Subgroups — #28384063

Order 14272.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₂₃	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₂₂₃	C₂	1	6	yes	maximal, complemented
3	#20297263	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊Dic₂₂₃	C₂	1	3	yes	maximal, complemented
5	C₂×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂₃	C₂²	1	6	yes	—
7	C₄⋊Dic₂₂₃	C₂²	1	3	yes	—
8	C₄⋊Dic₂₂₃	C₂²	1	6	yes	—
9	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
10	C₂²⋊Dic₂₂₃	C₂²	1	6	yes	—
11	C₄×Dic₂₂₃	C₂²	1	6	yes	—
12	#41×C₂₂₃	C₄	1	8	yes	complemented
13	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₂₂₃	C₂²	1	3	yes	—
15	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
16	C₂×Dic₂₂₃	—	4	6	no	—
17	C₂×Dic₂₂₃	C₂³	1	2	yes	—
18	C₂×Dic₂₂₃	C₂³	1	6	yes	—
19	C₂³×C₂₂₃	C₂³	1	3	yes	—
20	Q₈×C₂₂₃	C₂×C₄	1	4	yes	complemented
21	D₄×C₂₂₃	C₂×C₄	1	12	yes	complemented
22	C₂×C₄×C₂₂₃	C₂×C₄	1	12	yes	complemented
23	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
25	C₂²×C₂₂₃	—	2	3	no	—
26	C₂²×C₂₂₃	—	2	3	no	—
27	C₂²×C₂₂₃	C₂²×C₄	1	6	yes	—
28	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
29	C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
30	C₄×C₂₂₃	C₂²×C₄	1	6	yes	—
31	Dic₂₂₃	—	2	6	no	—
32	Dic₂₂₃	—	2	2	no	—
33	C₂×C₂₂₃	—	2	6	no	—
34	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
35	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
36	C₂×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
37	C₂₂₃	#520	1	1	yes	characteristic, complemented, Sylow
38	#520	—	223	1	no	maximal, Sylow
39	C₂×#41	D₂₂₃	1	1	yes	characteristic
40	C₄×Q₈	—	223	2	no	—
41	C₄×D₄	—	223	6	no	—
42	#117	—	223	3	no	—
43	C₂×C₄⋊C₄	—	223	3	no	—
44	#41	Dic₂₂₃	1	8	yes	complemented
45	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
46	C₂×D₄	C₂×D₂₂₃	1	3	yes	—
47	C₂²×C₄	—	223	6	no	—
48	C₂²×C₄	C₂×D₂₂₃	1	3	yes	—
49	C₄⋊C₄	—	223	3	no	—
50	C₄⋊C₄	—	223	6	no	—
51	C₄⋊C₄	—	223	1	no	—
52	C₂²⋊C₄	—	223	6	no	—
53	C₄²	—	223	6	no	—
54	C₂³	C₂²×D₂₂₃	1	3	yes	—
55	Q₈	C₂×Dic₂₂₃	1	4	yes	complemented
56	D₄	C₂×Dic₂₂₃	1	12	yes	complemented
57	C₂×C₄	—	892	6	no	—
58	C₂×C₄	—	223	2	no	—
59	C₂×C₄	—	223	6	no	—
60	C₂×C₄	C₂×Dic₂₂₃	1	12	yes	complemented
61	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
62	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
63	C₂²	—	2	3	no	—
64	C₂²	—	2	3	no	—
65	C₂²	C₂²×Dic₂₂₃	1	6	yes	—
66	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
67	C₄	—	446	6	no	—
68	C₄	—	446	2	no	—
69	C₄	C₂²×Dic₂₂₃	1	2	yes	—
70	C₄	C₂²×Dic₂₂₃	1	6	yes	—
71	C₂	—	2	6	no	—
72	C₂	C₂³×Dic₂₂₃	1	1	yes	characteristic
73	C₂	#20297350	1	1	yes	characteristic
74	C₂	#20297351	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator	C₂×C₂₂₃	C₂³×C₄
Fitting, prime radical	C₂×#41×C₂₂₃	C₂
socle, cosocle	C₂²×C₂₂₃	C₂⁴
hypercenter	C₂×#41	D₂₂₃
hypocenter	C₂₂₃	#520
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₂₃	C₂	semidirect
2. D₄×Dic₂₂₃	C₂	semidirect
3. #20297263	C₂	semidirect
4. C₂×C₄⋊Dic₂₂₃	C₂	semidirect
12. #41×C₂₂₃	C₄	semidirect
20. Q₈×C₂₂₃	C₂×C₄	semidirect
21. D₄×C₂₂₃	C₂×C₄	semidirect
22. C₂×C₄×C₂₂₃	C₂×C₄	semidirect
37. C₂₂₃	#520	semidirect
44. #41	Dic₂₂₃	semidirect
55. Q₈	C₂×Dic₂₂₃	semidirect
56. D₄	C₂×Dic₂₂₃	semidirect
60. C₂×C₄	C₂×Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₂₃	2
D₄×Dic₂₂₃	2
#20297263	2
C₂×C₄⋊Dic₂₂₃	2
C₂×#41×C₂₂₃	2
#520	223