Subgroups — C₂²×#66045 #13006170

Order 1280.

9606 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#66045	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#4040	C₂	1	3	yes	maximal, complemented
4	C₂²×#4041	C₂	1	3	yes	maximal, complemented
5	C₂²×#4030	C₂	1	3	yes	maximal, complemented
6	C₂²×#4028	C₂	1	3	yes	maximal, complemented
7	C₂²×#4020	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#4022	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₅	C₂²	1	1	yes	characteristic
10	C₂³×C₂²⋊D₅	C₂²	1	3	yes	—
11	C₂³×D₂₀	C₂²	1	3	yes	—
12	C₂³×D₂₀	C₂²	1	3	yes	—
13	C₂³×D₂₀	C₂²	1	3	yes	complemented
14	C₂³×C₄×D₅	C₂²	1	3	yes	complemented
15	C₂×C₅×#126	C₂²	1	24	yes	complemented
16	C₂²×C₅×C₄⋊C₄	C₂²	1	3	yes	complemented
17	C₂²×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
18	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
19	C₂²×#685	C₂²	1	3	yes	—
20	C₂²×#685	C₂²	1	3	yes	—
21	C₂²×#685	C₂²	1	3	yes	complemented
22	C₂²×#683	C₂²	1	3	yes	complemented
23	#66045	C₂²	1	256	yes	direct product factor
24	C₂×#4041	C₂²	1	72	yes	complemented
25	C₂×#4040	C₂²	1	72	yes	complemented
26	C₂×#4030	C₂²	1	72	yes	complemented
27	C₂×#4028	C₂²	1	72	yes	complemented
28	C₂×#4022	C₂²	1	24	yes	complemented
29	C₂×#4020	C₂²	1	24	yes	complemented
30	C₂²×#560	—	5	1	no	maximal, Sylow
31	C₂⁵×C₅	C₂³	1	1	yes	characteristic
32	C₂⁴×D₅	—	4	12	no	—
33	C₂⁴×D₅	C₂³	1	1	yes	characteristic
34	C₂⁴×D₅	C₂³	1	1	yes	characteristic
35	C₂⁴×D₅	C₂³	1	3	yes	—
36	C₂⁴×D₅	C₂³	1	12	yes	—
37	C₂³×C₄×C₅	C₂³	1	3	yes	—
38	C₂³×C₄×C₅	C₂³	1	3	yes	—
39	C₂²×C₂²⋊D₅	—	4	12	no	—
40	C₂²×C₂²⋊D₅	C₂³	1	36	yes	—
41	C₂³×Dic₅	C₂³	1	3	yes	—
42	C₂²×D₂₀	—	4	12	no	—
43	C₂²×D₂₀	—	4	12	no	—
44	C₂²×D₂₀	—	4	12	no	—
45	C₂²×D₂₀	C₂³	1	36	yes	—
46	C₂²×D₂₀	C₂³	1	36	yes	—
47	C₂²×D₂₀	C₂³	1	36	yes	complemented
48	C₂²×C₄×D₅	—	4	12	no	—
49	C₂²×C₄×D₅	C₂³	1	36	yes	complemented
50	C₅×#126	C₂³	1	64	yes	complemented
51	C₂×C₅×C₄⋊C₄	C₂³	1	36	yes	complemented
52	C₂×C₅×C₂²⋊C₄	C₂³	1	36	yes	—
53	C₂×C₄²×C₅	C₂³	1	12	yes	complemented
54	C₂×#685	C₂³	1	36	yes	—
55	C₂×#685	C₂³	1	36	yes	—
56	C₂×#685	C₂³	1	36	yes	complemented
