Subgroups — Dic₁₁×#585 #15660388

Order 2816.

11346 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×Dic₁₁	C₂	1	12	yes	maximal, complemented
2	H(5,𝔽₂)×Dic₁₁	C₂	1	20	yes	maximal, complemented
3	#13081819	C₂	1	15	yes	maximal, complemented
4	C₂×#41×Dic₁₁	C₂	1	15	yes	maximal, complemented
5	C₂×C₁₁×#585	C₂	1	1	yes	maximal, characteristic
6	C₁₁×#585	C₄	1	32	yes	complemented
7	C₂×C₁₁×2₋¹⁺⁴	C₂²	1	6	yes	—
8	C₂×C₁₁×H(5,𝔽₂)	C₂²	1	10	yes	—
9	C₂²×C₁₁×#41	C₂²	1	15	yes	—
10	#10623358	C₂²	1	120	yes	complemented
11	#41×Dic₁₁	C₂²	1	20	yes	complemented
12	#41×Dic₁₁	C₂²	1	60	yes	complemented
13	#41×Dic₁₁	C₂²	1	120	yes	complemented
14	#10623343	C₂²	1	30	yes	complemented
15	C₂×Q₈×Dic₁₁	C₂²	1	30	yes	complemented
16	#10623329	C₂²	1	90	yes	complemented
17	C₂×D₄×Dic₁₁	C₂²	1	90	yes	complemented
18	C₂×#66979	C₂²	1	45	yes	complemented
19	C₂²×C₄×Dic₁₁	C₂²	1	15	yes	complemented
20	C₁₁×2₋¹⁺⁴	C₂×C₄	1	96	yes	complemented
21	C₁₁×H(5,𝔽₂)	C₂×C₄	1	160	yes	complemented
22	C₂×C₁₁×#41	C₂×C₄	1	240	yes	complemented
23	C₂×C₁₁×#41	C₂³	1	20	yes	—
24	C₂×C₁₁×#41	C₂³	1	60	yes	—
25	C₂²×Q₈×C₁₁	C₂³	1	15	yes	—
26	C₂²×D₄×C₁₁	C₂³	1	45	yes	—
27	C₂³×C₄×C₁₁	C₂³	1	15	yes	—
28	C₂³×Dic₁₁	C₂³	1	30	yes	—
29	C₂×C₂²⋊Dic₁₁	C₂³	1	90	yes	—
30	Q₈×Dic₁₁	C₂³	1	40	yes	—
31	Q₈×Dic₁₁	C₂³	1	120	yes	—
32	D₄×Dic₁₁	C₂³	1	120	yes	—
33	D₄×Dic₁₁	C₂³	1	360	yes	—
34	#66979	C₂³	1	60	yes	—
35	#66979	C₂³	1	180	yes	—
36	C₂×C₄⋊Dic₁₁	C₂³	1	90	yes	—
37	C₂×C₄×Dic₁₁	C₂³	1	15	yes	—
38	C₂×C₄×Dic₁₁	C₂³	1	45	yes	—
39	C₂×C₄×Dic₁₁	C₂³	1	90	yes	—
40	C₄×#585	—	11	1	no	maximal, Sylow
41	C₂⁴×C₁₁	C₂⁴	1	15	yes	—
42	C₁₁×#41	C₂²×C₄	1	160	yes	complemented
43	C₁₁×#41	C₂²×C₄	1	480	yes	complemented
44	C₂×Q₈×C₁₁	C₂²×C₄	1	120	yes	complemented
45	C₂×Q₈×C₁₁	C₂⁴	1	20	yes	—
46	C₂×D₄×C₁₁	C₂²×C₄	1	360	yes	complemented
47	C₂×D₄×C₁₁	C₂⁴	1	60	yes	—
48	C₂²×C₄×C₁₁	C₂²×C₄	1	120	yes	complemented
49	C₂²×C₄×C₁₁	C₂⁴	1	15	yes	—
50	C₂²×C₄×C₁₁	C₂⁴	1	45	yes	—
51	C₂²×Dic₁₁	—	4	60	no	—
52	C₂²×Dic₁₁	C₂⁴	1	30	yes	—
53	C₂²×Dic₁₁	C₂⁴	1	90	yes	—
54	C₂²⋊Dic₁₁	C₂⁴	1	120	yes	—
55	C₄⋊Dic₁₁	C₂⁴	1	120	yes	—
56	C₄×Dic₁₁	C₂⁴	1	1	yes	characteristic
57	C₄×Dic₁₁	C₂⁴	1	15	yes	—
58	C₄×Dic₁₁	C₂⁴	1	30	yes	—
59	C₄×Dic₁₁	C₂⁴	1	90	yes	—
60	C₂×#585	D₁₁	1	1	yes	characteristic
61	C₄×2₋¹⁺⁴	—	11	12	no	—
62	C₄×H(5,𝔽₂)	—	11	20	no	—
63	#3428	—	11	15	no	—
64	C₂×C₄×#41	—	11	15	no	—
