Subgroups — #41003229

Order 26944.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269760	C₂	1	2	yes	maximal, complemented
2	C₂×D₃₃₆₈	C₂	1	2	yes	maximal, complemented
3	#27212363	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212354	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
7	C₂×D₁₆₈₄	—	2	2	no	—
8	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
9	C₂×D₁₆₈₄	C₂²	1	2	yes	—
10	#20269760	—	2	4	no	—
11	D₃₃₆₈	—	2	4	no	—
12	#20269748	—	2	2	no	—
13	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
14	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄₂₁	—	2	1	no	—
16	C₂²×D₄₂₁	—	2	2	no	—
17	Dic₈₄₂	—	2	2	no	—
18	D₁₆₈₄	—	2	2	no	—
19	D₁₆₈₄	—	2	2	no	—
20	D₁₆₈₄	—	2	2	no	—
21	D₁₆₈₄	—	4	4	no	—
22	C₂×Dic₄₂₁	—	2	1	no	—
23	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₈×C₄₂₁	D₄	1	4	yes	complemented
26	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₄₂₁	—	4	1	no	—
28	C₂×D₄₂₁	—	4	2	no	—
29	C₂×D₄₂₁	—	4	4	no	—
30	C₂×D₄₂₁	—	4	2	no	—
31	C₄×C₄₂₁	—	2	2	no	—
32	C₄×C₄₂₁	C₂×D₄	1	2	yes	—
33	Dic₄₂₁	—	4	1	no	—
34	C₂×C₄₂₁	#127	1	1	yes	characteristic
35	C₂×C₄₂₁	AGL(1,ℤ/8)	1	2	yes	—
36	D₄₂₁	—	8	2	no	—
37	D₄₂₁	—	8	1	no	—
38	C₄₂₁	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	421	1	no	maximal, Sylow
40	C₂×SD₁₆	—	421	2	no	—
41	C₂×D₈	—	421	2	no	—
42	#127	—	421	1	no	—
43	#124	—	421	1	no	—
44	#97	D₄₂₁	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	421	1	no	—
46	C₂×D₄	—	842	2	no	—
47	C₂×D₄	—	421	1	no	—
48	C₂×D₄	—	421	2	no	—
49	SD₁₆	—	842	4	no	—
50	D₈	—	842	4	no	—
51	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
52	C₂²⋊C₄	—	842	2	no	—
53	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
54	C₂³	—	842	1	no	—
55	C₂³	—	842	2	no	—
56	Q₈	—	842	2	no	—
57	D₄	—	842	2	no	—
58	D₄	—	842	2	no	—
59	D₄	—	842	2	no	—
60	D₄	—	1684	4	no	—
61	C₂×C₄	—	842	1	no	—
62	C₂×C₄	D₁₆₈₄	1	2	yes	—
63	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
64	C₈	D₁₆₈₄	1	4	yes	complemented
65	C₂²	—	1684	1	no	—
66	C₂²	—	1684	2	no	—
67	C₂²	—	1684	4	no	—
68	C₂²	—	1684	2	no	—
69	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	1684	1	no	—
72	C₄	C₂×D₁₆₈₄	1	2	yes	—
73	C₂	—	3368	2	no	—
74	C₂	—	3368	1	no	—
75	C₂	#27212363	1	1	yes	characteristic
76	C₂	#27212394	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#97×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	#97	D₄₂₁
hypocenter	C₄₂₁	#496
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. C₂×#20269760	C₂	semidirect
2. C₂×D₃₃₆₈	C₂	semidirect
3. #27212363	C₂	semidirect
4. #27212354	C₂	semidirect
5. #97×C₄₂₁	C₂	semidirect
25. C₈×C₄₂₁	D₄	semidirect
38. C₄₂₁	#496	semidirect
44. #97	D₄₂₁	semidirect
64. C₈	D₁₆₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269760	2
C₂×D₃₃₆₈	2
#27212363	2
#27212354	2
#97×C₄₂₁	2
#496	421