Group #21424920 — subgroups

Subgroups — #21424920

Order 7552.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14273724	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273649	C₂	1	4	yes	maximal, complemented
3	#16837268	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#97×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₃₆	—	2	2	no	—
6	C₂²×D₂₃₆	C₂²	1	2	yes	—
7	#14273724	—	2	2	no	—
8	#14273724	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	#14273649	—	2	8	no	—
11	C₅₉⋊#95	—	2	2	no	—
12	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
14	#97×C₅₉	—	2	2	no	—
15	C₂³×D₅₉	—	2	2	no	—
16	C₂²×C₄×C₅₉	D₄	1	2	yes	—
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₉	—	2	2	no	—
19	C₂×D₂₃₆	—	2	4	no	—
20	C₂×D₂₃₆	—	2	4	no	—
21	C₂×D₂₃₆	—	2	8	no	—
22	C₂×D₂₃₆	—	2	2	no	—
23	C₂×D₂₃₆	—	4	8	no	—
24	C₂×D₂₃₆	C₂×C₄	1	4	yes	—
25	C₂×C₈×C₅₉	—	2	2	no	—
26	C₂×C₈×C₅₉	—	2	2	no	—
27	C₂×C₈×C₅₉	D₄	1	4	yes	complemented
28	C₄²×C₅₉	C₂×C₄	1	2	yes	—
29	C₄²×C₅₉	D₄	1	2	yes	—
30	C₄⋊Dic₅₉	—	2	4	no	—
31	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₅₉	—	2	4	no	—
33	C₂²×D₅₉	—	4	2	no	—
34	C₂²×D₅₉	—	4	2	no	—
35	C₂²×D₅₉	—	4	8	no	—
36	C₂×C₄×C₅₉	—	2	4	no	—
37	C₂×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	#41	1	2	yes	—
42	C₂×Dic₅₉	—	4	2	no	—
43	C₂×Dic₅₉	—	4	4	no	—
44	D₂₃₆	—	2	8	no	—
45	D₂₃₆	—	4	16	no	—
46	D₂₃₆	—	4	4	no	—
47	C₈×C₅₉	—	2	4	no	—
48	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₅₉	#124	1	1	yes	characteristic
50	C₂²×C₅₉	#127	1	1	yes	characteristic
51	C₂²×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
52	C₂×D₅₉	—	4	4	no	—
53	C₂×D₅₉	—	4	8	no	—
54	C₂×D₅₉	—	8	4	no	—
55	C₂×D₅₉	—	8	4	no	—
56	C₄×C₅₉	—	2	4	no	—
57	C₄×C₅₉	C₄×D₄	1	2	yes	—
58	C₄×C₅₉	#121	1	2	yes	—
59	Dic₅₉	—	8	2	no	—
60	#1961	—	59	1	no	maximal, Sylow
61	C₂×C₅₉	#390	1	1	yes	characteristic
62	C₂×C₅₉	#418	1	2	yes	—
63	C₂×C₅₉	#490	1	2	yes	—
64	C₂×C₅₉	#496	1	2	yes	—
65	D₅₉	—	8	4	no	—
66	C₂×#127	—	59	1	no	—
67	C₂×#102	—	59	4	no	—
68	C₂×#97	D₅₉	1	1	yes	characteristic, complemented
69	#382	—	59	1	no	—
70	C₅₉	#1961	1	1	yes	characteristic, complemented, Sylow
71	C₂²×D₄	—	59	2	no	—
72	C₂²×D₄	—	118	2	no	—
73	C₂²×C₈	C₂×D₅₉	1	2	yes	—
74	#127	—	59	4	no	—
75	#127	—	118	2	no	—
76	C₂×C₄⋊C₄	—	59	2	no	—
77	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
78	#102	—	118	8	no	—
79	#97	—	2	2	no	—
80	#95	—	118	2	no	—
81	C₂⁴	—	118	2	no	—
82	C₂×D₄	—	59	4	no	—
83	C₂×D₄	—	118	8	no	—
84	C₂×D₄	—	118	4	no	—
85	C₂×D₄	—	118	4	no	—
86	C₂×D₄	—	118	2	no	—
87	C₂×D₄	—	236	8	no	—
88	C₂²×C₄	—	118	2	no	—
89	C₂²×C₄	D₂₃₆	1	2	yes	—
90	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₂₃₆	1	4	yes	complemented
94	C₄⋊C₄	—	118	4	no	—
95	C₄²	C₄×D₅₉	1	2	yes	—
96	C₄²	C₂²⋊D₅₉	1	2	yes	—
97	C₂³	—	118	4	no	—
98	C₂³	—	236	2	no	—
99	C₂³	—	236	2	no	—
100	C₂³	—	236	8	no	—
101	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
102	D₄	—	118	8	no	—
103	D₄	—	236	16	no	—
104	D₄	—	236	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	236	2	no	—
107	C₂×C₄	—	236	4	no	—
108	C₂×C₄	(C₂×D₅₉)⋊C₄	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
110	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
111	C₂×C₄	#11745685	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
113	C₈	—	2	4	no	—
114	C₂²	—	472	4	no	—
115	C₂²	—	472	4	no	—
116	C₂²	—	236	8	no	—
117	C₂²	—	236	4	no	—
118	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
119	C₂²	#14273715	1	1	yes	characteristic
120	C₂²	#14273724	1	1	yes	characteristic
121	C₂²	#14273755	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	472	2	no	—
124	C₄	C₄×D₂₃₆	1	2	yes	—
125	C₄	#14273701	1	2	yes	—
126	C₂	—	472	4	no	—
127	C₂	#16837304	1	1	yes	characteristic
128	C₂	#16837408	1	2	yes	—
129	C₂	#16837767	1	2	yes	—
130	C₂	#16837791	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273724

