Group #21434619 — subgroups

Subgroups — #21434619

Order 7552.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16837955	C₂	1	2	yes	maximal
2	#16837810	C₂	1	4	yes	maximal, complemented
3	#16837809	C₂	1	4	yes	maximal, complemented
4	#16837440	C₂	1	1	yes	maximal, characteristic
5	#16837438	C₂	1	2	yes	maximal
6	C₂×#14273637	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#427	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	#14273726	C₂²	1	4	yes	—
10	#14273718	C₂²	1	4	yes	—
11	#14273704	—	2	4	no	—
12	C₄×Dic₁₁₈	—	2	4	no	—
13	#14273686	C₂²	1	2	yes	—
14	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
15	#14273661	C₂²	1	2	yes	—
16	#14273660	C₂²	1	4	yes	—
17	#14273659	C₂²	1	2	yes	—
18	#14273658	C₂²	1	4	yes	—
19	#14273637	C₂²	1	2	yes	—
20	#14273637	C₂²	1	2	yes	—
21	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	#117×C₅₉	C₂²	1	2	yes	—
23	#107×C₅₉	C₂²	1	2	yes	—
24	#106×C₅₉	C₂²	1	2	yes	—
25	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₉	D₄	1	2	yes	—
27	C₂×Dic₁₁₈	—	2	4	no	—
28	C₂×C₈×C₅₉	Q₈	1	4	yes	complemented
29	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₅₉	—	2	2	no	—
33	C₄²×C₅₉	—	2	2	no	—
34	C₂²⋊Dic₅₉	—	2	2	no	—
35	C₄⋊Dic₅₉	—	2	4	no	—
36	C₄⋊Dic₅₉	C₂³	1	4	yes	—
37	Dic₅₉⋊C₄	—	2	4	no	—
38	Dic₅₉⋊C₄	—	2	4	no	—
39	C₄×Dic₅₉	—	2	2	no	—
40	C₄×Dic₅₉	D₄	1	2	yes	—
41	C₄×Dic₅₉	C₂³	1	2	yes	—
42	C₂×C₅₉⋊C₈	Q₈	1	4	yes	complemented
43	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
44	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	—	2	4	no	—
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	4	no	—
51	C₂×Dic₅₉	—	4	2	no	—
52	C₂×Dic₅₉	C₂×D₄	1	4	yes	—
53	Dic₁₁₈	—	4	8	no	—
54	C₈×C₅₉	C₂×Q₈	1	2	yes	—
55	C₈×C₅₉	C₂×Q₈	1	2	yes	—
56	C₅₉⋊C₈	C₂×Q₈	1	4	yes	—
57	C₂²×C₅₉	—	2	2	no	—
58	C₂²×C₅₉	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₅₉	—	4	4	no	—
61	C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
62	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
64	Dic₅₉	—	2	4	no	—
65	Dic₅₉	—	4	4	no	—
66	#3162	—	59	1	no	maximal, Sylow
67	C₂×C₅₉	—	2	2	no	—
68	C₂×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₅₉	#575	1	2	yes	—
70	#533	—	59	2	no	—
71	#501	—	59	4	no	—
72	#501	—	59	4	no	—
73	#427	—	59	1	no	—
74	#427	—	59	2	no	—
75	#427	D₅₉	1	1	yes	characteristic
76	C₂×#97	—	59	1	no	—
77	C₅₉	#3162	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₈	—	59	1	no	—
79	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
80	C₄⋊Q₈	—	59	4	no	—
81	#125	—	59	4	no	—
82	C₂²⋊Q₈	—	118	4	no	—
83	C₄×Q₈	—	118	4	no	—
84	#117	—	59	2	no	—
85	#117	C₂×D₅₉	1	2	yes	—
86	C₂×C₄²	—	59	1	no	—
87	#107	—	59	2	no	—
88	#107	—	59	4	no	—
89	#107	C₂×D₅₉	1	2	yes	—
90	#106	—	59	2	no	—
91	#106	—	59	4	no	—
92	#106	C₂×D₅₉	1	2	yes	—
93	#97	—	59	2	no	—
94	#97	—	59	2	no	—
95	C₂×Q₈	—	118	4	no	—
96	C₂²×C₄	—	59	2	no	—
97	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
98	C₂×C₈	—	59	4	no	—
99	C₂×C₈	—	59	2	no	—
100	C₂×C₈	Dic₁₁₈	1	4	yes	complemented
101	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
103	C₄⋊C₄	—	59	4	no	—
104	C₄⋊C₄	—	118	4	no	—
105	C₄⋊C₄	—	118	4	no	—
106	C₄⋊C₄	—	118	4	no	—
107	C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	118	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	59	2	no	—
112	C₄²	—	59	2	no	—
113	C₄²	—	118	2	no	—
114	C₂³	D₄×D₅₉	1	1	yes	characteristic
115	Q₈	—	236	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	59	4	no	—
118	C₂×C₄	—	118	4	no	—
119	C₂×C₄	—	236	2	no	—
120	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
121	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₅₉	1	2	yes	—
123	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
124	C₈	—	59	4	no	—
125	C₈	C₂×Dic₁₁₈	1	2	yes	—
126	C₈	C₂×Dic₁₁₈	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#14273726	1	2	yes	—
129	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	118	4	no	—
132	C₄	—	236	4	no	—
133	C₄	#14273726	1	2	yes	—
134	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
135	C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#14273726	1	1	yes	characteristic
138	C₂	#16838199	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837955

