Group #21426254 — subgroups

Subgroups — #21426254

Order 7552.

136 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16837804	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
3	#16837230	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837228	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837204	C₂	1	1	yes	maximal, characteristic
6	#16837204	C₂	1	1	yes	maximal, characteristic
7	C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
9	#14273719	—	2	1	no	—
10	C₄×D₂₃₆	—	2	1	no	—
11	#14273661	C₂²	1	1	yes	characteristic
12	#14273661	C₂²	1	1	yes	characteristic
13	#14273661	C₂²	1	1	yes	characteristic
14	#14273659	—	2	1	no	—
15	#14273649	—	2	1	no	—
16	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×D₂₃₆	—	2	1	no	—
20	C₂×C₄×D₅₉	—	4	1	no	—
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
23	C₄²×C₅₉	D₄	1	1	yes	characteristic
24	(C₂×D₅₉)⋊C₄	—	4	1	no	—
25	C₄⋊Dic₅₉	—	2	1	no	—
26	C₄⋊Dic₅₉	—	2	1	no	—
27	C₄⋊Dic₅₉	—	2	1	no	—
28	Dic₅₉⋊C₄	—	4	1	no	—
29	Dic₅₉⋊C₄	—	4	1	no	—
30	D₄₇₂	—	2	1	no	—
31	D₄₇₂	Q₈	1	2	yes	—
32	C₁₆×C₅₉	—	2	1	no	—
33	C₁₆×C₅₉	—	2	1	no	—
34	C₂²×D₅₉	—	4	1	no	—
35	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
36	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₅₉	—	4	1	no	—
39	C₂×Dic₅₉	—	4	1	no	—
40	C₂×Dic₅₉	—	4	1	no	—
41	D₂₃₆	—	2	2	no	—
42	D₂₃₆	—	4	1	no	—
43	C₄×D₅₉	—	8	2	no	—
44	C₈×C₅₉	—	2	1	no	—
45	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₅₉	#41	1	1	yes	characteristic
47	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
48	C₂×D₅₉	—	4	2	no	—
49	C₂×D₅₉	—	8	1	no	—
50	C₂×D₅₉	—	8	1	no	—
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	—	2	1	no	—
53	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₅₉	#137	1	1	yes	characteristic
55	Dic₅₉	—	8	1	no	—
56	Dic₅₉	—	8	1	no	—
57	Dic₅₉	—	8	1	no	—
58	#2228	—	59	1	no	maximal, Sylow
59	C₂×C₅₉	#474	1	1	yes	characteristic
60	C₂×C₅₉	#508	1	1	yes	characteristic
61	C₂×C₅₉	#509	1	1	yes	characteristic
62	D₅₉	—	8	2	no	—
63	#499	—	59	1	no	—
64	C₄×D₈	—	59	1	no	—
65	#367	—	59	1	no	—
66	#366	—	59	1	no	—
67	C₄⋊C₁₆	D₅₉	1	1	yes	characteristic, complemented
68	#357	—	59	1	no	—
69	#357	—	59	1	no	—
70	C₅₉	#2228	1	1	yes	characteristic, complemented, Sylow
71	C₂×D₈	—	59	1	no	—
72	#125	—	118	1	no	—
73	C₄×D₄	—	118	1	no	—
74	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
75	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
76	#107	—	59	1	no	—
77	#107	—	59	1	no	—
78	#107	—	59	1	no	—
79	#106	—	118	1	no	—
80	#102	—	118	1	no	—
81	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
82	C₂×D₄	—	118	1	no	—
83	C₂²×C₄	—	236	1	no	—
84	D₈	—	59	2	no	—
85	D₈	—	118	1	no	—
86	C₂×C₈	D₂₃₆	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
88	C₄⋊C₄	—	118	1	no	—
89	C₄⋊C₄	—	118	1	no	—
90	C₄⋊C₄	—	118	1	no	—
91	C₄⋊C₄	—	236	1	no	—
92	C₄⋊C₄	—	236	1	no	—
93	C₂²⋊C₄	—	236	1	no	—
94	C₄²	D₂₃₆	1	1	yes	characteristic
95	C₁₆	—	2	1	no	—
96	C₁₆	—	2	1	no	—
97	C₂³	—	236	1	no	—
98	D₄	—	118	2	no	—
99	D₄	—	236	1	no	—
100	C₂×C₄	—	472	2	no	—
101	C₂×C₄	—	236	1	no	—
102	C₂×C₄	—	236	1	no	—
103	C₂×C₄	—	236	1	no	—
104	C₂×C₄	D₄₇₂	1	1	yes	characteristic
105	C₂×C₄	D₄₇₂	1	1	yes	characteristic
106	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	#11745687	1	1	yes	characteristic
109	C₈	Q₈×D₅₉	1	1	yes	characteristic
110	C₂²	—	472	1	no	—
111	C₂²	—	472	1	no	—
112	C₂²	—	236	2	no	—
113	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	472	1	no	—
117	C₄	—	472	1	no	—
118	C₄	—	472	1	no	—
119	C₄	#14273707	1	1	yes	characteristic
120	C₄	#14273760	1	1	yes	characteristic
121	C₂	—	472	2	no	—
122	C₂	#16837683	1	1	yes	characteristic
123	C₂	#16837834	1	1	yes	characteristic
124	C₂	#16837840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837804

