Group #21422853 — subgroups

Subgroups — #21422853

Order 7552.

184 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 81 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	#16837424	C₂	1	2	yes	maximal
2	C₄×C₈×D₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837155	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837154	C₂	1	2	yes	maximal, complemented
5	C₅₉×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745659	—	2	2	no	—
7	C₂×C₈×D₅₉	C₄	1	1	yes	characteristic
8	C₂×C₈×D₅₉	C₄	1	1	yes	characteristic
9	C₄²×D₅₉	C₂²	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
11	Dic₅₉⋊C₈	—	2	2	no	—
12	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₈×Dic₅₉	C₄	1	1	yes	characteristic
14	C₈×Dic₅₉	C₄	1	1	yes	characteristic
15	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
19	C₂×C₄×D₅₉	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₅₉	—	2	2	no	—
21	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₉	C₂³	1	1	yes	characteristic
24	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₅₉	D₄	1	1	yes	characteristic
26	C₄×Dic₅₉	Q₈	1	1	yes	characteristic
27	C₂×C₅₉⋊C₈	—	2	2	no	—
28	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	#11745659	—	4	4	no	—
31	C₈×D₅₉	—	2	2	no	—
32	C₈×D₅₉	—	2	2	no	—
33	C₂²×D₅₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄×D₅₉	—	2	1	no	—
41	C₄×D₅₉	—	2	1	no	—
42	C₄×D₅₉	—	2	2	no	—
43	C₄×D₅₉	D₈	1	2	yes	—
44	C₄×D₅₉	Q₁₆	1	2	yes	—
45	C₈×C₅₉	—	2	1	no	—
46	C₈×C₅₉	—	4	2	no	—
47	C₈×C₅₉	M₁₆	1	2	yes	complemented
48	C₅₉⋊C₈	—	2	1	no	—
49	C₅₉⋊C₈	—	4	2	no	—
50	C₅₉⋊C₈	M₁₆	1	2	yes	complemented
51	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×D₅₉	—	2	1	no	—
53	C₂×D₅₉	—	2	1	no	—
54	C₂×D₅₉	#107	1	2	yes	—
55	C₄×C₅₉	—	2	1	no	—
56	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
57	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
58	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
59	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
60	Dic₅₉	—	2	1	no	—
61	Dic₅₉	#107	1	2	yes	—
62	Dic₅₉	#108	1	2	yes	—
63	#1567	—	59	1	no	maximal, Sylow
64	C₂×C₅₉	#423	1	1	yes	characteristic
65	C₂×C₅₉	C₂×#107	1	1	yes	characteristic
66	C₂×C₅₉	C₂×#108	1	1	yes	characteristic
67	D₅₉	—	2	2	no	—
68	#423	—	59	2	no	—
69	C₂×C₄×C₈	—	59	1	no	—
70	#335	—	59	2	no	—
71	#335	—	59	1	no	—
72	#335	D₅₉	1	1	yes	characteristic, complemented
73	C₅₉	#1567	1	1	yes	characteristic, complemented, Sylow
74	C₂×M₁₆	—	118	2	no	—
75	C₂²×C₈	—	59	1	no	—
76	C₂²×C₈	—	59	1	no	—
77	C₂×C₄²	—	59	1	no	—
78	C₄⋊C₈	—	59	2	no	—
79	C₄⋊C₈	—	118	2	no	—
80	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
81	C₄×C₈	—	59	1	no	—
82	C₄×C₈	—	59	1	no	—
83	C₄×C₈	—	59	1	no	—
84	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
85	C₂²×C₄	—	59	1	no	—
86	C₂²×C₄	—	59	1	no	—
87	C₂²×C₄	—	59	1	no	—
88	M₁₆	—	236	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	59	1	no	—
91	C₂×C₈	—	59	1	no	—
92	C₂×C₈	—	118	2	no	—
93	C₂×C₈	—	118	2	no	—
94	C₂×C₈	—	118	2	no	—
95	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
96	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
97	C₄²	—	59	1	no	—
98	C₄²	—	59	1	no	—
99	C₄²	—	59	1	no	—
100	C₄²	C₂²×D₅₉	1	1	yes	characteristic
101	C₂³	—	59	1	no	—
102	C₂×C₄	—	59	1	no	—
103	C₂×C₄	—	59	1	no	—
104	C₂×C₄	—	59	2	no	—
105	C₂×C₄	—	59	2	no	—
106	C₂×C₄	—	59	1	no	—
107	C₂×C₄	—	118	2	no	—
108	C₂×C₄	—	118	1	no	—
109	C₂×C₄	—	118	1	no	—
110	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
111	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
112	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
113	C₈	—	2	1	no	—
114	C₈	—	4	2	no	—
115	C₈	—	59	2	no	—
116	C₈	—	118	1	no	—
117	C₈	—	236	2	no	—
118	C₈	#11745659	1	2	yes	complemented
119	C₂²	—	59	2	no	—
120	C₂²	—	118	1	no	—
121	C₂²	—	118	1	no	—
122	C₂²	C₄⋊C₄×D₅₉	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	59	2	no	—
125	C₄	—	59	2	no	—
126	C₄	—	118	1	no	—
127	C₄	M₁₆×D₅₉	1	1	yes	characteristic
128	C₄	C₂×#11745659	1	1	yes	characteristic
129	C₄	D₈×D₅₉	1	1	yes	characteristic
130	C₄	Q₁₆×D₅₉	1	1	yes	characteristic
131	C₂	—	118	2	no	—
132	C₂	#16837424	1	1	yes	characteristic
133	C₂	#107×D₅₉	1	1	yes	characteristic
134	C₂	#108×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837424

