Subgroups — #42121852

Order 27712.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20294080	C₂	1	1	yes	maximal, characteristic
2	#28357178	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×D₄₃₃	C₂	1	1	yes	maximal, characteristic
4	#28357153	C₂	1	1	yes	maximal, characteristic, complemented
5	#28357152	C₂	1	1	yes	maximal, characteristic, complemented
6	#28357151	C₂	1	1	yes	maximal, characteristic, complemented
7	#107×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄₃₃	—	2	1	no	—
9	C₂×C₄×D₄₃₃	C₂²	1	1	yes	characteristic
10	#20294080	C₄	1	4	yes	complemented
11	C₂×#16812358	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₃₃	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₄₃₃	C₂²	1	1	yes	characteristic
14	#20294076	—	2	1	no	—
15	#20294074	—	2	1	no	—
16	C₄×Dic₄₃₃	—	2	1	no	—
17	C₂×C₈×C₄₃₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₃₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₄₃₃	C₂²	1	1	yes	characteristic
20	C₂²×D₄₃₃	D₄	1	1	yes	characteristic
21	C₂×Dic₄₃₃	—	2	1	no	—
22	C₂×Dic₄₃₃	—	2	1	no	—
23	C₂×Dic₄₃₃	D₄	1	1	yes	characteristic
24	C₄×D₄₃₃	—	4	2	no	—
25	C₄×D₄₃₃	C₂×C₄	1	2	yes	—
26	C₄×D₄₃₃	Q₈	1	2	yes	—
27	#16812358	C₂×C₄	1	2	yes	complemented
28	C₂×C₄×C₄₃₃	—	2	1	no	—
29	C₂×C₄×C₄₃₃	—	2	1	no	—
30	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
31	C₈×C₄₃₃	C₂×C₄	1	2	yes	complemented
32	C₂²×C₄₃₃	C₂×D₄	1	1	yes	characteristic
33	C₂×D₄₃₃	—	2	1	no	—
34	C₂×D₄₃₃	—	2	1	no	—
35	C₂×D₄₃₃	C₄⋊C₄	1	2	yes	—
36	C₄×C₄₃₃	—	4	1	no	—
37	C₄×C₄₃₃	—	4	1	no	—
38	C₄×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₄₃₃	C₂×Q₈	1	1	yes	characteristic
40	Dic₄₃₃	—	4	1	no	—
41	Dic₄₃₃	—	4	1	no	—
42	Dic₄₃₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₄₃₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄₃₃	AGL(1,ℤ/8)	1	1	yes	characteristic
45	C₂×C₄₃₃	#137	1	1	yes	characteristic
46	D₄₃₃	—	2	2	no	—
47	C₄₃₃	#428	1	1	yes	characteristic, complemented, Sylow
48	#428	—	433	1	no	maximal, Sylow
49	C₂×M₁₆	—	433	1	no	—
50	#117	—	433	1	no	—
51	C₂×C₄⋊C₄	—	433	1	no	—
52	#107	—	433	1	no	—
53	#107	D₄₃₃	1	1	yes	characteristic, complemented
54	#106	—	433	1	no	—
55	#106	—	433	1	no	—
56	C₂²×C₄	—	866	1	no	—
57	C₂²×C₄	—	433	1	no	—
58	M₁₆	—	433	4	no	—
59	C₂×C₈	—	433	1	no	—
60	C₂×C₈	C₂×D₄₃₃	1	1	yes	characteristic
61	C₄⋊C₄	—	866	1	no	—
62	C₄⋊C₄	—	433	1	no	—
63	C₄⋊C₄	—	433	1	no	—
64	C₄⋊C₄	C₂×D₄₃₃	1	1	yes	characteristic
65	C₄⋊C₄	C₂×D₄₃₃	1	1	yes	characteristic
66	C₂²⋊C₄	—	866	1	no	—
67	C₄²	—	866	1	no	—
68	C₂³	—	433	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	866	1	no	—
72	C₂×C₄	—	866	1	no	—
73	C₂×C₄	—	433	1	no	—
74	C₂×C₄	—	433	2	no	—
75	C₂×C₄	—	433	2	no	—
76	C₂×C₄	—	1732	2	no	—
77	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
78	C₈	—	433	2	no	—
79	C₈	C₄×D₄₃₃	1	2	yes	complemented
80	C₂²	—	866	1	no	—
81	C₂²	—	866	1	no	—
82	C₂²	—	433	2	no	—
83	C₂²	D₄×D₄₃₃	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	433	2	no	—
87	C₄	—	1732	1	no	—
88	C₄	—	1732	1	no	—
89	C₄	C₂×C₄×D₄₃₃	1	1	yes	characteristic
90	C₄	Q₈×D₄₃₃	1	1	yes	characteristic
91	C₂	—	866	2	no	—
92	C₂	C₄⋊C₄×D₄₃₃	1	1	yes	characteristic
93	C₂	#28357244	1	1	yes	characteristic
94	C₂	#28357252	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	C₂²×D₄₃₃	D₄
cosocle	C₂×C₄×C₄₃₃	C₂³
hypercenter	#107	D₄₃₃
hypocenter	C₄₃₃	#428
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
4. #28357153	C₂	semidirect
5. #28357152	C₂	semidirect
6. #28357151	C₂	semidirect
7. #107×C₄₃₃	C₂	semidirect
10. #20294080	C₄	semidirect
27. #16812358	C₂×C₄	semidirect
31. C₈×C₄₃₃	C₂×C₄	semidirect
47. C₄₃₃	#428	semidirect
53. #107	D₄₃₃	semidirect
79. 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×#20294080	2
#28357178	2
C₄⋊C₄×D₄₃₃	2
#28357153	2
#28357152	2
#28357151	2
#107×C₄₃₃	2
#428	433