Subgroups — #2153×C₂₂₃ #43251168

Order 28544.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×C₂₂₃	C₂	1	2	yes	maximal
2	#405×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₂₃	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
6	#117×C₂₂₃	C₄	1	1	yes	characteristic
7	M₃₂×C₂₂₃	—	2	2	no	—
8	M₃₂×C₂₂₃	C₂²	1	4	yes	—
9	C₂×C₁₆×C₂₂₃	—	2	2	no	—
10	#97×C₂₂₃	C₄	1	2	yes	—
11	C₄×C₈×C₂₂₃	C₂²	1	2	yes	—
12	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	M₁₆×C₂₂₃	C₈	1	4	yes	—
14	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₂₂₃	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
17	C₂×C₈×C₂₂₃	Q₈	1	2	yes	—
18	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₂₂₃	—	2	1	no	—
20	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
21	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
22	C₁₆×C₂₂₃	—	2	4	no	—
23	C₂³×C₂₂₃	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	—	2	2	no	—
25	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₂₂₃	M₁₆	1	1	yes	characteristic
28	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₈×C₂₂₃	C₄⋊C₄	1	2	yes	—
30	C₈×C₂₂₃	C₂×C₈	1	4	yes	—
31	C₈×C₂₂₃	C₂×D₄	1	1	yes	characteristic
32	C₈×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₂₂₃	—	4	1	no	—
34	C₂²×C₂₂₃	C₄⋊C₈	1	2	yes	—
35	C₂²×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₂₂₃	—	4	2	no	—
37	C₄×C₂₂₃	C₄⋊C₈	1	2	yes	—
38	C₄×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₂₂₃	—	2	1	no	—
41	C₂×C₂₂₃	—	2	2	no	—
42	C₂×C₂₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
43	C₂₂₃	#2153	1	1	yes	characteristic, direct product factor, Sylow
44	#2153	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×M₃₂	C₂×C₂₂₃	1	2	yes	—
46	#405	C₂×C₂₂₃	1	1	yes	characteristic
47	#364	C₂×C₂₂₃	1	4	yes	complemented
48	C₂×M₁₆	C₄×C₂₂₃	1	1	yes	characteristic
49	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
50	#117	C₄×C₂₂₃	1	1	yes	characteristic
51	M₃₂	—	2	2	no	—
52	M₃₂	C₂²×C₂₂₃	1	4	yes	—
53	C₂×C₁₆	—	2	2	no	—
54	#97	C₄×C₂₂₃	1	2	yes	—
55	C₄×C₈	C₂²×C₂₂₃	1	2	yes	—
56	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
57	M₁₆	C₈×C₂₂₃	1	4	yes	—
58	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
59	C₂×C₈	C₂×C₄×C₂₂₃	1	2	yes	—
60	C₂×C₈	D₄×C₂₂₃	1	2	yes	—
61	C₂×C₈	Q₈×C₂₂₃	1	2	yes	—
62	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
63	C₄⋊C₄	—	2	1	no	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
66	C₁₆	—	2	4	no	—
67	C₂³	M₁₆×C₂₂₃	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	C₄⋊C₄×C₂₂₃	1	2	yes	—
70	C₂×C₄	C₂×C₈×C₂₂₃	1	2	yes	—
71	C₂×C₄	M₁₆×C₂₂₃	1	1	yes	characteristic
72	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
73	C₈	C₄⋊C₄×C₂₂₃	1	2	yes	—
74	C₈	C₂×C₈×C₂₂₃	1	4	yes	—
75	C₈	C₂×D₄×C₂₂₃	1	1	yes	characteristic
76	C₈	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
77	C₂²	—	4	1	no	—
78	C₂²	C₄⋊C₈×C₂₂₃	1	2	yes	—
79	C₂²	C₂×M₁₆×C₂₂₃	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	C₄⋊C₈×C₂₂₃	1	2	yes	—
82	C₄	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
83	C₄	C₂²×C₈×C₂₂₃	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×C₄⋊C₈×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₂₃	C₂×D₄
commutator	C₄	C₂²×C₈×C₂₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
socle	C₂×C₂₂₃	C₂×C₄⋊C₈
prime radical	C₂³×C₂₂₃	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₂₃ ◁ C₂²×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. #364×C₂₂₃	C₂	semidirect
43. C₂₂₃	#2153	direct (N × H)
44. #2153	C₂₂₃	direct (N × H)
47. #364	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
#405×C₂₂₃	2
#364×C₂₂₃	2
#2153	223

Subgroups — #2153×C223 #43251168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2153×C₂₂₃ #43251168