Subgroups — C₂×H(5,𝔽₂)×C₅⋊D₅ #72221011

Order 3200.

26240 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	H(5,𝔽₂)×C₅⋊D₅	C₂	1	32	yes	maximal, direct product factor
3	C₂×#11717522	C₂	1	6	yes	maximal, complemented
4	C₂×#41×C₅⋊D₅	C₂	1	6	yes	maximal, complemented
5	C₂×#11717517	C₂	1	9	yes	maximal, complemented
6	C₂²×D₄×C₅⋊D₅	C₂	1	9	yes	maximal, complemented
7	C₂⁴×C₅⋊D₅	C₂²	1	6	yes	complemented
8	C₅²×H(5,𝔽₂)	C₂²	1	16	yes	complemented
9	C₂×C₅²×#41	C₂²	1	6	yes	complemented
10	C₂²×C₅²×D₄	C₂²	1	9	yes	complemented
11	C₂²×C₂²⋊(C₅⋊D₅)	C₂²	1	18	yes	complemented
12	#11717522	C₂²	1	96	yes	complemented
13	#41×C₅⋊D₅	C₂²	1	96	yes	complemented
14	C₂×#87904	C₂²	1	6	yes	complemented
15	C₂×Q₈×C₅⋊D₅	C₂²	1	2	yes	complemented
16	#11717517	C₂²	1	144	yes	complemented
17	C₂×#87902	C₂²	1	18	yes	complemented
18	C₂×D₄×C₅⋊D₅	C₂²	1	18	yes	complemented
19	C₂×D₄×C₅⋊D₅	C₂²	1	36	yes	complemented
20	C₂×D₄×C₅⋊D₅	C₂²	1	144	yes	complemented
21	C₂×#87900	C₂²	1	18	yes	complemented
22	C₂²×C₄⋊(C₅⋊D₅)	C₂²	1	9	yes	complemented
23	C₂²×C₄×C₅⋊D₅	C₂²	1	9	yes	complemented
24	C₂×D₅×H(5,𝔽₂)	—	5	6	no	maximal
25	C₂⁴×C₅²	C₂³	1	6	yes	complemented
26	C₂³×C₅⋊D₅	—	4	24	no	—
27	C₂³×C₅⋊D₅	C₂³	1	9	yes	—
28	C₂³×C₅⋊D₅	C₂³	1	48	yes	complemented
29	C₂³×C₅⋊D₅	C₂³	1	18	yes	complemented
30	C₅²×#41	C₂³	1	48	yes	complemented
31	C₂×C₅²×Q₈	C₂³	1	2	yes	complemented
32	C₂×C₅²×D₄	C₂³	1	18	yes	complemented
33	C₂×C₅²×D₄	C₂³	1	72	yes	complemented
34	C₂²×C₄×C₅²	C₂³	1	9	yes	complemented
35	C₂×C₂²⋊(C₅⋊D₅)	C₂³	1	36	yes	complemented
36	C₂×C₂²⋊(C₅⋊D₅)	C₂³	1	144	yes	complemented
37	C₂²×#1039	C₂³	1	9	yes	complemented
38	#87904	C₂³	1	48	yes	complemented
39	Q₈×C₅⋊D₅	C₂³	1	16	yes	complemented
40	#87902	C₂³	1	144	yes	complemented
41	D₄×C₅⋊D₅	C₂³	1	144	yes	complemented
42	D₄×C₅⋊D₅	C₂³	1	288	yes	complemented
43	#87900	C₂³	1	144	yes	complemented
44	C₂×C₄⋊(C₅⋊D₅)	C₂³	1	18	yes	complemented
45	C₂×C₄⋊(C₅⋊D₅)	C₂³	1	72	yes	complemented
