₉

yes

characteristic, complemented

#332

—

C₅₉

#1673

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂²×C₈

—

118

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

118

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

—

#103

—

118

—

C₂²⋊C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×D₄

—

C₂²×C₄

—

118

—

SD₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

236

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

118

—

Q₈

—

236

—

Q₈

—

236

—

D₄

—

D₄

—

118

—

C₂×C₄

—

118

—

100

C₂×C₄

—

236

—

101

C₂×C₄

—

236

—

102

C₂×C₄

—

236

—

103

C₂×C₄

C₂×D₂₃₆

yes

characteristic

104

C₂×C₄

D₄×D₅₉

yes

characteristic

105

C₂×C₄

#11745689

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

236

—

109

C₈

#11745660

yes

complemented

110

C₂²

—

118

—

111

C₂²

—

236

—

112

C₂²

—

236

—

113

C₂²

#14273702

yes

characteristic

114

C₄

—

115

C₄

—

472

—

116

C₄

—

236

—

117

C₄

D₈×D₅₉

yes

characteristic

118

C₄

C₂×#11745660

yes

characteristic

119

C₄

#14273750

yes

characteristic

120

C₄

#14273760

yes

characteristic

121

C₂

—

236

—

122

C₂

#16837608

yes

characteristic

123

C₂

#16837639

yes

characteristic

124

C₂

#16837678

yes

characteristic

	subgroup	G/H
center	C₂²	#14273702
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#335	C₂
socle	C₂²×C₅₉	#121
hypercenter	#335	D₅₉
hypocenter	C₅₉	#1673
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14273702

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#335

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #16837807	C₂	semidirect
4. #16837149	C₂	semidirect
5. C₅₉×#335	C₂	semidirect
45. C₈×C₅₉	SD₁₆	semidirect
66. #335	D₅₉	semidirect
68. C₅₉	#1673	semidirect
109. C₈	#11745660	semidirect

H ≅ G/N

type

1. #16837807

C₂

semidirect

4. #16837149

C₂

semidirect

5. C₅₉×#335

C₂

semidirect

45. C₈×C₅₉

SD₁₆

semidirect

66. #335

D₅₉

semidirect

68. C₅₉

#1673

semidirect

109. C₈

#11745660

semidirect

Subgroups — #21423486

Conjugacy classes of subgroups

Complements