Group #21434616 — subgroups

Subgroups — C₂×#16837812 #21434616

Order 7552.

384 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14273727	C₂	1	1	yes	maximal, characteristic
2	#16837812	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14273660	C₂	1	2	yes	maximal
4	C₂×#14273658	C₂	1	2	yes	maximal
5	C₂×#14273638	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₁₈	—	2	1	no	—
8	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
9	#14273727	—	2	2	no	—
10	#14273727	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₅₉	—	2	1	no	—
12	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
13	#14273660	C₂²	1	8	yes	—
14	#14273658	C₂²	1	8	yes	—
15	#14273638	C₂²	1	4	yes	—
16	#125×C₅₉	—	2	2	no	—
17	#125×C₅₉	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
19	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
20	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	—	2	1	no	—
24	C₂²×C₄×C₅₉	—	2	1	no	—
25	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
28	C₂²×Dic₅₉	—	2	2	no	—
29	C₂×Dic₁₁₈	—	2	4	no	—
30	C₂×Dic₁₁₈	—	4	4	no	—
31	C₄⋊C₄×C₅₉	—	2	4	no	—
32	C₄⋊C₄×C₅₉	—	2	4	no	—
33	C₄⋊C₄×C₅₉	—	2	1	no	—
34	C₄⋊C₄×C₅₉	—	2	1	no	—
35	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
36	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
37	C₄²×C₅₉	D₄	1	2	yes	—
38	C₄²×C₅₉	C₂³	1	2	yes	—
39	C₄⋊Dic₅₉	—	2	4	no	—
40	C₄⋊Dic₅₉	—	2	2	no	—
41	C₄⋊Dic₅₉	C₂³	1	4	yes	—
42	C₂×C₅₉⋊C₈	Q₈	1	8	yes	complemented
43	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
44	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	2	no	—
48	C₂×C₄×C₅₉	—	2	2	no	—
49	C₂×C₄×C₅₉	—	4	2	no	—
50	C₂×C₄×C₅₉	—	4	2	no	—
51	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×Q₈	1	4	yes	—
55	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₅₉	—	2	4	no	—
57	C₂×Dic₅₉	—	4	4	no	—
58	Dic₁₁₈	—	4	8	no	—
59	C₅₉⋊C₈	C₂×Q₈	1	8	yes	complemented
60	C₂²×C₅₉	C₄⋊Q₈	1	2	yes	—
61	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₂²×C₅₉	#137	1	2	yes	—
63	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₅₉	—	2	2	no	—
65	C₄×C₅₉	—	2	2	no	—
66	C₄×C₅₉	—	4	2	no	—
67	C₄×C₅₉	—	4	2	no	—
68	C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
69	C₄×C₅₉	C₂²×Q₈	1	2	yes	—
70	Dic₅₉	—	4	4	no	—
71	C₂×#501	—	59	1	no	maximal, Sylow
72	C₂×C₅₉	#501	1	4	yes	complemented
73	C₂×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	—	59	1	no	—
77	C₂×#125	D₅₉	1	1	yes	characteristic
78	#501	—	59	8	no	—
79	C₂×#107	—	59	2	no	—
80	C₂×#106	—	59	2	no	—
81	C₂×#97	—	59	1	no	—
82	C₅₉	C₂×#501	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	118	1	no	—
84	C₂²×C₈	—	59	2	no	—
85	C₄⋊Q₈	—	59	4	no	—
86	C₄⋊Q₈	—	118	2	no	—
87	#125	—	2	2	no	—
88	#125	C₂×D₅₉	1	4	yes	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	59	2	no	—
92	C₂×C₄⋊C₄	—	118	1	no	—
93	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
96	#107	—	59	8	no	—
97	#106	—	59	8	no	—
98	#97	—	59	4	no	—
99	C₂×Q₈	—	118	4	no	—
100	C₂×Q₈	—	236	4	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	118	2	no	—
104	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
107	C₂×C₈	—	59	8	no	—
108	C₂×C₈	—	59	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	59	4	no	—
114	C₄⋊C₄	—	118	2	no	—
115	C₄⋊C₄	—	118	4	no	—
116	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
117	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
118	C₄²	C₂²⋊D₅₉	1	2	yes	—
119	C₄²	C₂²×D₅₉	1	2	yes	—
120	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
121	Q₈	—	236	8	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	118	4	no	—
129	C₂×C₄	—	236	4	no	—
130	C₂×C₄	Q₈×D₅₉	1	4	yes	—
131	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
135	C₈	—	59	8	no	—
136	C₂²	#14273725	1	2	yes	—
137	C₂²	#14273752	1	2	yes	—
138	C₂²	#14273757	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	4	2	no	—
144	C₄	—	236	4	no	—
145	C₄	#14273725	1	2	yes	—
146	C₄	C₂×Q₈×D₅₉	1	2	yes	—
147	C₂	#16837812	1	4	yes	direct product factor
148	C₂	C₂×#14273725	1	1	yes	characteristic
149	C₂	C₂×#14273752	1	1	yes	characteristic
150	C₂	C₂×#14273757	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273727

