Group #28503360 — subgroups

Subgroups — #28503360

Order 15104.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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	#21426346	C₂	1	1	yes	maximal, characteristic
2	#21426321	C₂	1	2	yes	maximal, complemented
3	#21423689	C₂	1	2	yes	maximal, complemented
4	#21422012	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2254	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₉₄₄	—	2	2	no	—
7	#16837779	C₂²	1	2	yes	—
8	#16837776	—	2	2	no	—
9	#16837750	—	2	2	no	—
10	#16837202	—	2	4	no	—
11	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₅₉×#500	C₂²	1	2	yes	—
14	C₅₉×#367	—	2	2	no	—
15	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#11745671	—	4	4	no	—
17	C₂×D₄₇₂	—	2	2	no	—
18	C₂×D₄₇₂	—	2	2	no	—
19	#14273724	—	2	2	no	—
20	D₉₄₄	—	4	4	no	—
21	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
22	#14273647	—	4	4	no	—
23	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
24	C₄⋊Q₈×C₅₉	—	2	2	no	—
25	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
26	#107×C₅₉	—	2	2	no	—
27	#107×C₅₉	—	2	2	no	—
28	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₅₉	—	4	2	no	—
30	C₂×D₂₃₆	—	4	2	no	—
31	C₂×D₂₃₆	—	4	2	no	—
32	C₂×D₂₃₆	—	4	2	no	—
33	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₉	—	4	2	no	—
36	C₄⋊C₄×C₅₉	—	4	2	no	—
37	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
38	#11745671	—	8	8	no	—
39	C₂×C₅₉⋊C₈	D₈	1	4	yes	—
40	D₄₇₂	—	4	4	no	—
41	D₄₇₂	—	4	4	no	—
42	C₁₆×C₅₉	SD₁₆	1	4	yes	complemented
43	C₅₉⋊C₁₆	—	2	2	no	—
44	C₂²×D₅₉	—	8	2	no	—
45	Q₈×C₅₉	—	8	4	no	—
46	C₂×C₄×C₅₉	—	8	2	no	—
47	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
49	D₂₃₆	—	8	2	no	—
50	D₂₃₆	—	8	2	no	—
51	D₂₃₆	—	8	4	no	—
52	D₂₃₆	—	8	4	no	—
53	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
54	C₈×C₅₉	#135	1	2	yes	—
55	C₅₉⋊C₈	—	2	4	no	—
56	#12506	—	59	1	no	maximal, Sylow
57	C₂²×C₅₉	#493	1	1	yes	characteristic
58	C₂×D₅₉	—	16	2	no	—
59	C₂×D₅₉	—	16	2	no	—
60	C₂×D₅₉	—	16	2	no	—
61	C₄×C₅₉	—	16	2	no	—
62	C₄×C₅₉	#492	1	1	yes	characteristic
63	C₄×C₅₉	#495	1	1	yes	characteristic
64	C₄×C₅₉	#509	1	4	yes	—
65	#2254	D₅₉	1	1	yes	characteristic, complemented
66	#2246	—	59	1	no	—
67	#2240	—	59	2	no	—
68	#1706	—	59	2	no	—
69	#1311	—	59	1	no	—
70	C₂×C₅₉	#1706	1	1	yes	characteristic
71	C₂×C₅₉	#2250	1	2	yes	—
72	D₅₉	—	32	2	no	—
73	C₂×D₁₆	—	118	2	no	—
74	#500	C₂×D₅₉	1	2	yes	—
75	#493	—	59	2	no	—
76	#492	—	118	2	no	—
77	#486	—	118	2	no	—
78	#367	—	2	2	no	—
79	#357	—	118	4	no	—
80	C₄×C₁₆	—	59	1	no	—
81	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
82	C₈²	—	59	1	no	—
83	C₅₉	#12506	1	1	yes	characteristic, complemented, Sylow
84	C₂×SD₁₆	—	236	4	no	—
85	C₂×D₈	—	118	2	no	—
86	C₂×D₈	—	118	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	#127	—	118	2	no	—
89	D₁₆	—	236	4	no	—
90	C₂×C₁₆	—	59	2	no	—
91	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
92	#107	—	2	2	no	—
93	#107	—	2	2	no	—
94	#102	—	236	4	no	—
95	C₄×C₈	—	59	2	no	—
96	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
97	C₂×Q₈	—	4	2	no	—
98	C₂×D₄	—	236	2	no	—
99	C₂×D₄	—	236	2	no	—
100	C₂×D₄	—	236	2	no	—
101	SD₁₆	—	472	8	no	—
102	D₈	—	236	4	no	—
103	D₈	—	236	4	no	—
104	C₂×C₈	—	59	4	no	—
105	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	D₄×D₅₉	1	1	yes	characteristic
110	C₁₆	—	118	2	no	—
111	C₁₆	#11745671	1	4	yes	complemented
112	C₂³	—	472	2	no	—
113	Q₈	—	8	4	no	—
114	D₄	—	472	2	no	—
115	D₄	—	472	2	no	—
116	D₄	—	472	4	no	—
117	D₄	—	472	4	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	#14273723	1	1	yes	characteristic
120	C₂×C₄	D₈×D₅₉	1	2	yes	—
121	C₈	—	118	4	no	—
122	C₈	C₂×#11745671	1	2	yes	—
123	C₈	#14273750	1	2	yes	—
124	C₂²	—	944	2	no	—
125	C₂²	—	944	2	no	—
126	C₂²	—	944	2	no	—
127	C₂²	#16837777	1	1	yes	characteristic
128	C₄	—	16	2	no	—
129	C₄	#16837776	1	1	yes	characteristic
130	C₄	#16837787	1	1	yes	characteristic
131	C₄	#16837839	1	4	yes	—
132	C₂	—	1888	2	no	—
133	C₂	#21423689	1	1	yes	characteristic
134	C₂	#21426361	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21426346

