Subgroups — C₂×#10601495 #12097309

Order 1280.

1102 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#65411	C₂	1	2	yes	maximal
3	C₂×#65395	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂×#64775	C₂	1	2	yes	maximal
6	#10601495	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#4072	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
11	C₂³×#168	C₂²	1	1	yes	characteristic
12	C₂×C₅×#108	C₄	1	8	yes	complemented
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	#65411	—	2	4	no	—
15	#65411	C₂²	1	8	yes	—
16	#65395	C₄	1	4	yes	complemented
17	#65395	C₂²	1	4	yes	—
18	C₂×#3950	C₄	1	2	yes	complemented
19	C₂×#3949	C₄	1	2	yes	complemented
20	C₂×#3947	—	2	2	no	—
21	C₂×C₈×Dic₅	—	2	2	no	—
22	C₂×C₈×Dic₅	C₄	1	8	yes	complemented
23	C₂×C₈×Dic₅	C₂²	1	4	yes	—
24	#64775	C₂²	1	8	yes	—
25	C₂×#1772	—	5	1	no	maximal, Sylow
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂³×Dic₅	—	2	1	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	4	no	—
30	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
31	C₂×C₅×M₁₆	C₂³	1	4	yes	—
32	C₂²×C₅×C₈	—	2	2	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂×C₂²⋊Dic₅	—	2	2	no	—
40	#4072	—	2	2	no	—
41	#4072	—	2	2	no	—
42	#4072	C₂×C₄	1	4	yes	—
43	#4072	C₂³	1	4	yes	—
44	C₂×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
45	C₂×C₄×Dic₅	—	2	2	no	—
46	C₂×C₄×Dic₅	—	2	2	no	—
47	C₂×C₄×Dic₅	—	2	1	no	—
48	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
49	C₂×C₄×Dic₅	C₂³	1	2	yes	—
50	C₂²×#168	C₂×C₄	1	4	yes	—
51	C₂²×#168	D₄	1	1	yes	characteristic
52	C₂²×#168	D₄	1	1	yes	characteristic
53	C₂²×#168	D₄	1	2	yes	—
54	C₂²×#168	Q₈	1	4	yes	—
55	C₂²×#168	C₂³	1	2	yes	—
56	C₅×#108	C₂×C₄	1	16	yes	complemented
57	#3950	C₂×C₄	1	8	yes	complemented
58	#3949	C₂×C₄	1	8	yes	complemented
59	#3947	—	2	8	no	—
60	C₈×Dic₅	—	2	8	no	—
61	C₈×Dic₅	C₂×C₄	1	16	yes	complemented
62	C₂²×#108	D₅	1	1	yes	characteristic
63	C₂×#427	—	5	1	no	—
64	C₂×#405	—	5	2	no	—
65	C₂²×C₄×C₈	—	5	1	no	—
66	#1772	—	5	8	no	—
67	C₂×#339	—	5	2	no	—
68	C₂⁴×C₅	#41	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₅	#41	1	2	yes	—
77	C₂²×C₄×C₅	#41	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₅	—	2	8	no	—
80	C₂²×Dic₅	—	2	4	no	—
81	C₂²×Dic₅	—	2	2	no	—
82	C₂²×Dic₅	—	2	4	no	—
83	C₅×M₁₆	—	2	8	no	—
84	C₅×M₁₆	—	2	4	no	—
85	C₅×M₁₆	C₂²×C₄	1	8	yes	—
86	C₂×C₅×C₈	—	2	8	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	C₄²	1	4	yes	—
89	C₂×C₅×C₈	C₄²	1	4	yes	—
90	C₂×C₅×C₈	C₄²	1	8	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
92	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
95	C₂²⋊Dic₅	—	2	4	no	—
96	C₂²⋊Dic₅	—	2	4	no	—
97	C₄⋊Dic₅	—	2	4	no	—
98	C₄⋊Dic₅	C₂²×C₄	1	8	yes	—
99	C₄×Dic₅	—	2	4	no	—
100	C₄×Dic₅	—	2	4	no	—
101	C₄×Dic₅	—	2	4	no	—
102	C₄×Dic₅	—	2	2	no	—
103	C₄×Dic₅	C₂²×C₄	1	4	yes	—
104	C₂×#168	C₄⋊C₄	1	4	yes	—
105	C₂×#168	C₄⋊C₄	1	4	yes	—
106	C₂×#168	C₄⋊C₄	1	8	yes	—
107	C₂×#168	C₂²×C₄	1	4	yes	—
