Group #20299167 — subgroups

Subgroups — #20299167

Order 7232.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂²×D₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813619	C₂	1	2	yes	maximal, complemented
4	#16813610	C₂	1	2	yes	maximal, complemented
5	#16813597	C₂	1	4	yes	maximal, complemented
6	#16813592	C₂	1	4	yes	maximal, complemented
7	#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₁₃	C₂²	1	2	yes	—
9	#14268647	—	2	4	no	—
10	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄₅₂	—	2	4	no	—
12	C₂×D₄₅₂	C₂²	1	2	yes	—
13	C₂×D₄₅₂	C₂²	1	2	yes	—
14	C₂×D₄₅₂	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₄₅₂	C₂²	1	8	yes	complemented
16	C₂×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
17	C₂×C₄×D₁₁₃	C₂²	1	2	yes	complemented
18	#14268624	C₂²	1	8	yes	complemented
19	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	complemented
21	C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
22	C₄²×C₁₁₃	C₂²	1	2	yes	complemented
23	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
24	C₂²×D₁₁₃	—	2	4	no	—
25	C₂²×D₁₁₃	—	4	4	no	—
26	C₂²×D₁₁₃	C₂³	1	4	yes	—
27	C₂²×D₁₁₃	C₂³	1	2	yes	—
28	C₂×C₄×C₁₁₃	D₄	1	4	yes	complemented
29	C₂×C₄×C₁₁₃	C₂³	1	4	yes	—
30	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
32	C₂²⋊D₁₁₃	—	2	4	no	—
33	C₂×Dic₁₁₃	C₂³	1	2	yes	—
34	D₄₅₂	—	2	4	no	—
35	D₄₅₂	—	2	4	no	—
36	D₄₅₂	—	2	2	no	—
37	D₄₅₂	—	4	8	no	—
38	C₄×D₁₁₃	—	2	4	no	—
39	Dic₂₂₆	—	2	2	no	—
40	C₂²×C₁₁₃	—	2	2	no	—
41	C₂²×C₁₁₃	C₂×D₄	1	2	yes	—
42	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₁₃	—	2	2	no	—
44	C₂×D₁₁₃	—	2	4	no	—
45	C₂×D₁₁₃	—	2	4	no	—
46	C₂×D₁₁₃	—	2	8	no	—
47	C₂×D₁₁₃	—	4	8	no	—
48	C₄×C₁₁₃	—	2	4	no	—
49	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
50	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
51	Dic₁₁₃	—	2	2	no	—
52	C₂×C₁₁₃	—	2	2	no	—
53	C₂×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
55	D₁₁₃	—	4	2	no	—
56	D₁₁₃	—	4	4	no	—
57	C₁₁₃	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	113	1	no	maximal, Sylow
59	C₂×#41	—	113	1	no	—
60	C₂²×D₄	—	113	1	no	—
61	#127	—	113	2	no	—
62	#124	—	113	2	no	—
63	#121	—	113	4	no	—
64	#120	—	113	4	no	—
65	#117	D₁₁₃	1	1	yes	characteristic, complemented
66	C₂⁴	—	113	2	no	—
67	#41	—	226	4	no	—
68	C₂×Q₈	—	113	1	no	—
69	C₂×D₄	—	113	2	no	—
70	C₂×D₄	—	113	8	no	—
71	C₂×D₄	—	113	2	no	—
72	C₂×D₄	—	113	2	no	—
73	C₂×D₄	—	113	1	no	—
74	C₂×D₄	—	226	4	no	—
75	C₂²×C₄	—	113	2	no	—
76	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	complemented
78	C₂²⋊C₄	—	113	8	no	—
79	C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
80	C₄²	C₂×D₁₁₃	1	2	yes	complemented
81	C₂³	—	113	4	no	—
82	C₂³	—	113	2	no	—
83	C₂³	—	452	4	no	—
84	C₂³	—	226	4	no	—
85	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
86	Q₈	—	226	2	no	—
87	D₄	—	452	8	no	—
88	D₄	—	226	4	no	—
89	D₄	—	226	4	no	—
90	D₄	—	226	4	no	—
91	D₄	—	226	2	no	—
92	C₂×C₄	—	113	2	no	—
93	C₂×C₄	—	226	4	no	—
94	C₂×C₄	D₄₅₂	1	4	yes	complemented
95	C₂×C₄	C₂²×D₁₁₃	1	4	yes	—
96	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	452	8	no	—
100	C₂²	—	226	2	no	—
101	C₂²	—	226	4	no	—
102	C₂²	—	226	4	no	—
103	C₂²	—	226	8	no	—
104	C₂²	C₂×D₄₅₂	1	2	yes	—
105	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	226	2	no	—
108	C₄	C₂×D₄₅₂	1	2	yes	—
109	C₄	C₂×D₄₅₂	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	452	4	no	—
112	C₂	—	452	2	no	—
113	C₂	C₂²×D₄₅₂	1	1	yes	characteristic
114	C₂	#16813668	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14268647

