Subgroups — #28383972

Order 14272.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813160	C₂	1	2	yes	maximal, complemented
2	C₂×#16813157	C₂	1	2	yes	maximal
3	#20297292	C₂	1	1	yes	maximal, characteristic
4	#20297215	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₄₆	C₂²	1	1	yes	characteristic
7	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
8	#16813160	—	2	4	no	—
9	#16813157	—	2	4	no	—
10	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
11	(C₂×D₂₂₃)⋊C₄	—	2	2	no	—
12	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
13	C₄⋊C₄×C₂₂₃	—	2	2	no	—
14	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂²×D₂₂₃	—	2	1	no	—
16	Dic₄₄₆	—	2	2	no	—
17	D₈₉₂	—	2	2	no	—
18	C₂×Dic₂₂₃	—	2	1	no	—
19	C₂₂₃⋊C₈	D₄	1	4	yes	—
20	Q₈×C₂₂₃	—	2	2	no	—
21	Q₈×C₂₂₃	—	2	2	no	—
22	C₂×C₄×C₂₂₃	—	2	2	no	—
23	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
24	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
26	C₂×D₂₂₃	—	4	1	no	—
27	C₂×D₂₂₃	—	4	2	no	—
28	C₄×C₂₂₃	—	2	2	no	—
29	C₄×C₂₂₃	—	4	2	no	—
30	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
31	Dic₂₂₃	—	4	1	no	—
32	C₂×C₂₂₃	#127	1	1	yes	characteristic
33	C₂×C₂₂₃	#137	1	2	yes	—
34	D₂₂₃	—	8	1	no	—
35	C₂₂₃	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	223	1	no	maximal, Sylow
37	C₂×Q₁₆	—	223	2	no	—
38	C₂×SD₁₆	—	223	2	no	—
39	C₄⋊Q₈	D₂₂₃	1	1	yes	characteristic, complemented
40	#124	—	223	1	no	—
41	#97	—	223	1	no	—
42	C₂×Q₈	—	223	1	no	—
43	C₂×Q₈	C₂×D₂₂₃	1	2	yes	—
44	C₂×D₄	—	223	1	no	—
45	Q₁₆	—	446	4	no	—
46	SD₁₆	—	446	4	no	—
47	C₂×C₈	—	223	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₂²⋊C₄	—	446	2	no	—
50	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂³	—	446	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	Q₈	—	446	2	no	—
55	D₄	—	446	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	446	1	no	—
58	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
59	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
60	C₈	—	223	4	no	—
61	C₂²	—	892	1	no	—
62	C₂²	—	892	2	no	—
63	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	2	no	—
66	C₄	—	892	1	no	—
67	C₄	D₄×D₂₂₃	1	2	yes	—
68	C₂	—	1784	1	no	—
69	C₂	#20297300	1	1	yes	characteristic
70	C₂	#20297335	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₄⋊Q₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂×D₈₉₂	C₂²
hypercenter	C₄⋊Q₈	D₂₂₃
hypocenter	C₂₂₃	#497
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16813160	C₂	semidirect
4. #20297215	C₂	semidirect
5. C₄⋊Q₈×C₂₂₃	C₂	semidirect
35. C₂₂₃	#497	semidirect
39. C₄⋊Q₈	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813160	2
C₂×#16813157	2
#20297292	2
#20297215	2
C₄⋊Q₈×C₂₂₃	2
#497	223