Subgroups — C₂×#10617626 #13006144

Order 1280.

7022 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂²×#4072	C₂	1	2	yes	maximal
3	#10617626	C₂	1	32	yes	maximal, direct product factor
4	C₂×#65853	C₂	1	8	yes	maximal, complemented
5	C₂×#65851	C₂	1	8	yes	maximal, complemented
6	C₂×#65834	C₂	1	4	yes	maximal, complemented
7	C₂×#65817	C₂	1	4	yes	maximal
8	C₂×#65804	C₂	1	4	yes	maximal, complemented
9	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	4	yes	—
11	C₂×C₅×#117	C₂²	1	4	yes	—
12	C₂×C₅×#117	C₂²	1	16	yes	—
13	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂²×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅	C₂²	1	2	yes	—
16	C₂²×C₂²⋊Dic₅	C₂²	1	4	yes	—
17	C₂×#4072	C₂²	1	4	yes	—
18	C₂×#4072	C₂²	1	4	yes	—
19	C₂×#4072	C₂²	1	4	yes	—
20	C₂×#4072	C₂²	1	32	yes	—
21	C₂×#4072	C₂²	1	4	yes	—
22	C₂²×C₄⋊Dic₅	C₂²	1	4	yes	—
23	C₂×#4070	C₂²	1	16	yes	—
24	C₂²×C₄×Dic₅	C₂²	1	4	yes	—
25	#65853	C₂²	1	128	yes	complemented
26	#65851	C₂²	1	128	yes	complemented
27	#65834	C₂²	1	64	yes	complemented
28	C₂×#4036	C₂²	1	16	yes	complemented
29	C₂×#4034	C₂²	1	16	yes	complemented
30	#65817	C₂²	1	64	yes	—
31	C₂×#4024	C₂²	1	32	yes	—
32	#65804	C₂²	1	64	yes	complemented
33	C₂×#4016	C₂²	1	8	yes	complemented
34	C₂×#4015	C₂²	1	8	yes	complemented
35	C₂×C₄×Dic₁₀	C₂²	1	16	yes	complemented
36	C₂×#3458	—	5	1	no	maximal, Sylow
37	C₂⁵×C₅	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₅	Q₈	1	16	yes	complemented
39	C₂³×C₄×C₅	C₂³	1	4	yes	—
40	C₂³×C₄×C₅	C₂³	1	2	yes	—
41	C₂³×C₄×C₅	C₂³	1	2	yes	—
42	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅	C₂³	1	8	yes	—
45	C₂³×Dic₅	C₂³	1	8	yes	—
46	C₂²×Dic₁₀	C₂³	1	16	yes	—
47	C₅×#117	C₂³	1	32	yes	—
48	C₅×#117	C₂³	1	32	yes	—
49	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
50	C₂×C₅×C₄⋊C₄	C₂³	1	16	yes	—
51	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
52	C₂×C₅×C₂²⋊C₄	C₂³	1	16	yes	—
53	C₂×C₄²×C₅	C₂³	1	4	yes	—
54	C₂×C₄²×C₅	C₂³	1	4	yes	—
55	C₂×C₄²×C₅	C₂³	1	16	yes	—
56	C₂×C₂²⋊Dic₅	C₂³	1	8	yes	—
57	C₂×C₂²⋊Dic₅	C₂³	1	32	yes	—
58	C₂×C₂²⋊Dic₅	C₂³	1	8	yes	—
59	#4072	C₂³	1	32	yes	—
60	#4072	C₂³	1	32	yes	—
61	#4072	C₂³	1	32	yes	—
62	#4072	C₂³	1	32	yes	—
63	C₂×C₄⋊Dic₅	C₂³	1	16	yes	—