C₂

yes

maximal

C₄×C₈×D₅₉

C₂

yes

maximal, characteristic

#16837155

C₂

yes

maximal, characteristic, complemented

#16837154

C₂

yes

maximal, complemented

C₅₉×#335

C₂

yes

maximal, characteristic, complemented

C₂×#11745659

—

C₂×C₈×D₅₉

C₄

yes

characteristic

C₂×C₈×D₅₉

C₄

yes

characteristic

C₄²×D₅₉

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

Dic₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₅₉

C₄

yes

characteristic

C₈×Dic₅₉

C₄

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

Q₈

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

#11745659

—

C₈×D₅₉

—

C₈×D₅₉

—

C₂²×D₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×Dic₅₉

C₄⋊C₄

yes

characteristic

C₂×Dic₅₉

C₄⋊C₄

yes

characteristic

C₂×Dic₅₉

C₄⋊C₄

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

D₈

yes

—

C₄×D₅₉

Q₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

M₁₆

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

M₁₆

yes

complemented

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

#107

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

#107

yes

—

Dic₅₉

#108

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#423

—

C₂×C₄×C₈

—

#335

—

#335

—

#335

D₅₉

yes

characteristic, complemented

C₅₉

#1567

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₂²×D₅₉

yes

characteristic

101

C₂³

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

118

—

108

C₂×C₄

—

118

—

109

C₂×C₄

—

118

—

110

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

111

C₂×C₄

D₄×D₅₉

yes

characteristic

112

C₂×C₄

Q₈×D₅₉

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

118

—

117

C₈

—

236

—

118

C₈

#11745659

yes

complemented

119

C₂²

—

120

C₂²

—

118

—

121

C₂²

—

118

—

122

C₂²

C₄⋊C₄×D₅₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

118

—

127

C₄

M₁₆×D₅₉

yes

characteristic

128

C₄

C₂×#11745659

yes

characteristic

129

C₄

D₈×D₅₉

yes

characteristic

130

C₄

Q₁₆×D₅₉

yes

characteristic

131

C₂

—

118

—

132

C₂

#16837424

yes

characteristic

133

C₂

#107×D₅₉

yes

characteristic

134

C₂

#108×D₅₉

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#335	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#335	D₅₉
hypocenter	C₅₉	#1567
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#335

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
3. #16837155	C₂	semidirect
4. #16837154	C₂	semidirect
5. C₅₉×#335	C₂	semidirect
47. C₈×C₅₉	M₁₆	semidirect
50. C₅₉⋊C₈	M₁₆	semidirect
72. #335	D₅₉	semidirect
73. C₅₉	#1567	semidirect
118. C₈	#11745659	semidirect

H ≅ G/N

type

3. #16837155

C₂

semidirect

4. #16837154

C₂

semidirect

5. C₅₉×#335

C₂

semidirect

47. C₈×C₅₉

M₁₆

semidirect

50. C₅₉⋊C₈

M₁₆

semidirect

72. #335

D₅₉

semidirect

73. C₅₉

#1567

semidirect

118. C₈

#11745659

semidirect

Subgroups — #21422853

Conjugacy classes of subgroups

Complements