Subgroups — C₂×#20297214 #28383528

Order 14272.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#20297214	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	C₄	1	2	yes	—
6	C₂×C₂₂₃⋊C₈	C₄	1	4	yes	complemented
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
8	C₄×Dic₂₂₃	C₄	1	2	yes	—
9	C₄×Dic₂₂₃	C₂²	1	2	yes	—
10	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	C₄	1	4	yes	complemented
12	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
13	C₂×Dic₂₂₃	—	2	4	no	—
14	C₂×Dic₂₂₃	C₂×C₄	1	4	yes	—
15	C₂₂₃⋊C₈	C₂×C₄	1	4	yes	complemented
16	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₈×C₂₂₃	C₂×C₄	1	4	yes	complemented
22	C₂²×C₂₂₃	C₄²	1	2	yes	—
23	C₂²×C₂₂₃	M₁₆	1	4	yes	—
24	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₂₂₃	C₄²	1	2	yes	—
26	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
28	Dic₂₂₃	—	2	4	no	—
29	C₂×C₂₂₃	#97	1	4	yes	complemented
30	C₂×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₂₂₃	C₂×M₁₆	1	2	yes	—
32	C₂₂₃	C₂×#97	1	1	yes	characteristic, complemented, Sylow
33	C₂×#97	—	223	1	no	maximal, Sylow
34	C₂²×C₈	—	223	1	no	—
35	C₂²×C₈	D₂₂₃	1	1	yes	characteristic
36	C₂×C₄²	—	223	1	no	—
37	#97	—	223	4	no	—
38	C₂²×C₄	—	223	2	no	—
39	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
40	C₂×C₈	—	223	4	no	—
41	C₂×C₈	—	223	2	no	—
42	C₂×C₈	Dic₂₂₃	1	4	yes	complemented
43	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
44	C₄²	—	223	2	no	—
45	C₄²	—	223	2	no	—
46	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
47	C₂×C₄	—	446	4	no	—
48	C₂×C₄	—	223	4	no	—
49	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
50	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
51	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
52	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
53	C₈	—	223	4	no	—
54	C₈	C₂×Dic₂₂₃	1	4	yes	complemented
55	C₂²	C₄×Dic₂₂₃	1	2	yes	—
56	C₂²	#16813153	1	4	yes	—
57	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
58	C₄	—	446	4	no	—
59	C₄	C₄×Dic₂₂₃	1	2	yes	—
60	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
61	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
62	C₂	#20297214	1	4	yes	direct product factor
63	C₂	C₂×C₄×Dic₂₂₃	1	1	yes	characteristic
64	C₂	C₂×#16813153	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂²×C₈×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂²×C₈	D₂₂₃
hypocenter	C₂₂₃	C₂×#97
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #20297214	C₂	direct (N × H)
6. C₂×C₂₂₃⋊C₈	C₄	semidirect
11. C₂×C₈×C₂₂₃	C₄	semidirect
15. C₂₂₃⋊C₈	C₂×C₄	semidirect
21. C₈×C₂₂₃	C₂×C₄	semidirect
29. C₂×C₂₂₃	#97	semidirect
32. C₂₂₃	C₂×#97	semidirect
42. C₂×C₈	Dic₂₂₃	semidirect
54. C₈	C₂×Dic₂₂₃	semidirect
62. C₂	#20297214	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₂₃⋊C₈	2
C₂×C₄×Dic₂₂₃	2
#20297214	2
C₂²×C₈×C₂₂₃	2
C₂×#97	223

Subgroups — C2×#20297214 #28383528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20297214 #28383528