Group #21426231 — subgroups

Subgroups — #21426231

Order 7552.

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16837806	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837228	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837227	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837203	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#357	C₂	1	1	yes	maximal, characteristic
8	#14273726	—	2	1	no	—
9	D₄×Dic₅₉	—	2	1	no	—
10	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14273661	C₂²	1	1	yes	characteristic
12	#14273661	C₂²	1	1	yes	characteristic
13	#14273660	—	2	1	no	—
14	#14273648	—	2	1	no	—
15	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	#107×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₅₉	—	2	1	no	—
20	C₂²×Dic₅₉	—	4	1	no	—
21	C₂×Dic₁₁₈	—	4	1	no	—
22	D₈×C₅₉	—	2	1	no	—
23	D₈×C₅₉	Q₈	1	2	yes	complemented
24	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₅₉	—	2	1	no	—
26	C₂²⋊Dic₅₉	—	4	1	no	—
27	C₄⋊Dic₅₉	—	2	1	no	—
28	C₄⋊Dic₅₉	—	2	1	no	—
29	Dic₅₉⋊C₄	—	4	1	no	—
30	C₄×Dic₅₉	D₄	1	1	yes	characteristic
31	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
32	C₁₆×C₅₉	—	2	1	no	—
33	C₅₉⋊C₁₆	—	2	1	no	—
34	C₂³×C₅₉	—	4	1	no	—
35	D₄×C₅₉	—	2	2	no	—
36	D₄×C₅₉	—	4	1	no	—
37	C₂×C₄×C₅₉	—	4	1	no	—
38	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₅₉	—	4	1	no	—
40	C₂×Dic₅₉	—	4	1	no	—
41	C₂×Dic₅₉	—	8	2	no	—
42	C₂×Dic₅₉	D₈	1	1	yes	characteristic
43	C₂×Dic₅₉	D₈	1	1	yes	characteristic
44	Dic₁₁₈	—	8	2	no	—
45	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₅₉	#41	1	1	yes	characteristic
47	C₅₉⋊C₈	—	2	1	no	—
48	C₂²×C₅₉	—	4	2	no	—
49	C₂²×C₅₉	—	8	1	no	—
50	C₂²×C₅₉	—	8	1	no	—
51	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
52	C₄×C₅₉	—	8	1	no	—
53	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₅₉	#137	1	1	yes	characteristic
55	Dic₅₉	—	2	1	no	—
56	Dic₅₉	—	8	1	no	—
57	Dic₅₉	—	8	1	no	—
58	Dic₅₉	D₁₆	1	2	yes	—
59	#2224	—	59	1	no	maximal, Sylow
60	C₂×C₅₉	—	8	2	no	—
61	C₂×C₅₉	#474	1	1	yes	characteristic
62	C₂×C₅₉	C₂×D₁₆	1	1	yes	characteristic
63	C₂×C₅₉	#510	1	1	yes	characteristic
64	#500	—	59	1	no	—
65	C₄×D₈	—	59	1	no	—
66	#366	—	59	1	no	—
67	#366	—	59	1	no	—
68	C₄⋊C₁₆	—	59	1	no	—
69	#357	—	59	1	no	—
70	#357	D₅₉	1	1	yes	characteristic
71	C₅₉	#2224	1	1	yes	characteristic, complemented, Sylow
72	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
73	C₄⋊Q₈	—	118	1	no	—
74	C₄×D₄	—	118	1	no	—
75	C₂×C₁₆	—	59	1	no	—
76	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
77	#107	—	59	1	no	—
78	#107	—	59	1	no	—
79	#107	—	118	1	no	—
80	#107	C₂×D₅₉	1	1	yes	characteristic
81	#102	—	118	1	no	—
82	C₄×C₈	—	59	1	no	—
83	C₂×Q₈	—	236	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂²×C₄	—	236	1	no	—
86	D₈	—	2	1	no	—
87	D₈	Dic₁₁₈	1	2	yes	complemented
88	C₂×C₈	—	59	1	no	—
89	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	118	1	no	—
92	C₄⋊C₄	—	118	1	no	—
93	C₄⋊C₄	—	236	1	no	—
94	C₂²⋊C₄	—	236	1	no	—
95	C₄²	—	59	1	no	—
96	C₁₆	—	2	1	no	—
97	C₁₆	—	118	1	no	—
98	C₂³	—	4	1	no	—
99	Q₈	—	472	2	no	—
100	D₄	—	2	2	no	—
101	D₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	59	1	no	—
104	C₂×C₄	—	59	1	no	—
105	C₂×C₄	—	472	2	no	—
106	C₂×C₄	—	236	1	no	—
107	C₂×C₄	—	236	1	no	—
108	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
109	C₈	—	118	1	no	—
110	C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
111	C₈	#11745687	1	1	yes	characteristic
112	C₂²	—	4	2	no	—
113	C₂²	—	8	1	no	—
114	C₂²	—	8	1	no	—
115	C₂²	D₈×D₅₉	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	59	2	no	—
118	C₄	—	118	1	no	—
119	C₄	—	472	1	no	—
120	C₄	—	472	1	no	—
121	C₄	#14273704	1	1	yes	characteristic
122	C₄	#14273756	1	1	yes	characteristic
123	C₂	—	8	2	no	—
124	C₂	#16837680	1	1	yes	characteristic
125	C₂	D₁₆×D₅₉	1	1	yes	characteristic
126	C₂	#16837841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837806

