Group #21423172 — subgroups

Subgroups — #21423172

Order 7552.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16838132	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837599	C₂	1	2	yes	maximal
3	#16837151	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837134	C₂	1	2	yes	maximal, complemented
5	C₅₉×#492	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745670	—	2	2	no	—
7	#14273726	—	2	2	no	—
8	#14273725	—	2	2	no	—
9	C₄×Dic₁₁₈	C₂²	1	2	yes	—
10	#14273692	—	2	2	no	—
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
12	#14273648	—	2	2	no	—
13	C₂×SD₁₆×C₅₉	—	2	2	no	—
14	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	#127×C₅₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	D₄	1	2	yes	—
18	C₂×D₄×C₅₉	—	2	1	no	—
19	C₂×D₄×C₅₉	—	2	2	no	—
20	C₂×Dic₁₁₈	—	2	2	no	—
21	C₂×Dic₁₁₈	D₄	1	2	yes	—
22	SD₁₆×C₅₉	—	4	4	no	—
23	C₂×C₈×C₅₉	—	2	1	no	—
24	C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₄²×C₅₉	C₂³	1	1	yes	characteristic
26	#11745670	—	4	4	no	—
27	C₄⋊Dic₅₉	D₄	1	2	yes	—
28	Dic₅₉⋊C₄	—	2	4	no	—
29	Dic₅₉⋊C₄	—	2	2	no	—
30	C₄×Dic₅₉	—	2	2	no	—
31	C₄×Dic₅₉	—	2	2	no	—
32	C₂×C₅₉⋊C₈	—	2	2	no	—
33	C₂³×C₅₉	—	4	1	no	—
34	Q₈×C₅₉	—	4	2	no	—
35	D₄×C₅₉	—	4	2	no	—
36	D₄×C₅₉	—	4	2	no	—
37	D₄×C₅₉	—	4	2	no	—
38	C₂×C₄×C₅₉	—	2	2	no	—
39	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₅₉	—	2	2	no	—
42	C₂×Dic₅₉	—	2	2	no	—
43	Dic₁₁₈	—	2	2	no	—
44	Dic₁₁₈	—	8	2	no	—
45	Dic₁₁₈	SD₁₆	1	4	yes	complemented
46	C₈×C₅₉	—	2	2	no	—
47	C₅₉⋊C₈	—	4	2	no	—
48	C₂²×C₅₉	—	8	2	no	—
49	C₂²×C₅₉	—	8	1	no	—
50	C₂²×C₅₉	#120	1	1	yes	characteristic
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	—	4	2	no	—
53	C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
54	C₄×C₅₉	#137	1	2	yes	—
55	Dic₅₉	—	2	4	no	—
56	Dic₅₉	—	4	2	no	—
57	#1623	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	—	16	1	no	—
59	C₂×C₅₉	#448	1	2	yes	—
60	C₂×C₅₉	#453	1	1	yes	characteristic
61	Q₈²	—	59	1	no	—
62	#492	D₅₉	1	1	yes	characteristic
63	#460	—	59	2	no	—
64	#333	—	59	1	no	—
65	Q₈⋊C₈	—	59	2	no	—
66	C₅₉	#1623	1	1	yes	characteristic, complemented, Sylow
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	118	2	no	—
69	C₄⋊Q₈	—	118	2	no	—
70	C₄⋊Q₈	—	118	2	no	—
71	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
72	#127	C₂×D₅₉	1	1	yes	characteristic
73	C₄×Q₈	—	59	2	no	—
74	C₄×Q₈	—	118	2	no	—
75	C₄⋊C₈	—	59	2	no	—
76	#102	—	118	2	no	—
77	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
78	C₂×Q₈	—	59	2	no	—
79	C₂×Q₈	—	118	2	no	—
80	C₂×Q₈	C₂²⋊D₅₉	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	236	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	118	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	59	2	no	—
89	C₄⋊C₄	—	118	4	no	—
90	C₄⋊C₄	—	118	2	no	—
91	C₄²	—	118	2	no	—
92	C₄²	—	118	2	no	—
93	C₄²	C₂²×D₅₉	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	59	4	no	—
97	Q₈	—	118	2	no	—
98	Q₈	—	472	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	118	2	no	—
104	C₂×C₄	—	118	2	no	—
105	C₂×C₄	D₄×D₅₉	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	236	2	no	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	1	no	—
111	C₂²	#14273695	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	118	4	no	—
115	C₄	—	236	2	no	—
116	C₄	SD₁₆×D₅₉	1	2	yes	—
117	C₄	#14273756	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#16837539	1	2	yes	—
120	C₂	#16837569	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838132

C₂

yes

maximal, characteristic, complemented

#16837599

C₂

yes

maximal

#16837151

C₂

yes

maximal, characteristic, complemented

#16837134

C₂

yes

maximal, complemented

C₅₉×#492

C₂

yes

maximal, characteristic

C₂×#11745670

—

#14273726

—

#14273725

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273692

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273648

—

C₂×SD₁₆×C₅₉

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#127×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

D₄

yes

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₄

yes

—

SD₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745670

—

C₄⋊Dic₅₉

D₄

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

SD₁₆

yes

complemented

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#120

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

—

C₄×C₅₉

#137

yes

—

Dic₅₉

—

Dic₅₉

—

#1623

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#448

yes

—

C₂×C₅₉

#453

yes

characteristic

Q₈²

—

#492

D₅₉

yes

characteristic

#460

—

#333

—

Q₈⋊C₈

—

C₅₉

#1623

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

118

—

C₄⋊Q₈

—

118

—

C₄⋊Q₈

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

#127

C₂×D₅₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

118

—

C₄⋊C₈

—

#102

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

C₂²⋊D₅₉

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄²

—

118

—

C₄²

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

118

—

Q₈

—

472

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

118

—

104

C₂×C₄

—

118

—

105

C₂×C₄

D₄×D₅₉

yes

—

106

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

107

C₈

—

108

C₈

—

236

—

109

C₂²

—

110

C₂²

—

111

C₂²

#14273695

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

118

—

115

C₄

—

236

—

116

C₄

SD₁₆×D₅₉

yes

—

117

C₄

#14273756

yes

—

118

C₂

—

119

C₂

#16837539

yes

—

120

C₂

#16837569

yes

characteristic

	subgroup	G/H
center	C₂²	#14273695
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#492	C₂
socle	C₂²×C₅₉	#120
hypercenter	#492	D₅₉
hypocenter	C₅₉	#1623
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#14273695

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#492

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #16838132	C₂	semidirect
3. #16837151	C₂	semidirect
4. #16837134	C₂	semidirect
45. Dic₁₁₈	SD₁₆	semidirect
66. C₅₉	#1623	semidirect

H ≅ G/N

type

1. #16838132

C₂

semidirect

3. #16837151

C₂

semidirect

4. #16837134

C₂

semidirect

45. Dic₁₁₈

SD₁₆

semidirect

66. C₅₉

#1623

semidirect

Subgroups — #21423172

Conjugacy classes of subgroups

Complements