Subgroups — C₂×#10602218 #12251957

Order 1280.

738 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#65390	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3944	C₂	1	1	yes	maximal, characteristic
4	#10602218	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64732	C₂	1	4	yes	maximal, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#4072	C₄	1	1	yes	characteristic
8	C₂²×#681	C₄	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₄	1	2	yes	complemented
10	C₂×C₅×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₂×C₅×M₃₂	C₂²	1	4	yes	—
13	C₂²×C₅×C₁₆	C₄	1	2	yes	complemented
14	#65390	C₄	1	4	yes	—
15	#65390	C₂²	1	4	yes	—
16	C₂×#3947	C₂²	1	2	yes	—
17	C₂×C₈×Dic₅	C₄	1	2	yes	—
18	C₂×#3944	C₄	1	2	yes	complemented
19	C₂×#3944	C₄	1	4	yes	complemented
20	C₂×#3944	C₂²	1	2	yes	—
21	C₂×#3944	C₂²	1	4	yes	—
22	C₂²×#672	C₄	1	2	yes	complemented
23	#64732	C₂²	1	16	yes	complemented
24	C₂×#2110	—	5	1	no	maximal, Sylow
25	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	2	yes	—
36	C₂×C₂²⋊Dic₅	—	2	1	no	—
37	#4072	—	2	2	no	—
38	#4072	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊Dic₅	—	2	1	no	—
40	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
41	C₂×#681	—	2	1	no	—
42	C₂×#681	—	2	1	no	—
43	C₂×#681	—	2	2	no	—
44	C₂×#681	C₂×C₄	1	4	yes	—
45	C₂²×#168	C₂×C₄	1	2	yes	—
46	C₅×M₃₂	C₂×C₄	1	4	yes	—
47	C₅×M₃₂	C₂×C₄	1	8	yes	complemented
48	C₅×M₃₂	C₂³	1	4	yes	—
49	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
50	C₂×C₅×C₁₆	C₂×C₄	1	8	yes	complemented
51	#3947	C₂×C₄	1	4	yes	—
52	#3947	C₂³	1	4	yes	—
53	C₈×Dic₅	C₂×C₄	1	8	yes	—
54	#3944	C₂×C₄	1	4	yes	—
55	#3944	C₂×C₄	1	8	yes	complemented
56	#3944	C₂³	1	4	yes	—
57	C₂×#672	C₂×C₄	1	4	yes	—
58	C₂×#672	C₂×C₄	1	8	yes	complemented
59	C₂²×M₃₂	—	5	1	no	—
60	C₂²×M₃₂	D₅	1	1	yes	characteristic
61	C₂×#405	—	5	1	no	—
62	#2110	—	5	8	no	—
63	C₂×#347	—	5	4	no	—
64	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₄²	1	2	yes	—
68	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₅	M₁₆	1	2	yes	—
70	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₅	M₁₆	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
75	C₂²×Dic₅	—	2	2	no	—
76	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
77	C₂×C₅×C₈	C₄²	1	2	yes	—
78	C₂×C₅×C₈	C₄²	1	2	yes	—
79	C₂×C₅×C₈	C₄²	1	2	yes	—
80	C₂×C₅×C₈	C₄²	1	2	yes	—
81	C₂×C₅×C₈	C₄²	1	2	yes	—
82	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
94	C₂²⋊Dic₅	—	2	4	no	—
95	C₄⋊Dic₅	—	2	4	no	—
96	C₄×Dic₅	—	2	2	no	—
97	C₄×Dic₅	C₂²×C₄	1	4	yes	—
98	#681	—	2	4	no	—
99	#681	—	2	4	no	—
100	C₂×#168	—	2	2	no	—
101	C₂×#168	—	2	2	no	—
102	C₂×#168	C₂²×C₄	1	4	yes	—
103	C₅×C₁₆	C₂²×C₄	1	8	yes	—
104	#672	C₂²×C₄	1	8	yes	—
105	C₂²×M₁₆	—	5	1	no	—
106	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
107	C₂×#117	—	5	1	no	—
108	C₂×M₃₂	—	5	2	no	—
109	C₂×M₃₂	—	5	4	no	—
110	C₂×M₃₂	—	5	2	no	—
111	C₂×M₃₂	—	5	4	no	—
112	C₂×M₃₂	Dic₅	1	2	yes	complemented
113	C₂×M₃₂	Dic₅	1	4	yes	complemented
114	C₂×M₃₂	C₂×D₅	1	2	yes	—
115	C₂×M₃₂	C₂×D₅	1	4	yes	—
116	C₂²×C₁₆	—	5	2	no	—
117	C₂²×C₁₆	Dic₅	1	2	yes	complemented
118	#405	—	5	4	no	—
119	#405	—	5	4	no	—
120	C₂×#97	—	5	2	no	—
121	C₂×C₄×C₈	—	5	2	no	—
122	#347	—	5	16	no	—
123	C₂³×C₅	—	4	2	no	—
124	C₂³×C₅	#97	1	1	yes	characteristic
125	C₂³×C₅	#97	1	1	yes	characteristic
