Subgroups — #28383487

Order 14272.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₂₃⋊#95	C₂	1	4	yes	maximal
4	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	C₂²	1	2	yes	—
6	C₂²×Dic₂₂₃	C₂²	1	2	yes	—
7	C₄⋊Dic₂₂₃	—	2	2	no	—
8	C₄×Dic₂₂₃	—	2	2	no	—
9	C₂×Q₈×C₂₂₃	—	2	2	no	—
10	C₂×Q₈×C₂₂₃	—	2	2	no	—
11	C₂×Q₈×C₂₂₃	C₄	1	4	yes	complemented
12	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
13	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂×Dic₂₂₃	—	2	2	no	—
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂×Dic₂₂₃	—	2	4	no	—
18	C₂×Dic₂₂₃	Q₈	1	4	yes	complemented
19	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
20	Q₈×C₂₂₃	—	2	8	no	—
21	C₂×C₄×C₂₂₃	—	2	2	no	—
22	C₂×C₄×C₂₂₃	—	2	2	no	—
23	C₂×C₄×C₂₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
27	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₂₂₃	C₂×Q₈	1	2	yes	—
31	C₂²×C₂₂₃	#41	1	2	yes	—
32	C₄×C₂₂₃	—	2	4	no	—
33	C₄×C₂₂₃	C₂²⋊C₄	1	4	yes	—
34	Dic₂₂₃	—	2	4	no	—
35	Dic₂₂₃	—	4	2	no	—
36	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₂₂₃	C₄×Q₈	1	2	yes	—
38	C₂×C₂₂₃	C₂²⋊Q₈	1	2	yes	—
39	C₂×C₂₂₃	#124	1	1	yes	characteristic
40	C₂×C₂₂₃	C₄⋊Q₈	1	1	yes	characteristic
41	C₂₂₃	#391	1	1	yes	characteristic, complemented, Sylow
42	#391	—	223	1	no	maximal, Sylow
43	C₂²×Q₈	D₂₂₃	1	1	yes	characteristic
44	C₂×C₄⋊C₄	—	223	1	no	—
45	C₂×C₄²	—	223	1	no	—
46	#95	—	223	4	no	—
47	C₂×Q₈	—	2	2	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	Dic₂₂₃	1	4	yes	complemented
50	C₂²×C₄	—	223	2	no	—
51	C₂²×C₄	—	223	2	no	—
52	C₂²×C₄	C₂×D₂₂₃	1	2	yes	—
53	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
54	C₄⋊C₄	—	446	2	no	—
55	C₄²	—	446	2	no	—
56	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
57	Q₈	—	2	8	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	446	2	no	—
61	C₂×C₄	—	446	4	no	—
62	C₂×C₄	—	446	2	no	—
63	C₂×C₄	—	446	2	no	—
64	C₂×C₄	—	223	4	no	—
65	C₂×C₄	C₂×Dic₂₂₃	1	4	yes	—
66	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
67	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
68	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
69	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
70	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
71	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
72	C₂²	Q₈×D₂₂₃	1	2	yes	—
73	C₂²	#16813171	1	2	yes	—
74	C₄	—	2	4	no	—
75	C₄	—	892	2	no	—
76	C₄	—	446	4	no	—
77	C₄	C₂²⋊Dic₂₂₃	1	4	yes	—
78	C₂	C₂×C₂²⋊Dic₂₂₃	1	1	yes	characteristic
79	C₂	Q₈×Dic₂₂₃	1	2	yes	—
80	C₂	#20297280	1	2	yes	—
81	C₂	#20297293	1	1	yes	characteristic
82	C₂	#20297302	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₂₃
commutator	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₂²×Q₈×C₂₂₃	C₂
socle, prime radical, cosocle	C₂³×C₂₂₃	C₂³
hypercenter	C₂²×Q₈	D₂₂₃
hypocenter	C₂₂₃	#391
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
11. C₂×Q₈×C₂₂₃	C₄	semidirect
18. C₂×Dic₂₂₃	Q₈	semidirect
41. C₂₂₃	#391	semidirect
49. C₂×Q₈	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₂₃	2
C₂×C₄×Dic₂₂₃	2
C₂₂₃⋊#95	2
C₂²×Q₈×C₂₂₃	2
#391	223