Group #20299174 — subgroups

Subgroups — #20299174

Order 7232.

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

normal subgroups only

This group has 270 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
2	C₂²×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813622	C₂	1	2	yes	maximal, complemented
4	#16813610	C₂	1	2	yes	maximal, complemented
5	#16813606	C₂	1	4	yes	maximal, complemented
6	#16813602	C₂	1	4	yes	maximal, complemented
7	#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14268647	—	2	4	no	—
9	C₂×Dic₂₂₆	—	2	4	no	—
10	C₂×Dic₂₂₆	C₂²	1	2	yes	—
11	C₂×Dic₂₂₆	C₂²	1	2	yes	—
12	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
13	C₂×D₄₅₂	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
15	C₂²×Dic₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄×D₁₁₃	C₂²	1	2	yes	—
17	C₄⋊Dic₁₁₃	C₂²	1	8	yes	—
18	#14268624	C₂²	1	8	yes	—
19	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
22	C₄²×C₁₁₃	C₂²	1	2	yes	—
23	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
24	C₂²×D₁₁₃	C₂³	1	2	yes	—
25	C₂×C₄×C₁₁₃	D₄	1	4	yes	—
26	C₂×C₄×C₁₁₃	C₂³	1	4	yes	—
27	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
29	C₂²⋊D₁₁₃	—	2	4	no	—
30	C₂×Dic₁₁₃	—	2	4	no	—
31	C₂×Dic₁₁₃	C₂³	1	4	yes	—
32	C₂×Dic₁₁₃	C₂³	1	2	yes	—
33	D₄₅₂	—	2	2	no	—
34	C₄×D₁₁₃	—	2	4	no	—
35	Dic₂₂₆	—	2	4	no	—
36	Dic₂₂₆	—	2	4	no	—
37	Dic₂₂₆	—	2	2	no	—
38	C₂²×C₁₁₃	—	2	2	no	—
39	C₂²×C₁₁₃	C₂×D₄	1	2	yes	—
40	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
41	C₂×D₁₁₃	—	2	2	no	—
42	C₂×D₁₁₃	—	2	4	no	—
43	C₄×C₁₁₃	—	2	4	no	—
44	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
45	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
46	Dic₁₁₃	—	2	4	no	—
47	Dic₁₁₃	—	2	2	no	—
48	C₂×C₁₁₃	—	2	2	no	—
49	C₂×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
51	D₁₁₃	—	4	2	no	—
52	C₁₁₃	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	113	1	no	maximal, Sylow
54	C₂×#41	—	113	1	no	—
55	C₂²×Q₈	—	113	1	no	—
56	C₄⋊Q₈	—	113	2	no	—
57	#124	—	113	2	no	—
58	#123	—	113	4	no	—
59	C₂²⋊Q₈	—	113	4	no	—
60	#117	D₁₁₃	1	1	yes	characteristic, complemented
61	#41	—	226	4	no	—
62	C₂×Q₈	—	113	2	no	—
63	C₂×Q₈	—	113	2	no	—
64	C₂×Q₈	—	113	1	no	—
65	C₂×Q₈	—	226	4	no	—
66	C₂×D₄	—	113	2	no	—
67	C₂×D₄	—	113	1	no	—
68	C₂²×C₄	—	113	2	no	—
69	C₂²×C₄	—	113	2	no	—
70	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
71	C₄⋊C₄	—	113	8	no	—
72	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
73	C₂²⋊C₄	—	113	8	no	—
74	C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
75	C₄²	C₂×D₁₁₃	1	2	yes	—
76	C₂³	—	113	2	no	—
77	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
78	Q₈	—	226	4	no	—
79	Q₈	—	226	4	no	—
80	Q₈	—	226	2	no	—
81	D₄	—	226	4	no	—
82	D₄	—	226	2	no	—
83	C₂×C₄	—	113	4	no	—
84	C₂×C₄	—	113	2	no	—
85	C₂×C₄	—	226	4	no	—
86	C₂×C₄	—	226	4	no	—
87	C₂×C₄	D₄₅₂	1	4	yes	—
88	C₂×C₄	C₂²×D₁₁₃	1	4	yes	—
89	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	226	2	no	—
93	C₂²	—	226	4	no	—
94	C₂²	C₂×D₄₅₂	1	2	yes	—
95	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	226	2	no	—
98	C₄	—	226	4	no	—
99	C₄	C₂×D₄₅₂	1	2	yes	—
100	C₄	C₂×D₄₅₂	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	452	2	no	—
103	C₂	C₂²×D₄₅₂	1	1	yes	characteristic
104	C₂	#16813669	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14268647

