Subgroups — C₂×H(5,𝔽₂)×Dic₁₁ #15660377

Order 2816.

12378 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×Dic₁₁	C₂	1	32	yes	maximal, direct product factor
2	C₂×#10623358	C₂	1	6	yes	maximal, complemented
3	C₂×#10623329	C₂	1	9	yes	maximal, complemented
4	C₂×#41×Dic₁₁	C₂	1	6	yes	maximal, complemented
5	C₂²×D₄×Dic₁₁	C₂	1	9	yes	maximal, complemented
6	C₂²×C₁₁×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₁×H(5,𝔽₂)	C₄	1	32	yes	complemented
8	C₂×C₁₁×H(5,𝔽₂)	C₂²	1	16	yes	—
9	C₂²×C₁₁×#41	C₂²	1	6	yes	—
10	C₂³×D₄×C₁₁	C₂²	1	9	yes	—
11	C₂⁴×Dic₁₁	C₂²	1	6	yes	complemented
12	C₂²×C₂²⋊Dic₁₁	C₂²	1	18	yes	complemented
13	#10623358	C₂²	1	96	yes	complemented
14	#41×Dic₁₁	C₂²	1	96	yes	complemented
15	C₂×Q₈×Dic₁₁	C₂²	1	2	yes	complemented
16	C₂×Q₈×Dic₁₁	C₂²	1	6	yes	complemented
17	#10623329	C₂²	1	144	yes	complemented
18	C₂×D₄×Dic₁₁	C₂²	1	18	yes	complemented
19	C₂×D₄×Dic₁₁	C₂²	1	36	yes	complemented
20	C₂×D₄×Dic₁₁	C₂²	1	18	yes	complemented
21	C₂×D₄×Dic₁₁	C₂²	1	144	yes	complemented
22	C₂×#66979	C₂²	1	18	yes	complemented
23	C₂²×C₄⋊Dic₁₁	C₂²	1	9	yes	complemented
24	C₂²×C₄×Dic₁₁	C₂²	1	9	yes	complemented
25	C₂⁵×C₁₁	C₂³	1	6	yes	—
26	C₁₁×H(5,𝔽₂)	C₂×C₄	1	256	yes	complemented
27	C₂×C₁₁×#41	C₂×C₄	1	96	yes	complemented
28	C₂×C₁₁×#41	C₂³	1	48	yes	—
29	C₂²×Q₈×C₁₁	C₂³	1	2	yes	—
30	C₂²×D₄×C₁₁	C₂×C₄	1	144	yes	complemented
31	C₂²×D₄×C₁₁	C₂³	1	18	yes	—
32	C₂²×D₄×C₁₁	C₂³	1	72	yes	—
33	C₂³×C₄×C₁₁	C₂³	1	9	yes	—
34	C₂³×Dic₁₁	—	4	24	no	—
35	C₂³×Dic₁₁	C₂³	1	9	yes	—
36	C₂³×Dic₁₁	C₂³	1	9	yes	—
37	C₂³×Dic₁₁	C₂³	1	18	yes	—
38	C₂³×Dic₁₁	C₂³	1	48	yes	complemented
39	C₂×C₂²⋊Dic₁₁	C₂³	1	36	yes	—
40	C₂×C₂²⋊Dic₁₁	C₂³	1	144	yes	complemented
41	Q₈×Dic₁₁	C₂³	1	16	yes	complemented
42	Q₈×Dic₁₁	C₂³	1	48	yes	complemented
43	D₄×Dic₁₁	C₂³	1	144	yes	complemented
44	D₄×Dic₁₁	C₂³	1	288	yes	complemented
45	D₄×Dic₁₁	C₂³	1	144	yes	complemented
46	#66979	C₂³	1	144	yes	complemented
47	C₂×C₄⋊Dic₁₁	C₂³	1	18	yes	—
48	C₂×C₄⋊Dic₁₁	C₂³	1	6	yes	—
49	C₂×C₄⋊Dic₁₁	C₂³	1	72	yes	complemented
50	C₂×C₄×Dic₁₁	C₂³	1	6	yes	—
51	C₂×C₄×Dic₁₁	C₂³	1	18	yes	—
52	C₂×C₄×Dic₁₁	C₂³	1	72	yes	complemented
