Subgroups — #41003172

Order 26944.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212346	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₄₂₁	C₂	1	1	yes	maximal, characteristic
3	#27212297	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212291	C₂	1	1	yes	maximal, characteristic, complemented
5	#27212289	C₂	1	1	yes	maximal, characteristic, complemented
6	#27212278	C₂	1	1	yes	maximal, characteristic
7	#106×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄₂₁	—	2	1	no	—
10	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
11	C₂×#16809890	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₂₁	—	2	1	no	—
13	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
14	#20269750	—	2	1	no	—
15	#20269748	—	2	1	no	—
16	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
19	C₂²×D₄₂₁	D₄	1	1	yes	characteristic
20	Dic₈₄₂	—	2	1	no	—
21	Dic₈₄₂	—	2	1	no	—
22	C₂×Dic₄₂₁	—	2	1	no	—
23	C₂×Dic₄₂₁	—	2	1	no	—
24	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
25	C₄×D₄₂₁	—	2	1	no	—
26	C₄×D₄₂₁	—	2	1	no	—
27	C₄×D₄₂₁	—	4	2	no	—
28	#16809890	—	2	1	no	—
29	C₂×C₄×C₄₂₁	—	2	1	no	—
30	C₂×C₄×C₄₂₁	—	2	1	no	—
31	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
32	C₈×C₄₂₁	—	2	1	no	—
33	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
34	C₂×D₄₂₁	—	2	1	no	—
35	C₂×D₄₂₁	—	2	1	no	—
36	C₂×D₄₂₁	SD₁₆	1	2	yes	—
37	C₄×C₄₂₁	—	4	1	no	—
38	C₄×C₄₂₁	—	4	1	no	—
39	C₄×C₄₂₁	#41	1	1	yes	characteristic
40	C₄×C₄₂₁	#41	1	1	yes	characteristic
41	Dic₄₂₁	—	2	1	no	—
42	Dic₄₂₁	—	4	1	no	—
43	Dic₄₂₁	—	4	1	no	—
44	C₂×C₄₂₁	#123	1	1	yes	characteristic
45	C₂×C₄₂₁	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₄₂₁	#137	1	1	yes	characteristic
47	D₄₂₁	—	2	2	no	—
48	C₄₂₁	#483	1	1	yes	characteristic, complemented, Sylow
49	#483	—	421	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	421	1	no	—
51	C₂×C₄⋊C₄	—	421	1	no	—
52	#106	—	421	1	no	—
53	#106	D₄₂₁	1	1	yes	characteristic, complemented
54	#103	—	421	1	no	—
55	#103	—	421	1	no	—
56	C₂²⋊C₈	—	421	1	no	—
57	C₂×Q₈	—	421	1	no	—
58	C₂²×C₄	—	842	1	no	—
59	C₂²×C₄	—	421	1	no	—
60	C₂×C₈	—	421	1	no	—
61	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
62	C₄⋊C₄	—	842	1	no	—
63	C₄⋊C₄	—	842	1	no	—
64	C₄⋊C₄	—	421	1	no	—
65	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
66	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
67	C₂²⋊C₄	—	842	1	no	—
68	C₂³	—	421	1	no	—
69	Q₈	—	842	1	no	—
70	Q₈	—	842	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	842	1	no	—
74	C₂×C₄	—	842	1	no	—
75	C₂×C₄	—	842	1	no	—
76	C₂×C₄	—	842	1	no	—
77	C₂×C₄	—	1684	2	no	—
78	C₂×C₄	—	421	1	no	—
79	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	842	1	no	—
82	C₂²	—	842	1	no	—
83	C₂²	—	842	1	no	—
84	C₂²	—	421	2	no	—
85	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	842	1	no	—
89	C₄	—	1684	1	no	—
90	C₄	—	1684	1	no	—
91	C₄	#20269771	1	1	yes	characteristic
92	C₄	#20269772	1	1	yes	characteristic
93	C₂	—	842	2	no	—
94	C₂	#27212351	1	1	yes	characteristic
95	C₂	SD₁₆×D₄₂₁	1	1	yes	characteristic
96	C₂	#27212395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#106×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂²×D₄₂₁	D₄
hypercenter	#106	D₄₂₁
hypocenter	C₄₂₁	#483
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #27212297	C₂	semidirect
4. #27212291	C₂	semidirect
5. #27212289	C₂	semidirect
7. #106×C₄₂₁	C₂	semidirect
48. C₄₂₁	#483	semidirect
53. #106	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212346	2
C₄⋊C₄×D₄₂₁	2
#27212297	2
#27212291	2
#27212289	2
#27212278	2
#106×C₄₂₁	2
#483	421