Group #21426282 — subgroups

Subgroups — #21426282

Order 7552.

150 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16837638	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273661	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837230	C₂	1	2	yes	maximal, complemented
4	#16837204	C₂	1	2	yes	maximal, complemented
5	C₅₉×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
7	#14273701	—	2	1	no	—
8	C₂×C₄⋊Dic₅₉	—	2	1	no	—
9	#14273661	—	2	1	no	—
10	#14273661	C₂²	1	1	yes	characteristic
11	#14273661	C₂²	1	2	yes	—
12	#14273649	—	2	1	no	—
13	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
16	C₂×C₂²⋊D₅₉	—	4	1	no	—
17	C₂²×Dic₅₉	—	4	1	no	—
18	C₂×D₂₃₆	—	2	1	no	—
19	C₂×C₈×C₅₉	—	2	2	no	—
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	(C₂×D₅₉)⋊C₄	—	4	1	no	—
23	C₄⋊Dic₅₉	—	2	1	no	—
24	C₄⋊Dic₅₉	—	2	2	no	—
25	C₄⋊Dic₅₉	—	4	1	no	—
26	D₄₇₂	—	2	2	no	—
27	C₁₆×C₅₉	—	2	2	no	—
28	C₂³×C₅₉	D₈	1	1	yes	characteristic
29	C₂²×D₅₉	—	4	1	no	—
30	C₂×C₄×C₅₉	—	2	1	no	—
31	C₂×C₄×C₅₉	—	2	1	no	—
32	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²⋊D₅₉	—	8	2	no	—
35	C₂×Dic₅₉	—	4	2	no	—
36	C₂×Dic₅₉	—	4	1	no	—
37	C₂×Dic₅₉	—	8	2	no	—
38	D₂₃₆	—	4	1	no	—
39	D₂₃₆	—	4	1	no	—
40	C₈×C₅₉	—	2	1	no	—
41	C₈×C₅₉	#41	1	2	yes	—
42	C₂²×C₅₉	—	2	1	no	—
43	C₂²×C₅₉	—	2	1	no	—
44	C₂²×C₅₉	SD₃₂	1	2	yes	—
45	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
46	C₂×D₅₉	—	8	1	no	—
47	C₂×D₅₉	—	8	1	no	—
48	C₂×D₅₉	—	8	1	no	—
49	C₄×C₅₉	—	2	1	no	—
50	C₄×C₅₉	#123	1	1	yes	characteristic
51	C₄×C₅₉	#137	1	1	yes	characteristic
52	Dic₅₉	—	8	1	no	—
53	Dic₅₉	—	8	2	no	—
54	#2233	—	59	1	no	maximal, Sylow
55	C₂×C₅₉	—	2	2	no	—
56	C₂×C₅₉	#480	1	1	yes	characteristic
57	C₂×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
58	C₂×C₅₉	#509	1	1	yes	characteristic
59	D₅₉	—	16	1	no	—
60	#466	—	59	1	no	—
61	C₂×#107	—	59	1	no	—
62	#367	—	59	2	no	—
63	#357	—	59	2	no	—
64	C₂²⋊C₁₆	D₅₉	1	1	yes	characteristic, complemented
65	C₅₉	#2233	1	1	yes	characteristic, complemented, Sylow
66	C₂×D₈	—	59	1	no	—
67	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
68	#121	—	118	1	no	—
69	C₂×C₄⋊C₄	—	118	1	no	—
70	C₂×C₁₆	C₂×D₅₉	1	2	yes	—
71	#107	—	59	2	no	—
72	#107	—	59	1	no	—
73	#107	—	118	1	no	—
74	#102	—	118	1	no	—
75	C₂×D₄	—	118	1	no	—
76	C₂×D₄	—	236	1	no	—
77	C₂²×C₄	—	236	1	no	—
78	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
79	D₈	—	118	2	no	—
80	C₂×C₈	—	2	2	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	2	no	—
85	C₄⋊C₄	—	236	1	no	—
86	C₂²⋊C₄	—	236	1	no	—
87	C₁₆	—	2	2	no	—
88	C₂³	—	236	1	no	—
89	C₂³	D₄₇₂	1	1	yes	characteristic
90	D₄	—	472	2	no	—
91	D₄	—	236	1	no	—
92	D₄	—	236	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	472	2	no	—
96	C₂×C₄	—	236	1	no	—
97	C₂×C₄	—	236	2	no	—
98	C₂×C₄	D₄₇₂	1	1	yes	characteristic
99	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
100	C₈	—	2	1	no	—
101	C₈	#11745687	1	2	yes	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	472	1	no	—
105	C₂²	—	472	1	no	—
106	C₂²	—	472	1	no	—
107	C₂²	#14273672	1	2	yes	—
108	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	472	1	no	—
111	C₄	—	472	2	no	—
112	C₄	#14273711	1	1	yes	characteristic
113	C₄	#14273760	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	—	944	1	no	—
116	C₂	#16837716	1	1	yes	characteristic
117	C₂	C₂×#14273672	1	1	yes	characteristic
118	C₂	#16837840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837638

