Subgroups — C₂²×#65006 #12963598

Order 1280.

2710 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4050	C₂	1	1	yes	maximal, characteristic
2	C₂²×#4047	C₂	1	2	yes	maximal, complemented
3	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#3951	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#3937	C₂	1	2	yes	maximal, complemented
6	C₂×#65006	C₂	1	24	yes	maximal, direct product factor
7	C₂²×#709	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×#681	C₂²	1	2	yes	—
12	C₂²×Dic₂₀	C₄	1	1	yes	characteristic, complemented
13	C₂×#4050	C₄	1	12	yes	complemented
14	C₂×#4050	C₂²	1	12	yes	—
15	C₂²×D₄₀	C₄	1	1	yes	characteristic, complemented
16	C₂²×#677	C₄	1	2	yes	complemented
17	C₂×#4047	—	2	12	no	—
18	C₂×#4047	C₂²	1	24	yes	complemented
19	C₂²×#676	—	2	2	no	—
20	C₂²×C₈×D₅	—	2	2	no	—
21	C₂×C₅×#97	C₂²	1	12	yes	complemented
22	C₂×#3951	C₂²	1	12	yes	complemented
23	C₂×#3937	C₂²	1	24	yes	complemented
24	#65006	C₂²	1	64	yes	direct product factor
25	C₂²×#444	—	5	1	no	maximal, Sylow
26	C₂³×C₄×C₅	—	2	1	no	—
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₂²⋊D₅	—	2	2	no	—
29	C₂×#709	—	2	6	no	—
30	C₂×#709	—	2	6	no	—
31	C₂×#709	C₂×C₄	1	12	yes	—
32	C₂×#709	C₂³	1	12	yes	—
33	C₂²×D₂₀	C₂×C₄	1	2	yes	—
34	C₂²×C₄×D₅	—	2	2	no	—
35	C₂²×Dic₁₀	C₂×C₄	1	2	yes	—
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
39	C₂²×C₅×C₈	D₄	1	2	yes	—
40	C₂²×C₅×C₈	D₄	1	6	yes	—
41	C₂²×C₅×C₈	C₂³	1	6	yes	—
42	C₂²×C₅×C₈	C₂³	1	6	yes	—
43	C₂×C₄²×C₅	—	2	3	no	—
44	C₂×C₄²×C₅	C₂³	1	6	yes	—
45	C₂×#681	—	2	6	no	—
46	C₂×#681	C₂³	1	12	yes	—
47	C₂²×#168	—	2	2	no	—
48	C₂×Dic₂₀	C₂×C₄	1	12	yes	complemented
49	#4050	C₂×C₄	1	48	yes	complemented
50	#4050	C₂³	1	16	yes	—
51	C₂×D₄₀	C₂×C₄	1	12	yes	complemented
52	C₂×#677	C₂×C₄	1	24	yes	complemented
53	#4047	—	2	48	no	—
54	#4047	C₂³	1	32	yes	complemented
55	C₂×#676	—	2	24	no	—
56	C₂×C₈×D₅	—	2	24	no	—
57	C₅×#97	C₂³	1	16	yes	complemented
58	#3951	C₂³	1	16	yes	complemented
59	#3937	C₂³	1	32	yes	complemented
60	C₂²×#135	—	5	1	no	—
61	C₂²×#131	—	5	2	no	—
62	C₂×#444	—	5	24	no	—
63	C₂²×#108	—	5	1	no	—
64	C₂²×#104	—	5	2	no	—
65	C₂²×#97	D₅	1	1	yes	characteristic, complemented
66	C₂⁴×C₅	#41	1	1	yes	characteristic
67	C₂³×D₅	—	2	2	no	—
68	C₂²×C₄×C₅	—	2	6	no	—
69	C₂²×C₄×C₅	—	4	4	no	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	#41	1	3	yes	—
75	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
76	C₂×C₂²⋊D₅	—	2	12	no	—
77	C₂×C₂²⋊D₅	—	2	12	no	—
78	C₂²×Dic₅	—	2	2	no	—
79	#709	—	2	24	no	—
80	#709	—	2	12	no	—
81	#709	—	2	12	no	—
82	#709	C₂²×C₄	1	24	yes	—
