Subgroups — #2151×C₂₂₃ #43251166

Order 28544.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#405×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₁₆×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	C₂×M₁₆×C₂₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
7	#117×C₂₂₃	C₄	1	1	yes	characteristic
8	M₃₂×C₂₂₃	—	2	2	no	—
9	C₂×C₁₆×C₂₂₃	—	2	2	no	—
10	C₂×C₁₆×C₂₂₃	C₂²	1	2	yes	—
11	C₂×C₁₆×C₂₂₃	C₂²	1	2	yes	—
12	#97×C₂₂₃	C₄	1	2	yes	—
13	C₄×C₈×C₂₂₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
15	M₁₆×C₂₂₃	—	2	2	no	—
16	C₂×C₈×C₂₂₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
19	C₂×C₈×C₂₂₃	Q₈	1	2	yes	—
20	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₂₂₃	C₈	1	2	yes	—
22	C₂²⋊C₄×C₂₂₃	C₈	1	2	yes	—
23	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
24	C₁₆×C₂₂₃	—	2	2	no	—
25	C₁₆×C₂₂₃	—	2	2	no	—
26	C₂³×C₂₂₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₂₂₃	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₂₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₂₂₃	—	2	2	no	—
33	C₈×C₂₂₃	C₄⋊C₄	1	2	yes	—
34	C₈×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	C₈×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₂₂₃	—	2	2	no	—
37	C₂²×C₂₂₃	C₄⋊C₈	1	2	yes	—
38	C₂²×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₂₂₃	—	2	4	no	—
40	C₄×C₂₂₃	C₄⋊C₈	1	2	yes	—
41	C₄×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₂₂₃	—	2	2	no	—
44	C₂×C₂₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₂₂₃	#504	1	2	yes	—
46	C₂₂₃	#2151	1	1	yes	characteristic, direct product factor, Sylow
47	#2151	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×M₃₂	C₂×C₂₂₃	1	1	yes	characteristic
49	C₂²×C₁₆	C₂×C₂₂₃	1	1	yes	characteristic
50	#405	C₂×C₂₂₃	1	1	yes	characteristic
51	C₄⋊C₁₆	C₂×C₂₂₃	1	4	yes	complemented
52	C₂×M₁₆	C₄×C₂₂₃	1	1	yes	characteristic
53	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
54	#117	C₄×C₂₂₃	1	1	yes	characteristic
55	M₃₂	—	2	2	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	C₂²×C₂₂₃	1	2	yes	—
58	C₂×C₁₆	C₂²×C₂₂₃	1	2	yes	—
59	#97	C₄×C₂₂₃	1	2	yes	—
60	C₄×C₈	C₂²×C₂₂₃	1	2	yes	—
61	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	C₂×C₈	C₂×C₄×C₂₂₃	1	2	yes	—
64	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
65	C₂×C₈	D₄×C₂₂₃	1	2	yes	—
66	C₂×C₈	Q₈×C₂₂₃	1	2	yes	—
67	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
68	C₄⋊C₄	C₈×C₂₂₃	1	2	yes	—
69	C₂²⋊C₄	C₈×C₂₂₃	1	2	yes	—
70	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	2	2	no	—
73	C₂³	C₂×C₈×C₂₂₃	1	1	yes	characteristic
74	C₂×C₄	C₄⋊C₄×C₂₂₃	1	2	yes	—
75	C₂×C₄	C₂×C₈×C₂₂₃	1	4	yes	—
76	C₂×C₄	C₂×C₈×C₂₂₃	1	1	yes	characteristic
77	C₂×C₄	M₁₆×C₂₂₃	1	2	yes	—
78	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	C₄⋊C₄×C₂₂₃	1	2	yes	—
81	C₈	C₂×D₄×C₂₂₃	1	1	yes	characteristic
82	C₈	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊C₈×C₂₂₃	1	2	yes	—
85	C₂²	C₂²×C₈×C₂₂₃	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₄⋊C₈×C₂₂₃	1	2	yes	—
88	C₄	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
89	C₄	C₂×M₁₆×C₂₂₃	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊C₈×C₂₂₃	1	1	yes	characteristic
92	C₂	#504×C₂₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₂₃	C₂³
commutator	C₂²	C₂²×C₈×C₂₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
socle	C₂²×C₂₂₃	C₂²×C₈
prime radical	C₂³×C₂₂₃	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₁₆×C₂₂₃	C₂	semidirect
46. C₂₂₃	#2151	direct (N × H)
47. #2151	C₂₂₃	direct (N × H)
51. C₄⋊C₁₆	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×C₂₂₃	2
C₂²×C₁₆×C₂₂₃	2
#405×C₂₂₃	2
C₄⋊C₁₆×C₂₂₃	2
#2151	223

Subgroups — #2151×C223 #43251166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2151×C₂₂₃ #43251166