Group #21426261 — subgroups

Subgroups — #21426261

Order 7552.

124 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16837804	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#16837230	C₂	1	1	yes	maximal, characteristic
4	#16837228	C₂	1	1	yes	maximal, characteristic
5	#16837207	C₂	1	1	yes	maximal, characteristic
6	#16837207	C₂	1	1	yes	maximal, characteristic
7	C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	#14273719	—	2	1	no	—
10	C₄×Dic₁₁₈	—	2	1	no	—
11	#14273661	C₂²	1	1	yes	characteristic
12	#14273661	C₂²	1	1	yes	characteristic
13	#14273661	C₂²	1	1	yes	characteristic
14	#14273659	—	2	1	no	—
15	#14273652	—	2	1	no	—
16	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₁₈	—	2	1	no	—
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	C₄²×C₅₉	D₄	1	1	yes	characteristic
23	C₄⋊Dic₅₉	—	2	1	no	—
24	C₄⋊Dic₅₉	—	2	1	no	—
25	C₄⋊Dic₅₉	—	2	1	no	—
26	Dic₅₉⋊C₄	—	4	1	no	—
27	Dic₅₉⋊C₄	—	4	1	no	—
28	Dic₅₉⋊C₄	—	4	1	no	—
29	C₄×Dic₅₉	—	4	1	no	—
30	Dic₂₃₆	—	2	1	no	—
31	Dic₂₃₆	Q₈	1	2	yes	—
32	C₁₆×C₅₉	—	2	1	no	—
33	C₁₆×C₅₉	—	2	1	no	—
34	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
35	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₅₉	—	4	1	no	—
38	C₂×Dic₅₉	—	4	1	no	—
39	C₂×Dic₅₉	—	4	1	no	—
40	C₂×Dic₅₉	—	4	1	no	—
41	Dic₁₁₈	—	2	2	no	—
42	Dic₁₁₈	—	4	1	no	—
43	C₈×C₅₉	—	2	1	no	—
44	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₅₉	#41	1	1	yes	characteristic
46	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
47	C₄×C₅₉	—	2	1	no	—
48	C₄×C₅₉	—	2	1	no	—
49	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₄×C₅₉	#137	1	1	yes	characteristic
51	Dic₅₉	—	4	2	no	—
52	Dic₅₉	—	8	1	no	—
53	Dic₅₉	—	8	1	no	—
54	Dic₅₉	—	8	1	no	—
55	#2229	—	59	1	no	maximal, Sylow
56	C₂×C₅₉	#474	1	1	yes	characteristic
57	C₂×C₅₉	#508	1	1	yes	characteristic
58	C₂×C₅₉	#510	1	1	yes	characteristic
59	#499	—	59	1	no	—
60	C₄×Q₁₆	—	59	1	no	—
61	#367	—	59	1	no	—
62	#366	—	59	1	no	—
63	C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
64	#358	—	59	1	no	—
65	#358	—	59	1	no	—
66	C₅₉	#2229	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₁₆	—	59	1	no	—
68	#125	—	118	1	no	—
69	C₄×Q₈	—	118	1	no	—
70	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
71	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
72	#107	—	59	1	no	—
73	#107	—	59	1	no	—
74	#107	—	59	1	no	—
75	#106	—	118	1	no	—
76	#103	—	118	1	no	—
77	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
78	C₂×Q₈	—	118	1	no	—
79	Q₁₆	—	59	2	no	—
80	Q₁₆	—	118	1	no	—
81	C₂×C₈	D₂₃₆	1	1	yes	characteristic
82	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
83	C₄⋊C₄	—	118	1	no	—
84	C₄⋊C₄	—	118	1	no	—
85	C₄⋊C₄	—	118	1	no	—
86	C₄⋊C₄	—	236	1	no	—
87	C₄⋊C₄	—	236	1	no	—
88	C₄⋊C₄	—	236	1	no	—
89	C₄²	—	236	1	no	—
90	C₄²	D₂₃₆	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	2	1	no	—
93	Q₈	—	118	2	no	—
94	Q₈	—	236	1	no	—
95	C₂×C₄	—	236	1	no	—
96	C₂×C₄	—	236	1	no	—
97	C₂×C₄	—	236	1	no	—
98	C₂×C₄	—	236	1	no	—
99	C₂×C₄	D₄₇₂	1	1	yes	characteristic
100	C₂×C₄	D₄₇₂	1	1	yes	characteristic
101	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	#11745687	1	1	yes	characteristic
104	C₈	Q₈×D₅₉	1	1	yes	characteristic
105	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	472	1	no	—
109	C₄	—	472	1	no	—
110	C₄	—	472	1	no	—
111	C₄	—	236	2	no	—
112	C₄	#14273707	1	1	yes	characteristic
113	C₄	#14273760	1	1	yes	characteristic
114	C₂	#16837683	1	1	yes	characteristic
115	C₂	#16837834	1	1	yes	characteristic
116	C₂	#16837845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837804

C₂

yes

maximal, characteristic

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic

#16837230

C₂

yes

maximal, characteristic

#16837228

C₂

yes

maximal, characteristic

#16837207

C₂

yes

maximal, characteristic

#16837207

C₂

yes

maximal, characteristic

C₅₉×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

#14273719

—

C₄×Dic₁₁₈

—

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273659

—

#14273652

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

Q₈

yes

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#137

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₄×Q₁₆

—

#367

—

#366

—

C₄⋊C₁₆

D₅₉

yes

characteristic

#358

—

#358

—

C₅₉

#2229

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

#125

—

118

—

C₄×Q₈

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₂×C₁₆

C₂×D₅₉

yes

—

—

—

—

118

—

#103

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

Q₁₆

—

Q₁₆

—

118

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄²

—

236

—

C₄²

D₂₃₆

yes

characteristic

C₁₆

—

C₁₆

—

Q₈

—

118

—

Q₈

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

D₄₇₂

yes

characteristic

100

C₂×C₄

D₄₇₂

yes

characteristic

101

C₂×C₄

C₂×D₂₃₆

yes

characteristic

102

C₈

—

103

C₈

#11745687

yes

characteristic

104

C₈

Q₈×D₅₉

yes

characteristic

105

C₂²

C₂×D₄₇₂

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

472

—

109

C₄

—

472

—

110

C₄

—

472

—

111

C₄

—

236

—

112

C₄

#14273707

yes

characteristic

113

C₄

#14273760

yes

characteristic

114

C₂

#16837683

yes

characteristic

115

C₂

#16837834

yes

characteristic

116

C₂

#16837845

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×C₄⋊C₁₆	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#2229
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×C₄⋊C₁₆

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₄⋊C₁₆

D₅₉

hypocenter

C₅₉

#2229

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
66. C₅₉	#2229	semidirect

H ≅ G/N

type

66. C₅₉

#2229

semidirect

Subgroups — #21426261

Conjugacy classes of subgroups

Complements