57	C₂×#683	C₂³	1	36	yes	complemented
58	#4041	C₂³	1	192	yes	complemented
59	#4040	C₂³	1	192	yes	complemented
60	#4030	C₂³	1	192	yes	complemented
61	#4028	C₂³	1	192	yes	complemented
62	#4022	C₂³	1	64	yes	complemented
63	#4020	C₂³	1	64	yes	complemented
64	C₂×#560	—	5	48	no	—
65	C₂²×#127	—	5	1	no	—
66	C₂²×#126	—	5	1	no	—
67	C₂²×#126	D₅	1	1	yes	characteristic, complemented
68	C₂²×#123	—	5	3	no	—
69	C₂²×#121	—	5	3	no	—
70	C₂²×#121	—	5	3	no	—
71	C₂²×#120	—	5	3	no	—
72	C₂⁴×C₅	—	2	12	no	—
73	C₂⁴×C₅	C₂⁴	1	6	yes	—
74	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
75	C₂³×D₅	—	2	12	no	—
76	C₂³×D₅	—	2	12	no	—
77	C₂³×D₅	—	2	12	no	—
78	C₂³×D₅	—	2	12	no	—
79	C₂³×D₅	—	2	12	no	—
80	C₂³×D₅	—	4	72	no	—
81	C₂³×D₅	—	4	64	no	—
82	C₂³×D₅	C₂⁴	1	18	yes	—
83	C₂³×D₅	C₂⁴	1	6	yes	—
84	C₂³×D₅	C₂⁴	1	6	yes	—
85	C₂³×D₅	C₂⁴	1	16	yes	—
86	C₂²×C₄×C₅	—	2	12	no	—
87	C₂²×C₄×C₅	—	2	12	no	—
88	C₂²×C₄×C₅	C₂⁴	1	18	yes	—
89	C₂²×C₄×C₅	C₂⁴	1	18	yes	—
90	C₂×C₂²⋊D₅	—	4	72	no	—
91	C₂×C₂²⋊D₅	C₂⁴	1	48	yes	—
92	C₂²×Dic₅	—	2	12	no	—
93	C₂²×Dic₅	C₂⁴	1	18	yes	—
94	C₂×D₂₀	—	4	72	no	—
95	C₂×D₂₀	—	4	72	no	—
96	C₂×D₂₀	—	4	72	no	—
97	C₂×D₂₀	C₂⁴	1	48	yes	—
98	C₂×D₂₀	C₂⁴	1	48	yes	—
99	C₂×D₂₀	C₂⁴	1	48	yes	complemented
100	C₂×C₄×D₅	—	4	72	no	—
101	C₂×C₄×D₅	C₂⁴	1	48	yes	complemented
102	C₅×C₄⋊C₄	C₂⁴	1	48	yes	complemented
103	C₅×C₂²⋊C₄	C₂⁴	1	48	yes	—
104	C₄²×C₅	C₂⁴	1	16	yes	complemented
105	#685	C₂⁴	1	48	yes	—
106	#685	C₂⁴	1	48	yes	—
107	#685	C₂⁴	1	48	yes	complemented
108	#683	C₂⁴	1	48	yes	complemented
109	C₂⁶	—	5	1	no	—
110	C₂³×D₄	—	5	3	no	—
111	C₂³×D₄	—	5	3	no	—
112	C₂³×D₄	—	5	3	no	—
113	C₂³×D₄	—	5	3	no	—
114	C₂⁴×C₄	—	5	3	no	—
115	#560	—	5	256	no	—
116	C₂×#127	—	5	24	no	—
117	C₂×#126	—	5	24	no	—
118	C₂×#126	C₂×D₅	1	24	yes	complemented
119	C₂×#123	—	5	72	no	—
120	C₂×#121	—	5	72	no	—
121	C₂×#121	—	5	72	no	—
122	C₂×#120	—	5	72	no	—
123	C₂²×C₄⋊C₄	—	5	3	no	—
124	C₂²×C₄⋊C₄	C₂×D₅	1	3	yes	complemented
125	C₂²×C₂²⋊C₄	—	5	3	no	—
126	C₂²×C₂²⋊C₄	—	5	3	no	—
127	C₂²×C₂²⋊C₄	—	5	3	no	—
128	C₂²×C₂²⋊C₄	C₂×D₅	1	3	yes	—
129	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
130	C₂³×C₅	—	2	36	no	—
131	C₂³×C₅	—	4	16	no	—
