Subgroups — #41003076

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	#27212340	C₂	1	2	yes	maximal, complemented
3	#27212300	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212288	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₈₄	C₂²	1	2	yes	—
7	C₂×C₂²⋊D₄₂₁	—	2	2	no	—
8	D₃₃₆₈	—	2	2	no	—
9	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
10	#20269748	—	2	2	no	—
11	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₄₂₁	—	2	2	no	—
13	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄₂₁	—	2	2	no	—
16	D₁₆₈₄	—	2	2	no	—
17	D₁₆₈₄	—	2	2	no	—
18	C₂²⋊D₄₂₁	—	4	4	no	—
19	C₂×Dic₄₂₁	—	2	2	no	—
20	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₂₁	—	2	1	no	—
22	C₂×C₄×C₄₂₁	—	2	1	no	—
23	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₈×C₄₂₁	—	2	1	no	—
26	C₈×C₄₂₁	D₄	1	2	yes	complemented
27	C₂²×C₄₂₁	—	2	1	no	—
28	C₂²×C₄₂₁	—	2	1	no	—
29	C₂²×C₄₂₁	D₈	1	2	yes	complemented
30	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
31	C₂×D₄₂₁	—	4	2	no	—
32	C₂×D₄₂₁	—	4	2	no	—
33	C₂×D₄₂₁	—	4	2	no	—
34	C₄×C₄₂₁	—	2	1	no	—
35	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₂₁	#41	1	1	yes	characteristic
37	Dic₄₂₁	—	4	2	no	—
38	C₂×C₄₂₁	—	2	2	no	—
39	C₂×C₄₂₁	#121	1	1	yes	characteristic
40	C₂×C₄₂₁	C₂×D₈	1	1	yes	characteristic
41	C₂×C₄₂₁	#135	1	1	yes	characteristic
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	D₄₂₁	1	1	yes	characteristic, complemented
47	#121	—	421	2	no	—
48	#107	—	421	1	no	—
49	#102	—	421	2	no	—
50	C₂×D₄	—	842	2	no	—
51	C₂×D₄	—	421	2	no	—
52	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
53	D₈	—	842	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
56	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
57	C₄⋊C₄	—	421	2	no	—
58	C₂²⋊C₄	—	842	2	no	—
59	C₂³	—	842	2	no	—
60	C₂³	D₁₆₈₄	1	1	yes	characteristic
61	D₄	—	842	2	no	—
62	D₄	—	842	2	no	—
63	D₄	—	1684	4	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	842	2	no	—
67	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂²⋊D₄₂₁	1	2	yes	complemented
71	C₂²	—	2	1	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	1684	2	no	—
74	C₂²	—	1684	2	no	—
75	C₂²	—	1684	2	no	—
76	C₂²	D₃₃₆₈	1	2	yes	complemented
77	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	1684	2	no	—
80	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
81	C₄	#20269771	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	—	3368	2	no	—
84	C₂	#27212340	1	1	yes	characteristic
85	C₂	C₂×D₃₃₆₈	1	1	yes	characteristic
86	C₂	#27212388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₂²×C₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₂²×C₈	D₄₂₁
hypocenter	C₄₂₁	#466
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₃₃₆₈	C₂	semidirect
2. #27212340	C₂	semidirect
3. #27212300	C₂	semidirect
4. #27212288	C₂	semidirect
5. C₂²×C₈×C₄₂₁	C₂	semidirect
26. C₈×C₄₂₁	D₄	semidirect
29. C₂²×C₄₂₁	D₈	semidirect
43. C₄₂₁	#466	semidirect
46. C₂²×C₈	D₄₂₁	semidirect
70. C₈	C₂²⋊D₄₂₁	semidirect
76. C₂²	D₃₃₆₈	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
#27212340	2
#27212300	2
#27212288	2
C₂²×C₈×C₄₂₁	2
#466	421