Group #21424928 — subgroups

Subgroups — #21424928

Order 7552.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16837952	C₂	1	1	yes	maximal, characteristic
2	#16837426	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14273655	C₂	1	4	yes	maximal, complemented
4	C₅₉×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11745685	C₂²	1	2	yes	—
6	C₂×#11745664	C₂²	1	2	yes	—
7	#14273727	C₄	1	1	yes	characteristic, complemented
8	#14273724	C₄	1	1	yes	characteristic, complemented
9	#14273715	C₄	1	2	yes	complemented
10	#14273701	—	2	2	no	—
11	C₄×D₂₃₆	—	2	2	no	—
12	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
13	#14273655	—	2	8	no	—
14	C₂×C₄²×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	#95×C₅₉	—	2	2	no	—
17	C₂²×C₄×C₅₉	—	2	2	no	—
18	C₂²×C₄×C₅₉	D₄	1	2	yes	—
19	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂×C₂²⋊D₅₉	—	2	2	no	—
21	#11745685	—	2	4	no	—
22	#11745685	—	2	4	no	—
23	C₂×D₂₃₆	—	2	2	no	—
24	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₅₉	—	2	2	no	—
26	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
27	C₄²×C₅₉	—	2	2	no	—
28	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₉	D₄	1	2	yes	—
31	C₄²×C₅₉	D₄	1	4	yes	complemented
32	(C₂×D₅₉)⋊C₄	—	2	4	no	—
33	C₄⋊Dic₅₉	—	2	2	no	—
34	#11745664	—	2	4	no	—
35	C₂×C₅₉⋊C₈	—	2	2	no	—
36	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₅₉	—	2	2	no	—
38	C₂×C₄×C₅₉	—	2	4	no	—
39	C₂×C₄×C₅₉	—	4	2	no	—
40	C₂×C₄×C₅₉	—	4	4	no	—
41	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	2	yes	—
47	C₂²⋊D₅₉	—	4	4	no	—
48	C₂×Dic₅₉	—	2	2	no	—
49	D₂₃₆	—	2	4	no	—
50	D₂₃₆	—	4	4	no	—
51	C₄×D₅₉	—	4	2	no	—
52	C₄×D₅₉	—	4	2	no	—
53	Dic₁₁₈	—	2	4	no	—
54	C₅₉⋊C₈	—	4	2	no	—
55	C₂²×C₅₉	—	2	2	no	—
56	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₅₉	#121	1	2	yes	—
58	C₂×D₅₉	—	4	2	no	—
59	C₂×D₅₉	—	4	4	no	—
60	C₄×C₅₉	—	2	2	no	—
61	C₄×C₅₉	—	2	2	no	—
62	C₄×C₅₉	—	4	4	no	—
63	C₄×C₅₉	C₄×D₄	1	2	yes	—
64	C₄×C₅₉	#124	1	1	yes	characteristic
65	C₄×C₅₉	#127	1	1	yes	characteristic
66	Dic₅₉	—	4	2	no	—
67	#1963	—	59	1	no	maximal, Sylow
68	C₂×C₅₉	—	2	2	no	—
69	C₂×C₅₉	#390	1	1	yes	characteristic
70	C₂×C₅₉	#421	1	2	yes	—
71	D₅₉	—	8	2	no	—
72	#532	—	59	1	no	—
73	#423	—	59	1	no	—
74	C₂×#104	—	59	4	no	—
75	#376	D₅₉	1	1	yes	characteristic, complemented
76	C₅₉	#1963	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	59	2	no	—
78	C₂×M₁₆	—	59	2	no	—
79	C₄⋊Q₈	—	59	1	no	—
80	#127	—	59	1	no	—
81	#124	—	59	2	no	—
82	#121	—	118	2	no	—
83	C₄×D₄	—	118	2	no	—
84	C₂×C₄²	C₂×D₅₉	1	2	yes	—
85	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
86	C₄⋊C₈	—	118	2	no	—
87	#104	—	118	8	no	—
88	#95	—	2	2	no	—
89	#41	—	118	4	no	—
90	#41	—	118	4	no	—
91	C₂×Q₈	—	59	2	no	—
92	C₂×D₄	—	59	2	no	—
93	C₂×D₄	—	118	2	no	—
94	C₂×D₄	—	118	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	118	2	no	—
97	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
98	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
99	M₁₆	—	118	4	no	—
100	C₂×C₈	—	118	2	no	—
101	C₄⋊C₄	—	118	2	no	—
102	C₂²⋊C₄	—	118	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₄×D₅₉	1	1	yes	characteristic
105	C₄²	C₄×D₅₉	1	1	yes	characteristic
106	C₄²	D₂₃₆	1	2	yes	—
107	C₄²	D₂₃₆	1	4	yes	complemented
108	C₂³	—	118	2	no	—
109	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
110	Q₈	—	118	4	no	—
111	D₄	—	118	4	no	—
112	D₄	—	236	4	no	—
113	D₄	—	236	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	118	2	no	—
118	C₂×C₄	—	236	2	no	—
119	C₂×C₄	—	236	2	no	—
120	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
121	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
123	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
124	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
125	C₂×C₄	#11745685	1	2	yes	—
126	C₈	—	236	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	236	4	no	—
129	C₂²	—	236	2	no	—
130	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
131	C₂²	#14273701	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	236	2	no	—
136	C₄	C₄×D₂₃₆	1	2	yes	—
137	C₄	#14273715	1	1	yes	characteristic
138	C₄	#14273724	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	472	2	no	—
141	C₂	#16837304	1	1	yes	characteristic
142	C₂	#16837420	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837952

