Subgroups — #28384328

Order 14272.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297292	C₂	1	2	yes	maximal, complemented
2	#20297289	C₂	1	4	yes	maximal, complemented
3	#20297282	C₂	1	4	yes	maximal, complemented
4	#20297279	C₂	1	4	yes	maximal, complemented
5	#125×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₄₆	C₂²	1	2	yes	—
7	C₂×D₈₉₂	C₂²	1	4	yes	—
8	C₂×D₈₉₂	C₂²	1	2	yes	—
9	C₂×C₄×D₂₂₃	C₂²	1	4	yes	—
10	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
11	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
12	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
13	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
15	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
16	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂²×D₂₂₃	C₂³	1	4	yes	—
18	Dic₄₄₆	—	4	2	no	—
19	D₈₉₂	—	4	4	no	—
20	D₈₉₂	—	4	2	no	—
21	C₂×Dic₂₂₃	C₂³	1	4	yes	—
22	C₄×D₂₂₃	—	4	4	no	—
23	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
25	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
26	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
27	C₂×D₂₂₃	—	2	4	no	—
28	C₂×D₂₂₃	—	2	4	no	—
29	C₂×D₂₂₃	—	2	4	no	—
30	C₄×C₂₂₃	—	2	4	no	—
31	C₄×C₂₂₃	—	2	2	no	—
32	C₄×C₂₂₃	—	2	1	no	—
33	Dic₂₂₃	—	2	4	no	—
34	C₂×C₂₂₃	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
36	D₂₂₃	—	4	4	no	—
37	C₂₂₃	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	223	1	no	maximal, Sylow
39	#125	D₂₂₃	1	1	yes	characteristic, complemented
40	#124	—	223	2	no	—
41	#123	—	223	4	no	—
42	C₂²⋊Q₈	—	223	4	no	—
43	#121	—	223	4	no	—
44	C₂×Q₈	—	223	2	no	—
45	C₂×D₄	—	223	4	no	—
46	C₂×D₄	—	223	2	no	—
47	C₂²×C₄	—	223	4	no	—
48	C₄⋊C₄	—	223	4	no	—
49	C₄⋊C₄	C₂×D₂₂₃	1	4	yes	—
50	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
51	C₂²⋊C₄	—	223	4	no	—
52	C₂²⋊C₄	—	223	4	no	—
53	C₂²⋊C₄	—	223	4	no	—
54	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
55	C₂³	—	223	4	no	—
56	Q₈	—	892	2	no	—
57	D₄	—	892	4	no	—
58	D₄	—	892	2	no	—
59	C₂×C₄	—	892	4	no	—
60	C₂×C₄	—	223	4	no	—
61	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
63	C₂×C₄	C₂²×D₂₂₃	1	4	yes	—
64	C₂²	—	446	4	no	—
65	C₂²	—	446	4	no	—
66	C₂²	—	446	4	no	—
67	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	446	4	no	—
72	C₂	—	892	4	no	—
73	C₂	#20297350	1	2	yes	—
74	C₂	#20297352	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	#125×C₂₂₃	C₂
hypercenter	#125	D₂₂₃
hypocenter	C₂₂₃	#562
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #20297292	C₂	semidirect
2. #20297289	C₂	semidirect
3. #20297282	C₂	semidirect
4. #20297279	C₂	semidirect
5. #125×C₂₂₃	C₂	semidirect
37. C₂₂₃	#562	semidirect
39. #125	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297292	2
#20297289	2
#20297282	2
#20297279	2
#125×C₂₂₃	2
#562	223