Subgroups — #43252620

Order 28544.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₂₃⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₂₃⋊C₈	C₄	1	4	yes	—
4	C₄×C₂₂₃⋊C₈	—	2	2	no	—
5	C₄×C₂₂₃⋊C₈	—	2	2	no	—
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	C₂×C₂₂₃⋊C₈	—	2	4	no	—
11	C₂×C₂₂₃⋊C₈	C₈	1	8	yes	complemented
12	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₂₃	—	2	2	no	—
15	C₂×C₈×C₂₂₃	—	2	2	no	—
16	C₂×C₈×C₂₂₃	C₈	1	4	yes	complemented
17	C₄²×C₂₂₃	D₄	1	2	yes	—
18	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
19	C₄²×C₂₂₃	Q₈	1	1	yes	characteristic
20	C₂₂₃⋊C₈	—	2	8	no	—
21	C₂³×C₂₂₃	C₄²	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	C₂×C₈	1	4	yes	—
27	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₂₂₃	M₁₆	1	4	yes	—
29	C₂×C₄×C₂₂₃	M₁₆	1	2	yes	—
30	C₈×C₂₂₃	—	2	4	no	—
31	C₂²×C₂₂₃	#95	1	1	yes	characteristic
32	C₂²×C₂₂₃	C₄×C₈	1	2	yes	—
33	C₂²×C₂₂₃	C₄×C₈	1	1	yes	characteristic
34	C₂²×C₂₂₃	#97	1	2	yes	—
35	C₂²×C₂₂₃	#97	1	1	yes	characteristic
36	C₄×C₂₂₃	C₂²⋊C₈	1	4	yes	—
37	C₄×C₂₂₃	C₂²⋊C₈	1	2	yes	—
38	C₄×C₂₂₃	C₄⋊C₈	1	4	yes	—
39	C₄×C₂₂₃	C₄⋊C₈	1	2	yes	—
40	C₂×C₂₂₃	C₈²	1	1	yes	characteristic
41	C₂×C₂₂₃	#322	1	1	yes	characteristic
42	C₂×C₂₂₃	#322	1	2	yes	—
43	C₂×C₂₂₃	#336	1	2	yes	—
44	C₂×C₂₂₃	#336	1	1	yes	characteristic
45	C₂₂₃	#1273	1	1	yes	characteristic, complemented, Sylow
46	#1273	—	223	1	no	maximal, Sylow
47	C₂×C₄×C₈	—	223	2	no	—
48	C₂×C₄×C₈	D₂₂₃	1	1	yes	characteristic
49	C₂²×C₈	—	223	4	no	—
50	C₂²×C₈	Dic₂₂₃	1	2	yes	—
51	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	446	2	no	—
54	C₄×C₈	—	446	2	no	—
55	C₂²×C₄	C₄×D₂₂₃	1	2	yes	—
56	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	446	4	no	—
60	C₂×C₈	—	446	4	no	—
61	C₂×C₈	—	223	8	no	—
62	C₂×C₈	C₂₂₃⋊C₈	1	4	yes	complemented
63	C₄²	C₂²⋊D₂₂₃	1	2	yes	—
64	C₄²	D₈₉₂	1	1	yes	characteristic
65	C₄²	Dic₄₄₆	1	1	yes	characteristic
66	C₂³	C₄×Dic₂₂₃	1	1	yes	characteristic
67	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
68	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
69	C₂×C₄	Dic₂₂₃⋊C₄	1	2	yes	—
70	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
71	C₂×C₄	C₈×D₂₂₃	1	4	yes	—
72	C₂×C₄	C₂×C₂₂₃⋊C₈	1	2	yes	—
73	C₂×C₄	#16813153	1	4	yes	—
74	C₂×C₄	#16813154	1	2	yes	—
75	C₈	—	2	4	no	—
76	C₈	—	446	8	no	—
77	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
78	C₂²	C₈×Dic₂₂₃	1	2	yes	—
79	C₂²	C₄×C₂₂₃⋊C₈	1	1	yes	characteristic
80	C₂²	#20297214	1	2	yes	—
81	C₂²	#20297215	1	1	yes	characteristic
82	C₄	(C₂×D₂₂₃)⋊C₈	1	4	yes	—
83	C₄	C₂²⋊(C₂₂₃⋊C₈)	1	2	yes	—
84	C₄	Dic₂₂₃⋊C₈	1	4	yes	—
85	C₄	C₄⋊(C₂₂₃⋊C₈)	1	2	yes	—
86	C₂	C₈×C₂₂₃⋊C₈	1	1	yes	characteristic
87	C₂	#28383301	1	1	yes	characteristic
88	C₂	#28383302	1	2	yes	—
89	C₂	#28383334	1	2	yes	—
90	C₂	#28383335	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₈²
Fitting	C₂×C₄×C₈×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	C₄²
cosocle	C₂×C₄²×C₂₂₃	C₂²
hypercenter	C₂×C₄×C₈	D₂₂₃
hypocenter	C₂₂₃	#1273
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
11. C₂×C₂₂₃⋊C₈	C₈	semidirect
16. C₂×C₈×C₂₂₃	C₈	semidirect
45. C₂₂₃	#1273	semidirect
62. C₂×C₈	C₂₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂₂₃⋊C₈	2
C₂×C₄×C₈×C₂₂₃	2
#1273	223