64	C₂×C₄⋊Dic₅	C₂³	1	8	yes	—
65	C₂×C₄⋊Dic₅	C₂³	1	8	yes	—
66	C₂×C₄⋊Dic₅	C₂³	1	32	yes	—
67	#4070	C₂³	1	128	yes	—
68	C₂×#683	C₂³	1	32	yes	—
69	C₂×C₄×Dic₅	C₂³	1	8	yes	—
70	C₂×C₄×Dic₅	C₂³	1	8	yes	—
71	C₂×C₄×Dic₅	C₂³	1	32	yes	—
72	#4036	C₂³	1	128	yes	complemented
73	#4034	C₂³	1	128	yes	complemented
74	#4024	C₂³	1	256	yes	—
75	#4016	C₂³	1	64	yes	complemented
76	#4015	C₂³	1	64	yes	complemented
77	C₄×Dic₁₀	C₂³	1	128	yes	complemented
78	#3458	—	5	32	no	—
79	C₂×#544	—	5	8	no	—
80	C₂×#544	—	5	4	no	—
81	C₂×#543	—	5	4	no	—
82	C₂×#533	—	5	4	no	—
83	C₂×#533	—	5	8	no	—
84	C₂²×#117	—	5	2	no	—
85	C₂²×#117	D₅	1	1	yes	characteristic
86	C₂⁴×C₅	—	4	4	no	—
87	C₂⁴×C₅	C₂×Q₈	1	8	yes	—
88	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
89	C₂⁴×C₅	C₂⁴	1	2	yes	—
90	C₂⁴×C₅	C₂⁴	1	4	yes	—
91	C₂²×C₄×C₅	—	4	8	no	—
92	C₂²×C₄×C₅	C₂×Q₈	1	16	yes	—
93	C₂²×C₄×C₅	C₂×Q₈	1	16	yes	—
94	C₂²×C₄×C₅	C₂×Q₈	1	8	yes	—
95	C₂²×C₄×C₅	C₂×Q₈	1	8	yes	—
96	C₂²×C₄×C₅	C₂×Q₈	1	64	yes	complemented
97	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
98	C₂²×C₄×C₅	C₂⁴	1	16	yes	—
99	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
101	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
102	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
103	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
104	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
105	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
106	C₂²×Dic₅	—	4	16	no	—
107	C₂²×Dic₅	C₂⁴	1	16	yes	—
108	C₂²×Dic₅	C₂⁴	1	32	yes	—
109	C₂×Dic₁₀	—	4	32	no	—
110	C₂×Dic₁₀	C₂⁴	1	64	yes	—
111	C₅×C₄⋊C₄	C₂⁴	1	32	yes	—
112	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	—
113	C₄²×C₅	C₂⁴	1	16	yes	—
114	C₄²×C₅	C₂⁴	1	16	yes	—
115	C₂²⋊Dic₅	C₂⁴	1	32	yes	—
116	C₂²⋊Dic₅	C₂⁴	1	32	yes	—
117	C₄⋊Dic₅	C₂⁴	1	64	yes	—
118	C₄⋊Dic₅	C₂⁴	1	32	yes	—
119	C₄⋊Dic₅	C₂⁴	1	32	yes	—
120	#683	C₂⁴	1	128	yes	—
121	C₄×Dic₅	C₂⁴	1	32	yes	—
122	C₄×Dic₅	C₂⁴	1	32	yes	—
123	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
124	#544	—	5	128	no	—
125	#544	—	5	64	no	—
126	#543	—	5	64	no	—
127	#533	—	5	128	no	—
128	#533	—	5	64	no	—
129	C₂×C₄⋊Q₈	—	5	16	no	—
130	C₂×C₄⋊Q₈	—	5	8	no	—
131	C₂×#125	—	5	16	no	—
132	C₂×#125	—	5	8	no	—