C₂

yes

maximal, characteristic

#16837426

C₂

yes

maximal, characteristic, complemented

C₂×#14273655

C₂

yes

maximal, complemented

C₅₉×#376

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

—

C₂×#11745664

C₂²

yes

—

#14273727

C₄

yes

characteristic, complemented

#14273724

C₄

yes

characteristic, complemented

#14273715

C₄

yes

complemented

#14273701

—

C₄×D₂₃₆

—

C₄⋊(C₅₉⋊C₈)

—

#14273655

—

C₂×C₄²×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

—

#11745685

—

#11745685

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

D₄

yes

complemented

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

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

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

#121

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1963

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#390

yes

characteristic

C₂×C₅₉

#421

yes

—

D₅₉

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#1963

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

118

—

C₄×D₄

—

118

—

C₂×C₄²

C₂×D₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

118

—

#104

—

118

—

#95

—

#41

—

118

—

#41

—

118

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

118

—

100

C₂×C₈

—

118

—

101

C₄⋊C₄

—

118

—

102

C₂²⋊C₄

—

118

—

103

C₄²

—

104

C₄²

C₄×D₅₉

yes

characteristic

105

C₄²

C₄×D₅₉

yes

characteristic

106

C₄²

D₂₃₆

yes

—

107

C₄²

D₂₃₆

yes

complemented

108

C₂³

—

118

—

109

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

110

Q₈

—

118

—

111

D₄

—

118

—

112

D₄

—

236

—

113

D₄

—

236

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

—

118

C₂×C₄

—

236

—

119

C₂×C₄

—

236

—

120

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

121

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

122

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

123

C₂×C₄

C₂×D₂₃₆

yes

—

124

C₂×C₄

C₂×D₂₃₆

yes

characteristic

125

C₂×C₄

#11745685

yes

—

126

C₈

—

236

—

127

C₂²

—

128

C₂²

—

236

—

129

C₂²

—

236

—

130

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

131

C₂²

#14273701

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

236

—

136

C₄

C₄×D₂₃₆

yes

—

137

C₄

#14273715

yes

characteristic

138

C₄

#14273724

yes

characteristic

139

C₂

—

140

C₂

—

472

—

141

C₂

#16837304

yes

characteristic

142

C₂

#16837420

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#376	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#376	D₅₉
hypocenter	C₅₉	#1963
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#376

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
2. #16837426	C₂	semidirect
3. C₂×#14273655	C₂	semidirect
4. C₅₉×#376	C₂	semidirect
7. #14273727	C₄	semidirect
8. #14273724	C₄	semidirect
9. #14273715	C₄	semidirect
31. C₄²×C₅₉	D₄	semidirect
75. #376	D₅₉	semidirect
76. C₅₉	#1963	semidirect
107. C₄²	D₂₃₆	semidirect

H ≅ G/N

type

2. #16837426

C₂

semidirect

3. C₂×#14273655

C₂

semidirect

4. C₅₉×#376

C₂

semidirect

7. #14273727

C₄

semidirect

8. #14273724

C₄

semidirect

9. #14273715

C₄

semidirect

31. C₄²×C₅₉

D₄

semidirect

75. #376

D₅₉

semidirect

76. C₅₉

#1963

semidirect

107. C₄²

D₂₃₆

semidirect

Subgroups — #21424928

Conjugacy classes of subgroups

Complements