Group #20299313 — subgroups

Subgroups — #20299313

Order 7232.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#16813622	C₂	1	1	yes	maximal, characteristic, complemented
2	#16813608	C₂	1	4	yes	maximal, complemented
3	#16813596	C₂	1	2	yes	maximal, complemented
4	#16813593	C₂	1	2	yes	maximal, complemented
5	C₄×D₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813581	C₂	1	2	yes	maximal, complemented
7	#16813580	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₂₆	C₂²	1	2	yes	—
10	C₂×D₄₅₂	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₁₃	C₂²	1	4	yes	—
12	C₂×C₄×D₁₁₃	C₂²	1	2	yes	—
13	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
14	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
16	C₂²⋊Dic₁₁₃	C₂²	1	4	yes	—
17	#14268624	C₂²	1	4	yes	—
18	#14268624	C₂²	1	2	yes	—
19	C₄×Dic₁₁₃	C₂²	1	4	yes	—
20	C₂×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₁₃	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
24	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
25	C₂³×C₁₁₃	C₂³	1	2	yes	—
26	C₂²×D₁₁₃	C₂³	1	2	yes	—
27	D₄×C₁₁₃	—	2	2	no	—
28	C₂×C₄×C₁₁₃	—	2	4	no	—
29	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
31	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
33	C₂²⋊D₁₁₃	—	4	4	no	—
34	C₂×Dic₁₁₃	C₂³	1	2	yes	—
35	C₂×Dic₁₁₃	C₂³	1	4	yes	—
36	D₄₅₂	—	2	2	no	—
37	C₄×D₁₁₃	—	2	4	no	—
38	Dic₂₂₆	—	4	2	no	—
39	C₂²×C₁₁₃	—	2	2	no	—
40	C₂²×C₁₁₃	—	2	2	no	—
41	C₂²×C₁₁₃	—	2	2	no	—
42	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₁₃	—	2	2	no	—
44	C₂×D₁₁₃	—	2	2	no	—
45	C₂×D₁₁₃	—	2	2	no	—
46	C₄×C₁₁₃	—	2	2	no	—
47	C₄×C₁₁₃	—	2	1	no	—
48	C₄×C₁₁₃	#41	1	2	yes	—
49	C₄×C₁₁₃	#41	1	2	yes	—
50	Dic₁₁₃	—	2	2	no	—
51	Dic₁₁₃	—	2	4	no	—
52	C₂×C₁₁₃	—	4	2	no	—
53	C₂×C₁₁₃	C₂×#41	1	1	yes	characteristic
54	C₂×C₁₁₃	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
56	D₁₁₃	—	4	2	no	—
57	C₁₁₃	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	113	1	no	maximal, Sylow
59	C₄⋊Q₈	—	113	1	no	—
60	#124	—	113	4	no	—
61	#121	—	113	2	no	—
62	#121	—	113	2	no	—
63	C₄×D₄	—	113	1	no	—
64	C₄×D₄	D₁₁₃	1	1	yes	characteristic, complemented
65	#117	—	113	2	no	—
66	#117	—	113	2	no	—
67	C₂×Q₈	—	113	2	no	—
68	C₂×D₄	—	113	4	no	—
69	C₂×D₄	—	113	1	no	—
70	C₂×D₄	C₂×D₁₁₃	1	1	yes	characteristic
71	C₂²×C₄	—	113	2	no	—
72	C₂²×C₄	C₂×D₁₁₃	1	2	yes	—
73	C₄⋊C₄	—	113	1	no	—
74	C₄⋊C₄	—	113	2	no	—
75	C₄⋊C₄	—	113	2	no	—
76	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
77	C₂²⋊C₄	—	113	4	no	—
78	C₂²⋊C₄	—	113	4	no	—
79	C₂²⋊C₄	—	113	2	no	—
80	C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
81	C₄²	—	113	4	no	—
82	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
83	C₂³	—	113	2	no	—
84	C₂³	C₂²×D₁₁₃	1	2	yes	—
85	Q₈	—	452	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	452	4	no	—
88	D₄	—	226	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	113	2	no	—
91	C₂×C₄	—	113	4	no	—
92	C₂×C₄	—	226	4	no	—
93	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
94	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	226	2	no	—
101	C₂²	—	226	2	no	—
102	C₂²	—	226	2	no	—
103	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	226	2	no	—
107	C₄	—	226	4	no	—
108	C₄	#14268646	1	2	yes	—
109	C₄	#14268647	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	452	2	no	—
112	C₂	C₂×#14268646	1	1	yes	characteristic
113	C₂	C₂×#14268647	1	1	yes	characteristic
114	C₂	#16813668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16813622

