Group #28475860 — subgroups

Subgroups — #28475860

Order 15104.

238 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16837223	C₂	1	2	yes	maximal
2	C₅₉×#1844	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273667	—	2	2	no	—
4	#16837223	—	2	2	no	—
5	#16837223	C₄	1	4	yes	complemented
6	C₂×#106×C₅₉	—	2	2	no	—
7	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₅₉×#384	C₄	1	2	yes	—
9	#14273667	—	2	4	no	—
10	#14273667	—	4	2	no	—
11	C₂×C₅₉⋊C₁₆	—	4	2	no	—
12	C₂²×C₈×C₅₉	D₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
14	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
15	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
16	#106×C₅₉	—	4	2	no	—
17	#106×C₅₉	—	4	2	no	—
18	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
20	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	—	2	1	no	—
22	C₂²×C₄×C₅₉	—	4	2	no	—
23	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₉	D₈	1	2	yes	—
27	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
29	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
30	C₄⋊C₄×C₅₉	—	4	4	no	—
31	C₄⋊C₄×C₅₉	—	8	4	no	—
32	C₄²×C₅₉	—	2	1	no	—
33	C₄²×C₅₉	C₄²	1	1	yes	characteristic
34	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
35	C₅₉⋊C₁₆	—	4	4	no	—
36	C₂³×C₅₉	#100	1	1	yes	characteristic
37	C₂×C₄×C₅₉	—	2	4	no	—
38	C₂×C₄×C₅₉	—	2	1	no	—
39	C₂×C₄×C₅₉	—	2	1	no	—
40	C₂×C₄×C₅₉	—	8	4	no	—
41	C₂×C₄×C₅₉	—	8	2	no	—
42	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
43	C₂×C₄×C₅₉	#101	1	1	yes	characteristic
44	C₂×C₄×C₅₉	#102	1	2	yes	—
45	C₂×C₄×C₅₉	#103	1	2	yes	—
46	C₈×C₅₉	#102	1	2	yes	—
47	C₈×C₅₉	#103	1	2	yes	—
48	C₈×C₅₉	#106	1	2	yes	—
49	C₈×C₅₉	#107	1	2	yes	—
50	#7268	—	59	1	no	maximal, Sylow
51	C₂²×C₅₉	—	2	2	no	—
52	C₂²×C₅₉	#331	1	1	yes	characteristic
53	C₂²×C₅₉	#333	1	1	yes	characteristic
54	C₂²×C₅₉	#343	1	1	yes	characteristic
55	C₄×C₅₉	—	2	2	no	—
56	C₄×C₅₉	—	2	2	no	—
57	C₄×C₅₉	—	16	2	no	—
58	C₄×C₅₉	#332	1	2	yes	—
59	C₄×C₅₉	#340	1	2	yes	—
60	C₂×#364	—	59	2	no	—
61	#1844	D₅₉	1	1	yes	characteristic
62	C₂×C₅₉	—	2	1	no	—
63	C₂×C₅₉	—	2	1	no	—
64	C₂×C₅₉	#1296	1	1	yes	characteristic
65	C₂×C₅₉	#1359	1	2	yes	—
66	C₂×M₃₂	—	118	2	no	—
67	C₂×#106	—	2	2	no	—
68	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
69	#384	Dic₅₉	1	2	yes	—
70	#364	—	59	4	no	—
71	#364	—	118	2	no	—
72	C₅₉	#7268	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
74	C₂×C₄⋊C₄	—	2	2	no	—
75	C₂×C₄⋊C₄	—	4	2	no	—
76	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
77	M₃₂	—	118	4	no	—
78	M₃₂	—	236	2	no	—
79	C₂×C₁₆	—	236	2	no	—
80	#106	—	4	2	no	—
81	#106	—	4	2	no	—
82	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
83	C₄×C₈	C₄×D₅₉	1	2	yes	—
84	C₄×C₈	D₂₃₆	1	1	yes	characteristic
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
88	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
89	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
90	C₂×C₈	#11745669	1	2	yes	—
91	C₂×C₈	#11745670	1	2	yes	—
92	C₂×C₈	#11745671	1	2	yes	—
93	C₂×C₈	#11745672	1	2	yes	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄²	—	2	1	no	—
97	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
98	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
99	C₁₆	—	236	4	no	—
100	C₂³	#14273644	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
107	C₂×C₄	#14273646	1	1	yes	characteristic
108	C₂×C₄	#14273648	1	2	yes	—
109	C₂×C₄	#14273651	1	2	yes	—
110	C₈	#14273647	1	2	yes	—
111	C₈	#14273650	1	2	yes	—
112	C₈	#14273658	1	2	yes	—
113	C₈	#14273660	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	#16837146	1	1	yes	characteristic
116	C₂²	#16837151	1	1	yes	characteristic
117	C₂²	#16837171	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#16837148	1	2	yes	—
122	C₄	#16837164	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#21421979	1	1	yes	characteristic
126	C₂	#21422137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837223

