Group #21425957 — subgroups

Subgroups — #21425957

Order 7552.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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×#14273671	C₂	1	1	yes	maximal, characteristic
2	#16837496	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837227	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837207	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837177	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#357	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	#14273692	—	2	1	no	—
10	D₄×Dic₅₉	—	2	1	no	—
11	#14273671	C₄	1	4	yes	complemented
12	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14273661	C₂²	1	1	yes	characteristic
14	#14273650	—	2	1	no	—
15	#14273648	—	2	1	no	—
16	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	#107×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×D₄×C₅₉	—	2	1	no	—
21	C₂²×Dic₅₉	—	4	1	no	—
22	C₂×Dic₁₁₈	—	2	1	no	—
23	D₈×C₅₉	—	2	1	no	—
24	D₈×C₅₉	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₉	—	2	1	no	—
27	C₂²⋊Dic₅₉	—	4	1	no	—
28	C₄⋊Dic₅₉	—	2	1	no	—
29	Dic₅₉⋊C₄	—	4	1	no	—
30	C₄×Dic₅₉	—	4	1	no	—
31	C₄×Dic₅₉	D₄	1	1	yes	characteristic
32	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
33	Dic₂₃₆	—	2	1	no	—
34	Dic₂₃₆	C₂×C₄	1	2	yes	complemented
35	C₁₆×C₅₉	—	2	1	no	—
36	C₅₉⋊C₁₆	C₂×C₄	1	2	yes	complemented
37	C₂³×C₅₉	—	4	1	no	—
38	D₄×C₅₉	—	2	2	no	—
39	D₄×C₅₉	—	4	1	no	—
40	C₂×C₄×C₅₉	—	4	1	no	—
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₅₉	—	4	1	no	—
43	C₂×Dic₅₉	—	4	1	no	—
44	C₂×Dic₅₉	—	8	2	no	—
45	C₂×Dic₅₉	D₈	1	1	yes	characteristic
46	C₂×Dic₅₉	D₈	1	1	yes	characteristic
47	Dic₁₁₈	—	2	2	no	—
48	Dic₁₁₈	—	4	1	no	—
49	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₈×C₅₉	#41	1	1	yes	characteristic
51	C₅₉⋊C₈	—	2	1	no	—
52	C₂²×C₅₉	—	4	2	no	—
53	C₂²×C₅₉	—	8	1	no	—
54	C₂²×C₅₉	—	8	1	no	—
55	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
56	C₄×C₅₉	—	8	1	no	—
57	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
58	C₄×C₅₉	#135	1	1	yes	characteristic
59	Dic₅₉	—	2	1	no	—
60	Dic₅₉	—	2	1	no	—
61	Dic₅₉	—	4	2	no	—
62	Dic₅₉	—	8	1	no	—
63	#2175	—	59	1	no	maximal, Sylow
64	C₂×C₅₉	—	8	2	no	—
65	C₂×C₅₉	C₄×D₈	1	1	yes	characteristic
66	C₂×C₅₉	#509	1	1	yes	characteristic
67	C₂×C₅₉	#510	1	1	yes	characteristic
68	C₂×SD₃₂	—	59	1	no	—
69	C₄×Q₁₆	—	59	1	no	—
70	C₄×D₈	—	59	1	no	—
71	#366	—	59	1	no	—
72	#358	—	59	1	no	—
73	#357	D₅₉	1	1	yes	characteristic
74	#346	—	59	1	no	—
75	C₅₉	#2175	1	1	yes	characteristic, complemented, Sylow
76	C₂×Q₁₆	—	59	1	no	—
77	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
78	C₄×Q₈	—	118	1	no	—
79	C₄×D₄	—	118	1	no	—
80	SD₃₂	—	59	4	no	—
81	C₂×C₁₆	—	59	1	no	—
82	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
83	#107	—	59	1	no	—
84	#107	C₂×D₅₉	1	1	yes	characteristic
85	#103	—	118	1	no	—
86	#102	—	118	1	no	—
87	C₄×C₈	—	59	1	no	—
88	C₂×Q₈	—	118	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂²×C₄	—	236	1	no	—
91	Q₁₆	—	59	2	no	—
92	Q₁₆	—	118	1	no	—
93	D₈	—	2	1	no	—
94	D₈	C₄×D₅₉	1	2	yes	—
95	C₂×C₈	—	59	1	no	—
96	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	118	1	no	—
99	C₄⋊C₄	—	236	1	no	—
100	C₂²⋊C₄	—	236	1	no	—
101	C₄²	—	59	1	no	—
102	C₄²	—	236	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	59	2	no	—
105	C₂³	—	4	1	no	—
106	Q₈	—	118	2	no	—
107	Q₈	—	236	1	no	—
108	D₄	—	2	2	no	—
109	D₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	59	1	no	—
112	C₂×C₄	—	59	1	no	—
113	C₂×C₄	—	472	2	no	—
114	C₂×C₄	—	236	1	no	—
115	C₂×C₄	—	236	1	no	—
116	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
117	C₈	—	118	1	no	—
118	C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
119	C₈	#11745687	1	1	yes	characteristic
120	C₂²	—	4	2	no	—
121	C₂²	—	8	1	no	—
122	C₂²	—	8	1	no	—
123	C₂²	D₈×D₅₉	1	1	yes	characteristic
124	C₄	—	8	1	no	—
125	C₄	—	118	1	no	—
126	C₄	—	118	1	no	—
127	C₄	—	472	1	no	—
128	C₄	—	236	2	no	—
129	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
130	C₄	#14273748	1	1	yes	characteristic
131	C₂	—	8	2	no	—
132	C₂	#16837482	1	1	yes	characteristic
133	C₂	#16837836	1	1	yes	characteristic
134	C₂	#16837841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273671

