Group #28503364 — subgroups

Subgroups — #28503364

Order 15104.

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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	#21426378	C₂	1	1	yes	maximal, characteristic
2	#21426349	C₂	1	1	yes	maximal, characteristic
3	#21426324	C₂	1	2	yes	maximal
4	#21423693	C₂	1	2	yes	maximal
5	C₅₉×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇₂	—	2	2	no	—
7	#16837807	C₂²	1	2	yes	—
8	#16837800	—	2	2	no	—
9	#16837782	C₂²	1	2	yes	—
10	#16837753	—	2	2	no	—
11	#16837230	—	2	2	no	—
12	#16837207	—	2	4	no	—
13	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₈²×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₃₆	—	2	2	no	—
17	C₂×Dic₂₃₆	—	2	2	no	—
18	#14273727	—	2	2	no	—
19	#14273727	—	2	2	no	—
20	Dic₄₇₂	—	4	4	no	—
21	#14273661	—	2	2	no	—
22	#14273661	—	2	2	no	—
23	#14273659	—	4	4	no	—
24	#14273652	—	4	4	no	—
25	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
26	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
27	C₄×C₈×C₅₉	D₄	1	2	yes	—
28	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₁₈	—	4	2	no	—
30	C₂×Dic₁₁₈	—	4	2	no	—
31	C₂×C₈×C₅₉	D₈	1	4	yes	—
32	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₅₉	—	4	2	no	—
36	C₄⋊Dic₅₉	—	4	2	no	—
37	C₄⋊Dic₅₉	—	4	2	no	—
38	C₄⋊Dic₅₉	—	4	2	no	—
39	Dic₂₃₆	—	4	4	no	—
40	Dic₂₃₆	—	4	4	no	—
41	C₁₆×C₅₉	—	2	2	no	—
42	C₁₆×C₅₉	SD₁₆	1	4	yes	—
43	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
45	C₂×Dic₅₉	—	8	2	no	—
46	C₂×Dic₅₉	—	8	2	no	—
47	Dic₁₁₈	—	8	2	no	—
48	Dic₁₁₈	—	8	2	no	—
49	Dic₁₁₈	—	8	4	no	—
50	C₈×C₅₉	—	2	4	no	—
51	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
52	C₈×C₅₉	#135	1	2	yes	—
53	#12507	—	59	1	no	maximal, Sylow
54	C₂²×C₅₉	#493	1	1	yes	characteristic
55	C₄×C₅₉	#492	1	1	yes	characteristic
56	C₄×C₅₉	#495	1	1	yes	characteristic
57	C₄×C₅₉	#510	1	4	yes	—
58	Dic₅₉	—	16	2	no	—
59	Dic₅₉	—	16	2	no	—
60	#2254	—	59	1	no	—
61	#2247	—	59	1	no	—
62	#2241	—	59	2	no	—
63	#1707	—	59	2	no	—
64	#1311	D₅₉	1	1	yes	characteristic
65	C₂×C₅₉	#1706	1	1	yes	characteristic
66	C₂×C₅₉	#2251	1	2	yes	—
67	C₂×Q₃₂	—	118	2	no	—
68	#500	—	59	2	no	—
69	#498	—	118	2	no	—
70	#494	—	59	2	no	—
71	#487	—	118	2	no	—
72	#367	—	118	2	no	—
73	#358	—	118	4	no	—
74	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
75	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
76	C₈²	C₂×D₅₉	1	1	yes	characteristic
77	C₅₉	#12507	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	118	2	no	—
79	C₂×Q₁₆	—	118	2	no	—
80	C₄⋊Q₈	—	118	2	no	—
81	C₄⋊Q₈	—	118	2	no	—
82	Q₃₂	—	236	4	no	—
83	C₂×C₁₆	D₂₃₆	1	2	yes	—
84	C₂×C₁₆	D₂₃₆	1	2	yes	—
85	#107	—	118	2	no	—
86	#107	—	118	2	no	—
87	#106	—	236	4	no	—
88	#103	—	236	4	no	—
89	C₄×C₈	D₂₃₆	1	2	yes	—
90	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
91	C₂×Q₈	—	236	2	no	—
92	C₂×Q₈	—	236	2	no	—
93	Q₁₆	—	236	4	no	—
94	Q₁₆	—	236	4	no	—
95	C₂×C₈	D₄₇₂	1	4	yes	—
96	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
97	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
98	C₄⋊C₄	—	236	2	no	—
99	C₄⋊C₄	—	236	2	no	—
100	C₄⋊C₄	—	236	2	no	—
101	C₄⋊C₄	—	236	2	no	—
102	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	#11745660	1	4	yes	—
105	Q₈	—	472	2	no	—
106	Q₈	—	472	2	no	—
107	Q₈	—	472	4	no	—
108	C₂×C₄	—	472	2	no	—
109	C₂×C₄	—	472	2	no	—
110	C₂×C₄	#14273724	1	1	yes	characteristic
111	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	C₂×#11745660	1	2	yes	—
114	C₈	#14273749	1	2	yes	—
115	C₂²	#16837779	1	1	yes	characteristic
116	C₄	—	944	2	no	—
117	C₄	—	944	2	no	—
118	C₄	#16837775	1	1	yes	characteristic
119	C₄	#16837786	1	1	yes	characteristic
120	C₄	#16837845	1	4	yes	—
121	C₂	#21423688	1	1	yes	characteristic
122	C₂	#21426367	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21426378