65	C₂³×C₁₁	—	4	30	no	—
66	C₂³×C₁₁	C₂³×C₄	1	60	yes	—
67	C₂³×C₁₁	C₂⁵	1	15	yes	—
68	Q₈×C₁₁	C₂³×C₄	1	80	yes	—
69	D₄×C₁₁	C₂³×C₄	1	240	yes	—
70	C₂×C₄×C₁₁	C₂³×C₄	1	60	yes	—
71	C₂×C₄×C₁₁	C₂³×C₄	1	180	yes	—
72	C₂×C₄×C₁₁	C₂⁵	1	15	yes	—
73	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
74	C₂×Dic₁₁	—	2	60	no	—
75	C₂×Dic₁₁	—	4	90	no	—
76	C₂×Dic₁₁	C₂⁵	1	30	yes	—
77	C₂×Dic₁₁	C₂⁵	1	2	yes	—
78	#585	Dic₁₁	1	32	yes	direct product factor
79	C₂×2₋¹⁺⁴	C₂×D₁₁	1	6	yes	—
80	C₂×H(5,𝔽₂)	C₂×D₁₁	1	10	yes	—
81	C₂²×#41	C₂×D₁₁	1	15	yes	—
82	#520	—	11	120	no	—
83	#519	—	11	30	no	—
84	#518	—	11	90	no	—
85	C₄×#41	—	11	20	no	—
86	C₄×#41	—	11	60	no	—
87	C₄×#41	—	11	120	no	—
88	C₂×C₄×Q₈	—	11	30	no	—
89	C₂×C₄×D₄	—	11	90	no	—
90	C₂×#117	—	11	45	no	—
91	C₂²×C₄²	—	11	15	no	—
92	C₂²×C₁₁	—	2	30	no	—
93	C₂²×C₁₁	—	4	60	no	—
94	C₂²×C₁₁	C₂⁴×C₄	1	30	yes	—
95	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
96	C₄×C₁₁	C₂⁴×C₄	1	2	yes	—
97	C₄×C₁₁	C₂⁴×C₄	1	30	yes	—
98	Dic₁₁	—	2	30	no	—
99	Dic₁₁	#585	1	4	yes	direct product factor
100	2₋¹⁺⁴	C₂×Dic₁₁	1	96	yes	complemented
101	H(5,𝔽₂)	C₂×Dic₁₁	1	160	yes	complemented
102	C₂×#41	C₂×Dic₁₁	1	240	yes	complemented
103	C₂×#41	C₂²×D₁₁	1	20	yes	—
104	C₂×#41	C₂²×D₁₁	1	60	yes	—
105	C₂²×Q₈	C₂²×D₁₁	1	15	yes	—
106	C₂²×D₄	C₂²×D₁₁	1	45	yes	—
107	C₂³×C₄	—	11	30	no	—
108	C₂³×C₄	C₂²×D₁₁	1	15	yes	—
109	C₄×Q₈	—	11	40	no	—
110	C₄×Q₈	—	11	120	no	—
111	C₄×D₄	—	11	120	no	—
112	C₄×D₄	—	11	360	no	—
113	#117	—	11	60	no	—
114	#117	—	11	180	no	—
115	C₂×C₄⋊C₄	—	11	90	no	—
116	C₂×C₂²⋊C₄	—	11	90	no	—
117	C₂×C₄²	—	11	45	no	—
118	C₂×C₄²	—	11	90	no	—
119	C₂×C₄²	—	11	15	no	—
120	C₂×C₁₁	—	2	30	no	—
121	C₂×C₁₁	C₂⁵×C₄	1	1	yes	characteristic
122	C₂×C₁₁	C₂×#585	1	2	yes	—
123	C₂⁴	C₂³×D₁₁	1	15	yes	—
124	#41	C₂²×Dic₁₁	1	160	yes	complemented
125	#41	C₂²×Dic₁₁	1	480	yes	complemented
126	C₂×Q₈	C₂²×Dic₁₁	1	120	yes	complemented
127	C₂×Q₈	C₂³×D₁₁	1	20	yes	—
128	C₂×D₄	C₂²×Dic₁₁	1	360	yes	complemented
129	C₂×D₄	C₂³×D₁₁	1	60	yes	—
130	C₂²×C₄	—	11	30	no	—
131	C₂²×C₄	—	11	90	no	—
132	C₂²×C₄	—	44	60	no	—
133	C₂²×C₄	C₂²×Dic₁₁	1	120	yes	complemented
134	C₂²×C₄	C₂³×D₁₁	1	45	yes	—
135	C₂²×C₄	C₂³×D₁₁	1	15	yes	—
136	C₄⋊C₄	—	11	120	no	—
137	C₂²⋊C₄	—	11	120	no	—
138	C₄²	—	11	30	no	—
139	C₄²	—	11	90	no	—
140	C₄²	—	11	1	no	—
141	C₄²	—	11	15	no	—
142	C₁₁	C₄×#585	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	4	30	no	—
