Subgroups — #21418521

Order 7456.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₆₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₂₃₃	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Dic₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#16836571	C₂	1	1	yes	maximal, characteristic, complemented
5	#16836571	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×Dic₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₃₃	C₂²	1	1	yes	characteristic
9	Dic₄₆₆	—	2	2	no	—
10	C₂²⋊D₂₃₃	—	2	2	no	—
11	C₂×Dic₂₃₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₃₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₃₃	C₂²	1	1	yes	characteristic
14	C₂³×C₂₃₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
17	C₂²×C₂₃₃	—	2	1	no	—
18	C₂²×C₂₃₃	—	2	1	no	—
19	C₂²×C₂₃₃	—	2	1	no	—
20	C₂²×C₂₃₃	C₂³	1	1	yes	characteristic
21	C₂×D₂₃₃	—	2	1	no	—
22	C₂×D₂₃₃	—	2	1	no	—
23	C₂×D₂₃₃	—	2	1	no	—
24	C₄×C₂₃₃	—	2	1	no	—
25	C₄×C₂₃₃	—	2	1	no	—
26	Dic₂₃₃	—	2	1	no	—
27	Dic₂₃₃	—	2	1	no	—
28	Dic₂₃₃	D₄	1	2	yes	—
29	C₂×C₂₃₃	—	4	1	no	—
30	C₂×C₂₃₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₂₃₃	#41	1	1	yes	characteristic
32	C₂×C₂₃₃	#41	1	1	yes	characteristic
33	D₂₃₃	—	4	1	no	—
34	C₂₃₃	#124	1	1	yes	characteristic, complemented, Sylow
35	#124	—	233	1	no	maximal, Sylow
36	C₂×Q₈	—	233	1	no	—
37	C₂×D₄	—	233	1	no	—
38	C₂²⋊C₄	—	233	1	no	—
39	C₂²⋊C₄	—	233	1	no	—
40	C₂²⋊C₄	—	233	1	no	—
41	C₂²⋊C₄	D₂₃₃	1	1	yes	characteristic, complemented
42	C₄²	—	233	1	no	—
43	C₂³	—	233	1	no	—
44	C₂³	C₂×D₂₃₃	1	1	yes	characteristic
45	Q₈	—	466	2	no	—
46	D₄	—	466	2	no	—
47	C₂×C₄	—	233	1	no	—
48	C₂×C₄	—	233	1	no	—
49	C₂×C₄	—	233	1	no	—
50	C₂×C₄	C₂×D₂₃₃	1	1	yes	characteristic
51	C₂×C₄	C₂×D₂₃₃	1	1	yes	characteristic
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	466	1	no	—
56	C₂²	—	466	1	no	—
57	C₂²	—	466	1	no	—
58	C₂²	C₂²×D₂₃₃	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	466	1	no	—
62	C₄	—	466	1	no	—
63	C₄	—	233	2	no	—
64	C₂	—	4	1	no	—
65	C₂	—	932	1	no	—
66	C₂	D₄×D₂₃₃	1	1	yes	characteristic
67	C₂	#16836593	1	1	yes	characteristic
68	C₂	#16836594	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₂₃₃
commutator, socle	C₂²×C₂₃₃	C₂³
Fitting	C₂²⋊C₄×C₂₃₃	C₂
hypercenter	C₂²⋊C₄	D₂₃₃
hypocenter	C₂₃₃	#124
LCS	G ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×Dic₄₆₆	C₂	semidirect
3. C₂²⋊Dic₂₃₃	C₂	semidirect
4. #16836571	C₂	semidirect
5. #16836571	C₂	semidirect
6. C₄×Dic₂₃₃	C₂	semidirect
7. C₂²⋊C₄×C₂₃₃	C₂	semidirect
34. C₂₃₃	#124	semidirect
41. C₂²⋊C₄	D₂₃₃	semidirect