Group #21432473 — subgroups

Subgroups — #21432473

Order 7552.

504 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14273727	C₂	1	1	yes	maximal, characteristic
2	#16837309	C₂	1	4	yes	maximal
3	#16837307	C₂	1	4	yes	maximal
4	#16837267	C₂	1	4	yes	maximal
5	C₄²×Dic₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₁₈	C₂²	1	2	yes	—
8	#14273727	—	2	4	no	—
9	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
11	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
12	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
13	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
14	C₅₉⋊#95	C₂²	1	16	yes	—
15	C₂²×Q₈×C₅₉	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₅₉	—	2	4	no	—
17	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₅₉	—	2	4	no	—
21	C₂×Q₈×C₅₉	—	2	8	no	—
22	C₂²×C₄×C₅₉	C₂³	1	4	yes	—
23	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₉	C₂³	1	4	yes	—
27	C₂²×Dic₅₉	C₂³	1	4	yes	—
28	C₂×Dic₁₁₈	—	2	4	no	—
29	C₂×Dic₁₁₈	—	2	8	no	—
30	C₄⋊C₄×C₅₉	—	2	4	no	—
31	C₄⋊C₄×C₅₉	—	2	4	no	—
32	C₄²×C₅₉	D₄	1	4	yes	—
33	C₄⋊Dic₅₉	—	2	4	no	—
34	C₄⋊Dic₅₉	—	2	4	no	—
35	C₄×Dic₅₉	—	2	4	no	—
36	C₄×Dic₅₉	—	2	4	no	—
37	C₄×Dic₅₉	Q₈	1	8	yes	complemented
38	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
39	Q₈×C₅₉	—	4	8	no	—
40	C₂×C₄×C₅₉	—	2	4	no	—
41	C₂×C₄×C₅₉	—	2	8	no	—
42	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅₉	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₅₉	#41	1	4	yes	—
47	C₂×C₄×C₅₉	#41	1	4	yes	—
48	C₂×Dic₅₉	—	2	4	no	—
49	C₂×Dic₅₉	—	2	4	no	—
50	C₂×Dic₅₉	—	2	4	no	—
51	C₂×Dic₅₉	—	2	8	no	—
52	C₂×Dic₅₉	C₂×Q₈	1	8	yes	—
53	Dic₁₁₈	—	4	8	no	—
54	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
55	C₂²×C₅₉	C₂²×Q₈	1	2	yes	—
56	C₂²×C₅₉	C₂×#41	1	2	yes	—
57	C₂²×C₅₉	C₂×#41	1	2	yes	—
58	C₄×C₅₉	—	4	4	no	—
59	C₄×C₅₉	#124	1	4	yes	—
60	C₄×C₅₉	#124	1	4	yes	—
61	C₄×C₅₉	C₄⋊Q₈	1	4	yes	—
62	Dic₅₉	—	2	8	no	—
63	Dic₅₉	—	4	4	no	—
64	#2862	—	59	1	no	maximal, Sylow
65	C₂×C₅₉	C₂×#124	1	1	yes	characteristic
66	C₂×C₅₉	C₂×#124	1	1	yes	characteristic
67	C₂×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₅₉	#533	1	4	yes	—
69	C₂×C₄⋊Q₈	—	59	1	no	—
70	C₂×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
71	#391	—	59	4	no	—
72	#391	—	59	4	no	—
73	#382	—	59	4	no	—
74	C₄³	—	59	1	no	—
75	C₅₉	#2862	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	59	2	no	—
77	C₂²×Q₈	C₂×D₅₉	1	2	yes	—
78	C₄⋊Q₈	—	2	4	no	—
79	C₄⋊Q₈	—	118	4	no	—
80	C₂×C₄⋊C₄	—	59	2	no	—
81	C₂×C₄⋊C₄	—	59	2	no	—
82	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
83	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
84	C₂×C₄²	—	59	2	no	—
85	C₂×C₄²	—	59	2	no	—
86	C₂×C₄²	—	59	2	no	—
87	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
88	#95	—	59	16	no	—
89	C₂×Q₈	—	2	8	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×Q₈	—	118	8	no	—
92	C₂×Q₈	—	118	4	no	—
93	C₂²×C₄	—	59	4	no	—
94	C₂²×C₄	—	59	4	no	—
95	C₂²×C₄	C₂²×D₅₉	1	4	yes	—
96	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	118	4	no	—
102	C₄⋊C₄	—	118	4	no	—
103	C₄²	—	59	8	no	—
104	C₄²	—	118	4	no	—
105	C₄²	—	118	4	no	—
106	C₄²	C₂²⋊D₅₉	1	4	yes	—
107	C₂³	C₂³×D₅₉	1	1	yes	characteristic
108	Q₈	—	4	8	no	—
109	Q₈	—	236	8	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	59	8	no	—
113	C₂×C₄	—	118	4	no	—
114	C₂×C₄	—	118	4	no	—
115	C₂×C₄	—	118	4	no	—
116	C₂×C₄	—	118	8	no	—
117	C₂×C₄	#11745687	1	4	yes	—
118	C₂×C₄	Q₈×D₅₉	1	4	yes	—
119	C₂×C₄	#11745689	1	4	yes	—
120	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
122	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
123	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
124	C₂²	C₂×Q₈×D₅₉	1	2	yes	—
125	C₂²	C₂×#11745687	1	2	yes	—
126	C₂²	C₂×#11745689	1	2	yes	—
127	C₄	—	4	4	no	—
128	C₄	—	118	8	no	—
129	C₄	—	236	4	no	—
130	C₄	#14273714	1	4	yes	—
131	C₄	#14273716	1	4	yes	—
132	C₄	#14273725	1	4	yes	—
133	C₂	C₂×#14273714	1	1	yes	characteristic
134	C₂	C₂×#14273716	1	1	yes	characteristic
135	C₂	C₂×#14273725	1	1	yes	characteristic
136	C₂	#16837957	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273727

