Subgroups — #14580538

Order 2624.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10620284	C₂	1	2	yes	maximal
2	C₄×D₁₆₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#66491	C₂	1	1	yes	maximal, characteristic, complemented
4	#13050088	C₂	1	2	yes	maximal, complemented
5	C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₁	C₂²	1	2	yes	—
7	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄₁	—	2	1	no	—
9	C₄⋊C₄×C₄₁	—	2	1	no	—
10	C₄⋊C₄×C₄₁	—	2	1	no	—
11	C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	#10620284	—	2	4	no	—
13	#10620281	—	2	1	no	—
14	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
15	C₂×#66491	C₂²	1	2	yes	—
16	C₂²×D₄₁	—	2	1	no	—
17	Q₈×C₄₁	—	2	2	no	—
18	Q₈×C₄₁	—	2	2	no	—
19	C₂×C₄×C₄₁	—	2	2	no	—
20	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
23	C₂×Dic₄₁	—	2	1	no	—
24	D₁₆₄	—	2	1	no	—
25	D₁₆₄	D₄	1	2	yes	—
26	C₄×D₄₁	—	4	2	no	—
27	#66491	—	2	2	no	—
28	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
29	C₂×D₄₁	—	2	2	no	—
30	C₂×D₄₁	—	4	1	no	—
31	C₂×D₄₁	—	4	1	no	—
32	C₄×C₄₁	—	2	1	no	—
33	C₄×C₄₁	—	4	2	no	—
34	C₄×C₄₁	SD₁₆	1	2	yes	complemented
35	C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₁	#41	1	1	yes	characteristic
37	Dic₄₁	—	4	1	no	—
38	C₂×C₄₁	#121	1	1	yes	characteristic
39	C₂×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₄₁	#137	1	1	yes	characteristic
41	D₄₁	—	4	2	no	—
42	#461	—	41	1	no	maximal, Sylow
43	C₄₁	#461	1	1	yes	characteristic, complemented, Sylow
44	C₂×SD₁₆	—	41	2	no	—
45	C₄⋊Q₈	D₄₁	1	1	yes	characteristic, complemented
46	C₄×D₄	—	41	1	no	—
47	C₄⋊C₈	—	41	1	no	—
48	#103	—	41	2	no	—
49	C₂×Q₈	C₂×D₄₁	1	2	yes	—
50	C₂×D₄	—	41	1	no	—
51	C₂²×C₄	—	82	1	no	—
52	SD₁₆	—	82	4	no	—
53	C₂×C₈	—	41	2	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	41	1	no	—
57	C₂²⋊C₄	—	82	1	no	—
58	C₄²	C₂×D₄₁	1	1	yes	characteristic
59	C₂³	—	82	1	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	2	2	no	—
62	D₄	—	41	2	no	—
63	D₄	—	82	1	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	82	1	no	—
66	C₂×C₄	—	164	2	no	—
67	C₂×C₄	C₂²⋊D₄₁	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₄₁	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
70	C₈	—	82	2	no	—
71	C₂²	—	82	2	no	—
72	C₂²	—	164	1	no	—
73	C₂²	—	164	1	no	—
74	C₂²	C₂×C₂²⋊D₄₁	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	2	no	—
77	C₄	—	164	1	no	—
78	C₄	#10620284	1	2	yes	complemented
79	C₄	D₄×D₄₁	1	1	yes	characteristic
80	C₄	#10620308	1	1	yes	characteristic
81	C₂	—	164	2	no	—
82	C₂	#13050135	1	1	yes	characteristic
83	C₂	C₂×#10620284	1	1	yes	characteristic
84	C₂	#13050195	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₁
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₄⋊Q₈×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
prime radical	C₂×D₁₆₄	C₂²
hypercenter	C₄⋊Q₈	D₄₁
hypocenter	C₄₁	#461
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊#66491	C₂	semidirect
4. #13050088	C₂	semidirect
5. C₄⋊Q₈×C₄₁	C₂	semidirect
34. C₄×C₄₁	SD₁₆	semidirect
43. C₄₁	#461	semidirect
45. C₄⋊Q₈	D₄₁	semidirect
78. C₄	#10620284	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10620284	2
C₄×D₁₆₄	2
C₄⋊#66491	2
#13050088	2
C₄⋊Q₈×C₄₁	2
#461	41