Subgroups — #28384206

Order 14272.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297283	C₂	1	4	yes	maximal, complemented
2	#20297281	C₂	1	8	yes	maximal, complemented
3	C₂×C₂²⋊Dic₂₂₃	C₂	1	2	yes	maximal
4	C₂×C₂²⋊C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₄₆	C₂²	1	4	yes	complemented
6	C₂²×Dic₂₂₃	C₂²	1	4	yes	—
7	C₄⋊Dic₂₂₃	C₂²	1	4	yes	complemented
8	Dic₂₂₃⋊C₄	C₂²	1	8	yes	complemented
9	C₂²⋊Dic₂₂₃	C₂²	1	4	yes	—
10	C₂²⋊Dic₂₂₃	C₂²	1	4	yes	—
11	C₂⁴×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
13	C₂²⋊C₄×C₂₂₃	C₂²	1	4	yes	—
14	Dic₄₄₆	—	4	4	no	—
15	C₂×Dic₂₂₃	—	4	4	no	—
16	C₂×Dic₂₂₃	C₂³	1	8	yes	—
17	C₂³×C₂₂₃	—	4	2	no	—
18	C₂³×C₂₂₃	Q₈	1	4	yes	complemented
19	C₂³×C₂₂₃	C₂³	1	2	yes	—
20	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	—	4	2	no	—
22	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
23	C₂²×C₂₂₃	—	2	2	no	—
24	C₂²×C₂₂₃	—	2	4	no	—
25	C₂²×C₂₂₃	—	4	4	no	—
26	C₂²×C₂₂₃	C₂×Q₈	1	4	yes	—
27	C₂²×C₂₂₃	C₂×Q₈	1	2	yes	—
28	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
29	C₄×C₂₂₃	—	2	4	no	—
30	Dic₂₂₃	—	2	8	no	—
31	C₂×C₂₂₃	—	2	2	no	—
32	C₂×C₂₂₃	—	2	4	no	—
33	C₂×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
34	C₂×C₂₂₃	H(5,𝔽₂)	1	2	yes	—
35	C₂₂₃	#543	1	1	yes	characteristic, complemented, Sylow
36	#543	—	223	1	no	maximal, Sylow
37	C₂²⋊Q₈	—	223	4	no	—
38	C₂²⋊Q₈	—	223	8	no	—
39	C₂×C₂²⋊C₄	—	223	2	no	—
40	C₂×C₂²⋊C₄	D₂₂₃	1	1	yes	characteristic
41	C₂⁴	C₂×D₂₂₃	1	1	yes	characteristic
42	C₂×Q₈	—	223	4	no	—
43	C₂²×C₄	—	223	4	no	—
44	C₂²×C₄	C₂×D₂₂₃	1	2	yes	—
45	C₄⋊C₄	—	223	8	no	—
46	C₄⋊C₄	—	223	4	no	—
47	C₂²⋊C₄	—	223	4	no	—
48	C₂²⋊C₄	—	223	4	no	—
49	C₂²⋊C₄	C₂×D₂₂₃	1	4	yes	—
50	C₂³	—	4	2	no	—
51	C₂³	Dic₄₄₆	1	4	yes	complemented
52	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
53	C₂³	C₂²×D₂₂₃	1	2	yes	—
54	Q₈	—	892	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	892	4	no	—
57	C₂×C₄	—	223	8	no	—
58	C₂×C₄	C₂²×D₂₂₃	1	4	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	—	2	4	no	—
61	C₂²	—	4	4	no	—
62	C₂²	C₂×Dic₄₄₆	1	2	yes	—
63	C₂²	C₂×Dic₄₄₆	1	4	yes	—
64	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	446	8	no	—
67	C₂	—	2	2	no	—
68	C₂	—	2	4	no	—
69	C₂	C₂²×Dic₄₄₆	1	1	yes	characteristic
70	C₂	#20297349	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	C₂×C₂²⋊C₄×C₂₂₃	C₂
prime radical	C₂⁴×C₂₂₃	C₂²
hypercenter	C₂×C₂²⋊C₄	D₂₂₃
hypocenter	C₂₂₃	#543
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #20297283	C₂	semidirect
2. #20297281	C₂	semidirect
5. C₂×Dic₄₄₆	C₂²	semidirect
7. C₄⋊Dic₂₂₃	C₂²	semidirect
8. Dic₂₂₃⋊C₄	C₂²	semidirect
18. C₂³×C₂₂₃	Q₈	semidirect
35. C₂₂₃	#543	semidirect
51. C₂³	Dic₄₄₆	semidirect

Maximal subgroups

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

subgroup	index
#20297283	2
#20297281	2
C₂×C₂²⋊Dic₂₂₃	2
C₂×C₂²⋊C₄×C₂₂₃	2
#543	223