C₂

yes

maximal, characteristic

#16837309

C₂

yes

maximal

#16837307

C₂

yes

maximal

#16837267

C₂

yes

maximal

C₄²×Dic₅₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

#14273727

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₂²×Q₈×C₅₉

C₂²

yes

—

C₄⋊Q₈×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

Q₈

yes

complemented

C₂³×C₅₉

C₂⁴

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×Q₈

yes

—

Dic₁₁₈

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂²×C₅₉

C₂²×Q₈

yes

—

C₂²×C₅₉

C₂×#41

yes

—

C₂²×C₅₉

C₂×#41

yes

—

C₄×C₅₉

—

C₄×C₅₉

#124

yes

—

C₄×C₅₉

#124

yes

—

C₄×C₅₉

C₄⋊Q₈

yes

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₉

#533

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₅₉

yes

—

—

—

C₄³

—

C₅₉

#2862

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

118

—

102

C₄⋊C₄

—

118

—

103

C₄²

—

104

C₄²

—

118

—

105

C₄²

—

118

—

106

C₄²

C₂²⋊D₅₉

yes

—

107

C₂³

C₂³×D₅₉

yes

characteristic

108

Q₈

—

109

Q₈

—

236

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

118

—

114

C₂×C₄

—

118

—

115

C₂×C₄

—

118

—

116

C₂×C₄

—

118

—

117

C₂×C₄

#11745687

yes

—

118

C₂×C₄

Q₈×D₅₉

yes

—

119

C₂×C₄

#11745689

yes

—

120

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

121

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

122

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

123

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

124

C₂²

C₂×Q₈×D₅₉

yes

—

125

C₂²

C₂×#11745687

yes

—

126

C₂²

C₂×#11745689

yes

—

127

C₄

—

128

C₄

—

118

—

129

C₄

—

236

—

130

C₄

#14273714

yes

—

131

C₄

#14273716

yes

—

132

C₄

#14273725

yes

—

133

C₂

C₂×#14273714

yes

characteristic

134

C₂

C₂×#14273716

yes

characteristic

135

C₂

C₂×#14273725

yes

characteristic

136

C₂

#16837957

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator, socle	C₂³×C₅₉	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₅₉	C₂
hypercenter	C₂×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#2862
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator, socle

C₂³×C₅₉

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₅₉

C₂

hypercenter

C₂×C₄⋊Q₈

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
37. C₄×Dic₅₉	Q₈	semidirect
75. C₅₉	#2862	semidirect

H ≅ G/N

type

37. C₄×Dic₅₉

Q₈

semidirect

75. C₅₉

#2862

semidirect

Subgroups — #21432473

Conjugacy classes of subgroups

Complements