Group #20299342 — subgroups

Subgroups — #20299342

Order 7232.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#16813614	C₂	1	1	yes	maximal, characteristic, complemented
2	#16813610	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813606	C₂	1	2	yes	maximal, complemented
4	#16813604	C₂	1	2	yes	maximal, complemented
5	#16813599	C₂	1	2	yes	maximal, complemented
6	#16813598	C₂	1	2	yes	maximal, complemented
7	#16813595	C₂	1	2	yes	maximal, complemented
8	#16813594	C₂	1	2	yes	maximal, complemented
9	#124×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
11	C₂×D₄₅₂	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
14	C₂²×Dic₁₁₃	C₂²	1	2	yes	—
15	C₂×C₄×D₁₁₃	C₂²	1	2	yes	—
16	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
17	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
19	#14268626	C₂²	1	2	yes	—
20	#14268626	C₂²	1	2	yes	—
21	#14268626	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
23	#14268624	C₂²	1	2	yes	—
24	#14268624	C₂²	1	2	yes	—
25	#14268624	C₂²	1	2	yes	—
26	C₂×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
30	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
31	C₂³×C₁₁₃	C₂³	1	2	yes	—
32	C₂²×D₁₁₃	C₂³	1	2	yes	—
33	Q₈×C₁₁₃	—	4	1	no	—
34	D₄×C₁₁₃	—	4	1	no	—
35	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
39	C₂²⋊D₁₁₃	—	4	2	no	—
40	C₂²⋊D₁₁₃	—	4	2	no	—
41	C₂×Dic₁₁₃	—	4	2	no	—
42	C₂×Dic₁₁₃	C₂³	1	2	yes	—
43	C₂×Dic₁₁₃	C₂³	1	2	yes	—
44	C₂×Dic₁₁₃	C₂³	1	2	yes	—
45	D₄₅₂	—	4	1	no	—
46	C₄×D₁₁₃	—	4	2	no	—
47	Dic₂₂₆	—	4	1	no	—
48	C₂²×C₁₁₃	—	2	2	no	—
49	C₂²×C₁₁₃	—	2	2	no	—
50	C₂²×C₁₁₃	—	2	2	no	—
51	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
52	C₂×D₁₁₃	—	2	2	no	—
53	C₂×D₁₁₃	—	2	2	no	—
54	C₂×D₁₁₃	—	2	2	no	—
55	C₄×C₁₁₃	—	2	1	no	—
56	C₄×C₁₁₃	—	2	2	no	—
57	C₄×C₁₁₃	—	2	1	no	—
58	C₄×C₁₁₃	—	2	1	no	—
59	Dic₁₁₃	—	2	2	no	—
60	Dic₁₁₃	—	2	2	no	—
61	Dic₁₁₃	—	2	2	no	—
62	C₂×C₁₁₃	—	4	2	no	—
63	C₂×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
66	D₁₁₃	—	4	2	no	—
67	C₁₁₃	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	113	1	no	maximal, Sylow
69	#125	—	113	1	no	—
70	#124	—	113	1	no	—
71	#124	D₁₁₃	1	1	yes	characteristic, complemented
72	#123	—	113	2	no	—
73	#123	—	113	2	no	—
74	C₂²⋊Q₈	—	113	2	no	—
75	C₂²⋊Q₈	—	113	2	no	—
76	#121	—	113	2	no	—
77	#121	—	113	2	no	—
78	C₂×Q₈	—	113	1	no	—
79	C₂×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
80	C₂×D₄	—	113	2	no	—
81	C₂×D₄	—	113	2	no	—
82	C₂×D₄	—	113	1	no	—
83	C₂×D₄	C₂×D₁₁₃	1	1	yes	characteristic
84	C₂²×C₄	—	113	2	no	—
85	C₂²×C₄	—	113	2	no	—
86	C₄⋊C₄	—	113	2	no	—
87	C₄⋊C₄	—	113	2	no	—
88	C₄⋊C₄	—	113	2	no	—
89	C₄⋊C₄	—	113	2	no	—
90	C₄⋊C₄	—	113	1	no	—
91	C₄⋊C₄	—	113	1	no	—
92	C₂²⋊C₄	—	113	2	no	—
93	C₂²⋊C₄	—	113	2	no	—
94	C₂²⋊C₄	—	113	2	no	—
95	C₂²⋊C₄	—	113	2	no	—
96	C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
97	C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
98	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
99	C₂³	—	113	2	no	—
100	C₂³	C₂²×D₁₁₃	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	452	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	452	1	no	—
105	D₄	—	452	2	no	—
106	D₄	—	452	2	no	—
107	C₂×C₄	—	113	2	no	—
108	C₂×C₄	—	113	2	no	—
109	C₂×C₄	—	113	2	no	—
110	C₂×C₄	—	452	2	no	—
111	C₂×C₄	—	452	2	no	—
112	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	226	2	no	—
120	C₂²	—	226	2	no	—
121	C₂²	—	226	2	no	—
122	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	226	2	no	—
128	C₄	—	226	2	no	—
129	C₄	—	226	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	452	2	no	—
132	C₂	#16813667	1	1	yes	characteristic
133	C₂	#16813668	1	1	yes	characteristic
134	C₂	#16813669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16813614

