Subgroups — #43252737

Order 28544.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383306	C₂	1	2	yes	maximal
2	#382×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₂₂₃⋊C₈)	—	2	2	no	—
4	C₂×C₄⋊C₄×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₂₂₃	C₄	1	1	yes	characteristic
7	#95×C₂₂₃	—	2	2	no	—
8	C₂×C₂₂₃⋊C₈	—	4	2	no	—
9	C₂²×C₄×C₂₂₃	—	2	1	no	—
10	C₂²×C₄×C₂₂₃	—	2	1	no	—
11	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	D₄	1	2	yes	—
13	C₄⋊C₄×C₂₂₃	—	4	1	no	—
14	C₄⋊C₄×C₂₂₃	—	4	1	no	—
15	C₄²×C₂₂₃	—	2	1	no	—
16	C₄²×C₂₂₃	C₈	1	2	yes	complemented
17	C₂₂₃⋊C₈	—	8	2	no	—
18	C₂³×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	—	2	1	no	—
20	C₂×C₄×C₂₂₃	—	2	1	no	—
21	C₂×C₄×C₂₂₃	—	2	2	no	—
22	C₂×C₄×C₂₂₃	—	2	2	no	—
23	C₂×C₄×C₂₂₃	—	4	1	no	—
24	C₂×C₄×C₂₂₃	—	4	1	no	—
25	C₂×C₄×C₂₂₃	—	4	2	no	—
26	C₂×C₄×C₂₂₃	—	4	2	no	—
27	C₂×C₄×C₂₂₃	—	4	1	no	—
28	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	M₁₆	1	1	yes	characteristic
30	C₂²×C₂₂₃	—	2	1	no	—
31	C₂²×C₂₂₃	—	2	1	no	—
32	C₂²×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₂²×C₂₂₃	#99	1	1	yes	characteristic
34	C₂²×C₂₂₃	#101	1	1	yes	characteristic
35	C₄×C₂₂₃	—	2	1	no	—
36	C₄×C₂₂₃	—	2	1	no	—
37	C₄×C₂₂₃	—	4	2	no	—
38	C₄×C₂₂₃	—	4	2	no	—
39	C₄×C₂₂₃	—	8	1	no	—
40	C₂×C₂₂₃	—	2	1	no	—
41	C₂×C₂₂₃	—	2	1	no	—
42	C₂×C₂₂₃	#324	1	1	yes	characteristic
43	C₂×C₂₂₃	#354	1	1	yes	characteristic
44	C₂×C₂₂₃	#355	1	1	yes	characteristic
45	C₂₂₃	#1324	1	1	yes	characteristic, complemented, Sylow
46	#1324	—	223	1	no	maximal, Sylow
47	#382	D₂₂₃	1	1	yes	characteristic
48	#324	—	223	2	no	—
49	C₂×C₄⋊C₄	Dic₂₂₃	1	1	yes	characteristic
50	C₂×C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂×C₄²	Dic₂₂₃	1	1	yes	characteristic
52	C₂²⋊C₈	—	446	2	no	—
53	#95	—	2	2	no	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
57	C₂²×C₄	C₂²⋊D₂₂₃	1	2	yes	—
58	C₂×C₈	—	892	2	no	—
59	C₄⋊C₄	—	4	1	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄²	—	2	1	no	—
62	C₄²	C₂₂₃⋊C₈	1	2	yes	complemented
63	C₂³	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
74	C₂×C₄	#16813154	1	1	yes	characteristic
75	C₈	—	1784	2	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	C₂²⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic
79	C₂²	#20297219	1	1	yes	characteristic
80	C₂²	#20297223	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	2	no	—
84	C₄	—	4	2	no	—
85	C₄	—	8	1	no	—
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#28383306	1	1	yes	characteristic
89	C₂	#28383375	1	1	yes	characteristic
90	C₂	#28383377	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20297219
commutator	C₂×C₄×C₂₂₃	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₂₂₃
Fitting	#382×C₂₂₃	C₂
socle	C₂²×C₂₂₃	#99
prime radical	C₂³×C₂₂₃	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₂₂₃	C₂²
hypercenter	#382	D₂₂₃
hypocenter	C₂₂₃	#1324
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #382

Complements

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

N	H ≅ G/N	type
16. C₄²×C₂₂₃	C₈	semidirect
45. C₂₂₃	#1324	semidirect
62. 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
#28383306	2
#382×C₂₂₃	2
#1324	223