Subgroups — #43252668

Order 28544.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383604	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16813154	—	2	2	no	—
4	C₄⋊(C₂₂₃⋊C₈)	—	2	2	no	—
5	C₄⋊(C₂₂₃⋊C₈)	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	#16813154	—	4	2	no	—
10	#16813154	—	4	2	no	—
11	C₂×C₂₂₃⋊C₈	—	2	4	no	—
12	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₂₂₃	—	2	2	no	—
16	C₂×C₈×C₂₂₃	—	4	2	no	—
17	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
18	C₄²×C₂₂₃	D₄	1	2	yes	—
19	C₂₂₃⋊C₈	—	4	4	no	—
20	C₂³×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	—	2	1	no	—
22	C₂×C₄×C₂₂₃	—	2	1	no	—
23	C₂×C₄×C₂₂₃	—	4	1	no	—
24	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	SD₁₆	1	2	yes	—
30	C₂×C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
31	C₈×C₂₂₃	—	4	2	no	—
32	C₂²×C₂₂₃	—	2	1	no	—
33	C₂²×C₂₂₃	—	2	1	no	—
34	C₂²×C₂₂₃	#95	1	1	yes	characteristic
35	C₂²×C₂₂₃	#106	1	1	yes	characteristic
36	C₂²×C₂₂₃	#107	1	1	yes	characteristic
37	C₄×C₂₂₃	—	2	2	no	—
38	C₄×C₂₂₃	—	2	1	no	—
39	C₄×C₂₂₃	#100	1	1	yes	characteristic
40	C₄×C₂₂₃	#101	1	1	yes	characteristic
41	C₄×C₂₂₃	#102	1	2	yes	—
42	C₄×C₂₂₃	#103	1	2	yes	—
43	C₂×C₂₂₃	—	2	1	no	—
44	C₂×C₂₂₃	—	2	1	no	—
45	C₂×C₂₂₃	#338	1	1	yes	characteristic
46	C₂×C₂₂₃	#340	1	1	yes	characteristic
47	C₂×C₂₂₃	#343	1	1	yes	characteristic
48	C₂₂₃	#1292	1	1	yes	characteristic, complemented, Sylow
49	#1292	—	223	1	no	maximal, Sylow
50	#423	—	223	2	no	—
51	C₂×C₄⋊C₈	D₂₂₃	1	1	yes	characteristic
52	C₂×M₁₆	—	446	2	no	—
53	C₂²×C₈	Dic₂₂₃	1	2	yes	—
54	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	446	2	no	—
57	C₄⋊C₈	—	223	4	no	—
58	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
59	C₂²×C₄	D₈₉₂	1	1	yes	characteristic
60	C₂²×C₄	Dic₄₄₆	1	1	yes	characteristic
61	M₁₆	—	892	2	no	—
62	M₁₆	—	892	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	446	4	no	—
66	C₄²	C₄×D₂₂₃	1	2	yes	—
67	C₄²	C₂²⋊D₂₂₃	1	2	yes	—
68	C₂³	C₄⋊Dic₂₂₃	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
73	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
74	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
75	C₂×C₄	Dic₂₂₃⋊C₄	1	2	yes	—
76	C₂×C₄	D₁₇₈₄	1	1	yes	characteristic
77	C₂×C₄	#16813159	1	2	yes	—
78	C₂×C₄	Dic₈₉₂	1	1	yes	characteristic
79	C₈	—	4	2	no	—
80	C₈	—	892	4	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
84	C₂²	#20297236	1	1	yes	characteristic
85	C₂²	#20297238	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	#20297221	1	1	yes	characteristic
89	C₄	#20297223	1	1	yes	characteristic
90	C₄	#20297226	1	2	yes	—
91	C₄	#20297229	1	2	yes	—
92	C₂	—	2	1	no	—
93	C₂	—	2	1	no	—
94	C₂	C₂₂₃⋊#338	1	1	yes	characteristic
95	C₂	#28383343	1	1	yes	characteristic
96	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	#1292
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
48. C₂₂₃	#1292	semidirect

Maximal subgroups

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

subgroup	index
#28383604	2
C₂×C₄⋊C₈×C₂₂₃	2
#1292	223