C₂

yes

maximal, characteristic

C₂×#14273649

C₂

yes

maximal, complemented

#16837268

C₂

yes

maximal, characteristic, complemented

C₂×#97×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

—

C₂²×D₂₃₆

C₂²

yes

—

#14273724

—

#14273724

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273649

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#97×C₅₉

—

C₂³×D₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

complemented

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

#121

yes

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#490

yes

—

C₂×C₅₉

#496

yes

—

D₅₉

—

C₂×#127

—

C₂×#102

—

C₂×#97

D₅₉

yes

characteristic, complemented

#382

—

C₅₉

#1961

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

118

—

C₂²×C₈

C₂×D₅₉

yes

—

#127

—

#127

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#102

—

118

—

#97

—

#95

—

118

—

C₂⁴

—

118

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂²×C₄

—

118

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₂₃₆

yes

complemented

C₄⋊C₄

—

118

—

C₄²

C₄×D₅₉

yes

—

C₄²

C₂²⋊D₅₉

yes

—

C₂³

—

118

—

C₂³

—

236

—

C₂³

—

236

—

100

C₂³

—

236

—

101

C₂³

C₂×D₂₃₆

yes

characteristic

102

D₄

—

118

—

103

D₄

—

236

—

104

D₄

—

236

—

105

C₂×C₄

—

106

C₂×C₄

—

236

—

107

C₂×C₄

—

236

—

108

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

109

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

110

C₂×C₄

C₂×D₂₃₆

yes

—

111

C₂×C₄

#11745685

yes

—

112

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

113

C₈

—

114

C₂²

—

472

—

115

C₂²

—

472

—

116

C₂²

—

236

—

117

C₂²

—

236

—

118

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

119

C₂²

#14273715

yes

characteristic

120

C₂²

#14273724

yes

characteristic

121

C₂²

#14273755

yes

—

122

C₄

—

123

C₄

—

472

—

124

C₄

C₄×D₂₃₆

yes

—

125

C₄

#14273701

yes

—

126

C₂

—

472

—

127

C₂

#16837304

yes

characteristic

128

C₂

#16837408

yes

—

129

C₂

#16837767

yes

—

130

C₂

#16837791

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#97×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#97	D₅₉
hypocenter	C₅₉	#1961
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#97×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×#97

D₅₉

hypocenter

C₅₉

#1961

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#14273649	C₂	semidirect
3. #16837268	C₂	semidirect
4. C₂×#97×C₅₉	C₂	semidirect
8. #14273724	C₄	semidirect
27. C₂×C₈×C₅₉	D₄	semidirect
68. C₂×#97	D₅₉	semidirect
70. C₅₉	#1961	semidirect
93. C₂×C₈	D₂₃₆	semidirect

H ≅ G/N

type

2. C₂×#14273649

C₂

semidirect

3. #16837268

C₂

semidirect

4. C₂×#97×C₅₉

C₂

semidirect

8. #14273724

C₄

semidirect

27. C₂×C₈×C₅₉

D₄

semidirect

68. C₂×#97

D₅₉

semidirect

70. C₅₉

#1961

semidirect

93. C₂×C₈

D₂₃₆

semidirect

Subgroups — #21424920

Conjugacy classes of subgroups

Complements