Subgroups — #14580532

Order 2624.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₂₈	C₂	1	2	yes	maximal, complemented
2	#13050161	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#13050086	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₄	—	2	1	no	—
7	C₂×D₁₆₄	—	2	1	no	—
8	C₂×D₁₆₄	C₂²	1	2	yes	—
9	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄₁	—	2	1	no	—
11	C₂×C₈×C₄₁	C₂²	1	2	yes	—
12	C₄²×C₄₁	C₂²	1	1	yes	characteristic
13	#10620281	—	2	1	no	—
14	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
15	D₃₂₈	—	2	4	no	—
16	C₂²×D₄₁	—	2	1	no	—
17	C₂²×D₄₁	—	2	2	no	—
18	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₄₁	—	2	1	no	—
22	D₁₆₄	—	2	2	no	—
23	D₁₆₄	—	2	1	no	—
24	D₁₆₄	—	2	2	no	—
25	D₁₆₄	—	4	2	no	—
26	D₁₆₄	—	4	2	no	—
27	D₁₆₄	D₄	1	2	yes	complemented
28	C₄×D₄₁	—	4	2	no	—
29	C₈×C₄₁	—	2	2	no	—
30	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
31	C₂×D₄₁	—	2	2	no	—
32	C₂×D₄₁	—	4	1	no	—
33	C₂×D₄₁	—	4	1	no	—
34	C₂×D₄₁	—	4	2	no	—
35	C₂×D₄₁	—	4	2	no	—
36	C₂×D₄₁	—	4	2	no	—
37	C₄×C₄₁	—	2	1	no	—
38	C₄×C₄₁	D₈	1	2	yes	complemented
39	C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
40	C₄×C₄₁	#41	1	1	yes	characteristic
41	Dic₄₁	—	4	1	no	—
42	C₂×C₄₁	#121	1	1	yes	characteristic
43	C₂×C₄₁	C₂×D₈	1	1	yes	characteristic
44	C₂×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₄₁	—	4	2	no	—
46	D₄₁	—	8	2	no	—
47	#459	—	41	1	no	maximal, Sylow
48	C₄₁	#459	1	1	yes	characteristic, complemented, Sylow
49	C₂×D₈	—	41	2	no	—
50	#127	—	41	1	no	—
51	C₄×D₄	—	41	1	no	—
52	C₄⋊C₈	D₄₁	1	1	yes	characteristic, complemented
53	#102	—	41	2	no	—
54	C₂×D₄	—	41	1	no	—
55	C₂×D₄	—	41	2	no	—
56	C₂×D₄	—	82	1	no	—
57	C₂×D₄	—	82	1	no	—
58	C₂²×C₄	—	82	1	no	—
59	D₈	—	82	4	no	—
60	C₂×C₈	C₂×D₄₁	1	2	yes	—
61	C₄⋊C₄	—	41	1	no	—
62	C₂²⋊C₄	—	82	1	no	—
63	C₄²	C₂×D₄₁	1	1	yes	characteristic
64	C₂³	—	82	1	no	—
65	C₂³	—	82	2	no	—
66	D₄	—	41	2	no	—
67	D₄	—	82	2	no	—
68	D₄	—	82	1	no	—
69	D₄	—	82	2	no	—
70	D₄	—	164	2	no	—
71	D₄	—	164	2	no	—
72	C₂×C₄	—	82	1	no	—
73	C₂×C₄	—	164	2	no	—
74	C₂×C₄	D₁₆₄	1	1	yes	characteristic
75	C₂×C₄	D₁₆₄	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₂²	—	82	2	no	—
79	C₂²	—	164	2	no	—
80	C₂²	—	164	2	no	—
81	C₂²	—	164	2	no	—
82	C₂²	—	164	1	no	—
83	C₂²	—	164	1	no	—
84	C₂²	C₂×D₁₆₄	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	164	1	no	—
87	C₄	D₃₂₈	1	2	yes	complemented
88	C₄	D₄×D₄₁	1	1	yes	characteristic
89	C₄	#10620310	1	1	yes	characteristic
90	C₂	—	328	2	no	—
91	C₂	—	164	2	no	—
92	C₂	#13050139	1	1	yes	characteristic
93	C₂	C₂×D₃₂₈	1	1	yes	characteristic
94	C₂	#13050192	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₄
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₄⋊C₈×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
hypercenter	C₄⋊C₈	D₄₁
hypocenter	C₄₁	#459
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₃₂₈	C₂	semidirect
2. #13050161	C₂	semidirect
3. C₄×D₁₆₄	C₂	semidirect
4. #13050086	C₂	semidirect
5. C₄⋊C₈×C₄₁	C₂	semidirect
27. D₁₆₄	D₄	semidirect
38. C₄×C₄₁	D₈	semidirect
48. C₄₁	#459	semidirect
52. C₄⋊C₈	D₄₁	semidirect
87. C₄	D₃₂₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₂₈	2
#13050161	2
C₄×D₁₆₄	2
#13050086	2
C₄⋊C₈×C₄₁	2
#459	41