Subgroups — C₂×#65939 #10617841

Order 640.

1984 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₅	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#4085	C₂	1	2	yes	maximal, complemented
4	C₂×#4083	C₂	1	2	yes	maximal, complemented
5	C₂×#4076	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#685	C₂	1	1	yes	maximal, characteristic, complemented
7	#65939	C₂	1	16	yes	maximal, direct product factor
8	C₂×#4040	C₂	1	2	yes	maximal, complemented
9	C₂×#4029	C₂	1	2	yes	maximal, complemented
10	C₂×#4028	C₂	1	2	yes	maximal, complemented
11	C₂⁴×D₅	C₂²	1	1	yes	characteristic
12	C₂⁴×D₅	C₂²	1	2	yes	complemented
13	C₂²×C₅×D₄	C₂²	1	2	yes	complemented
14	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
15	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂²×C₂²⋊D₅	C₂²	1	2	yes	complemented
18	C₂²×C₂²⋊D₅	C₂²	1	2	yes	complemented
19	C₂³×Dic₅	C₂²	1	1	yes	characteristic, complemented
20	C₂×D₄×D₅	—	2	16	no	—
21	C₂×D₄×D₅	C₂²	1	16	yes	complemented
22	C₂²×D₂₀	C₂²	1	2	yes	—
23	C₂²×D₂₀	C₂²	1	1	yes	characteristic, complemented
24	C₂²×C₄×D₅	C₂²	1	2	yes	complemented
25	C₅×#121	C₂²	1	8	yes	complemented
26	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
29	#4085	C₂²	1	16	yes	complemented
30	#4083	C₂²	1	16	yes	complemented
31	#4076	C₂²	1	8	yes	complemented
32	C₂×#685	C₂²	1	2	yes	complemented
33	C₂×#685	C₂²	1	2	yes	complemented
34	C₂×#685	C₂²	1	2	yes	complemented
35	C₂×#685	C₂²	1	2	yes	complemented
36	C₂×#685	C₂²	1	8	yes	complemented
37	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
38	C₂×#683	C₂²	1	2	yes	complemented
39	#4040	C₂²	1	16	yes	complemented
40	#4029	C₂²	1	16	yes	complemented
41	#4028	C₂²	1	16	yes	complemented
42	C₂×#534	—	5	1	no	maximal, Sylow
43	C₂⁴×C₅	C₂³	1	2	yes	—
44	C₂⁴×C₅	C₂³	1	1	yes	characteristic
45	C₂³×D₅	—	2	8	no	—
46	C₂³×D₅	—	4	2	no	—
47	C₂³×D₅	—	4	2	no	—
48	C₂³×D₅	—	4	4	no	—
49	C₂³×D₅	—	4	4	no	—
50	C₂³×D₅	D₄	1	8	yes	complemented
51	C₂³×D₅	C₂³	1	2	yes	—
52	C₂³×D₅	C₂³	1	2	yes	—
53	C₂³×D₅	C₂³	1	4	yes	—
54	C₂³×D₅	C₂³	1	8	yes	complemented
55	C₂×C₅×D₄	—	2	8	no	—
56	C₂×C₅×D₄	—	4	2	no	—
57	C₂×C₅×D₄	C₂³	1	8	yes	complemented
58	C₂×C₅×D₄	C₂³	1	4	yes	complemented
59	C₂²×C₄×C₅	—	4	2	no	—
60	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂³	1	2	yes	—
62	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂³	1	4	yes	complemented
