Subgroups — #43252663

Order 28544.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383637	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₂₃⋊C₈)	—	2	2	no	—
4	C₂²⋊(C₂₂₃⋊C₈)	—	2	2	no	—
5	#20297215	C₄	1	4	yes	—
6	C₂²×C₈×C₂₂₃	C₄	1	2	yes	—
7	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₂₂₃	—	2	2	no	—
9	C₂×C₂₂₃⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₂₂₃	—	2	2	no	—
14	C₂×C₈×C₂₂₃	—	4	2	no	—
15	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
16	C₄²×C₂₂₃	D₄	1	2	yes	—
17	C₂₂₃⋊C₈	—	4	4	no	—
18	C₂³×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	—	2	2	no	—
20	C₂×C₄×C₂₂₃	—	2	1	no	—
21	C₂×C₄×C₂₂₃	—	2	1	no	—
22	C₂×C₄×C₂₂₃	—	4	1	no	—
23	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
27	C₈×C₂₂₃	—	4	2	no	—
28	C₂²×C₂₂₃	—	2	1	no	—
29	C₂²×C₂₂₃	—	2	1	no	—
30	C₂²×C₂₂₃	#95	1	1	yes	characteristic
31	C₂²×C₂₂₃	#108	1	2	yes	—
32	C₄×C₂₂₃	—	2	2	no	—
33	C₄×C₂₂₃	—	2	2	no	—
34	C₄×C₂₂₃	—	2	1	no	—
35	C₄×C₂₂₃	#100	1	1	yes	characteristic
36	C₄×C₂₂₃	#101	1	1	yes	characteristic
37	C₂×C₂₂₃	—	2	2	no	—
38	C₂×C₂₂₃	H(3,ℤ/4)	1	1	yes	characteristic
39	C₂×C₂₂₃	#341	1	1	yes	characteristic
40	C₂×C₂₂₃	#343	1	1	yes	characteristic
41	C₂₂₃	#1290	1	1	yes	characteristic, complemented, Sylow
42	#1290	—	223	1	no	maximal, Sylow
43	#432	—	223	2	no	—
44	C₂×C₄⋊C₈	D₂₂₃	1	1	yes	characteristic
45	C₂²×C₈	Dic₂₂₃	1	2	yes	—
46	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	446	2	no	—
49	C₂²⋊C₈	—	446	2	no	—
50	#97	—	223	4	no	—
51	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
52	C₂²×C₄	D₈₉₂	1	1	yes	characteristic
53	C₂²×C₄	Dic₄₄₆	1	1	yes	characteristic
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	—	446	4	no	—
57	C₄²	C₄×D₂₂₃	1	2	yes	—
58	C₄²	C₂²⋊D₂₂₃	1	2	yes	—
59	C₂³	C₄⋊Dic₂₂₃	1	1	yes	characteristic
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
65	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
66	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
67	C₂×C₄	Dic₂₂₃⋊C₄	1	2	yes	—
68	C₈	—	4	2	no	—
69	C₈	—	892	4	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
73	C₂²	#20297240	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	2	1	no	—
77	C₄	#20297221	1	1	yes	characteristic
78	C₄	#20297223	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	#28383337	1	1	yes	characteristic
81	C₂	#28383345	1	1	yes	characteristic
82	C₂	#28383349	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂₂₃⋊#95
commutator	C₂×C₄×C₂₂₃	C₄²
Frattini	C₂×C₄²	C₂×D₂₂₃
Fitting	C₂×C₄⋊C₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	#95
prime radical	C₂³×C₂₂₃	C₄⋊C₄
cosocle	C₂×C₄²×C₂₂₃	C₂²
hypercenter	C₂×C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#1290
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
41. C₂₂₃	#1290	semidirect

Maximal subgroups

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

subgroup	index
#28383637	2
C₂×C₄⋊C₈×C₂₂₃	2
#1290	223