Subgroups — C₂×#66102 #10619084

Order 640.

1484 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×#41	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#4096	C₂	1	2	yes	maximal, complemented
4	#66102	C₂	1	16	yes	maximal, direct product factor
5	C₂×#4064	C₂	1	2	yes	maximal, complemented
6	C₂×#4061	C₂	1	2	yes	maximal, complemented
7	C₂×#4060	C₂	1	2	yes	maximal, complemented
8	C₂×#4057	C₂	1	2	yes	maximal, complemented
9	C₂×#4050	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#676	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#41	C₂²	1	2	yes	complemented
12	D₅×#41	—	2	8	no	—
13	D₅×#41	C₂²	1	16	yes	complemented
14	C₂×#713	C₂²	1	2	yes	—
15	C₂×Q₈×D₅	C₂²	1	2	yes	complemented
16	C₂×#711	—	2	2	no	—
17	C₂×#711	C₂²	1	2	yes	—
18	C₂×D₄×D₅	—	2	2	no	—
19	C₂×D₄×D₅	C₂²	1	2	yes	complemented
20	C₂×#709	—	2	2	no	—
21	C₂×#709	C₂²	1	2	yes	complemented
22	C₂²×C₄×D₅	—	2	2	no	—
23	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
24	C₅×#135	C₂²	1	8	yes	complemented
25	C₂×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₅×SD₁₆	C₂²	1	2	yes	complemented
27	C₂×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
29	#4096	C₂²	1	16	yes	complemented
30	C₂×#689	C₂²	1	2	yes	complemented
31	C₂×#688	C₂²	1	2	yes	complemented
32	C₂×#687	C₂²	1	2	yes	complemented
33	C₂×#686	C₂²	1	2	yes	complemented
34	C₂²×#168	C₂²	1	1	yes	characteristic, complemented
35	#4064	C₂²	1	16	yes	complemented
36	#4061	C₂²	1	16	yes	complemented
37	#4060	C₂²	1	16	yes	complemented
38	#4057	C₂²	1	16	yes	complemented
39	C₂×Dic₂₀	C₂²	1	1	yes	characteristic, complemented
40	#4050	C₂²	1	8	yes	complemented
41	C₂×D₄₀	C₂²	1	1	yes	characteristic, complemented
42	C₂×#677	C₂²	1	2	yes	complemented
43	C₂×#676	C₂²	1	2	yes	complemented
44	C₂×#676	C₂²	1	2	yes	complemented
45	C₂×#676	C₂²	1	8	yes	complemented
46	C₂×#575	—	5	1	no	maximal, Sylow
47	C₂³×D₅	—	2	2	no	—
48	C₂³×D₅	D₄	1	1	yes	characteristic
49	C₅×#41	—	2	4	no	—
50	C₅×#41	C₂³	1	8	yes	complemented
51	C₂×C₅×Q₈	C₂³	1	2	yes	—
52	C₂×C₅×D₄	—	2	2	no	—
53	C₂×C₅×D₄	C₂³	1	2	yes	—
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₂²⋊D₅	—	2	2	no	—
57	C₂×C₂²⋊D₅	—	2	2	no	—
58	C₂×C₂²⋊D₅	—	2	2	no	—
59	C₂²×Dic₅	—	2	2	no	—
60	C₂²×Dic₅	D₄	1	1	yes	characteristic
61	#713	—	2	4	no	—
62	#713	C₂³	1	8	yes	—
63	Q₈×D₅	—	2	4	no	—