64	C₂×C₂²⋊D₅	—	2	8	no	—
65	C₂×C₂²⋊D₅	—	2	8	no	—
66	C₂×C₂²⋊D₅	—	4	4	no	—
67	C₂×C₂²⋊D₅	C₂³	1	8	yes	—
68	C₂×C₂²⋊D₅	C₂³	1	8	yes	complemented
69	C₂×C₂²⋊D₅	C₂³	1	8	yes	complemented
70	C₂²×Dic₅	—	4	2	no	—
71	C₂²×Dic₅	C₂³	1	2	yes	—
72	C₂²×Dic₅	C₂³	1	2	yes	—
73	C₂²×Dic₅	C₂³	1	4	yes	complemented
74	D₄×D₅	—	2	64	no	—
75	C₂×D₂₀	—	2	8	no	—
76	C₂×D₂₀	—	4	2	no	—
77	C₂×D₂₀	C₂³	1	8	yes	—
78	C₂×D₂₀	C₂³	1	4	yes	complemented
79	C₂×C₄×D₅	—	2	8	no	—
80	C₂×C₄×D₅	C₂³	1	8	yes	complemented
81	C₅×C₄⋊C₄	C₂³	1	4	yes	complemented
82	C₅×C₂²⋊C₄	C₂³	1	8	yes	complemented
83	C₂²⋊Dic₅	C₂³	1	8	yes	complemented
84	#685	C₂³	1	8	yes	complemented
85	#685	C₂³	1	8	yes	complemented
86	#685	C₂³	1	8	yes	complemented
87	#685	C₂³	1	8	yes	complemented
88	C₄⋊Dic₅	C₂³	1	4	yes	complemented
89	#683	C₂³	1	8	yes	complemented
90	C₂³×D₄	—	5	2	no	—
91	#534	—	5	16	no	—
92	C₂×#123	—	5	2	no	—
93	C₂×#123	—	5	2	no	—
94	C₂×#121	—	5	2	no	—
95	C₂×#121	—	5	1	no	—
96	C₂×#121	D₅	1	1	yes	characteristic, complemented
97	C₂×#120	—	5	2	no	—
98	C₂×#120	—	5	2	no	—
99	C₂²×C₂²⋊C₄	—	5	1	no	—
100	C₂³×C₅	—	2	2	no	—
101	C₂³×C₅	—	2	2	no	—
102	C₂³×C₅	—	2	4	no	—
103	C₂³×C₅	—	2	4	no	—
104	C₂³×C₅	—	2	4	no	—
105	C₂³×C₅	C₂×D₄	1	4	yes	—
106	C₂³×C₅	C₂⁴	1	4	yes	—
107	C₂³×C₅	C₂⁴	1	2	yes	—
108	C₂³×C₅	C₂⁴	1	1	yes	characteristic
109	C₂²×D₅	—	2	4	no	—
110	C₂²×D₅	—	2	4	no	—
111	C₂²×D₅	—	2	4	no	—
112	C₂²×D₅	—	2	4	no	—
113	C₂²×D₅	—	2	4	no	—
114	C₂²×D₅	—	2	16	no	—
115	C₂²×D₅	—	4	16	no	—
116	C₂²×D₅	—	4	4	no	—
117	C₂²×D₅	—	4	4	no	—
118	C₂²×D₅	—	4	8	no	—
119	C₂²×D₅	—	4	8	no	—
120	C₂²×D₅	—	4	32	no	—
121	C₂²×D₅	C₂×D₄	1	8	yes	—
122	C₂²×D₅	C₂×D₄	1	16	yes	complemented
123	C₂²×D₅	C₂⁴	1	4	yes	—
124	C₂²×D₅	C₂⁴	1	4	yes	—
125	C₅×D₄	—	2	16	no	—
126	C₅×D₄	—	4	4	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	C₂⁴	1	4	yes	—
132	C₂×C₄×C₅	C₂⁴	1	2	yes	—
133	C₂×C₄×C₅	C₂⁴	1	2	yes	—
134	C₂²⋊D₅	—	2	16	no	—
135	C₂²⋊D₅	—	2	16	no	—
136	C₂²⋊D₅	—	4	8	no	—
137	C₂×Dic₅	—	2	4	no	—
138	C₂×Dic₅	—	2	4	no	—
139	C₂×Dic₅	—	4	4	no	—
140	C₂×Dic₅	C₂⁴	1	4	yes	—
141	C₂×Dic₅	C₂⁴	1	4	yes	—
142	D₂₀	—	2	16	no	—
143	D₂₀	—	4	4	no	—
144	C₄×D₅	—	2	16	no	—
145	C₂⁵	—	5	1	no	—
146	C₂⁵	—	5	2	no	—
147	C₂²×D₄	—	5	2	no	—
148	C₂²×D₄	—	5	2	no	—
149	C₂²×D₄	—	5	2	no	—
150	C₂²×D₄	—	5	16	no	—
151	C₂²×D₄	—	5	2	no	—
152	C₂²×D₄	—	5	1	no	—
153	C₂²×D₄	—	10	16	no	—
154	C₂²×D₄	C₂×D₅	1	2	yes	complemented