133	C₂×C₂²⋊Q₈	—	5	16	no	—
134	C₂×C₂²⋊Q₈	—	5	32	no	—
135	C₂×C₄×Q₈	—	5	16	no	—
136	C₂×#117	—	5	4	no	—
137	C₂×#117	—	5	4	no	—
138	C₂×#117	—	5	4	no	—
139	C₂×#117	—	5	4	no	—
140	C₂×#117	—	5	32	no	—
141	C₂×#117	C₂×D₅	1	4	yes	—
142	C₂×#117	C₂×D₅	1	4	yes	—
143	C₂×#117	C₂×D₅	1	16	yes	—
144	C₂²×C₄⋊C₄	—	5	4	no	—
145	C₂²×C₄⋊C₄	C₂×D₅	1	2	yes	—
146	C₂²×C₂²⋊C₄	—	5	4	no	—
147	C₂²×C₂²⋊C₄	C₂×D₅	1	2	yes	—
148	C₂²×C₄²	—	5	4	no	—
149	C₂²×C₄²	C₂×D₅	1	2	yes	—
150	C₂³×C₅	—	2	8	no	—
151	C₂³×C₅	—	2	4	no	—
152	C₂³×C₅	—	4	16	no	—
153	C₂³×C₅	—	4	8	no	—
154	C₂³×C₅	C₂²×Q₈	1	4	yes	—
155	C₂³×C₅	C₂²×Q₈	1	8	yes	—
156	C₂³×C₅	C₂²×Q₈	1	16	yes	—
157	C₂³×C₅	C₂⁵	1	2	yes	—
158	C₂³×C₅	C₂⁵	1	4	yes	—
159	C₂³×C₅	C₂⁵	1	1	yes	characteristic
160	C₂×C₄×C₅	—	2	16	no	—
161	C₂×C₄×C₅	—	4	16	no	—
162	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
163	C₂×C₄×C₅	C₂²×Q₈	1	8	yes	—
164	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
165	C₂×C₄×C₅	C₂²×Q₈	1	16	yes	—
166	C₂×C₄×C₅	C₂²×Q₈	1	16	yes	—
167	C₂×C₄×C₅	C₂²×Q₈	1	32	yes	—
168	C₂×C₄×C₅	C₂²×Q₈	1	32	yes	—
169	C₂×C₄×C₅	C₂⁵	1	16	yes	—
170	C₂×C₄×C₅	C₂⁵	1	2	yes	—
171	C₂×C₄×C₅	C₂⁵	1	4	yes	—
172	C₂×C₄×C₅	C₂⁵	1	2	yes	—
173	C₂×Dic₅	—	2	32	no	—
174	C₂×Dic₅	—	4	32	no	—
175	C₂×Dic₅	C₂⁵	1	32	yes	—
176	Dic₁₀	—	4	64	no	—
177	C₂⁵	C₂²×D₅	1	1	yes	characteristic
178	C₂²×Q₈	—	5	16	no	—
179	C₂³×C₄	—	5	8	no	—
180	C₂³×C₄	Dic₁₀	1	16	yes	complemented
181	C₂³×C₄	C₂²×D₅	1	4	yes	—
182	C₂³×C₄	C₂²×D₅	1	2	yes	—
183	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
184	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
185	C₂³×C₄	C₂²×D₅	1	2	yes	—
186	C₂³×C₄	C₂²×D₅	1	8	yes	—
187	C₄⋊Q₈	—	5	128	no	—
188	C₄⋊Q₈	—	5	64	no	—
189	#125	—	5	128	no	—
190	#125	—	5	64	no	—
191	C₂²⋊Q₈	—	5	128	no	—
192	C₂²⋊Q₈	—	5	256	no	—
193	C₄×Q₈	—	5	128	no	—
194	#117	—	5	32	no	—
195	#117	—	5	32	no	—
196	#117	—	5	32	no	—
197	#117	—	5	32	no	—
198	#117	C₂²×D₅	1	32	yes	—
199	#117	C₂²×D₅	1	32	yes	—
200	C₂×C₄⋊C₄	—	5	8	no	—
201	C₂×C₄⋊C₄	—	5	8	no	—
202	C₂×C₄⋊C₄	—	5	32	no	—
203	C₂×C₄⋊C₄	—	5	32	no	—
204	C₂×C₄⋊C₄	—	5	16	no	—
205	C₂×C₄⋊C₄	C₂²×D₅	1	8	yes	—
206	C₂×C₄⋊C₄	C₂²×D₅	1	16	yes	—
207	C₂×C₂²⋊C₄	—	5	8	no	—
208	C₂×C₂²⋊C₄	—	5	8	no	—
209	C₂×C₂²⋊C₄	—	5	32	no	—
