Group #21425244 — subgroups

Subgroups — #21425244

Order 7552.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂×#14273650	C₂	1	2	yes	maximal
2	#16837308	C₂	1	1	yes	maximal, characteristic
3	#16837164	C₂	1	2	yes	maximal
4	#16837157	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
9	#14273650	—	2	4	no	—
10	C₅₉⋊#95	—	2	2	no	—
11	#125×C₅₉	—	2	4	no	—
12	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	—	2	2	no	—
16	C₂²×C₄×C₅₉	D₄	1	2	yes	—
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₉	—	2	1	no	—
19	C₂²×Dic₅₉	—	2	1	no	—
20	C₂×Dic₁₁₈	—	2	2	no	—
21	C₂×Dic₁₁₈	—	2	2	no	—
22	C₂×Dic₁₁₈	—	2	1	no	—
23	C₂×Dic₁₁₈	Q₈	1	2	yes	—
24	C₄⋊C₄×C₅₉	—	2	8	no	—
25	C₄⋊C₄×C₅₉	—	2	2	no	—
26	C₄⋊C₄×C₅₉	D₄	1	4	yes	—
27	C₄²×C₅₉	—	2	2	no	—
28	C₄⋊Dic₅₉	—	2	2	no	—
29	C₂×C₅₉⋊C₈	—	2	2	no	—
30	C₂×C₅₉⋊C₈	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	2	2	no	—
32	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	—	2	2	no	—
34	C₂×C₄×C₅₉	—	2	4	no	—
35	C₂×C₄×C₅₉	—	2	2	no	—
36	C₂×C₄×C₅₉	—	2	2	no	—
37	C₂×C₄×C₅₉	—	4	2	no	—
38	C₂×C₄×C₅₉	—	4	2	no	—
39	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	#41	1	2	yes	—
42	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
43	C₂×Dic₅₉	—	2	2	no	—
44	C₂×Dic₅₉	—	4	1	no	—
45	C₂×Dic₅₉	—	4	1	no	—
46	C₂×Dic₅₉	—	4	1	no	—
47	C₂×Dic₅₉	—	4	2	no	—
48	Dic₁₁₈	—	2	4	no	—
49	Dic₁₁₈	—	4	2	no	—
50	C₅₉⋊C₈	—	4	2	no	—
51	C₂²×C₅₉	#120	1	1	yes	characteristic
52	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₅₉	#124	1	1	yes	characteristic
54	C₂²×C₅₉	#135	1	2	yes	—
55	C₂²×C₅₉	#137	1	2	yes	—
56	C₄×C₅₉	—	4	2	no	—
57	C₄×C₅₉	—	4	4	no	—
58	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₅₉	#124	1	2	yes	—
60	Dic₅₉	—	4	2	no	—
61	Dic₅₉	—	8	1	no	—
62	#2026	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	#393	1	1	yes	characteristic
64	C₂×C₅₉	#452	1	2	yes	—
65	C₂×C₅₉	#479	1	2	yes	—
66	C₂×C₅₉	#488	1	1	yes	characteristic
67	C₂×C₅₉	#491	1	1	yes	characteristic
68	C₂×#125	D₅₉	1	1	yes	characteristic
69	C₂×#103	—	59	2	no	—
70	#391	—	59	1	no	—
71	#340	—	59	2	no	—
72	#336	—	59	1	no	—
73	C₅₉	#2026	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₈	—	59	1	no	—
75	C₂²×C₈	—	59	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	59	1	no	—
79	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
80	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
81	#103	—	118	4	no	—
82	#95	—	118	2	no	—
83	C₂×Q₈	—	59	2	no	—
84	C₂×Q₈	—	118	1	no	—
85	C₂×Q₈	—	118	2	no	—
86	C₂×Q₈	—	118	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	118	1	no	—
89	C₂²×C₄	—	118	1	no	—
90	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
91	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
92	C₂×C₈	—	118	2	no	—
93	C₂×C₈	—	118	2	no	—
94	C₂×C₈	—	118	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	118	2	no	—
98	C₄⋊C₄	C₂²⋊D₅₉	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
101	Q₈	—	118	4	no	—
102	Q₈	—	236	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	118	2	no	—
110	C₂×C₄	—	236	1	no	—
111	C₂×C₄	—	236	2	no	—
112	C₂×C₄	—	236	1	no	—
113	C₂×C₄	—	236	1	no	—
114	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
115	C₂×C₄	#11745689	1	2	yes	—
116	C₂×C₄	#11745689	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
118	C₈	—	236	2	no	—
119	C₂²	#14273696	1	1	yes	characteristic
120	C₂²	#14273703	1	1	yes	characteristic
121	C₂²	#14273716	1	1	yes	characteristic
122	C₂²	#14273747	1	2	yes	—
123	C₂²	#14273757	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	472	1	no	—
127	C₄	—	236	2	no	—
128	C₄	#14273703	1	2	yes	—
129	C₄	#14273716	1	2	yes	—
130	C₂	#16837313	1	1	yes	characteristic
131	C₂	#16837564	1	2	yes	—
132	C₂	#16837706	1	2	yes	—
133	C₂	#16837759	1	1	yes	characteristic
134	C₂	#16837772	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273650

C₂

yes

maximal

#16837308

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal

#16837157

C₂

yes

maximal, characteristic

C₂×#125×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

#14273650

—

C₅₉⋊#95

—

#125×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

Q₈

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×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

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

#120

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

#124

yes

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#479

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

—

—

—

—

C₅₉

#2026

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#103

—

118

—

#95

—

118

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

C₂²⋊D₅₉

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

101

Q₈

—

118

—

102

Q₈

—

236

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

118

—

110

C₂×C₄

—

236

—

111

C₂×C₄

—

236

—

112

C₂×C₄

—

236

—

113

C₂×C₄

—

236

—

114

C₂×C₄

Q₈×D₅₉

yes

characteristic

115

C₂×C₄

#11745689

yes

—

116

C₂×C₄

#11745689

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

118

C₈

—

236

—

119

C₂²

#14273696

yes

characteristic

120

C₂²

#14273703

yes

characteristic

121

C₂²

#14273716

yes

characteristic

122

C₂²

#14273747

yes

—

123

C₂²

#14273757

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

472

—

127

C₄

—

236

—

128

C₄

#14273703

yes

—

129

C₄

#14273716

yes

—

130

C₂

#16837313

yes

characteristic

131

C₂

#16837564

yes

—

132

C₂

#16837706

yes

—

133

C₂

#16837759

yes

characteristic

134

C₂

#16837772

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#125×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#125	D₅₉
hypocenter	C₅₉	#2026
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#125×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
73. C₅₉	#2026	semidirect

H ≅ G/N

type

73. C₅₉

#2026

semidirect

Subgroups — #21425244

Conjugacy classes of subgroups

Complements