Subgroups — #28383936

Order 14272.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297295	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297293	C₂	1	1	yes	maximal, characteristic
3	#20297228	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297226	C₂	1	1	yes	maximal, characteristic
5	#20297224	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
9	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₂₃	—	2	1	no	—
11	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
12	Dic₂₂₃⋊C₄	—	2	1	no	—
13	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
14	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
15	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₂²×D₂₂₃	—	2	1	no	—
20	D₈₉₂	—	2	1	no	—
21	D₈₉₂	—	2	1	no	—
22	C₂×Dic₂₂₃	—	2	1	no	—
23	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
24	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
25	C₂₂₃⋊C₈	—	2	1	no	—
26	Q₈×C₂₂₃	—	2	1	no	—
27	Q₈×C₂₂₃	—	2	1	no	—
28	C₂×C₄×C₂₂₃	—	2	1	no	—
29	C₂×C₄×C₂₂₃	—	2	1	no	—
30	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
31	C₈×C₂₂₃	—	2	1	no	—
32	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×D₂₂₃	—	4	1	no	—
34	C₂×D₂₂₃	—	4	1	no	—
35	C₂×D₂₂₃	—	4	1	no	—
36	C₄×C₂₂₃	—	4	1	no	—
37	C₄×C₂₂₃	—	4	1	no	—
38	C₄×C₂₂₃	#41	1	1	yes	characteristic
39	C₄×C₂₂₃	#41	1	1	yes	characteristic
40	Dic₂₂₃	—	2	1	no	—
41	Dic₂₂₃	—	2	1	no	—
42	Dic₂₂₃	—	4	1	no	—
43	C₂×C₂₂₃	#124	1	1	yes	characteristic
44	C₂×C₂₂₃	#135	1	1	yes	characteristic
45	C₂×C₂₂₃	#135	1	1	yes	characteristic
46	D₂₂₃	—	8	1	no	—
47	C₂₂₃	#488	1	1	yes	characteristic, complemented, Sylow
48	#488	—	223	1	no	maximal, Sylow
49	#125	—	223	1	no	—
50	#124	—	223	1	no	—
51	#103	—	223	1	no	—
52	#103	D₂₂₃	1	1	yes	characteristic, complemented
53	#102	—	223	1	no	—
54	#102	—	223	1	no	—
55	C₄×C₈	—	223	1	no	—
56	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
57	C₂×D₄	—	223	1	no	—
58	C₂×C₈	—	223	1	no	—
59	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
60	C₄⋊C₄	—	446	1	no	—
61	C₄⋊C₄	—	446	1	no	—
62	C₄⋊C₄	—	223	1	no	—
63	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	446	1	no	—
65	C₂²⋊C₄	—	446	1	no	—
66	C₄²	—	223	1	no	—
67	C₂³	—	446	1	no	—
68	Q₈	—	2	1	no	—
69	Q₈	—	2	1	no	—
70	D₄	—	446	1	no	—
71	D₄	—	446	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	446	1	no	—
75	C₂×C₄	—	223	1	no	—
76	C₂×C₄	—	223	1	no	—
77	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	446	1	no	—
80	C₂²	—	892	1	no	—
81	C₂²	—	892	1	no	—
82	C₂²	—	892	1	no	—
83	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	892	1	no	—
87	C₄	—	446	1	no	—
88	C₄	—	446	1	no	—
89	C₄	#16813169	1	1	yes	characteristic
90	C₄	#16813170	1	1	yes	characteristic
91	C₂	—	1784	1	no	—
92	C₂	#20297290	1	1	yes	characteristic
93	C₂	#20297325	1	1	yes	characteristic
94	C₂	#20297327	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#103×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂×D₈₉₂	C₂²
hypercenter	#103	D₂₂₃
hypocenter	C₂₂₃	#488
LCS	G ▷ C₂×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
1. #20297295	C₂	semidirect
3. #20297228	C₂	semidirect
6. C₈×Dic₂₂₃	C₂	semidirect
7. #103×C₂₂₃	C₂	semidirect
47. C₂₂₃	#488	semidirect
52. #103	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297295	2
#20297293	2
#20297228	2
#20297226	2
#20297224	2
C₈×Dic₂₂₃	2
#103×C₂₂₃	2
#488	223