46	C₂×C₄×C₅⋊D₅	C₂³	1	6	yes	—
47	C₂×C₄×C₅⋊D₅	C₂³	1	72	yes	complemented
48	C₂×C₄×C₅⋊D₅	C₂³	1	18	yes	complemented
49	C₂×#6045	C₂³	1	6	yes	complemented
50	C₂×C₅×H(5,𝔽₂)	D₅	1	6	yes	complemented
51	D₅×H(5,𝔽₂)	—	5	192	no	—
52	C₂×#4149	—	5	36	no	—
53	C₂×D₅×#41	—	5	36	no	—
54	C₂×#4144	—	5	54	no	—
55	C₂²×D₄×D₅	—	5	54	no	—
56	C₂³×C₅²	—	4	12	no	—
57	C₂³×C₅²	C₂⁴	1	9	yes	—
58	C₂³×C₅²	C₂⁴	1	24	yes	complemented
59	C₂²×C₅⋊D₅	—	2	36	no	—
60	C₂²×C₅⋊D₅	—	4	96	no	—
61	C₂²×C₅⋊D₅	—	4	36	no	—
62	C₂²×C₅⋊D₅	C₂⁴	1	9	yes	—
63	C₂²×C₅⋊D₅	C₂⁴	1	36	yes	—
64	C₂²×C₅⋊D₅	C₂⁴	1	1	yes	characteristic
65	C₂²×C₅⋊D₅	C₂⁴	1	72	yes	complemented
66	C₅²×Q₈	C₂⁴	1	8	yes	complemented
67	C₅²×D₄	C₂⁴	1	72	yes	complemented
68	C₂×C₄×C₅²	C₂⁴	1	6	yes	—
69	C₂×C₄×C₅²	C₂⁴	1	36	yes	complemented
70	C₂²⋊(C₅⋊D₅)	C₂⁴	1	144	yes	complemented
71	C₂×#1039	C₂⁴	1	6	yes	—
72	C₂×#1039	C₂⁴	1	36	yes	complemented
73	C₄⋊(C₅⋊D₅)	C₂⁴	1	72	yes	complemented
74	C₄×C₅⋊D₅	C₂⁴	1	24	yes	—
75	C₄×C₅⋊D₅	C₂⁴	1	72	yes	complemented
76	#6045	C₂⁴	1	24	yes	complemented
77	C₂⁴×D₅	—	5	36	no	—
78	C₅×H(5,𝔽₂)	C₂×D₅	1	96	yes	complemented
79	C₂×C₅×#41	C₂×D₅	1	36	yes	complemented
80	C₂²×C₅×D₄	C₂×D₅	1	54	yes	complemented
81	C₂²×C₂²⋊D₅	—	5	108	no	—
82	#4149	—	5	576	no	—
83	D₅×#41	—	5	576	no	—
84	C₂×#713	—	5	36	no	—
85	C₂×Q₈×D₅	—	5	12	no	—
86	#4144	—	5	864	no	—
87	C₂×#711	—	5	108	no	—
88	C₂×D₄×D₅	—	5	108	no	—
89	C₂×D₄×D₅	—	5	216	no	—
90	C₂×D₄×D₅	—	5	864	no	—
91	C₂×#709	—	5	108	no	—
92	C₂²×D₂₀	—	5	54	no	—
93	C₂²×C₄×D₅	—	5	54	no	—
94	C₂²×H(5,𝔽₂)	—	25	1	no	Sylow
95	C₂²×C₅²	—	2	18	no	—
96	C₂²×C₅²	—	4	24	no	—
97	C₂²×C₅²	C₂⁵	1	18	yes	—
98	C₂²×C₅²	C₂⁵	1	1	yes	characteristic
99	C₂×C₅⋊D₅	—	2	18	no	—
100	C₂×C₅⋊D₅	—	2	72	no	—
101	C₂×C₅⋊D₅	—	4	72	no	—
102	C₂×C₅⋊D₅	H(5,𝔽₂)	1	4	yes	direct product factor
103	C₂×C₅⋊D₅	C₂⁵	1	18	yes	—
104	C₂×C₅⋊D₅	C₂⁵	1	2	yes	—
105	C₄×C₅²	C₂⁵	1	12	yes	—
106	#1039	C₂⁵	1	12	yes	—
107	C₂⁴×C₅	C₂²×D₅	1	36	yes	complemented