64	Q₈×D₅	C₂³	1	8	yes	complemented
65	#711	—	2	16	no	—
66	#711	—	2	4	no	—
67	#711	C₂³	1	8	yes	—
68	D₄×D₅	—	2	16	no	—
69	D₄×D₅	—	2	4	no	—
70	D₄×D₅	C₂³	1	8	yes	complemented
71	#709	—	2	4	no	—
72	#709	—	2	16	no	—
73	#709	C₂³	1	8	yes	complemented
74	C₂×D₂₀	—	2	2	no	—
75	C₂×D₂₀	C₂³	1	2	yes	—
76	C₂×C₄×D₅	—	2	2	no	—
77	C₂×C₄×D₅	—	2	2	no	—
78	C₂×C₄×D₅	—	2	2	no	—
79	C₂×C₄×D₅	—	2	16	no	—
80	C₂×C₄×D₅	D₄	1	2	yes	—
81	C₂×C₄×D₅	D₄	1	4	yes	—
82	C₂×C₄×D₅	C₂³	1	2	yes	—
83	C₂×C₄×D₅	C₂³	1	4	yes	—
84	C₂×Dic₁₀	—	2	2	no	—
85	C₂×Dic₁₀	C₂³	1	2	yes	—
86	C₅×Q₁₆	C₂³	1	4	yes	complemented
87	C₅×SD₁₆	C₂³	1	8	yes	complemented
88	C₅×D₈	C₂³	1	4	yes	complemented
89	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂³	1	4	yes	complemented
92	#689	C₂³	1	8	yes	complemented
93	#688	C₂³	1	8	yes	complemented
94	#687	C₂³	1	8	yes	complemented
95	#686	C₂³	1	8	yes	complemented
96	C₂×#168	C₂³	1	2	yes	—
97	C₂×#168	C₂³	1	4	yes	complemented
98	Dic₂₀	C₂³	1	4	yes	complemented
99	D₄₀	C₂³	1	4	yes	complemented
100	#677	C₂³	1	8	yes	complemented
101	#676	C₂³	1	8	yes	complemented
102	#676	C₂³	1	8	yes	complemented
103	C₂²×#41	—	5	2	no	—
104	#575	—	5	16	no	—
105	C₂×#137	—	5	2	no	—
106	C₂×#137	—	5	2	no	—
107	C₂×AGL(1,ℤ/8)	—	5	2	no	—
108	C₂×AGL(1,ℤ/8)	—	5	2	no	—
109	C₂×#135	—	5	2	no	—
110	C₂×#135	—	5	1	no	—
111	C₂×#135	D₅	1	1	yes	characteristic, complemented
112	C₂²×M₁₆	—	5	1	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₅	—	2	2	no	—
116	C₂²×D₅	—	2	8	no	—
117	C₂²×D₅	—	4	2	no	—
118	C₂²×D₅	—	4	8	no	—
119	C₂²×D₅	C₂×D₄	1	2	yes	—
120	C₂²×D₅	C₂×D₄	1	4	yes	—
121	C₅×Q₈	—	2	2	no	—
122	C₅×Q₈	C₂⁴	1	4	yes	—
123	C₅×D₄	—	2	8	no	—
124	C₅×D₄	—	2	2	no	—
125	C₅×D₄	C₂⁴	1	4	yes	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	8	no	—
128	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
129	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂⁴	1	2	yes	—
132	C₂²⋊D₅	—	2	8	no	—
133	C₂²⋊D₅	—	2	8	no	—
134	C₂²⋊D₅	—	2	8	no	—
135	C₂×Dic₅	—	2	2	no	—
136	C₂×Dic₅	—	2	8	no	—
137	C₂×Dic₅	C₂×D₄	1	2	yes	—
138	C₂×Dic₅	C₂×D₄	1	4	yes	—
139	D₂₀	—	2	8	no	—
140	D₂₀	—	2	2	no	—
141	D₂₀	C₂⁴	1	4	yes	—
142	C₄×D₅	—	2	8	no	—
143	C₄×D₅	—	2	8	no	—
144	C₄×D₅	—	2	8	no	—
145	C₄×D₅	C₂×D₄	1	8	yes	—
146	C₄×D₅	C₂×D₄	1	4	yes	—
147	C₄×D₅	C₂⁴	1	4	yes	—
148	Dic₁₀	—	2	2	no	—
149	Dic₁₀	—	2	8	no	—
150	Dic₁₀	C₂⁴	1	4	yes	—
151	C₅×C₈	C₂⁴	1	2	yes	—