C₂

yes

maximal, characteristic, complemented

C₄×D₄₇₂

C₂

yes

maximal, characteristic

#16837230

C₂

yes

maximal, characteristic, complemented

#16837228

C₂

yes

maximal, characteristic, complemented

#16837204

C₂

yes

maximal, characteristic

#16837204

C₂

yes

maximal, characteristic

C₅₉×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₂²

yes

characteristic

#14273719

—

C₄×D₂₃₆

—

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273659

—

#14273649

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

D₄₇₂

—

D₄₇₂

Q₈

yes

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#137

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#499

—

C₄×D₈

—

#367

—

#366

—

C₄⋊C₁₆

D₅₉

yes

characteristic, complemented

#357

—

#357

—

C₅₉

#2228

yes

characteristic, complemented, Sylow

C₂×D₈

—

#125

—

118

—

C₄×D₄

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₂×C₁₆

C₂×D₅₉

yes

—

—

—

—

118

—

#102

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×D₄

—

118

—

C₂²×C₄

—

236

—

D₈

—

D₈

—

118

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₂²⋊C₄

—

236

—

C₄²

D₂₃₆

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

236

—

D₄

—

118

—

D₄

—

236

—

100

C₂×C₄

—

472

—

101

C₂×C₄

—

236

—

102

C₂×C₄

—

236

—

103

C₂×C₄

—

236

—

104

C₂×C₄

D₄₇₂

yes

characteristic

105

C₂×C₄

D₄₇₂

yes

characteristic

106

C₂×C₄

C₂×D₂₃₆

yes

characteristic

107

C₈

—

108

C₈

#11745687

yes

characteristic

109

C₈

Q₈×D₅₉

yes

characteristic

110

C₂²

—

472

—

111

C₂²

—

472

—

112

C₂²

—

236

—

113

C₂²

C₂×D₄₇₂

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

472

—

117

C₄

—

472

—

118

C₄

—

472

—

119

C₄

#14273707

yes

characteristic

120

C₄

#14273760

yes

characteristic

121

C₂

—

472

—

122

C₂

#16837683

yes

characteristic

123

C₂

#16837834

yes

characteristic

124

C₂

#16837840

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×C₄⋊C₁₆	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#2228
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×C₄⋊C₁₆

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₄⋊C₁₆

D₅₉

hypocenter

C₅₉

#2228

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
1. #16837804	C₂	semidirect
3. #16837230	C₂	semidirect
4. #16837228	C₂	semidirect
7. C₅₉×C₄⋊C₁₆	C₂	semidirect
67. C₄⋊C₁₆	D₅₉	semidirect
70. C₅₉	#2228	semidirect

H ≅ G/N

type

1. #16837804

C₂

semidirect

3. #16837230

C₂

semidirect

4. #16837228

C₂

semidirect

7. C₅₉×C₄⋊C₁₆

C₂

semidirect

67. C₄⋊C₁₆

D₅₉

semidirect

70. C₅₉

#2228

semidirect

Subgroups — #21426254

Conjugacy classes of subgroups

Complements