Subgroups — #42121821

Order 27712.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₈₆₆	C₂	1	1	yes	maximal, characteristic
2	#28357179	C₂	1	1	yes	maximal, characteristic
3	#28357145	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₄₃₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₈₆₆	—	2	2	no	—
6	C₂×Dic₈₆₆	—	2	1	no	—
7	C₂×Dic₈₆₆	C₄	1	4	yes	complemented
8	C₂×Dic₈₆₆	C₂²	1	2	yes	—
9	C₂²×Dic₄₃₃	—	2	1	no	—
10	C₄⋊Dic₄₃₃	C₂²	1	2	yes	—
11	C₂²⋊Dic₄₃₃	—	2	1	no	—
12	C₄×Dic₄₃₃	—	2	1	no	—
13	C₂²×C₄×C₄₃₃	C₂²	1	1	yes	characteristic
14	M₁₆×C₄₃₃	—	2	2	no	—
15	C₂×C₈×C₄₃₃	C₂²	1	2	yes	—
16	Dic₈₆₆	—	2	4	no	—
17	Dic₈₆₆	—	2	2	no	—
18	Dic₈₆₆	C₂×C₄	1	4	yes	—
19	C₂×Dic₄₃₃	—	2	4	no	—
20	C₂×Dic₄₃₃	—	2	2	no	—
21	C₂×Dic₄₃₃	—	2	2	no	—
22	C₂³×C₄₃₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₃₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₄₃₃	D₄	1	2	yes	—
25	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
27	C₈×C₄₃₃	—	2	2	no	—
28	C₂²×C₄₃₃	—	2	2	no	—
29	C₂²×C₄₃₃	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₄₃₃	C₂×D₄	1	1	yes	characteristic
31	C₄×C₄₃₃	C₂²⋊C₄	1	2	yes	—
32	C₄×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₄₃₃	C₂×D₄	1	1	yes	characteristic
34	Dic₄₃₃	—	2	4	no	—
35	Dic₄₃₃	—	4	2	no	—
36	C₂×C₄₃₃	—	2	2	no	—
37	C₂×C₄₃₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄₃₃	#137	1	2	yes	—
39	C₄₃₃	#419	1	1	yes	characteristic, complemented, Sylow
40	#419	—	433	1	no	maximal, Sylow
41	C₂²×Q₈	—	433	1	no	—
42	C₂×M₁₆	D₄₃₃	1	1	yes	characteristic
43	#117	—	433	1	no	—
44	#103	—	433	4	no	—
45	C₂×Q₈	—	866	2	no	—
46	C₂×Q₈	—	866	1	no	—
47	C₂×Q₈	—	433	4	no	—
48	C₂×Q₈	—	433	2	no	—
49	C₂²×C₄	—	866	1	no	—
50	C₂²×C₄	C₂×D₄₃₃	1	1	yes	characteristic
51	M₁₆	—	2	2	no	—
52	C₂×C₈	C₂×D₄₃₃	1	2	yes	—
53	C₄⋊C₄	—	433	2	no	—
54	C₂²⋊C₄	—	866	1	no	—
55	C₄²	—	866	1	no	—
56	C₂³	D₁₇₃₂	1	1	yes	characteristic
57	Q₈	—	866	4	no	—
58	Q₈	—	866	2	no	—
59	Q₈	—	433	4	no	—
60	C₂×C₄	—	866	4	no	—
61	C₂×C₄	—	866	2	no	—
62	C₂×C₄	—	866	2	no	—
63	C₂×C₄	C₄×D₄₃₃	1	2	yes	—
64	C₂×C₄	C₂²⋊D₄₃₃	1	2	yes	—
65	C₂×C₄	D₁₇₃₂	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	#20294074	1	2	yes	—
70	C₂²	C₂×D₁₇₃₂	1	1	yes	characteristic
71	C₄	—	866	4	no	—
72	C₄	—	1732	2	no	—
73	C₄	#20294074	1	2	yes	—
74	C₄	C₂×C₄×D₄₃₃	1	1	yes	characteristic
75	C₄	C₂×C₂²⋊D₄₃₃	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×#20294074	1	1	yes	characteristic
78	C₂	#28357253	1	2	yes	—

	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	#419
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. #28357145	C₂	semidirect
7. C₂×Dic₈₆₆	C₄	semidirect
39. C₄₃₃	#419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₈₆₆	2
#28357179	2
#28357145	2
C₂×M₁₆×C₄₃₃	2
#419	433