Subgroups — #41002882

Order 26944.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212325	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₄₂₁	C₂	1	1	yes	maximal, characteristic
3	#27212300	C₂	1	2	yes	maximal, complemented
4	#27212298	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₁	—	2	1	no	—
7	C₄⋊Dic₄₂₁	—	2	1	no	—
8	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
10	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₄₂₁	—	2	1	no	—
12	C₄×Dic₄₂₁	—	2	1	no	—
13	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
14	M₁₆×C₄₂₁	C₄	1	4	yes	complemented
15	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
16	C₂×Dic₄₂₁	—	2	1	no	—
17	C₂×Dic₄₂₁	—	2	1	no	—
18	C₂×Dic₄₂₁	—	2	2	no	—
19	C₂×Dic₄₂₁	—	4	2	no	—
20	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	Q₈	1	2	yes	—
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₈×C₄₂₁	C₂×C₄	1	4	yes	complemented
26	C₂²×C₄₂₁	—	2	1	no	—
27	C₂²×C₄₂₁	—	2	1	no	—
28	C₂²×C₄₂₁	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₂₁	C₄⋊C₄	1	2	yes	—
31	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
33	Dic₄₂₁	—	4	1	no	—
34	Dic₄₂₁	—	4	2	no	—
35	Dic₄₂₁	—	4	1	no	—
36	C₂×C₄₂₁	—	2	2	no	—
37	C₂×C₄₂₁	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₄₂₁	#137	1	1	yes	characteristic
40	C₄₂₁	#428	1	1	yes	characteristic, complemented, Sylow
41	#428	—	421	1	no	maximal, Sylow
42	C₂×M₁₆	D₄₂₁	1	1	yes	characteristic
43	#117	—	421	1	no	—
44	C₂×C₄⋊C₄	—	421	1	no	—
45	#107	—	421	2	no	—
46	#106	—	421	2	no	—
47	C₂²×C₄	—	842	1	no	—
48	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
49	M₁₆	Dic₄₂₁	1	4	yes	complemented
50	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
51	C₄⋊C₄	—	842	1	no	—
52	C₄⋊C₄	—	421	1	no	—
53	C₄⋊C₄	—	421	2	no	—
54	C₄⋊C₄	—	421	1	no	—
55	C₂²⋊C₄	—	842	1	no	—
56	C₄²	—	842	1	no	—
57	C₂³	D₁₆₈₄	1	1	yes	characteristic
58	C₂×C₄	—	842	1	no	—
59	C₂×C₄	—	842	1	no	—
60	C₂×C₄	—	842	2	no	—
61	C₂×C₄	—	1684	2	no	—
62	C₂×C₄	C₂×Dic₄₂₁	1	2	yes	—
63	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
64	C₂×C₄	Dic₈₄₂	1	2	yes	—
65	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
66	C₈	C₂×Dic₄₂₁	1	4	yes	complemented
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	C₄⋊Dic₄₂₁	1	2	yes	—
70	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
71	C₄	—	1684	2	no	—
72	C₄	—	1684	1	no	—
73	C₄	—	1684	1	no	—
74	C₄	C₄⋊Dic₄₂₁	1	2	yes	—
75	C₄	C₂²×Dic₄₂₁	1	1	yes	characteristic
76	C₄	C₂×Dic₈₄₂	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×C₄⋊Dic₄₂₁	1	1	yes	characteristic
79	C₂	#27212394	1	1	yes	characteristic
80	C₂	#27212399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₂×M₁₆×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂³×C₄₂₁	D₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	C₂×M₁₆	D₄₂₁
hypocenter	C₄₂₁	#428
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #27212300	C₂	semidirect
4. #27212298	C₂	semidirect
14. M₁₆×C₄₂₁	C₄	semidirect
25. C₈×C₄₂₁	C₂×C₄	semidirect
40. C₄₂₁	#428	semidirect
49. M₁₆	Dic₄₂₁	semidirect
66. C₈	C₂×Dic₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212325	2
C₂×C₄⋊Dic₄₂₁	2
#27212300	2
#27212298	2
C₂×M₁₆×C₄₂₁	2
#428	421