Group #28491766 — subgroups

Subgroups — #28491766

Order 15104.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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	#21424919	C₂	1	3	yes	maximal
2	#21421977	C₂	1	3	yes	maximal, complemented
3	C₅₉×#2845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14273724	C₂²	1	1	yes	characteristic
5	C₂×#14273647	—	2	6	no	—
6	C₂×#14273638	C₂²	1	3	yes	—
7	C₅₉×#398	—	2	6	no	—
8	C₅₉×#382	C₂²	1	3	yes	—
9	C₂²×D₂₃₆	—	2	3	no	—
10	C₂²×C₅₉⋊C₈	—	2	3	no	—
11	#14273724	—	2	3	no	—
12	#14273724	Q₈	1	2	yes	—
13	#14273647	—	4	12	no	—
14	#14273638	—	2	6	no	—
15	C₂×C₄⋊C₄×C₅₉	—	2	6	no	—
16	C₂×C₄⋊C₄×C₅₉	D₄	1	6	yes	—
17	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
18	#95×C₅₉	—	2	2	no	—
19	#95×C₅₉	—	2	6	no	—
20	C₂³×D₅₉	—	4	1	no	—
21	C₂²×C₄×C₅₉	—	2	6	no	—
22	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
23	C₂×D₂₃₆	—	2	6	no	—
24	C₂×D₂₃₆	—	4	6	no	—
25	C₂×D₂₃₆	—	4	6	no	—
26	C₂×D₂₃₆	—	4	3	no	—
27	C₄⋊C₄×C₅₉	—	4	6	no	—
28	C₄⋊C₄×C₅₉	—	8	6	no	—
29	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₅₉	#41	1	3	yes	—
31	C₂×C₅₉⋊C₈	—	2	6	no	—
32	C₂×C₅₉⋊C₈	—	4	3	no	—
33	C₂×C₅₉⋊C₈	—	4	3	no	—
34	C₂³×C₅₉	#120	1	1	yes	characteristic
35	C₂²×D₅₉	—	4	2	no	—
36	C₂²×D₅₉	—	8	3	no	—
37	C₂²×D₅₉	—	8	3	no	—
38	C₂×C₄×C₅₉	—	2	3	no	—
39	C₂×C₄×C₅₉	—	2	3	no	—
40	C₂×C₄×C₅₉	—	4	6	no	—
41	C₂×C₄×C₅₉	—	4	6	no	—
42	C₂×C₄×C₅₉	—	4	6	no	—
43	C₂×C₄×C₅₉	C₂²⋊Q₈	1	3	yes	—
44	C₂×C₄×C₅₉	#124	1	3	yes	—
45	D₂₃₆	—	4	12	no	—
46	D₂₃₆	—	8	6	no	—
47	C₅₉⋊C₈	—	4	6	no	—
48	#10360	—	59	1	no	maximal, Sylow
49	C₂²×C₅₉	#393	1	1	yes	characteristic
50	C₂²×C₅₉	#453	1	3	yes	—
51	C₂²×C₅₉	#454	1	3	yes	—
52	C₂×D₅₉	—	8	6	no	—
53	C₂×D₅₉	—	16	3	no	—
54	C₂×D₅₉	—	16	1	no	—
55	C₄×C₅₉	—	2	3	no	—
56	C₄×C₅₉	—	2	3	no	—
57	C₄×C₅₉	—	8	6	no	—
58	#2845	D₅₉	1	1	yes	characteristic, complemented
59	#1961	—	59	3	no	—
60	#1295	—	59	3	no	—
61	C₂×C₅₉	#1656	1	3	yes	—
62	C₂×C₅₉	#1658	1	1	yes	characteristic
63	C₂×C₅₉	#2027	1	3	yes	—
64	D₅₉	—	16	2	no	—
65	C₂×#127	—	59	1	no	—
66	C₂×#102	—	118	6	no	—
67	C₂×#97	—	59	3	no	—
68	#398	—	2	6	no	—
69	#382	C₂×D₅₉	1	3	yes	—
70	C₅₉	#10360	1	1	yes	characteristic, complemented, Sylow
71	C₂²×D₄	—	118	3	no	—
72	C₂²×C₈	—	118	3	no	—
73	#127	—	59	2	no	—
74	#127	—	118	3	no	—
75	C₂×C₄⋊C₄	—	2	6	no	—
76	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	6	yes	—
77	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
78	#102	—	236	12	no	—
79	#97	—	118	6	no	—
80	#95	—	2	2	no	—
81	#95	—	2	6	no	—
82	C₂⁴	—	236	1	no	—
83	C₂×D₄	—	118	6	no	—
84	C₂×D₄	—	236	6	no	—
85	C₂×D₄	—	236	6	no	—
86	C₂×D₄	—	236	3	no	—
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
89	C₂×C₈	—	118	6	no	—
90	C₂×C₈	—	236	3	no	—
91	C₂×C₈	—	236	3	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄⋊C₄	—	8	6	no	—
94	C₄²	Q₈×D₅₉	1	1	yes	characteristic
95	C₄²	#11745689	1	3	yes	—
96	C₂³	—	472	3	no	—
97	C₂³	—	472	3	no	—
98	C₂³	—	236	2	no	—
99	C₂³	#14273696	1	1	yes	characteristic
100	D₄	—	472	6	no	—
101	D₄	—	236	12	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	#14273703	1	3	yes	—
108	C₂×C₄	#14273716	1	3	yes	—
109	C₈	—	236	6	no	—
110	C₂²	—	944	3	no	—
111	C₂²	—	944	1	no	—
112	C₂²	—	472	6	no	—
113	C₂²	#16837313	1	1	yes	characteristic
114	C₂²	#16837570	1	3	yes	—
115	C₂²	#16837575	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	944	2	no	—
120	C₂	#21423389	1	3	yes	—
121	C₂	#21423398	1	1	yes	characteristic
122	C₂	#21425249	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424919

