Group #21422948 — subgroups

Subgroups — #21422948

Order 7552.

152 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16837532	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
4	#16837155	C₂	1	1	yes	maximal, characteristic
5	#16837134	C₂	1	1	yes	maximal, characteristic
6	#16837134	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₃₆	C₄	1	1	yes	characteristic
9	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
11	#14273661	C₄	1	1	yes	characteristic
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₅₉⋊C₈	—	2	1	no	—
15	Dic₅₉⋊C₈	—	2	1	no	—
16	#14273652	C₄	1	1	yes	characteristic
17	#14273652	C₄	1	1	yes	characteristic
18	#14273637	—	2	1	no	—
19	C₈×Dic₅₉	—	2	1	no	—
20	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₉	—	2	1	no	—
26	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
33	Dic₅₉⋊C₄	—	2	1	no	—
34	Dic₅₉⋊C₄	—	2	1	no	—
35	C₄×Dic₅₉	—	2	1	no	—
36	C₄×Dic₅₉	—	2	1	no	—
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	Dic₂₃₆	—	2	2	no	—
40	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
43	C₂×Dic₅₉	—	2	1	no	—
44	C₂×Dic₅₉	—	2	1	no	—
45	C₂×Dic₅₉	—	2	1	no	—
46	C₂×Dic₅₉	—	2	1	no	—
47	Dic₁₁₈	—	2	1	no	—
48	Dic₁₁₈	—	2	1	no	—
49	Dic₁₁₈	—	2	1	no	—
50	Dic₁₁₈	M₁₆	1	2	yes	—
51	C₈×C₅₉	—	2	1	no	—
52	C₈×C₅₉	—	2	1	no	—
53	C₈×C₅₉	—	2	1	no	—
54	C₈×C₅₉	—	4	1	no	—
55	C₈×C₅₉	Q₁₆	1	2	yes	complemented
56	C₅₉⋊C₈	—	4	1	no	—
57	C₅₉⋊C₈	—	4	1	no	—
58	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
59	C₄×C₅₉	—	2	1	no	—
60	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
61	C₄×C₅₉	#131	1	1	yes	characteristic
62	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₅₉	#135	1	1	yes	characteristic
64	Dic₅₉	—	2	2	no	—
65	Dic₅₉	—	4	1	no	—
66	Dic₅₉	—	4	1	no	—
67	Dic₅₉	—	4	1	no	—
68	#1584	—	59	1	no	maximal, Sylow
69	C₂×C₅₉	#435	1	1	yes	characteristic
70	C₂×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
71	C₂×C₅₉	#444	1	1	yes	characteristic
72	#446	—	59	1	no	—
73	C₈×Q₈	—	59	1	no	—
74	C₄×Q₁₆	—	59	1	no	—
75	#335	—	59	1	no	—
76	Q₈⋊C₈	—	59	1	no	—
77	Q₈⋊C₈	—	59	1	no	—
78	#322	D₅₉	1	1	yes	characteristic
79	C₅₉	#1584	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	59	1	no	—
81	C₄×Q₈	—	59	1	no	—
82	C₄×Q₈	—	59	1	no	—
83	#107	—	59	1	no	—
84	C₄⋊C₈	—	59	1	no	—
85	C₄⋊C₈	—	59	1	no	—
86	C₄⋊C₈	—	118	1	no	—
87	C₄⋊C₈	—	118	1	no	—
88	#103	—	59	1	no	—
89	#103	—	59	1	no	—
90	#97	—	118	1	no	—
91	C₄×C₈	—	118	1	no	—
92	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
93	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
94	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
95	C₂×Q₈	—	59	1	no	—
96	C₂×Q₈	—	59	1	no	—
97	Q₁₆	—	118	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	118	1	no	—
100	C₂×C₈	—	118	1	no	—
101	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
102	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
103	C₂×C₈	D₂₃₆	1	1	yes	characteristic
104	C₂×C₈	D₂₃₆	1	1	yes	characteristic
105	C₄⋊C₄	—	59	1	no	—
106	C₄⋊C₄	—	59	1	no	—
107	C₄⋊C₄	—	118	1	no	—
108	C₄⋊C₄	—	118	1	no	—
109	C₄²	—	118	1	no	—
110	C₄²	—	118	1	no	—
111	C₄²	C₂²×D₅₉	1	1	yes	characteristic
112	Q₈	—	59	2	no	—
113	Q₈	—	118	1	no	—
114	Q₈	—	118	1	no	—
115	Q₈	—	118	1	no	—
116	C₂×C₄	—	118	1	no	—
117	C₂×C₄	—	118	1	no	—
118	C₂×C₄	—	118	1	no	—
119	C₂×C₄	—	118	1	no	—
120	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
121	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
122	C₂×C₄	#11745685	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	2	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	236	1	no	—
128	C₈	—	236	1	no	—
129	C₈	Dic₂₃₆	1	2	yes	complemented
130	C₂²	C₄×D₂₃₆	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	118	2	no	—
133	C₄	—	236	1	no	—
134	C₄	—	236	1	no	—
135	C₄	—	236	1	no	—
136	C₄	M₁₆×D₅₉	1	1	yes	characteristic
137	C₄	#14273735	1	1	yes	characteristic
138	C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
139	C₄	#14273749	1	1	yes	characteristic
140	C₂	#16837475	1	1	yes	characteristic
141	C₂	C₄×Dic₂₃₆	1	1	yes	characteristic
142	C₂	#16837525	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837532

