Subgroups — #28383585

Order 14272.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813154	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297263	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297225	C₂	1	4	yes	maximal, complemented
4	C₂²×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	#16813154	—	2	2	no	—
6	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	complemented
7	C₄⋊Dic₂₂₃	C₂²	1	2	yes	complemented
8	C₂²⋊Dic₂₂₃	—	2	1	no	—
9	C₄×Dic₂₂₃	—	2	1	no	—
10	C₂⁴×C₂₂₃	—	2	1	no	—
11	C₂×D₄×C₂₂₃	—	2	1	no	—
12	C₂×D₄×C₂₂₃	—	2	2	no	—
13	C₂×D₄×C₂₂₃	C₄	1	4	yes	complemented
14	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂₂₃⋊C₈	—	2	2	no	—
18	C₂³×C₂₂₃	—	2	4	no	—
19	C₂³×C₂₂₃	—	2	2	no	—
20	C₂³×C₂₂₃	—	4	4	no	—
21	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
22	D₄×C₂₂₃	—	2	4	no	—
23	D₄×C₂₂₃	—	2	2	no	—
24	D₄×C₂₂₃	C₂×C₄	1	4	yes	—
25	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
28	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₂₂₃	—	2	2	no	—
30	C₂²×C₂₂₃	—	2	4	no	—
31	C₂²×C₂₂₃	—	4	8	no	—
32	C₂²×C₂₂₃	—	4	4	no	—
33	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
36	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
38	Dic₂₂₃	—	4	2	no	—
39	C₂×C₂₂₃	—	2	2	no	—
40	C₂×C₂₂₃	—	4	4	no	—
41	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₂₂₃	AGL(1,ℤ/8)	1	2	yes	—
43	C₂₂₃	#418	1	1	yes	characteristic, complemented, Sylow
44	#418	—	223	1	no	maximal, Sylow
45	C₂²×D₄	D₂₂₃	1	1	yes	characteristic
46	C₂×M₁₆	—	223	1	no	—
47	#117	—	223	1	no	—
48	#102	—	223	4	no	—
49	C₂⁴	—	2	1	no	—
50	C₂×D₄	—	2	1	no	—
51	C₂×D₄	—	2	2	no	—
52	C₂×D₄	Dic₂₂₃	1	4	yes	complemented
53	C₂×D₄	C₂×D₂₂₃	1	2	yes	—
54	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
55	M₁₆	—	446	2	no	—
56	C₂×C₈	—	223	2	no	—
57	C₄⋊C₄	—	223	2	no	—
58	C₂²⋊C₄	—	446	1	no	—
59	C₄²	—	446	1	no	—
60	C₂³	—	2	4	no	—
61	C₂³	—	2	2	no	—
62	C₂³	—	4	4	no	—
63	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
64	D₄	—	2	4	no	—
65	D₄	—	2	2	no	—
66	D₄	C₂×Dic₂₂₃	1	4	yes	—
67	C₂×C₄	—	446	2	no	—
68	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
69	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
70	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
71	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
72	C₈	—	446	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	4	8	no	—
76	C₂²	—	4	4	no	—
77	C₂²	C₂²⋊Dic₂₂₃	1	2	yes	—
78	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
79	C₄	—	892	2	no	—
80	C₄	C₂²⋊Dic₂₂₃	1	2	yes	—
81	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
82	C₄	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	4	4	no	—
85	C₂	C₂×C₂²⋊Dic₂₂₃	1	1	yes	characteristic
86	C₂	#20297330	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting, prime radical	C₂²×D₄×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂²×D₄	D₂₂₃
hypocenter	C₂₂₃	#418
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16813154	C₂	semidirect
2. #20297263	C₂	semidirect
3. #20297225	C₂	semidirect
6. C₂×C₂₂₃⋊C₈	C₂²	semidirect
7. C₄⋊Dic₂₂₃	C₂²	semidirect
13. C₂×D₄×C₂₂₃	C₄	semidirect
43. C₂₂₃	#418	semidirect
52. C₂×D₄	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813154	2
#20297263	2
#20297225	2
C₂²×D₄×C₂₂₃	2
#418	223