132	C₂³×C₅	H(5,𝔽₂)	1	12	yes	—
133	C₂³×C₅	C₂⁵	1	4	yes	—
134	C₂³×C₅	C₂⁵	1	3	yes	—
135	C₂²×D₅	—	2	36	no	—
136	C₂²×D₅	—	2	36	no	—
137	C₂²×D₅	—	2	36	no	—
138	C₂²×D₅	—	2	36	no	—
139	C₂²×D₅	—	2	36	no	—
140	C₂²×D₅	—	4	48	no	—
141	C₂²×D₅	—	4	192	no	—
142	C₂²×D₅	—	4	48	no	—
143	C₂²×D₅	—	4	16	no	—
144	C₂²×D₅	—	4	16	no	—
145	C₂²×D₅	C₂⁵	1	4	yes	—
146	C₂²×D₅	C₂⁵	1	4	yes	—
147	C₂²×D₅	C₂⁵	1	12	yes	—
148	C₂×C₄×C₅	—	2	36	no	—
149	C₂×C₄×C₅	—	2	36	no	—
150	C₂×C₄×C₅	C₂⁵	1	12	yes	—
151	C₂×C₄×C₅	C₂⁵	1	12	yes	—
152	C₂²⋊D₅	—	4	48	no	—
153	C₂×Dic₅	—	2	36	no	—
154	C₂×Dic₅	C₂⁵	1	12	yes	—
155	D₂₀	—	4	48	no	—
156	D₂₀	—	4	48	no	—
157	D₂₀	—	4	48	no	—
158	C₄×D₅	—	4	48	no	—
159	C₂⁵	—	5	1	no	—
160	C₂⁵	—	5	1	no	—
161	C₂⁵	—	5	3	no	—
162	C₂⁵	—	5	12	no	—
163	C₂⁵	—	20	12	no	—
164	C₂⁵	C₂²×D₅	1	1	yes	characteristic
165	C₂²×D₄	—	5	36	no	—
166	C₂²×D₄	—	5	36	no	—
167	C₂²×D₄	—	5	36	no	—
168	C₂²×D₄	—	5	36	no	—
169	C₂²×D₄	—	20	12	no	—
170	C₂²×D₄	—	20	12	no	—
171	C₂²×D₄	—	20	12	no	—
172	C₂²×D₄	—	20	12	no	—
173	C₂³×C₄	—	5	3	no	—
174	C₂³×C₄	—	5	36	no	—
175	C₂³×C₄	—	20	12	no	—
176	C₂³×C₄	C₂²×D₅	1	3	yes	—
177	C₂³×C₄	C₂²×D₅	1	3	yes	—
178	#127	—	5	64	no	—
179	#126	—	5	64	no	—
180	#126	C₂²×D₅	1	64	yes	complemented
181	#123	—	5	192	no	—
182	#121	—	5	192	no	—
183	#121	—	5	192	no	—
184	#120	—	5	192	no	—
185	C₂×C₄⋊C₄	—	5	36	no	—
186	C₂×C₄⋊C₄	C₂²×D₅	1	36	yes	complemented
187	C₂×C₂²⋊C₄	—	5	36	no	—
188	C₂×C₂²⋊C₄	—	5	36	no	—
189	C₂×C₂²⋊C₄	—	5	36	no	—
190	C₂×C₂²⋊C₄	C₂²×D₅	1	36	yes	—
191	C₂×C₄²	C₂²×D₅	1	12	yes	complemented
192	C₂²×C₅	—	2	12	no	—
193	C₂²×C₅	—	4	24	no	—
194	C₂²×C₅	#560	1	16	yes	complemented
195	C₂²×C₅	C₂×H(5,𝔽₂)	1	18	yes	—
196	C₂²×C₅	C₂⁶	1	1	yes	characteristic
197	C₂×D₅	—	2	12	no	—
198	C₂×D₅	—	2	12	no	—
199	C₂×D₅	—	2	12	no	—
200	C₂×D₅	—	2	12	no	—
201	C₂×D₅	—	2	12	no	—
202	C₂×D₅	—	4	64	no	—
203	C₂×D₅	—	4	72	no	—
204	C₂×D₅	—	4	24	no	—
205	C₂×D₅	—	4	24	no	—
206	C₄×C₅	—	2	12	no	—
207	C₄×C₅	—	2	12	no	—
208	Dic₅	—	2	12	no	—
209	C₂⁴	—	2	12	no	—
210	C₂⁴	—	5	6	no	—
211	C₂⁴	—	5	6	no	—
212	C₂⁴	—	5	18	no	—
213	C₂⁴	—	5	16	no	—
214	C₂⁴	—	10	12	no	—
215	C₂⁴	—	10	12	no	—
216	C₂⁴	—	10	12	no	—
217	C₂⁴	—	10	12	no	—
