Subgroups — #41003077

Order 26944.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₃₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212341	C₂	1	2	yes	maximal, complemented
4	#27212286	C₂	1	2	yes	maximal, complemented
5	#107×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₈₄	—	2	2	no	—
7	C₂×D₁₆₈₄	C₂²	1	2	yes	—
8	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
9	D₃₃₆₈	—	2	2	no	—
10	C₂×#16809890	C₂²	1	1	yes	characteristic
11	C₈×D₄₂₁	—	2	2	no	—
12	#20269748	—	2	2	no	—
13	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
15	C₂²×D₄₂₁	—	2	2	no	—
16	C₂²×D₄₂₁	D₄	1	1	yes	characteristic
17	D₁₆₈₄	—	2	2	no	—
18	D₁₆₈₄	—	2	2	no	—
19	D₁₆₈₄	—	4	4	no	—
20	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
21	C₄×D₄₂₁	—	2	1	no	—
22	C₄×D₄₂₁	—	2	1	no	—
23	#16809890	—	2	1	no	—
24	C₂×C₄×C₄₂₁	—	2	2	no	—
25	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
26	C₈×C₄₂₁	D₄	1	2	yes	complemented
27	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
28	C₂×D₄₂₁	—	2	1	no	—
29	C₂×D₄₂₁	—	2	1	no	—
30	C₂×D₄₂₁	—	4	2	no	—
31	C₂×D₄₂₁	—	4	2	no	—
32	C₂×D₄₂₁	—	4	2	no	—
33	C₂×D₄₂₁	D₈	1	2	yes	complemented
34	C₄×C₄₂₁	—	4	2	no	—
35	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₂₁	#41	1	1	yes	characteristic
37	Dic₄₂₁	—	2	1	no	—
38	C₂×C₄₂₁	#121	1	1	yes	characteristic
39	C₂×C₄₂₁	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄₂₁	#135	1	1	yes	characteristic
41	D₄₂₁	—	2	2	no	—
42	D₄₂₁	—	8	2	no	—
43	C₄₂₁	#466	1	1	yes	characteristic, complemented, Sylow
44	#466	—	421	1	no	maximal, Sylow
45	C₂×D₈	—	421	1	no	—
46	C₂²×C₈	—	421	1	no	—
47	#121	—	421	2	no	—
48	#107	D₄₂₁	1	1	yes	characteristic, complemented
49	#102	—	421	2	no	—
50	C₂×D₄	—	842	2	no	—
51	C₂×D₄	—	421	2	no	—
52	C₂²×C₄	—	421	1	no	—
53	D₈	—	842	2	no	—
54	C₂×C₈	—	842	2	no	—
55	C₂×C₈	—	421	1	no	—
56	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
57	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
58	C₂²⋊C₄	—	842	2	no	—
59	C₂³	—	842	2	no	—
60	C₂³	—	421	1	no	—
61	D₄	—	842	2	no	—
62	D₄	—	842	2	no	—
63	D₄	—	1684	4	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	842	1	no	—
66	C₂×C₄	—	842	1	no	—
67	C₂×C₄	—	421	1	no	—
68	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
69	C₈	—	842	1	no	—
70	C₈	D₁₆₈₄	1	2	yes	complemented
71	C₂²	—	842	1	no	—
72	C₂²	—	842	1	no	—
73	C₂²	—	1684	2	no	—
74	C₂²	—	1684	2	no	—
75	C₂²	—	1684	2	no	—
76	C₂²	—	421	2	no	—
77	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
78	C₄	—	4	2	no	—
79	C₄	—	842	1	no	—
80	C₄	C₂×D₁₆₈₄	1	1	yes	characteristic
81	C₄	#20269772	1	1	yes	characteristic
82	C₂	—	3368	2	no	—
83	C₂	—	842	2	no	—
84	C₂	#27212341	1	1	yes	characteristic
85	C₂	D₈×D₄₂₁	1	1	yes	characteristic
86	C₂	#27212389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#107×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	#107	D₄₂₁
hypocenter	C₄₂₁	#466
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
1. C₂×D₃₃₆₈	C₂	semidirect
2. C₂×C₈×D₄₂₁	C₂	semidirect
3. #27212341	C₂	semidirect
4. #27212286	C₂	semidirect
5. #107×C₄₂₁	C₂	semidirect
26. C₈×C₄₂₁	D₄	semidirect
33. C₂×D₄₂₁	D₈	semidirect
43. C₄₂₁	#466	semidirect
48. #107	D₄₂₁	semidirect
70. 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×D₃₃₆₈	2
C₂×C₈×D₄₂₁	2
#27212341	2
#27212286	2
#107×C₄₂₁	2
#466	421