Subgroups — #28417309

Order 14912.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11745400	C₂	1	2	yes	maximal
2	C₂×C₈×D₂₃₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11745400	—	2	2	no	—
4	C₄⋊#11745400	—	2	2	no	—
5	C₄⋊#11745400	C₄	1	4	yes	complemented
6	C₂×C₄×D₂₃₃	C₂²	1	1	yes	characteristic
7	C₂×#14272338	C₄	1	1	yes	characteristic, complemented
8	C₈×D₂₃₃	—	2	2	no	—
9	C₂×C₈×C₂₃₃	C₄	1	1	yes	characteristic, complemented
10	C₂×#11745400	—	2	4	no	—
11	C₂×#11745400	—	4	2	no	—
12	C₂×#11745400	—	4	2	no	—
13	C₂²×D₂₃₃	D₄	1	1	yes	characteristic
14	C₂×Dic₂₃₃	Q₈	1	1	yes	characteristic
15	C₄×D₂₃₃	C₂×C₄	1	2	yes	—
16	C₄×D₂₃₃	D₄	1	2	yes	—
17	#14272338	—	2	1	no	—
18	C₂×C₄×C₂₃₃	C₂×C₄	1	1	yes	characteristic
19	C₈×C₂₃₃	—	2	1	no	—
20	C₂²×C₂₃₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×D₂₃₃	C₂²⋊C₄	1	2	yes	—
22	C₂×D₂₃₃	D₈	1	1	yes	characteristic
23	C₂×D₂₃₃	SD₁₆	1	2	yes	—
24	C₂×D₂₃₃	Q₁₆	1	1	yes	characteristic
25	#11745400	—	4	4	no	—
26	C₄×C₂₃₃	C₄²	1	1	yes	characteristic
27	C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
28	Dic₂₃₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₂₃₃	#95	1	1	yes	characteristic
30	C₂×C₂₃₃	#106	1	1	yes	characteristic
31	C₂×C₂₃₃	#107	1	1	yes	characteristic
32	D₂₃₃	#102	1	2	yes	—
33	D₂₃₃	#103	1	2	yes	—
34	C₂₃₃	#340	1	1	yes	characteristic, complemented, Sylow
35	#340	—	233	1	no	maximal, Sylow
36	C₂²×C₈	—	233	1	no	—
37	C₂×C₄⋊C₄	—	233	2	no	—
38	C₂²×C₄	—	466	2	no	—
39	C₂²×C₄	—	233	1	no	—
40	C₂×C₈	—	466	2	no	—
41	C₂×C₈	—	233	1	no	—
42	C₂×C₈	#11745400	1	1	yes	characteristic, complemented
43	C₄⋊C₄	—	466	2	no	—
44	C₄⋊C₄	—	233	4	no	—
45	C₂³	—	233	1	no	—
46	C₂×C₄	—	932	2	no	—
47	C₂×C₄	—	932	2	no	—
48	C₂×C₄	—	466	4	no	—
49	C₂×C₄	—	233	1	no	—
50	C₂×C₄	—	233	2	no	—
51	C₂×C₄	—	233	2	no	—
52	C₂×C₄	C₂×#11745400	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	466	1	no	—
55	C₂²	—	233	2	no	—
56	C₂²	—	233	2	no	—
57	C₂²	—	233	1	no	—
58	C₂²	—	233	1	no	—
59	C₂²	C₄⋊#11745400	1	1	yes	characteristic
60	C₄	—	932	4	no	—
61	C₄	—	233	2	no	—
62	C₄	C₄×#11745400	1	1	yes	characteristic
63	C₄	C₂²⋊#11745400	1	1	yes	characteristic
64	C₂	—	233	2	no	—
65	C₂	—	233	2	no	—
66	C₂	(C₂×#11745400)⋊C₄	1	1	yes	characteristic
67	C₂	#21418603	1	1	yes	characteristic
68	C₂	#21418604	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#11745400
commutator	C₄×C₂₃₃	C₄²
Frattini	C₂×C₄	C₂×#11745400
Fitting	C₂×C₈×C₂₃₃	C₄
socle	C₂²×C₂₃₃	C₄⋊C₄
prime radical	C₂²×D₂₃₃	D₄
cosocle	C₂×C₄×D₂₃₃	C₂²
hypercenter	C₂×C₈	#11745400
hypocenter	C₂₃₃	#340
LCS	G ▷ C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. C₄⋊#11745400	C₄	semidirect
7. C₂×#14272338	C₄	semidirect
9. C₂×C₈×C₂₃₃	C₄	semidirect
34. C₂₃₃	#340	semidirect
42. C₂×C₈	#11745400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11745400	2
C₂×C₈×D₂₃₃	2
#340	233