108	C₂³×D₅	—	5	54	no	—
109	C₂³×D₅	—	5	108	no	—
110	C₂³×D₅	—	5	288	no	—
111	C₂³×D₅	—	20	144	no	—
112	C₅×#41	C₂²×D₅	1	288	yes	complemented
113	C₂×C₅×Q₈	C₂²×D₅	1	12	yes	complemented
114	C₂×C₅×D₄	C₂²×D₅	1	108	yes	complemented
115	C₂×C₅×D₄	C₂²×D₅	1	432	yes	complemented
116	C₂²×C₄×C₅	C₂²×D₅	1	54	yes	complemented
117	C₂×C₂²⋊D₅	—	5	216	no	—
118	C₂×C₂²⋊D₅	—	5	864	no	—
119	C₂²×Dic₅	—	5	54	no	—
120	#713	—	5	288	no	—
121	Q₈×D₅	—	5	96	no	—
122	#711	—	5	864	no	—
123	D₄×D₅	—	5	864	no	—
124	D₄×D₅	—	5	1728	no	—
125	#709	—	5	864	no	—
126	C₂×D₂₀	—	5	108	no	—
127	C₂×D₂₀	—	5	432	no	—
128	C₂×C₄×D₅	—	5	36	no	—
129	C₂×C₄×D₅	—	5	108	no	—
130	C₂×C₄×D₅	—	5	432	no	—
131	C₂×Dic₁₀	—	5	36	no	—
132	C₂×H(5,𝔽₂)	—	25	32	no	—
133	C₂×H(5,𝔽₂)	—	25	6	no	—
134	C₂×H(5,𝔽₂)	—	25	9	no	—
135	C₂×H(5,𝔽₂)	C₅⋊D₅	1	1	yes	characteristic, direct product factor
136	C₂²×#41	—	25	6	no	—
137	C₂³×D₄	—	25	9	no	—
138	C₂×C₅²	—	2	18	no	—
139	C₂×C₅²	C₂×H(5,𝔽₂)	1	2	yes	complemented
140	C₂×C₅²	C₂⁶	1	1	yes	characteristic
141	C₅⋊D₅	—	2	18	no	—
142	C₅⋊D₅	C₂×H(5,𝔽₂)	1	4	yes	direct product factor
143	C₂³×C₅	—	4	72	no	—
144	C₂³×C₅	C₂³×D₅	1	54	yes	—
145	C₂³×C₅	C₂³×D₅	1	144	yes	complemented
146	C₂²×D₅	—	5	6	no	—
147	C₂²×D₅	—	5	54	no	—
148	C₂²×D₅	—	5	216	no	—
149	C₂²×D₅	—	5	432	no	—
150	C₂²×D₅	—	10	216	no	—
151	C₂²×D₅	—	20	576	no	—
152	C₂²×D₅	—	20	216	no	—
153	C₅×Q₈	C₂³×D₅	1	48	yes	complemented
154	C₅×D₄	C₂³×D₅	1	432	yes	complemented
155	C₂×C₄×C₅	C₂³×D₅	1	36	yes	—
156	C₂×C₄×C₅	C₂³×D₅	1	216	yes	complemented
157	C₂²⋊D₅	—	5	864	no	—
158	C₂×Dic₅	—	5	36	no	—
159	C₂×Dic₅	—	5	216	no	—
160	D₂₀	—	5	432	no	—
161	C₄×D₅	—	5	144	no	—
162	C₄×D₅	—	5	432	no	—
163	Dic₁₀	—	5	144	no	—
164	C₂⁵	—	25	6	no	—
165	H(5,𝔽₂)	—	25	96	no	—
166	H(5,𝔽₂)	—	25	144	no	—
167	H(5,𝔽₂)	C₂×C₅⋊D₅	1	16	yes	direct product factor
168	C₂×#41	—	25	96	no	—
169	C₂×#41	—	25	6	no	—
170	C₂×#41	—	25	18	no	—
171	C₂×#41	—	25	18	no	—
172	C₂×#41	C₂×C₅⋊D₅	1	6	yes	complemented