C₂

yes

maximal, characteristic

C₂²×Dic₂₂₆

C₂

yes

maximal, characteristic, complemented

#16813622

C₂

yes

maximal, complemented

#16813610

C₂

yes

maximal, complemented

#16813606

C₂

yes

maximal, complemented

#16813602

C₂

yes

maximal, complemented

#117×C₁₁₃

C₂

yes

maximal, characteristic, complemented

#14268647

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

C₂²

yes

—

C₂×Dic₂₂₆

C₂²

yes

—

C₂×Dic₂₂₆

C₂²

yes

characteristic

C₂×D₄₅₂

C₂²

yes

characteristic

C₂×C₂²⋊D₁₁₃

C₂²

yes

—

C₂²×Dic₁₁₃

C₂²

yes

—

C₂×C₄×D₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

—

#14268624

C₂²

yes

—

C₂²×C₄×C₁₁₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₁₃

C₂²

yes

—

C₂²⋊C₄×C₁₁₃

C₂²

yes

—

C₄²×C₁₁₃

C₂²

yes

—

C₂³×C₁₁₃

C₂³

yes

characteristic

C₂²×D₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

D₄

yes

—

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

—

Dic₂₂₆

—

Dic₂₂₆

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×D₄

yes

—

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

C₂²×D₄

yes

characteristic

C₂×C₁₁₃

2₋¹⁺⁴

yes

—

D₁₁₃

—

C₁₁₃

#536

yes

characteristic, complemented, Sylow

#536

—

113

maximal, Sylow

C₂×#41

—

113

—

C₂²×Q₈

—

113

—

C₄⋊Q₈

—

113

—

#124

—

113

—

#123

—

113

—

C₂²⋊Q₈

—

113

—

#117

D₁₁₃

yes

characteristic, complemented

#41

—

226

—

C₂×Q₈

—

113

—

C₂×Q₈

—

113

—

C₂×Q₈

—

113

—

C₂×Q₈

—

226

—

C₂×D₄

—

113

—

C₂×D₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂×D₁₁₃

yes

characteristic

C₄⋊C₄

—

113

—

C₄⋊C₄

C₂×D₁₁₃

yes

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

C₂×D₁₁₃

yes

—

C₄²

C₂×D₁₁₃

yes

—

C₂³

—

113

—

C₂³

C₂²×D₁₁₃

yes

characteristic

Q₈

—

226

—

Q₈

—

226

—

Q₈

—

226

—

D₄

—

226

—

D₄

—

226

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

D₄₅₂

yes

—

C₂×C₄

C₂²×D₁₁₃

yes

—

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂²

—

C₂²

—

226

—

C₂²

—

226

—

C₂²

C₂×D₄₅₂

yes

—

C₂²

C₂³×D₁₁₃

yes

characteristic

C₄

—

C₄

—

226

—

C₄

—

226

—

C₄

C₂×D₄₅₂

yes

—

100

C₄

C₂×D₄₅₂

yes

—

101

C₂

—

102

C₂

—

452

—

103

C₂

C₂²×D₄₅₂

yes

characteristic

104

C₂

#16813669

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	#117×C₁₁₃	C₂
hypercenter	#117	D₁₁₃
hypocenter	C₁₁₃	#536
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
2. C₂²×Dic₂₂₆	C₂	semidirect
3. #16813622	C₂	semidirect
4. #16813610	C₂	semidirect
5. #16813606	C₂	semidirect
6. #16813602	C₂	semidirect
7. #117×C₁₁₃	C₂	semidirect
52. C₁₁₃	#536	semidirect
60. #117	D₁₁₃	semidirect

H ≅ G/N

type

2. C₂²×Dic₂₂₆

C₂

semidirect

3. #16813622

C₂

semidirect

4. #16813610

C₂

semidirect

5. #16813606

C₂

semidirect

6. #16813602

C₂

semidirect

7. #117×C₁₁₃

C₂

semidirect

52. C₁₁₃

#536

semidirect

60. #117

D₁₁₃

semidirect

Subgroups — #20299174

Conjugacy classes of subgroups

Complements