C₂

yes

maximal, characteristic

#16837812

C₂

yes

maximal, direct product factor

C₂×#14273660

C₂

yes

maximal

C₂×#14273658

C₂

yes

maximal

C₂×#14273638

C₂

yes

maximal, characteristic

C₂×#125×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

—

C₂²×C₅₉⋊C₈

C₂²

yes

—

#14273727

—

#14273727

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273660

C₂²

yes

—

#14273658

C₂²

yes

—

#14273638

C₂²

yes

—

#125×C₅₉

—

#125×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

Q₈

yes

complemented

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₅₉⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₅₉

C₄⋊Q₈

yes

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂²×C₅₉

#137

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄⋊Q₈

yes

—

C₄×C₅₉

C₂²×Q₈

yes

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×C₄⋊Q₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₄⋊Q₈

—

C₂×#125

D₅₉

yes

—

—

—

C₂×#97

—

C₅₉

C₂×#501

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

118

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

#125

—

#125

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄²

C₂×D₅₉

yes

—

—

—

C₂×Q₈

—

118

—

100

C₂×Q₈

—

236

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

118

—

104

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

105

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

106

C₂²×C₄

C₂²×D₅₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

118

—

115

C₄⋊C₄

—

118

—

116

C₄⋊C₄

C₂²×D₅₉

yes

—

117

C₄⋊C₄

C₂²×D₅₉

yes

—

118

C₄²

C₂²⋊D₅₉

yes

—

119

C₄²

C₂²×D₅₉

yes

—

120

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

121

Q₈

—

236

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

118

—

129

C₂×C₄

—

236

—

130

C₂×C₄

Q₈×D₅₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

132

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

133

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

134

C₂×C₄

C₂³×D₅₉

yes

characteristic

135

C₈

—

136

C₂²

#14273725

yes

—

137

C₂²

#14273752

yes

—

138

C₂²

#14273757

yes

—

139

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

236

—

145

C₄

#14273725

yes

—

146

C₄

C₂×Q₈×D₅₉

yes

—

147

C₂

#16837812

yes

direct product factor

148

C₂

C₂×#14273725

yes

characteristic

149

C₂

C₂×#14273752

yes

characteristic

150

C₂

C₂×#14273757

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×#125×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#125	D₅₉
hypocenter	C₅₉	C₂×#501
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂×#125×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #16837812	C₂	direct (N × H)
42. C₂×C₅₉⋊C₈	Q₈	semidirect
59. C₅₉⋊C₈	C₂×Q₈	semidirect
72. C₂×C₅₉	#501	semidirect
82. C₅₉	C₂×#501	semidirect
147. C₂	#16837812	direct (N × H)

H ≅ G/N

type

2. #16837812

C₂

direct (N × H)

42. C₂×C₅₉⋊C₈

Q₈

semidirect

59. C₅₉⋊C₈

C₂×Q₈

semidirect

72. C₂×C₅₉

#501

semidirect

82. C₅₉

C₂×#501

semidirect

147. C₂

#16837812

direct (N × H)

Subgroups — C₂×#16837812 #21434616

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16837812 #21434616

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837812 #21434616