173	C₂²×Q₈	—	25	2	no	—
174	C₂²×D₄	—	25	18	no	—
175	C₂²×D₄	—	25	18	no	—
176	C₂²×D₄	—	25	36	no	—
177	C₂²×D₄	—	25	144	no	—
178	C₂²×D₄	—	25	9	no	—
179	C₂²×D₄	C₂×C₅⋊D₅	1	9	yes	complemented
180	C₂³×C₄	—	25	9	no	—
181	C₅²	C₂²×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
182	C₂²×C₅	—	2	108	no	—
183	C₂²×C₅	—	4	144	no	—
184	C₂²×C₅	C₂⁴×D₅	1	108	yes	—
185	C₂²×C₅	C₂⁴×D₅	1	6	yes	—
186	C₂×D₅	—	5	24	no	—
187	C₂×D₅	—	5	12	no	—
188	C₂×D₅	—	5	108	no	—
189	C₂×D₅	—	10	432	no	—
190	C₂×D₅	—	10	108	no	—
191	C₂×D₅	—	20	432	no	—
192	C₄×C₅	C₂⁴×D₅	1	72	yes	—
193	Dic₅	—	5	72	no	—
194	C₂⁴	—	25	9	no	—
195	C₂⁴	—	25	18	no	—
196	C₂⁴	—	25	48	no	—
197	C₂⁴	—	100	24	no	—
198	C₂⁴	C₂²×C₅⋊D₅	1	6	yes	complemented
199	#41	—	25	48	no	—
200	#41	—	25	144	no	—
201	#41	—	25	144	no	—
202	#41	C₂²×C₅⋊D₅	1	48	yes	complemented
203	C₂×Q₈	—	25	16	no	—
204	C₂×Q₈	—	25	6	no	—
205	C₂×Q₈	C₂²×C₅⋊D₅	1	2	yes	complemented
206	C₂×D₄	—	25	36	no	—
207	C₂×D₄	—	25	144	no	—
208	C₂×D₄	—	25	144	no	—
209	C₂×D₄	—	25	288	no	—
210	C₂×D₄	—	25	18	no	—
211	C₂×D₄	—	25	72	no	—
212	C₂×D₄	C₂²×C₅⋊D₅	1	18	yes	complemented
213	C₂×D₄	C₂²×C₅⋊D₅	1	72	yes	complemented
214	C₂²×C₄	—	25	9	no	—
215	C₂²×C₄	—	25	6	no	—
216	C₂²×C₄	—	25	18	no	—
217	C₂²×C₄	—	25	72	no	—
218	C₂²×C₄	C₂²×C₅⋊D₅	1	9	yes	complemented
219	C₂×C₅	—	2	108	no	—
220	C₂×C₅	D₅×H(5,𝔽₂)	1	12	yes	complemented
221	C₂×C₅	C₂⁵×D₅	1	6	yes	—
222	D₅	—	5	24	no	—
223	D₅	—	10	108	no	—
224	C₂³	—	4	12	no	—
225	C₂³	—	25	1	no	—
226	C₂³	—	25	36	no	—
227	C₂³	—	25	9	no	—
228	C₂³	—	25	72	no	—
229	C₂³	—	50	36	no	—
230	C₂³	—	100	36	no	—
231	C₂³	—	100	96	no	—
232	C₂³	C₂³×C₅⋊D₅	1	9	yes	—
233	C₂³	C₂³×C₅⋊D₅	1	24	yes	complemented
234	Q₈	—	25	24	no	—
235	Q₈	C₂³×C₅⋊D₅	1	8	yes	complemented
236	D₄	—	25	144	no	—
237	D₄	—	25	72	no	—
238	D₄	C₂³×C₅⋊D₅	1	72	yes	complemented
239	C₂×C₄	—	25	36	no	—
240	C₂×C₄	—	25	6	no	—
241	C₂×C₄	—	25	24	no	—
242	C₂×C₄	—	25	72	no	—
243	C₂×C₄	C₂³×C₅⋊D₅	1	6	yes	—
