Group #28500713 — subgroups

Subgroups — #28500713

Order 15104.

226 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 69 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	#21425973	C₂	1	1	yes	maximal, characteristic
2	#21423009	C₂	1	2	yes	maximal, complemented
3	#21422062	C₂	1	2	yes	maximal, complemented
4	#21422016	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₉₄₄	C₄	1	1	yes	characteristic
7	C₄×D₄₇₂	C₂²	1	2	yes	—
8	#16837479	—	2	2	no	—
9	#16837230	C₄	1	1	yes	characteristic
10	#16837204	C₄	1	2	yes	—
11	#16837178	C₂²	1	1	yes	characteristic
12	#16837129	—	2	2	no	—
13	#16837123	C₂²	1	1	yes	characteristic
14	C₅₉×#346	C₂²	1	1	yes	characteristic
15	C₅₉×#334	C₂²	1	2	yes	—
16	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
17	C₂×#11745659	—	4	2	no	—
18	C₄×D₂₃₆	—	2	2	no	—
19	D₉₄₄	—	2	2	no	—
20	C₂×C₅₉⋊C₁₆	—	2	1	no	—
21	#14273661	C₂×C₄	1	2	yes	—
22	#14273649	—	2	2	no	—
23	(C₂×D₅₉)⋊C₈	—	4	2	no	—
24	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₅₉	—	2	2	no	—
28	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×D₂₃₆	—	2	2	no	—
30	C₂×C₄×D₅₉	—	4	2	no	—
31	C₂×C₈×C₅₉	—	4	2	no	—
32	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
34	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
35	(C₂×D₅₉)⋊C₄	—	4	2	no	—
36	C₄⋊Dic₅₉	—	2	2	no	—
37	C₂×C₅₉⋊C₈	—	2	2	no	—
38	D₄₇₂	—	2	2	no	—
39	D₄₇₂	—	2	2	no	—
40	#11745659	—	8	4	no	—
41	C₁₆×C₅₉	—	2	1	no	—
42	C₁₆×C₅₉	M₁₆	1	2	yes	complemented
43	C₅₉⋊C₁₆	—	4	1	no	—
44	C₂²×D₅₉	—	4	2	no	—
45	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×Dic₅₉	—	4	2	no	—
48	D₂₃₆	—	4	2	no	—
49	D₂₃₆	—	4	2	no	—
50	C₄×D₅₉	—	8	2	no	—
51	C₄×D₅₉	—	8	2	no	—
52	C₈×C₅₉	—	2	1	no	—
53	C₈×C₅₉	—	8	2	no	—
54	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₅₉	#131	1	1	yes	characteristic
56	C₅₉⋊C₈	—	4	2	no	—
57	#12089	—	59	1	no	maximal, Sylow
58	C₂²×C₅₉	#442	1	1	yes	characteristic
59	C₂×D₅₉	—	8	2	no	—
60	C₂×D₅₉	—	8	2	no	—
61	C₂×D₅₉	—	8	2	no	—
62	C₄×C₅₉	—	2	2	no	—
63	C₄×C₅₉	#436	1	1	yes	characteristic
64	C₄×C₅₉	#444	1	1	yes	characteristic
65	Dic₅₉	—	8	2	no	—
66	#2177	—	59	1	no	—
67	#1594	—	59	2	no	—
68	#1368	D₅₉	1	1	yes	characteristic, complemented
69	#1333	—	59	2	no	—
70	#1313	—	59	1	no	—
71	C₂×C₅₉	#1594	1	1	yes	characteristic
72	C₂×C₅₉	AGL(1,ℤ/16)	1	2	yes	—
73	D₅₉	—	16	2	no	—
74	C₂×D₁₆	—	59	1	no	—
75	C₄×D₈	—	59	2	no	—
76	#436	—	118	2	no	—
77	#367	—	59	1	no	—
78	#357	—	59	2	no	—
79	#346	—	59	1	no	—
80	#346	C₂×D₅₉	1	1	yes	characteristic
81	#334	C₂×D₅₉	1	2	yes	—
82	D₄⋊C₈	—	118	2	no	—
83	#322	—	59	1	no	—
84	C₅₉	#12089	1	1	yes	characteristic, complemented, Sylow
85	C₂×D₈	—	59	2	no	—
86	C₂×M₁₆	—	236	2	no	—
87	C₄×D₄	—	118	2	no	—
88	D₁₆	—	118	2	no	—
89	C₂×C₁₆	—	118	1	no	—
90	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
91	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
92	#107	—	59	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	#102	—	118	2	no	—
95	C₂²⋊C₈	—	236	2	no	—
96	C₄×C₈	—	59	2	no	—
97	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
98	C₂×D₄	—	118	2	no	—
99	C₂²×C₄	—	236	2	no	—
100	D₈	—	118	2	no	—
101	D₈	—	118	2	no	—
102	M₁₆	—	472	4	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	118	2	no	—
105	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
106	C₂×C₈	#11745689	1	1	yes	characteristic
107	C₄⋊C₄	—	118	2	no	—
108	C₂²⋊C₄	—	236	2	no	—
109	C₄²	D₄×D₅₉	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	236	1	no	—
112	C₁₆	#11745659	1	2	yes	complemented
113	C₂³	—	236	2	no	—
114	D₄	—	236	2	no	—
115	D₄	—	236	2	no	—
116	C₂×C₄	—	472	2	no	—
117	C₂×C₄	—	472	2	no	—
118	C₂×C₄	—	236	2	no	—
119	C₂×C₄	#14273689	1	1	yes	characteristic
120	C₂×C₄	#14273754	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	8	2	no	—
123	C₈	—	236	2	no	—
124	C₈	C₂×#11745659	1	1	yes	characteristic
125	C₈	#14273735	1	1	yes	characteristic
126	C₂²	—	472	2	no	—
127	C₂²	—	472	2	no	—
128	C₂²	—	472	2	no	—
129	C₂²	#16837513	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	472	2	no	—
132	C₄	#16837479	1	1	yes	characteristic
133	C₄	#16837523	1	1	yes	characteristic
134	C₂	—	944	2	no	—
135	C₂	#21423009	1	1	yes	characteristic
136	C₂	#21425993	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425973

