Subgroups — C₂×#20297352 #28384414

Order 14272.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297352	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16813171	C₂	1	4	yes	maximal, complemented
3	C₂×#16813170	C₂	1	6	yes	maximal, complemented
4	C₂×Q₈×D₂₂₃	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813171	C₂²	1	32	yes	complemented
7	#16813170	C₂²	1	48	yes	complemented
8	C₂×Dic₄₄₆	C₂²	1	6	yes	—
9	Q₈×D₂₂₃	C₂²	1	32	yes	complemented
10	C₂×D₈₉₂	C₂²	1	6	yes	—
11	C₂×C₂²⋊D₂₂₃	C₂²	1	4	yes	—
12	C₂×C₄×D₂₂₃	C₂²	1	12	yes	—
13	C₂×Q₈×C₂₂₃	C₂²	1	4	yes	—
14	C₂×Q₈×C₂₂₃	C₂²	1	8	yes	complemented
15	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
16	C₂²×D₂₂₃	C₂³	1	4	yes	—
17	Dic₄₄₆	C₂³	1	24	yes	—
18	D₈₉₂	C₂³	1	24	yes	—
19	C₂²⋊D₂₂₃	C₂³	1	16	yes	—
20	C₂×Dic₂₂₃	C₂³	1	4	yes	—
21	C₄×D₂₂₃	C₂³	1	48	yes	—
22	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
23	Q₈×C₂₂₃	C₂³	1	16	yes	—
24	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
25	C₂×C₄×C₂₂₃	C₂³	1	12	yes	—
26	C₂²×C₂₂₃	—	2	2	no	—
27	C₂²×C₂₂₃	C₂⁴	1	2	yes	—
28	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
29	C₂×D₂₂₃	—	2	8	no	—
30	C₂×D₂₂₃	C₂⁴	1	8	yes	—
31	C₄×C₂₂₃	C₂⁴	1	12	yes	—
32	Dic₂₂₃	C₂⁴	1	8	yes	—
33	C₂×C₂₂₃	—	2	2	no	—
34	C₂×C₂₂₃	2₋¹⁺⁴	1	2	yes	complemented
35	C₂×C₂₂₃	C₂⁵	1	1	yes	characteristic
36	D₂₂₃	—	2	8	no	—
37	C₂₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
38	C₂×2₋¹⁺⁴	—	223	1	no	maximal, Sylow
39	2₋¹⁺⁴	—	223	16	no	—
40	C₂×#41	—	223	4	no	—
41	C₂×#41	—	223	6	no	—
42	C₂²×Q₈	—	223	4	no	—
43	C₂²×Q₈	D₂₂₃	1	1	yes	characteristic, complemented
44	#41	—	223	32	no	—
45	#41	—	223	48	no	—
46	C₂×Q₈	—	223	32	no	—
47	C₂×Q₈	—	223	6	no	—
48	C₂×Q₈	C₂×D₂₂₃	1	4	yes	—
49	C₂×Q₈	C₂×D₂₂₃	1	8	yes	complemented
50	C₂×D₄	—	223	4	no	—
51	C₂×D₄	—	223	6	no	—
52	C₂²×C₄	—	223	12	no	—
53	C₂²×C₄	C₂×D₂₂₃	1	3	yes	—
54	C₂³	—	223	4	no	—
55	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
56	Q₈	—	223	24	no	—
57	Q₈	C₂²×D₂₂₃	1	16	yes	—
58	D₄	—	223	16	no	—
59	D₄	—	223	24	no	—
60	C₂×C₄	—	223	4	no	—
61	C₂×C₄	—	223	48	no	—
62	C₂×C₄	C₂²×D₂₂₃	1	12	yes	—
63	C₂×C₄	C₂²×D₂₂₃	1	6	yes	—
64	C₂²	—	2	2	no	—
65	C₂²	—	446	8	no	—
66	C₂²	—	223	8	no	—
67	C₂²	C₂³×D₂₂₃	1	2	yes	—
68	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
69	C₄	—	223	8	no	—
70	C₄	C₂³×D₂₂₃	1	12	yes	—
71	C₂	—	2	2	no	—
72	C₂	—	446	8	no	—
73	C₂	#20297352	1	2	yes	direct product factor
74	C₂	C₂⁴×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₂₃
commutator, cosocle	C₂×C₂₂₃	C₂⁵
Frattini	C₂	C₂⁴×D₂₂₃
Fitting	C₂²×Q₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂⁴
hypercenter	C₂²×Q₈	D₂₂₃
hypocenter	C₂₂₃	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. #20297352	C₂	direct (N × H)
2. C₂×#16813171	C₂	semidirect
3. C₂×#16813170	C₂	semidirect
4. C₂×Q₈×D₂₂₃	C₂	semidirect
5. C₂²×Q₈×C₂₂₃	C₂	semidirect
6. #16813171	C₂²	semidirect
7. #16813170	C₂²	semidirect
9. Q₈×D₂₂₃	C₂²	semidirect
14. C₂×Q₈×C₂₂₃	C₂²	semidirect
34. C₂×C₂₂₃	2₋¹⁺⁴	semidirect
37. C₂₂₃	C₂×2₋¹⁺⁴	semidirect
43. C₂²×Q₈	D₂₂₃	semidirect
49. C₂×Q₈	C₂×D₂₂₃	semidirect
73. C₂	#20297352	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20297352	2
C₂×#16813171	2
C₂×#16813170	2
C₂×Q₈×D₂₂₃	2
C₂²×Q₈×C₂₂₃	2
C₂×2₋¹⁺⁴	223

Subgroups — C2×#20297352 #28384414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20297352 #28384414