Subgroups — #28383709

Order 14272.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₄₄₆	C₂	1	1	yes	maximal, characteristic
2	Dic₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	Dic₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#20297215	C₂	1	1	yes	maximal, characteristic
5	#20297214	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₄₆	C₄	1	1	yes	characteristic
9	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₂₃	C₄	1	1	yes	characteristic
12	Dic₂₂₃⋊C₄	C₄	1	1	yes	characteristic
13	Dic₂₂₃⋊C₄	C₄	1	1	yes	characteristic
14	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
19	Dic₄₄₆	—	2	2	no	—
20	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂₂₃⋊C₈	—	2	1	no	—
25	C₂₂₃⋊C₈	Q₈	1	2	yes	complemented
26	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
29	C₈×C₂₂₃	—	2	1	no	—
30	C₈×C₂₂₃	—	2	1	no	—
31	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₂₂₃	—	2	1	no	—
33	C₄×C₂₂₃	—	2	1	no	—
34	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₂₂₃	#41	1	1	yes	characteristic
36	Dic₂₂₃	—	2	1	no	—
37	Dic₂₂₃	—	2	1	no	—
38	Dic₂₂₃	—	2	1	no	—
39	Dic₂₂₃	M₁₆	1	2	yes	—
40	C₂×C₂₂₃	C₄×Q₈	1	1	yes	characteristic
41	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₂₂₃	#131	1	1	yes	characteristic
43	C₂₂₃	#446	1	1	yes	characteristic, complemented, Sylow
44	#446	—	223	1	no	maximal, Sylow
45	C₄×Q₈	—	223	1	no	—
46	C₄⋊C₈	—	223	1	no	—
47	C₄⋊C₈	—	223	1	no	—
48	C₄⋊C₈	D₂₂₃	1	1	yes	characteristic
49	#97	—	223	1	no	—
50	#97	—	223	1	no	—
51	C₄×C₈	—	223	1	no	—
52	C₂×Q₈	—	223	1	no	—
53	C₂×C₈	—	223	1	no	—
54	C₂×C₈	—	223	1	no	—
55	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
56	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
57	C₄⋊C₄	—	223	1	no	—
58	C₄⋊C₄	—	223	1	no	—
59	C₄⋊C₄	—	223	1	no	—
60	C₄²	—	223	1	no	—
61	C₄²	—	223	1	no	—
62	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
63	Q₈	—	446	2	no	—
64	C₂×C₄	—	223	1	no	—
65	C₂×C₄	—	223	1	no	—
66	C₂×C₄	—	223	1	no	—
67	C₂×C₄	—	223	1	no	—
68	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
69	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	—	446	1	no	—
74	C₈	—	223	2	no	—
75	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	446	1	no	—
79	C₄	—	446	1	no	—
80	C₄	—	446	1	no	—
81	C₄	—	223	2	no	—
82	C₄	Q₈×D₂₂₃	1	1	yes	characteristic
83	C₄	#16813169	1	1	yes	characteristic
84	C₂	#20297269	1	1	yes	characteristic
85	C₂	M₁₆×D₂₂₃	1	1	yes	characteristic
86	C₂	#20297310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator, socle, prime radical	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₄⋊C₈×C₂₂₃	C₂
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#446
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
25. C₂₂₃⋊C₈	Q₈	semidirect
43. C₂₂₃	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₄₄₆	2
Dic₂₂₃⋊C₈	2
#20297215	2
#20297214	2
C₈×Dic₂₂₃	2
C₄⋊C₈×C₂₂₃	2
#446	223