Group #21423417 — subgroups

Subgroups — #21423417

Order 7552.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16838150	C₂	1	1	yes	maximal, characteristic
2	#16837764	C₂	1	2	yes	maximal, complemented
3	#16837144	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837141	C₂	1	2	yes	maximal, complemented
5	C₅₉×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	#14273715	C₂²	1	2	yes	—
7	#14273712	—	2	2	no	—
8	#14273705	—	2	2	no	—
9	#14273702	—	2	2	no	—
10	#14273650	—	2	2	no	—
11	#14273647	—	2	2	no	—
12	#14273638	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	#125×C₅₉	C₂²	1	1	yes	characteristic
15	#103×C₅₉	—	2	2	no	—
16	#97×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	—	2	1	no	—
18	C₂×D₂₃₆	—	2	2	no	—
19	C₂×D₂₃₆	D₄	1	2	yes	—
20	C₂×C₄×D₅₉	—	2	2	no	—
21	C₂×Dic₁₁₈	D₄	1	2	yes	—
22	C₂×C₈×C₅₉	—	2	1	no	—
23	C₄⋊C₄×C₅₉	—	2	2	no	—
24	C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
26	C₄²×C₅₉	C₂³	1	1	yes	characteristic
27	(C₂×D₅₉)⋊C₄	—	2	2	no	—
28	(C₂×D₅₉)⋊C₄	—	2	2	no	—
29	(C₂×D₅₉)⋊C₄	—	2	2	no	—
30	Dic₅₉⋊C₄	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	2	2	no	—
32	C₂²×D₅₉	—	2	2	no	—
33	Q₈×C₅₉	—	4	2	no	—
34	C₂×C₄×C₅₉	—	2	2	no	—
35	C₂×C₄×C₅₉	—	4	1	no	—
36	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₅₉	—	2	2	no	—
39	D₂₃₆	—	4	2	no	—
40	D₂₃₆	—	8	2	no	—
41	C₄×D₅₉	—	8	2	no	—
42	Dic₁₁₈	—	4	2	no	—
43	C₈×C₅₉	—	4	1	no	—
44	C₅₉⋊C₈	—	4	2	no	—
45	C₂²×C₅₉	#120	1	1	yes	characteristic
46	C₂×D₅₉	—	4	2	no	—
47	C₂×D₅₉	—	4	2	no	—
48	C₂×D₅₉	—	4	2	no	—
49	C₄×C₅₉	—	2	1	no	—
50	C₄×C₅₉	—	2	2	no	—
51	C₄×C₅₉	—	4	2	no	—
52	C₄×C₅₉	—	8	1	no	—
53	Dic₅₉	—	4	2	no	—
54	#1662	—	59	1	no	maximal, Sylow
55	C₂×C₅₉	#454	1	1	yes	characteristic
56	C₂×C₅₉	#455	1	2	yes	—
57	D₅₉	—	8	2	no	—
58	#562	—	59	1	no	—
59	#491	D₅₉	1	1	yes	characteristic, complemented
60	#489	—	59	2	no	—
61	#330	—	59	1	no	—
62	#329	—	59	2	no	—
63	C₅₉	#1662	1	1	yes	characteristic, complemented, Sylow
64	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
65	#125	C₂×D₅₉	1	1	yes	characteristic
66	#124	—	59	2	no	—
67	#123	—	118	2	no	—
68	C₂²⋊Q₈	—	118	2	no	—
69	#121	—	118	2	no	—
70	#103	—	2	2	no	—
71	#103	—	118	2	no	—
72	#102	—	118	2	no	—
73	#97	—	59	2	no	—
74	#97	C₂×D₅₉	1	1	yes	characteristic
75	C₂×Q₈	—	2	1	no	—
76	C₂×Q₈	—	59	2	no	—
77	C₂×D₄	—	59	2	no	—
78	C₂×D₄	—	118	2	no	—
79	C₂²×C₄	—	118	2	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	118	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	118	2	no	—
85	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
86	C₂²⋊C₄	—	118	2	no	—
87	C₂²⋊C₄	—	118	2	no	—
88	C₂²⋊C₄	—	118	2	no	—
89	C₄²	C₂²×D₅₉	1	1	yes	characteristic
90	C₂³	—	118	2	no	—
91	Q₈	—	4	2	no	—
92	Q₈	—	236	2	no	—
93	D₄	—	472	2	no	—
94	D₄	—	236	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	118	2	no	—
98	C₂×C₄	—	472	2	no	—
99	C₂×C₄	D₄×D₅₉	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
101	C₈	—	4	1	no	—
102	C₈	—	236	2	no	—
103	C₂²	—	236	2	no	—
104	C₂²	—	236	2	no	—
105	C₂²	—	236	2	no	—
106	C₂²	#14273695	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	2	no	—
109	C₄	—	4	2	no	—
110	C₄	—	8	1	no	—
111	C₄	—	236	2	no	—
112	C₂	—	472	2	no	—
113	C₂	#16837574	1	1	yes	characteristic
114	C₂	#16837578	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16838150

C₂

yes

maximal, characteristic

#16837764

C₂

yes

maximal, complemented

#16837144

C₂

yes

maximal, characteristic, complemented

#16837141

C₂

yes

maximal, complemented

C₅₉×#491

C₂

yes

maximal, characteristic, complemented

#14273715

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#125×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

#97×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

D₄

yes

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

D₄

yes

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

Dic₅₉⋊C₄

—

C₂×C₅₉⋊C₈

—

C₂²×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

#120

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

—

#562

—

#491

D₅₉

yes

characteristic, complemented

#489

—

#330

—

#329

—

C₅₉

#1662

yes

characteristic, complemented, Sylow

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

#125

C₂×D₅₉

yes

characteristic

#124

—

#123

—

118

—

C₂²⋊Q₈

—

118

—

#121

—

118

—

#103

—

#103

—

118

—

#102

—

118

—

#97

—

#97

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂²×C₄

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

C₂²⋊D₅₉

yes

—

C₂²⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

118

—

Q₈

—

Q₈

—

236

—

D₄

—

472

—

D₄

—

236

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

472

—

C₂×C₄

D₄×D₅₉

yes

—

100

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

101

C₈

—

102

C₈

—

236

—

103

C₂²

—

236

—

104

C₂²

—

236

—

105

C₂²

—

236

—

106

C₂²

#14273695

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

236

—

112

C₂

—

472

—

113

C₂

#16837574

yes

characteristic

114

C₂

#16837578

yes

—

	subgroup	G/H
center	C₂²	#14273695
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#491	C₂
socle	C₂²×C₅₉	#120
hypercenter	#491	D₅₉
hypocenter	C₅₉	#1662
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

subgroup

G/H

center

C₂²

#14273695

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#491

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #491

N	H ≅ G/N	type
2. #16837764	C₂	semidirect
3. #16837144	C₂	semidirect
4. #16837141	C₂	semidirect
5. C₅₉×#491	C₂	semidirect
59. #491	D₅₉	semidirect
63. C₅₉	#1662	semidirect

H ≅ G/N

type

2. #16837764

C₂

semidirect

3. #16837144

C₂

semidirect

4. #16837141

C₂

semidirect

5. C₅₉×#491

C₂

semidirect

59. #491

D₅₉

semidirect

63. C₅₉

#1662

semidirect

Subgroups — #21423417

Conjugacy classes of subgroups

Complements