Subgroups — #42122665

Order 27712.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357277	C₂	1	1	yes	maximal, characteristic
2	(C₂×#13181884)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#28357196	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13181884	—	2	1	no	—
5	C₂×#16812365	—	2	1	no	—
6	C₂×C₄×D₄₃₃	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₄₃₃	C₄	1	1	yes	characteristic, complemented
8	#20294076	—	2	1	no	—
9	#20294074	—	2	1	no	—
10	C₂×Q₈×C₄₃₃	C₄	1	1	yes	characteristic, complemented
11	C₂×#13181884	—	2	2	no	—
12	C₂×#13181884	—	4	1	no	—
13	C₂×#13181884	—	4	1	no	—
14	#16812365	—	4	1	no	—
15	C₂²×D₄₃₃	D₄	1	1	yes	characteristic
16	C₂×Dic₄₃₃	—	2	1	no	—
17	C₂×Dic₄₃₃	D₄	1	1	yes	characteristic
18	C₄×D₄₃₃	—	4	1	no	—
19	Q₈×C₄₃₃	—	4	1	no	—
20	C₂×C₄×C₄₃₃	—	2	1	no	—
21	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₄₃₃	—	2	1	no	—
24	C₂×D₄₃₃	—	2	1	no	—
25	C₂×D₄₃₃	SD₁₆	1	1	yes	characteristic
26	C₂×D₄₃₃	Q₁₆	1	1	yes	characteristic
27	#13181884	—	4	2	no	—
28	C₄×C₄₃₃	—	2	1	no	—
29	C₄×C₄₃₃	—	4	1	no	—
30	Dic₄₃₃	—	2	1	no	—
31	Dic₄₃₃	—	4	1	no	—
32	C₂×C₄₃₃	#99	1	1	yes	characteristic
33	C₂×C₄₃₃	#103	1	1	yes	characteristic
34	C₂×C₄₃₃	#104	1	1	yes	characteristic
35	D₄₃₃	—	2	1	no	—
36	D₄₃₃	—	2	1	no	—
37	C₄₃₃	#328	1	1	yes	characteristic, complemented, Sylow
38	#328	—	433	1	no	maximal, Sylow
39	C₂²⋊Q₈	—	433	1	no	—
40	C₂²⋊C₈	—	433	1	no	—
41	#95	—	433	1	no	—
42	C₂×Q₈	#13181884	1	1	yes	characteristic, complemented
43	C₂²×C₄	—	866	1	no	—
44	C₂²×C₄	—	433	1	no	—
45	C₂×C₈	—	866	1	no	—
46	C₄⋊C₄	—	866	1	no	—
47	C₄⋊C₄	—	433	1	no	—
48	C₂²⋊C₄	—	866	1	no	—
49	C₂³	—	433	1	no	—
50	Q₈	—	4	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	866	2	no	—
53	C₂×C₄	—	866	1	no	—
54	C₂×C₄	—	433	1	no	—
55	C₂×C₄	—	1732	1	no	—
56	C₂×C₄	—	1732	1	no	—
57	C₂×C₄	—	1732	1	no	—
58	C₂×C₄	C₂×#13181884	1	1	yes	characteristic
59	C₈	—	1732	1	no	—
60	C₂²	—	866	1	no	—
61	C₂²	—	866	1	no	—
62	C₂²	—	433	1	no	—
63	C₂²	—	433	1	no	—
64	C₂²	C₂²⋊#13181884	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	866	1	no	—
68	C₄	—	1732	2	no	—
69	C₄	—	1732	1	no	—
70	C₂	—	866	1	no	—
71	C₂	—	866	1	no	—
72	C₂	#28357278	1	1	yes	characteristic
73	C₂	#28357283	1	1	yes	characteristic
74	C₂	#28357285	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13181884
commutator	C₂×C₄×C₄₃₃	C₂×C₄
Frattini	C₂×C₄	C₂×#13181884
Fitting	C₂×Q₈×C₄₃₃	C₄
socle	C₂²×C₄₃₃	C₂²⋊C₄
prime radical	C₂²×D₄₃₃	D₄
cosocle	C₂×C₄×D₄₃₃	C₂²
hypercenter	C₂×Q₈	#13181884
hypocenter	C₄₃₃	#328
LCS	G ▷ C₂×C₄×C₄₃₃ ▷ C₂²×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
7. C₄⋊Dic₄₃₃	C₄	semidirect
10. C₂×Q₈×C₄₃₃	C₄	semidirect
37. C₄₃₃	#328	semidirect
42. C₂×Q₈	#13181884	semidirect

Maximal subgroups

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

subgroup	index
#28357277	2
(C₂×#13181884)⋊C₄	2
#28357196	2
#328	433