Subgroups — #28383589

Order 14272.

150 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
2	#20297263	C₂	1	1	yes	maximal, characteristic
3	#20297228	C₂	1	4	yes	maximal, complemented
4	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	#16813154	—	2	2	no	—
6	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
8	C₂²⋊Dic₂₂₃	—	2	1	no	—
9	C₄×Dic₂₂₃	—	2	1	no	—
10	C₂×Q₈×C₂₂₃	—	2	2	no	—
11	C₂×Q₈×C₂₂₃	—	2	1	no	—
12	C₂×Q₈×C₂₂₃	C₄	1	4	yes	complemented
13	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₂₂₃	—	2	1	no	—
15	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂₂₃⋊C₈	—	2	2	no	—
18	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
19	Q₈×C₂₂₃	—	2	4	no	—
20	Q₈×C₂₂₃	—	2	2	no	—
21	Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₂₂₃	—	2	4	no	—
23	C₂×C₄×C₂₂₃	—	2	2	no	—
24	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
27	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₂₂₃	—	2	2	no	—
29	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₂₃	—	2	4	no	—
32	C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
33	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	Dic₂₂₃	—	4	2	no	—
36	C₂×C₂₂₃	—	2	2	no	—
37	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₂₂₃	#137	1	2	yes	—
39	C₂₂₃	#419	1	1	yes	characteristic, complemented, Sylow
40	#419	—	223	1	no	maximal, Sylow
41	C₂²×Q₈	D₂₂₃	1	1	yes	characteristic
42	C₂×M₁₆	—	223	1	no	—
43	#117	—	223	1	no	—
44	#103	—	223	4	no	—
45	C₂×Q₈	—	2	2	no	—
46	C₂×Q₈	—	2	1	no	—
47	C₂×Q₈	Dic₂₂₃	1	4	yes	complemented
48	C₂×Q₈	C₂×D₂₂₃	1	2	yes	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
51	M₁₆	—	446	2	no	—
52	C₂×C₈	—	223	2	no	—
53	C₄⋊C₄	—	223	2	no	—
54	C₂²⋊C₄	—	446	1	no	—
55	C₄²	—	446	1	no	—
56	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
57	Q₈	—	2	4	no	—
58	Q₈	—	2	2	no	—
59	Q₈	C₂×Dic₂₂₃	1	4	yes	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	446	2	no	—
63	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
64	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
65	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
66	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
67	C₈	—	446	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²⋊Dic₂₂₃	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
71	C₄	—	2	4	no	—
72	C₄	—	892	2	no	—
73	C₄	C₂²⋊Dic₂₂₃	1	2	yes	—
74	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
75	C₄	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×C₂²⋊Dic₂₂₃	1	1	yes	characteristic
78	C₂	#20297335	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂²×Q₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂²×Q₈	D₂₂₃
hypocenter	C₂₂₃	#419
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
3. #20297228	C₂	semidirect
12. C₂×Q₈×C₂₂₃	C₄	semidirect
39. C₂₂₃	#419	semidirect
47. 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×#16813154	2
#20297263	2
#20297228	2
C₂²×Q₈×C₂₂₃	2
#419	223