Subgroups — #28383970

Order 14272.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813157	C₂	1	2	yes	maximal, complemented
2	C₂×#16813155	C₂	1	2	yes	maximal, complemented
3	#20297301	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297215	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₉₂	—	2	2	no	—
7	C₂×D₈₉₂	C₂²	1	2	yes	—
8	#16813157	—	2	4	no	—
9	#16813155	—	2	4	no	—
10	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
11	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₂₂₃	—	2	2	no	—
14	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂²×D₂₂₃	—	2	2	no	—
16	D₈₉₂	—	2	2	no	—
17	D₈₉₂	—	2	2	no	—
18	D₈₉₂	—	4	4	no	—
19	C₂₂₃⋊C₈	D₄	1	4	yes	complemented
20	C₂³×C₂₂₃	—	2	1	no	—
21	Q₈×C₂₂₃	—	2	2	no	—
22	D₄×C₂₂₃	—	2	2	no	—
23	C₂×C₄×C₂₂₃	—	2	1	no	—
24	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
25	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₂₂₃	—	4	1	no	—
27	C₂²×C₂₂₃	—	4	2	no	—
28	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₂×D₂₂₃	—	4	2	no	—
30	C₂×D₂₂₃	—	4	4	no	—
31	C₄×C₂₂₃	—	2	2	no	—
32	C₄×C₂₂₃	—	4	1	no	—
33	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
34	C₂×C₂₂₃	—	8	1	no	—
35	C₂×C₂₂₃	#127	1	1	yes	characteristic
36	C₂×C₂₂₃	AGL(1,ℤ/8)	1	2	yes	—
37	D₂₂₃	—	8	2	no	—
38	C₂₂₃	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	223	1	no	maximal, Sylow
40	C₂×SD₁₆	—	223	2	no	—
41	C₂×D₈	—	223	2	no	—
42	#127	—	223	1	no	—
43	#124	D₂₂₃	1	1	yes	characteristic, complemented
44	#97	—	223	1	no	—
45	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
46	C₂×D₄	—	446	2	no	—
47	C₂×D₄	—	223	2	no	—
48	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
49	SD₁₆	—	446	4	no	—
50	D₈	—	446	4	no	—
51	C₂×C₈	—	223	2	no	—
52	C₂²⋊C₄	—	2	2	no	—
53	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	C₂³	—	446	2	no	—
56	Q₈	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	892	4	no	—
59	D₄	—	446	2	no	—
60	D₄	—	446	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
63	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
64	C₈	—	223	4	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	892	2	no	—
68	C₂²	—	892	4	no	—
69	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	4	1	no	—
72	C₄	D₄×D₂₂₃	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	—	1784	2	no	—
75	C₂	#20297300	1	1	yes	characteristic
76	C₂	#20297329	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#124×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
hypercenter	#124	D₂₂₃
hypocenter	C₂₂₃	#496
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#16813157	C₂	semidirect
2. C₂×#16813155	C₂	semidirect
3. #20297301	C₂	semidirect
4. #20297215	C₂	semidirect
5. #124×C₂₂₃	C₂	semidirect
19. C₂₂₃⋊C₈	D₄	semidirect
38. C₂₂₃	#496	semidirect
43. #124	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813157	2
C₂×#16813155	2
#20297301	2
#20297215	2
#124×C₂₂₃	2
#496	223