Subgroups — #28383672

Order 14272.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813160	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#20297269	C₂	1	1	yes	maximal, characteristic
4	#20297237	C₂	1	1	yes	maximal, characteristic
5	#20297229	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
7	#103×C₂₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₄₆	C₂²	1	1	yes	characteristic
9	#16813160	C₄	1	4	yes	complemented
10	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₂₃	—	2	1	no	—
12	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
13	Dic₂₂₃⋊C₄	—	2	1	no	—
14	C₄×Dic₂₂₃	—	2	1	no	—
15	C₄×Dic₂₂₃	—	2	1	no	—
16	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
20	Dic₄₄₆	—	2	1	no	—
21	Dic₄₄₆	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₂₃	—	2	1	no	—
23	C₂×Dic₂₂₃	—	2	1	no	—
24	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
25	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
26	C₂₂₃⋊C₈	C₂×C₄	1	2	yes	—
27	Q₈×C₂₂₃	—	2	1	no	—
28	Q₈×C₂₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₂₂₃	—	2	1	no	—
30	C₂×C₄×C₂₂₃	—	2	1	no	—
31	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
32	C₈×C₂₂₃	—	2	1	no	—
33	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₂₃	—	2	2	no	—
35	C₄×C₂₂₃	—	4	1	no	—
36	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₂₂₃	#41	1	1	yes	characteristic
38	Dic₂₂₃	—	2	2	no	—
39	Dic₂₂₃	—	2	1	no	—
40	Dic₂₂₃	—	4	1	no	—
41	Dic₂₂₃	Q₁₆	1	2	yes	complemented
42	C₂×C₂₂₃	C₄×D₄	1	1	yes	characteristic
43	C₂×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₂₂₃	#135	1	1	yes	characteristic
45	C₂₂₃	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	223	1	no	maximal, Sylow
47	C₂×Q₁₆	—	223	1	no	—
48	C₄×Q₈	—	223	1	no	—
49	C₄×Q₈	—	223	1	no	—
50	#107	—	223	1	no	—
51	#103	—	223	1	no	—
52	#103	D₂₂₃	1	1	yes	characteristic
53	C₄×C₈	—	223	1	no	—
54	C₂×Q₈	—	223	1	no	—
55	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
56	Q₁₆	—	223	4	no	—
57	C₂×C₈	—	223	1	no	—
58	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
59	C₄⋊C₄	—	446	1	no	—
60	C₄⋊C₄	—	446	1	no	—
61	C₄⋊C₄	—	223	1	no	—
62	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
63	C₄²	—	446	1	no	—
64	C₄²	—	446	1	no	—
65	C₄²	—	223	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	446	1	no	—
68	Q₈	—	223	2	no	—
69	Q₈	C₄×D₂₂₃	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	446	1	no	—
73	C₂×C₄	—	446	1	no	—
74	C₂×C₄	—	223	1	no	—
75	C₂×C₄	—	223	1	no	—
76	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	223	2	no	—
79	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	—	892	1	no	—
83	C₄	—	446	2	no	—
84	C₄	—	446	1	no	—
85	C₄	—	223	2	no	—
86	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
87	C₄	#16813169	1	1	yes	characteristic
88	C₂	(C₂²⋊D₂₂₃)⋊C₄	1	1	yes	characteristic
89	C₂	Q₁₆×D₂₂₃	1	1	yes	characteristic
90	C₂	#20297325	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#103×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	#103	D₂₂₃
hypocenter	C₂₂₃	C₄×Q₁₆
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
9. #16813160	C₄	semidirect
41. Dic₂₂₃	Q₁₆	semidirect
45. C₂₂₃	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813160	2
Q₈×Dic₂₂₃	2
#20297269	2
#20297237	2
#20297229	2
C₈×Dic₂₂₃	2
#103×C₂₂₃	2
C₄×Q₁₆	223