Subgroups — #42121853

Order 27712.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357179	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₄₃₃	C₂	1	1	yes	maximal, characteristic
3	#28357154	C₂	1	2	yes	maximal, complemented
4	#28357152	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₄₃₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₃₃	—	2	1	no	—
7	C₄⋊Dic₄₃₃	—	2	1	no	—
8	C₄⋊Dic₄₃₃	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₄₃₃	C₂²	1	2	yes	—
10	C₄⋊Dic₄₃₃	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₄₃₃	—	2	1	no	—
12	C₄×Dic₄₃₃	—	2	1	no	—
13	C₂²×C₄×C₄₃₃	C₂²	1	1	yes	characteristic
14	M₁₆×C₄₃₃	C₄	1	4	yes	complemented
15	C₂×C₈×C₄₃₃	C₂²	1	2	yes	—
16	C₂×Dic₄₃₃	—	2	1	no	—
17	C₂×Dic₄₃₃	—	2	1	no	—
18	C₂×Dic₄₃₃	—	2	2	no	—
19	C₂×Dic₄₃₃	—	4	2	no	—
20	C₂³×C₄₃₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₃₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₃₃	Q₈	1	2	yes	—
24	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
25	C₈×C₄₃₃	C₂×C₄	1	4	yes	complemented
26	C₂²×C₄₃₃	—	2	1	no	—
27	C₂²×C₄₃₃	—	2	1	no	—
28	C₂²×C₄₃₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄₃₃	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₃₃	C₄⋊C₄	1	2	yes	—
31	C₄×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₄₃₃	C₂×Q₈	1	1	yes	characteristic
33	Dic₄₃₃	—	4	1	no	—
34	Dic₄₃₃	—	4	2	no	—
35	Dic₄₃₃	—	4	1	no	—
36	C₂×C₄₃₃	—	2	2	no	—
37	C₂×C₄₃₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄₃₃	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₄₃₃	#137	1	1	yes	characteristic
40	C₄₃₃	#428	1	1	yes	characteristic, complemented, Sylow
41	#428	—	433	1	no	maximal, Sylow
42	C₂×M₁₆	D₄₃₃	1	1	yes	characteristic
43	#117	—	433	1	no	—
44	C₂×C₄⋊C₄	—	433	1	no	—
45	#107	—	433	2	no	—
46	#106	—	433	2	no	—
47	C₂²×C₄	—	866	1	no	—
48	C₂²×C₄	C₂×D₄₃₃	1	1	yes	characteristic
49	M₁₆	Dic₄₃₃	1	4	yes	complemented
50	C₂×C₈	C₂×D₄₃₃	1	2	yes	—
51	C₄⋊C₄	—	866	1	no	—
52	C₄⋊C₄	—	433	1	no	—
53	C₄⋊C₄	—	433	2	no	—
54	C₄⋊C₄	—	433	1	no	—
55	C₂²⋊C₄	—	866	1	no	—
56	C₄²	—	866	1	no	—
57	C₂³	D₁₇₃₂	1	1	yes	characteristic
58	C₂×C₄	—	866	1	no	—
59	C₂×C₄	—	866	1	no	—
60	C₂×C₄	—	866	2	no	—
61	C₂×C₄	—	1732	2	no	—
62	C₂×C₄	C₂×Dic₄₃₃	1	2	yes	—
63	C₂×C₄	D₁₇₃₂	1	1	yes	characteristic
64	C₂×C₄	Dic₈₆₆	1	2	yes	—
65	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
66	C₈	C₂×Dic₄₃₃	1	4	yes	complemented
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	C₄⋊Dic₄₃₃	1	2	yes	—
70	C₂²	C₂×D₁₇₃₂	1	1	yes	characteristic
71	C₄	—	1732	2	no	—
72	C₄	—	1732	1	no	—
73	C₄	—	1732	1	no	—
74	C₄	C₄⋊Dic₄₃₃	1	2	yes	—
75	C₄	C₂²×Dic₄₃₃	1	1	yes	characteristic
76	C₄	C₂×Dic₈₆₆	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×C₄⋊Dic₄₃₃	1	1	yes	characteristic
79	C₂	#28357248	1	1	yes	characteristic
80	C₂	#28357253	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₇₃₂
commutator	C₄×C₄₃₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₃₃
Fitting	C₂×M₁₆×C₄₃₃	C₂
socle	C₂²×C₄₃₃	C₂×D₄
prime radical	C₂³×C₄₃₃	D₄
cosocle	C₂×C₄×C₄₃₃	C₂³
hypercenter	C₂×M₁₆	D₄₃₃
hypocenter	C₄₃₃	#428
LCS	G ▷ C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #28357154	C₂	semidirect
4. #28357152	C₂	semidirect
14. M₁₆×C₄₃₃	C₄	semidirect
25. C₈×C₄₃₃	C₂×C₄	semidirect
40. C₄₃₃	#428	semidirect
49. M₁₆	Dic₄₃₃	semidirect
66. C₈	C₂×Dic₄₃₃	semidirect

Maximal subgroups

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

subgroup	index
#28357179	2
C₂×C₄⋊Dic₄₃₃	2
#28357154	2
#28357152	2
C₂×M₁₆×C₄₃₃	2
#428	433