C₂

yes

maximal, characteristic, complemented

#16813608

C₂

yes

maximal, complemented

#16813596

C₂

yes

maximal, complemented

#16813593

C₂

yes

maximal, complemented

C₄×D₄₅₂

C₂

yes

maximal, characteristic, complemented

#16813581

C₂

yes

maximal, complemented

#16813580

C₂

yes

maximal, complemented

C₄×D₄×C₁₁₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₂₆

C₂²

yes

—

C₂×D₄₅₂

C₂²

yes

characteristic

C₂×C₂²⋊D₁₁₃

C₂²

yes

—

C₂×C₄×D₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

characteristic

C₄⋊Dic₁₁₃

C₂²

yes

—

C₂²⋊Dic₁₁₃

C₂²

yes

—

#14268624

C₂²

yes

—

#14268624

C₂²

yes

—

C₄×Dic₁₁₃

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

—

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

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

#41

yes

—

C₄×C₁₁₃

#41

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂×#41

yes

characteristic

C₂×C₁₁₃

C₂×#41

yes

characteristic

C₂×C₁₁₃

H(5,𝔽₂)

yes

characteristic

D₁₁₃

—

C₁₁₃

#555

yes

characteristic, complemented, Sylow

#555

—

113

maximal, Sylow

C₄⋊Q₈

—

113

—

#124

—

113

—

#121

—

113

—

#121

—

113

—

C₄×D₄

—

113

—

C₄×D₄

D₁₁₃

yes

characteristic, complemented

#117

—

113

—

#117

—

113

—

C₂×Q₈

—

113

—

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂×D₄

C₂×D₁₁₃

yes

characteristic

C₂²×C₄

—

113

—

C₂²×C₄

C₂×D₁₁₃

yes

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

C₂×D₁₁₃

yes

characteristic

C₂²⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

C₂×D₁₁₃

yes

—

C₄²

—

113

—

C₄²

C₂×D₁₁₃

yes

characteristic

C₂³

—

113

—

C₂³

C₂²×D₁₁₃

yes

—

Q₈

—

452

—

D₄

—

D₄

—

452

—

D₄

—

226

—

C₂×C₄

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

226

—

C₂×C₄

C₂²×D₁₁₃

yes

—

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

226

—

101

C₂²

—

226

—

102

C₂²

—

226

—

103

C₂²

C₂³×D₁₁₃

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

226

—

107

C₄

—

226

—

108

C₄

#14268646

yes

—

109

C₄

#14268647

yes

—

110

C₂

—

111

C₂

—

452

—

112

C₂

C₂×#14268646

yes

characteristic

113

C₂

C₂×#14268647

yes

characteristic

114

C₂

#16813668

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	#555
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. #16813622	C₂	semidirect
2. #16813608	C₂	semidirect
3. #16813596	C₂	semidirect
4. #16813593	C₂	semidirect
5. C₄×D₄₅₂	C₂	semidirect
6. #16813581	C₂	semidirect
7. #16813580	C₂	semidirect
8. C₄×D₄×C₁₁₃	C₂	semidirect
57. C₁₁₃	#555	semidirect
64. C₄×D₄	D₁₁₃	semidirect

H ≅ G/N

type

1. #16813622

C₂

semidirect

2. #16813608

C₂

semidirect

3. #16813596

C₂

semidirect

4. #16813593

C₂

semidirect

5. C₄×D₄₅₂

C₂

semidirect

6. #16813581

C₂

semidirect

7. #16813580

C₂

semidirect

8. C₄×D₄×C₁₁₃

C₂

semidirect

57. C₁₁₃

#555

semidirect

64. C₄×D₄

D₁₁₃

semidirect

Subgroups — #20299313

Conjugacy classes of subgroups

Complements