C₂

yes

maximal, characteristic

C₈×Dic₁₁₈

C₂

yes

maximal, characteristic

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic

#16837155

C₂

yes

maximal, characteristic

#16837134

C₂

yes

maximal, characteristic

#16837134

C₂

yes

maximal, characteristic

C₅₉×#322

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₄

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273661

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

Dic₅₉⋊C₈

—

#14273652

C₄

yes

characteristic

#14273652

C₄

yes

characteristic

#14273637

—

C₈×Dic₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

C₂×C₄

yes

characteristic

C₂×Dic₁₁₈

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

M₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

Q₁₆

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

#131

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₄×C₅₉

#135

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₄×Q₁₆

yes

C₂×C₅₉

yes

—

C₈×Q₈

—

C₄×Q₁₆

—

#335

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

D₅₉

yes

characteristic

C₅₉

#1584

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

118

—

#103

—

#103

—

#97

—

118

—

C₄×C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

100

C₂×C₈

—

118

—

101

C₂×C₈

C₄×D₅₉

yes

characteristic

102

C₂×C₈

C₄×D₅₉

yes

characteristic

103

C₂×C₈

D₂₃₆

yes

characteristic

104

C₂×C₈

D₂₃₆

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

118

—

108

C₄⋊C₄

—

118

—

109

C₄²

—

118

—

110

C₄²

—

118

—

111

C₄²

C₂²×D₅₉

yes

characteristic

112

Q₈

—

113

Q₈

—

118

—

114

Q₈

—

118

—

115

Q₈

—

118

—

116

C₂×C₄

—

118

—

117

C₂×C₄

—

118

—

118

C₂×C₄

—

118

—

119

C₂×C₄

—

118

—

120

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

121

C₂×C₄

C₂×D₂₃₆

yes

characteristic

122

C₂×C₄

#11745685

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

236

—

128

C₈

—

236

—

129

C₈

Dic₂₃₆

yes

complemented

130

C₂²

C₄×D₂₃₆

yes

characteristic

131

C₄

—

132

C₄

—

118

—

133

C₄

—

236

—

134

C₄

—

236

—

135

C₄

—

236

—

136

C₄

M₁₆×D₅₉

yes

characteristic

137

C₄

#14273735

yes

characteristic

138

C₄

C₂×Dic₂₃₆

yes

characteristic

139

C₄

#14273749

yes

characteristic

140

C₂

#16837475

yes

characteristic

141

C₂

C₄×Dic₂₃₆

yes

characteristic

142

C₂

#16837525

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#322	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	#322	D₅₉
hypocenter	C₅₉	#1584
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#322

C₂

socle

C₂²×C₅₉

C₄×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
55. C₈×C₅₉	Q₁₆	semidirect
79. C₅₉	#1584	semidirect
129. C₈	Dic₂₃₆	semidirect

H ≅ G/N

type

55. C₈×C₅₉

Q₁₆

semidirect

79. C₅₉

#1584

semidirect

129. C₈

Dic₂₃₆

semidirect

Subgroups — #21422948

Conjugacy classes of subgroups

Complements