210	C₂×C₂²⋊C₄	C₂²×D₅	1	8	yes	—
211	C₂×C₂²⋊C₄	C₂²×D₅	1	16	yes	—
212	C₂×C₄²	—	5	8	no	—
213	C₂×C₄²	—	5	32	no	—
214	C₂×C₄²	—	5	8	no	—
215	C₂×C₄²	C₂²×D₅	1	4	yes	—
216	C₂×C₄²	C₂²×D₅	1	16	yes	—
217	C₂×C₄²	C₂²×D₅	1	4	yes	—
218	C₂²×C₅	—	2	8	no	—
219	C₂²×C₅	—	2	4	no	—
220	C₂²×C₅	—	2	16	no	—
221	C₂²×C₅	—	2	8	no	—
222	C₂²×C₅	—	4	16	no	—
223	C₂²×C₅	C₂³×Q₈	1	4	yes	—
224	C₂²×C₅	C₂³×Q₈	1	8	yes	—
225	C₂²×C₅	C₂³×Q₈	1	2	yes	—
226	C₂²×C₅	#585	1	4	yes	—
227	C₂²×C₅	C₂⁶	1	1	yes	characteristic
228	C₄×C₅	—	2	16	no	—
229	C₄×C₅	C₂³×Q₈	1	4	yes	—
230	C₄×C₅	C₂³×Q₈	1	8	yes	—
231	C₄×C₅	C₂³×Q₈	1	4	yes	—
232	Dic₅	—	2	32	no	—
233	C₂⁴	—	4	4	no	—
234	C₂⁴	C₂×Dic₁₀	1	8	yes	—
235	C₂⁴	C₂³×D₅	1	1	yes	characteristic
236	C₂⁴	C₂³×D₅	1	2	yes	—
237	C₂⁴	C₂³×D₅	1	4	yes	—
238	C₂×Q₈	—	5	64	no	—
239	C₂×Q₈	—	20	32	no	—
240	C₂²×C₄	—	4	8	no	—
241	C₂²×C₄	—	5	16	no	—
242	C₂²×C₄	—	5	32	no	—
243	C₂²×C₄	—	20	16	no	—
244	C₂²×C₄	C₂×Dic₁₀	1	16	yes	—
245	C₂²×C₄	C₂×Dic₁₀	1	8	yes	—
246	C₂²×C₄	C₂×Dic₁₀	1	8	yes	—
247	C₂²×C₄	C₂×Dic₁₀	1	16	yes	—
248	C₂²×C₄	C₂×Dic₁₀	1	64	yes	complemented
249	C₂²×C₄	C₂³×D₅	1	8	yes	—
250	C₂²×C₄	C₂³×D₅	1	16	yes	—
251	C₂²×C₄	C₂³×D₅	1	2	yes	—
252	C₂²×C₄	C₂³×D₅	1	8	yes	—
253	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
254	C₂²×C₄	C₂³×D₅	1	4	yes	—
255	C₂²×C₄	C₂³×D₅	1	8	yes	—
256	C₂²×C₄	C₂³×D₅	1	4	yes	—
257	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
258	C₄⋊C₄	—	5	32	no	—
259	C₄⋊C₄	—	5	32	no	—
260	C₄⋊C₄	—	5	128	no	—
261	C₄⋊C₄	—	5	64	no	—
262	C₄⋊C₄	C₂³×D₅	1	32	yes	—
263	C₂²⋊C₄	—	5	32	no	—
264	C₂²⋊C₄	—	5	32	no	—
265	C₂²⋊C₄	C₂³×D₅	1	32	yes	—
266	C₄²	—	5	32	no	—
267	C₄²	—	5	32	no	—
268	C₄²	C₂³×D₅	1	16	yes	—
269	C₄²	C₂³×D₅	1	16	yes	—
270	C₂×C₅	—	2	8	no	—
271	C₂×C₅	—	2	4	no	—
272	C₂×C₅	#3458	1	4	yes	complemented
273	C₂×C₅	C₂⁴×Q₈	1	1	yes	characteristic
274	C₂×C₅	C₂×#585	1	2	yes	—
275	C₂³	—	2	4	no	—
276	C₂³	—	2	8	no	—
277	C₂³	—	4	16	no	—
278	C₂³	—	4	8	no	—
279	C₂³	C₂²×Dic₁₀	1	4	yes	—
280	C₂³	C₂²×Dic₁₀	1	8	yes	—
281	C₂³	C₂²×Dic₁₀	1	16	yes	—
282	C₂³	C₂⁴×D₅	1	2	yes	—
283	C₂³	C₂⁴×D₅	1	4	yes	—
284	C₂³	C₂⁴×D₅	1	1	yes	characteristic
285	Q₈	—	20	64	no	—
286	C₂×C₄	—	2	16	no	—
287	C₂×C₄	—	4	16	no	—
288	C₂×C₄	—	5	32	no	—
