Group #20299109 — subgroups

Subgroups — #20299109

Order 7232.

330 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14268647	C₂	1	1	yes	maximal, characteristic, complemented
2	#16813605	C₂	1	2	yes	maximal, complemented
3	#16813601	C₂	1	2	yes	maximal, complemented
4	#16813596	C₂	1	2	yes	maximal, complemented
5	#16813591	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊D₁₁₃	C₂	1	4	yes	maximal, complemented
7	#16813581	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#14268647	—	2	4	no	—
10	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄₅₂	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₂²⋊D₁₁₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
14	C₂×C₄×D₁₁₃	C₂²	1	2	yes	complemented
15	C₄⋊Dic₁₁₃	C₂²	1	2	yes	complemented
16	#14268626	C₂²	1	4	yes	complemented
17	C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
18	C₂²⋊Dic₁₁₃	C₂²	1	4	yes	—
19	#14268624	C₂²	1	4	yes	complemented
20	C₄×Dic₁₁₃	C₂²	1	2	yes	complemented
21	C₂⁴×C₁₁₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₁₁₃	—	2	4	no	—
23	C₂²×C₄×C₁₁₃	C₂²	1	2	yes	—
24	C₂²×C₄×C₁₁₃	C₂²	1	2	yes	—
25	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
27	C₂³×C₁₁₃	—	2	2	no	—
28	C₂³×C₁₁₃	—	2	4	no	—
29	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
30	C₂³×C₁₁₃	C₂³	1	2	yes	—
31	C₂²×D₁₁₃	C₂³	1	2	yes	—
32	C₂×C₄×C₁₁₃	—	2	2	no	—
33	C₂×C₄×C₁₁₃	—	2	4	no	—
34	C₂×C₄×C₁₁₃	—	2	4	no	—
35	C₂×C₄×C₁₁₃	D₄	1	4	yes	complemented
36	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
37	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
39	C₂²⋊D₁₁₃	—	2	4	no	—
40	C₂²⋊D₁₁₃	—	2	8	no	—
41	C₂×Dic₁₁₃	C₂³	1	4	yes	—
42	C₂×Dic₁₁₃	C₂³	1	2	yes	—
43	D₄₅₂	—	2	2	no	—
44	C₄×D₁₁₃	—	2	4	no	—
45	Dic₂₂₆	—	2	2	no	—
46	C₂²×C₁₁₃	—	2	8	no	—
47	C₂²×C₁₁₃	—	2	2	no	—
48	C₂²×C₁₁₃	—	2	2	no	—
49	C₂²×C₁₁₃	—	2	2	no	—
50	C₂²×C₁₁₃	C₂×D₄	1	2	yes	—
51	C₂²×C₁₁₃	#41	1	4	yes	complemented
52	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
53	C₂×D₁₁₃	—	2	2	no	—
54	C₂×D₁₁₃	—	2	4	no	—
55	C₄×C₁₁₃	—	2	2	no	—
56	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
57	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
58	Dic₁₁₃	—	2	2	no	—
59	Dic₁₁₃	—	2	4	no	—
60	C₂×C₁₁₃	—	2	2	no	—
61	C₂×C₁₁₃	—	2	4	no	—
62	C₂×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₁₁₃	C₂×#41	1	2	yes	—
64	D₁₁₃	—	4	2	no	—
65	C₁₁₃	#525	1	1	yes	characteristic, complemented, Sylow
66	#525	—	113	1	no	maximal, Sylow
67	C₂×#41	—	113	1	no	—
68	C₂³×C₄	D₁₁₃	1	1	yes	characteristic, complemented
69	#123	—	113	2	no	—
70	C₂²⋊Q₈	—	113	2	no	—
71	#121	—	113	2	no	—
72	#120	—	113	2	no	—
73	C₄×D₄	—	113	4	no	—
74	#117	—	113	1	no	—
75	C₂⁴	C₂×D₁₁₃	1	1	yes	characteristic
76	#41	—	226	4	no	—
77	C₂×Q₈	—	113	1	no	—
78	C₂×D₄	—	113	4	no	—
79	C₂×D₄	—	113	2	no	—
80	C₂×D₄	—	113	1	no	—
81	C₂²×C₄	—	2	4	no	—
82	C₂²×C₄	—	113	2	no	—
83	C₂²×C₄	C₂×D₁₁₃	1	2	yes	—
84	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₁₁₃	1	2	yes	—
86	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
87	C₄⋊C₄	—	113	2	no	—
88	C₄⋊C₄	—	113	4	no	—
89	C₂²⋊C₄	—	113	2	no	—
90	C₂²⋊C₄	—	113	4	no	—
91	C₂²⋊C₄	—	113	4	no	—
92	C₄²	—	113	2	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	113	2	no	—
96	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
97	C₂³	C₂²×D₁₁₃	1	2	yes	—
98	Q₈	—	226	2	no	—
99	D₄	—	226	4	no	—
100	D₄	—	226	8	no	—
101	D₄	—	226	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	113	4	no	—
106	C₂×C₄	—	113	2	no	—
107	C₂×C₄	—	226	4	no	—
108	C₂×C₄	C₂²⋊D₁₁₃	1	4	yes	complemented
109	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
110	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
112	C₂²	—	2	8	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	226	2	no	—
117	C₂²	—	226	4	no	—
118	C₂²	C₂×C₂²⋊D₁₁₃	1	2	yes	—
119	C₂²	#14268647	1	4	yes	complemented
120	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	226	2	no	—
123	C₄	—	226	4	no	—
124	C₄	C₂×C₂²⋊D₁₁₃	1	2	yes	—
125	C₄	C₂×C₂²⋊D₁₁₃	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	2	4	no	—
128	C₂	—	452	2	no	—
129	C₂	C₂²×C₂²⋊D₁₁₃	1	1	yes	characteristic
130	C₂	C₂×#14268647	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14268647