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₅₂

C₂

yes

maximal, characteristic, complemented

#16813619

C₂

yes

maximal, complemented

#16813610

C₂

yes

maximal, complemented

#16813597

C₂

yes

maximal, complemented

#16813592

C₂

yes

maximal, complemented

#117×C₁₁₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₁₃

C₂²

yes

—

#14268647

—

C₂×Dic₂₂₆

C₂²

yes

characteristic, complemented

C₂×D₄₅₂

—

C₂×D₄₅₂

C₂²

yes

—

C₂×D₄₅₂

C₂²

yes

—

C₂×D₄₅₂

C₂²

yes

characteristic, complemented

C₂×D₄₅₂

C₂²

yes

complemented

C₂×C₂²⋊D₁₁₃

C₂²

yes

—

C₂×C₄×D₁₁₃

C₂²

yes

complemented

#14268624

C₂²

yes

complemented

C₂²×C₄×C₁₁₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₁₃

C₂²

yes

complemented

C₂²⋊C₄×C₁₁₃

C₂²

yes

—

C₄²×C₁₁₃

C₂²

yes

complemented

C₂³×C₁₁₃

C₂³

yes

characteristic

C₂²×D₁₁₃

—

C₂²×D₁₁₃

—

C₂²×D₁₁₃

C₂³

yes

—

C₂²×D₁₁₃

C₂³

yes

—

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

—

C₂×Dic₁₁₃

C₂³

yes

—

D₄₅₂

—

D₄₅₂

—

D₄₅₂

—

D₄₅₂

—

C₄×D₁₁₃

—

Dic₂₂₆

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×D₄

yes

—

C₂²×C₁₁₃

C₂⁴

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×D₄

yes

—

C₄×C₁₁₃

C₂×D₄

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂²×D₄

yes

characteristic

C₂×C₁₁₃

H(5,𝔽₂)

yes

—

D₁₁₃

—

D₁₁₃

—

C₁₁₃

#535

yes

characteristic, complemented, Sylow

#535

—

113

maximal, Sylow

C₂×#41

—

113

—

C₂²×D₄

—

113

—

#127

—

113

—

#124

—

113

—

#121

—

113

—

#120

—

113

—

#117

D₁₁₃

yes

characteristic, complemented

C₂⁴

—

113

—

#41

—

226

—

C₂×Q₈

—

113

—

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂×D₄

—

226

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂×D₁₁₃

yes

characteristic

C₄⋊C₄

C₂×D₁₁₃

yes

complemented

C₂²⋊C₄

—

113

—

C₂²⋊C₄

C₂×D₁₁₃

yes

—

C₄²

C₂×D₁₁₃

yes

complemented

C₂³

—

113

—

C₂³

—

113

—

C₂³

—

452

—

C₂³

—

226

—

C₂³

C₂²×D₁₁₃

yes

characteristic

Q₈

—

226

—

D₄

—

452

—

D₄

—

226

—

D₄

—

226

—

D₄

—

226

—

D₄

—

226

—

C₂×C₄

—

113

—

C₂×C₄

—

226

—

C₂×C₄

D₄₅₂

yes

complemented

C₂×C₄

C₂²×D₁₁₃

yes

—

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂²

—

C₂²

—

452

—

100

C₂²

—

226

—

101

C₂²

—

226

—

102

C₂²

—

226

—

103

C₂²

—

226

—

104

C₂²

C₂×D₄₅₂

yes

—

105

C₂²

C₂³×D₁₁₃

yes

characteristic

106

C₄

—

107

C₄

—

226

—

108

C₄

C₂×D₄₅₂

yes

—

109

C₄

C₂×D₄₅₂

yes

—

110

C₂

—

111

C₂

—

452

—

112

C₂

—

452

—

113

C₂

C₂²×D₄₅₂

yes

characteristic

114

C₂

#16813668

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	#117×C₁₁₃	C₂
hypercenter	#117	D₁₁₃
hypocenter	C₁₁₃	#535
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₁₃

commutator, socle

C₂²×C₁₁₃

C₂⁴

Fitting

#117×C₁₁₃

C₂

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂×#14268647	C₂	semidirect
2. C₂²×D₄₅₂	C₂	semidirect
3. #16813619	C₂	semidirect
4. #16813610	C₂	semidirect
5. #16813597	C₂	semidirect
6. #16813592	C₂	semidirect
7. #117×C₁₁₃	C₂	semidirect
10. C₂×Dic₂₂₆	C₂²	semidirect
14. C₂×D₄₅₂	C₂²	semidirect
15. C₂×D₄₅₂	C₂²	semidirect
17. C₂×C₄×D₁₁₃	C₂²	semidirect
18. #14268624	C₂²	semidirect
20. C₄⋊C₄×C₁₁₃	C₂²	semidirect
22. C₄²×C₁₁₃	C₂²	semidirect
28. C₂×C₄×C₁₁₃	D₄	semidirect
57. C₁₁₃	#535	semidirect
65. #117	D₁₁₃	semidirect
77. C₄⋊C₄	C₂×D₁₁₃	semidirect
80. C₄²	C₂×D₁₁₃	semidirect
94. C₂×C₄	D₄₅₂	semidirect

H ≅ G/N

type

1. C₂×#14268647

C₂

semidirect

2. C₂²×D₄₅₂

C₂

semidirect

3. #16813619

C₂

semidirect

4. #16813610

C₂

semidirect

5. #16813597

C₂

semidirect

6. #16813592

C₂

semidirect

7. #117×C₁₁₃

C₂

semidirect

10. C₂×Dic₂₂₆

C₂²

semidirect

14. C₂×D₄₅₂

C₂²

semidirect

15. C₂×D₄₅₂

C₂²

semidirect

17. C₂×C₄×D₁₁₃

C₂²

semidirect

18. #14268624

C₂²

semidirect

20. C₄⋊C₄×C₁₁₃

C₂²

semidirect

22. C₄²×C₁₁₃

C₂²

semidirect

28. C₂×C₄×C₁₁₃

D₄

semidirect

57. C₁₁₃

#535

semidirect

65. #117

D₁₁₃

semidirect

77. C₄⋊C₄

C₂×D₁₁₃

semidirect

80. C₄²

C₂×D₁₁₃

semidirect

94. C₂×C₄

D₄₅₂

semidirect

Subgroups — #20299167

Conjugacy classes of subgroups

Complements