Subgroups — #28417304

Order 14912.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₂₃₃⋊M₁₆	C₂	1	2	yes	maximal
2	C₂×#16836577	C₂	1	1	yes	maximal, characteristic
3	#16836604	—	2	2	no	—
4	#16836603	—	2	2	no	—
5	#16836602	C₄	1	4	yes	—
6	C₂×C₄×D₂₃₃	C₂²	1	1	yes	characteristic
7	#16836577	—	2	2	no	—
8	C₂×#14272338	C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₃₃	C₄	1	1	yes	characteristic
10	#14272345	—	2	4	no	—
11	C₂²×D₂₃₃	Q₈	1	1	yes	characteristic
12	C₂×Dic₂₃₃	D₄	1	1	yes	characteristic
13	C₄×D₂₃₃	C₂×C₄	1	2	yes	—
14	C₄×D₂₃₃	D₄	1	2	yes	—
15	#14272338	—	2	1	no	—
16	C₂×C₄×C₂₃₃	C₂×C₄	1	1	yes	characteristic
17	C₈×C₂₃₃	—	2	1	no	—
18	C₂²×C₂₃₃	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₂₃₃	—	4	1	no	—
20	C₂×D₂₃₃	C₄⋊C₄	1	2	yes	—
21	C₄×C₂₃₃	C₄²	1	1	yes	characteristic
22	C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₂₃₃	C₂²⋊C₄	1	2	yes	—
24	C₂×C₂₃₃	—	2	1	no	—
25	C₂×C₂₃₃	#95	1	1	yes	characteristic
26	D₂₃₃	—	2	2	no	—
27	C₂₃₃	#338	1	1	yes	characteristic, complemented, Sylow
28	#338	—	233	1	no	maximal, Sylow
29	C₂×M₁₆	—	233	2	no	—
30	C₂×M₁₆	—	233	1	no	—
31	C₂²×C₄	—	233	1	no	—
32	M₁₆	—	466	2	no	—
33	M₁₆	—	466	2	no	—
34	M₁₆	—	466	2	no	—
35	C₂×C₈	—	233	4	no	—
36	C₂×C₈	—	233	1	no	—
37	C₂×C₈	#11745400	1	1	yes	characteristic
38	C₂³	—	233	1	no	—
39	C₂×C₄	—	233	1	no	—
40	C₂×C₄	—	233	2	no	—
41	C₂×C₄	—	233	2	no	—
42	C₂×C₄	C₂×#11745400	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₈	—	466	4	no	—
45	C₈	—	466	1	no	—
46	C₂²	—	932	1	no	—
47	C₂²	—	233	2	no	—
48	C₂²	C₄⋊#11745400	1	1	yes	characteristic
49	C₄	—	233	2	no	—
50	C₄	C₄×#11745400	1	1	yes	characteristic
51	C₄	C₂²⋊#11745400	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	466	2	no	—
54	C₂	(C₂×#11745400)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#11745400
commutator	C₄×C₂₃₃	C₄²
Frattini	C₂×C₄	C₂×#11745400
Fitting	C₂×C₈×C₂₃₃	C₄
socle	C₂×C₂₃₃	#95
prime radical	C₂²×D₂₃₃	Q₈
cosocle	C₂×C₄×D₂₃₃	C₂²
hypercenter	C₂×C₈	#11745400
hypocenter	C₂₃₃	#338
LCS	G ▷ C₄×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
27. C₂₃₃	#338	semidirect

Maximal subgroups

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

subgroup	index
D₂₃₃⋊M₁₆	2
C₂×#16836577	2
#338	233