C₂

yes

maximal, characteristic

#21423009

C₂

yes

maximal, complemented

#21422062

C₂

yes

maximal, complemented

#21422016

C₂

yes

maximal, characteristic, complemented

C₅₉×#1368

C₂

yes

maximal, characteristic, complemented

C₂×D₉₄₄

C₄

yes

characteristic

C₄×D₄₇₂

C₂²

yes

—

#16837479

—

#16837230

C₄

yes

characteristic

#16837204

C₄

yes

—

#16837178

C₂²

yes

characteristic

#16837129

—

#16837123

C₂²

yes

characteristic

C₅₉×#346

C₂²

yes

characteristic

C₅₉×#334

C₂²

yes

—

C₂×D₄₇₂

C₂×C₄

yes

—

C₂×#11745659

—

C₄×D₂₃₆

—

D₉₄₄

—

C₂×C₅₉⋊C₁₆

—

#14273661

C₂×C₄

yes

—

#14273649

—

(C₂×D₅₉)⋊C₈

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

D₄₇₂

—

D₄₇₂

—

#11745659

—

C₁₆×C₅₉

—

C₁₆×C₅₉

M₁₆

yes

complemented

C₅₉⋊C₁₆

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₈×C₅₉

#131

yes

characteristic

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#2177

—

#1594

—

#1368

D₅₉

yes

characteristic, complemented

#1333

—

#1313

—

C₂×C₅₉

#1594

yes

characteristic

C₂×C₅₉

AGL(1,ℤ/16)

yes

—

D₅₉

—

C₂×D₁₆

—

C₄×D₈

—

#436

—

118

—

—

—

—

C₂×D₅₉

yes

characteristic

#334

C₂×D₅₉

yes

—

D₄⋊C₈

—

118

—

#322

—

C₅₉

#12089

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

236

—

C₄×D₄

—

118

—

D₁₆

—

118

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₂×C₁₆

C₄×D₅₉

yes

characteristic

#107

—

C₄⋊C₈

—

#102

—

118

—

C₂²⋊C₈

—

236

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×D₄

—

118

—

C₂²×C₄

—

236

—

100

D₈

—

118

—

101

D₈

—

118

—

102

M₁₆

—

472

—

103

C₂×C₈

—

104

C₂×C₈

—

118

—

105

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

106

C₂×C₈

#11745689

yes

characteristic

107

C₄⋊C₄

—

118

—

108

C₂²⋊C₄

—

236

—

109

C₄²

D₄×D₅₉

yes

characteristic

110

C₁₆

—

111

C₁₆

—

236

—

112

C₁₆

#11745659

yes

complemented

113

C₂³

—

236

—

114

D₄

—

236

—

115

D₄

—

236

—

116

C₂×C₄

—

472

—

117

C₂×C₄

—

472

—

118

C₂×C₄

—

236

—

119

C₂×C₄

#14273689

yes

characteristic

120

C₂×C₄

#14273754

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

236

—

124

C₈

C₂×#11745659

yes

characteristic

125

C₈

#14273735

yes

characteristic

126

C₂²

—

472

—

127

C₂²

—

472

—

128

C₂²

—

472

—

129

C₂²

#16837513

yes

characteristic

130

C₄

—

131

C₄

—

472

—

132

C₄

#16837479

yes

characteristic

133

C₄

#16837523

yes

characteristic

134

C₂

—

944

—

135

C₂

#21423009

yes

characteristic

136

C₂

#21425993

yes

—

	subgroup	G/H
center	C₂²	#16837513
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1368	C₂
socle	C₂²×C₅₉	#442
prime radical	C₂×D₉₄₄	C₄
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1368	D₅₉
hypocenter	C₅₉	#12089
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#16837513

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1368

C₂

socle

C₂²×C₅₉

#442

prime radical

C₂×D₉₄₄

C₄

cosocle

C₄×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
2. #21423009	C₂	semidirect
3. #21422062	C₂	semidirect
4. #21422016	C₂	semidirect
5. C₅₉×#1368	C₂	semidirect
42. C₁₆×C₅₉	M₁₆	semidirect
68. #1368	D₅₉	semidirect
84. C₅₉	#12089	semidirect
112. C₁₆	#11745659	semidirect

H ≅ G/N

type

2. #21423009

C₂

semidirect

3. #21422062

C₂

semidirect

4. #21422016

C₂

semidirect

5. C₅₉×#1368

C₂

semidirect

42. C₁₆×C₅₉

M₁₆

semidirect

68. #1368

D₅₉

semidirect

84. C₅₉

#12089

semidirect

112. C₁₆

#11745659

semidirect

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

subgroup	index
#21425973	2
#21423009	2
#21422062	2
#21422016	2
C₅₉×#1368	2
#12089	59

subgroup

index

C₅₉×#1368

Subgroups — #28500713

Conjugacy classes of subgroups

Complements

Maximal subgroups