152	C₅×C₈	C₂⁴	1	2	yes	—
153	#168	C₂⁴	1	4	yes	—
154	C₂×#41	—	5	16	no	—
155	C₂×#41	—	5	2	no	—
156	C₂×#41	—	5	2	no	—
157	C₂×#41	—	5	2	no	—
158	C₂×#41	—	10	2	no	—
159	C₂×#41	—	10	2	no	—
160	C₂×#41	—	10	8	no	—
161	C₂×#41	C₂×D₅	1	2	yes	complemented
162	C₂²×Q₈	—	5	2	no	—
163	C₂²×D₄	—	5	2	no	—
164	C₂²×D₄	—	10	2	no	—
165	C₂³×C₄	—	5	1	no	—
166	C₂³×C₄	—	10	2	no	—
167	#137	—	5	16	no	—
168	#137	—	5	16	no	—
169	AGL(1,ℤ/8)	—	5	16	no	—
170	AGL(1,ℤ/8)	—	5	16	no	—
171	#135	—	5	16	no	—
172	#135	—	5	8	no	—
173	#135	C₂×D₅	1	8	yes	complemented
174	C₂×Q₁₆	—	5	2	no	—
175	C₂×Q₁₆	—	5	1	no	—
176	C₂×Q₁₆	C₂×D₅	1	1	yes	characteristic, complemented
177	C₂×SD₁₆	—	5	2	no	—
178	C₂×SD₁₆	—	5	2	no	—
179	C₂×SD₁₆	—	5	2	no	—
180	C₂×SD₁₆	C₂×D₅	1	2	yes	complemented
181	C₂×D₈	—	5	2	no	—
182	C₂×D₈	—	5	1	no	—
183	C₂×D₈	C₂×D₅	1	1	yes	characteristic, complemented
184	C₂×M₁₆	—	5	2	no	—
185	C₂×M₁₆	—	5	2	no	—
186	C₂×M₁₆	—	5	8	no	—
187	C₂²×C₈	—	5	1	no	—
188	C₂²×C₈	C₂×D₅	1	1	yes	characteristic, complemented
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	4	no	—
191	C₂²×C₅	—	4	4	no	—
192	C₂²×C₅	C₂²×D₄	1	2	yes	—
193	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
194	C₂×D₅	—	2	4	no	—
195	C₂×D₅	—	2	4	no	—
196	C₂×D₅	—	4	4	no	—
197	C₂×D₅	—	4	16	no	—
198	C₂×D₅	—	4	4	no	—
199	C₂×D₅	C₂²×D₄	1	4	yes	—
200	C₄×C₅	—	2	4	no	—
201	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
202	C₄×C₅	C₂²×D₄	1	2	yes	—
203	C₄×C₅	C₂⁵	1	1	yes	characteristic
204	Dic₅	—	2	4	no	—
205	Dic₅	C₂²×D₄	1	4	yes	—
206	C₂⁴	—	5	1	no	—
207	C₂⁴	—	10	2	no	—
208	#41	—	2	4	no	—
209	#41	—	5	8	no	—
210	#41	—	5	8	no	—
211	#41	—	5	8	no	—
212	#41	—	10	16	no	—
213	#41	—	10	16	no	—
214	#41	—	10	4	no	—
215	#41	—	10	4	no	—
216	#41	—	10	4	no	—
217	#41	C₂²×D₅	1	8	yes	complemented
218	C₂×Q₈	—	5	8	no	—
219	C₂×Q₈	—	5	2	no	—
220	C₂×Q₈	—	10	2	no	—
221	C₂×Q₈	—	10	4	no	—
222	C₂×Q₈	C₂²×D₅	1	2	yes	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	5	8	no	—
225	C₂×D₄	—	5	2	no	—
226	C₂×D₄	—	10	2	no	—
227	C₂×D₄	—	10	4	no	—
228	C₂×D₄	—	10	2	no	—
229	C₂×D₄	—	10	16	no	—
230	C₂×D₄	—	10	2	no	—
231	C₂×D₄	—	10	2	no	—
232	C₂×D₄	C₂²×D₅	1	2	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	5	1	no	—
235	C₂²×C₄	—	5	4	no	—
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	—	5	2	no	—
238	C₂²×C₄	—	5	4	no	—
239	C₂²×C₄	—	10	2	no	—
