Subgroups — #28417314

Order 14912.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₂₃₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11745400	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₂₃₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11745400	C₄	1	2	yes	—
5	#16836604	—	2	1	no	—
6	#16836603	C₄	1	2	yes	complemented
7	#16836602	—	2	1	no	—
8	C₄⋊#11745400	—	2	1	no	—
9	C₄⋊#11745400	—	2	1	no	—
10	C₂×C₄×D₂₃₃	C₂²	1	1	yes	characteristic
11	#16836578	C₄	1	1	yes	characteristic, complemented
12	#16836577	—	2	1	no	—
13	C₈×D₂₃₃	—	2	1	no	—
14	M₁₆×C₂₃₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#11745400	—	2	2	no	—
16	C₂×#11745400	—	4	2	no	—
17	#14272345	—	2	2	no	—
18	C₂²×D₂₃₃	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₂₃₃	C₂×C₄	1	1	yes	characteristic
20	C₄×D₂₃₃	D₄	1	1	yes	characteristic
21	C₄×D₂₃₃	D₄	1	1	yes	characteristic
22	C₄×D₂₃₃	D₄	1	1	yes	characteristic
23	C₄×D₂₃₃	Q₈	1	1	yes	characteristic
24	#14272338	—	2	1	no	—
25	C₂×C₄×C₂₃₃	C₂×C₄	1	1	yes	characteristic
26	C₈×C₂₃₃	—	2	1	no	—
27	C₂²×C₂₃₃	C₄²	1	1	yes	characteristic
28	C₂×D₂₃₃	—	2	1	no	—
29	C₂×D₂₃₃	—	2	1	no	—
30	C₂×D₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×D₂₃₃	C₄⋊C₄	1	1	yes	characteristic
32	#11745400	—	4	2	no	—
33	C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₂₃₃	C₄⋊C₄	1	1	yes	characteristic
35	Dic₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₂₃₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₂₃₃	—	2	1	no	—
38	C₂×C₂₃₃	#95	1	1	yes	characteristic
39	D₂₃₃	—	2	1	no	—
40	D₂₃₃	#100	1	1	yes	characteristic
41	D₂₃₃	#101	1	1	yes	characteristic
42	C₂₃₃	#343	1	1	yes	characteristic, complemented, Sylow
43	#343	—	233	1	no	maximal, Sylow
44	C₂×M₁₆	—	233	1	no	—
45	C₂×M₁₆	—	233	1	no	—
46	C₂×C₄⋊C₄	—	233	1	no	—
47	C₂²×C₄	—	233	2	no	—
48	C₂²×C₄	—	233	1	no	—
49	M₁₆	—	466	1	no	—
50	M₁₆	—	466	1	no	—
51	M₁₆	—	233	2	no	—
52	M₁₆	—	233	1	no	—
53	M₁₆	#11745400	1	1	yes	characteristic, complemented
54	C₂×C₈	—	466	1	no	—
55	C₂×C₈	—	466	1	no	—
56	C₄⋊C₄	—	466	1	no	—
57	C₄⋊C₄	—	466	1	no	—
58	C₂³	—	233	1	no	—
59	C₂×C₄	—	932	2	no	—
60	C₂×C₄	—	466	2	no	—
61	C₂×C₄	—	233	1	no	—
62	C₂×C₄	—	233	1	no	—
63	C₂×C₄	—	233	1	no	—
64	C₂×C₄	—	233	1	no	—
65	C₂×C₄	—	233	1	no	—
66	C₂×C₄	C₂×#11745400	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	466	2	no	—
69	C₈	—	466	1	no	—
70	C₂²	—	466	1	no	—
71	C₂²	—	466	1	no	—
72	C₂²	—	233	1	no	—
73	C₂²	—	233	1	no	—
74	C₂²	C₄×#11745400	1	1	yes	characteristic
75	C₄	—	932	2	no	—
76	C₄	—	233	1	no	—
77	C₄	—	233	1	no	—
78	C₄	C₂²⋊#11745400	1	1	yes	characteristic
79	C₄	C₄⋊#11745400	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	466	1	no	—
82	C₂	—	233	1	no	—
83	C₂	—	233	1	no	—
84	C₂	(C₂×#11745400)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#11745400)⋊C₄
commutator	C₂²×C₂₃₃	C₄²
Frattini	C₂×C₄	C₂×#11745400
Fitting	M₁₆×C₂₃₃	C₄
socle	C₂×C₂₃₃	#95
prime radical	C₂²×D₂₃₃	C₂×C₄
cosocle	C₂×C₄×D₂₃₃	C₂²
hypercenter	M₁₆	#11745400
hypocenter	C₂₃₃	#343
LCS	G ▷ C₂²×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
6. #16836603	C₄	semidirect
11. #16836578	C₄	semidirect
14. M₁₆×C₂₃₃	C₄	semidirect
42. C₂₃₃	#343	semidirect
53. M₁₆	#11745400	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×C₄⋊#11745400	2
M₁₆×D₂₃₃	2
#343	233