Subgroups — #28383323

Order 14272.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297301	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₉₂	—	2	1	no	—
5	C₂×D₈₉₂	—	2	1	no	—
6	C₂×D₈₉₂	C₄	1	2	yes	—
7	C₂×C₂₂₃⋊C₈	—	2	1	no	—
8	C₂×C₈×C₂₂₃	—	2	1	no	—
9	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂²×D₂₂₃	—	2	2	no	—
11	D₈₉₂	—	4	2	no	—
12	D₈₉₂	—	4	1	no	—
13	D₈₉₂	—	4	1	no	—
14	D₈₉₂	—	4	1	no	—
15	D₈₉₂	—	4	1	no	—
16	C₂₂₃⋊C₈	—	4	1	no	—
17	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
20	C₈×C₂₂₃	—	4	1	no	—
21	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₂₂₃	—	4	2	no	—
23	C₂×D₂₂₃	—	4	2	no	—
24	C₂×D₂₂₃	—	4	2	no	—
25	C₄×C₂₂₃	—	2	1	no	—
26	C₄×C₂₂₃	D₈	1	1	yes	characteristic
27	C₄×C₂₂₃	D₈	1	1	yes	characteristic
28	C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
29	C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
30	C₂×C₂₂₃	#100	1	1	yes	characteristic
31	C₂×C₂₂₃	#102	1	1	yes	characteristic
32	C₂×C₂₂₃	#102	1	1	yes	characteristic
33	D₂₂₃	—	8	2	no	—
34	C₂₂₃	#331	1	1	yes	characteristic, complemented, Sylow
35	#331	—	223	1	no	maximal, Sylow
36	#127	—	223	1	no	—
37	C₄⋊C₈	—	223	1	no	—
38	C₄⋊C₈	D₂₂₃	1	1	yes	characteristic, complemented
39	C₂×D₄	—	446	1	no	—
40	C₂×D₄	—	446	1	no	—
41	C₂×D₄	—	223	2	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	446	1	no	—
44	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
45	C₂³	—	446	2	no	—
46	D₄	—	892	2	no	—
47	D₄	—	892	1	no	—
48	D₄	—	892	1	no	—
49	D₄	—	892	1	no	—
50	D₄	—	892	1	no	—
51	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
53	C₂×C₄	D₈₉₂	1	1	yes	characteristic
54	C₈	—	4	1	no	—
55	C₈	—	892	1	no	—
56	C₂²	—	892	2	no	—
57	C₂²	—	892	2	no	—
58	C₂²	—	892	2	no	—
59	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#16813155	1	1	yes	characteristic
62	C₄	D₁₇₈₄	1	1	yes	characteristic
63	C₄	#16813157	1	1	yes	characteristic
64	C₄	#16813159	1	1	yes	characteristic
65	C₂	—	1784	2	no	—
66	C₂	#20297220	1	1	yes	characteristic
67	C₂	#20297224	1	1	yes	characteristic
68	C₂	#20297226	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₂₂₃)⋊C₄
commutator	C₂×C₄×C₂₂₃	C₂×C₄
Frattini	C₄²	C₂×D₂₂₃
Fitting	C₄⋊C₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₄
prime radical	#20297301	C₂
cosocle	C₄²×C₂₂₃	C₂²
hypercenter	C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#331
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₂₂₃⋊C₈)	C₂	semidirect
3. C₄⋊C₈×C₂₂₃	C₂	semidirect
34. C₂₂₃	#331	semidirect
38. C₄⋊C₈	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297301	2
C₄⋊(C₂₂₃⋊C₈)	2
C₄⋊C₈×C₂₂₃	2
#331	223