83	#709	C₂⁴	1	8	yes	—
84	C₂×D₂₀	—	2	6	no	—
85	C₂×D₂₀	C₂²×C₄	1	12	yes	—
86	C₂×C₄×D₅	—	2	12	no	—
87	C₂×C₄×D₅	—	2	12	no	—
88	C₂×Dic₁₀	—	2	6	no	—
89	C₂×Dic₁₀	C₂²×C₄	1	12	yes	—
90	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
92	C₂×C₅×C₈	C₂²×C₄	1	12	yes	—
93	C₂×C₅×C₈	C₂×D₄	1	12	yes	—
94	C₂×C₅×C₈	C₂×D₄	1	12	yes	—
95	C₂×C₅×C₈	C₂⁴	1	4	yes	—
96	C₂×C₅×C₈	C₂⁴	1	4	yes	—
97	C₄²×C₅	—	2	6	no	—
98	C₄²×C₅	C₂⁴	1	4	yes	—
99	#681	—	2	12	no	—
100	#681	C₂⁴	1	8	yes	—
101	C₂×#168	—	2	12	no	—
102	Dic₂₀	C₂²×C₄	1	16	yes	complemented
103	D₄₀	C₂²×C₄	1	16	yes	complemented
104	#677	C₂²×C₄	1	32	yes	complemented
105	#676	—	2	32	no	—
106	C₈×D₅	—	2	32	no	—
107	C₂²×#41	—	5	2	no	—
108	C₂×#135	—	5	12	no	—
109	C₂×#135	—	5	12	no	—
110	C₂²×Q₁₆	—	5	1	no	—
111	C₂²×SD₁₆	—	5	2	no	—
112	C₂²×D₈	—	5	1	no	—
113	C₂×#131	—	5	24	no	—
114	C₂×#131	—	10	12	no	—
115	C₂²×M₁₆	—	5	2	no	—
116	C₂²×M₁₆	—	10	2	no	—
117	C₂³×C₈	—	10	2	no	—
118	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
119	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
120	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
121	#444	—	5	64	no	—
122	C₂×#108	—	5	12	no	—
123	C₂×#104	—	5	24	no	—
124	C₂×#97	C₂×D₅	1	12	yes	complemented
125	C₂³×C₅	—	2	4	no	—
126	C₂³×C₅	C₄×D₄	1	3	yes	—
127	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
128	C₂³×C₅	C₂×#41	1	3	yes	—
129	C₂²×D₅	—	2	12	no	—
130	C₂²×D₅	—	4	8	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	4	12	no	—
133	C₂×C₄×C₅	C₄×D₄	1	3	yes	—
134	C₂×C₄×C₅	C₄×D₄	1	3	yes	—
135	C₂×C₄×C₅	C₄×D₄	1	6	yes	—
136	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
137	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
138	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
139	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
140	C₂×C₄×C₅	C₂×#41	1	3	yes	—
141	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
142	C₂²⋊D₅	—	2	24	no	—
143	C₂²⋊D₅	—	2	8	no	—
144	C₂×Dic₅	—	2	12	no	—
145	D₂₀	—	2	12	no	—
146	D₂₀	C₂³×C₄	1	8	yes	—
147	C₄×D₅	—	2	24	no	—
148	C₄×D₅	—	2	8	no	—
149	Dic₁₀	—	2	12	no	—
150	Dic₁₀	C₂³×C₄	1	8	yes	—
151	C₅×C₈	C₂³×C₄	1	4	yes	—
152	C₅×C₈	C₂³×C₄	1	4	yes	—
153	C₅×C₈	C₂²×D₄	1	8	yes	—
154	#168	—	2	8	no	—
155	C₂×#41	—	5	12	no	—
156	C₂×#41	—	5	12	no	—
157	C₂×#41	—	10	6	no	—
158	C₂×#41	—	10	6	no	—
159	C₂²×Q₈	—	5	2	no	—
160	C₂²×D₄	—	5	2	no	—
161	C₂²×D₄	—	10	2	no	—
162	C₂³×C₄	—	2	1	no	—
163	C₂³×C₄	—	10	2	no	—
164	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
165	#135	—	5	48	no	—
166	#135	—	5	16	no	—
167	C₂×Q₁₆	—	5	12	no	—
168	C₂×SD₁₆	—	5	24	no	—
