Group #20299252 — subgroups

Subgroups — #20299252

Order 7232.

292 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14268646	C₂	1	2	yes	maximal, complemented
2	#16813622	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813606	C₂	1	4	yes	maximal, complemented
4	#16813602	C₂	1	2	yes	maximal, complemented
5	#16813596	C₂	1	2	yes	maximal, complemented
6	C₄×D₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Dic₁₁₃	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#14268646	—	2	8	no	—
10	C₂×Dic₂₂₆	C₂²	1	2	yes	—
11	C₂×D₄₅₂	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₁₃	C₂²	1	4	yes	—
13	C₂²×Dic₁₁₃	C₂²	1	4	yes	—
14	C₂×C₄×D₁₁₃	C₂²	1	2	yes	—
15	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
17	C₄⋊Dic₁₁₃	C₂²	1	4	yes	—
18	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
19	#14268624	C₂²	1	2	yes	—
20	#14268624	C₂²	1	4	yes	—
21	C₂×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₁₁₃	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
25	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
26	C₂³×C₁₁₃	C₂³	1	2	yes	—
27	C₂²×D₁₁₃	C₂³	1	2	yes	—
28	D₄×C₁₁₃	D₄	1	4	yes	complemented
29	C₂×C₄×C₁₁₃	—	2	4	no	—
30	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
32	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
34	C₂²⋊D₁₁₃	—	2	8	no	—
35	C₂×Dic₁₁₃	—	2	8	no	—
36	C₂×Dic₁₁₃	C₂³	1	2	yes	—
37	C₂×Dic₁₁₃	C₂³	1	4	yes	—
38	D₄₅₂	—	2	2	no	—
39	C₄×D₁₁₃	—	2	4	no	—
40	Dic₂₂₆	—	2	4	no	—
41	C₂²×C₁₁₃	—	2	2	no	—
42	C₂²×C₁₁₃	—	2	2	no	—
43	C₂²×C₁₁₃	C₂×D₄	1	4	yes	—
44	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₁₃	—	2	2	no	—
46	C₂×D₁₁₃	—	2	2	no	—
47	C₂×D₁₁₃	—	2	2	no	—
48	C₄×C₁₁₃	—	2	2	no	—
49	C₄×C₁₁₃	—	2	1	no	—
50	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
51	C₄×C₁₁₃	#41	1	2	yes	complemented
52	Dic₁₁₃	—	2	4	no	—
53	Dic₁₁₃	—	2	2	no	—
54	C₂×C₁₁₃	—	2	4	no	—
55	C₂×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₁₁₃	C₂×#41	1	1	yes	characteristic
57	C₂×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
58	D₁₁₃	—	4	2	no	—
59	C₁₁₃	#547	1	1	yes	characteristic, complemented, Sylow
60	#547	—	113	1	no	maximal, Sylow
61	C₂×#41	—	113	2	no	—
62	C₄⋊Q₈	—	113	1	no	—
63	#123	—	113	4	no	—
64	C₂²⋊Q₈	—	113	2	no	—
65	#121	—	113	2	no	—
66	C₄×D₄	—	113	1	no	—
67	C₄×D₄	D₁₁₃	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₄	—	113	2	no	—
69	#41	—	226	8	no	—
70	C₂×Q₈	—	113	2	no	—
71	C₂×D₄	—	113	4	no	—
72	C₂×D₄	—	113	1	no	—
73	C₂×D₄	C₂×D₁₁₃	1	1	yes	characteristic
74	C₂²×C₄	—	113	4	no	—
75	C₂²×C₄	—	113	2	no	—
76	C₂²×C₄	C₂×D₁₁₃	1	2	yes	—
77	C₄⋊C₄	—	113	1	no	—
78	C₄⋊C₄	—	113	2	no	—
79	C₄⋊C₄	—	113	4	no	—
80	C₄⋊C₄	—	113	2	no	—
81	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
82	C₂²⋊C₄	—	113	2	no	—
83	C₂²⋊C₄	—	113	4	no	—
84	C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
85	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
86	C₂³	—	113	2	no	—
87	C₂³	C₂²×D₁₁₃	1	2	yes	—
88	Q₈	—	226	4	no	—
89	D₄	—	226	8	no	—
90	D₄	—	226	2	no	—
91	D₄	D₄₅₂	1	4	yes	complemented
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	113	2	no	—
94	C₂×C₄	—	113	4	no	—
95	C₂×C₄	—	226	8	no	—
96	C₂×C₄	—	226	4	no	—
97	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
99	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
101	C₂²	—	2	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	226	2	no	—
104	C₂²	—	226	2	no	—
105	C₂²	—	226	2	no	—
106	C₂²	C₂×D₄₅₂	1	4	yes	—
107	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	226	2	no	—
111	C₄	—	226	4	no	—
112	C₄	C₂×D₄₅₂	1	2	yes	—
113	C₄	#14268646	1	2	yes	complemented
114	C₂	—	2	4	no	—
115	C₂	—	452	2	no	—
116	C₂	C₂²×D₄₅₂	1	1	yes	characteristic
117	C₂	C₂×#14268646	1	1	yes	characteristic
118	C₂	#16813669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14268646