108	C₂×#168	C₂×D₄	1	2	yes	—
109	C₂×#168	C₂×D₄	1	2	yes	—
110	C₂×#168	C₂×Q₈	1	4	yes	—
111	C₂²×M₁₆	C₂×D₅	1	2	yes	—
112	C₂³×C₈	—	5	1	no	—
113	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
114	C₂×#117	—	5	2	no	—
115	C₂²×C₄²	—	5	1	no	—
116	C₂×#108	Dic₅	1	8	yes	complemented
117	C₂×#108	C₂×D₅	1	4	yes	—
118	#427	—	5	4	no	—
119	#427	—	5	4	no	—
120	C₂×#107	—	5	2	no	—
121	C₂×#106	—	5	2	no	—
122	#405	—	5	8	no	—
123	#405	—	10	4	no	—
124	C₂×#97	—	10	2	no	—
125	C₂×C₄×C₈	—	5	8	no	—
126	C₂×C₄×C₈	—	5	4	no	—
127	C₂×C₄×C₈	—	10	2	no	—
128	#339	—	5	8	no	—
129	C₂³×C₅	—	2	4	no	—
130	C₂³×C₅	#117	1	2	yes	—
131	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
132	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
133	C₂³×C₅	C₄×Q₈	1	2	yes	—
134	C₂³×C₅	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₅	#117	1	2	yes	—
140	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
145	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
146	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
147	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
153	C₂×Dic₅	—	2	8	no	—
154	C₂×Dic₅	—	2	16	no	—
155	C₂×Dic₅	—	2	4	no	—
156	C₂×Dic₅	—	2	8	no	—
157	C₂×Dic₅	—	4	8	no	—
158	C₅×C₈	—	2	8	no	—
159	C₅×C₈	C₂×C₄²	1	4	yes	—
160	C₅×C₈	C₂×C₄²	1	4	yes	—
161	#168	C₂×C₄⋊C₄	1	4	yes	—
162	#168	C₂×C₄⋊C₄	1	4	yes	—
163	C₂³×C₄	—	10	1	no	—
164	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	C₂×Dic₅	1	8	yes	—
168	C₂×M₁₆	C₂²×D₅	1	4	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	5	4	no	—
171	C₂²×C₈	—	5	2	no	—
172	C₂²×C₈	—	5	1	no	—
173	C₂²×C₈	—	5	1	no	—
174	C₂²×C₈	—	5	4	no	—
175	C₂²×C₈	—	5	2	no	—
176	C₂²×C₈	C₄×D₅	1	4	yes	—
177	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
178	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₅	1	2	yes	—
180	C₂²×C₈	C₂×Dic₅	1	4	yes	—
181	C₂²×C₈	C₂²×D₅	1	2	yes	—
182	#117	—	5	4	no	—
183	#117	—	5	4	no	—
184	#117	—	10	2	no	—
185	#117	—	10	2	no	—
186	C₂×C₄⋊C₄	—	5	4	no	—
187	C₂×C₂²⋊C₄	—	10	2	no	—
188	C₂×C₄²	—	5	4	no	—
189	C₂×C₄²	—	5	2	no	—
190	C₂×C₄²	—	10	2	no	—
191	C₂×C₄²	—	10	2	no	—
192	C₂×C₄²	—	10	1	no	—
193	#108	C₂×Dic₅	1	16	yes	complemented
194	#107	—	5	8	no	—
195	#106	—	5	8	no	—
196	#97	—	10	8	no	—
197	C₄×C₈	—	5	16	no	—
198	C₄×C₈	—	10	8	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	—	2	4	no	—
202	C₂²×C₅	—	2	4	no	—
203	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
204	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
205	C₂²×C₅	#443	1	4	yes	—
206	C₂²×C₅	C₂×#117	1	1	yes	characteristic
207	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
209	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
210	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
211	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
212	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
213	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
