Group #21426312 — subgroups

Subgroups — #21426312

Order 7552.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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	#16837656	C₂	1	1	yes	maximal, characteristic
2	#16837449	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837224	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837216	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837214	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837183	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#365	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
9	C₂×#11745659	C₂²	1	1	yes	characteristic
10	M₁₆×D₅₉	—	2	1	no	—
11	#14273702	—	2	1	no	—
12	#14273667	C₂²	1	1	yes	characteristic
13	#14273663	C₂²	1	1	yes	characteristic
14	#14273662	—	2	1	no	—
15	#14273647	—	2	1	no	—
16	M₃₂×C₅₉	C₂²	1	1	yes	characteristic
17	#108×C₅₉	C₂²	1	1	yes	characteristic
18	#106×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×D₂₃₆	—	2	1	no	—
20	C₂×D₂₃₆	—	4	1	no	—
21	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
22	M₁₆×C₅₉	—	2	1	no	—
23	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₅₉	—	2	1	no	—
25	(C₂×D₅₉)⋊C₄	—	4	1	no	—
26	#11745664	—	2	1	no	—
27	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
28	D₄₇₂	—	2	1	no	—
29	D₄₇₂	—	2	1	no	—
30	#11745659	—	2	1	no	—
31	#11745659	—	2	1	no	—
32	#11745659	—	4	1	no	—
33	C₈×D₅₉	—	4	1	no	—
34	C₁₆×C₅₉	—	2	1	no	—
35	C₅₉⋊C₁₆	—	2	1	no	—
36	C₂²×D₅₉	—	4	1	no	—
37	C₂²×D₅₉	SD₁₆	1	1	yes	characteristic
38	C₂×C₄×C₅₉	—	4	1	no	—
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₅₉	SD₁₆	1	1	yes	characteristic
41	D₂₃₆	—	4	1	no	—
42	D₂₃₆	—	4	1	no	—
43	D₂₃₆	—	8	2	no	—
44	C₄×D₅₉	—	2	1	no	—
45	C₄×D₅₉	—	2	1	no	—
46	C₈×C₅₉	—	4	1	no	—
47	C₈×C₅₉	#41	1	1	yes	characteristic
48	C₈×C₅₉	#41	1	1	yes	characteristic
49	C₅₉⋊C₈	—	2	1	no	—
50	C₅₉⋊C₈	—	4	1	no	—
51	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×D₅₉	—	2	1	no	—
53	C₂×D₅₉	—	4	1	no	—
54	C₂×D₅₉	—	8	1	no	—
55	C₂×D₅₉	—	8	1	no	—
56	C₂×D₅₉	—	8	1	no	—
57	C₄×C₅₉	—	8	1	no	—
58	C₄×C₅₉	#123	1	1	yes	characteristic
59	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	Dic₅₉	—	2	1	no	—
61	#2238	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	—	2	1	no	—
63	C₂×C₅₉	#481	1	1	yes	characteristic
64	D₅₉	—	4	1	no	—
65	D₅₉	—	16	1	no	—
66	#469	—	59	1	no	—
67	#430	—	59	1	no	—
68	#365	—	59	1	no	—
69	#365	D₅₉	1	1	yes	characteristic, complemented
70	#361	—	59	1	no	—
71	#361	—	59	1	no	—
72	#349	—	59	1	no	—
73	C₅₉	#2238	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₈	—	59	1	no	—
75	C₂×M₁₆	—	59	1	no	—
76	C₂×M₁₆	—	118	1	no	—
77	#121	—	118	1	no	—
78	M₃₂	—	59	1	no	—
79	M₃₂	C₂×D₅₉	1	1	yes	characteristic
80	#108	—	59	1	no	—
81	#108	—	118	1	no	—
82	#108	C₂×D₅₉	1	1	yes	characteristic
83	#106	C₂×D₅₉	1	1	yes	characteristic
84	#102	—	118	1	no	—
85	C₂×D₄	—	118	1	no	—
86	C₂×D₄	—	236	1	no	—
87	C₂²×C₄	—	59	1	no	—
88	D₈	—	118	1	no	—
89	D₈	—	118	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	118	1	no	—
92	M₁₆	—	118	1	no	—
93	M₁₆	—	118	1	no	—
94	M₁₆	—	236	1	no	—
95	C₂×C₈	—	59	1	no	—
96	C₂×C₈	—	236	1	no	—
97	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
98	C₄⋊C₄	—	2	1	no	—
99	C₂²⋊C₄	—	236	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	118	1	no	—
102	C₂³	—	59	1	no	—
103	C₂³	—	236	1	no	—
104	D₄	—	472	2	no	—
105	D₄	—	236	1	no	—
106	D₄	—	236	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	59	1	no	—
109	C₂×C₄	—	118	1	no	—
110	C₂×C₄	—	118	1	no	—
111	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
112	C₈	—	4	1	no	—
113	C₈	—	118	1	no	—
114	C₈	—	236	1	no	—
115	C₈	#11745685	1	1	yes	characteristic
116	C₈	#11745689	1	1	yes	characteristic
117	C₂²	—	118	1	no	—
118	C₂²	—	472	1	no	—
119	C₂²	—	472	1	no	—
120	C₂²	—	472	1	no	—
121	C₂²	—	236	1	no	—
122	C₂²	SD₁₆×D₅₉	1	1	yes	characteristic
123	C₄	—	8	1	no	—
124	C₄	—	118	1	no	—
125	C₄	#14273712	1	1	yes	characteristic
126	C₄	#14273754	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	944	1	no	—
129	C₂	—	236	1	no	—
130	C₂	#16837722	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837656

