Subgroups — #10598868

Order 640.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#676	C₂	1	2	yes	maximal
2	C₄×#676	C₂	1	1	yes	maximal, characteristic
3	C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	#64674	C₂	1	2	yes	maximal, complemented
5	C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#676	C₄	1	2	yes	—
7	C₂×#676	C₄	1	4	yes	—
8	C₄²×D₅	C₂²	1	1	yes	characteristic
9	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₄×C₅×C₈	C₂²	1	2	yes	—
11	#3947	C₄	1	2	yes	—
12	#3947	C₄	1	4	yes	—
13	C₄×#168	C₂²	1	2	yes	—
14	C₄×#168	C₂²	1	1	yes	characteristic
15	#1455	—	5	1	no	maximal, Sylow
16	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
18	C₂×C₅×C₈	C₂×C₄	1	4	yes	—
19	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₄²×C₅	C₂³	1	1	yes	characteristic
21	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₅	C₂×C₄	1	2	yes	—
23	C₂×#168	C₂×C₄	1	2	yes	—
24	C₂×#168	C₂×C₄	1	4	yes	—
25	#676	—	2	4	no	—
26	#676	—	2	8	no	—
27	C₄×M₁₆	—	5	2	no	—
28	C₄×M₁₆	—	5	1	no	—
29	#322	—	5	2	no	—
30	#322	D₅	1	1	yes	characteristic, complemented
31	C₈²	—	5	1	no	—
32	C₂²×D₅	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
35	C₂×Dic₅	C₄²	1	1	yes	characteristic
36	C₂×Dic₅	C₄²	1	2	yes	—
37	C₄×D₅	—	2	2	no	—
38	C₄×D₅	—	2	4	no	—
39	C₅×C₈	—	2	4	no	—
40	C₅×C₈	M₁₆	1	4	yes	complemented
41	#168	—	2	2	no	—
42	#168	M₁₆	1	8	yes	complemented
43	C₂×M₁₆	—	5	4	no	—
44	C₂×M₁₆	—	5	2	no	—
45	C₂×C₄²	—	5	1	no	—
46	#97	—	5	4	no	—
47	#97	—	5	2	no	—
48	C₄×C₈	—	5	2	no	—
49	C₄×C₈	—	5	1	no	—
50	C₄×C₈	C₂×D₅	1	1	yes	characteristic
51	C₄×C₈	C₂×D₅	1	2	yes	—
52	C₂²×C₅	C₂×C₄²	1	1	yes	characteristic
53	C₂×D₅	—	2	1	no	—
54	C₂×D₅	—	2	2	no	—
55	C₄×C₅	C₂×M₁₆	1	2	yes	—
56	C₄×C₅	C₂×M₁₆	1	4	yes	—
57	Dic₅	—	2	1	no	—
58	Dic₅	—	2	2	no	—
59	C₂²×C₄	—	5	1	no	—
60	C₂²×C₄	—	5	2	no	—
61	M₁₆	—	10	8	no	—
62	M₁₆	—	10	4	no	—
63	C₂×C₈	—	5	4	no	—
64	C₂×C₈	—	5	2	no	—
65	C₂×C₈	C₄×D₅	1	4	yes	—
66	C₂×C₈	C₄×D₅	1	2	yes	—
67	C₄²	—	5	1	no	—
68	C₄²	—	5	2	no	—
69	C₄²	C₂²×D₅	1	1	yes	characteristic
70	C₂×C₅	C₄×M₁₆	1	1	yes	characteristic
71	C₂×C₅	C₄×M₁₆	1	2	yes	—
72	D₅	—	4	1	no	—
73	C₂³	—	5	1	no	—
74	C₂×C₄	—	5	1	no	—
75	C₂×C₄	—	5	2	no	—
76	C₂×C₄	—	10	2	no	—
77	C₂×C₄	—	10	4	no	—
78	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
79	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
80	C₈	—	2	4	no	—
81	C₈	—	5	8	no	—
82	C₈	—	10	2	no	—
83	C₈	#676	1	4	yes	complemented
84	C₅	#1455	1	1	yes	characteristic, complemented, Sylow
85	C₂²	—	10	1	no	—
86	C₂²	—	10	2	no	—
87	C₂²	C₄²×D₅	1	1	yes	characteristic
88	C₄	—	10	1	no	—
89	C₄	—	10	2	no	—
90	C₄	C₂×#676	1	2	yes	—
91	C₄	D₅×M₁₆	1	4	yes	—
92	C₂	—	20	1	no	—
93	C₂	C₄×#676	1	1	yes	characteristic
94	C₂	#64996	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅
commutator, socle	C₂²×C₅	C₂×C₄²
Fitting	C₅×#322	C₂
prime radical	C₂²×D₅	C₄²
cosocle	C₄²×C₅	C₂³
hypercenter	#322	D₅
hypocenter	C₅	#1455
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. C₅×#322	C₂	semidirect
4. #64674	C₂	semidirect
5. C₈×#168	C₂	semidirect
30. #322	D₅	semidirect
40. C₅×C₈	M₁₆	semidirect
42. #168	M₁₆	semidirect
83. C₈	#676	semidirect
84. C₅	#1455	semidirect

Maximal subgroups

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

subgroup	index
C₄×#676	2
C₅×#322	2
#64674	2
C₈×#168	2
#1455	5