Group #21423742 — subgroups

Subgroups — #21423742

Order 7552.

164 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 61 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
2	#16837800	C₂	1	1	yes	maximal, characteristic
3	#16837753	C₂	1	1	yes	maximal, characteristic
4	#16837753	C₂	1	1	yes	maximal, characteristic
5	#16837753	C₂	1	1	yes	maximal, characteristic
6	#16837753	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#322	C₂	1	1	yes	maximal, characteristic
8	#14273727	C₂²	1	1	yes	characteristic
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273727	C₂²	1	1	yes	characteristic
11	#14273727	C₂²	1	1	yes	characteristic
12	#14273661	—	2	2	no	—
13	#14273659	—	2	2	no	—
14	#14273652	—	2	2	no	—
15	#14273652	—	2	2	no	—
16	#14273652	—	2	2	no	—
17	#14273652	—	2	2	no	—
18	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₁₈	—	2	1	no	—
22	C₂×Dic₁₁₈	—	2	1	no	—
23	C₂×Dic₁₁₈	—	2	1	no	—
24	C₂×Dic₁₁₈	—	2	1	no	—
25	C₂×C₈×C₅₉	D₄	1	2	yes	—
26	C₂×C₈×C₅₉	D₄	1	2	yes	—
27	C₂×C₈×C₅₉	D₄	1	2	yes	—
28	C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₅₉	—	2	1	no	—
30	C₄⋊Dic₅₉	—	2	1	no	—
31	C₄⋊Dic₅₉	—	2	1	no	—
32	C₄⋊Dic₅₉	—	2	1	no	—
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₅₉	—	4	1	no	—
41	C₂×Dic₅₉	—	4	1	no	—
42	C₂×Dic₅₉	—	4	1	no	—
43	C₂×Dic₅₉	—	4	1	no	—
44	Dic₁₁₈	—	4	2	no	—
45	Dic₁₁₈	—	4	2	no	—
46	Dic₁₁₈	—	4	2	no	—
47	Dic₁₁₈	—	4	2	no	—
48	C₈×C₅₉	—	2	2	no	—
49	C₈×C₅₉	—	2	2	no	—
50	C₈×C₅₉	—	2	2	no	—
51	C₂²×C₅₉	#127	1	1	yes	characteristic
52	C₄×C₅₉	#135	1	2	yes	—
53	C₄×C₅₉	#137	1	2	yes	—
54	C₄×C₅₉	#137	1	2	yes	—
55	Dic₅₉	—	8	1	no	—
56	Dic₅₉	—	8	1	no	—
57	Dic₅₉	—	8	1	no	—
58	Dic₅₉	—	8	1	no	—
59	#1718	—	59	1	no	maximal, Sylow
60	C₂×C₅₉	#495	1	1	yes	characteristic
61	C₂×C₅₉	#497	1	1	yes	characteristic
62	C₂×C₅₉	#497	1	1	yes	characteristic
63	#500	—	59	1	no	—
64	#498	—	59	1	no	—
65	#487	—	59	1	no	—
66	#487	—	59	1	no	—
67	#487	—	59	1	no	—
68	#487	—	59	1	no	—
69	#322	D₅₉	1	1	yes	characteristic
70	C₅₉	#1718	1	1	yes	characteristic, complemented, Sylow
71	C₄⋊Q₈	—	59	1	no	—
72	C₄⋊Q₈	—	59	1	no	—
73	C₄⋊Q₈	—	59	1	no	—
74	C₄⋊Q₈	—	59	1	no	—
75	#107	—	118	2	no	—
76	#106	—	118	2	no	—
77	#103	—	118	2	no	—
78	#103	—	118	2	no	—
79	#103	—	118	2	no	—
80	#103	—	118	2	no	—
81	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
82	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
83	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
84	C₂×Q₈	—	118	1	no	—
85	C₂×Q₈	—	118	1	no	—
86	C₂×Q₈	—	118	1	no	—
87	C₂×Q₈	—	118	1	no	—
88	C₂×C₈	D₂₃₆	1	2	yes	—
89	C₂×C₈	D₂₃₆	1	2	yes	—
90	C₂×C₈	D₂₃₆	1	2	yes	—
91	C₄⋊C₄	—	118	1	no	—
92	C₄⋊C₄	—	118	1	no	—
93	C₄⋊C₄	—	118	1	no	—
94	C₄⋊C₄	—	118	1	no	—
95	C₄⋊C₄	—	118	1	no	—
96	C₄⋊C₄	—	118	1	no	—
97	C₄⋊C₄	—	118	1	no	—
98	C₄⋊C₄	—	118	1	no	—
99	C₄²	C₂²×D₅₉	1	1	yes	characteristic
100	Q₈	—	236	2	no	—
101	Q₈	—	236	2	no	—
102	Q₈	—	236	2	no	—
103	Q₈	—	236	2	no	—
104	C₂×C₄	—	236	1	no	—
105	C₂×C₄	—	236	1	no	—
106	C₂×C₄	—	236	1	no	—
107	C₂×C₄	—	236	1	no	—
108	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
109	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
110	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	2	2	no	—
114	C₂²	#14273724	1	1	yes	characteristic
115	C₄	—	472	1	no	—
116	C₄	—	472	1	no	—
117	C₄	—	472	1	no	—
118	C₄	—	472	1	no	—
119	C₄	#14273749	1	2	yes	—
120	C₄	#14273760	1	2	yes	—
121	C₄	#14273760	1	2	yes	—
122	C₂	#16837786	1	1	yes	characteristic
123	C₂	#16837796	1	1	yes	characteristic
124	C₂	#16837796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837807