C₂

yes

maximal, characteristic, complemented

#16813610

C₂

yes

maximal, characteristic, complemented

#16813606

C₂

yes

maximal, complemented

#16813604

C₂

yes

maximal, complemented

#16813599

C₂

yes

maximal, complemented

#16813598

C₂

yes

maximal, complemented

#16813595

C₂

yes

maximal, complemented

#16813594

C₂

yes

maximal, complemented

#124×C₁₁₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₂₆

C₂²

yes

characteristic

C₂×D₄₅₂

C₂²

yes

characteristic

C₂×C₂²⋊D₁₁₃

C₂²

yes

—

C₂×C₂²⋊D₁₁₃

C₂²

yes

—

C₂²×Dic₁₁₃

C₂²

yes

—

C₂×C₄×D₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

characteristic

C₄⋊Dic₁₁₃

C₂²

yes

characteristic

#14268626

C₂²

yes

—

#14268626

C₂²

yes

—

#14268626

C₂²

yes

—

C₂²⋊Dic₁₁₃

C₂²

yes

—

#14268624

C₂²

yes

—

#14268624

C₂²

yes

—

#14268624

C₂²

yes

—

C₂×Q₈×C₁₁₃

C₂²

yes

characteristic

C₂×D₄×C₁₁₃

C₂²

yes

characteristic

C₂²⋊C₄×C₁₁₃

C₂²

yes

—

C₂²⋊C₄×C₁₁₃

C₂²

yes

—

C₄²×C₁₁₃

C₂²

yes

characteristic

C₂³×C₁₁₃

C₂³

yes

—

C₂²×D₁₁₃

C₂³

yes

—

Q₈×C₁₁₃

—

D₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂³

yes

—

C₂²⋊D₁₁₃

—

C₂²⋊D₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

D₄₅₂

—

C₄×D₁₁₃

—

Dic₂₂₆

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂⁴

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

2₋¹⁺⁴

yes

characteristic

D₁₁₃

—

C₁₁₃

#562

yes

characteristic, complemented, Sylow

#562

—

113

maximal, Sylow

#125

—

113

—

#124

—

113

—

#124

D₁₁₃

yes

characteristic, complemented

#123

—

113

—

#123

—

113

—

C₂²⋊Q₈

—

113

—

C₂²⋊Q₈

—

113

—

#121

—

113

—

#121

—

113

—

C₂×Q₈

—

113

—

C₂×Q₈

C₂×D₁₁₃

yes

characteristic

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂×D₄

C₂×D₁₁₃

yes

characteristic

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

C₂×D₁₁₃

yes

—

C₂²⋊C₄

C₂×D₁₁₃

yes

—

C₄²

C₂×D₁₁₃

yes

characteristic

C₂³

—

113

—

100

C₂³

C₂²×D₁₁₃

yes

—

101

Q₈

—

102

Q₈

—

452

—

103

D₄

—

104

D₄

—

452

—

105

D₄

—

452

—

106

D₄

—

452

—

107

C₂×C₄

—

113

—

108

C₂×C₄

—

113

—

109

C₂×C₄

—

113

—

110

C₂×C₄

—

452

—

111

C₂×C₄

—

452

—

112

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

113

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

114

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

115

C₂×C₄

C₂²×D₁₁₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

226

—

120

C₂²

—

226

—

121

C₂²

—

226

—

122

C₂²

C₂³×D₁₁₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

226

—

128

C₄

—

226

—

129

C₄

—

226

—

130

C₂

—

131

C₂

—

452

—

132

C₂

#16813667

yes

characteristic

133

C₂

#16813668

yes

characteristic

134

C₂

#16813669

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	#124×C₁₁₃	C₂
hypercenter	#124	D₁₁₃
hypocenter	C₁₁₃	#562
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₁₃

commutator, socle

C₂²×C₁₁₃

C₂⁴

Fitting

#124×C₁₁₃

C₂

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #16813614	C₂	semidirect
2. #16813610	C₂	semidirect
3. #16813606	C₂	semidirect
4. #16813604	C₂	semidirect
5. #16813599	C₂	semidirect
6. #16813598	C₂	semidirect
7. #16813595	C₂	semidirect
8. #16813594	C₂	semidirect
9. #124×C₁₁₃	C₂	semidirect
67. C₁₁₃	#562	semidirect
71. #124	D₁₁₃	semidirect

H ≅ G/N

type

1. #16813614

C₂

semidirect

2. #16813610

C₂

semidirect

3. #16813606

C₂

semidirect

4. #16813604

C₂

semidirect

5. #16813599

C₂

semidirect

6. #16813598

C₂

semidirect

7. #16813595

C₂

semidirect

8. #16813594

C₂

semidirect

9. #124×C₁₁₃

C₂

semidirect

67. C₁₁₃

#562

semidirect

71. #124

D₁₁₃

semidirect

Subgroups — #20299342

Conjugacy classes of subgroups

Complements