155	C₂²×D₄	C₂×D₅	1	1	yes	characteristic, complemented
156	C₂³×C₄	—	5	1	no	—
157	C₂³×C₄	—	5	2	no	—
158	C₂³×C₄	C₂×D₅	1	1	yes	characteristic, complemented
159	#123	—	5	16	no	—
160	#123	—	5	16	no	—
161	#121	—	5	16	no	—
162	#121	—	5	8	no	—
163	#121	C₂×D₅	1	8	yes	complemented
164	#120	—	5	16	no	—
165	#120	—	5	16	no	—
166	C₂×C₄⋊C₄	—	5	2	no	—
167	C₂×C₄⋊C₄	—	5	1	no	—
168	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
169	C₂×C₂²⋊C₄	—	5	2	no	—
170	C₂×C₂²⋊C₄	—	5	2	no	—
171	C₂×C₂²⋊C₄	—	5	2	no	—
172	C₂×C₂²⋊C₄	—	5	2	no	—
173	C₂×C₂²⋊C₄	—	5	2	no	—
174	C₂×C₂²⋊C₄	—	5	8	no	—
175	C₂×C₂²⋊C₄	C₂×D₅	1	2	yes	complemented
176	C₂²×C₅	—	2	8	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	2	4	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	4	8	no	—
183	C₂²×C₅	C₂²×D₄	1	4	yes	—
184	C₂²×C₅	C₂²×D₄	1	2	yes	—
185	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
186	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
187	C₂²×C₅	C₂⁵	1	1	yes	characteristic
188	C₂×D₅	—	2	16	no	—
189	C₂×D₅	—	2	4	no	—
190	C₂×D₅	—	2	4	no	—
191	C₂×D₅	—	2	4	no	—
192	C₂×D₅	—	2	4	no	—
193	C₂×D₅	—	2	4	no	—
194	C₂×D₅	—	4	16	no	—
195	C₂×D₅	—	4	32	no	—
196	C₂×D₅	—	4	8	no	—
197	C₂×D₅	C₂²×D₄	1	8	yes	—
198	C₄×C₅	—	2	4	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	2	2	no	—
201	Dic₅	—	2	4	no	—
202	Dic₅	—	2	4	no	—
203	C₂⁴	—	5	2	no	—
204	C₂⁴	—	5	8	no	—
205	C₂⁴	—	5	4	no	—
206	C₂⁴	—	5	2	no	—
207	C₂⁴	—	5	8	no	—
208	C₂⁴	—	10	8	no	—
209	C₂⁴	—	20	2	no	—
210	C₂⁴	—	20	2	no	—
211	C₂⁴	—	20	4	no	—
212	C₂⁴	—	20	4	no	—
213	C₂⁴	C₂²×D₅	1	2	yes	—
214	C₂⁴	C₂²×D₅	1	1	yes	characteristic
215	C₂×D₄	—	2	8	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	5	8	no	—
218	C₂×D₄	—	5	8	no	—
219	C₂×D₄	—	5	8	no	—
220	C₂×D₄	—	5	8	no	—
221	C₂×D₄	—	5	4	no	—
222	C₂×D₄	—	10	8	no	—
223	C₂×D₄	—	10	64	no	—
224	C₂×D₄	—	10	8	no	—
225	C₂×D₄	—	10	8	no	—
226	C₂×D₄	—	20	2	no	—
227	C₂×D₄	—	20	4	no	—
228	C₂×D₄	C₂²×D₅	1	8	yes	complemented
229	C₂×D₄	C₂²×D₅	1	4	yes	complemented
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	5	2	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	—	5	4	no	—
234	C₂²×C₄	—	5	8	no	—
235	C₂²×C₄	—	10	8	no	—
236	C₂²×C₄	—	20	2	no	—
237	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
238	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
239	C₂²×C₄	C₂²×D₅	1	2	yes	—
240	C₂²×C₄	C₂²×D₅	1	4	yes	complemented
