Group #21426132 — subgroups

Subgroups — #21426132

Order 7552.

186 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₂×D₉₄₄	C₂	1	2	yes	maximal, complemented
2	#16837779	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837204	C₂	1	2	yes	maximal, complemented
5	C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₇₂	—	2	1	no	—
7	C₂×D₄₇₂	C₂²	1	2	yes	—
8	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
9	#14273724	—	2	1	no	—
10	C₄×D₂₃₆	—	2	1	no	—
11	D₉₄₄	—	2	4	no	—
12	#14273661	C₂²	1	1	yes	characteristic
13	#14273649	—	2	1	no	—
14	C₂×C₁₆×C₅₉	C₂²	1	2	yes	—
15	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×D₂₃₆	—	2	1	no	—
17	C₂×D₂₃₆	—	2	2	no	—
18	C₂×D₂₃₆	—	4	1	no	—
19	C₂×D₂₃₆	—	4	1	no	—
20	C₂×C₄×D₅₉	—	4	1	no	—
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
23	C₄²×C₅₉	D₄	1	1	yes	characteristic
24	(C₂×D₅₉)⋊C₄	—	4	1	no	—
25	C₄⋊Dic₅₉	—	2	1	no	—
26	D₄₇₂	—	2	2	no	—
27	D₄₇₂	—	2	1	no	—
28	D₄₇₂	—	2	2	no	—
29	D₄₇₂	—	4	2	no	—
30	D₄₇₂	D₄	1	2	yes	complemented
31	C₁₆×C₅₉	—	2	2	no	—
32	C₂²×D₅₉	—	4	1	no	—
33	C₂²×D₅₉	—	4	2	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	D₂₃₆	—	2	2	no	—
39	D₂₃₆	—	4	2	no	—
40	D₂₃₆	—	4	1	no	—
41	D₂₃₆	—	4	2	no	—
42	D₂₃₆	—	8	2	no	—
43	D₂₃₆	—	8	2	no	—
44	C₄×D₅₉	—	8	2	no	—
45	C₈×C₅₉	—	2	1	no	—
46	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₈×C₅₉	#41	1	1	yes	characteristic
48	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
49	C₂×D₅₉	—	4	2	no	—
50	C₂×D₅₉	—	8	1	no	—
51	C₂×D₅₉	—	8	1	no	—
52	C₂×D₅₉	—	8	2	no	—
53	C₂×D₅₉	—	8	2	no	—
54	C₂×D₅₉	—	8	2	no	—
55	C₄×C₅₉	—	2	1	no	—
56	C₄×C₅₉	D₁₆	1	2	yes	complemented
57	C₄×C₅₉	#121	1	1	yes	characteristic
58	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₅₉	—	8	1	no	—
60	#2206	—	59	1	no	maximal, Sylow
61	C₂×C₅₉	#459	1	1	yes	characteristic
62	C₂×C₅₉	C₂×D₁₆	1	1	yes	characteristic
63	C₂×C₅₉	#509	1	1	yes	characteristic
64	D₅₉	—	8	2	no	—
65	D₅₉	—	16	2	no	—
66	C₂×D₁₆	—	59	2	no	—
67	#493	—	59	1	no	—
68	C₄×D₈	—	59	1	no	—
69	C₄⋊C₁₆	D₅₉	1	1	yes	characteristic, complemented
70	#357	—	59	2	no	—
71	C₅₉	#2206	1	1	yes	characteristic, complemented, Sylow
72	C₂×D₈	—	59	1	no	—
73	C₂×D₈	—	59	2	no	—
74	C₂×D₈	—	118	1	no	—
75	#127	—	118	1	no	—
76	C₄×D₄	—	118	1	no	—
77	D₁₆	—	118	4	no	—
78	C₂×C₁₆	C₂×D₅₉	1	2	yes	—
79	#107	—	59	1	no	—
80	#102	—	118	1	no	—
81	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
82	C₂×D₄	—	118	1	no	—
83	C₂×D₄	—	118	2	no	—
84	C₂×D₄	—	236	1	no	—
85	C₂×D₄	—	236	1	no	—
86	C₂²×C₄	—	236	1	no	—
87	D₈	—	59	2	no	—
88	D₈	—	118	2	no	—
89	D₈	—	118	1	no	—
90	D₈	—	118	2	no	—
91	D₈	—	236	2	no	—
92	C₂×C₈	D₂₃₆	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
94	C₄⋊C₄	—	118	1	no	—
95	C₂²⋊C₄	—	236	1	no	—
96	C₄²	D₂₃₆	1	1	yes	characteristic
97	C₁₆	—	2	2	no	—
98	C₂³	—	236	2	no	—
99	C₂³	—	236	1	no	—
100	D₄	—	118	2	no	—
101	D₄	—	472	2	no	—
102	D₄	—	472	2	no	—
103	D₄	—	236	2	no	—
104	D₄	—	236	2	no	—
105	D₄	—	236	1	no	—
106	C₂×C₄	—	472	2	no	—
107	C₂×C₄	—	236	1	no	—
108	C₂×C₄	D₄₇₂	1	1	yes	characteristic
109	C₂×C₄	D₄₇₂	1	1	yes	characteristic
110	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
111	C₈	—	2	1	no	—
112	C₈	D₄×D₅₉	1	1	yes	characteristic
113	C₈	#11745689	1	1	yes	characteristic
114	C₂²	—	472	2	no	—
115	C₂²	—	472	2	no	—
116	C₂²	—	472	2	no	—
117	C₂²	—	472	1	no	—
118	C₂²	—	472	1	no	—
119	C₂²	—	236	2	no	—
120	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	472	1	no	—
123	C₄	D₉₄₄	1	2	yes	complemented
124	C₄	#14273702	1	1	yes	characteristic
125	C₄	#14273755	1	1	yes	characteristic
126	C₂	—	944	2	no	—
127	C₂	—	472	2	no	—
128	C₂	#16837598	1	1	yes	characteristic
129	C₂	C₂×D₉₄₄	1	1	yes	characteristic
130	C₂	#16837840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₉₄₄

