Group #28505039 — subgroups

Subgroups — #28505039

Order 15104.

322 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 171 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×C₄⋊(C₅₉⋊C₁₆)	C₂	1	2	yes	maximal
2	#21422174	C₂	1	4	yes	maximal
3	C₅₉×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅₉⋊C₁₆	C₂²	1	4	yes	—
5	C₄⋊(C₅₉⋊C₁₆)	—	2	4	no	—
6	C₂×#97×C₅₉	C₄	1	2	yes	—
7	C₂×#97×C₅₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₅₉×#376	C₄	1	1	yes	characteristic
11	C₂×C₅₉⋊C₁₆	—	2	4	no	—
12	C₂×C₅₉⋊C₁₆	—	2	8	no	—
13	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
18	#97×C₅₉	—	2	4	no	—
19	#97×C₅₉	—	2	2	no	—
20	C₄×C₈×C₅₉	—	2	4	no	—
21	C₄×C₈×C₅₉	D₄	1	2	yes	—
22	C₄×C₈×C₅₉	Q₈	1	2	yes	—
23	#95×C₅₉	C₈	1	4	yes	—
24	C₂²×C₄×C₅₉	C₂×C₈	1	4	yes	—
25	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
26	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	—	2	2	no	—
28	C₂×C₈×C₅₉	—	2	2	no	—
29	C₂×C₈×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
32	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₅₉	#41	1	4	yes	—
35	C₄²×C₅₉	—	2	2	no	—
36	C₄²×C₅₉	—	2	2	no	—
37	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
38	C₄²×C₅₉	M₁₆	1	2	yes	—
39	C₅₉⋊C₁₆	—	4	4	no	—
40	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	—	2	4	no	—
42	C₂×C₄×C₅₉	—	2	4	no	—
43	C₂×C₄×C₅₉	—	2	8	no	—
44	C₂×C₄×C₅₉	C₄⋊C₈	1	4	yes	—
45	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#117	1	2	yes	—
47	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
48	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
49	C₈×C₅₉	—	2	2	no	—
50	C₈×C₅₉	—	2	4	no	—
51	C₈×C₅₉	#124	1	2	yes	—
52	C₈×C₅₉	#125	1	2	yes	—
53	#12880	—	59	1	no	maximal, Sylow
54	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₂²×C₅₉	#431	1	2	yes	—
56	C₂²×C₅₉	#504	1	4	yes	—
57	C₄×C₅₉	—	2	4	no	—
58	C₄×C₅₉	—	4	4	no	—
59	C₄×C₅₉	#382	1	1	yes	characteristic
60	C₄×C₅₉	#423	1	1	yes	characteristic
61	C₄×C₅₉	#432	1	2	yes	—
62	C₂×C₄⋊C₁₆	—	59	2	no	—
63	#1726	D₅₉	1	1	yes	characteristic
64	#1374	—	59	4	no	—
65	C₂×C₅₉	#1831	1	1	yes	characteristic
66	C₂×C₅₉	#2151	1	2	yes	—
67	C₂×C₅₉	#2164	1	4	yes	—
68	C₂²×C₁₆	—	59	4	no	—
69	C₂×#97	Dic₅₉	1	2	yes	—
70	C₂×#97	Dic₅₉	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
72	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
73	#376	Dic₅₉	1	1	yes	characteristic
74	C₄⋊C₁₆	—	118	4	no	—
75	C₅₉	#12880	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
77	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
79	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
80	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
81	C₂×C₁₆	—	118	4	no	—
82	C₂×C₁₆	—	118	8	no	—
83	#97	—	2	4	no	—
84	#97	—	2	2	no	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	Dic₁₁₈	1	2	yes	—
87	C₄×C₈	D₂₃₆	1	2	yes	—
88	#95	C₅₉⋊C₈	1	4	yes	—
89	C₂²×C₄	C₂×C₅₉⋊C₈	1	4	yes	—
90	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
91	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	#11745664	1	4	yes	—
96	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
97	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
98	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
99	C₂×C₈	#11745685	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	#11745664	1	2	yes	—
103	C₄²	C₄⋊Dic₅₉	1	2	yes	—
104	C₁₆	—	236	4	no	—
105	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	8	no	—
109	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
111	C₂×C₄	#14273686	1	2	yes	—
112	C₂×C₄	C₂×#11745664	1	2	yes	—
113	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	#14273715	1	2	yes	—
117	C₈	#14273719	1	2	yes	—
118	C₂²	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
119	C₂²	#16837455	1	2	yes	—
120	C₂²	#16837819	1	4	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#16837268	1	1	yes	characteristic
124	C₄	#16837426	1	1	yes	characteristic
125	C₄	#16837459	1	2	yes	—
126	C₂	#21424213	1	1	yes	characteristic
127	C₂	#21425855	1	2	yes	—
128	C₂	#21425903	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₅₉⋊C₁₆)