169	C₂×D₈	—	5	12	no	—
170	#131	—	5	32	no	—
171	#131	—	10	48	no	—
172	C₂×M₁₆	—	5	12	no	—
173	C₂×M₁₆	—	10	6	no	—
174	C₂×M₁₆	—	10	24	no	—
175	C₂²×C₈	—	10	2	no	—
176	C₂²×C₈	—	10	24	no	—
177	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
178	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₅	1	6	yes	—
180	C₂²×C₈	D₂₀	1	6	yes	—
181	C₂²×C₈	D₂₀	1	2	yes	—
182	C₂²×C₈	C₂²×D₅	1	6	yes	—
183	C₂²×C₈	C₂²×D₅	1	6	yes	—
184	C₂×C₄²	—	2	3	no	—
185	C₂×C₄²	C₂²×D₅	1	6	yes	—
186	#108	—	5	16	no	—
187	#104	—	5	32	no	—
188	#97	C₂²×D₅	1	16	yes	complemented
189	C₂²×C₅	—	2	6	no	—
190	C₂²×C₅	—	2	6	no	—
191	C₂²×C₅	#444	1	4	yes	complemented
192	C₂²×C₅	C₂×C₄×D₄	1	3	yes	—
193	C₂²×C₅	C₂×C₄×D₄	1	3	yes	—
194	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
195	C₂×D₅	—	2	8	no	—
196	C₂×D₅	—	4	24	no	—
197	C₄×C₅	—	4	4	no	—
198	C₄×C₅	C₂×C₄×D₄	1	3	yes	—
199	C₄×C₅	C₂×C₄×D₄	1	3	yes	—
200	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
201	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
202	Dic₅	—	2	8	no	—
203	C₂⁴	—	10	2	no	—
204	C₂⁴	#709	1	1	yes	characteristic
205	#41	—	5	24	no	—
206	#41	—	5	8	no	—
207	#41	—	10	12	no	—
208	#41	—	10	12	no	—
209	#41	—	10	24	no	—
210	C₂×Q₈	—	5	12	no	—
211	C₂×Q₈	—	10	6	no	—
212	C₂×D₄	—	5	12	no	—
213	C₂×D₄	—	10	12	no	—
214	C₂×D₄	—	10	12	no	—
215	C₂×D₄	—	10	6	no	—
216	C₂²×C₄	—	2	6	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	—	10	2	no	—
219	C₂²×C₄	—	10	12	no	—
220	C₂²×C₄	—	10	12	no	—
221	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
223	C₂²×C₄	C₂×D₂₀	1	3	yes	—
224	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
225	C₂²×C₄	#709	1	3	yes	—
226	C₂²×C₄	C₂³×D₅	1	3	yes	—
227	Q₁₆	—	5	16	no	—
228	SD₁₆	—	5	32	no	—
229	D₈	—	5	16	no	—
230	M₁₆	—	5	8	no	—
231	M₁₆	—	10	12	no	—
232	M₁₆	—	10	32	no	—
233	C₂×C₈	—	10	12	no	—
234	C₂×C₈	—	10	32	no	—
235	C₂×C₈	C₂×C₄×D₅	1	6	yes	—
236	C₂×C₈	C₂×C₄×D₅	1	6	yes	—
237	C₂×C₈	C₂×C₄×D₅	1	12	yes	—
238	C₂×C₈	C₂×D₂₀	1	12	yes	—
239	C₂×C₈	C₂×D₂₀	1	12	yes	—
240	C₂×C₈	C₂³×D₅	1	4	yes	—
241	C₂×C₈	C₂³×D₅	1	4	yes	—
242	C₄²	—	2	6	no	—
243	C₄²	C₂³×D₅	1	4	yes	—
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	2	3	no	—
246	C₂×C₅	C₂×#444	1	6	yes	complemented
247	C₂×C₅	C₂²×C₄×D₄	1	1	yes	characteristic
248	D₅	—	4	8	no	—
249	C₂³	—	2	4	no	—
250	C₂³	—	10	12	no	—
251	C₂³	—	20	8	no	—
252	C₂³	C₄×D₂₀	1	3	yes	—
253	C₂³	C₄×D₂₀	1	1	yes	characteristic
254	C₂³	C₂×#709	1	3	yes	—
255	Q₈	—	5	8	no	—
256	Q₈	—	10	12	no	—
257	D₄	—	5	8	no	—
258	D₄	—	10	24	no	—
259	D₄	—	10	8	no	—
260	D₄	—	10	12	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	12	no	—
