Subgroups — #10574097

Order 512.

9993 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#63076	C₂	1	2	yes	maximal
2	C₂×#3573	C₂	1	1	yes	maximal, characteristic
3	#60113	C₂	1	30	yes	maximal, complemented
4	C₂×#2916	C₂	1	30	yes	maximal, complemented
5	C₂×#585	C₂²	1	1	yes	characteristic
6	#3573	—	2	32	no	—
7	#3573	C₄	1	32	yes	—
8	C₂×#568	C₂²	1	60	yes	—
9	C₂×#568	C₂²	1	15	yes	—
10	C₂²×#131	C₂²	1	15	yes	—
11	#2916	C₂²	1	80	yes	complemented
12	#2916	C₂²	1	240	yes	complemented
13	C₂×#430	C₂²	1	180	yes	complemented
14	C₂²×#108	C₂²	1	60	yes	complemented
15	#585	C₂×C₄	1	16	yes	—
16	#585	D₄	1	16	yes	—
17	C₂×2₋¹⁺⁴	Q₈	1	6	yes	—
18	C₂×H(5,𝔽₂)	Q₈	1	10	yes	—
19	C₂²×#41	C₂³	1	15	yes	—
20	#568	—	2	240	no	—
21	#568	C₂×C₄	1	240	yes	—
22	#568	C₂³	1	160	yes	—
23	C₂×#131	—	2	240	no	—
24	C₂×#131	C₂×C₄	1	240	yes	—
25	C₂×#131	C₂³	1	20	yes	—
26	C₂×#131	C₂³	1	60	yes	—
27	C₂²×M₁₆	C₂³	1	120	yes	—
28	C₂²×M₁₆	C₂³	1	45	yes	—
29	C₂³×C₈	C₂³	1	15	yes	—
30	#430	C₂³	1	480	yes	—
31	C₂×#108	C₂³	1	120	yes	—
32	C₂×#108	C₂³	1	360	yes	—
33	2₋¹⁺⁴	C₄⋊C₄	1	96	yes	—
34	H(5,𝔽₂)	C₄⋊C₄	1	160	yes	—
35	C₂×#41	C₂²×C₄	1	120	yes	—
36	C₂×#41	C₂×D₄	1	120	yes	—
37	C₂×#41	C₂×Q₈	1	60	yes	—
38	C₂×#41	C₂⁴	1	20	yes	—
39	C₂²×Q₈	C₂×Q₈	1	15	yes	—
40	C₂²×D₄	C₂×Q₈	1	45	yes	—
41	C₂³×C₄	C₂⁴	1	15	yes	—
42	#131	—	2	640	no	—
43	#131	C₂²×C₄	1	160	yes	—
44	#131	C₂²×C₄	1	480	yes	—
45	C₂×M₁₆	—	2	360	no	—
46	C₂×M₁₆	C₂²×C₄	1	360	yes	—
47	C₂×M₁₆	C₂⁴	1	240	yes	—
48	C₂×M₁₆	C₂⁴	1	60	yes	—
49	C₂²×C₈	—	2	120	no	—
50	C₂²×C₈	C₂²×C₄	1	120	yes	—
51	C₂²×C₈	C₂⁴	1	15	yes	—
52	C₂²×C₈	C₂⁴	1	45	yes	—
53	#108	C₂⁴	1	16	yes	—
54	#108	C₂⁴	1	240	yes	—
55	C₂⁴	C₂²×Q₈	1	15	yes	—
56	#41	C₂×C₄⋊C₄	1	480	yes	—
57	#41	C₂³×C₄	1	80	yes	—
58	#41	C₂²×D₄	1	80	yes	—
59	C₂×Q₈	C₂×C₄⋊C₄	1	120	yes	—
60	C₂×Q₈	C₂²×Q₈	1	20	yes	—
61	C₂×D₄	C₂×C₄⋊C₄	1	360	yes	—
62	C₂×D₄	C₂²×Q₈	1	60	yes	—
63	C₂²×C₄	C₂³×C₄	1	60	yes	—
64	C₂²×C₄	C₂²×D₄	1	60	yes	—
65	C₂²×C₄	C₂²×Q₈	1	45	yes	—
66	C₂²×C₄	C₂⁵	1	15	yes	—
67	M₁₆	—	2	240	no	—
68	M₁₆	C₂³×C₄	1	240	yes	—
69	M₁₆	C₂⁵	1	32	yes	—
70	C₂×C₈	—	2	240	no	—
71	C₂×C₈	C₂³×C₄	1	60	yes	—
72	C₂×C₈	C₂³×C₄	1	180	yes	—
73	C₂×C₈	C₂⁵	1	1	yes	characteristic
74	C₂×C₈	C₂⁵	1	15	yes	—
75	C₂³	—	8	15	no	—
76	C₂³	C₂²×C₄⋊C₄	1	60	yes	—
77	C₂³	C₂³×Q₈	1	15	yes	—
78	Q₈	C₂²×C₄⋊C₄	1	80	yes	—
79	D₄	C₂²×C₄⋊C₄	1	240	yes	—
80	C₂×C₄	C₂²×C₄⋊C₄	1	180	yes	—
81	C₂×C₄	C₂⁴×C₄	1	30	yes	—
82	C₂×C₄	C₂³×D₄	1	30	yes	—
83	C₂×C₄	C₂³×Q₈	1	15	yes	—
84	C₂×C₄	C₂⁶	1	1	yes	characteristic
85	C₈	—	2	32	no	—
86	C₈	C₂⁴×C₄	1	2	yes	—
87	C₈	C₂⁴×C₄	1	30	yes	—
88	C₂²	—	4	15	no	—
89	C₂²	—	4	60	no	—
90	C₂²	C₂³×C₄⋊C₄	1	30	yes	—
91	C₂²	C₂⁴×Q₈	1	1	yes	characteristic
92	C₄	C₂³×C₄⋊C₄	1	30	yes	—
93	C₄	C₂⁵×C₄	1	1	yes	characteristic
94	C₄	C₂⁴×D₄	1	1	yes	characteristic
95	C₂	—	2	1	no	—
96	C₂	—	2	30	no	—
97	C₂	C₂⁴×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂⁴×D₄
commutator	C₄	C₂⁵×C₄
Frattini, cosocle	C₂×C₄	C₂⁶
socle	C₂	C₂⁴×C₄⋊C₄
prime radical	C₂×#585	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#585 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #60113	C₂	semidirect
4. C₂×#2916	C₂	semidirect
11. #2916	C₂²	semidirect
12. #2916	C₂²	semidirect
13. C₂×#430	C₂²	semidirect
14. C₂²×#108	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#63076	2
C₂×#3573	2
#60113	2
C₂×#2916	2