Subgroups — #14580813

Order 2624.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₈₂	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₈₂	C₂	1	4	yes	maximal, complemented
3	#13050129	C₂	1	4	yes	maximal, complemented
4	#13050123	C₂	1	1	yes	maximal, characteristic
5	#13050120	C₂	1	4	yes	maximal
6	#117×C₄₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×C₄₁	C₂²	1	1	yes	characteristic
8	C₂²×Dic₄₁	C₂²	1	2	yes	—
9	C₂×Dic₈₂	C₄	1	8	yes	complemented
10	C₂×Dic₈₂	C₂²	1	2	yes	—
11	C₂×Dic₈₂	C₂²	1	2	yes	—
12	C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
13	C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
14	C₄²×C₄₁	C₂²	1	2	yes	—
15	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
16	C₄⋊Dic₄₁	C₂²	1	2	yes	—
17	#10620279	C₂²	1	8	yes	—
18	C₄×Dic₄₁	C₂²	1	8	yes	—
19	C₄×Dic₄₁	C₂²	1	2	yes	—
20	C₂³×C₄₁	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₄₁	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₄₁	C₂³	1	4	yes	—
23	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₄₁	C₂×C₄	1	8	yes	—
26	C₂×Dic₄₁	C₂³	1	4	yes	—
27	C₂×Dic₄₁	C₂³	1	4	yes	—
28	Dic₈₂	C₂×C₄	1	8	yes	—
29	Dic₈₂	C₂×C₄	1	8	yes	—
30	C₂²×C₄₁	—	2	2	no	—
31	C₂²×C₄₁	C₂²×C₄	1	2	yes	—
32	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
33	C₄×C₄₁	—	2	4	no	—
34	C₄×C₄₁	C₂²×C₄	1	2	yes	—
35	C₄×C₄₁	C₂²×C₄	1	2	yes	—
36	Dic₄₁	—	2	4	no	—
37	Dic₄₁	C₂²×C₄	1	8	yes	—
38	C₂×C₄₁	—	2	2	no	—
39	C₂×C₄₁	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₄₁	2₋¹⁺⁴	1	2	yes	—
41	#519	—	41	1	no	maximal, Sylow
42	C₄₁	#519	1	1	yes	characteristic, complemented, Sylow
43	C₂²×Q₈	—	41	1	no	—
44	C₄×Q₈	—	41	4	no	—
45	C₄×Q₈	—	41	4	no	—
46	#117	—	41	1	no	—
47	#117	—	41	4	no	—
48	#117	D₄₁	1	1	yes	characteristic
49	C₂×Q₈	—	41	8	no	—
50	C₂×Q₈	—	41	2	no	—
51	C₂×Q₈	—	41	2	no	—
52	C₂²×C₄	—	41	2	no	—
53	C₂²×C₄	C₂×D₄₁	1	1	yes	characteristic
54	C₄⋊C₄	—	41	8	no	—
55	C₄⋊C₄	—	41	2	no	—
56	C₄⋊C₄	C₂×D₄₁	1	2	yes	—
57	C₂²⋊C₄	—	41	2	no	—
58	C₂²⋊C₄	C₂×D₄₁	1	2	yes	—
59	C₄²	—	41	8	no	—
60	C₄²	—	41	2	no	—
61	C₄²	C₂×D₄₁	1	2	yes	—
62	C₂³	C₂²×D₄₁	1	1	yes	characteristic
63	Q₈	—	41	8	no	—
64	Q₈	—	41	8	no	—
65	C₂×C₄	—	41	8	no	—
66	C₂×C₄	—	41	4	no	—
67	C₂×C₄	—	41	4	no	—
68	C₂×C₄	C₄×D₄₁	1	4	yes	—
69	C₂×C₄	C₂²×D₄₁	1	4	yes	—
70	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
72	C₂²	—	2	2	no	—
73	C₂²	C₂×C₄×D₄₁	1	2	yes	—
74	C₂²	C₂³×D₄₁	1	1	yes	characteristic
75	C₄	—	2	4	no	—
76	C₄	—	41	8	no	—
77	C₄	—	82	4	no	—
78	C₄	C₂×C₄×D₄₁	1	2	yes	—
79	C₄	C₂×C₄×D₄₁	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	C₂²×C₄×D₄₁	1	1	yes	characteristic
82	C₂	#13050211	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator	C₂×C₄₁	C₂³×C₄
Fitting	#117×C₄₁	C₂
socle, cosocle	C₂²×C₄₁	C₂⁴
prime radical	C₂³×C₄₁	C₂³
hypercenter	#117	D₄₁
hypocenter	C₄₁	#519
LCS	G ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₈₂	C₂	semidirect
3. #13050129	C₂	semidirect
9. C₂×Dic₈₂	C₄	semidirect
42. C₄₁	#519	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₈₂	2
C₄×Dic₈₂	2
#13050129	2
#13050123	2
#13050120	2
#117×C₄₁	2
#519	41