Subgroups — #1446×C₂₂₃ #43250461

Order 28544.

736 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#95×C₂₂₃	C₂	1	6	yes	maximal
3	C₂²×Q₈×C₂₂₃	—	2	12	no	—
4	C₂²×Q₈×C₂₂₃	C₄	1	4	yes	complemented
5	C₂³×C₄×C₂₂₃	C₂²	1	4	yes	—
6	C₂³×C₄×C₂₂₃	C₂²	1	3	yes	—
7	#95×C₂₂₃	—	2	12	no	—
8	C₂⁴×C₂₂₃	C₂³	1	1	yes	characteristic
9	C₂×Q₈×C₂₂₃	—	2	32	no	—
10	C₂×Q₈×C₂₂₃	—	2	24	no	—
11	C₂²×C₄×C₂₂₃	—	2	12	no	—
12	C₂²×C₄×C₂₂₃	—	2	12	no	—
13	C₂²×C₄×C₂₂₃	—	2	12	no	—
14	C₂²×C₄×C₂₂₃	C₂×C₄	1	6	yes	—
15	C₂²×C₄×C₂₂₃	D₄	1	12	yes	—
16	C₂²×C₄×C₂₂₃	Q₈	1	8	yes	complemented
17	C₂³×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
18	C₂³×C₂₂₃	C₂×D₄	1	6	yes	—
19	C₂³×C₂₂₃	C₂×Q₈	1	4	yes	—
20	C₂³×C₂₂₃	#41	1	4	yes	—
21	Q₈×C₂₂₃	—	2	32	no	—
22	C₂×C₄×C₂₂₃	—	2	12	no	—
23	C₂×C₄×C₂₂₃	—	2	24	no	—
24	C₂×C₄×C₂₂₃	—	2	24	no	—
25	C₂×C₄×C₂₂₃	—	4	4	no	—
26	C₂×C₄×C₂₂₃	—	4	12	no	—
27	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	12	yes	—
28	C₂²×C₂₂₃	C₂×C₂²⋊C₄	1	3	yes	—
29	C₂²×C₂₂₃	C₄×Q₈	1	4	yes	—
30	C₂²×C₂₂₃	#120	1	4	yes	—
31	C₂²×C₂₂₃	C₂²⋊Q₈	1	12	yes	—
32	C₂²×C₂₂₃	#124	1	6	yes	—
33	C₂²×C₂₂₃	C₄⋊Q₈	1	6	yes	—
34	C₄×C₂₂₃	—	2	12	no	—
35	C₄×C₂₂₃	—	4	8	no	—
36	C₂×C₂₂₃	#379	1	1	yes	characteristic
37	C₂×C₂₂₃	#391	1	6	yes	—
38	C₂×C₂₂₃	#393	1	4	yes	—
39	C₂×C₂₂₃	#395	1	4	yes	—
40	C₂₂₃	#1446	1	1	yes	characteristic, direct product factor, Sylow
41	#1446	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂³×Q₈	C₂×C₂₂₃	1	1	yes	characteristic
43	C₂×#95	C₂×C₂₂₃	1	6	yes	—
44	C₂²×Q₈	—	2	12	no	—
45	C₂²×Q₈	C₄×C₂₂₃	1	4	yes	complemented
46	C₂³×C₄	C₂²×C₂₂₃	1	4	yes	—
47	C₂³×C₄	C₂²×C₂₂₃	1	3	yes	—
48	#95	—	2	12	no	—
49	C₂⁴	C₂³×C₂₂₃	1	1	yes	characteristic
50	C₂×Q₈	—	2	32	no	—
51	C₂×Q₈	—	2	24	no	—
52	C₂²×C₄	—	2	12	no	—
53	C₂²×C₄	—	2	12	no	—
54	C₂²×C₄	—	2	12	no	—
55	C₂²×C₄	C₂×C₄×C₂₂₃	1	6	yes	—
56	C₂²×C₄	D₄×C₂₂₃	1	12	yes	—
57	C₂²×C₄	Q₈×C₂₂₃	1	8	yes	complemented
58	C₂³	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
59	C₂³	C₂×D₄×C₂₂₃	1	6	yes	—
60	C₂³	C₂×Q₈×C₂₂₃	1	4	yes	—
61	C₂³	#41×C₂₂₃	1	4	yes	—
62	Q₈	—	2	32	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	—	2	24	no	—
65	C₂×C₄	—	2	24	no	—
66	C₂×C₄	—	4	4	no	—
67	C₂×C₄	—	4	12	no	—
68	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	12	yes	—
69	C₂²	C₂×C₂²⋊C₄×C₂₂₃	1	3	yes	—
70	C₂²	C₄×Q₈×C₂₂₃	1	4	yes	—
71	C₂²	#120×C₂₂₃	1	4	yes	—
72	C₂²	C₂²⋊Q₈×C₂₂₃	1	12	yes	—
73	C₂²	#124×C₂₂₃	1	6	yes	—
74	C₂²	C₄⋊Q₈×C₂₂₃	1	6	yes	—
75	C₄	—	2	12	no	—
76	C₄	—	4	8	no	—
77	C₂	#379×C₂₂₃	1	1	yes	characteristic
78	C₂	#391×C₂₂₃	1	6	yes	—
79	C₂	#393×C₂₂₃	1	4	yes	—
80	C₂	#395×C₂₂₃	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₂₃	C₂³
commutator	C₂³	C₂²×C₄×C₂₂₃
Frattini, cosocle	C₂⁴	C₂³×C₂₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂²×Q₈×C₂₂₃	C₄	semidirect
16. C₂²×C₄×C₂₂₃	Q₈	semidirect
40. C₂₂₃	#1446	direct (N × H)
41. #1446	C₂₂₃	direct (N × H)
45. C₂²×Q₈	C₄×C₂₂₃	semidirect
57. C₂²×C₄	Q₈×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₈×C₂₂₃	2
C₂×#95×C₂₂₃	2
#1446	223

Subgroups — #1446×C223 #43250461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1446×C₂₂₃ #43250461