218	C₂⁴	—	10	12	no	—
219	C₂⁴	—	20	72	no	—
220	C₂⁴	—	20	64	no	—
221	C₂⁴	C₂³×D₅	1	6	yes	—
222	C₂⁴	C₂³×D₅	1	1	yes	characteristic
223	C₂×D₄	—	5	48	no	—
224	C₂×D₄	—	5	48	no	—
225	C₂×D₄	—	5	48	no	—
226	C₂×D₄	—	5	48	no	—
227	C₂×D₄	—	20	72	no	—
228	C₂×D₄	—	20	72	no	—
229	C₂×D₄	—	20	72	no	—
230	C₂×D₄	—	20	72	no	—
231	C₂²×C₄	—	2	12	no	—
232	C₂²×C₄	—	2	12	no	—
233	C₂²×C₄	—	5	18	no	—
234	C₂²×C₄	—	5	48	no	—
235	C₂²×C₄	—	10	12	no	—
236	C₂²×C₄	—	20	72	no	—
237	C₂²×C₄	C₂³×D₅	1	18	yes	—
238	C₂²×C₄	C₂³×D₅	1	18	yes	—
239	C₄⋊C₄	—	5	48	no	—
240	C₄⋊C₄	C₂³×D₅	1	48	yes	complemented
241	C₂²⋊C₄	—	5	48	no	—
242	C₂²⋊C₄	—	5	48	no	—
243	C₂²⋊C₄	—	5	48	no	—
244	C₂²⋊C₄	C₂³×D₅	1	48	yes	—
245	C₄²	C₂³×D₅	1	16	yes	complemented
246	C₂×C₅	—	4	4	no	—
247	C₂×C₅	C₂×#560	1	12	yes	complemented
248	C₂×C₅	C₂²×H(5,𝔽₂)	1	3	yes	—
249	D₅	—	4	12	no	—
250	D₅	—	4	4	no	—
251	D₅	—	4	4	no	—
252	C₂³	—	2	36	no	—
253	C₂³	—	4	16	no	—
254	C₂³	—	5	4	no	—
255	C₂³	—	5	4	no	—
256	C₂³	—	5	12	no	—
257	C₂³	—	10	36	no	—
258	C₂³	—	10	36	no	—
259	C₂³	—	10	36	no	—
260	C₂³	—	10	36	no	—
261	C₂³	—	10	36	no	—
262	C₂³	—	20	48	no	—
263	C₂³	—	20	192	no	—
264	C₂³	—	20	16	no	—
265	C₂³	—	20	16	no	—
266	C₂³	—	20	48	no	—
267	C₂³	#4149	1	12	yes	—
268	C₂³	C₂⁴×D₅	1	4	yes	—
269	C₂³	C₂⁴×D₅	1	3	yes	—
270	D₄	—	20	48	no	—
271	D₄	—	20	48	no	—
272	D₄	—	20	48	no	—
273	D₄	—	20	48	no	—
274	C₂×C₄	—	2	36	no	—
275	C₂×C₄	—	2	36	no	—
276	C₂×C₄	—	5	12	no	—
277	C₂×C₄	—	10	36	no	—
278	C₂×C₄	—	20	48	no	—
279	C₂×C₄	C₂⁴×D₅	1	12	yes	—
280	C₂×C₄	C₂⁴×D₅	1	12	yes	—
281	C₅	C₂²×#560	1	1	yes	characteristic, complemented, Sylow
282	C₂²	—	2	12	no	—
283	C₂²	—	4	24	no	—
284	C₂²	—	10	12	no	—
285	C₂²	—	10	12	no	—
286	C₂²	—	10	12	no	—
287	C₂²	—	10	12	no	—
288	C₂²	—	10	12	no	—
289	C₂²	—	20	64	no	—
290	C₂²	—	20	24	no	—
291	C₂²	—	20	24	no	—
292	C₂²	—	20	72	no	—
293	C₂²	#66045	1	16	yes	direct product factor
294	C₂²	C₂×#4149	1	18	yes	—
295	C₂²	C₂⁵×D₅	1	1	yes	characteristic
296	C₄	—	2	12	no	—
297	C₄	—	2	12	no	—
298	C₄	—	10	12	no	—
299	C₂	—	4	4	no	—
300	C₂	—	20	4	no	—
301	C₂	—	20	4	no	—
302	C₂	—	20	12	no	—
303	C₂	C₂×#66045	1	12	yes	direct product factor
