Group #21425035 — subgroups

Subgroups — #21425035

Order 7552.

280 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273638	C₂	1	1	yes	maximal, characteristic
3	#16837156	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₁₈	C₄	1	1	yes	characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₄×Dic₁₁₈	—	2	2	no	—
8	C₄×Dic₁₁₈	—	2	2	no	—
9	C₂×C₄⋊Dic₅₉	C₄	1	1	yes	characteristic
10	C₂×Dic₅₉⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
12	#14273638	—	2	2	no	—
13	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₉	—	2	2	no	—
16	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
20	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₁₈	—	2	2	no	—
22	C₂×Dic₁₁₈	—	2	4	no	—
23	C₂×C₈×C₅₉	—	2	2	no	—
24	C₂×C₈×C₅₉	—	2	4	no	—
25	C₄²×C₅₉	D₄	1	2	yes	—
26	C₄²×C₅₉	D₄	1	2	yes	—
27	C₄⋊Dic₅₉	—	2	2	no	—
28	Dic₅₉⋊C₄	—	2	4	no	—
29	C₄×Dic₅₉	—	2	2	no	—
30	C₄×Dic₅₉	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	2	2	no	—
32	C₂×C₅₉⋊C₈	—	2	2	no	—
33	C₂×C₅₉⋊C₈	Q₈	1	4	yes	complemented
34	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	—	2	2	no	—
36	C₂×C₄×C₅₉	—	2	2	no	—
37	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₅₉	#41	1	2	yes	—
43	C₂×Dic₅₉	—	2	2	no	—
44	C₂×Dic₅₉	—	2	2	no	—
45	C₂×Dic₅₉	—	2	2	no	—
46	C₂×Dic₅₉	—	2	4	no	—
47	C₂×Dic₅₉	M₁₆	1	4	yes	—
48	Dic₁₁₈	—	2	8	no	—
49	C₈×C₅₉	—	4	2	no	—
50	C₅₉⋊C₈	—	2	4	no	—
51	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₅₉	C₄×Q₈	1	2	yes	—
53	C₂²×C₅₉	C₂×M₁₆	1	2	yes	—
54	C₂²×C₅₉	#131	1	2	yes	—
55	C₄×C₅₉	—	2	2	no	—
56	C₄×C₅₉	—	2	2	no	—
57	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₅₉	#124	1	1	yes	characteristic
59	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₅₉	—	2	4	no	—
61	Dic₅₉	—	4	2	no	—
62	#1985	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	#391	1	1	yes	characteristic
64	C₂×C₅₉	#407	1	1	yes	characteristic
65	C₂×C₅₉	#408	1	1	yes	characteristic
66	C₂×C₅₉	#446	1	4	yes	—
67	C₂×C₄×Q₈	—	59	1	no	—
68	C₂×C₄⋊C₈	D₅₉	1	1	yes	characteristic
69	C₂×#97	—	59	1	no	—
70	#336	—	59	4	no	—
71	C₅₉	#1985	1	1	yes	characteristic, complemented, Sylow
72	C₂²×Q₈	—	59	1	no	—
73	C₂²×C₈	—	59	2	no	—
74	C₂²×C₈	C₂×D₅₉	1	2	yes	—
75	C₄×Q₈	—	118	2	no	—
76	C₄×Q₈	—	118	2	no	—
77	C₂×C₄⋊C₄	—	59	2	no	—
78	C₂×C₄⋊C₄	—	59	1	no	—
79	C₂×C₄²	—	59	2	no	—
80	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#97	—	118	2	no	—
83	C₂×Q₈	—	118	4	no	—
84	C₂×Q₈	—	118	2	no	—
85	C₂²×C₄	—	59	2	no	—
86	C₂²×C₄	—	59	2	no	—
87	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	59	4	no	—
93	C₂×C₈	—	118	2	no	—
94	C₂×C₈	—	118	2	no	—
95	C₄⋊C₄	—	118	4	no	—
96	C₄⋊C₄	—	118	2	no	—
97	C₄²	—	118	2	no	—
98	C₄²	—	118	2	no	—
99	C₄²	D₂₃₆	1	2	yes	—
100	C₄²	C₂²⋊D₅₉	1	2	yes	—
101	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
102	Q₈	—	118	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	59	4	no	—
106	C₂×C₄	—	118	2	no	—
107	C₂×C₄	—	118	2	no	—
108	C₂×C₄	—	118	2	no	—
109	C₂×C₄	—	118	4	no	—
110	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
111	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
112	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
113	C₂×C₄	#11745687	1	2	yes	—
114	C₂×C₄	Q₈×D₅₉	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
116	C₈	—	4	2	no	—
117	C₈	—	118	4	no	—
118	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
119	C₂²	#14273692	1	2	yes	—
120	C₂²	M₁₆×D₅₉	1	2	yes	—
121	C₂²	#14273733	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	118	4	no	—
125	C₄	—	236	2	no	—
126	C₄	#14273707	1	2	yes	—
127	C₄	#14273714	1	1	yes	characteristic
128	C₄	#14273725	1	1	yes	characteristic
129	C₂	#16837307	1	1	yes	characteristic
130	C₂	#16837365	1	1	yes	characteristic
131	C₂	#16837368	1	1	yes	characteristic
132	C₂	#16837531	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₁₈

C₂

yes

maximal, characteristic

C₂×#14273638

C₂

yes

maximal, characteristic

#16837156

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

C₂×C₄⋊Dic₅₉

C₄

yes

characteristic

C₂×Dic₅₉⋊C₄

C₄

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

#14273638

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₈

yes

complemented

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

M₁₆

yes

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

C₄×Q₈

yes

—

C₂²×C₅₉

C₂×M₁₆

yes

—

C₂²×C₅₉

#131

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

#124

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

D₅₉

yes

characteristic

C₂×#97

—

#336

—

C₅₉

#1985

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

—

C₄×Q₈

—

118

—

C₄×Q₈

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

#97

—

118

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄²

—

118

—

C₄²

—

118

—

C₄²

D₂₃₆

yes

—

100

C₄²

C₂²⋊D₅₉

yes

—

101

C₂³

C₂×C₄×D₅₉

yes

characteristic

102

Q₈

—

118

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

118

—

107

C₂×C₄

—

118

—

108

C₂×C₄

—

118

—

109

C₂×C₄

—

118

—

110

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

111

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

112

C₂×C₄

C₂×D₂₃₆

yes

characteristic

113

C₂×C₄

#11745687

yes

—

114

C₂×C₄

Q₈×D₅₉

yes

—

115

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

116

C₈

—

117

C₈

—

118

—

118

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

119

C₂²

#14273692

yes

—

120

C₂²

M₁₆×D₅₉

yes

—

121

C₂²

#14273733

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

118

—

125

C₄

—

236

—

126

C₄

#14273707

yes

—

127

C₄

#14273714

yes

characteristic

128

C₄

#14273725

yes

characteristic

129

C₂

#16837307

yes

characteristic

130

C₂

#16837365

yes

characteristic

131

C₂

#16837368

yes

characteristic

132

C₂

#16837531

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₉
commutator, socle	C₂³×C₅₉	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₅₉	C₂
hypercenter	C₂×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#1985
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₉

commutator, socle

C₂³×C₅₉

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₅₉

C₂

hypercenter

C₂×C₄⋊C₈

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
33. C₂×C₅₉⋊C₈	Q₈	semidirect
71. C₅₉	#1985	semidirect

H ≅ G/N

type

33. C₂×C₅₉⋊C₈

Q₈

semidirect

71. C₅₉

#1985

semidirect

Subgroups — #21425035

Conjugacy classes of subgroups

Complements