Subgroups — C₂²×#125×C₂₂₃ #43252452

Order 28544.

1192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₂₂₃	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊C₄×C₂₂₃	C₂	1	4	yes	maximal
3	C₂²×C₄⋊C₄×C₂₂₃	C₂	1	2	yes	maximal
4	C₂²×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₂₃	C₂²	1	2	yes	—
6	C₂³×C₄×C₂₂₃	C₂²	1	4	yes	—
7	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
8	#125×C₂₂₃	C₂²	1	64	yes	direct product factor
9	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	48	yes	—
10	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	24	yes	—
11	C₂×C₄²×C₂₂₃	C₂²	1	12	yes	—
12	C₂⁴×C₂₂₃	C₂³	1	1	yes	characteristic
13	C₂²×C₄×C₂₂₃	—	2	8	no	—
14	C₂²×C₄×C₂₂₃	—	2	16	no	—
15	C₂²×C₄×C₂₂₃	Q₈	1	8	yes	—
16	C₂²×C₄×C₂₂₃	C₂³	1	12	yes	—
17	C₂²×C₄×C₂₂₃	C₂³	1	24	yes	—
18	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
19	C₄⋊C₄×C₂₂₃	C₂³	1	64	yes	—
20	C₄⋊C₄×C₂₂₃	C₂³	1	32	yes	—
21	C₄²×C₂₂₃	C₂³	1	16	yes	—
22	C₂³×C₂₂₃	C₂×Q₈	1	4	yes	—
23	C₂³×C₂₂₃	#41	1	8	yes	—
24	C₂³×C₂₂₃	C₂⁴	1	3	yes	—
25	C₂×C₄×C₂₂₃	—	2	24	no	—
26	C₂×C₄×C₂₂₃	—	2	48	no	—
27	C₂×C₄×C₂₂₃	C₂×Q₈	1	24	yes	—
28	C₂×C₄×C₂₂₃	C₂⁴	1	8	yes	—
29	C₂×C₄×C₂₂₃	C₂⁴	1	16	yes	—
30	C₂×C₄×C₂₂₃	C₂⁴	1	4	yes	—
31	C₂²×C₂₂₃	#125	1	16	yes	direct product factor
32	C₂²×C₂₂₃	C₂²×Q₈	1	6	yes	—
33	C₂²×C₂₂₃	C₂×#41	1	12	yes	—
34	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
35	C₄×C₂₂₃	—	2	8	no	—
36	C₄×C₂₂₃	—	2	16	no	—
37	C₄×C₂₂₃	C₂²×Q₈	1	8	yes	—
38	C₂×C₂₂₃	C₂×#125	1	12	yes	direct product factor
39	C₂×C₂₂₃	C₂³×Q₈	1	1	yes	characteristic
40	C₂×C₂₂₃	C₂²×#41	1	2	yes	—
41	C₂₂₃	C₂²×#125	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×#125	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂×#125	C₂×C₂₂₃	1	24	yes	direct product factor
44	C₂²×C₄⋊C₄	C₂×C₂₂₃	1	4	yes	—
45	C₂²×C₄⋊C₄	C₂×C₂₂₃	1	2	yes	—
46	C₂²×C₄²	C₂×C₂₂₃	1	1	yes	characteristic
47	C₂³×C₄	C₂²×C₂₂₃	1	4	yes	—
48	C₂³×C₄	C₂²×C₂₂₃	1	2	yes	—
49	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
50	#125	C₂²×C₂₂₃	1	64	yes	direct product factor
51	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	48	yes	—
52	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	24	yes	—
53	C₂×C₄²	C₂²×C₂₂₃	1	12	yes	—
54	C₂⁴	C₂³×C₂₂₃	1	1	yes	characteristic
55	C₂²×C₄	—	2	16	no	—
56	C₂²×C₄	—	2	8	no	—
57	C₂²×C₄	Q₈×C₂₂₃	1	8	yes	—
58	C₂²×C₄	C₂³×C₂₂₃	1	12	yes	—
59	C₂²×C₄	C₂³×C₂₂₃	1	24	yes	—
60	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
61	C₄⋊C₄	C₂³×C₂₂₃	1	64	yes	—
62	C₄⋊C₄	C₂³×C₂₂₃	1	32	yes	—
63	C₄²	C₂³×C₂₂₃	1	16	yes	—
64	C₂³	C₂×Q₈×C₂₂₃	1	4	yes	—
65	C₂³	#41×C₂₂₃	1	8	yes	—
66	C₂³	C₂⁴×C₂₂₃	1	3	yes	—
67	C₂×C₄	—	2	24	no	—
68	C₂×C₄	—	2	48	no	—
69	C₂×C₄	C₂×Q₈×C₂₂₃	1	24	yes	—
70	C₂×C₄	C₂⁴×C₂₂₃	1	8	yes	—
71	C₂×C₄	C₂⁴×C₂₂₃	1	16	yes	—
72	C₂×C₄	C₂⁴×C₂₂₃	1	4	yes	—
73	C₂²	#125×C₂₂₃	1	16	yes	direct product factor
74	C₂²	C₂²×Q₈×C₂₂₃	1	6	yes	—
75	C₂²	C₂×#41×C₂₂₃	1	12	yes	—
76	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
77	C₄	—	2	8	no	—
78	C₄	—	2	16	no	—
79	C₄	C₂²×Q₈×C₂₂₃	1	8	yes	—
80	C₂	C₂×#125×C₂₂₃	1	12	yes	direct product factor
81	C₂	C₂³×Q₈×C₂₂₃	1	1	yes	characteristic
82	C₂	C₂²×#41×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₂₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#125×C₂₂₃	C₂	direct (N × H)
8. #125×C₂₂₃	C₂²	direct (N × H)
31. C₂²×C₂₂₃	#125	direct (N × H)
38. C₂×C₂₂₃	C₂×#125	direct (N × H)
41. C₂₂₃	C₂²×#125	direct (N × H)
42. C₂²×#125	C₂₂₃	direct (N × H)
43. C₂×#125	C₂×C₂₂₃	direct (N × H)
50. #125	C₂²×C₂₂₃	direct (N × H)
73. C₂²	#125×C₂₂₃	direct (N × H)
80. C₂	C₂×#125×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
C₂×#125×C₂₂₃	2
C₂²×C₄⋊C₄×C₂₂₃	2
C₂²×C₄²×C₂₂₃	2
C₂²×#125	223

Subgroups — C22×#125×C223 #43252452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#125×C₂₂₃ #43252452