240	C₂²×C₄	—	10	16	no	—
241	C₂²×C₄	—	10	2	no	—
242	C₂²×C₄	—	10	2	no	—
243	C₂²×C₄	—	10	2	no	—
244	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
245	Q₁₆	—	5	8	no	—
246	Q₁₆	—	5	4	no	—
247	Q₁₆	C₂²×D₅	1	4	yes	complemented
248	SD₁₆	—	5	8	no	—
249	SD₁₆	—	5	8	no	—
250	SD₁₆	—	5	8	no	—
251	SD₁₆	C₂²×D₅	1	8	yes	complemented
252	D₈	—	5	8	no	—
253	D₈	—	5	4	no	—
254	D₈	C₂²×D₅	1	4	yes	complemented
255	M₁₆	—	5	8	no	—
256	M₁₆	—	5	8	no	—
257	C₂×C₈	—	5	2	no	—
258	C₂×C₈	—	5	4	no	—
259	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₅	1	4	yes	complemented
262	C₂×C₅	—	2	2	no	—
263	C₂×C₅	—	4	4	no	—
264	C₂×C₅	#575	1	2	yes	complemented
265	C₂×C₅	C₂³×D₄	1	1	yes	characteristic
266	D₅	—	2	4	no	—
267	D₅	—	4	4	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	5	2	no	—
270	C₂³	—	5	4	no	—
271	C₂³	—	10	8	no	—
272	C₂³	—	10	2	no	—
273	C₂³	—	20	2	no	—
274	C₂³	—	20	8	no	—
275	C₂³	D₄×D₅	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	—	5	4	no	—
278	Q₈	—	10	8	no	—
279	Q₈	—	10	2	no	—
280	Q₈	C₂³×D₅	1	4	yes	—
281	D₄	—	2	8	no	—
282	D₄	—	2	2	no	—
283	D₄	—	5	4	no	—
284	D₄	—	10	8	no	—
285	D₄	—	10	8	no	—
286	D₄	—	10	8	no	—
287	D₄	—	10	8	no	—
288	D₄	—	10	2	no	—
289	D₄	C₂³×D₅	1	4	yes	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	5	2	no	—
293	C₂×C₄	—	5	4	no	—
294	C₂×C₄	—	5	8	no	—
295	C₂×C₄	—	5	4	no	—
296	C₂×C₄	—	5	4	no	—
297	C₂×C₄	—	10	8	no	—
298	C₂×C₄	—	10	8	no	—
299	C₂×C₄	—	10	8	no	—
300	C₂×C₄	—	10	2	no	—
301	C₂×C₄	—	10	8	no	—
302	C₂×C₄	D₄×D₅	1	2	yes	—
303	C₂×C₄	D₄×D₅	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₅	1	2	yes	—
306	C₈	—	5	4	no	—
307	C₈	C₂³×D₅	1	2	yes	—
308	C₈	C₂³×D₅	1	2	yes	—
309	C₅	C₂×#575	1	1	yes	characteristic, complemented, Sylow
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	5	4	no	—
314	C₂²	—	10	4	no	—
315	C₂²	—	10	4	no	—
316	C₂²	—	20	4	no	—
317	C₂²	—	20	16	no	—
318	C₂²	—	20	4	no	—
319	C₂²	C₂×D₄×D₅	1	2	yes	—
320	C₂²	C₂×D₄×D₅	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	5	4	no	—
323	C₄	—	10	4	no	—
324	C₄	C₂×D₄×D₅	1	2	yes	—
325	C₄	C₂×D₄×D₅	1	1	yes	characteristic
326	C₄	C₂⁴×D₅	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	4	4	no	—
329	C₂	—	10	4	no	—
330	C₂	—	20	4	no	—
331	C₂	#66102	1	2	yes	direct product factor
332	C₂	C₂²×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁵
Frattini	C₄	C₂⁴×D₅
