Subgroups — #41002956

Order 26944.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212388	C₂	1	1	yes	maximal, characteristic
2	#27212374	C₂	1	2	yes	maximal, complemented
3	#27212292	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269771	C₂²	1	2	yes	—
7	Dic₁₆₈₄	C₄	1	1	yes	characteristic, complemented
8	#20269760	C₄	1	2	yes	complemented
9	D₃₃₆₈	C₄	1	1	yes	characteristic, complemented
10	#20269754	—	2	2	no	—
11	C₂×#16809890	C₂²	1	1	yes	characteristic
12	C₈×D₄₂₁	—	2	2	no	—
13	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
14	M₁₆×C₄₂₁	C₂²	1	2	yes	—
15	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
16	Dic₈₄₂	C₂×C₄	1	2	yes	—
17	D₁₆₈₄	C₂×C₄	1	2	yes	—
18	C₂²⋊D₄₂₁	—	2	2	no	—
19	C₂×Dic₄₂₁	—	2	1	no	—
20	C₄×D₄₂₁	—	2	2	no	—
21	#16809890	D₄	1	1	yes	characteristic
22	#16809890	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
24	C₈×C₄₂₁	—	2	2	no	—
25	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
26	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄₂₁	#41	1	1	yes	characteristic
28	C₂×D₄₂₁	—	2	2	no	—
29	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
31	Dic₄₂₁	—	2	2	no	—
32	Dic₄₂₁	—	2	2	no	—
33	C₂×C₄₂₁	—	2	1	no	—
34	C₂×C₄₂₁	C₄×D₄	1	1	yes	characteristic
35	D₄₂₁	—	4	2	no	—
36	C₄₂₁	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	421	1	no	maximal, Sylow
38	#135	—	421	1	no	—
39	#131	—	421	2	no	—
40	#108	D₄₂₁	1	1	yes	characteristic, complemented
41	#104	—	421	2	no	—
42	C₄×C₈	—	421	1	no	—
43	#41	—	421	2	no	—
44	Q₁₆	—	421	1	no	—
45	SD₁₆	—	421	2	no	—
46	D₈	—	421	1	no	—
47	M₁₆	—	842	2	no	—
48	M₁₆	C₂×D₄₂₁	1	2	yes	—
49	C₂×C₈	—	842	2	no	—
50	C₂×C₈	—	421	1	no	—
51	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
52	C₄²	—	421	1	no	—
53	Q₈	—	421	2	no	—
54	D₄	—	842	2	no	—
55	D₄	—	421	2	no	—
56	C₂×C₄	—	842	1	no	—
57	C₂×C₄	—	842	2	no	—
58	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	421	1	no	—
61	C₈	—	421	1	no	—
62	C₈	C₄×D₄₂₁	1	1	yes	characteristic
63	C₈	C₄×D₄₂₁	1	1	yes	characteristic
64	C₂²	—	842	2	no	—
65	C₂²	#20269772	1	1	yes	characteristic
66	C₄	—	842	2	no	—
67	C₄	—	842	2	no	—
68	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
69	C₄	D₄×D₄₂₁	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	1684	2	no	—
72	C₂	#27212328	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	C₄×D₄
prime radical	#27212388	C₂
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#108	D₄₂₁
hypocenter	C₄₂₁	#443
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
2. #27212374	C₂	semidirect
3. #27212292	C₂	semidirect
4. C₈×Dic₄₂₁	C₂	semidirect
5. #108×C₄₂₁	C₂	semidirect
7. Dic₁₆₈₄	C₄	semidirect
8. #20269760	C₄	semidirect
9. D₃₃₆₈	C₄	semidirect
36. C₄₂₁	#443	semidirect
40. #108	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212388	2
#27212374	2
#27212292	2
C₈×Dic₄₂₁	2
#108×C₄₂₁	2
#443	421