241	C₄⋊C₄	—	5	8	no	—
242	C₄⋊C₄	—	5	4	no	—
243	C₄⋊C₄	C₂²×D₅	1	4	yes	complemented
244	C₂²⋊C₄	—	5	8	no	—
245	C₂²⋊C₄	—	5	8	no	—
246	C₂²⋊C₄	—	5	8	no	—
247	C₂²⋊C₄	—	5	8	no	—
248	C₂²⋊C₄	—	5	8	no	—
249	C₂²⋊C₄	C₂²×D₅	1	8	yes	complemented
250	C₂×C₅	—	2	4	no	—
251	C₂×C₅	—	4	4	no	—
252	C₂×C₅	#534	1	4	yes	complemented
253	C₂×C₅	C₂³×D₄	1	1	yes	characteristic
254	C₂×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
255	C₂×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
256	D₅	—	2	8	no	—
257	D₅	—	4	4	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	5	8	no	—
264	C₂³	—	5	4	no	—
265	C₂³	—	5	16	no	—
266	C₂³	—	5	4	no	—
267	C₂³	—	10	16	no	—
268	C₂³	—	10	4	no	—
269	C₂³	—	10	4	no	—
270	C₂³	—	10	4	no	—
271	C₂³	—	10	4	no	—
272	C₂³	—	10	4	no	—
273	C₂³	—	20	16	no	—
274	C₂³	—	20	4	no	—
275	C₂³	—	20	4	no	—
276	C₂³	—	20	8	no	—
277	C₂³	—	20	8	no	—
278	C₂³	—	20	32	no	—
279	C₂³	D₄×D₅	1	4	yes	—
280	C₂³	C₂³×D₅	1	4	yes	—
281	C₂³	C₂³×D₅	1	2	yes	—
282	C₂³	C₂³×D₅	1	1	yes	characteristic
283	D₄	—	2	16	no	—
284	D₄	—	4	4	no	—
285	D₄	—	10	16	no	—
286	D₄	—	10	16	no	—
287	D₄	—	10	16	no	—
288	D₄	—	20	4	no	—
289	D₄	—	20	8	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	5	4	no	—
295	C₂×C₄	—	5	4	no	—
296	C₂×C₄	—	10	16	no	—
297	C₂×C₄	—	10	4	no	—
298	C₂×C₄	—	10	4	no	—
299	C₂×C₄	—	20	4	no	—
300	C₂×C₄	C₂³×D₅	1	2	yes	—
301	C₂×C₄	C₂³×D₅	1	2	yes	—
302	C₂×C₄	C₂³×D₅	1	4	yes	—
303	C₅	C₂×#534	1	1	yes	characteristic, complemented, Sylow
304	C₂²	—	2	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	4	8	no	—
311	C₂²	—	5	8	no	—
312	C₂²	—	10	16	no	—
313	C₂²	—	10	4	no	—
314	C₂²	—	10	4	no	—
315	C₂²	—	10	4	no	—
316	C₂²	—	10	4	no	—
317	C₂²	—	10	4	no	—
318	C₂²	—	20	16	no	—
319	C₂²	—	20	32	no	—
320	C₂²	—	20	8	no	—
321	C₂²	C₂×D₄×D₅	1	4	yes	—
322	C₂²	C₂×D₄×D₅	1	2	yes	—
323	C₂²	#4144	1	2	yes	—
324	C₂²	#4149	1	2	yes	—
325	C₂²	C₂⁴×D₅	1	1	yes	characteristic
326	C₄	—	2	4	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	10	4	no	—
330	C₄	—	10	4	no	—
331	C₂	—	2	4	no	—
332	C₂	—	4	4	no	—
333	C₂	—	10	8	no	—
334	C₂	—	20	4	no	—
335	C₂	#65939	1	4	yes	direct product factor
336	C₂	C₂²×D₄×D₅	1	1	yes	characteristic
337	C₂	C₂×#4144	1	1	yes	characteristic
338	C₂	C₂×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₅
Fitting	C₂×C₅×#121	C₂
socle	C₂³×C₅	C₂⁴
hypercenter	C₂×#121	D₅
hypocenter	C₅	C₂×#534