C₂

yes

maximal, characteristic

#16837800

C₂

yes

maximal, characteristic

#16837753

C₂

yes

maximal, characteristic

#16837753

C₂

yes

maximal, characteristic

#16837753

C₂

yes

maximal, characteristic

#16837753

C₂

yes

maximal, characteristic

C₅₉×#322

C₂

yes

maximal, characteristic

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

—

—

—

—

—

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

#127

yes

characteristic

C₄×C₅₉

#135

yes

—

C₄×C₅₉

#137

yes

—

C₄×C₅₉

#137

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

—

—

—

D₅₉

yes

characteristic

C₅₉

#1718

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#107

—

118

—

#106

—

118

—

#103

—

118

—

#103

—

118

—

#103

—

118

—

#103

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂×C₈

D₂₃₆

yes

—

C₂×C₈

D₂₃₆

yes

—

C₂×C₈

D₂₃₆

yes

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

100

Q₈

—

236

—

101

Q₈

—

236

—

102

Q₈

—

236

—

103

Q₈

—

236

—

104

C₂×C₄

—

236

—

105

C₂×C₄

—

236

—

106

C₂×C₄

—

236

—

107

C₂×C₄

—

236

—

108

C₂×C₄

C₂×D₂₃₆

yes

characteristic

109

C₂×C₄

C₂×D₂₃₆

yes

characteristic

110

C₂×C₄

C₂×D₂₃₆

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

#14273724

yes

characteristic

115

C₄

—

472

—

116

C₄

—

472

—

117

C₄

—

472

—

118

C₄

—

472

—

119

C₄

#14273749

yes

—

120

C₄

#14273760

yes

—

121

C₄

#14273760

yes

—

122

C₂

#16837786

yes

characteristic

123

C₂

#16837796

yes

characteristic

124

C₂

#16837796

yes

characteristic

	subgroup	G/H
center	C₂²	#14273724
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#322	C₂
socle	C₂²×C₅₉	#127
hypercenter	#322	D₅₉
hypocenter	C₅₉	#1718
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#14273724

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#322

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
70. C₅₉	#1718	semidirect

H ≅ G/N

type

70. C₅₉

#1718

semidirect

Subgroups — #21423742

Conjugacy classes of subgroups

Complements