Subgroups — #42121893

Order 27712.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₄₆₄	C₂	1	1	yes	maximal, characteristic
2	#28357182	C₂	1	2	yes	maximal, complemented
3	#28357140	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#107×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₇₃₂	C₂²	1	2	yes	—
7	C₂×C₄×D₄₃₃	—	2	2	no	—
8	D₃₄₆₄	C₄	1	4	yes	complemented
9	C₂×#16812358	C₂²	1	1	yes	characteristic
10	#20294074	—	2	2	no	—
11	C₄×Dic₄₃₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₄₃₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₃₃	C₂²	1	2	yes	—
14	C₂²×D₄₃₃	—	2	2	no	—
15	D₁₇₃₂	—	2	2	no	—
16	D₁₇₃₂	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₄₃₃	D₄	1	1	yes	characteristic
18	C₂×Dic₄₃₃	D₄	1	1	yes	characteristic
19	C₄×D₄₃₃	—	4	4	no	—
20	#16812358	—	2	1	no	—
21	C₂×C₄×C₄₃₃	—	2	2	no	—
22	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
23	C₈×C₄₃₃	C₂×C₄	1	2	yes	complemented
24	C₂²×C₄₃₃	C₂×D₄	1	1	yes	characteristic
25	C₂×D₄₃₃	—	2	4	no	—
26	C₂×D₄₃₃	—	4	2	no	—
27	C₂×D₄₃₃	—	4	2	no	—
28	C₄×C₄₃₃	—	4	2	no	—
29	C₄×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₄₃₃	#41	1	1	yes	characteristic
31	Dic₄₃₃	—	2	1	no	—
32	Dic₄₃₃	D₈	1	2	yes	complemented
33	C₂×C₄₃₃	C₄×D₄	1	1	yes	characteristic
34	C₂×C₄₃₃	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄₃₃	#135	1	1	yes	characteristic
36	D₄₃₃	—	4	4	no	—
37	C₄₃₃	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
38	C₄×D₈	—	433	1	no	maximal, Sylow
39	C₂×D₈	—	433	1	no	—
40	C₄×D₄	—	433	2	no	—
41	#107	D₄₃₃	1	1	yes	characteristic, complemented
42	#102	—	433	2	no	—
43	C₄×C₈	—	433	1	no	—
44	C₂×D₄	—	433	2	no	—
45	C₂²×C₄	—	866	2	no	—
46	D₈	—	433	4	no	—
47	C₂×C₈	—	433	1	no	—
48	C₂×C₈	C₂×D₄₃₃	1	1	yes	characteristic
49	C₄⋊C₄	C₂×D₄₃₃	1	2	yes	—
50	C₂²⋊C₄	—	866	2	no	—
51	C₄²	—	433	1	no	—
52	C₂³	—	866	2	no	—
53	D₄	—	866	2	no	—
54	D₄	—	433	4	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	433	1	no	—
57	C₂×C₄	—	433	1	no	—
58	C₂×C₄	—	1732	4	no	—
59	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
60	C₈	—	866	1	no	—
61	C₈	C₄×D₄₃₃	1	2	yes	complemented
62	C₂²	—	866	4	no	—
63	C₂²	—	1732	2	no	—
64	C₂²	—	1732	2	no	—
65	C₂²	D₄×D₄₃₃	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	866	1	no	—
68	C₄	—	433	2	no	—
69	C₄	C₂×C₄×D₄₃₃	1	1	yes	characteristic
70	C₄	#20294098	1	1	yes	characteristic
71	C₂	—	1732	4	no	—
72	C₂	#28357182	1	1	yes	characteristic
73	C₂	D₈×D₄₃₃	1	1	yes	characteristic
74	C₂	#28357243	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₃₃
commutator	C₄×C₄₃₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₃₃
Fitting	#107×C₄₃₃	C₂
socle	C₂²×C₄₃₃	C₂×D₄
prime radical	C₂×D₃₄₆₄	C₂
cosocle	C₂×C₄×C₄₃₃	C₂³
hypercenter	#107	D₄₃₃
hypocenter	C₄₃₃	C₄×D₈
LCS	G ▷ C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
2. #28357182	C₂	semidirect
3. #28357140	C₂	semidirect
4. C₈×Dic₄₃₃	C₂	semidirect
5. #107×C₄₃₃	C₂	semidirect
8. D₃₄₆₄	C₄	semidirect
16. D₁₇₃₂	C₂×C₄	semidirect
23. C₈×C₄₃₃	C₂×C₄	semidirect
32. Dic₄₃₃	D₈	semidirect
37. C₄₃₃	C₄×D₈	semidirect
41. #107	D₄₃₃	semidirect
61. C₈	C₄×D₄₃₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₄₆₄	2
#28357182	2
#28357140	2
C₈×Dic₄₃₃	2
#107×C₄₃₃	2
C₄×D₈	433