144	C₂³	C₂³×Dic₁₁	1	60	yes	—
145	C₂³	C₂⁴×D₁₁	1	15	yes	—
146	Q₈	C₂³×Dic₁₁	1	80	yes	—
147	D₄	C₂³×Dic₁₁	1	240	yes	—
148	C₂×C₄	—	11	2	no	—
149	C₂×C₄	—	11	30	no	—
150	C₂×C₄	—	22	60	no	—
151	C₂×C₄	—	44	90	no	—
152	C₂×C₄	C₂³×Dic₁₁	1	60	yes	—
153	C₂×C₄	C₂³×Dic₁₁	1	180	yes	—
154	C₂×C₄	C₂⁴×D₁₁	1	15	yes	—
155	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
156	C₂²	—	2	30	no	—
157	C₂²	—	4	60	no	—
158	C₂²	C₂⁴×Dic₁₁	1	30	yes	—
159	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
160	C₄	—	11	4	no	—
161	C₄	—	22	30	no	—
162	C₄	C₂⁴×Dic₁₁	1	2	yes	—
163	C₄	C₂⁴×Dic₁₁	1	30	yes	—
164	C₂	—	2	30	no	—
165	C₂	C₂⁵×Dic₁₁	1	1	yes	characteristic
166	C₂	D₁₁×#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₁₁
commutator	C₂×C₁₁	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₁
Fitting, prime radical	C₂×C₁₁×#585	C₂
socle, cosocle	C₂²×C₁₁	C₂⁶
hypercenter	C₂×#585	D₁₁
hypocenter	C₁₁	C₄×#585
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#585

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×Dic₁₁	C₂	semidirect
2. H(5,𝔽₂)×Dic₁₁	C₂	semidirect
3. #13081819	C₂	semidirect
4. C₂×#41×Dic₁₁	C₂	semidirect
6. C₁₁×#585	C₄	semidirect
10. #10623358	C₂²	semidirect
11. #41×Dic₁₁	C₂²	semidirect
12. #41×Dic₁₁	C₂²	semidirect
13. #41×Dic₁₁	C₂²	semidirect
14. #10623343	C₂²	semidirect
15. C₂×Q₈×Dic₁₁	C₂²	semidirect
16. #10623329	C₂²	semidirect
17. C₂×D₄×Dic₁₁	C₂²	semidirect
18. C₂×#66979	C₂²	semidirect
19. C₂²×C₄×Dic₁₁	C₂²	semidirect
20. C₁₁×2₋¹⁺⁴	C₂×C₄	semidirect
21. C₁₁×H(5,𝔽₂)	C₂×C₄	semidirect
22. C₂×C₁₁×#41	C₂×C₄	semidirect
42. C₁₁×#41	C₂²×C₄	semidirect
43. C₁₁×#41	C₂²×C₄	semidirect
44. C₂×Q₈×C₁₁	C₂²×C₄	semidirect
46. C₂×D₄×C₁₁	C₂²×C₄	semidirect
48. C₂²×C₄×C₁₁	C₂²×C₄	semidirect
78. #585	Dic₁₁	direct (N × H)
99. Dic₁₁	#585	direct (N × H)
100. 2₋¹⁺⁴	C₂×Dic₁₁	semidirect
101. H(5,𝔽₂)	C₂×Dic₁₁	semidirect
102. C₂×#41	C₂×Dic₁₁	semidirect
124. #41	C₂²×Dic₁₁	semidirect
125. #41	C₂²×Dic₁₁	semidirect
126. C₂×Q₈	C₂²×Dic₁₁	semidirect
128. C₂×D₄	C₂²×Dic₁₁	semidirect
133. C₂²×C₄	C₂²×Dic₁₁	semidirect
142. C₁₁	C₄×#585	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×Dic₁₁	2
H(5,𝔽₂)×Dic₁₁	2
#13081819	2
C₂×#41×Dic₁₁	2
C₂×C₁₁×#585	2
C₄×#585	11

Subgroups — Dic11×#585 #15660388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₁×#585 #15660388