Group #20298512 — subgroups

Subgroups — #20298512

Order 7232.

238 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 65 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×#14268626	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Dic₁₁₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Dic₁₁₃	C₂	1	2	yes	maximal, complemented
4	#16813529	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₁₃	C₂²	1	2	yes	—
8	C₂²×Dic₁₁₃	C₂²	1	1	yes	characteristic
9	#14268626	—	2	2	no	—
10	C₂²⋊Dic₁₁₃	—	2	4	no	—
11	C₂²⋊Dic₁₁₃	—	2	4	no	—
12	C₂⁴×C₁₁₃	C₂²	1	2	yes	—
13	C₂×D₄×C₁₁₃	—	2	4	no	—
14	C₂×D₄×C₁₁₃	—	2	2	no	—
15	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₂³×C₁₁₃	—	2	2	no	—
17	C₂³×C₁₁₃	—	2	2	no	—
18	C₂³×C₁₁₃	—	2	2	no	—
19	C₂³×C₁₁₃	—	2	2	no	—
20	C₂³×C₁₁₃	—	2	2	no	—
21	C₂³×C₁₁₃	—	2	2	no	—
22	C₂³×C₁₁₃	D₄	1	4	yes	—
23	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
24	D₄×C₁₁₃	—	2	8	no	—
25	C₂×C₄×C₁₁₃	—	2	2	no	—
26	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
27	C₂×Dic₁₁₃	—	2	1	no	—
28	C₂×Dic₁₁₃	—	2	1	no	—
29	C₂×Dic₁₁₃	—	2	2	no	—
30	C₂×Dic₁₁₃	—	2	2	no	—
31	C₂×Dic₁₁₃	—	2	2	no	—
32	C₂×Dic₁₁₃	—	2	1	no	—
33	C₂×Dic₁₁₃	—	2	2	no	—
34	C₂×Dic₁₁₃	D₄	1	2	yes	complemented
35	C₂²×C₁₁₃	—	2	4	no	—
36	C₂²×C₁₁₃	—	2	4	no	—
37	C₂²×C₁₁₃	—	2	4	no	—
38	C₂²×C₁₁₃	—	4	2	no	—
39	C₂²×C₁₁₃	—	4	2	no	—
40	C₂²×C₁₁₃	—	4	2	no	—
41	C₂²×C₁₁₃	—	4	2	no	—
42	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₁₁₃	C₂×D₄	1	2	yes	—
44	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₁₁₃	#41	1	1	yes	characteristic
46	C₂²×C₁₁₃	#41	1	2	yes	—
47	C₄×C₁₁₃	—	2	2	no	—
48	Dic₁₁₃	—	2	2	no	—
49	Dic₁₁₃	—	4	1	no	—
50	Dic₁₁₃	—	4	2	no	—
51	C₂×C₁₁₃	—	4	4	no	—
52	C₂×C₁₁₃	#120	1	1	yes	characteristic
53	C₂×C₁₁₃	#121	1	1	yes	characteristic
54	C₂×C₁₁₃	#121	1	2	yes	—
55	C₂×C₁₁₃	#123	1	2	yes	—
56	C₂×C₁₁₃	#124	1	1	yes	characteristic
57	C₁₁₃	#394	1	1	yes	characteristic, complemented, Sylow
58	#394	—	113	1	no	maximal, Sylow
59	C₂²×D₄	D₁₁₃	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	113	1	no	—
61	C₂×C₂²⋊C₄	—	113	2	no	—
62	C₂×C₂²⋊C₄	—	113	2	no	—
63	#95	—	113	1	no	—
64	C₂⁴	C₂×D₁₁₃	1	2	yes	—
65	C₂×D₄	—	2	4	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂²×C₄	—	113	1	no	—
68	C₂²×C₄	—	113	2	no	—
69	C₂²×C₄	—	113	1	no	—
70	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
71	C₄⋊C₄	—	226	2	no	—
72	C₂²⋊C₄	—	226	4	no	—
73	C₂²⋊C₄	—	226	4	no	—
74	C₂³	—	2	2	no	—
75	C₂³	—	2	2	no	—
76	C₂³	—	2	2	no	—
77	C₂³	—	2	2	no	—
78	C₂³	—	2	2	no	—
79	C₂³	—	2	2	no	—
80	C₂³	C₂²⋊D₁₁₃	1	4	yes	—
81	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
82	D₄	—	2	8	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	113	2	no	—
85	C₂×C₄	—	226	2	no	—
86	C₂×C₄	—	226	2	no	—
87	C₂×C₄	—	226	2	no	—
88	C₂×C₄	—	226	1	no	—
89	C₂×C₄	—	226	2	no	—
90	C₂×C₄	—	226	1	no	—
91	C₂×C₄	—	226	1	no	—
92	C₂×C₄	C₂²⋊D₁₁₃	1	2	yes	—
93	C₂²	—	2	4	no	—
94	C₂²	—	2	4	no	—
95	C₂²	—	2	4	no	—
96	C₂²	—	4	2	no	—
97	C₂²	—	4	2	no	—
98	C₂²	—	4	2	no	—
99	C₂²	—	4	2	no	—
100	C₂²	D₄×D₁₁₃	1	1	yes	characteristic
101	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
102	C₂²	C₂×C₂²⋊D₁₁₃	1	2	yes	—
103	C₂²	#14268646	1	1	yes	characteristic
104	C₂²	#14268646	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	452	1	no	—
107	C₄	—	452	2	no	—
108	C₄	—	226	2	no	—
109	C₂	—	4	4	no	—
110	C₂	#16813591	1	1	yes	characteristic
111	C₂	#16813593	1	1	yes	characteristic
112	C₂	#16813594	1	2	yes	—
113	C₂	#16813603	1	2	yes	—
114	C₂	#16813609	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14268626