126	C₂³×C₅	#97	1	2	yes	—
127	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
128	C₂³×C₅	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₅	#97	1	2	yes	—
130	C₂×C₄×C₅	#97	1	2	yes	—
131	C₂×C₄×C₅	#97	1	2	yes	—
132	C₂×C₄×C₅	#97	1	2	yes	—
133	C₂×C₄×C₅	#97	1	2	yes	—
134	C₂×C₄×C₅	#97	1	1	yes	characteristic
135	C₂×C₄×C₅	#97	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
139	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
143	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
144	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
147	C₂×Dic₅	—	2	4	no	—
148	C₂×Dic₅	—	4	4	no	—
149	C₅×C₈	C₂×C₄²	1	2	yes	—
150	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
151	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
152	C₅×C₈	C₂×C₄²	1	2	yes	—
153	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
154	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
155	#168	—	2	4	no	—
156	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
157	C₂×M₁₆	—	5	4	no	—
158	C₂×M₁₆	—	10	1	no	—
159	C₂×M₁₆	—	10	1	no	—
160	C₂×M₁₆	—	10	2	no	—
161	C₂²×C₈	—	5	2	no	—
162	C₂²×C₈	C₄×D₅	1	2	yes	—
163	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
165	C₂²×C₈	C₄×D₅	1	2	yes	—
166	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
167	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
168	C₂²×C₈	C₂×Dic₅	1	2	yes	—
169	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₅	1	2	yes	—
172	#117	—	5	4	no	—
173	#117	—	10	2	no	—
174	C₂×C₄⋊C₄	—	10	1	no	—
175	C₂×C₂²⋊C₄	—	10	1	no	—
176	C₂×C₄²	—	5	2	no	—
177	M₃₂	—	5	4	no	—
178	M₃₂	—	5	8	no	—
179	M₃₂	—	5	4	no	—
180	M₃₂	C₂×Dic₅	1	4	yes	—
181	M₃₂	C₂×Dic₅	1	8	yes	complemented
182	M₃₂	C₂²×D₅	1	4	yes	—
183	C₂×C₁₆	—	5	4	no	—
184	C₂×C₁₆	—	5	8	no	—
185	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
186	C₂×C₁₆	C₂×Dic₅	1	8	yes	complemented
187	#97	—	5	4	no	—
188	#97	—	5	4	no	—
189	C₄×C₈	—	5	8	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	4	2	no	—
194	C₂²×C₅	—	4	2	no	—
195	C₂²×C₅	C₂×#97	1	1	yes	characteristic
196	C₂²×C₅	C₂×#97	1	2	yes	—
197	C₂²×C₅	C₂×#97	1	2	yes	—
198	C₂²×C₅	C₂×#97	1	1	yes	characteristic
199	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
200	C₄×C₅	C₂×#97	1	1	yes	characteristic
201	C₄×C₅	C₂×#97	1	1	yes	characteristic
202	C₄×C₅	C₂×#97	1	2	yes	—
203	C₄×C₅	C₂×#97	1	2	yes	—
204	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
205	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
206	Dic₅	—	4	4	no	—
207	C₂⁴	#676	1	1	yes	characteristic
208	C₂²×C₄	—	10	2	no	—
209	C₂²×C₄	#676	1	1	yes	characteristic
210	C₂²×C₄	#676	1	2	yes	—
211	C₂²×C₄	#676	1	1	yes	characteristic
212	C₂²×C₄	#676	1	1	yes	characteristic
213	C₂²×C₄	#676	1	2	yes	—
214	C₂²×C₄	C₄×Dic₅	1	2	yes	—
215	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
216	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
219	M₁₆	—	10	4	no	—
220	M₁₆	—	10	4	no	—
221	C₂×C₈	—	5	4	no	—
222	C₂×C₈	—	10	2	no	—
223	C₂×C₈	—	10	2	no	—
224	C₂×C₈	C₄×Dic₅	1	2	yes	—
225	C₂×C₈	C₄×Dic₅	1	2	yes	—
226	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
227	C₂×C₈	C₄×Dic₅	1	2	yes	—
228	C₂×C₈	C₄×Dic₅	1	2	yes	—
229	C₂×C₈	C₄×Dic₅	1	2	yes	—
230	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
