Subgroups — #28383965

Order 14272.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813157	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₇₈₄	C₂	1	1	yes	maximal, characteristic
3	#20297293	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₉₂	C₂²	1	2	yes	—
7	#16813157	—	2	4	no	—
8	D₁₇₈₄	—	2	2	no	—
9	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
10	(C₂×D₂₂₃)⋊C₄	—	2	4	no	—
11	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
13	Q₁₆×C₂₂₃	—	2	2	no	—
14	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂²×D₂₂₃	—	2	2	no	—
16	D₈₉₂	—	2	2	no	—
17	D₈₉₂	—	2	2	no	—
18	C₂×Dic₂₂₃	D₄	1	2	yes	—
19	C₂₂₃⋊C₈	D₄	1	2	yes	—
20	Q₈×C₂₂₃	—	2	2	no	—
21	Q₈×C₂₂₃	—	2	2	no	—
22	C₂×C₄×C₂₂₃	—	2	2	no	—
23	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₈×C₂₂₃	D₄	1	2	yes	—
25	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
26	C₂×D₂₂₃	—	4	2	no	—
27	C₂×D₂₂₃	—	4	4	no	—
28	C₄×C₂₂₃	—	4	2	no	—
29	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
31	Dic₂₂₃	—	2	2	no	—
32	C₂×C₂₂₃	#127	1	1	yes	characteristic
33	C₂×C₂₂₃	#135	1	2	yes	—
34	D₂₂₃	—	8	2	no	—
35	C₂₂₃	#495	1	1	yes	characteristic, complemented, Sylow
36	#495	—	223	1	no	maximal, Sylow
37	C₂×Q₁₆	D₂₂₃	1	1	yes	characteristic, complemented
38	C₂×SD₁₆	—	223	2	no	—
39	C₂×D₈	—	223	1	no	—
40	#124	—	223	2	no	—
41	C₄×C₈	—	223	1	no	—
42	C₂×Q₈	C₂×D₂₂₃	1	2	yes	—
43	C₂×D₄	—	223	2	no	—
44	Q₁₆	—	2	2	no	—
45	SD₁₆	—	446	4	no	—
46	D₈	—	446	2	no	—
47	C₂×C₈	—	223	1	no	—
48	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
49	C₂²⋊C₄	—	446	4	no	—
50	C₄²	—	223	1	no	—
51	C₂³	—	446	2	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	D₄	—	446	2	no	—
55	D₄	—	446	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	223	2	no	—
58	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
59	C₈	—	223	2	no	—
60	C₈	C₂²⋊D₂₂₃	1	2	yes	—
61	C₂²	—	892	2	no	—
62	C₂²	—	892	4	no	—
63	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	446	2	no	—
66	C₄	D₄×D₂₂₃	1	1	yes	characteristic
67	C₄	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
68	C₂	—	1784	2	no	—
69	C₂	#20297300	1	1	yes	characteristic
70	C₂	#20297327	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂×Q₁₆×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂×D₁₇₈₄	C₂
hypercenter	C₂×Q₁₆	D₂₂₃
hypocenter	C₂₂₃	#495
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16813157	C₂	semidirect
3. #20297293	C₂	semidirect
4. C₈×Dic₂₂₃	C₂	semidirect
5. C₂×Q₁₆×C₂₂₃	C₂	semidirect
35. C₂₂₃	#495	semidirect
37. C₂×Q₁₆	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813157	2
C₂×D₁₇₈₄	2
#20297293	2
C₈×Dic₂₂₃	2
C₂×Q₁₆×C₂₂₃	2
#495	223