244	C₂×C₄	C₂³×C₅⋊D₅	1	36	yes	complemented
245	C₅	C₂×D₅×H(5,𝔽₂)	1	6	yes	complemented
246	C₂²	—	2	18	no	—
247	C₂²	—	4	24	no	—
248	C₂²	—	25	4	no	—
249	C₂²	—	25	2	no	—
250	C₂²	—	25	18	no	—
251	C₂²	—	50	72	no	—
252	C₂²	—	50	18	no	—
253	C₂²	—	100	72	no	—
254	C₂²	C₂⁴×C₅⋊D₅	1	18	yes	—
255	C₂²	C₂⁴×C₅⋊D₅	1	1	yes	characteristic
256	C₄	—	25	12	no	—
257	C₄	C₂⁴×C₅⋊D₅	1	12	yes	—
258	C₂	—	2	18	no	—
259	C₂	—	25	4	no	—
260	C₂	—	50	18	no	—
261	C₂	H(5,𝔽₂)×C₅⋊D₅	1	2	yes	direct product factor
262	C₂	C₂⁵×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁴×C₅⋊D₅
commutator, cosocle	C₂×C₅²	C₂⁶
Frattini	C₂	C₂⁵×C₅⋊D₅
Fitting	C₂×C₅²×H(5,𝔽₂)	C₂
socle	C₂²×C₅²	C₂⁵
hypercenter	C₂×H(5,𝔽₂)	C₅⋊D₅
hypocenter	C₅²	C₂²×H(5,𝔽₂)
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×H(5,𝔽₂)	C₂	semidirect
2. H(5,𝔽₂)×C₅⋊D₅	C₂	direct (N × H)
3. C₂×#11717522	C₂	semidirect
4. C₂×#41×C₅⋊D₅	C₂	semidirect
5. C₂×#11717517	C₂	semidirect
6. C₂²×D₄×C₅⋊D₅	C₂	semidirect
7. C₂⁴×C₅⋊D₅	C₂²	semidirect
8. C₅²×H(5,𝔽₂)	C₂²	semidirect
9. C₂×C₅²×#41	C₂²	semidirect
10. C₂²×C₅²×D₄	C₂²	semidirect
11. C₂²×C₂²⋊(C₅⋊D₅)	C₂²	semidirect
12. #11717522	C₂²	semidirect
13. #41×C₅⋊D₅	C₂²	semidirect
14. C₂×#87904	C₂²	semidirect
15. C₂×Q₈×C₅⋊D₅	C₂²	semidirect
16. #11717517	C₂²	semidirect
17. C₂×#87902	C₂²	semidirect
18. C₂×D₄×C₅⋊D₅	C₂²	semidirect
19. C₂×D₄×C₅⋊D₅	C₂²	semidirect
20. C₂×D₄×C₅⋊D₅	C₂²	semidirect
21. C₂×#87900	C₂²	semidirect
22. C₂²×C₄⋊(C₅⋊D₅)	C₂²	semidirect
23. C₂²×C₄×C₅⋊D₅	C₂²	semidirect
25. C₂⁴×C₅²	C₂³	semidirect
28. C₂³×C₅⋊D₅	C₂³	semidirect
29. C₂³×C₅⋊D₅	C₂³	semidirect
30. C₅²×#41	C₂³	semidirect
31. C₂×C₅²×Q₈	C₂³	semidirect
32. C₂×C₅²×D₄	C₂³	semidirect
33. C₂×C₅²×D₄	C₂³	semidirect
34. C₂²×C₄×C₅²	C₂³	semidirect
35. C₂×C₂²⋊(C₅⋊D₅)	C₂³	semidirect
36. C₂×C₂²⋊(C₅⋊D₅)	C₂³	semidirect
37. C₂²×#1039	C₂³	semidirect
38. #87904	C₂³	semidirect
39. Q₈×C₅⋊D₅	C₂³	semidirect
40. #87902	C₂³	semidirect
41. D₄×C₅⋊D₅	C₂³	semidirect
