Subgroups — #42122083

Order 27712.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357251	C₂	1	2	yes	maximal
2	#28357226	C₂	1	1	yes	maximal, characteristic
3	#28357156	C₂	1	1	yes	maximal, characteristic, complemented
4	#28357139	C₂	1	2	yes	maximal, complemented
5	C₂×Q₁₆×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294097	C₂²	1	1	yes	characteristic
7	#20294087	—	2	2	no	—
8	#20294084	—	2	2	no	—
9	#20294081	C₂²	1	1	yes	characteristic
10	#20294081	C₂²	1	2	yes	—
11	#20294080	—	2	1	no	—
12	C₈×D₄₃₃	—	2	1	no	—
13	C₂×Q₈×C₄₃₃	C₂²	1	2	yes	—
14	Q₁₆×C₄₃₃	—	2	1	no	—
15	Q₁₆×C₄₃₃	—	2	1	no	—
16	C₂×C₈×C₄₃₃	C₂²	1	1	yes	characteristic
17	Dic₈₆₆	D₄	1	1	yes	characteristic
18	D₁₇₃₂	D₄	1	1	yes	characteristic
19	C₂²⋊D₄₃₃	—	2	1	no	—
20	C₄×D₄₃₃	—	2	1	no	—
21	#16812358	—	2	1	no	—
22	#16812358	—	2	2	no	—
23	Q₈×C₄₃₃	—	2	2	no	—
24	Q₈×C₄₃₃	—	2	2	no	—
25	C₂×C₄×C₄₃₃	—	2	2	no	—
26	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
27	C₈×C₄₃₃	D₄	1	1	yes	characteristic
28	C₈×C₄₃₃	D₄	1	1	yes	characteristic
29	C₂²×C₄₃₃	#41	1	1	yes	characteristic
30	C₂×D₄₃₃	—	2	1	no	—
31	C₄×C₄₃₃	—	4	2	no	—
32	C₄×C₄₃₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₄₃₃	C₂×D₄	1	1	yes	characteristic
34	Dic₄₃₃	—	2	1	no	—
35	C₂×C₄₃₃	—	2	1	no	—
36	C₂×C₄₃₃	#121	1	1	yes	characteristic
37	D₄₃₃	—	4	1	no	—
38	C₄₃₃	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	433	1	no	maximal, Sylow
40	#137	—	433	2	no	—
41	C₂×Q₁₆	D₄₃₃	1	1	yes	characteristic, complemented
42	#131	—	433	1	no	—
43	#108	—	433	1	no	—
44	#101	—	433	2	no	—
45	#41	—	433	1	no	—
46	C₂×Q₈	C₂×D₄₃₃	1	2	yes	—
47	Q₁₆	—	2	1	no	—
48	Q₁₆	—	2	1	no	—
49	Q₁₆	—	866	2	no	—
50	SD₁₆	—	866	2	no	—
51	M₁₆	—	866	1	no	—
52	M₁₆	—	433	1	no	—
53	M₁₆	—	433	2	no	—
54	C₂×C₈	—	866	1	no	—
55	C₂×C₈	C₂×D₄₃₃	1	1	yes	characteristic
56	Q₈	—	2	2	no	—
57	Q₈	—	2	2	no	—
58	Q₈	—	433	1	no	—
59	D₄	—	866	1	no	—
60	D₄	—	433	1	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	866	1	no	—
63	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
64	C₈	—	866	2	no	—
65	C₈	—	866	1	no	—
66	C₈	C₂²⋊D₄₃₃	1	1	yes	characteristic
67	C₈	C₂²⋊D₄₃₃	1	1	yes	characteristic
68	C₂²	—	866	1	no	—
69	C₂²	#20294096	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	866	1	no	—
72	C₄	D₄×D₄₃₃	1	1	yes	characteristic
73	C₄	C₂×C₂²⋊D₄₃₃	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	1732	1	no	—
76	C₂	#28357191	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28357191
commutator, cosocle	C₂×C₄×C₄₃₃	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₃
Fitting	C₂×Q₁₆×C₄₃₃	C₂
socle	C₂×C₄₃₃	#121
prime radical	#20294097	C₂²
hypercenter	C₂×Q₁₆	D₄₃₃
hypocenter	C₄₃₃	#473
LCS	G ▷ C₂×C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #28357156	C₂	semidirect
4. #28357139	C₂	semidirect
5. C₂×Q₁₆×C₄₃₃	C₂	semidirect
38. C₄₃₃	#473	semidirect
41. C₂×Q₁₆	D₄₃₃	semidirect

Maximal subgroups

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

subgroup	index
#28357251	2
#28357226	2
#28357156	2
#28357139	2
C₂×Q₁₆×C₄₃₃	2
#473	433