Subgroups — #28383887

Order 14272.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297296	C₂	1	1	yes	maximal, characteristic
2	#20297237	C₂	1	1	yes	maximal, characteristic
3	#20297235	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#20297228	C₂	1	1	yes	maximal, characteristic
6	#20297228	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
12	Dic₂₂₃⋊C₄	—	2	1	no	—
13	Dic₂₂₃⋊C₄	—	2	1	no	—
14	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₂₂₃	—	2	1	no	—
16	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₄²×C₂₂₃	—	2	1	no	—
18	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₂₃	—	2	1	no	—
20	C₂×Dic₂₂₃	—	2	1	no	—
21	C₂₂₃⋊C₈	—	2	1	no	—
22	C₂₂₃⋊C₈	—	2	1	no	—
23	Q₈×C₂₂₃	—	2	1	no	—
24	Q₈×C₂₂₃	Q₈	1	2	yes	—
25	C₂×C₄×C₂₂₃	—	2	1	no	—
26	C₂×C₄×C₂₂₃	—	2	1	no	—
27	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₂₃	—	2	2	no	—
32	C₄×C₂₂₃	—	2	1	no	—
33	C₄×C₂₂₃	—	2	1	no	—
34	C₄×C₂₂₃	—	4	1	no	—
35	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₂₂₃	#41	1	1	yes	characteristic
37	Dic₂₂₃	—	4	1	no	—
38	Dic₂₂₃	—	4	1	no	—
39	C₂×C₂₂₃	C₂²⋊Q₈	1	1	yes	characteristic
40	C₂×C₂₂₃	#135	1	1	yes	characteristic
41	C₂×C₂₂₃	#137	1	1	yes	characteristic
42	C₂₂₃	#479	1	1	yes	characteristic, complemented, Sylow
43	#479	—	223	1	no	maximal, Sylow
44	#125	—	223	1	no	—
45	C₄×Q₈	D₂₂₃	1	1	yes	characteristic
46	#107	—	223	1	no	—
47	#106	—	223	1	no	—
48	C₄⋊C₈	—	223	1	no	—
49	#103	—	223	1	no	—
50	#103	—	223	1	no	—
51	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
52	C₂×C₈	—	223	1	no	—
53	C₂×C₈	—	223	1	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	446	1	no	—
56	C₄⋊C₄	—	446	1	no	—
57	C₄⋊C₄	—	223	1	no	—
58	C₄⋊C₄	—	223	1	no	—
59	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
60	C₄²	—	2	1	no	—
61	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
62	Q₈	—	2	1	no	—
63	Q₈	Dic₄₄₆	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	446	1	no	—
67	C₂×C₄	—	446	1	no	—
68	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
71	C₈	—	446	1	no	—
72	C₈	—	446	1	no	—
73	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	892	1	no	—
79	C₄	—	892	1	no	—
80	C₄	C₂×Dic₄₄₆	1	1	yes	characteristic
81	C₄	#16813170	1	1	yes	characteristic
82	C₂	#20297283	1	1	yes	characteristic
83	C₂	#20297324	1	1	yes	characteristic
84	C₂	#20297335	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₄×Q₈×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂×D₄
hypercenter	C₄×Q₈	D₂₂₃
hypocenter	C₂₂₃	#479
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
42. C₂₂₃	#479	semidirect

Maximal subgroups

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

subgroup	index
#20297296	2
#20297237	2
#20297235	2
C₄⋊(C₂₂₃⋊C₈)	2
#20297228	2
C₄×Q₈×C₂₂₃	2
#479	223