Subgroups — #43252626

Order 28544.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383633	C₂	1	2	yes	maximal
2	C₄³×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₂₃⋊C₈)	C₄	1	4	yes	complemented
4	C₂²⋊(C₂₂₃⋊C₈)	—	2	2	no	—
5	C₄×C₂₂₃⋊C₈	—	2	2	no	—
6	C₂×C₄²×C₂₂₃	—	2	2	no	—
7	C₂×C₄²×C₂₂₃	C₄	1	2	yes	—
8	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₂₃⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₂₂₃	—	2	2	no	—
11	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	D₄	1	2	yes	—
13	C₄²×C₂₂₃	—	2	8	no	—
14	C₄²×C₂₂₃	—	2	2	no	—
15	C₄²×C₂₂₃	C₈	1	4	yes	complemented
16	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
17	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
18	C₄²×C₂₂₃	Q₈	1	1	yes	characteristic
19	C₂₂₃⋊C₈	—	4	4	no	—
20	C₂³×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	—	2	8	no	—
22	C₂×C₄×C₂₂₃	—	2	4	no	—
23	C₂×C₄×C₂₂₃	—	2	4	no	—
24	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₂₂₃	M₁₆	1	2	yes	—
30	C₂²×C₂₂₃	—	2	2	no	—
31	C₂²×C₂₂₃	#95	1	1	yes	characteristic
32	C₂²×C₂₂₃	C₂²⋊C₈	1	2	yes	—
33	C₄×C₂₂₃	—	2	8	no	—
34	C₄×C₂₂₃	—	2	2	no	—
35	C₄×C₂₂₃	C₄×C₈	1	1	yes	characteristic
36	C₄×C₂₂₃	#97	1	1	yes	characteristic
37	C₄×C₂₂₃	#104	1	4	yes	—
38	C₄×C₂₂₃	C₄⋊C₈	1	2	yes	—
39	C₂×C₂₂₃	—	2	2	no	—
40	C₂×C₂₂₃	#336	1	1	yes	characteristic
41	C₂×C₂₂₃	#339	1	2	yes	—
42	C₂₂₃	#1276	1	1	yes	characteristic, complemented, Sylow
43	#1276	—	223	1	no	maximal, Sylow
44	#431	—	223	2	no	—
45	C₄³	D₂₂₃	1	1	yes	characteristic
46	C₂×C₄²	—	2	2	no	—
47	C₂×C₄²	Dic₂₂₃	1	2	yes	—
48	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
49	C₄⋊C₈	—	223	4	no	—
50	C₂²⋊C₈	—	446	2	no	—
51	C₄×C₈	—	446	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₂₂₃	1	2	yes	—
55	C₂×C₈	—	446	4	no	—
56	C₄²	—	2	8	no	—
57	C₄²	—	2	2	no	—
58	C₄²	C₂₂₃⋊C₈	1	4	yes	complemented
59	C₄²	C₄×D₂₂₃	1	2	yes	—
60	C₄²	D₈₉₂	1	1	yes	characteristic
61	C₄²	Dic₄₄₆	1	1	yes	characteristic
62	C₂³	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
63	C₂×C₄	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
67	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
68	C₂×C₄	Dic₂₂₃⋊C₄	1	2	yes	—
69	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
70	C₂×C₄	C₂×C₂₂₃⋊C₈	1	2	yes	—
71	C₂×C₄	#16813154	1	2	yes	—
72	C₈	—	892	4	no	—
73	C₂²	—	2	2	no	—
74	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
75	C₂²	C₂²⋊(C₂₂₃⋊C₈)	1	2	yes	—
76	C₄	—	2	8	no	—
77	C₄	—	2	2	no	—
78	C₄	C₄×C₂₂₃⋊C₈	1	1	yes	characteristic
79	C₄	#20297215	1	1	yes	characteristic
80	C₄	#20297232	1	4	yes	—
81	C₄	C₄⋊(C₂₂₃⋊C₈)	1	2	yes	—
82	C₂	—	2	2	no	—
83	C₂	#28383335	1	1	yes	characteristic
84	C₂	#28383340	1	2	yes	—

	subgroup	G/H
center	C₄²	D₈₉₂
commutator	C₄×C₂₂₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₂₃
Fitting	C₄³×C₂₂₃	C₂
socle	C₂²×C₂₂₃	#95
prime radical	C₂³×C₂₂₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₂₂₃	C₂²
hypercenter	C₄³	D₂₂₃
hypocenter	C₂₂₃	#1276
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₂₂₃⋊C₈)	C₄	semidirect
15. C₄²×C₂₂₃	C₈	semidirect
42. C₂₂₃	#1276	semidirect
58. C₄²	C₂₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#28383633	2
C₄³×C₂₂₃	2
#1276	223