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×D₅	C₂	semidirect
2. C₂×C₅×#121	C₂	semidirect
3. C₂×#4085	C₂	semidirect
4. C₂×#4083	C₂	semidirect
5. C₂×#4076	C₂	semidirect
6. C₂²×#685	C₂	semidirect
7. #65939	C₂	direct (N × H)
8. C₂×#4040	C₂	semidirect
9. C₂×#4029	C₂	semidirect
10. C₂×#4028	C₂	semidirect
12. C₂⁴×D₅	C₂²	semidirect
13. C₂²×C₅×D₄	C₂²	semidirect
14. C₂²×C₅×D₄	C₂²	semidirect
15. C₂³×C₄×C₅	C₂²	semidirect
17. C₂²×C₂²⋊D₅	C₂²	semidirect
18. C₂²×C₂²⋊D₅	C₂²	semidirect
19. C₂³×Dic₅	C₂²	semidirect
21. C₂×D₄×D₅	C₂²	semidirect
23. C₂²×D₂₀	C₂²	semidirect
24. C₂²×C₄×D₅	C₂²	semidirect
25. C₅×#121	C₂²	semidirect
26. C₂×C₅×C₄⋊C₄	C₂²	semidirect
27. C₂×C₅×C₂²⋊C₄	C₂²	semidirect
28. C₂×C₂²⋊Dic₅	C₂²	semidirect
29. #4085	C₂²	semidirect
30. #4083	C₂²	semidirect
31. #4076	C₂²	semidirect
32. C₂×#685	C₂²	semidirect
33. C₂×#685	C₂²	semidirect
34. C₂×#685	C₂²	semidirect
35. C₂×#685	C₂²	semidirect
36. C₂×#685	C₂²	semidirect
37. C₂×C₄⋊Dic₅	C₂²	semidirect
38. C₂×#683	C₂²	semidirect
39. #4040	C₂²	semidirect
40. #4029	C₂²	semidirect
41. #4028	C₂²	semidirect
50. C₂³×D₅	D₄	semidirect
54. C₂³×D₅	C₂³	semidirect
57. C₂×C₅×D₄	C₂³	semidirect
58. C₂×C₅×D₄	C₂³	semidirect
63. C₂²×C₄×C₅	C₂³	semidirect
68. C₂×C₂²⋊D₅	C₂³	semidirect
69. C₂×C₂²⋊D₅	C₂³	semidirect
73. C₂²×Dic₅	C₂³	semidirect
78. C₂×D₂₀	C₂³	semidirect
80. C₂×C₄×D₅	C₂³	semidirect
81. C₅×C₄⋊C₄	C₂³	semidirect
82. C₅×C₂²⋊C₄	C₂³	semidirect
83. C₂²⋊Dic₅	C₂³	semidirect
84. #685	C₂³	semidirect
85. #685	C₂³	semidirect
86. #685	C₂³	semidirect
87. #685	C₂³	semidirect
88. C₄⋊Dic₅	C₂³	semidirect
89. #683	C₂³	semidirect
96. C₂×#121	D₅	semidirect
122. C₂²×D₅	C₂×D₄	semidirect
154. C₂²×D₄	C₂×D₅	semidirect
155. C₂²×D₄	C₂×D₅	semidirect
158. C₂³×C₄	C₂×D₅	semidirect
163. #121	C₂×D₅	semidirect
168. C₂×C₄⋊C₄	C₂×D₅	semidirect
175. C₂×C₂²⋊C₄	C₂×D₅	semidirect
228. C₂×D₄	C₂²×D₅	semidirect
229. C₂×D₄	C₂²×D₅	semidirect
240. C₂²×C₄	C₂²×D₅	semidirect
243. C₄⋊C₄	C₂²×D₅	semidirect
249. C₂²⋊C₄	C₂²×D₅	semidirect
252. C₂×C₅	#534	semidirect
303. C₅	C₂×#534	semidirect
335. C₂	#65939	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×D₄×D₅	2
C₂×C₅×#121	2
C₂×#4085	2
C₂×#4083	2
C₂×#4076	2
C₂²×#685	2
#65939	2
C₂×#4040	2
C₂×#4029	2
C₂×#4028	2
C₂×#534	5

Subgroups — C2×#65939 #10617841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#65939 #10617841