C₂

yes

maximal

#16837810

C₂

yes

maximal, complemented

#16837809

C₂

yes

maximal, complemented

#16837440

C₂

yes

maximal, characteristic

#16837438

C₂

yes

maximal

C₂×#14273637

C₂

yes

maximal, characteristic

C₅₉×#427

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273726

C₂²

yes

—

#14273718

C₂²

yes

—

#14273704

—

C₄×Dic₁₁₈

—

#14273686

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273661

C₂²

yes

—

#14273660

C₂²

yes

—

#14273659

C₂²

yes

—

#14273658

C₂²

yes

—

#14273637

C₂²

yes

—

#14273637

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

#106×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

D₄

yes

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

Q₈

yes

complemented

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

—

C₄×Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

Q₈

yes

complemented

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

Dic₁₁₈

—

C₈×C₅₉

C₂×Q₈

yes

—

C₈×C₅₉

C₂×Q₈

yes

—

C₅₉⋊C₈

C₂×Q₈

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄⋊Q₈

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₄⋊Q₈

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

C₂²×Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#3162

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₉

#575

yes

—

—

—

—

—

—

D₅₉

yes

characteristic

C₂×#97

—

C₅₉

#3162

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

118

—

#117

—

#117

C₂×D₅₉

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₅₉

yes

—

#106

—

#106

—

#106

C₂×D₅₉

yes

—

#97

—

#97

—

C₂×Q₈

—

118

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₁₁₈

yes

complemented

101

C₂×C₈

C₂²×D₅₉

yes

characteristic

102

C₂×C₈

C₂²×D₅₉

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

118

—

105

C₄⋊C₄

—

118

—

106

C₄⋊C₄

—

118

—

107

C₄⋊C₄

C₂²×D₅₉

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

118

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

118

—

114

C₂³

D₄×D₅₉

yes

characteristic

115

Q₈

—

236

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

118

—

119

C₂×C₄

—

236

—

120

C₂×C₄

C₂×Dic₁₁₈

yes

—

121

C₂×C₄

D₄×D₅₉

yes

characteristic

122

C₂×C₄

Q₈×D₅₉

yes

—

123

C₂×C₄

C₂³×D₅₉

yes

characteristic

124

C₈

—

125

C₈

C₂×Dic₁₁₈

yes

—

126

C₈

C₂×Dic₁₁₈

yes

—

127

C₂²

—

128

C₂²

#14273726

yes

—

129

C₂²

C₂×D₄×D₅₉

yes

characteristic

130

C₄

—

131

C₄

—

118

—

132

C₄

—

236

—

133

C₄

#14273726

yes

—

134

C₄

C₂×Q₈×D₅₉

yes

characteristic

135

C₄

C₂²×Dic₁₁₈

yes

characteristic

136

C₂

—

137

C₂

C₂×#14273726

yes

characteristic

138

C₂

#16838199

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#427	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#427	D₅₉
hypocenter	C₅₉	#3162
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#427

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #16837810	C₂	semidirect
3. #16837809	C₂	semidirect
28. C₂×C₈×C₅₉	Q₈	semidirect
42. C₂×C₅₉⋊C₈	Q₈	semidirect
77. C₅₉	#3162	semidirect
100. C₂×C₈	Dic₁₁₈	semidirect

H ≅ G/N

type

2. #16837810

C₂

semidirect

3. #16837809

C₂

semidirect

28. C₂×C₈×C₅₉

Q₈

semidirect

42. C₂×C₅₉⋊C₈

Q₈

semidirect

77. C₅₉

#3162

semidirect

100. C₂×C₈

Dic₁₁₈

semidirect

Subgroups — #21434619

Conjugacy classes of subgroups

Complements