Subgroups — #2862×C₂₂₃ #43251877

Order 28544.

504 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₂₂₃	C₂	1	2	yes	maximal
2	#391×C₂₂₃	C₂	1	8	yes	maximal
3	#382×C₂₂₃	C₂	1	4	yes	maximal
4	C₄³×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₂₃	C₂²	1	4	yes	—
6	C₄⋊Q₈×C₂₂₃	—	2	8	no	—
7	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
8	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
9	C₂×C₄²×C₂₂₃	C₂²	1	4	yes	—
10	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	—
12	#95×C₂₂₃	C₂²	1	16	yes	—
13	C₂×Q₈×C₂₂₃	—	2	8	no	—
14	C₂×Q₈×C₂₂₃	—	2	16	no	—
15	C₂²×C₄×C₂₂₃	C₂³	1	8	yes	—
16	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
17	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
19	C₄⋊C₄×C₂₂₃	—	2	8	no	—
20	C₄⋊C₄×C₂₂₃	—	2	8	no	—
21	C₄²×C₂₂₃	—	2	8	no	—
22	C₄²×C₂₂₃	D₄	1	4	yes	—
23	C₄²×C₂₂₃	Q₈	1	8	yes	complemented
24	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
25	Q₈×C₂₂₃	—	4	16	no	—
26	C₂×C₄×C₂₂₃	—	2	8	no	—
27	C₂×C₄×C₂₂₃	—	2	8	no	—
28	C₂×C₄×C₂₂₃	—	2	16	no	—
29	C₂×C₄×C₂₂₃	C₂×D₄	1	4	yes	—
30	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₂₂₃	C₂×Q₈	1	8	yes	—
32	C₂×C₄×C₂₂₃	C₂×Q₈	1	4	yes	—
33	C₂×C₄×C₂₂₃	#41	1	8	yes	—
34	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
35	C₂²×C₂₂₃	C₂²×Q₈	1	2	yes	—
36	C₂²×C₂₂₃	C₂×#41	1	4	yes	—
37	C₄×C₂₂₃	—	2	8	no	—
38	C₄×C₂₂₃	—	4	8	no	—
39	C₄×C₂₂₃	#124	1	8	yes	—
40	C₄×C₂₂₃	C₄⋊Q₈	1	4	yes	—
41	C₂×C₂₂₃	C₂×#124	1	2	yes	—
42	C₂×C₂₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₂₂₃	#533	1	4	yes	—
44	C₂₂₃	#2862	1	1	yes	characteristic, direct product factor, Sylow
45	#2862	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₄⋊Q₈	C₂×C₂₂₃	1	2	yes	—
47	#391	C₂×C₂₂₃	1	8	yes	—
48	#382	C₂×C₂₂₃	1	4	yes	—
49	C₄³	C₂×C₂₂₃	1	1	yes	characteristic
50	C₂²×Q₈	C₂²×C₂₂₃	1	4	yes	—
51	C₄⋊Q₈	—	2	8	no	—
52	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	4	yes	—
53	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	4	yes	—
54	C₂×C₄²	C₂²×C₂₂₃	1	4	yes	—
55	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
56	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	—
57	#95	C₂²×C₂₂₃	1	16	yes	—
58	C₂×Q₈	—	2	8	no	—
59	C₂×Q₈	—	2	16	no	—
60	C₂²×C₄	C₂³×C₂₂₃	1	8	yes	—
61	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
62	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
63	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
64	C₄⋊C₄	—	2	8	no	—
65	C₄⋊C₄	—	2	8	no	—
66	C₄²	—	2	8	no	—
67	C₄²	D₄×C₂₂₃	1	4	yes	—
68	C₄²	Q₈×C₂₂₃	1	8	yes	complemented
69	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
70	Q₈	—	4	16	no	—
71	C₂×C₄	—	2	8	no	—
72	C₂×C₄	—	2	8	no	—
73	C₂×C₄	—	2	16	no	—
74	C₂×C₄	C₂×D₄×C₂₂₃	1	4	yes	—
75	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
76	C₂×C₄	C₂×Q₈×C₂₂₃	1	8	yes	—
77	C₂×C₄	C₂×Q₈×C₂₂₃	1	4	yes	—
78	C₂×C₄	#41×C₂₂₃	1	8	yes	—
79	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
80	C₂²	C₂²×Q₈×C₂₂₃	1	2	yes	—
81	C₂²	C₂×#41×C₂₂₃	1	4	yes	—
82	C₄	—	2	8	no	—
83	C₄	—	4	8	no	—
84	C₄	#124×C₂₂₃	1	8	yes	—
85	C₄	C₄⋊Q₈×C₂₂₃	1	4	yes	—
86	C₂	C₂×#124×C₂₂₃	1	2	yes	—
87	C₂	C₂×C₄⋊Q₈×C₂₂₃	1	1	yes	characteristic
88	C₂	#533×C₂₂₃	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
23. C₄²×C₂₂₃	Q₈	semidirect
44. C₂₂₃	#2862	direct (N × H)
45. #2862	C₂₂₃	direct (N × H)
68. C₄²	Q₈×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₂₂₃	2
#391×C₂₂₃	2
#382×C₂₂₃	2
C₄³×C₂₂₃	2
#2862	223

Subgroups — #2862×C223 #43251877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2862×C₂₂₃ #43251877