289	C₂×C₄	—	10	32	no	—
290	C₂×C₄	—	20	32	no	—
291	C₂×C₄	C₂²×Dic₁₀	1	8	yes	—
292	C₂×C₄	C₂²×Dic₁₀	1	4	yes	—
293	C₂×C₄	C₂²×Dic₁₀	1	4	yes	—
294	C₂×C₄	C₂²×Dic₁₀	1	32	yes	—
295	C₂×C₄	C₂²×Dic₁₀	1	16	yes	—
296	C₂×C₄	C₂²×Dic₁₀	1	32	yes	—
297	C₂×C₄	C₂²×Dic₁₀	1	16	yes	—
298	C₂×C₄	C₂⁴×D₅	1	16	yes	—
299	C₂×C₄	C₂⁴×D₅	1	2	yes	—
300	C₂×C₄	C₂⁴×D₅	1	4	yes	—
301	C₂×C₄	C₂⁴×D₅	1	2	yes	—
302	C₅	C₂×#3458	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	2	8	no	—
304	C₂²	—	2	16	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	4	16	no	—
308	C₂²	C₂³×Dic₁₀	1	4	yes	—
309	C₂²	C₂³×Dic₁₀	1	8	yes	—
310	C₂²	C₂³×Dic₁₀	1	2	yes	—
311	C₂²	#66283	1	4	yes	—
312	C₂²	C₂⁵×D₅	1	1	yes	characteristic
313	C₄	—	2	16	no	—
314	C₄	—	10	32	no	—
315	C₄	C₂³×Dic₁₀	1	8	yes	—
316	C₄	C₂³×Dic₁₀	1	4	yes	—
317	C₄	C₂³×Dic₁₀	1	4	yes	—
318	C₂	—	2	4	no	—
319	C₂	—	2	8	no	—
320	C₂	#10617626	1	4	yes	direct product factor
321	C₂	C₂⁴×Dic₁₀	1	1	yes	characteristic
322	C₂	C₂×#66283	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁶
Frattini	C₂²	C₂⁵×D₅
Fitting	C₂²×C₅×#117	C₂
socle	C₂³×C₅	C₂⁵
prime radical	C₂⁵×C₅	C₂³
hypercenter	C₂²×#117	D₅
hypocenter	C₅	C₂×#3458
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
3. #10617626	C₂	direct (N × H)
4. C₂×#65853	C₂	semidirect
5. C₂×#65851	C₂	semidirect
6. C₂×#65834	C₂	semidirect
8. C₂×#65804	C₂	semidirect
25. #65853	C₂²	semidirect
26. #65851	C₂²	semidirect
27. #65834	C₂²	semidirect
28. C₂×#4036	C₂²	semidirect
29. C₂×#4034	C₂²	semidirect
32. #65804	C₂²	semidirect
33. C₂×#4016	C₂²	semidirect
34. C₂×#4015	C₂²	semidirect
35. C₂×C₄×Dic₁₀	C₂²	semidirect
38. C₂³×C₄×C₅	Q₈	semidirect
72. #4036	C₂³	semidirect
73. #4034	C₂³	semidirect
75. #4016	C₂³	semidirect
76. #4015	C₂³	semidirect
77. C₄×Dic₁₀	C₂³	semidirect
96. C₂²×C₄×C₅	C₂×Q₈	semidirect
180. C₂³×C₄	Dic₁₀	semidirect
248. C₂²×C₄	C₂×Dic₁₀	semidirect
272. C₂×C₅	#3458	semidirect
302. C₅	C₂×#3458	semidirect
320. C₂	#10617626	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×#117	2
C₂²×#4072	2
#10617626	2
C₂×#65853	2
C₂×#65851	2
C₂×#65834	2
C₂×#65817	2
C₂×#65804	2
C₂×#3458	5

Subgroups — C2×#10617626 #13006144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10617626 #13006144