263	C₂×C₄	—	10	12	no	—
264	C₂×C₄	—	10	24	no	—
265	C₂×C₄	—	10	8	no	—
266	C₂×C₄	C₄×D₂₀	1	3	yes	—
267	C₂×C₄	C₄×D₂₀	1	3	yes	—
268	C₂×C₄	C₄×D₂₀	1	6	yes	—
269	C₂×C₄	C₂²×C₄×D₅	1	3	yes	—
270	C₂×C₄	C₂²×C₄×D₅	1	3	yes	—
271	C₂×C₄	C₂²×D₂₀	1	3	yes	—
272	C₂×C₄	C₂²×D₂₀	1	3	yes	—
273	C₂×C₄	C₂×#709	1	3	yes	—
274	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
275	C₈	—	10	8	no	—
276	C₈	C₂²×C₄×D₅	1	4	yes	—
277	C₈	C₂²×C₄×D₅	1	4	yes	—
278	C₈	C₂²×D₂₀	1	8	yes	—
279	C₅	C₂²×#444	1	1	yes	characteristic, complemented, Sylow
280	C₂²	—	2	6	no	—
281	C₂²	—	2	6	no	—
282	C₂²	—	10	8	no	—
283	C₂²	—	20	24	no	—
284	C₂²	#65006	1	4	yes	direct product factor
285	C₂²	C₂×C₄×D₂₀	1	3	yes	—
286	C₂²	C₂×C₄×D₂₀	1	3	yes	—
287	C₂²	C₂²×#709	1	1	yes	characteristic
288	C₄	—	4	4	no	—
289	C₄	—	10	8	no	—
290	C₄	C₂×C₄×D₂₀	1	3	yes	—
291	C₄	C₂×C₄×D₂₀	1	3	yes	—
292	C₄	C₂³×C₄×D₅	1	1	yes	characteristic
293	C₄	C₂³×D₂₀	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	3	no	—
296	C₂	—	20	8	no	—
297	C₂	C₂×#65006	1	6	yes	direct product factor
298	C₂	C₂²×C₄×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₄×C₅	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₅×#97	C₂
socle	C₂³×C₅	C₄×D₄
prime radical	C₂²×#4050	C₂
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂²×#97	D₅
hypocenter	C₅	C₂²×#444
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. C₂²×#4047	C₂	semidirect
3. C₂²×C₅×#97	C₂	semidirect
4. C₂²×#3951	C₂	semidirect
5. C₂²×#3937	C₂	semidirect
6. C₂×#65006	C₂	direct (N × H)
12. C₂²×Dic₂₀	C₄	semidirect
13. C₂×#4050	C₄	semidirect
15. C₂²×D₄₀	C₄	semidirect
16. C₂²×#677	C₄	semidirect
18. C₂×#4047	C₂²	semidirect
21. C₂×C₅×#97	C₂²	semidirect
22. C₂×#3951	C₂²	semidirect
23. C₂×#3937	C₂²	semidirect
24. #65006	C₂²	direct (N × H)
48. C₂×Dic₂₀	C₂×C₄	semidirect
49. #4050	C₂×C₄	semidirect
51. C₂×D₄₀	C₂×C₄	semidirect
52. C₂×#677	C₂×C₄	semidirect
54. #4047	C₂³	semidirect
57. C₅×#97	C₂³	semidirect
58. #3951	C₂³	semidirect
59. #3937	C₂³	semidirect
65. C₂²×#97	D₅	semidirect
102. Dic₂₀	C₂²×C₄	semidirect
103. D₄₀	C₂²×C₄	semidirect
104. #677	C₂²×C₄	semidirect
124. C₂×#97	C₂×D₅	semidirect
188. #97	C₂²×D₅	semidirect
191. C₂²×C₅	#444	semidirect
246. C₂×C₅	C₂×#444	semidirect
279. C₅	C₂²×#444	semidirect
284. C₂²	#65006	direct (N × H)
297. C₂	C₂×#65006	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×#4050	2
C₂²×#4047	2
C₂²×C₅×#97	2
C₂²×#3951	2
C₂²×#3937	2
C₂×#65006	2
C₂²×#444	5

Subgroups — C22×#65006 #12963598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65006 #12963598