C₂

yes

maximal, characteristic

#21426349

C₂

yes

maximal, characteristic

#21426324

C₂

yes

maximal

#21423693

C₂

yes

maximal

C₅₉×#1311

C₂

yes

maximal, characteristic

C₂×Dic₄₇₂

—

#16837807

C₂²

yes

—

#16837800

—

#16837782

C₂²

yes

—

#16837753

—

#16837230

—

#16837207

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₈²×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

#14273727

—

#14273727

—

Dic₄₇₂

—

—

—

—

—

C₂×C₁₆×C₅₉

D₄

yes

—

C₂×C₁₆×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₁₆×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×SD₁₆

yes

—

C₈×C₅₉

#135

yes

—

—

C₂²×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

Dic₅₉

—

—

—

—

—

D₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×Q₃₂

—

118

—

#500

—

#498

—

118

—

#494

—

#487

—

118

—

#367

—

118

—

#358

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#12507

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

118

—

C₄⋊Q₈

—

118

—

Q₃₂

—

236

—

C₂×C₁₆

D₂₃₆

yes

—

C₂×C₁₆

D₂₃₆

yes

—

#107

—

118

—

#107

—

118

—

#106

—

236

—

#103

—

236

—

C₄×C₈

D₂₃₆

yes

—

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

236

—

C₂×Q₈

—

236

—

Q₁₆

—

236

—

Q₁₆

—

236

—

C₂×C₈

D₄₇₂

yes

—

C₂×C₈

C₂×D₂₃₆

yes

characteristic

C₂×C₈

C₂×D₂₃₆

yes

characteristic

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

100

C₄⋊C₄

—

236

—

101

C₄⋊C₄

—

236

—

102

C₄²

C₂×D₂₃₆

yes

characteristic

103

C₁₆

—

104

C₁₆

#11745660

yes

—

105

Q₈

—

472

—

106

Q₈

—

472

—

107

Q₈

—

472

—

108

C₂×C₄

—

472

—

109

C₂×C₄

—

472

—

110

C₂×C₄

#14273724

yes

characteristic

111

C₂×C₄

C₂×D₄₇₂

yes

—

112

C₈

—

113

C₈

C₂×#11745660

yes

—

114

C₈

#14273749

yes

—

115

C₂²

#16837779

yes

characteristic

116

C₄

—

944

—

117

C₄

—

944

—

118

C₄

#16837775

yes

characteristic

119

C₄

#16837786

yes

characteristic

120

C₄

#16837845

yes

—

121

C₂

#21423688

yes

characteristic

122

C₂

#21426367

yes

—

	subgroup	G/H
center	C₂²	#16837779
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1311	C₂
socle, prime radical	C₂²×C₅₉	#493
hypercenter	#1311	D₅₉
hypocenter	C₅₉	#12507
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#16837779

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1311

C₂

C₂²×C₅₉

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
77. C₅₉	#12507	semidirect

H ≅ G/N

type

77. C₅₉

#12507

semidirect

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

subgroup	index
#21426378	2
#21426349	2
#21426324	2
#21423693	2
C₅₉×#1311	2
#12507	59

subgroup

index

C₅₉×#1311

Subgroups — #28503364

Conjugacy classes of subgroups

Complements

Maximal subgroups