Subgroups — C₂×#63076 #10586983

Order 512.

56326 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#585	C₂	1	1	yes	maximal, characteristic
2	#63076	C₂	1	64	yes	maximal, direct product factor
3	C₂×#3573	C₂	1	32	yes	maximal, complemented
4	C₂²×#568	C₂	1	30	yes	maximal, complemented
5	C₂×#585	C₄	1	16	yes	—
6	C₂×#585	C₄	1	32	yes	—
7	C₂×#585	C₂²	1	16	yes	—
8	C₂×#585	C₂²	1	32	yes	—
9	C₂²×2₋¹⁺⁴	C₄	1	6	yes	—
10	C₂²×H(5,𝔽₂)	C₄	1	10	yes	—
11	C₂³×#41	C₂²	1	15	yes	—
12	#3573	C₂²	1	1024	yes	complemented
13	C₂×#568	C₂²	1	80	yes	complemented
14	C₂×#568	C₂²	1	240	yes	complemented
15	C₂×#568	C₂²	1	960	yes	complemented
16	C₂²×#131	C₂²	1	240	yes	complemented
17	C₂³×M₁₆	C₂²	1	60	yes	complemented
18	#585	C₂×C₄	1	256	yes	—
19	#585	C₂×C₄	1	512	yes	—
20	#585	C₂³	1	256	yes	—
21	C₂×2₋¹⁺⁴	C₂×C₄	1	96	yes	—
22	C₂×2₋¹⁺⁴	C₂×C₄	1	192	yes	—
23	C₂×H(5,𝔽₂)	C₂×C₄	1	160	yes	—
24	C₂×H(5,𝔽₂)	C₂×C₄	1	320	yes	—
25	C₂²×#41	C₂×C₄	1	60	yes	—
26	C₂²×#41	C₂×C₄	1	120	yes	—
27	C₂²×#41	C₂×C₄	1	240	yes	—
28	C₂²×#41	C₂³	1	20	yes	—
29	C₂²×#41	C₂³	1	120	yes	—
30	C₂²×#41	C₂³	1	240	yes	—
31	C₂³×Q₈	C₂×C₄	1	15	yes	—
32	C₂³×D₄	C₂×C₄	1	45	yes	—
33	C₂⁴×C₄	C₂³	1	15	yes	—
34	#568	C₂³	1	1280	yes	complemented
35	#568	C₂³	1	3840	yes	complemented
36	C₂×#131	C₂³	1	640	yes	complemented
37	C₂×#131	C₂³	1	3840	yes	complemented
38	C₂²×M₁₆	C₂³	1	120	yes	—
39	C₂²×M₁₆	C₂³	1	360	yes	complemented
40	C₂²×M₁₆	C₂³	1	960	yes	complemented
41	C₂³×C₈	C₂³	1	120	yes	complemented
42	C₂⁵	C₂²×C₄	1	15	yes	—
43	2₋¹⁺⁴	C₂²×C₄	1	1536	yes	—
44	H(5,𝔽₂)	C₂²×C₄	1	2560	yes	—
45	C₂×#41	C₂²×C₄	1	80	yes	—
46	C₂×#41	C₂²×C₄	1	160	yes	—
47	C₂×#41	C₂²×C₄	1	480	yes	—
48	C₂×#41	C₂²×C₄	1	960	yes	—
49	C₂×#41	C₂²×C₄	1	960	yes	—
50	C₂×#41	C₂²×C₄	1	1920	yes	—
51	C₂×#41	C₂⁴	1	80	yes	—
52	C₂×#41	C₂⁴	1	960	yes	—
53	C₂×#41	C₂⁴	1	160	yes	—
54	C₂²×Q₈	C₂²×C₄	1	20	yes	—
55	C₂²×Q₈	C₂²×C₄	1	120	yes	—
56	C₂²×Q₈	C₂²×C₄	1	240	yes	—
57	C₂²×D₄	C₂²×C₄	1	60	yes	—
58	C₂²×D₄	C₂²×C₄	1	360	yes	—
59	C₂²×D₄	C₂²×C₄	1	720	yes	—
60	C₂³×C₄	C₂²×C₄	1	45	yes	—
61	C₂³×C₄	C₂²×C₄	1	60	yes	—
62	C₂³×C₄	C₂²×C₄	1	120	yes	—
63	C₂³×C₄	C₂⁴	1	60	yes	—
64	C₂³×C₄	C₂⁴	1	15	yes	—
65	C₂³×C₄	C₂⁴	1	120	yes	—
66	#131	C₂⁴	1	5120	yes	complemented
67	C₂×M₁₆	C₂⁴	1	16	yes	—
68	C₂×M₁₆	C₂⁴	1	240	yes	—
69	C₂×M₁₆	C₂⁴	1	960	yes	—
70	C₂×M₁₆	C₂⁴	1	2880	yes	complemented
71	C₂²×C₈	C₂⁴	1	240	yes	—
72	C₂²×C₈	C₂⁴	1	960	yes	complemented
73	C₂⁴	—	8	30	no	—
74	C₂⁴	C₂³×C₄	1	15	yes	—
75	C₂⁴	C₂³×C₄	1	60	yes	—
76	C₂⁴	C₂³×C₄	1	120	yes	—
77	#41	C₂³×C₄	1	320	yes	—
78	#41	C₂³×C₄	1	640	yes	—
79	#41	C₂³×C₄	1	3840	yes	—