C₂

yes

maximal, characteristic, complemented

D₈×Dic₅₉

C₂

yes

maximal, characteristic

#16837228

C₂

yes

maximal, characteristic, complemented

#16837227

C₂

yes

maximal, characteristic, complemented

Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#16837203

C₂

yes

maximal, characteristic

C₅₉×#357

C₂

yes

maximal, characteristic

#14273726

—

D₄×Dic₅₉

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273660

—

#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

Q₈

yes

complemented

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

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

—

C₈×C₅₉

C₂×Q₈

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

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#137

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

D₁₆

yes

—

#2224

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#474

yes

characteristic

C₂×C₅₉

C₂×D₁₆

yes

C₂×C₅₉

yes

—

C₄×D₈

—

#366

—

#366

—

C₄⋊C₁₆

—

#357

—

#357

D₅₉

yes

characteristic

C₅₉

#2224

yes

characteristic, complemented, Sylow

C₂×D₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

—

118

—

C₄×D₄

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

—

—

—

118

—

#107

C₂×D₅₉

yes

characteristic

#102

—

118

—

C₄×C₈

—

C₂×Q₈

—

236

—

C₂×D₄

—

C₂²×C₄

—

236

—

D₈

—

D₈

Dic₁₁₈

yes

complemented

C₂×C₈

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₂²⋊C₄

—

236

—

C₄²

—

C₁₆

—

C₁₆

—

118

—

C₂³

—

Q₈

—

472

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

472

—

106

C₂×C₄

—

236

—

107

C₂×C₄

—

236

—

108

C₂×C₄

D₄×D₅₉

yes

characteristic

109

C₈

—

118

—

110

C₈

C₂×Dic₁₁₈

yes

characteristic

111

C₈

#11745687

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

D₈×D₅₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

118

—

119

C₄

—

472

—

120

C₄

—

472

—

121

C₄

#14273704

yes

characteristic

122

C₄

#14273756

yes

characteristic

123

C₂

—

124

C₂

#16837680

yes

characteristic

125

C₂

D₁₆×D₅₉

yes

characteristic

126

C₂

#16837841

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#357	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#357	D₅₉
hypocenter	C₅₉	#2224
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#357

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. #16837806	C₂	semidirect
3. #16837228	C₂	semidirect
4. #16837227	C₂	semidirect
5. Dic₅₉⋊C₁₆	C₂	semidirect
23. D₈×C₅₉	Q₈	semidirect
71. C₅₉	#2224	semidirect
87. D₈	Dic₁₁₈	semidirect

H ≅ G/N

type

1. #16837806

C₂

semidirect

3. #16837228

C₂

semidirect

4. #16837227

C₂

semidirect

5. Dic₅₉⋊C₁₆

C₂

semidirect

23. D₈×C₅₉

Q₈

semidirect

71. C₅₉

#2224

semidirect

87. D₈

Dic₁₁₈

semidirect

Subgroups — #21426231

Conjugacy classes of subgroups

Complements