304	C₂	C₂²×#4149	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁶
Frattini	C₂²	C₂⁵×D₅
Fitting	C₂²×C₅×#126	C₂
socle	C₂⁴×C₅	C₂⁴
hypercenter	C₂²×#126	D₅
hypocenter	C₅	C₂²×#560
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#126

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#126	C₂	semidirect
2. C₂×#66045	C₂	direct (N × H)
3. C₂²×#4040	C₂	semidirect
4. C₂²×#4041	C₂	semidirect
5. C₂²×#4030	C₂	semidirect
6. C₂²×#4028	C₂	semidirect
7. C₂²×#4020	C₂	semidirect
8. C₂²×#4022	C₂	semidirect
13. C₂³×D₂₀	C₂²	semidirect
14. C₂³×C₄×D₅	C₂²	semidirect
15. C₂×C₅×#126	C₂²	semidirect
16. C₂²×C₅×C₄⋊C₄	C₂²	semidirect
18. C₂²×C₄²×C₅	C₂²	semidirect
21. C₂²×#685	C₂²	semidirect
22. C₂²×#683	C₂²	semidirect
23. #66045	C₂²	direct (N × H)
24. C₂×#4041	C₂²	semidirect
25. C₂×#4040	C₂²	semidirect
26. C₂×#4030	C₂²	semidirect
27. C₂×#4028	C₂²	semidirect
28. C₂×#4022	C₂²	semidirect
29. C₂×#4020	C₂²	semidirect
47. C₂²×D₂₀	C₂³	semidirect
49. C₂²×C₄×D₅	C₂³	semidirect
50. C₅×#126	C₂³	semidirect
51. C₂×C₅×C₄⋊C₄	C₂³	semidirect
53. C₂×C₄²×C₅	C₂³	semidirect
56. C₂×#685	C₂³	semidirect
57. C₂×#683	C₂³	semidirect
58. #4041	C₂³	semidirect
59. #4040	C₂³	semidirect
60. #4030	C₂³	semidirect
61. #4028	C₂³	semidirect
62. #4022	C₂³	semidirect
63. #4020	C₂³	semidirect
67. C₂²×#126	D₅	semidirect
99. C₂×D₂₀	C₂⁴	semidirect
101. C₂×C₄×D₅	C₂⁴	semidirect
102. C₅×C₄⋊C₄	C₂⁴	semidirect
104. C₄²×C₅	C₂⁴	semidirect
107. #685	C₂⁴	semidirect
108. #683	C₂⁴	semidirect
118. C₂×#126	C₂×D₅	semidirect
124. C₂²×C₄⋊C₄	C₂×D₅	semidirect
129. C₂²×C₄²	C₂×D₅	semidirect
180. #126	C₂²×D₅	semidirect
186. C₂×C₄⋊C₄	C₂²×D₅	semidirect
191. C₂×C₄²	C₂²×D₅	semidirect
194. C₂²×C₅	#560	semidirect
240. C₄⋊C₄	C₂³×D₅	semidirect
245. C₄²	C₂³×D₅	semidirect
247. C₂×C₅	C₂×#560	semidirect
281. C₅	C₂²×#560	semidirect
293. C₂²	#66045	direct (N × H)
303. C₂	C₂×#66045	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×C₅×#126	2
C₂×#66045	2
C₂²×#4040	2
C₂²×#4041	2
C₂²×#4030	2
C₂²×#4028	2
C₂²×#4020	2
C₂²×#4022	2
C₂²×#560	5

Subgroups — C22×#66045 #13006170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#66045 #13006170