C₂

yes

maximal

#21422174

C₂

yes

maximal

C₅₉×#1726

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

—

C₄⋊(C₅₉⋊C₁₆)

—

C₂×#97×C₅₉

C₄

yes

—

C₂×#97×C₅₉

C₄

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#376

C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

#97×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

#95×C₅₉

C₈

yes

—

C₂²×C₄×C₅₉

C₂×C₈

yes

—

C₂²×C₄×C₅₉

C₂×C₈

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

M₁₆

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

M₁₆

yes

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#124

yes

—

C₈×C₅₉

#125

yes

—

#12880

—

maximal, Sylow

C₂²×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₉

#431

yes

—

C₂²×C₅₉

#504

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₄⋊C₁₆

—

#1726

D₅₉

yes

characteristic

#1374

—

C₂×C₅₉

#1831

yes

characteristic

C₂×C₅₉

#2151

yes

—

C₂×C₅₉

#2164

yes

—

C₂²×C₁₆

—

C₂×#97

Dic₅₉

yes

—

C₂×#97

Dic₅₉

yes

characteristic

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#376

Dic₅₉

yes

characteristic

C₄⋊C₁₆

—

118

—

C₅₉

#12880

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₅₉

yes

—

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂×Dic₅₉

yes

characteristic

C₂×C₁₆

—

118

—

C₂×C₁₆

—

118

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

D₂₃₆

yes

—

#95

C₅₉⋊C₈

yes

—

C₂²×C₄

C₂×C₅₉⋊C₈

yes

—

C₂²×C₄

C₂×C₅₉⋊C₈

yes

—

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#11745664

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

C₂×C₈

C₂×D₂₃₆

yes

characteristic

C₂×C₈

#11745685

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

#11745664

yes

—

103

C₄²

C₄⋊Dic₅₉

yes

—

104

C₁₆

—

236

—

105

C₂³

C₂²×C₅₉⋊C₈

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊(C₅₉⋊C₈)

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

111

C₂×C₄

#14273686

yes

—

112

C₂×C₄

C₂×#11745664

yes

—

113

C₂×C₄

C₂×#11745664

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

#14273715

yes

—

117

C₈

#14273719

yes

—

118

C₂²

C₂×C₄⋊(C₅₉⋊C₈)

yes

characteristic

119

C₂²

#16837455

yes

—

120

C₂²

#16837819

yes

—

121

C₄

—

122

C₄

—

123

C₄

#16837268

yes

characteristic

124

C₄

#16837426

yes

characteristic

125

C₄

#16837459

yes

—

126

C₂

#21424213

yes

characteristic

127

C₂

#21425855

yes

—

128

C₂

#21425903

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator, socle, prime radical	C₂³×C₅₉	C₂²×C₈
Fitting	C₅₉×#1726	C₂
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1726	D₅₉
hypocenter	C₅₉	#12880
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₅₉

commutator, socle, prime radical

C₂³×C₅₉

C₂²×C₈

Fitting

C₅₉×#1726

C₂

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₈ ◁ #1726

N	H ≅ G/N	type
75. C₅₉	#12880	semidirect

H ≅ G/N

type

75. C₅₉

#12880

semidirect

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

subgroup	index
C₂×C₄⋊(C₅₉⋊C₁₆)	2
#21422174	2
C₅₉×#1726	2
#12880	59

subgroup

index

C₂×C₄⋊(C₅₉⋊C₁₆)

#21422174

C₅₉×#1726

#12880

Subgroups — #28505039

Conjugacy classes of subgroups

Complements

Maximal subgroups