Subgroups — #1838×C₂₂₃ #43250853

Order 28544.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#423×C₂₂₃	C₂	1	2	yes	maximal
2	C₂×#97×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#341×C₂₂₃	C₂	1	4	yes	maximal
4	C₂×M₁₆×C₂₂₃	C₂²	1	4	yes	—
5	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
6	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₄⋊C₈×C₂₂₃	—	2	4	no	—
8	#97×C₂₂₃	C₄	1	4	yes	—
9	C₂²×C₄×C₂₂₃	Q₈	1	2	yes	—
10	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
11	M₁₆×C₂₂₃	—	2	8	no	—
12	C₂×C₈×C₂₂₃	—	2	4	no	—
13	C₂×C₈×C₂₂₃	—	2	4	no	—
14	C₂×C₈×C₂₂₃	C₂×C₄	1	4	yes	—
15	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
16	C₄²×C₂₂₃	D₄	1	2	yes	—
17	C₂³×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	—	4	2	no	—
19	C₂×C₄×C₂₂₃	C₄⋊C₄	1	4	yes	—
20	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	#41	1	4	yes	—
23	C₈×C₂₂₃	—	2	4	no	—
24	C₈×C₂₂₃	—	4	4	no	—
25	C₂²×C₂₂₃	—	2	2	no	—
26	C₂²×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₂₃	#125	1	2	yes	—
28	C₄×C₂₂₃	—	2	4	no	—
29	C₄×C₂₂₃	#117	1	2	yes	—
30	C₄×C₂₂₃	#124	1	2	yes	—
31	C₂×C₂₂₃	—	2	2	no	—
32	C₂×C₂₂₃	#382	1	1	yes	characteristic
33	C₂×C₂₂₃	#430	1	2	yes	—
34	C₂₂₃	#1838	1	1	yes	characteristic, direct product factor, Sylow
35	#1838	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	#423	C₂×C₂₂₃	1	2	yes	—
37	C₂×#97	C₂×C₂₂₃	1	1	yes	characteristic
38	#341	C₂×C₂₂₃	1	4	yes	—
39	C₂×M₁₆	C₂²×C₂₂₃	1	4	yes	—
40	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
41	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
42	C₄⋊C₈	—	2	4	no	—
43	#97	C₄×C₂₂₃	1	4	yes	—
44	C₂²×C₄	Q₈×C₂₂₃	1	2	yes	—
45	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
46	M₁₆	—	2	8	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	2	4	no	—
49	C₂×C₈	C₂×C₄×C₂₂₃	1	4	yes	—
50	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
51	C₄²	D₄×C₂₂₃	1	2	yes	—
52	C₂³	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	C₄⋊C₄×C₂₂₃	1	4	yes	—
55	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
56	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
57	C₂×C₄	#41×C₂₂₃	1	4	yes	—
58	C₈	—	2	4	no	—
59	C₈	—	4	4	no	—
60	C₂²	—	2	2	no	—
61	C₂²	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
62	C₂²	#125×C₂₂₃	1	2	yes	—
63	C₄	—	2	4	no	—
64	C₄	#117×C₂₂₃	1	2	yes	—
65	C₄	#124×C₂₂₃	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	#382×C₂₂₃	1	1	yes	characteristic
68	C₂	#430×C₂₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₂₃
socle	C₂²×C₂₂₃	C₂×C₄⋊C₄
prime radical	C₂³×C₂₂₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ C₂×C₄²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
34. C₂₂₃	#1838	direct (N × H)
35. #1838	C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
#423×C₂₂₃	2
C₂×#97×C₂₂₃	2
#341×C₂₂₃	2
#1838	223

Subgroups — #1838×C223 #43250853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1838×C₂₂₃ #43250853