Subgroups — #41002803

Order 26944.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269755	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212278	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄₂₁	C₄	1	1	yes	characteristic
6	C₂²×Dic₄₂₁	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄₂₁	—	2	2	no	—
8	C₂×C₄×D₄₂₁	—	2	2	no	—
9	C₂×C₄×D₄₂₁	C₄	1	2	yes	—
10	C₂×C₄×D₄₂₁	C₂²	1	2	yes	—
11	#20269755	—	2	2	no	—
12	C₂×#16809890	C₂²	1	2	yes	complemented
13	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
14	M₁₆×C₄₂₁	—	2	2	no	—
15	C₂×C₈×C₄₂₁	C₂²	1	2	yes	complemented
16	C₂²×D₄₂₁	—	2	2	no	—
17	C₂²×D₄₂₁	—	2	4	no	—
18	C₂²×D₄₂₁	C₂×C₄	1	2	yes	—
19	C₂×Dic₄₂₁	—	2	2	no	—
20	C₂×Dic₄₂₁	C₂×C₄	1	2	yes	—
21	C₄×D₄₂₁	—	2	4	no	—
22	C₄×D₄₂₁	—	2	2	no	—
23	C₄×D₄₂₁	—	2	2	no	—
24	#16809890	—	2	2	no	—
25	C₂³×C₄₂₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
28	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
29	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
30	C₈×C₄₂₁	—	2	2	no	—
31	C₂²×C₄₂₁	—	2	2	no	—
32	C₂²×C₄₂₁	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×D₄₂₁	—	2	8	no	—
35	C₂×D₄₂₁	—	2	2	no	—
36	C₂×D₄₂₁	—	2	2	no	—
37	C₂×D₄₂₁	M₁₆	1	4	yes	complemented
38	C₄×C₄₂₁	C₂²⋊C₄	1	2	yes	—
39	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
40	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
41	Dic₄₂₁	—	2	2	no	—
42	C₂×C₄₂₁	—	2	2	no	—
43	C₂×C₄₂₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄₂₁	C₂×M₁₆	1	2	yes	—
45	D₄₂₁	—	2	4	no	—
46	C₄₂₁	#407	1	1	yes	characteristic, complemented, Sylow
47	#407	—	421	1	no	maximal, Sylow
48	C₂³×C₄	—	421	1	no	—
49	C₂×M₁₆	—	421	1	no	—
50	C₂×M₁₆	D₄₂₁	1	1	yes	characteristic, complemented
51	C₂²⋊C₈	—	421	4	no	—
52	C₂⁴	—	421	1	no	—
53	C₂²×C₄	—	842	2	no	—
54	C₂²×C₄	—	842	2	no	—
55	C₂²×C₄	—	421	1	no	—
56	C₂²×C₄	—	421	2	no	—
57	C₂²×C₄	—	421	2	no	—
58	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
59	M₁₆	—	2	2	no	—
60	M₁₆	—	842	2	no	—
61	C₂×C₈	—	421	2	no	—
62	C₂×C₈	C₂×D₄₂₁	1	2	yes	complemented
63	C₂³	—	842	4	no	—
64	C₂³	—	842	2	no	—
65	C₂³	—	421	2	no	—
66	C₂³	C₄×D₄₂₁	1	1	yes	characteristic
67	C₂×C₄	—	842	2	no	—
68	C₂×C₄	—	842	4	no	—
69	C₂×C₄	—	842	2	no	—
70	C₂×C₄	—	842	2	no	—
71	C₂×C₄	—	421	2	no	—
72	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₄₂₁	1	2	yes	—
74	C₂×C₄	D₁₆₈₄	1	2	yes	—
75	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₈	—	842	2	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	842	8	no	—
80	C₂²	—	842	2	no	—
81	C₂²	—	842	2	no	—
82	C₂²	—	421	4	no	—
83	C₂²	#20269748	1	2	yes	—
84	C₂²	C₂×C₄×D₄₂₁	1	1	yes	characteristic
85	C₄	—	842	2	no	—
86	C₄	#20269748	1	2	yes	—
87	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
88	C₄	C₂×D₁₆₈₄	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	—	842	4	no	—
91	C₂	C₂×#20269748	1	1	yes	characteristic
92	C₂	M₁₆×D₄₂₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₂₁
commutator, socle	C₂²×C₄₂₁	C₂²×C₄
Fitting	C₂×M₁₆×C₄₂₁	C₂
prime radical	C₂³×D₄₂₁	C₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	C₂×M₁₆	D₄₂₁
hypocenter	C₄₂₁	#407
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#20269755	C₂	semidirect
3. #27212278	C₂	semidirect
4. C₂×M₁₆×C₄₂₁	C₂	semidirect
12. C₂×#16809890	C₂²	semidirect
15. C₂×C₈×C₄₂₁	C₂²	semidirect
37. C₂×D₄₂₁	M₁₆	semidirect
46. C₄₂₁	#407	semidirect
50. C₂×M₁₆	D₄₂₁	semidirect
62. C₂×C₈	C₂×D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄₂₁	2
C₂×#20269755	2
#27212278	2
C₂×M₁₆×C₄₂₁	2
#407	421