C₂

yes

maximal

C₅₉×#1844

C₂

yes

maximal, characteristic

C₂×#14273667

—

#16837223

—

#16837223

C₄

yes

complemented

C₂×#106×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#384

C₄

yes

—

#14273667

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

#106×C₅₉

—

#106×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₄²

yes

characteristic

C₄²×C₅₉

C₄⋊C₄

yes

characteristic

C₅₉⋊C₁₆

—

C₂³×C₅₉

#100

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

#103

yes

—

C₈×C₅₉

#102

yes

—

C₈×C₅₉

#103

yes

—

C₈×C₅₉

#106

yes

—

C₈×C₅₉

#107

yes

—

#7268

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#332

yes

—

C₄×C₅₉

#340

yes

—

C₂×#364

—

#1844

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1296

yes

characteristic

C₂×C₅₉

#1359

yes

—

C₂×M₃₂

—

118

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#384

Dic₅₉

yes

—

#364

—

#364

—

118

—

C₅₉

#7268

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₅₉

yes

characteristic

M₃₂

—

118

—

M₃₂

—

236

—

C₂×C₁₆

—

236

—

#106

—

#106

—

C₄×C₈

Dic₁₁₈

yes

characteristic

C₄×C₈

C₄×D₅₉

yes

—

C₄×C₈

D₂₃₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₅₉

yes

characteristic

C₂×C₈

Dic₅₉⋊C₄

yes

—

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

C₂×C₈

#11745669

yes

—

C₂×C₈

#11745670

yes

—

C₂×C₈

#11745671

yes

—

C₂×C₈

#11745672

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₄×Dic₅₉

yes

characteristic

C₄²

C₄⋊Dic₅₉

yes

characteristic

C₁₆

—

236

—

100

C₂³

#14273644

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₅₉⋊#95

yes

characteristic

107

C₂×C₄

#14273646

yes

characteristic

108

C₂×C₄

#14273648

yes

—

109

C₂×C₄

#14273651

yes

—

110

C₈

#14273647

yes

—

111

C₈

#14273650

yes

—

112

C₈

#14273658

yes

—

113

C₈

#14273660

yes

—

114

C₂²

—

115

C₂²

#16837146

yes

characteristic

116

C₂²

#16837151

yes

characteristic

117

C₂²

#16837171

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#16837148

yes

—

122

C₄

#16837164

yes

—

123

C₂

—

124

C₂

—

125

C₂

#21421979

yes

characteristic

126

C₂

#21422137

yes

—

	subgroup	G/H
center	C₂²	#16837146
commutator	C₄²×C₅₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₅₉×#1844	C₂
socle	C₂²×C₅₉	#331
prime radical	C₂³×C₅₉	#100
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	#1844	D₅₉
hypocenter	C₅₉	#7268
LCS	G ▷ C₄²×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

subgroup

G/H

center

C₂²

#16837146

commutator

C₄²×C₅₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₉

Fitting

C₅₉×#1844

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄×C₈×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

N	H ≅ G/N	type
5. #16837223	C₄	semidirect
72. C₅₉	#7268	semidirect

H ≅ G/N

type

5. #16837223

C₄

semidirect

72. C₅₉

#7268

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16837223	2
C₅₉×#1844	2
#7268	59

subgroup

index

C₂×#16837223

C₅₉×#1844

#7268

Subgroups — #28475860

Conjugacy classes of subgroups

Complements

Maximal subgroups