C₂

yes

maximal, characteristic

#16837449

C₂

yes

maximal, characteristic, complemented

#16837224

C₂

yes

maximal, characteristic, complemented

#16837216

C₂

yes

maximal, characteristic, complemented

#16837214

C₂

yes

maximal, characteristic, complemented

#16837183

C₂

yes

maximal, characteristic, complemented

C₅₉×#365

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₂²

yes

characteristic

C₂×#11745659

C₂²

yes

characteristic

M₁₆×D₅₉

—

#14273702

—

#14273667

C₂²

yes

characteristic

#14273663

C₂²

yes

characteristic

#14273662

—

#14273647

—

M₃₂×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₂²

yes

characteristic

#106×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

D₄

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

(C₂×D₅₉)⋊C₄

—

#11745664

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

D₄₇₂

—

D₄₇₂

—

#11745659

—

#11745659

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

—

C₂²×D₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

SD₁₆

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#2238

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#481

yes

characteristic

D₅₉

—

D₅₉

—

—

—

—

D₅₉

yes

characteristic, complemented

#361

—

#361

—

#349

—

C₅₉

#2238

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

—

118

—

#121

—

118

—

M₃₂

—

M₃₂

C₂×D₅₉

yes

characteristic

#108

—

#108

—

118

—

#108

C₂×D₅₉

yes

characteristic

#106

C₂×D₅₉

yes

characteristic

#102

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂²×C₄

—

D₈

—

118

—

D₈

—

118

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

M₁₆

—

118

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

236

—

100

C₁₆

—

101

C₁₆

—

118

—

102

C₂³

—

103

C₂³

—

236

—

104

D₄

—

472

—

105

D₄

—

236

—

106

D₄

—

236

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

118

—

110

C₂×C₄

—

118

—

111

C₂×C₄

D₄×D₅₉

yes

characteristic

112

C₈

—

113

C₈

—

118

—

114

C₈

—

236

—

115

C₈

#11745685

yes

characteristic

116

C₈

#11745689

yes

characteristic

117

C₂²

—

118

—

118

C₂²

—

472

—

119

C₂²

—

472

—

120

C₂²

—

472

—

121

C₂²

—

236

—

122

C₂²

SD₁₆×D₅₉

yes

characteristic

123

C₄

—

124

C₄

—

118

—

125

C₄

#14273712

yes

characteristic

126

C₄

#14273754

yes

characteristic

127

C₂

—

128

C₂

—

944

—

129

C₂

—

236

—

130

C₂

#16837722

yes

characteristic

	subgroup	G/H
center	C₂	#16837722
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#365	C₂
socle	C₂×C₅₉	#481
hypercenter	#365	D₅₉
hypocenter	C₅₉	#2238
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#16837722

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#365

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
2. #16837449	C₂	semidirect
3. #16837224	C₂	semidirect
4. #16837216	C₂	semidirect
5. #16837214	C₂	semidirect
6. #16837183	C₂	semidirect
7. C₅₉×#365	C₂	semidirect
69. #365	D₅₉	semidirect
73. C₅₉	#2238	semidirect

H ≅ G/N

type

2. #16837449

C₂

semidirect

3. #16837224

C₂

semidirect

4. #16837216

C₂

semidirect

5. #16837214

C₂

semidirect

6. #16837183

C₂

semidirect

7. C₅₉×#365

C₂

semidirect

69. #365

D₅₉

semidirect

73. C₅₉

#2238

semidirect

Subgroups — #21426312

Conjugacy classes of subgroups

Complements