Subgroups — #42122912

Order 27712.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357317	C₂	1	1	yes	maximal, characteristic, complemented
2	#28357315	C₂	1	1	yes	maximal, characteristic, complemented
3	#28357308	C₂	1	1	yes	maximal, characteristic
4	#20294109	—	2	1	no	—
5	#20294109	—	2	1	no	—
6	#20294107	—	2	1	no	—
7	#20294107	C₄	1	1	yes	characteristic
8	#20294105	—	2	1	no	—
9	#20294105	C₂²	1	1	yes	characteristic
10	#20294098	C₄	1	1	yes	characteristic
11	#16812365	—	2	1	no	—
12	#16812365	D₄	1	1	yes	characteristic
13	#16812365	D₄	1	1	yes	characteristic
14	D₁₇₃₂	—	2	1	no	—
15	D₁₇₃₂	C₈	1	1	yes	characteristic
16	C₄×D₄₃₃	—	2	1	no	—
17	C₄×D₄₃₃	C₂×C₄	1	1	yes	characteristic
18	Q₈×C₄₃₃	C₈	1	1	yes	characteristic
19	C₂×D₄₃₃	—	2	1	no	—
20	C₂×D₄₃₃	M₁₆	1	1	yes	characteristic
21	C₄×C₄₃₃	—	2	1	no	—
22	C₄×C₄₃₃	C₂×C₈	1	1	yes	characteristic
23	Dic₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄₃₃	C₂²⋊C₈	1	1	yes	characteristic
25	D₄₃₃	—	2	1	no	—
26	D₄₃₃	—	4	1	no	—
27	C₄₃₃	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	433	1	no	maximal, Sylow
29	#131	—	433	1	no	—
30	M₃₂	—	433	1	no	—
31	C₂×C₁₆	—	433	1	no	—
32	#41	—	433	1	no	—
33	M₁₆	—	866	1	no	—
34	M₁₆	—	433	1	no	—
35	C₂×C₈	—	866	1	no	—
36	C₂×C₈	—	433	1	no	—
37	C₁₆	—	866	1	no	—
38	C₁₆	—	866	1	no	—
39	Q₈	#16812367	1	1	yes	characteristic
40	D₄	—	866	1	no	—
41	D₄	—	433	1	no	—
42	C₂×C₄	—	866	1	no	—
43	C₂×C₄	—	433	1	no	—
44	C₈	—	866	1	no	—
45	C₈	—	433	1	no	—
46	C₈	—	433	1	no	—
47	C₂²	—	866	1	no	—
48	C₂²	—	433	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	433	1	no	—
51	C₄	C₂×#16812367	1	1	yes	characteristic
52	C₂	—	866	1	no	—
53	C₂	—	1732	1	no	—
54	C₂	C₂²⋊#16812367	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#16812367
commutator	C₄×C₄₃₃	C₂×C₈
Frattini	C₄	C₂×#16812367
Fitting	Q₈×C₄₃₃	C₈
socle	C₂×C₄₃₃	C₂²⋊C₈
prime radical	#20294098	C₄
cosocle	#20294105	C₂²
hypercenter	Q₈	#16812367
hypocenter	C₄₃₃	#350
LCS	G ▷ C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. #28357317	C₂	semidirect
2. #28357315	C₂	semidirect
27. C₄₃₃	#350	semidirect

Maximal subgroups

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

subgroup	index
#28357317	2
#28357315	2
#28357308	2
#350	433