Subgroups — #20297289

Order 7136.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₂₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₂₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₂₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₂₃	C₂²	1	1	yes	characteristic
9	C₂²×D₂₂₃	C₂²	1	1	yes	characteristic
10	D₈₉₂	—	2	2	no	—
11	C₂×Dic₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₂₃	C₂²	1	1	yes	characteristic
13	C₄×D₂₂₃	—	2	2	no	—
14	C₂×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₂₂₃	C₂³	1	1	yes	characteristic
18	C₂×D₂₂₃	—	2	1	no	—
19	C₂×D₂₂₃	—	2	1	no	—
20	C₂×D₂₂₃	—	2	1	no	—
21	C₂×D₂₂₃	—	2	1	no	—
22	C₂×D₂₂₃	—	2	1	no	—
23	C₂×D₂₂₃	D₄	1	2	yes	—
24	C₄×C₂₂₃	—	2	1	no	—
25	C₄×C₂₂₃	—	2	1	no	—
26	C₄×C₂₂₃	—	2	1	no	—
27	Dic₂₂₃	—	2	1	no	—
28	Dic₂₂₃	—	2	1	no	—
29	C₂×C₂₂₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₂₂₃	#41	1	1	yes	characteristic
31	C₂×C₂₂₃	#41	1	1	yes	characteristic
32	D₂₂₃	—	2	2	no	—
33	D₂₂₃	—	4	1	no	—
34	C₂₂₃	#123	1	1	yes	characteristic, complemented, Sylow
35	#123	—	223	1	no	maximal, Sylow
36	C₂×D₄	—	223	1	no	—
37	C₂²×C₄	—	223	1	no	—
38	C₄⋊C₄	—	223	1	no	—
39	C₄⋊C₄	D₂₂₃	1	1	yes	characteristic, complemented
40	C₂²⋊C₄	—	223	1	no	—
41	C₂²⋊C₄	—	223	1	no	—
42	C₂²⋊C₄	—	223	1	no	—
43	C₂³	—	223	1	no	—
44	C₂³	—	223	1	no	—
45	D₄	—	446	2	no	—
46	C₂×C₄	—	446	2	no	—
47	C₂×C₄	—	223	1	no	—
48	C₂×C₄	—	223	1	no	—
49	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic
50	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic
52	C₂²	—	446	1	no	—
53	C₂²	—	446	1	no	—
54	C₂²	—	446	1	no	—
55	C₂²	—	446	1	no	—
56	C₂²	—	446	1	no	—
57	C₂²	—	223	2	no	—
58	C₂²	C₂²×D₂₂₃	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	2	1	no	—
62	C₄	—	446	1	no	—
63	C₄	—	446	1	no	—
64	C₂	—	892	1	no	—
65	C₂	—	446	2	no	—
66	C₂	D₄×D₂₂₃	1	1	yes	characteristic
67	C₂	#16813170	1	1	yes	characteristic
68	C₂	#16813171	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂³
Fitting	C₄⋊C₄×C₂₂₃	C₂
hypercenter	C₄⋊C₄	D₂₂₃
hypocenter	C₂₂₃	#123
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₂₂₃	C₂	semidirect
3. Dic₂₂₃⋊C₄	C₂	semidirect
4. (C₂×D₂₂₃)⋊C₄	C₂	semidirect
5. (C₂×D₂₂₃)⋊C₄	C₂	semidirect
6. (C₂×D₂₂₃)⋊C₄	C₂	semidirect
7. C₄⋊C₄×C₂₂₃	C₂	semidirect
34. C₂₂₃	#123	semidirect
39. C₄⋊C₄	D₂₂₃	semidirect