53	C₂×C₄×H(5,𝔽₂)	—	11	1	no	maximal, Sylow
54	C₂⁴×C₁₁	—	4	12	no	—
55	C₂⁴×C₁₁	—	4	12	no	—
56	C₂⁴×C₁₁	C₂²×C₄	1	48	yes	complemented
57	C₂⁴×C₁₁	C₂⁴	1	9	yes	—
58	C₂⁴×C₁₁	C₂⁴	1	24	yes	—
59	C₁₁×#41	C₂²×C₄	1	384	yes	complemented
60	C₂×Q₈×C₁₁	C₂²×C₄	1	16	yes	complemented
61	C₂×Q₈×C₁₁	C₂⁴	1	8	yes	—
62	C₂×D₄×C₁₁	C₂²×C₄	1	144	yes	complemented
63	C₂×D₄×C₁₁	C₂²×C₄	1	576	yes	complemented
64	C₂×D₄×C₁₁	C₂⁴	1	72	yes	—
65	C₂²×C₄×C₁₁	C₂²×C₄	1	72	yes	complemented
66	C₂²×C₄×C₁₁	C₂⁴	1	6	yes	—
67	C₂²×C₄×C₁₁	C₂⁴	1	36	yes	—
68	C₂²×Dic₁₁	—	2	36	no	—
69	C₂²×Dic₁₁	—	4	36	no	—
70	C₂²×Dic₁₁	—	4	18	no	—
71	C₂²×Dic₁₁	—	4	96	no	—
72	C₂²×Dic₁₁	C₂⁴	1	6	yes	—
73	C₂²×Dic₁₁	C₂⁴	1	9	yes	—
74	C₂²×Dic₁₁	C₂⁴	1	36	yes	—
75	C₂²×Dic₁₁	C₂⁴	1	36	yes	—
76	C₂²×Dic₁₁	C₂⁴	1	72	yes	—
77	C₂²×Dic₁₁	C₂⁴	1	1	yes	characteristic
78	C₂²⋊Dic₁₁	C₂⁴	1	144	yes	—
79	C₄⋊Dic₁₁	C₂⁴	1	72	yes	—
80	C₄⋊Dic₁₁	C₂⁴	1	24	yes	—
81	C₄×Dic₁₁	C₂⁴	1	24	yes	—
82	C₄×Dic₁₁	C₂⁴	1	72	yes	—
83	C₂²×H(5,𝔽₂)	D₁₁	1	1	yes	characteristic
84	C₄×H(5,𝔽₂)	—	11	32	no	—
85	C₂×#520	—	11	6	no	—
86	C₂×#518	—	11	9	no	—
87	C₂×C₄×#41	—	11	6	no	—
88	C₂²×C₄×D₄	—	11	9	no	—
89	C₂³×C₁₁	—	2	18	no	—
90	C₂³×C₁₁	—	2	18	no	—
91	C₂³×C₁₁	—	4	24	no	—
92	C₂³×C₁₁	—	4	96	no	—
93	C₂³×C₁₁	—	4	24	no	—
94	C₂³×C₁₁	C₂³×C₄	1	36	yes	—
95	C₂³×C₁₁	C₂³×C₄	1	96	yes	complemented
96	C₂³×C₁₁	C₂⁵	1	18	yes	—
97	C₂³×C₁₁	C₂⁵	1	1	yes	characteristic
98	Q₈×C₁₁	C₂³×C₄	1	32	yes	complemented
99	D₄×C₁₁	C₂³×C₄	1	288	yes	complemented
100	C₂×C₄×C₁₁	C₂³×C₄	1	24	yes	—
101	C₂×C₄×C₁₁	C₂³×C₄	1	144	yes	complemented
102	C₂×C₄×C₁₁	C₂⁵	1	12	yes	—
103	C₂×Dic₁₁	—	2	18	no	—
104	C₂×Dic₁₁	—	2	12	no	—
105	C₂×Dic₁₁	—	2	72	no	—
106	C₂×Dic₁₁	—	4	72	no	—
107	C₂×Dic₁₁	—	4	36	no	—
108	C₂×Dic₁₁	H(5,𝔽₂)	1	4	yes	direct product factor
109	C₂×Dic₁₁	C₂⁵	1	12	yes	—
110	C₂×Dic₁₁	C₂⁵	1	18	yes	—
111	C₂×Dic₁₁	C₂⁵	1	2	yes	—
112	C₂×H(5,𝔽₂)	Dic₁₁	1	32	yes	direct product factor
113	C₂×H(5,𝔽₂)	C₂×D₁₁	1	16	yes	—
114	C₂²×#41	C₂×D₁₁	1	6	yes	—
115	C₂³×D₄	C₂×D₁₁	1	9	yes	—
116	C₂⁴×C₄	—	11	6	no	—
117	#520	—	11	96	no	—
118	#518	—	11	144	no	—
119	C₄×#41	—	11	96	no	—
120	C₂×C₄×Q₈	—	11	2	no	—
