Group #21425815 — subgroups

Subgroups — #21425815

Order 7552.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273667	C₂	1	1	yes	maximal, characteristic
3	#16837205	C₂	1	4	yes	maximal, complemented
4	C₅₉×#427	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₁₈	—	2	1	no	—
6	C₂×Dic₂₃₆	—	2	1	no	—
7	C₂×Dic₂₃₆	—	2	2	no	—
8	C₂×Dic₂₃₆	C₄	1	4	yes	complemented
9	C₂×Dic₂₃₆	C₂²	1	2	yes	—
10	#14273667	—	2	2	no	—
11	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	#117×C₅₉	—	2	1	no	—
14	#107×C₅₉	C₂²	1	2	yes	—
15	#106×C₅₉	—	2	1	no	—
16	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₉	—	4	1	no	—
18	C₂×Dic₁₁₈	—	2	4	no	—
19	C₂×Dic₁₁₈	—	2	2	no	—
20	C₂×Dic₁₁₈	—	4	1	no	—
21	C₂×Dic₁₁₈	—	4	2	no	—
22	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₉	D₄	1	2	yes	—
25	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₉	—	2	2	no	—
27	C₂²⋊C₄×C₅₉	—	4	1	no	—
28	C₄²×C₅₉	—	4	1	no	—
29	Dic₂₃₆	—	2	4	no	—
30	Dic₂₃₆	—	2	2	no	—
31	Dic₂₃₆	C₂×C₄	1	4	yes	—
32	C₅₉⋊C₁₆	—	2	2	no	—
33	C₂³×C₅₉	D₈	1	1	yes	characteristic
34	C₂×C₄×C₅₉	—	4	2	no	—
35	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
37	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
38	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₅₉	—	4	4	no	—
40	C₂×Dic₅₉	—	4	2	no	—
41	Dic₁₁₈	—	2	4	no	—
42	Dic₁₁₈	—	4	4	no	—
43	Dic₁₁₈	—	4	2	no	—
44	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
45	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₅₉	—	2	2	no	—
48	C₂²×C₅₉	#102	1	2	yes	—
49	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
50	C₄×C₅₉	—	8	2	no	—
51	C₄×C₅₉	#102	1	2	yes	—
52	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₅₉	—	4	4	no	—
55	#2141	—	59	1	no	maximal, Sylow
56	C₂×C₅₉	—	2	2	no	—
57	C₂×C₅₉	C₂×#102	1	1	yes	characteristic
58	C₂×C₅₉	#510	1	2	yes	—
59	C₂²×Q₁₆	—	59	1	no	—
60	C₂×M₃₂	—	59	1	no	—
61	#427	D₅₉	1	1	yes	characteristic
62	#358	—	59	4	no	—
63	C₅₉	#2141	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	—	118	1	no	—
65	C₂×Q₁₆	—	59	4	no	—
66	C₂×Q₁₆	—	59	2	no	—
67	C₂×Q₁₆	—	118	2	no	—
68	C₂×Q₁₆	—	118	1	no	—
69	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
70	#117	—	2	1	no	—
71	M₃₂	—	118	2	no	—
72	C₂×C₁₆	—	59	2	no	—
73	#107	C₂×D₅₉	1	2	yes	—
74	#106	—	2	1	no	—
75	C₂×Q₈	—	118	4	no	—
76	C₂×Q₈	—	118	2	no	—
77	C₂×Q₈	—	236	1	no	—
78	C₂×Q₈	—	236	2	no	—
79	C₂²×C₄	—	236	1	no	—
80	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
81	Q₁₆	—	59	4	no	—
82	Q₁₆	—	118	4	no	—
83	Q₁₆	—	118	2	no	—
84	C₂×C₈	C₄×D₅₉	1	2	yes	—
85	C₂×C₈	D₂₃₆	1	2	yes	—
86	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	4	1	no	—
90	C₄²	—	4	1	no	—
91	C₁₆	—	118	2	no	—
92	C₂³	#11745669	1	1	yes	characteristic
93	Q₈	—	118	4	no	—
94	Q₈	—	236	4	no	—
95	Q₈	—	236	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	236	4	no	—
98	C₂×C₄	—	236	2	no	—
99	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
100	C₂×C₄	#11745669	1	1	yes	characteristic
101	C₂×C₄	#11745671	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
103	C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
104	C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
105	C₈	C₂×D₂₃₆	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#14273647	1	2	yes	—
108	C₂²	C₂×#11745669	1	1	yes	characteristic
109	C₄	—	8	2	no	—
110	C₄	—	236	4	no	—
111	C₄	#14273647	1	2	yes	—
112	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
113	C₄	C₂×#11745671	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#14273647	1	1	yes	characteristic
116	C₂	#16837842	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₃₆

C₂

yes

maximal, characteristic

C₂×#14273667

C₂

yes

maximal, characteristic

#16837205

C₂

yes

maximal, complemented

C₅₉×#427

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

C₄

yes

complemented

C₂×Dic₂₃₆

C₂²

yes

—

#14273667

—

C₂×C₅₉⋊C₁₆

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#117×C₅₉

—

#107×C₅₉

C₂²

yes

—

#106×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

C₂×C₄

yes

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

#102

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#102

yes

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

#2141

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×#102

yes

characteristic

C₂×C₅₉

#510

yes

—

C₂²×Q₁₆

—

C₂×M₃₂

—

#427

D₅₉

yes

characteristic

#358

—

C₅₉

#2141

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

118

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

M₃₂

—

118

—

C₂×C₁₆

—

#107

C₂×D₅₉

yes

—

#106

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂×Q₈

—

236

—

C₂×Q₈

—

236

—

C₂²×C₄

—

236

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

Q₁₆

—

Q₁₆

—

118

—

Q₁₆

—

118

—

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

D₂₃₆

yes

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

118

—

C₂³

#11745669

yes

characteristic

Q₈

—

118

—

Q₈

—

236

—

Q₈

—

236

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

100

C₂×C₄

#11745669

yes

characteristic

101

C₂×C₄

#11745671

yes

—

102

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

103

C₈

(C₂×D₅₉)⋊C₄

yes

—

104

C₈

C₂×C₄×D₅₉

yes

characteristic

105

C₈

C₂×D₂₃₆

yes

characteristic

106

C₂²

—

107

C₂²

#14273647

yes

—

108

C₂²

C₂×#11745669

yes

characteristic

109

C₄

—

110

C₄

—

236

—

111

C₄

#14273647

yes

—

112

C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

113

C₄

C₂×#11745671

yes

characteristic

114

C₂

—

115

C₂

C₂×#14273647

yes

characteristic

116

C₂

#16837842

yes

—

	subgroup	G/H
center	C₂²	C₂×#11745669
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#427	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#427	D₅₉
hypocenter	C₅₉	#2141
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#11745669

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#427

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
3. #16837205	C₂	semidirect
8. C₂×Dic₂₃₆	C₄	semidirect
63. C₅₉	#2141	semidirect

H ≅ G/N

type

3. #16837205

C₂

semidirect

8. C₂×Dic₂₃₆

C₄

semidirect

63. C₅₉

#2141

semidirect

Subgroups — #21425815

Conjugacy classes of subgroups

Complements