Subgroups — #28383305

Order 14272.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₂₃	C₄	1	1	yes	characteristic
5	C₂²×Dic₂₂₃	C₄	1	1	yes	characteristic
6	C₂×C₂₂₃⋊C₈	—	2	1	no	—
7	Dic₂₂₃⋊C₄	—	2	1	no	—
8	Dic₂₂₃⋊C₄	—	2	1	no	—
9	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₂₂₃	—	2	1	no	—
11	C₂×Dic₂₂₃	—	2	1	no	—
12	C₂×Dic₂₂₃	—	2	1	no	—
13	C₂×Dic₂₂₃	—	2	1	no	—
14	C₂×Dic₂₂₃	—	4	1	no	—
15	C₂×Dic₂₂₃	C₈	1	2	yes	complemented
16	C₂₂₃⋊C₈	—	4	1	no	—
17	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	—	4	1	no	—
19	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
21	C₈×C₂₂₃	—	4	1	no	—
22	C₂²×C₂₂₃	—	2	1	no	—
23	C₂²×C₂₂₃	—	2	1	no	—
24	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₂₂₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₂₂₃	M₁₆	1	1	yes	characteristic
27	C₄×C₂₂₃	—	2	1	no	—
28	C₄×C₂₂₃	—	2	1	no	—
29	Dic₂₂₃	—	2	2	no	—
30	Dic₂₂₃	—	4	1	no	—
31	C₂×C₂₂₃	—	2	1	no	—
32	C₂×C₂₂₃	—	2	1	no	—
33	C₂×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₂₂₃	#99	1	1	yes	characteristic
35	C₂×C₂₂₃	#101	1	1	yes	characteristic
36	C₂₂₃	#324	1	1	yes	characteristic, complemented, Sylow
37	#324	—	223	1	no	maximal, Sylow
38	C₂×C₄⋊C₄	—	223	1	no	—
39	C₂²⋊C₈	—	223	1	no	—
40	C₂²⋊C₈	D₂₂₃	1	1	yes	characteristic
41	C₂²×C₄	—	223	1	no	—
42	C₂²×C₄	—	223	1	no	—
43	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	446	1	no	—
46	C₄⋊C₄	—	446	1	no	—
47	C₄⋊C₄	—	446	1	no	—
48	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	892	1	no	—
51	C₂×C₄	—	446	1	no	—
52	C₂×C₄	—	446	1	no	—
53	C₂×C₄	—	446	1	no	—
54	C₂×C₄	—	223	2	no	—
55	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
56	C₂×C₄	D₈₉₂	1	1	yes	characteristic
57	C₈	—	4	1	no	—
58	C₈	—	892	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
62	C₂²	C₈×D₂₂₃	1	1	yes	characteristic
63	C₂²	#16813153	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	2	1	no	—
66	C₄	—	892	1	no	—
67	C₄	—	446	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	2	1	no	—
70	C₂	(C₂×D₂₂₃)⋊C₈	1	1	yes	characteristic
71	C₂	#20297218	1	1	yes	characteristic
72	C₂	#20297222	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₂₂₃)⋊C₄
commutator	C₂²×C₂₂₃	C₂×C₈
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	C₂²⋊C₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₄
prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂²⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#324
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
15. C₂×Dic₂₂₃	C₈	semidirect
36. C₂₂₃	#324	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₂₃⋊C₄	2
C₂²⋊(C₂₂₃⋊C₈)	2
C₂²⋊C₈×C₂₂₃	2
#324	223