237	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
238	C₂×C₈	C₂²×Dic₅	1	2	yes	—
239	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₅	1	2	yes	—
241	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
242	C₄⋊C₄	—	10	4	no	—
243	C₂²⋊C₄	—	10	4	no	—
244	C₄²	—	5	4	no	—
245	C₄²	—	10	2	no	—
246	C₁₆	—	5	8	no	—
247	C₁₆	C₂²×Dic₅	1	8	yes	—
248	C₂×C₅	—	2	1	no	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	2	2	no	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	#2110	1	2	yes	complemented
253	C₂×C₅	C₂²×#97	1	1	yes	characteristic
254	C₂³	—	4	2	no	—
255	C₂³	#3947	1	1	yes	characteristic
256	C₂³	#3947	1	1	yes	characteristic
257	C₂³	#3947	1	2	yes	—
258	C₂³	C₂×#676	1	1	yes	characteristic
259	C₂³	C₂×#676	1	2	yes	—
260	C₂×C₄	—	10	4	no	—
261	C₂×C₄	—	20	4	no	—
262	C₂×C₄	#3947	1	2	yes	—
263	C₂×C₄	#3947	1	2	yes	—
264	C₂×C₄	#3947	1	2	yes	—
265	C₂×C₄	#3947	1	2	yes	—
266	C₂×C₄	#3947	1	2	yes	—
267	C₂×C₄	#3947	1	1	yes	characteristic
268	C₂×C₄	#3947	1	1	yes	characteristic
269	C₂×C₄	C₂×#676	1	1	yes	characteristic
270	C₂×C₄	C₂×#676	1	1	yes	characteristic
271	C₂×C₄	C₂×#676	1	2	yes	—
272	C₂×C₄	C₂×#676	1	2	yes	—
273	C₂×C₄	C₂×#676	1	1	yes	characteristic
274	C₂×C₄	C₂×#676	1	2	yes	—
275	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
276	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
280	C₈	—	10	4	no	—
281	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
282	C₈	C₂×C₄×Dic₅	1	2	yes	—
283	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
284	C₈	C₂×C₄×Dic₅	1	2	yes	—
285	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
286	C₈	C₂³×Dic₅	1	1	yes	characteristic
287	C₅	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₂×#3947	1	1	yes	characteristic
294	C₂²	C₂×#3947	1	2	yes	—
295	C₂²	C₂×#3947	1	2	yes	—
296	C₂²	C₂×#3947	1	1	yes	characteristic
297	C₂²	C₂²×#676	1	1	yes	characteristic
298	C₄	—	20	4	no	—
299	C₄	C₂×#3947	1	1	yes	characteristic
300	C₄	C₂×#3947	1	1	yes	characteristic
301	C₄	C₂×#3947	1	2	yes	—
302	C₄	C₂×#3947	1	2	yes	—
303	C₄	C₂²×#676	1	1	yes	characteristic
304	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#10602218	1	2	yes	direct product factor
310	C₂	C₂²×#3947	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂²×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×#97
prime radical	C₂⁴×C₅	M₁₆
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂²×M₃₂	D₅
hypocenter	C₅	C₂×#2110
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #10602218	C₂	direct (N × H)
5. C₂×#64732	C₂	semidirect
9. C₂×C₅×M₃₂	C₄	semidirect
10. C₂×C₅×M₃₂	C₄	semidirect
13. C₂²×C₅×C₁₆	C₄	semidirect
18. C₂×#3944	C₄	semidirect
19. C₂×#3944	C₄	semidirect
22. C₂²×#672	C₄	semidirect
23. #64732	C₂²	semidirect
47. C₅×M₃₂	C₂×C₄	semidirect
50. C₂×C₅×C₁₆	C₂×C₄	semidirect
55. #3944	C₂×C₄	semidirect
58. C₂×#672	C₂×C₄	semidirect
112. C₂×M₃₂	Dic₅	semidirect
113. C₂×M₃₂	Dic₅	semidirect
117. C₂²×C₁₆	Dic₅	semidirect
181. M₃₂	C₂×Dic₅	semidirect
186. C₂×C₁₆	C₂×Dic₅	semidirect
252. C₂×C₅	#2110	semidirect
287. C₅	C₂×#2110	semidirect
309. C₂	#10602218	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×M₃₂	2
C₂×#65390	2
C₂²×#3944	2
#10602218	2
C₂×#64732	2
C₂×#2110	5

Subgroups — C2×#10602218 #12251957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10602218 #12251957