121	C₂×C₄×Q₈	—	11	6	no	—
122	C₂×C₄×D₄	—	11	18	no	—
123	C₂×C₄×D₄	—	11	36	no	—
124	C₂×C₄×D₄	—	11	144	no	—
125	C₂×C₄×D₄	—	11	18	no	—
126	C₂×#117	—	11	18	no	—
127	C₂²×C₄⋊C₄	—	11	9	no	—
128	C₂²×C₂²⋊C₄	—	11	18	no	—
129	C₂²×C₄²	—	11	9	no	—
130	C₂²×C₁₁	—	2	18	no	—
131	C₂²×C₁₁	—	2	18	no	—
132	C₂²×C₁₁	—	2	72	no	—
133	C₂²×C₁₁	—	4	96	no	—
134	C₂²×C₁₁	C₂⁴×C₄	1	36	yes	—
135	C₂²×C₁₁	C₂⁴×C₄	1	2	yes	—
136	C₂²×C₁₁	C₂×H(5,𝔽₂)	1	2	yes	—
137	C₂²×C₁₁	#585	1	2	yes	—
138	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
139	C₄×C₁₁	C₂⁴×C₄	1	24	yes	—
140	Dic₁₁	—	2	18	no	—
141	Dic₁₁	—	2	12	no	—
142	Dic₁₁	C₂×H(5,𝔽₂)	1	4	yes	direct product factor
143	C₂⁵	C₂²×D₁₁	1	6	yes	—
144	H(5,𝔽₂)	C₂×Dic₁₁	1	256	yes	direct product factor
145	C₂×#41	C₂×Dic₁₁	1	96	yes	complemented
146	C₂×#41	C₂²×D₁₁	1	48	yes	—
147	C₂²×Q₈	C₂²×D₁₁	1	2	yes	—
148	C₂²×D₄	C₂×Dic₁₁	1	144	yes	complemented
149	C₂²×D₄	C₂²×D₁₁	1	18	yes	—
150	C₂²×D₄	C₂²×D₁₁	1	72	yes	—
151	C₂³×C₄	—	11	9	no	—
152	C₂³×C₄	—	11	9	no	—
153	C₂³×C₄	—	11	18	no	—
154	C₂³×C₄	—	11	48	no	—
155	C₂³×C₄	—	44	24	no	—
156	C₂³×C₄	C₂²×D₁₁	1	9	yes	—
157	C₄×Q₈	—	11	16	no	—
158	C₄×Q₈	—	11	48	no	—
159	C₄×D₄	—	11	144	no	—
160	C₄×D₄	—	11	288	no	—
161	C₄×D₄	—	11	144	no	—
162	#117	—	11	144	no	—
163	C₂×C₄⋊C₄	—	11	18	no	—
164	C₂×C₄⋊C₄	—	11	72	no	—
165	C₂×C₄⋊C₄	—	11	6	no	—
166	C₂×C₂²⋊C₄	—	11	36	no	—
167	C₂×C₂²⋊C₄	—	11	144	no	—
168	C₂×C₄²	—	11	18	no	—
169	C₂×C₄²	—	11	6	no	—
170	C₂×C₄²	—	11	72	no	—
171	C₂×C₁₁	—	2	36	no	—
172	C₂×C₁₁	C₄×H(5,𝔽₂)	1	4	yes	complemented
173	C₂×C₁₁	C₂⁵×C₄	1	1	yes	characteristic
174	C₂×C₁₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₁₁	C₂×#585	1	1	yes	characteristic
176	C₂⁴	—	4	12	no	—
177	C₂⁴	—	4	12	no	—
178	C₂⁴	C₂²×Dic₁₁	1	48	yes	complemented
179	C₂⁴	C₂³×D₁₁	1	9	yes	—
180	C₂⁴	C₂³×D₁₁	1	24	yes	—
181	#41	C₂²×Dic₁₁	1	384	yes	complemented
182	C₂×Q₈	C₂²×Dic₁₁	1	16	yes	complemented
183	C₂×Q₈	C₂³×D₁₁	1	8	yes	—
184	C₂×D₄	C₂²×Dic₁₁	1	144	yes	complemented
185	C₂×D₄	C₂²×Dic₁₁	1	576	yes	complemented
186	C₂×D₄	C₂³×D₁₁	1	72	yes	—
187	C₂²×C₄	—	11	1	no	—
188	C₂²×C₄	—	11	36	no	—
189	C₂²×C₄	—	11	6	no	—