C₂

yes

maximal, complemented

#16813622

C₂

yes

maximal, characteristic, complemented

#16813606

C₂

yes

maximal, complemented

#16813602

C₂

yes

maximal, complemented

#16813596

C₂

yes

maximal, complemented

C₄×D₄₅₂

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Dic₁₁₃

C₂

yes

maximal, complemented

C₄×D₄×C₁₁₃

C₂

yes

maximal, characteristic, complemented

#14268646

—

C₂×Dic₂₂₆

C₂²

yes

—

C₂×D₄₅₂

C₂²

yes

characteristic

C₂×C₂²⋊D₁₁₃

C₂²

yes

—

C₂²×Dic₁₁₃

C₂²

yes

—

C₂×C₄×D₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

characteristic

C₄⋊Dic₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

—

#14268624

C₂²

yes

—

#14268624

C₂²

yes

—

C₂×D₄×C₁₁₃

C₂²

yes

characteristic

C₂²×C₄×C₁₁₃

C₂²

yes

—

C₄⋊C₄×C₁₁₃

C₂²

yes

characteristic

C₂²⋊C₄×C₁₁₃

C₂²

yes

—

C₄²×C₁₁₃

C₂²

yes

characteristic

C₂³×C₁₁₃

C₂³

yes

—

C₂²×D₁₁₃

C₂³

yes

—

D₄×C₁₁₃

D₄

yes

complemented

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂²⋊D₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

D₄₅₂

—

C₄×D₁₁₃

—

Dic₂₂₆

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×D₄

yes

—

C₂²×C₁₁₃

C₂⁴

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×D₄

yes

—

C₄×C₁₁₃

#41

yes

complemented

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂²×D₄

yes

characteristic

C₂×C₁₁₃

C₂×#41

yes

characteristic

C₂×C₁₁₃

2₋¹⁺⁴

yes

characteristic

D₁₁₃

—

C₁₁₃

#547

yes

characteristic, complemented, Sylow

#547

—

113

maximal, Sylow

C₂×#41

—

113

—

C₄⋊Q₈

—

113

—

#123

—

113

—

C₂²⋊Q₈

—

113

—

#121

—

113

—

C₄×D₄

—

113

—

C₄×D₄

D₁₁₃

yes

characteristic, complemented

C₂×C₄⋊C₄

—

113

—

#41

—

226

—

C₂×Q₈

—

113

—

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂×D₄

C₂×D₁₁₃

yes

characteristic

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂×D₁₁₃

yes

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

C₂×D₁₁₃

yes

characteristic

C₂²⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

C₂×D₁₁₃

yes

—

C₄²

C₂×D₁₁₃

yes

characteristic

C₂³

—

113

—

C₂³

C₂²×D₁₁₃

yes

—

Q₈

—

226

—

D₄

—

226

—

D₄

—

226

—

D₄

D₄₅₂

yes

complemented

C₂×C₄

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₄

C₂²×D₁₁₃

yes

—

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

100

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

—

226

—

104

C₂²

—

226

—

105

C₂²

—

226

—

106

C₂²

C₂×D₄₅₂

yes

—

107

C₂²

C₂³×D₁₁₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

226

—

111

C₄

—

226

—

112

C₄

C₂×D₄₅₂

yes

—

113

C₄

#14268646

yes

complemented

114

C₂

—

115

C₂

—

452

—

116

C₂

C₂²×D₄₅₂

yes

characteristic

117

C₂

C₂×#14268646

yes

characteristic

118

C₂

#16813669

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	C₄×D₄×C₁₁₃	C₂
hypercenter	C₄×D₄	D₁₁₃
hypocenter	C₁₁₃	#547
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₁₃

commutator, socle

C₂²×C₁₁₃

C₂⁴

Fitting

C₄×D₄×C₁₁₃

C₂

hypercenter

C₄×D₄

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#14268646	C₂	semidirect
2. #16813622	C₂	semidirect
3. #16813606	C₂	semidirect
4. #16813602	C₂	semidirect
5. #16813596	C₂	semidirect
6. C₄×D₄₅₂	C₂	semidirect
7. C₂×C₄⋊Dic₁₁₃	C₂	semidirect
8. C₄×D₄×C₁₁₃	C₂	semidirect
28. D₄×C₁₁₃	D₄	semidirect
51. C₄×C₁₁₃	#41	semidirect
59. C₁₁₃	#547	semidirect
67. C₄×D₄	D₁₁₃	semidirect
91. D₄	D₄₅₂	semidirect
113. C₄	#14268646	semidirect

H ≅ G/N

type

1. C₂×#14268646

C₂

semidirect

2. #16813622

C₂

semidirect

3. #16813606

C₂

semidirect

4. #16813602

C₂

semidirect

5. #16813596

C₂

semidirect

6. C₄×D₄₅₂

C₂

semidirect

7. C₂×C₄⋊Dic₁₁₃

C₂

semidirect

8. C₄×D₄×C₁₁₃

C₂

semidirect

28. D₄×C₁₁₃

D₄

semidirect

51. C₄×C₁₁₃

#41

semidirect

59. C₁₁₃

#547

semidirect

67. C₄×D₄

D₁₁₃

semidirect

91. D₄

D₄₅₂

semidirect

113. C₄

#14268646

semidirect

Subgroups — #20299252

Conjugacy classes of subgroups

Complements