80	#41	C₂⁵	1	320	yes	—
81	C₂×Q₈	C₂³×C₄	1	80	yes	—
82	C₂×Q₈	C₂³×C₄	1	160	yes	—
83	C₂×Q₈	C₂³×C₄	1	960	yes	—
84	C₂×D₄	C₂³×C₄	1	240	yes	—
85	C₂×D₄	C₂³×C₄	1	480	yes	—
86	C₂×D₄	C₂³×C₄	1	2880	yes	—
87	C₂²×C₄	C₂³×C₄	1	15	yes	—
88	C₂²×C₄	C₂³×C₄	1	30	yes	—
89	C₂²×C₄	C₂³×C₄	1	60	yes	—
90	C₂²×C₄	C₂³×C₄	1	180	yes	—
91	C₂²×C₄	C₂³×C₄	1	360	yes	—
92	C₂²×C₄	C₂³×C₄	1	240	yes	—
93	C₂²×C₄	C₂³×C₄	1	480	yes	—
94	C₂²×C₄	C₂⁵	1	240	yes	—
95	C₂²×C₄	C₂⁵	1	60	yes	—
96	C₂²×C₄	C₂⁵	1	30	yes	—
97	C₂²×C₄	C₂⁵	1	1	yes	characteristic
98	M₁₆	C₂⁵	1	64	yes	—
99	M₁₆	C₂⁵	1	960	yes	—
100	C₂×C₈	C₂⁵	1	960	yes	—
101	C₂×C₈	C₂⁵	1	32	yes	—
102	C₂³	—	4	30	no	—
103	C₂³	—	4	120	no	—
104	C₂³	—	8	120	no	—
105	C₂³	C₂⁴×C₄	1	1	yes	characteristic
106	C₂³	C₂⁴×C₄	1	30	yes	—
107	C₂³	C₂⁴×C₄	1	60	yes	—
108	C₂³	C₂⁴×C₄	1	240	yes	—
109	Q₈	C₂⁴×C₄	1	320	yes	—
110	D₄	C₂⁴×C₄	1	960	yes	—
111	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
112	C₂×C₄	C₂⁴×C₄	1	2	yes	—
113	C₂×C₄	C₂⁴×C₄	1	60	yes	—
114	C₂×C₄	C₂⁴×C₄	1	30	yes	—
115	C₂×C₄	C₂⁴×C₄	1	120	yes	—
116	C₂×C₄	C₂⁴×C₄	1	60	yes	—
117	C₂×C₄	C₂⁴×C₄	1	720	yes	—
118	C₂×C₄	C₂⁶	1	60	yes	—
119	C₂×C₄	C₂⁶	1	2	yes	—
120	C₂×C₄	C₂⁶	1	1	yes	characteristic
121	C₈	C₂⁶	1	64	yes	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	60	no	—
124	C₂²	—	4	60	no	—
125	C₂²	—	4	240	no	—
126	C₂²	C₂⁵×C₄	1	2	yes	—
127	C₂²	C₂⁵×C₄	1	60	yes	—
128	C₂²	C₂⁵×C₄	1	1	yes	characteristic
129	C₄	C₂⁵×C₄	1	2	yes	—
130	C₄	C₂⁵×C₄	1	1	yes	characteristic
131	C₄	C₂⁵×C₄	1	60	yes	—
132	C₄	C₂⁷	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	2	60	no	—
135	C₂	#63076	1	2	yes	direct product factor
136	C₂	C₂⁶×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂⁶
commutator	C₂	C₂⁶×C₄
Frattini, cosocle	C₄	C₂⁷
socle	C₂²	C₂⁵×C₄
prime radical	C₂²×#585	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #63076	C₂	direct (N × H)
3. C₂×#3573	C₂	semidirect
4. C₂²×#568	C₂	semidirect
12. #3573	C₂²	semidirect
13. C₂×#568	C₂²	semidirect
14. C₂×#568	C₂²	semidirect
15. C₂×#568	C₂²	semidirect
16. C₂²×#131	C₂²	semidirect
17. C₂³×M₁₆	C₂²	semidirect
34. #568	C₂³	semidirect
35. #568	C₂³	semidirect
36. C₂×#131	C₂³	semidirect
37. C₂×#131	C₂³	semidirect
39. C₂²×M₁₆	C₂³	semidirect
40. C₂²×M₁₆	C₂³	semidirect
41. C₂³×C₈	C₂³	semidirect
66. #131	C₂⁴	semidirect
70. C₂×M₁₆	C₂⁴	semidirect
72. C₂²×C₈	C₂⁴	semidirect
135. C₂	#63076	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#585	2
#63076	2
C₂×#3573	2
C₂²×#568	2

Subgroups — C2×#63076 #10586983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#63076 #10586983