190	C₂²×C₄	—	11	9	no	—
191	C₂²×C₄	—	11	36	no	—
192	C₂²×C₄	—	11	72	no	—
193	C₂²×C₄	—	22	36	no	—
194	C₂²×C₄	—	44	96	no	—
195	C₂²×C₄	—	44	36	no	—
196	C₂²×C₄	—	44	18	no	—
197	C₂²×C₄	C₂²×Dic₁₁	1	72	yes	complemented
198	C₂²×C₄	C₂³×D₁₁	1	6	yes	—
199	C₂²×C₄	C₂³×D₁₁	1	36	yes	—
200	C₄⋊C₄	—	11	72	no	—
201	C₄⋊C₄	—	11	24	no	—
202	C₂²⋊C₄	—	11	144	no	—
203	C₄²	—	11	24	no	—
204	C₄²	—	11	72	no	—
205	C₁₁	C₂×C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	2	18	no	—
207	C₂³	—	2	18	no	—
208	C₂³	—	4	24	no	—
209	C₂³	—	4	96	no	—
210	C₂³	—	4	24	no	—
211	C₂³	C₂³×Dic₁₁	1	36	yes	—
212	C₂³	C₂³×Dic₁₁	1	96	yes	complemented
213	C₂³	C₂⁴×D₁₁	1	18	yes	—
214	C₂³	C₂⁴×D₁₁	1	1	yes	characteristic
215	Q₈	C₂³×Dic₁₁	1	32	yes	complemented
216	D₄	C₂³×Dic₁₁	1	288	yes	complemented
217	C₂×C₄	—	11	4	no	—
218	C₂×C₄	—	11	2	no	—
219	C₂×C₄	—	11	12	no	—
220	C₂×C₄	—	11	18	no	—
221	C₂×C₄	—	22	72	no	—
222	C₂×C₄	—	22	18	no	—
223	C₂×C₄	—	22	12	no	—
224	C₂×C₄	—	44	72	no	—
225	C₂×C₄	—	44	36	no	—
226	C₂×C₄	C₂³×Dic₁₁	1	24	yes	—
227	C₂×C₄	C₂³×Dic₁₁	1	144	yes	complemented
228	C₂×C₄	C₂⁴×D₁₁	1	12	yes	—
229	C₂²	—	2	18	no	—
230	C₂²	—	2	18	no	—
231	C₂²	—	2	72	no	—
232	C₂²	—	4	96	no	—
233	C₂²	D₁₁×H(5,𝔽₂)	1	2	yes	—
234	C₂²	#10623455	1	2	yes	—
235	C₂²	C₂⁴×Dic₁₁	1	36	yes	—
236	C₂²	C₂⁴×Dic₁₁	1	2	yes	—
237	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
238	C₄	—	11	4	no	—
239	C₄	—	22	18	no	—
240	C₄	—	22	12	no	—
241	C₄	C₂⁴×Dic₁₁	1	24	yes	—
242	C₂	—	2	36	no	—
243	C₂	H(5,𝔽₂)×Dic₁₁	1	4	yes	direct product factor
244	C₂	C₂⁵×Dic₁₁	1	1	yes	characteristic
245	C₂	C₂×D₁₁×H(5,𝔽₂)	1	1	yes	characteristic
246	C₂	C₂×#10623455	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₁₁
commutator	C₂×C₁₁	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₁
Fitting, prime radical	C₂²×C₁₁×H(5,𝔽₂)	C₂
socle	C₂³×C₁₁	C₂⁵
cosocle	C₂²×C₁₁	C₂⁶
hypercenter	C₂²×H(5,𝔽₂)	D₁₁
hypocenter	C₁₁	C₂×C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×Dic₁₁	C₂	direct (N × H)
2. C₂×#10623358	C₂	semidirect
3. C₂×#10623329	C₂	semidirect
4. C₂×#41×Dic₁₁	C₂	semidirect
5. C₂²×D₄×Dic₁₁	C₂	semidirect