C₂

yes

maximal, characteristic, complemented

#16813605

C₂

yes

maximal, complemented

#16813601

C₂

yes

maximal, complemented

#16813596

C₂

yes

maximal, complemented

#16813591

C₂

yes

maximal, complemented

C₄×C₂²⋊D₁₁₃

C₂

yes

maximal, complemented

#16813581

C₂

yes

maximal, characteristic, complemented

C₂³×C₄×C₁₁₃

C₂

yes

maximal, characteristic, complemented

#14268647

—

C₂×Dic₂₂₆

C₂²

yes

characteristic, complemented

C₂×D₄₅₂

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₁₁₃

C₂²

yes

—

C₂×C₂²⋊D₁₁₃

C₂²

yes

—

C₂×C₄×D₁₁₃

C₂²

yes

complemented

C₄⋊Dic₁₁₃

C₂²

yes

complemented

#14268626

C₂²

yes

complemented

C₂²⋊Dic₁₁₃

C₂²

yes

—

C₂²⋊Dic₁₁₃

C₂²

yes

—

#14268624

C₂²

yes

complemented

C₄×Dic₁₁₃

C₂²

yes

complemented

C₂⁴×C₁₁₃

C₂²

yes

characteristic

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

C₂²

yes

—

C₂²×C₄×C₁₁₃

C₂²

yes

—

C₂²×C₄×C₁₁₃

C₂²

yes

characteristic

C₂²×C₄×C₁₁₃

C₂²

yes

characteristic

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

C₂³

yes

characteristic

C₂³×C₁₁₃

C₂³

yes

—

C₂²×D₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

D₄

yes

complemented

C₂×C₄×C₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂²⋊D₁₁₃

—

C₂²⋊D₁₁₃

—

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

D₄₅₂

—

C₄×D₁₁₃

—

Dic₂₂₆

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×D₄

yes

—

C₂²×C₁₁₃

#41

yes

complemented

C₂²×C₁₁₃

C₂⁴

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×D₄

yes

—

C₄×C₁₁₃

C₂×D₄

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂²×D₄

yes

characteristic

C₂×C₁₁₃

C₂×#41

yes

—

D₁₁₃

—

C₁₁₃

#525

yes

characteristic, complemented, Sylow

#525

—

113

maximal, Sylow

C₂×#41

—

113

—

C₂³×C₄

D₁₁₃

yes

characteristic, complemented

#123

—

113

—

C₂²⋊Q₈

—

113

—

#121

—

113

—

#120

—

113

—

C₄×D₄

—

113

—

#117

—

113

—

C₂⁴

C₂×D₁₁₃

yes

characteristic

#41

—

226

—

C₂×Q₈

—

113

—

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂²×C₄

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂×D₁₁₃

yes

—

C₂²×C₄

C₂×D₁₁₃

yes

characteristic

C₂²×C₄

C₂×D₁₁₃

yes

—

C₂²×C₄

C₂×D₁₁₃

yes

characteristic

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₄²

—

113

—

C₂³

—

C₂³

—

C₂³

—

113

—

C₂³

C₂²×D₁₁₃

yes

characteristic

C₂³

C₂²×D₁₁₃

yes

—

Q₈

—

226

—

D₄

—

226

—

100

D₄

—

226

—

101

D₄

—