C₂

yes

maximal, characteristic

C₂×#14273661

C₂

yes

maximal, characteristic, complemented

#16837230

C₂

yes

maximal, complemented

#16837204

C₂

yes

maximal, complemented

C₅₉×C₂²⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₂²

yes

characteristic

#14273701

—

C₂×C₄⋊Dic₅₉

—

#14273661

—

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

—

#14273649

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂×C₂²⋊D₅₉

—

C₂²×Dic₅₉

—

C₂×D₂₃₆

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

D₄₇₂

—

C₁₆×C₅₉

—

C₂³×C₅₉

D₈

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₈×C₅₉

#41

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

SD₃₂

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#2233

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#480

yes

characteristic

C₂×C₅₉

C₂×SD₃₂

yes

characteristic

C₂×C₅₉

#509

yes

characteristic

D₅₉

—

—

—

—

—

C₂²⋊C₁₆

D₅₉

yes

characteristic, complemented

C₅₉

#2233

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#121

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

—

#107

—

#107

—

#107

—

118

—

#102

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂²×C₄

—

236

—

C₂²×C₄

D₂₃₆

yes

characteristic

D₈

—

118

—

C₂×C₈

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₂²⋊C₄

—

236

—

C₁₆

—

C₂³

—

236

—

C₂³

D₄₇₂

yes

characteristic

D₄

—

472

—

D₄

—

236

—

D₄

—

236

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

472

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

D₄₇₂

yes

characteristic

C₂×C₄

C₂×D₂₃₆

yes

characteristic

100

C₈

—

101

C₈

#11745687

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

472

—

105

C₂²

—

472

—

106

C₂²

—

472

—

107

C₂²

#14273672

yes

—

108

C₂²

C₂×D₄₇₂

yes

characteristic

109

C₄

—

110

C₄

—

472

—

111

C₄

—

472

—

112

C₄

#14273711

yes

characteristic

113

C₄

#14273760

yes

characteristic

114

C₂

—

115

C₂

—

944

—

116

C₂

#16837716

yes

characteristic

117

C₂

C₂×#14273672

yes

characteristic

118

C₂

#16837840

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×C₂²⋊C₁₆	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₂²⋊C₁₆	D₅₉
hypocenter	C₅₉	#2233
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×C₂²⋊C₁₆

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₂²⋊C₁₆

D₅₉

hypocenter

C₅₉

#2233

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. C₂×#14273661	C₂	semidirect
3. #16837230	C₂	semidirect
4. #16837204	C₂	semidirect
5. C₅₉×C₂²⋊C₁₆	C₂	semidirect
64. C₂²⋊C₁₆	D₅₉	semidirect
65. C₅₉	#2233	semidirect

H ≅ G/N

type

2. C₂×#14273661

C₂

semidirect

3. #16837230

C₂

semidirect

4. #16837204

C₂

semidirect

5. C₅₉×C₂²⋊C₁₆

C₂

semidirect

64. C₂²⋊C₁₆

D₅₉

semidirect

65. C₅₉

#2233

semidirect

Subgroups — #21426282

Conjugacy classes of subgroups

Complements