C₂

yes

maximal, complemented

#16837779

C₂

yes

maximal, characteristic, complemented

C₄×D₄₇₂

C₂

yes

maximal, characteristic, complemented

#16837204

C₂

yes

maximal, complemented

C₅₉×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

—

C₂×D₄₇₂

C₂²

yes

—

C₂×D₄₇₂

C₂²

yes

characteristic

#14273724

—

C₄×D₂₃₆

—

D₉₄₄

—

#14273661

C₂²

yes

characteristic

#14273649

—

C₂×C₁₆×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

D₄₇₂

—

D₄₇₂

—

D₄₇₂

—

D₄₇₂

—

D₄₇₂

D₄

yes

complemented

C₁₆×C₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

D₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×D₁₆

yes

characteristic

C₂×C₅₉

#509

yes

characteristic

D₅₉

—

D₅₉

—

C₂×D₁₆

—

#493

—

C₄×D₈

—

C₄⋊C₁₆

D₅₉

yes

characteristic, complemented

#357

—

C₅₉

#2206

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

118

—

#127

—

118

—

C₄×D₄

—

118

—

D₁₆

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

—

#107

—

#102

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

C₂²×C₄

—

236

—

D₈

—

D₈

—

118

—

D₈

—

118

—

D₈

—

118

—

D₈

—

236

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

236

—

C₄²

D₂₃₆

yes

characteristic

C₁₆

—

C₂³

—

236

—

C₂³

—

236

—

100

D₄

—

118

—

101

D₄

—

472

—

102

D₄

—

472

—

103

D₄

—

236

—

104

D₄

—

236

—

105

D₄

—

236

—

106

C₂×C₄

—

472

—

107

C₂×C₄

—

236

—

108

C₂×C₄

D₄₇₂

yes

characteristic

109

C₂×C₄

D₄₇₂

yes

characteristic

110

C₂×C₄

C₂×D₂₃₆

yes

characteristic

111

C₈

—

112

C₈

D₄×D₅₉

yes

characteristic

113

C₈

#11745689

yes

characteristic

114

C₂²

—

472

—

115

C₂²

—

472

—

116

C₂²

—

472

—

117

C₂²

—

472

—

118

C₂²

—

472

—

119

C₂²

—

236

—

120

C₂²

C₂×D₄₇₂

yes

characteristic

121

C₄

—

122

C₄

—

472

—

123

C₄

D₉₄₄

yes

complemented

124

C₄

#14273702

yes

characteristic

125

C₄

#14273755

yes

characteristic

126

C₂

—

944

—

127

C₂

—

472

—

128

C₂

#16837598

yes

characteristic

129

C₂

C₂×D₉₄₄

yes

characteristic

130

C₂

#16837840

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	#2206
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

#2206

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
1. C₂×D₉₄₄	C₂	semidirect
2. #16837779	C₂	semidirect
3. C₄×D₄₇₂	C₂	semidirect
4. #16837204	C₂	semidirect
5. C₅₉×C₄⋊C₁₆	C₂	semidirect
30. D₄₇₂	D₄	semidirect
56. C₄×C₅₉	D₁₆	semidirect
69. C₄⋊C₁₆	D₅₉	semidirect
71. C₅₉	#2206	semidirect
123. C₄	D₉₄₄	semidirect

H ≅ G/N

type

1. C₂×D₉₄₄

C₂

semidirect

2. #16837779

C₂

semidirect

3. C₄×D₄₇₂

C₂

semidirect

4. #16837204

C₂

semidirect

5. C₅₉×C₄⋊C₁₆

C₂

semidirect

30. D₄₇₂

D₄

semidirect

56. C₄×C₅₉

D₁₆

semidirect

69. C₄⋊C₁₆

D₅₉

semidirect

71. C₅₉

#2206

semidirect

123. C₄

D₉₄₄

semidirect

Subgroups — #21426132

Conjugacy classes of subgroups

Complements