Subgroups — #108×D₄₂₁ #41002884

Order 26944.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×D₄₂₁	C₂	1	2	yes	maximal
2	C₂×C₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic
3	#27212302	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212301	C₂	1	2	yes	maximal, complemented
5	#108×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
7	#20269755	C₂²	1	2	yes	—
8	#20269754	—	2	2	no	—
9	C₂×#16809890	C₂²	1	1	yes	characteristic
10	C₈×D₄₂₁	—	2	2	no	—
11	C₈×D₄₂₁	C₄	1	2	yes	—
12	C₈×D₄₂₁	C₄	1	2	yes	—
13	M₁₆×C₄₂₁	C₂²	1	2	yes	—
14	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄₂₁	Q₈	1	1	yes	characteristic
16	C₂×Dic₄₂₁	Q₈	1	1	yes	characteristic
17	C₄×D₄₂₁	C₂×C₄	1	1	yes	characteristic
18	C₄×D₄₂₁	C₂×C₄	1	1	yes	characteristic
19	C₄×D₄₂₁	D₄	1	1	yes	characteristic
20	C₄×D₄₂₁	D₄	1	1	yes	characteristic
21	#16809890	—	2	2	no	—
22	#16809890	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
24	C₈×C₄₂₁	—	2	2	no	—
25	C₈×C₄₂₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
27	C₂×D₄₂₁	—	2	2	no	—
28	C₂×D₄₂₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×D₄₂₁	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
32	Dic₄₂₁	C₄⋊C₄	1	1	yes	characteristic
33	Dic₄₂₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄₂₁	—	2	1	no	—
35	C₂×C₄₂₁	C₂×C₄⋊C₄	1	1	yes	characteristic
36	D₄₂₁	—	2	1	no	—
37	D₄₂₁	#108	1	2	yes	direct product factor
38	C₄₂₁	C₂×#108	1	1	yes	characteristic, complemented, Sylow
39	C₂×#108	—	421	1	no	maximal, Sylow
40	C₂×M₁₆	—	421	2	no	—
41	C₂²×C₈	—	421	1	no	—
42	#108	—	421	2	no	—
43	#108	—	421	1	no	—
44	#108	D₄₂₁	1	1	yes	characteristic, direct product factor
45	C₂²×C₄	—	421	1	no	—
46	M₁₆	—	842	2	no	—
47	M₁₆	—	421	2	no	—
48	M₁₆	C₂×D₄₂₁	1	2	yes	—
49	C₂×C₈	—	842	2	no	—
50	C₂×C₈	—	421	1	no	—
51	C₂×C₈	—	421	2	no	—
52	C₂×C₈	—	421	2	no	—
53	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
54	C₂³	—	421	1	no	—
55	C₂×C₄	—	421	1	no	—
56	C₂×C₄	—	421	1	no	—
57	C₂×C₄	—	421	1	no	—
58	C₂×C₄	—	421	1	no	—
59	C₂×C₄	—	421	1	no	—
60	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₈	—	842	2	no	—
63	C₈	—	421	2	no	—
64	C₈	C₄×D₄₂₁	1	2	yes	—
65	C₂²	—	842	2	no	—
66	C₂²	—	421	1	no	—
67	C₂²	—	421	1	no	—
68	C₂²	Q₈×D₄₂₁	1	1	yes	characteristic
69	C₄	—	421	1	no	—
70	C₄	—	421	1	no	—
71	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
72	C₄	D₄×D₄₂₁	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	842	1	no	—
75	C₂	—	421	2	no	—
76	C₂	C₄⋊C₄×D₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₄₂₁
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#108×C₄₂₁	C₂
socle	C₂×C₄₂₁	C₂×C₄⋊C₄
prime radical	C₂²×D₄₂₁	Q₈
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#108	D₄₂₁
hypocenter	C₄₂₁	C₂×#108
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
3. #27212302	C₂	semidirect
4. #27212301	C₂	semidirect
5. #108×C₄₂₁	C₂	semidirect
37. D₄₂₁	#108	direct (N × H)
38. C₄₂₁	C₂×#108	semidirect
44. #108	D₄₂₁	direct (N × H)

Maximal subgroups

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

subgroup	index
M₁₆×D₄₂₁	2
C₂×C₈×D₄₂₁	2
#27212302	2
#27212301	2
#108×C₄₂₁	2
C₂×#108	421

Subgroups — #108×D421 #41002884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #108×D₄₂₁ #41002884