Subgroups — #28383753

Order 14272.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297329	C₂	1	2	yes	maximal, complemented
2	#20297293	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297231	C₂	1	2	yes	maximal, complemented
4	#20297223	C₂	1	1	yes	maximal, characteristic, complemented
5	2₋¹⁺⁴×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
7	#16813157	—	2	2	no	—
8	#16813155	—	2	2	no	—
9	#16813154	C₂²	1	2	yes	—
10	(C₂×D₂₂₃)⋊C₄	—	2	2	no	—
11	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
12	#41×C₂₂₃	—	2	2	no	—
13	#41×C₂₂₃	—	2	2	no	—
14	#41×C₂₂₃	C₂²	1	2	yes	—
15	C₂×Q₈×C₂₂₃	—	2	2	no	—
16	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂²×D₂₂₃	—	2	1	no	—
18	D₈₉₂	—	2	2	no	—
19	C₂×Dic₂₂₃	—	2	1	no	—
20	C₂₂₃⋊C₈	—	2	2	no	—
21	Q₈×C₂₂₃	—	2	2	no	—
22	Q₈×C₂₂₃	—	2	2	no	—
23	Q₈×C₂₂₃	D₄	1	2	yes	—
24	D₄×C₂₂₃	—	2	2	no	—
25	D₄×C₂₂₃	—	2	2	no	—
26	D₄×C₂₂₃	D₄	1	2	yes	—
27	C₂×C₄×C₂₂₃	—	2	4	no	—
28	C₂×C₄×C₂₂₃	—	2	2	no	—
29	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
30	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
31	C₂²×C₂₂₃	—	2	2	no	—
32	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×D₂₂₃	—	4	2	no	—
34	C₂×D₂₂₃	—	4	1	no	—
35	C₄×C₂₂₃	—	2	2	no	—
36	C₄×C₂₂₃	—	2	2	no	—
37	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
38	Dic₂₂₃	—	4	1	no	—
39	C₂×C₂₂₃	—	2	1	no	—
40	C₂×C₂₂₃	—	4	2	no	—
41	C₂×C₂₂₃	#120	1	1	yes	characteristic
42	D₂₂₃	—	8	1	no	—
43	C₂₂₃	#454	1	1	yes	characteristic, complemented, Sylow
44	#454	—	223	1	no	maximal, Sylow
45	2₋¹⁺⁴	D₂₂₃	1	1	yes	characteristic, complemented
46	AGL(1,ℤ/8)	—	223	2	no	—
47	#124	—	223	1	no	—
48	#104	—	223	2	no	—
49	#101	—	223	1	no	—
50	#41	—	2	2	no	—
51	#41	—	2	2	no	—
52	#41	C₂×D₂₂₃	1	2	yes	—
53	C₂×Q₈	—	2	2	no	—
54	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
55	C₂×D₄	—	223	1	no	—
56	SD₁₆	—	446	2	no	—
57	D₈	—	446	2	no	—
58	M₁₆	—	223	2	no	—
59	C₂²⋊C₄	—	446	2	no	—
60	C₄²	—	223	1	no	—
61	C₂³	—	446	1	no	—
62	Q₈	—	2	2	no	—
63	Q₈	—	2	2	no	—
64	Q₈	C₂²⋊D₂₂₃	1	2	yes	—
65	D₄	—	2	2	no	—
66	D₄	—	2	2	no	—
67	D₄	—	446	2	no	—
68	D₄	C₂²⋊D₂₂₃	1	2	yes	—
69	C₂×C₄	—	2	4	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	446	1	no	—
72	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
73	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
74	C₈	—	446	2	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	892	2	no	—
77	C₂²	—	892	1	no	—
78	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	2	no	—
81	C₄	—	892	1	no	—
82	C₄	C₂×C₂²⋊D₂₂₃	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	4	2	no	—
85	C₂	—	1784	1	no	—
86	C₂	#20297273	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20297273
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	2₋¹⁺⁴×C₂₂₃	C₂
socle	C₂×C₂₂₃	#120
hypercenter	2₋¹⁺⁴	D₂₂₃
hypocenter	C₂₂₃	#454
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #20297329	C₂	semidirect
2. #20297293	C₂	semidirect
3. #20297231	C₂	semidirect
4. #20297223	C₂	semidirect
5. 2₋¹⁺⁴×C₂₂₃	C₂	semidirect
43. C₂₂₃	#454	semidirect
45. 2₋¹⁺⁴	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297329	2
#20297293	2
#20297231	2
#20297223	2
2₋¹⁺⁴×C₂₂₃	2
#454	223