226

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

113

—

106

C₂×C₄

—

113

—

107

C₂×C₄

—

226

—

108

C₂×C₄

C₂²⋊D₁₁₃

yes

complemented

109

C₂×C₄

C₂²×D₁₁₃

yes

—

110

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

111

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

226

—

117

C₂²

—

226

—

118

C₂²

C₂×C₂²⋊D₁₁₃

yes

—

119

C₂²

#14268647

yes

complemented

120

C₂²

C₂³×D₁₁₃

yes

characteristic

121

C₄

—

122

C₄

—

226

—

123

C₄

—

226

—

124

C₄

C₂×C₂²⋊D₁₁₃

yes

—

125

C₄

C₂×C₂²⋊D₁₁₃

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

452

—

129

C₂

C₂²×C₂²⋊D₁₁₃

yes

characteristic

130

C₂

C₂×#14268647

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Frattini	C₂²	C₂³×D₁₁₃
Fitting	C₂³×C₄×C₁₁₃	C₂
hypercenter	C₂³×C₄	D₁₁₃
hypocenter	C₁₁₃	#525
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

subgroup

G/H

center

C₂×C₄

C₂²×D₁₁₃

commutator, socle

C₂²×C₁₁₃

C₂⁴

Frattini

C₂²

C₂³×D₁₁₃

Fitting

C₂³×C₄×C₁₁₃

C₂

hypercenter

C₂³×C₄

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂×C₄ ◁ C₂³×C₄

N	H ≅ G/N	type
1. C₂×#14268647	C₂	semidirect
2. #16813605	C₂	semidirect
3. #16813601	C₂	semidirect
4. #16813596	C₂	semidirect
5. #16813591	C₂	semidirect
6. C₄×C₂²⋊D₁₁₃	C₂	semidirect
7. #16813581	C₂	semidirect
8. C₂³×C₄×C₁₁₃	C₂	semidirect
10. C₂×Dic₂₂₆	C₂²	semidirect
11. C₂×D₄₅₂	C₂²	semidirect
14. C₂×C₄×D₁₁₃	C₂²	semidirect
15. C₄⋊Dic₁₁₃	C₂²	semidirect
16. #14268626	C₂²	semidirect
19. #14268624	C₂²	semidirect
20. C₄×Dic₁₁₃	C₂²	semidirect
35. C₂×C₄×C₁₁₃	D₄	semidirect
51. C₂²×C₁₁₃	#41	semidirect
65. C₁₁₃	#525	semidirect
68. C₂³×C₄	D₁₁₃	semidirect
108. C₂×C₄	C₂²⋊D₁₁₃	semidirect
119. C₂²	#14268647	semidirect

H ≅ G/N

type

1. C₂×#14268647

C₂

semidirect

2. #16813605

C₂

semidirect

3. #16813601

C₂

semidirect

4. #16813596

C₂

semidirect

5. #16813591

C₂

semidirect

6. C₄×C₂²⋊D₁₁₃

C₂

semidirect

7. #16813581

C₂

semidirect

8. C₂³×C₄×C₁₁₃

C₂

semidirect

10. C₂×Dic₂₂₆

C₂²

semidirect

11. C₂×D₄₅₂

C₂²

semidirect

14. C₂×C₄×D₁₁₃

C₂²

semidirect

15. C₄⋊Dic₁₁₃

C₂²

semidirect

16. #14268626

C₂²

semidirect

19. #14268624

C₂²

semidirect

20. C₄×Dic₁₁₃

C₂²

semidirect

35. C₂×C₄×C₁₁₃

D₄

semidirect

51. C₂²×C₁₁₃

#41

semidirect

65. C₁₁₃

#525

semidirect

68. C₂³×C₄

D₁₁₃

semidirect

108. C₂×C₄

C₂²⋊D₁₁₃

semidirect

119. C₂²

#14268647

semidirect

Subgroups — #20299109

Conjugacy classes of subgroups

Complements