C₂

yes

maximal, characteristic

#21426321

C₂

yes

maximal, complemented

#21423689

C₂

yes

maximal, complemented

#21422012

C₂

yes

maximal, characteristic, complemented

C₅₉×#2254

C₂

yes

maximal, characteristic, complemented

C₂×D₉₄₄

—

#16837779

C₂²

yes

—

#16837776

—

#16837750

—

#16837202

—

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₅₉⋊C₈

C₂²

yes

characteristic

C₅₉×#500

C₂²

yes

—

C₅₉×#367

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#11745671

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

#14273724

—

D₉₄₄

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

#14273647

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₄⋊Q₈×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

—

#107×C₅₉

—

#107×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

#11745671

—

C₂×C₅₉⋊C₈

D₈

yes

—

D₄₇₂

—

D₄₇₂

—

C₁₆×C₅₉

SD₁₆

yes

complemented

C₅₉⋊C₁₆

—

C₂²×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

C₂×SD₁₆

yes

—

C₈×C₅₉

#135

yes

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#2254

D₅₉

yes

characteristic, complemented

—

—

—

—

C₂×C₅₉

#1706

yes

characteristic

C₂×C₅₉

#2250

yes

—

D₅₉

—

C₂×D₁₆

—

118

—

#500

C₂×D₅₉

yes

—

#493

—

#492

—

118

—

#486

—

118

—

#367

—

#357

—

118

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₈²

—

C₅₉

#12506

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

236

—

C₂×D₈

—

118

—

C₂×D₈

—

118

—

C₄⋊Q₈

—

#127

—

118

—

D₁₆

—

236

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅₉

yes

—

#107

—

#107

—

#102

—

236

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

100

C₂×D₄

—

236

—

101

SD₁₆

—

472

—

102

D₈

—

236

—

103

D₈

—

236

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₅₉

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

D₄×D₅₉

yes

characteristic

110

C₁₆

—

118

—

111

C₁₆

#11745671

yes

complemented

112

C₂³

—

472

—

113

Q₈

—

114

D₄

—

472

—

115

D₄

—

472

—

116

D₄

—

472

—

117

D₄

—

472

—

118

C₂×C₄

—

119

C₂×C₄

#14273723

yes

characteristic

120

C₂×C₄

D₈×D₅₉

yes

—

121

C₈

—

118

—

122

C₈

C₂×#11745671

yes

—

123

C₈

#14273750

yes

—

124

C₂²

—

944

—

125

C₂²

—

944

—

126

C₂²

—

944

—

127

C₂²

#16837777

yes

characteristic

128

C₄

—

129

C₄

#16837776

yes

characteristic

130

C₄

#16837787

yes

characteristic

131

C₄

#16837839

yes

—

132

C₂

—

1888

—

133

C₂

#21423689

yes

characteristic

134

C₂

#21426361

yes

—

	subgroup	G/H
center	C₂²	#16837777
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2254	C₂
socle	C₂²×C₅₉	#493
prime radical	#21426346	C₂
hypercenter	#2254	D₅₉
hypocenter	C₅₉	#12506
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#16837777

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#2254

C₂

C₂²×C₅₉

C₂

D₅₉

C₅₉

G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
2. #21426321	C₂	semidirect
3. #21423689	C₂	semidirect
4. #21422012	C₂	semidirect
5. C₅₉×#2254	C₂	semidirect
42. C₁₆×C₅₉	SD₁₆	semidirect
65. #2254	D₅₉	semidirect
83. C₅₉	#12506	semidirect
111. C₁₆	#11745671	semidirect

H ≅ G/N

type

2. #21426321

C₂

semidirect

3. #21423689

C₂

semidirect

4. #21422012

C₂

semidirect

5. C₅₉×#2254

C₂

semidirect

42. C₁₆×C₅₉

SD₁₆

semidirect

65. #2254

D₅₉

semidirect

83. C₅₉

#12506

semidirect

111. C₁₆

#11745671

semidirect

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

subgroup	index
#21426346	2
#21426321	2
#21423689	2
#21422012	2
C₅₉×#2254	2
#12506	59

subgroup

index

C₅₉×#2254

Subgroups — #28503360

Conjugacy classes of subgroups

Complements

Maximal subgroups