7. C₂×C₁₁×H(5,𝔽₂)	C₄	semidirect
11. C₂⁴×Dic₁₁	C₂²	semidirect
12. C₂²×C₂²⋊Dic₁₁	C₂²	semidirect
13. #10623358	C₂²	semidirect
14. #41×Dic₁₁	C₂²	semidirect
15. C₂×Q₈×Dic₁₁	C₂²	semidirect
16. C₂×Q₈×Dic₁₁	C₂²	semidirect
17. #10623329	C₂²	semidirect
18. C₂×D₄×Dic₁₁	C₂²	semidirect
19. C₂×D₄×Dic₁₁	C₂²	semidirect
20. C₂×D₄×Dic₁₁	C₂²	semidirect
21. C₂×D₄×Dic₁₁	C₂²	semidirect
22. C₂×#66979	C₂²	semidirect
23. C₂²×C₄⋊Dic₁₁	C₂²	semidirect
24. C₂²×C₄×Dic₁₁	C₂²	semidirect
26. C₁₁×H(5,𝔽₂)	C₂×C₄	semidirect
27. C₂×C₁₁×#41	C₂×C₄	semidirect
30. C₂²×D₄×C₁₁	C₂×C₄	semidirect
38. C₂³×Dic₁₁	C₂³	semidirect
40. C₂×C₂²⋊Dic₁₁	C₂³	semidirect
41. Q₈×Dic₁₁	C₂³	semidirect
42. Q₈×Dic₁₁	C₂³	semidirect
43. D₄×Dic₁₁	C₂³	semidirect
44. D₄×Dic₁₁	C₂³	semidirect
45. D₄×Dic₁₁	C₂³	semidirect
46. #66979	C₂³	semidirect
49. C₂×C₄⋊Dic₁₁	C₂³	semidirect
52. C₂×C₄×Dic₁₁	C₂³	semidirect
56. C₂⁴×C₁₁	C₂²×C₄	semidirect
59. C₁₁×#41	C₂²×C₄	semidirect
60. C₂×Q₈×C₁₁	C₂²×C₄	semidirect
62. C₂×D₄×C₁₁	C₂²×C₄	semidirect
63. C₂×D₄×C₁₁	C₂²×C₄	semidirect
65. C₂²×C₄×C₁₁	C₂²×C₄	semidirect
95. C₂³×C₁₁	C₂³×C₄	semidirect
98. Q₈×C₁₁	C₂³×C₄	semidirect
99. D₄×C₁₁	C₂³×C₄	semidirect
101. C₂×C₄×C₁₁	C₂³×C₄	semidirect
108. C₂×Dic₁₁	H(5,𝔽₂)	direct (N × H)
112. C₂×H(5,𝔽₂)	Dic₁₁	direct (N × H)
142. Dic₁₁	C₂×H(5,𝔽₂)	direct (N × H)
144. H(5,𝔽₂)	C₂×Dic₁₁	direct (N × H)
145. C₂×#41	C₂×Dic₁₁	semidirect
148. C₂²×D₄	C₂×Dic₁₁	semidirect
172. C₂×C₁₁	C₄×H(5,𝔽₂)	semidirect
178. C₂⁴	C₂²×Dic₁₁	semidirect
181. #41	C₂²×Dic₁₁	semidirect
182. C₂×Q₈	C₂²×Dic₁₁	semidirect
184. C₂×D₄	C₂²×Dic₁₁	semidirect
185. C₂×D₄	C₂²×Dic₁₁	semidirect
197. C₂²×C₄	C₂²×Dic₁₁	semidirect
205. C₁₁	C₂×C₄×H(5,𝔽₂)	semidirect
212. C₂³	C₂³×Dic₁₁	semidirect
215. Q₈	C₂³×Dic₁₁	semidirect
216. D₄	C₂³×Dic₁₁	semidirect
227. C₂×C₄	C₂³×Dic₁₁	semidirect
243. C₂	H(5,𝔽₂)×Dic₁₁	direct (N × H)

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×Dic₁₁	2
C₂×#10623358	2
C₂×#10623329	2
C₂×#41×Dic₁₁	2
C₂²×D₄×Dic₁₁	2
C₂²×C₁₁×H(5,𝔽₂)	2
C₂×C₄×H(5,𝔽₂)	11

Subgroups — C2×H(5,𝔽2)×Dic11 #15660377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×H(5,𝔽₂)×Dic₁₁ #15660377