Subgroups — #41003112

Order 26944.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212397	C₂	1	2	yes	maximal
2	#27212372	C₂	1	1	yes	maximal, characteristic
3	#27212302	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212285	C₂	1	2	yes	maximal, complemented
5	C₂×Q₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269771	C₂²	1	1	yes	characteristic
7	#20269761	—	2	2	no	—
8	#20269758	—	2	2	no	—
9	#20269755	C₂²	1	2	yes	—
10	#20269755	C₂²	1	1	yes	characteristic
11	#20269754	—	2	1	no	—
12	C₈×D₄₂₁	—	2	1	no	—
13	C₂×Q₈×C₄₂₁	C₂²	1	2	yes	—
14	Q₁₆×C₄₂₁	—	2	1	no	—
15	Q₁₆×C₄₂₁	—	2	1	no	—
16	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
17	Dic₈₄₂	D₄	1	1	yes	characteristic
18	D₁₆₈₄	D₄	1	1	yes	characteristic
19	C₂²⋊D₄₂₁	—	2	1	no	—
20	C₄×D₄₂₁	—	2	1	no	—
21	#16809890	—	2	1	no	—
22	#16809890	—	2	2	no	—
23	Q₈×C₄₂₁	—	2	2	no	—
24	Q₈×C₄₂₁	—	2	2	no	—
25	C₂×C₄×C₄₂₁	—	2	2	no	—
26	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
27	C₈×C₄₂₁	D₄	1	1	yes	characteristic
28	C₈×C₄₂₁	D₄	1	1	yes	characteristic
29	C₂²×C₄₂₁	#41	1	1	yes	characteristic
30	C₂×D₄₂₁	—	2	1	no	—
31	C₄×C₄₂₁	—	4	2	no	—
32	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
34	Dic₄₂₁	—	2	1	no	—
35	C₂×C₄₂₁	—	2	1	no	—
36	C₂×C₄₂₁	#121	1	1	yes	characteristic
37	D₄₂₁	—	4	1	no	—
38	C₄₂₁	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	421	1	no	maximal, Sylow
40	#137	—	421	2	no	—
41	C₂×Q₁₆	D₄₂₁	1	1	yes	characteristic, complemented
42	#131	—	421	1	no	—
43	#108	—	421	1	no	—
44	#101	—	421	2	no	—
45	#41	—	421	1	no	—
46	C₂×Q₈	C₂×D₄₂₁	1	2	yes	—
47	Q₁₆	—	2	1	no	—
48	Q₁₆	—	2	1	no	—
49	Q₁₆	—	842	2	no	—
50	SD₁₆	—	842	2	no	—
51	M₁₆	—	842	1	no	—
52	M₁₆	—	421	1	no	—
53	M₁₆	—	421	2	no	—
54	C₂×C₈	—	842	1	no	—
55	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
56	Q₈	—	2	2	no	—
57	Q₈	—	2	2	no	—
58	Q₈	—	421	1	no	—
59	D₄	—	842	1	no	—
60	D₄	—	421	1	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	842	1	no	—
63	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
64	C₈	—	842	2	no	—
65	C₈	—	842	1	no	—
66	C₈	C₂²⋊D₄₂₁	1	1	yes	characteristic
67	C₈	C₂²⋊D₄₂₁	1	1	yes	characteristic
68	C₂²	—	842	1	no	—
69	C₂²	#20269770	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	842	1	no	—
72	C₄	D₄×D₄₂₁	1	1	yes	characteristic
73	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	1684	1	no	—
76	C₂	#27212337	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212337
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₂×Q₁₆×C₄₂₁	C₂
socle	C₂×C₄₂₁	#121
prime radical	#20269771	C₂²
hypercenter	C₂×Q₁₆	D₄₂₁
hypocenter	C₄₂₁	#473
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #27212302	C₂	semidirect
4. #27212285	C₂	semidirect
5. C₂×Q₁₆×C₄₂₁	C₂	semidirect
38. C₄₂₁	#473	semidirect
41. C₂×Q₁₆	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212397	2
#27212372	2
#27212302	2
#27212285	2
C₂×Q₁₆×C₄₂₁	2
#473	421