C₂

yes

maximal

#21421977

C₂

yes

maximal, complemented

C₅₉×#2845

C₂

yes

maximal, characteristic, complemented

C₂×#14273724

C₂²

yes

characteristic

C₂×#14273647

—

C₂×#14273638

C₂²

yes

—

C₅₉×#398

—

C₅₉×#382

C₂²

yes

—

C₂²×D₂₃₆

—

C₂²×C₅₉⋊C₈

—

#14273724

—

#14273724

Q₈

yes

—

#14273647

—

#14273638

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#95×C₅₉

—

#95×C₅₉

—

C₂³×D₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#120

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

#124

yes

—

D₂₃₆

—

D₂₃₆

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#454

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

#2845

D₅₉

yes

characteristic, complemented

#1961

—

#1295

—

C₂×C₅₉

#1656

yes

—

C₂×C₅₉

#1658

yes

characteristic

C₂×C₅₉

#2027

yes

—

D₅₉

—

C₂×#127

—

C₂×#102

—

118

—

C₂×#97

—

#398

—

#382

C₂×D₅₉

yes

—

C₅₉

#10360

yes

characteristic, complemented, Sylow

C₂²×D₄

—

118

—

C₂²×C₈

—

118

—

#127

—

#127

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₅₉

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#102

—

236

—

#97

—

118

—

#95

—

#95

—

C₂⁴

—

236

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

—

C₂×C₈

—

118

—

C₂×C₈

—

236

—

C₂×C₈

—

236

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₅₉

yes

characteristic

C₄²

#11745689

yes

—

C₂³

—

472

—

C₂³

—

472

—

C₂³

—

236

—

C₂³

#14273696

yes

characteristic

100

D₄

—

472

—

101

D₄

—

236

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#14273703

yes

—

108

C₂×C₄

#14273716

yes

—

109

C₈

—

236

—

110

C₂²

—

944

—

111

C₂²

—

944

—

112

C₂²

—

472

—

113

C₂²

#16837313

yes

characteristic

114

C₂²

#16837570

yes

—

115

C₂²

#16837575

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

944

—

120

C₂

#21423389

yes

—

121

C₂

#21423398

yes

characteristic

122

C₂

#21425249

yes

—

	subgroup	G/H
center	C₂³	#14273696
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2845	C₂
socle	C₂³×C₅₉	#120
prime radical	C₂×#14273724	C₂²
hypercenter	#2845	D₅₉
hypocenter	C₅₉	#10360
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#14273696

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#2845

C₂

C₂³×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
2. #21421977	C₂	semidirect
3. C₅₉×#2845	C₂	semidirect
58. #2845	D₅₉	semidirect
70. C₅₉	#10360	semidirect

H ≅ G/N

type

2. #21421977

C₂

semidirect

3. C₅₉×#2845

C₂

semidirect

58. #2845

D₅₉

semidirect

70. C₅₉

#10360

semidirect

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

subgroup	index
#21424919	2
#21421977	2
C₅₉×#2845	2
#10360	59

subgroup

index

#21424919

#21421977

C₅₉×#2845

#10360

Subgroups — #28491766

Conjugacy classes of subgroups

Complements

Maximal subgroups