42. D₄×C₅⋊D₅	C₂³	semidirect
43. #87900	C₂³	semidirect
44. C₂×C₄⋊(C₅⋊D₅)	C₂³	semidirect
45. C₂×C₄⋊(C₅⋊D₅)	C₂³	semidirect
47. C₂×C₄×C₅⋊D₅	C₂³	semidirect
48. C₂×C₄×C₅⋊D₅	C₂³	semidirect
49. C₂×#6045	C₂³	semidirect
50. C₂×C₅×H(5,𝔽₂)	D₅	semidirect
58. C₂³×C₅²	C₂⁴	semidirect
65. C₂²×C₅⋊D₅	C₂⁴	semidirect
66. C₅²×Q₈	C₂⁴	semidirect
67. C₅²×D₄	C₂⁴	semidirect
69. C₂×C₄×C₅²	C₂⁴	semidirect
70. C₂²⋊(C₅⋊D₅)	C₂⁴	semidirect
72. C₂×#1039	C₂⁴	semidirect
73. C₄⋊(C₅⋊D₅)	C₂⁴	semidirect
75. C₄×C₅⋊D₅	C₂⁴	semidirect
76. #6045	C₂⁴	semidirect
78. C₅×H(5,𝔽₂)	C₂×D₅	semidirect
79. C₂×C₅×#41	C₂×D₅	semidirect
80. C₂²×C₅×D₄	C₂×D₅	semidirect
102. C₂×C₅⋊D₅	H(5,𝔽₂)	direct (N × H)
107. C₂⁴×C₅	C₂²×D₅	semidirect
112. C₅×#41	C₂²×D₅	semidirect
113. C₂×C₅×Q₈	C₂²×D₅	semidirect
114. C₂×C₅×D₄	C₂²×D₅	semidirect
115. C₂×C₅×D₄	C₂²×D₅	semidirect
116. C₂²×C₄×C₅	C₂²×D₅	semidirect
135. C₂×H(5,𝔽₂)	C₅⋊D₅	direct (N × H)
139. C₂×C₅²	C₂×H(5,𝔽₂)	semidirect
142. C₅⋊D₅	C₂×H(5,𝔽₂)	direct (N × H)
145. C₂³×C₅	C₂³×D₅	semidirect
153. C₅×Q₈	C₂³×D₅	semidirect
154. C₅×D₄	C₂³×D₅	semidirect
156. C₂×C₄×C₅	C₂³×D₅	semidirect
167. H(5,𝔽₂)	C₂×C₅⋊D₅	direct (N × H)
172. C₂×#41	C₂×C₅⋊D₅	semidirect
179. C₂²×D₄	C₂×C₅⋊D₅	semidirect
181. C₅²	C₂²×H(5,𝔽₂)	semidirect
198. C₂⁴	C₂²×C₅⋊D₅	semidirect
202. #41	C₂²×C₅⋊D₅	semidirect
205. C₂×Q₈	C₂²×C₅⋊D₅	semidirect
212. C₂×D₄	C₂²×C₅⋊D₅	semidirect
213. C₂×D₄	C₂²×C₅⋊D₅	semidirect
218. C₂²×C₄	C₂²×C₅⋊D₅	semidirect
220. C₂×C₅	D₅×H(5,𝔽₂)	semidirect
233. C₂³	C₂³×C₅⋊D₅	semidirect
235. Q₈	C₂³×C₅⋊D₅	semidirect
238. D₄	C₂³×C₅⋊D₅	semidirect
244. C₂×C₄	C₂³×C₅⋊D₅	semidirect
245. C₅	C₂×D₅×H(5,𝔽₂)	semidirect
261. C₂	H(5,𝔽₂)×C₅⋊D₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²×H(5,𝔽₂)	2
H(5,𝔽₂)×C₅⋊D₅	2
C₂×#11717522	2
C₂×#41×C₅⋊D₅	2
C₂×#11717517	2
C₂²×D₄×C₅⋊D₅	2
C₂×D₅×H(5,𝔽₂)	5

Subgroups — C2×H(5,𝔽2)×C5⋊D5 #72221011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×H(5,𝔽₂)×C₅⋊D₅ #72221011