Subgroups — #65334

Order 320.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#4072	C₂	1	1	yes	maximal, characteristic
3	#3967	C₂	1	1	yes	maximal, characteristic, complemented
4	#3967	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊#168	C₂	1	1	yes	maximal, characteristic
6	#3940	C₂	1	1	yes	maximal, characteristic, complemented
7	#3939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊C₄	—	2	1	no	—
11	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₅×C₂²⋊C₄	—	2	1	no	—
13	C₂²⋊Dic₅	—	2	1	no	—
14	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
16	C₄×Dic₅	—	2	1	no	—
17	C₂×#168	C₂²	1	1	yes	characteristic
18	C₂×#168	C₂²	1	1	yes	characteristic
19	#485	—	5	1	no	maximal, Sylow
20	C₂³×C₅	D₄	1	1	yes	characteristic
21	C₅×Q₈	—	2	1	no	—
22	C₅×Q₈	—	2	1	no	—
23	C₂×C₄×C₅	—	2	1	no	—
24	C₂×C₄×C₅	—	2	1	no	—
25	C₂×C₄×C₅	—	2	1	no	—
26	C₂×C₄×C₅	—	2	1	no	—
27	C₂×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×Dic₅	—	2	1	no	—
30	C₂×Dic₅	—	2	1	no	—
31	#168	—	2	1	no	—
32	#168	—	2	1	no	—
33	C₂²⋊Q₈	D₅	1	1	yes	characteristic
34	#117	—	5	1	no	—
35	#107	—	5	1	no	—
36	#106	—	5	1	no	—
37	#103	—	5	1	no	—
38	#103	—	5	1	no	—
39	C₂²⋊C₈	—	5	1	no	—
40	C₂²×C₅	—	2	1	no	—
41	C₂²×C₅	—	2	1	no	—
42	C₂²×C₅	—	2	1	no	—
43	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
44	C₄×C₅	—	2	1	no	—
45	C₄×C₅	—	4	1	no	—
46	C₄×C₅	—	4	1	no	—
47	C₄×C₅	#41	1	1	yes	characteristic
48	C₄×C₅	#41	1	1	yes	characteristic
49	Dic₅	—	4	1	no	—
50	Dic₅	—	4	1	no	—
51	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
52	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
53	C₂×C₈	—	5	1	no	—
54	C₂×C₈	—	5	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	5	1	no	—
57	C₄⋊C₄	—	5	1	no	—
58	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
59	C₂²⋊C₄	—	2	1	no	—
60	C₂²⋊C₄	—	10	1	no	—
61	C₄²	—	10	1	no	—
62	C₂×C₅	—	4	1	no	—
63	C₂×C₅	#123	1	1	yes	characteristic
64	C₂×C₅	#135	1	1	yes	characteristic
65	C₂×C₅	#137	1	1	yes	characteristic
66	C₂³	C₂²⋊D₅	1	1	yes	characteristic
67	Q₈	—	2	1	no	—
68	Q₈	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	10	1	no	—
74	C₂×C₄	—	10	1	no	—
75	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
77	C₈	—	10	1	no	—
78	C₈	—	10	1	no	—
79	C₅	#485	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	20	1	no	—
88	C₄	—	20	1	no	—
89	C₄	#711	1	1	yes	characteristic
90	C₄	#711	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#4081	1	1	yes	characteristic
93	C₂	#4088	1	1	yes	characteristic
94	C₂	#4096	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₂²⋊Q₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂³×C₅	D₄
hypercenter	C₂²⋊Q₈	D₅
hypocenter	C₅	#485
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #3967	C₂	semidirect
4. #3967	C₂	semidirect
6. #3940	C₂	semidirect
7. #3939	C₂	semidirect
79. C₅	#485	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₂²⋊Q₈	2
#4072	2
#3967	2
C₂²⋊#168	2
#3940	2
#3939	2
#485	5