Subgroups — #28417338

Order 14912.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21418595	C₂	1	1	yes	maximal, characteristic
2	#21418593	C₂	1	1	yes	maximal, characteristic
3	#21418519	C₂	1	1	yes	maximal, characteristic
4	#16836603	—	2	1	no	—
5	C₂²⋊#11745400	—	2	1	no	—
6	C₂×C₂²⋊D₂₃₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₃₃	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₂₃₃	—	2	1	no	—
9	#16836571	—	2	1	no	—
10	C₂²⋊C₄×C₂₃₃	C₄	1	1	yes	characteristic, complemented
11	C₂×#11745400	—	4	1	no	—
12	#14272345	—	4	1	no	—
13	C₂²×D₂₃₃	D₄	1	1	yes	characteristic
14	C₂²⋊D₂₃₃	—	4	1	no	—
15	C₂×Dic₂₃₃	—	2	1	no	—
16	C₂×Dic₂₃₃	—	4	1	no	—
17	C₂×Dic₂₃₃	D₄	1	1	yes	characteristic
18	C₂³×C₂₃₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃₃	—	2	1	no	—
20	C₂²×C₂₃₃	—	2	1	no	—
21	C₂²×C₂₃₃	—	4	1	no	—
22	C₂²×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₂₃₃	—	2	1	no	—
24	C₂×D₂₃₃	—	4	1	no	—
25	#11745400	—	8	1	no	—
26	C₄×C₂₃₃	—	4	1	no	—
27	Dic₂₃₃	—	2	1	no	—
28	Dic₂₃₃	—	4	1	no	—
29	C₂×C₂₃₃	—	2	1	no	—
30	C₂×C₂₃₃	—	4	1	no	—
31	C₂×C₂₃₃	#99	1	1	yes	characteristic
32	D₂₃₃	—	4	1	no	—
33	C₂₃₃	#352	1	1	yes	characteristic, complemented, Sylow
34	#352	—	233	1	no	maximal, Sylow
35	#123	—	233	1	no	—
36	#100	—	233	1	no	—
37	#99	—	233	1	no	—
38	C₂×D₄	—	233	1	no	—
39	C₂²×C₄	—	233	1	no	—
40	M₁₆	—	466	1	no	—
41	C₄⋊C₄	—	466	1	no	—
42	C₂²⋊C₄	—	466	1	no	—
43	C₂²⋊C₄	—	466	1	no	—
44	C₂²⋊C₄	#11745400	1	1	yes	characteristic, complemented
45	C₂³	—	233	1	no	—
46	C₂³	C₂×#11745400	1	1	yes	characteristic
47	D₄	—	932	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	932	1	no	—
50	C₂×C₄	—	932	1	no	—
51	C₂×C₄	—	466	1	no	—
52	C₂×C₄	—	233	1	no	—
53	C₈	—	932	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	932	1	no	—
57	C₂²	—	466	1	no	—
58	C₂²	C₂²⋊#11745400	1	1	yes	characteristic
59	C₄	—	4	1	no	—
60	C₄	—	1864	1	no	—
61	C₄	—	932	1	no	—
62	C₄	—	466	1	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	932	1	no	—
66	C₂	#21418594	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21418594
commutator	C₂³×C₂₃₃	C₂×C₄
Frattini	C₂³	C₂×#11745400
Fitting	C₂²⋊C₄×C₂₃₃	C₄
socle	C₂×C₂₃₃	#99
prime radical, cosocle	C₂×C₂²⋊D₂₃₃	C₂²
hypercenter	C₂²⋊C₄	#11745400
hypocenter	C₂₃₃	#352
LCS	G ▷ C₂³×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₂²×Dic₂₃₃	C₄	semidirect
10. C₂²⋊C₄×C₂₃₃	C₄	semidirect
33. C₂₃₃	#352	semidirect
44. C₂²⋊C₄	#11745400	semidirect

Maximal subgroups

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

subgroup	index
#21418595	2
#21418593	2
#21418519	2
#352	233