Fitting	C₂×C₅×#135	C₂
socle	C₂²×C₅	C₂²×D₄
hypercenter	C₂×#135	D₅
hypocenter	C₅	C₂×#575
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×D₅×#41	C₂	semidirect
2. C₂×C₅×#135	C₂	semidirect
3. C₂×#4096	C₂	semidirect
4. #66102	C₂	direct (N × H)
5. C₂×#4064	C₂	semidirect
6. C₂×#4061	C₂	semidirect
7. C₂×#4060	C₂	semidirect
8. C₂×#4057	C₂	semidirect
9. C₂×#4050	C₂	semidirect
10. C₂²×#676	C₂	semidirect
11. C₂×C₅×#41	C₂²	semidirect
13. D₅×#41	C₂²	semidirect
15. C₂×Q₈×D₅	C₂²	semidirect
19. C₂×D₄×D₅	C₂²	semidirect
21. C₂×#709	C₂²	semidirect
24. C₅×#135	C₂²	semidirect
25. C₂×C₅×Q₁₆	C₂²	semidirect
26. C₂×C₅×SD₁₆	C₂²	semidirect
27. C₂×C₅×D₈	C₂²	semidirect
28. C₂²×C₅×C₈	C₂²	semidirect
29. #4096	C₂²	semidirect
30. C₂×#689	C₂²	semidirect
31. C₂×#688	C₂²	semidirect
32. C₂×#687	C₂²	semidirect
33. C₂×#686	C₂²	semidirect
34. C₂²×#168	C₂²	semidirect
35. #4064	C₂²	semidirect
36. #4061	C₂²	semidirect
37. #4060	C₂²	semidirect
38. #4057	C₂²	semidirect
39. C₂×Dic₂₀	C₂²	semidirect
40. #4050	C₂²	semidirect
41. C₂×D₄₀	C₂²	semidirect
42. C₂×#677	C₂²	semidirect
43. C₂×#676	C₂²	semidirect
44. C₂×#676	C₂²	semidirect
45. C₂×#676	C₂²	semidirect
50. C₅×#41	C₂³	semidirect
64. Q₈×D₅	C₂³	semidirect
70. D₄×D₅	C₂³	semidirect
73. #709	C₂³	semidirect
86. C₅×Q₁₆	C₂³	semidirect
87. C₅×SD₁₆	C₂³	semidirect
88. C₅×D₈	C₂³	semidirect
91. C₂×C₅×C₈	C₂³	semidirect
92. #689	C₂³	semidirect
93. #688	C₂³	semidirect
94. #687	C₂³	semidirect
95. #686	C₂³	semidirect
97. C₂×#168	C₂³	semidirect
98. Dic₂₀	C₂³	semidirect
99. D₄₀	C₂³	semidirect
100. #677	C₂³	semidirect
101. #676	C₂³	semidirect
102. #676	C₂³	semidirect
111. C₂×#135	D₅	semidirect
161. C₂×#41	C₂×D₅	semidirect
173. #135	C₂×D₅	semidirect
176. C₂×Q₁₆	C₂×D₅	semidirect
180. C₂×SD₁₆	C₂×D₅	semidirect
183. C₂×D₈	C₂×D₅	semidirect
188. C₂²×C₈	C₂×D₅	semidirect
217. #41	C₂²×D₅	semidirect
247. Q₁₆	C₂²×D₅	semidirect
251. SD₁₆	C₂²×D₅	semidirect
254. D₈	C₂²×D₅	semidirect
261. C₂×C₈	C₂²×D₅	semidirect
264. C₂×C₅	#575	semidirect
309. C₅	C₂×#575	semidirect
331. C₂	#66102	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×D₅×#41	2
C₂×C₅×#135	2
C₂×#4096	2
#66102	2
C₂×#4064	2
C₂×#4061	2
C₂×#4060	2
C₂×#4057	2
C₂×#4050	2
C₂²×#676	2
C₂×#575	5

Subgroups — C2×#66102 #10619084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#66102 #10619084