Group #21425443 — subgroups

Subgroups — #21425443

Order 7552.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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	#16837445	C₂	1	2	yes	maximal
2	#16837267	C₂	1	1	yes	maximal, characteristic
3	#16837171	C₂	1	1	yes	maximal, characteristic
4	#16837161	C₂	1	2	yes	maximal
5	C₅₉×#544	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
9	#14273660	—	2	2	no	—
10	#14273658	—	2	2	no	—
11	C₅₉⋊#95	—	2	2	no	—
12	C₄⋊Q₈×C₅₉	—	2	2	no	—
13	#125×C₅₉	—	2	2	no	—
14	C₂²⋊Q₈×C₅₉	—	2	2	no	—
15	#117×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	—	2	2	no	—
18	C₂²×C₄×C₅₉	D₄	1	2	yes	—
19	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₉	—	2	1	no	—
21	C₂²×Dic₅₉	—	2	1	no	—
22	C₄⋊C₄×C₅₉	—	2	2	no	—
23	C₄⋊C₄×C₅₉	—	2	4	no	—
24	C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
27	C₂²⋊C₄×C₅₉	—	2	2	no	—
28	C₄²×C₅₉	—	2	2	no	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	C₄×Dic₅₉	—	2	1	no	—
31	C₄×Dic₅₉	Q₈	1	2	yes	complemented
32	#11745664	—	2	2	no	—
33	#11745664	Q₈	1	4	yes	complemented
34	C₂×C₅₉⋊C₈	—	2	2	no	—
35	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₅₉	—	8	2	no	—
37	C₂×C₄×C₅₉	—	2	2	no	—
38	C₂×C₄×C₅₉	—	2	2	no	—
39	C₂×C₄×C₅₉	—	2	2	no	—
40	C₂×C₄×C₅₉	—	4	2	no	—
41	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
45	C₂×Dic₅₉	—	2	2	no	—
46	C₂×Dic₅₉	—	4	1	no	—
47	C₂×Dic₅₉	—	4	1	no	—
48	C₂×Dic₅₉	—	4	2	no	—
49	C₂×Dic₅₉	—	4	1	no	—
50	C₅₉⋊C₈	—	2	4	no	—
51	C₂²×C₅₉	—	2	1	no	—
52	C₂²×C₅₉	—	2	1	no	—
53	C₂²×C₅₉	#120	1	1	yes	characteristic
54	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₅₉	—	4	2	no	—
57	C₄×C₅₉	—	4	2	no	—
58	C₄×C₅₉	—	4	2	no	—
59	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
61	Dic₅₉	—	4	2	no	—
62	Dic₅₉	—	8	1	no	—
63	#2061	—	59	1	no	maximal, Sylow
64	C₂×C₅₉	—	2	1	no	—
65	C₂×C₅₉	—	2	1	no	—
66	C₂×C₅₉	#395	1	1	yes	characteristic
67	C₂×C₅₉	#454	1	1	yes	characteristic
68	C₂×C₅₉	#455	1	1	yes	characteristic
69	#544	D₅₉	1	1	yes	characteristic
70	#428	—	59	2	no	—
71	#382	—	59	1	no	—
72	#343	—	59	1	no	—
73	#339	—	59	2	no	—
74	C₅₉	#2061	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	59	2	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	#125	—	2	2	no	—
78	C₂²⋊Q₈	—	2	2	no	—
79	#117	C₂×D₅₉	1	2	yes	—
80	C₂×C₄⋊C₄	—	59	1	no	—
81	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
82	C₂×C₄²	—	59	1	no	—
83	#107	—	118	2	no	—
84	#106	—	118	2	no	—
85	#95	—	118	2	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂²×C₄	—	118	1	no	—
88	C₂²×C₄	—	118	1	no	—
89	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
90	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
91	M₁₆	—	59	4	no	—
92	M₁₆	—	118	2	no	—
93	C₂×C₈	—	118	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	118	2	no	—
98	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
99	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
100	C₂²⋊C₄	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	59	2	no	—
103	C₄²	—	118	1	no	—
104	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
105	Q₈	—	8	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	118	2	no	—
111	C₂×C₄	—	236	2	no	—
112	C₂×C₄	—	236	1	no	—
113	C₂×C₄	—	236	1	no	—
114	C₂×C₄	—	236	1	no	—
115	C₂×C₄	Q₈×D₅₉	1	2	yes	—
116	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
117	C₂×C₄	#11745689	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
119	C₈	—	118	4	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#14273696	1	1	yes	characteristic
123	C₂²	#14273703	1	1	yes	characteristic
124	C₂²	#14273725	1	1	yes	characteristic
125	C₄	—	4	2	no	—
126	C₄	—	4	2	no	—
127	C₄	—	4	2	no	—
128	C₄	—	472	1	no	—
129	C₄	—	236	2	no	—
130	C₄	#14273703	1	2	yes	—
131	C₄	#14273725	1	2	yes	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#16837321	1	1	yes	characteristic
135	C₂	#16837575	1	1	yes	characteristic
136	C₂	#16837580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837445

C₂

yes

maximal

#16837267

C₂

yes

maximal, characteristic

#16837171

C₂

yes

maximal, characteristic

#16837161

C₂

yes

maximal

C₅₉×#544

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273660

—

#14273658

—

C₅₉⋊#95

—

C₄⋊Q₈×C₅₉

—

#125×C₅₉

—

C₂²⋊Q₈×C₅₉

—

#117×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

Q₈

yes

complemented

#11745664

—

#11745664

Q₈

yes

complemented

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#120

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

C₄⋊Q₈

yes

—

Dic₅₉

—

Dic₅₉

—

#2061

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

—

—

—

—

C₅₉

#2061

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄²

—

#107

—

118

—

#106

—

118

—

#95

—

118

—

C₂×Q₈

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

118

—

C₂×C₈

—

118

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

C₂²⋊D₅₉

yes

—

C₄⋊C₄

C₂²⋊D₅₉

yes

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

118

—

104

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

118

—

111

C₂×C₄

—

236

—

112

C₂×C₄

—

236

—

113

C₂×C₄

—

236

—

114

C₂×C₄

—

236

—

115

C₂×C₄

Q₈×D₅₉

yes

—

116

C₂×C₄

Q₈×D₅₉

yes

characteristic

117

C₂×C₄

#11745689

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

119

C₈

—

118

—

120

C₂²

—

121

C₂²

—

122

C₂²

#14273696

yes

characteristic

123

C₂²

#14273703

yes

characteristic

124

C₂²

#14273725

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

472

—

129

C₄

—

236

—

130

C₄

#14273703

yes

—

131

C₄

#14273725

yes

—

132

C₂

—

133

C₂

—

134

C₂

#16837321

yes

characteristic

135

C₂

#16837575

yes

characteristic

136

C₂

#16837580

yes

characteristic

	subgroup	G/H
center	C₂²	#14273696
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#544	C₂
socle	C₂²×C₅₉	#120
hypercenter	#544	D₅₉
hypocenter	C₅₉	#2061
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#14273696

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#544

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
31. C₄×Dic₅₉	Q₈	semidirect
33. #11745664	Q₈	semidirect
74. C₅₉	#2061	semidirect

H ≅ G/N

type

31. C₄×Dic₅₉

Q₈

semidirect

33. #11745664

Q₈

semidirect

74. C₅₉

#2061

semidirect

Subgroups — #21425443

Conjugacy classes of subgroups

Complements