C₂

yes

maximal, characteristic

#16837496

C₂

yes

maximal, characteristic, complemented

D₈×Dic₅₉

C₂

yes

maximal, characteristic

#16837227

C₂

yes

maximal, characteristic, complemented

#16837207

C₂

yes

maximal, characteristic, complemented

#16837177

C₂

yes

maximal, characteristic, complemented

C₅₉×#357

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

#14273692

—

D₄×Dic₅₉

—

#14273671

C₄

yes

complemented

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273650

—

#14273648

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×D₈×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

D₈×C₅₉

—

D₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

C₂×C₄

yes

complemented

C₁₆×C₅₉

—

C₅₉⋊C₁₆

C₂×C₄

yes

complemented

C₂³×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

#135

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2175

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₄×D₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×SD₃₂

—

C₄×Q₁₆

—

C₄×D₈

—

#366

—

#358

—

#357

D₅₉

yes

characteristic

#346

—

C₅₉

#2175

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

C₂×D₅₉

yes

characteristic

C₄×Q₈

—

118

—

C₄×D₄

—

118

—

SD₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

—

#107

C₂×D₅₉

yes

characteristic

#103

—

118

—

#102

—

118

—

C₄×C₈

—

C₂×Q₈

—

118

—

C₂×D₄

—

C₂²×C₄

—

236

—

Q₁₆

—

Q₁₆

—

118

—

D₈

—

D₈

C₄×D₅₉

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

100

C₂²⋊C₄

—

236

—

101

C₄²

—

102

C₄²

—

236

—

103

C₁₆

—

104

C₁₆

—

105

C₂³

—

106

Q₈

—

118

—

107

Q₈

—

236

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

472

—

114

C₂×C₄

—

236

—

115

C₂×C₄

—

236

—

116

C₂×C₄

D₄×D₅₉

yes

characteristic

117

C₈

—

118

—

118

C₈

C₂×C₄×D₅₉

yes

characteristic

119

C₈

#11745687

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

D₈×D₅₉

yes

characteristic

124

C₄

—

125

C₄

—

118

—

126

C₄

—

118

—

127

C₄

—

472

—

128

C₄

—

236

—

129

C₄

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

130

C₄

#14273748

yes

characteristic

131

C₂

—

132

C₂

#16837482

yes

characteristic

133

C₂

#16837836

yes

characteristic

134

C₂

#16837841

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#357	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#357	D₅₉
hypocenter	C₅₉	#2175
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#357

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
2. #16837496	C₂	semidirect
4. #16837227	C₂	semidirect
5. #16837207	C₂	semidirect
6. #16837177	C₂	semidirect
11. #14273671	C₄	semidirect
34. Dic₂₃₆	C₂×C₄	semidirect
36. C₅₉⋊C₁₆	C₂×C₄	semidirect
75. C₅₉	#2175	semidirect

H ≅ G/N

type

2. #16837496

C₂

semidirect

4. #16837227

C₂

semidirect

5. #16837207

C₂

semidirect

6. #16837177

C₂

semidirect

11. #14273671

C₄

semidirect

34. Dic₂₃₆

C₂×C₄

semidirect

36. C₅₉⋊C₁₆

C₂×C₄

semidirect

75. C₅₉

#2175

semidirect

Subgroups — #21425957

Conjugacy classes of subgroups

Complements