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₁₃

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₁₃

C₂

yes

maximal, complemented

#16813529

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₃

C₂²

yes

characteristic

C₂²×Dic₁₁₃

C₂²

yes

—

C₂²×Dic₁₁₃

C₂²

yes

characteristic

#14268626

—

C₂²⋊Dic₁₁₃

—

C₂²⋊Dic₁₁₃

—

C₂⁴×C₁₁₃

C₂²

yes

—

C₂×D₄×C₁₁₃

—

C₂×D₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

C₂²

yes

characteristic

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

D₄

yes

—

C₂³×C₁₁₃

C₂³

yes

characteristic

D₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

D₄

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

D₄

yes

complemented

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×D₄

yes

characteristic

C₂²×C₁₁₃

C₂×D₄

yes

—

C₂²×C₁₁₃

C₂×D₄

yes

C₂²×C₁₁₃

yes

C₂²×C₁₁₃

yes

—

C₄×C₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

C₂×C₁₁₃

#123

yes

—

C₂×C₁₁₃

#124

yes

characteristic

C₁₁₃

#394

yes

characteristic, complemented, Sylow

#394

—

113

maximal, Sylow

C₂²×D₄

D₁₁₃

yes

characteristic

C₂×C₄⋊C₄

—

113

—

C₂×C₂²⋊C₄

—

113

—

C₂×C₂²⋊C₄

—

113

—

#95

—

113

—

C₂⁴

C₂×D₁₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂×D₁₁₃

yes

characteristic

C₄⋊C₄

—

226

—

C₂²⋊C₄

—

226

—

C₂²⋊C₄

—

226

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊D₁₁₃

yes

—

C₂³

C₂²×D₁₁₃

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

113

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

C₂²⋊D₁₁₃

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

D₄×D₁₁₃

yes

characteristic

101

C₂²

C₂×C₂²⋊D₁₁₃

yes

characteristic

102

C₂²

C₂×C₂²⋊D₁₁₃

yes

—

103

C₂²

#14268646

yes

characteristic

104

C₂²

#14268646

yes

—

105

C₄

—

106

C₄

—

452

—

107

C₄

—

452

—

108

C₄

—

226

—

109

C₂

—

110

C₂

#16813591

yes

characteristic

111

C₂

#16813593

yes

characteristic

112

C₂

#16813594

yes

—

113

C₂

#16813603

yes

—

114

C₂

#16813609

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₁₃
commutator, socle	C₂³×C₁₁₃	C₂³
Fitting	C₂²×D₄×C₁₁₃	C₂
hypercenter	C₂²×D₄	D₁₁₃
hypocenter	C₁₁₃	#394
LCS	G ▷ C₂³×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₁₃

commutator, socle

C₂³×C₁₁₃

C₂³

Fitting

C₂²×D₄×C₁₁₃

C₂

hypercenter

C₂²×D₄

D₁₁₃

C₁₁₃

G ▷ C₂³×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂×#14268626	C₂	semidirect
2. C₂×C₂²⋊Dic₁₁₃	C₂	semidirect
3. C₂×C₂²⋊Dic₁₁₃	C₂	semidirect
4. #16813529	C₂	semidirect
34. C₂×Dic₁₁₃	D₄	semidirect
57. C₁₁₃	#394	semidirect

H ≅ G/N

type

1. C₂×#14268626

C₂

semidirect

2. C₂×C₂²⋊Dic₁₁₃

C₂

semidirect

3. C₂×C₂²⋊Dic₁₁₃

C₂

semidirect

4. #16813529

C₂

semidirect

34. C₂×Dic₁₁₃

D₄

semidirect

57. C₁₁₃

#394

semidirect

Subgroups — #20298512

Conjugacy classes of subgroups

Complements