214	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
215	Dic₅	—	2	8	no	—
216	Dic₅	—	4	8	no	—
217	C₂⁴	#713	1	1	yes	characteristic
218	C₂²×C₄	—	10	8	no	—
219	C₂²×C₄	—	10	2	no	—
220	C₂²×C₄	—	10	4	no	—
221	C₂²×C₄	—	10	4	no	—
222	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
225	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
226	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
227	C₂²×C₄	#711	1	2	yes	—
228	C₂²×C₄	Q₈×D₅	1	2	yes	—
229	C₂²×C₄	#713	1	1	yes	characteristic
230	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
231	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	8	no	—
234	M₁₆	C₂²×Dic₅	1	8	yes	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	5	4	no	—
238	C₂×C₈	—	5	2	no	—
239	C₂×C₈	—	5	2	no	—
240	C₂×C₈	—	5	4	no	—
241	C₂×C₈	—	5	8	no	—
242	C₂×C₈	—	5	4	no	—
243	C₂×C₈	—	5	4	no	—
244	C₂×C₈	C₄×Dic₅	1	4	yes	—
245	C₂×C₈	C₄×Dic₅	1	4	yes	—
246	C₂×C₈	C₄×Dic₅	1	8	yes	—
247	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
248	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
250	C₂×C₈	C₂²×Dic₅	1	4	yes	—
251	C₄⋊C₄	—	5	8	no	—
252	C₄⋊C₄	—	10	4	no	—
253	C₂²⋊C₄	—	10	4	no	—
254	C₂²⋊C₄	—	10	4	no	—
255	C₄²	—	5	4	no	—
256	C₄²	—	10	4	no	—
257	C₄²	—	10	2	no	—
258	C₄²	—	10	4	no	—
259	C₄²	—	10	4	no	—
260	C₂×C₅	—	2	1	no	—
261	C₂×C₅	—	2	1	no	—
262	C₂×C₅	—	2	2	no	—
263	C₂×C₅	#1772	1	4	yes	complemented
264	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₅	C₂×#443	1	2	yes	—
266	C₂³	—	2	4	no	—
267	C₂³	#4038	1	2	yes	—
268	C₂³	#4039	1	1	yes	characteristic
269	C₂³	#4039	1	1	yes	characteristic
270	C₂³	Q₈×Dic₅	1	2	yes	—
271	C₂³	C₂×#713	1	1	yes	characteristic
272	C₂×C₄	—	10	16	no	—
273	C₂×C₄	—	10	8	no	—
274	C₂×C₄	—	10	4	no	—
275	C₂×C₄	—	10	8	no	—
276	C₂×C₄	—	20	8	no	—
277	C₂×C₄	#4033	1	2	yes	—
278	C₂×C₄	#4033	1	2	yes	—
279	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
280	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
281	C₂×C₄	#4038	1	2	yes	—
282	C₂×C₄	#4039	1	1	yes	characteristic
283	C₂×C₄	#4039	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
286	C₂×C₄	D₄×Dic₅	1	2	yes	—
287	C₂×C₄	D₄×Dic₅	1	2	yes	—
288	C₂×C₄	Q₈×Dic₅	1	2	yes	—
289	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
292	C₂×C₄	C₂×#711	1	1	yes	characteristic
293	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
294	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
295	C₈	—	2	8	no	—
296	C₈	—	5	4	no	—
297	C₈	—	5	4	no	—
298	C₈	C₂×C₄×Dic₅	1	4	yes	—
299	C₈	C₂×C₄×Dic₅	1	4	yes	—
300	C₅	C₂×#1772	1	1	yes	characteristic, complemented, Sylow
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	#65184	1	4	yes	—
306	C₂²	C₄⋊C₄×Dic₅	1	2	yes	—
307	C₂²	C₄⋊C₄×Dic₅	1	2	yes	—
308	C₂²	C₂×#4038	1	1	yes	characteristic
309	C₂²	C₂×#4039	1	1	yes	characteristic
310	C₂²	C₂×Q₈×Dic₅	1	1	yes	characteristic
311	C₄	—	10	8	no	—
312	C₄	—	20	8	no	—
313	C₄	C₄⋊C₄×Dic₅	1	2	yes	—
314	C₄	C₄⋊C₄×Dic₅	1	2	yes	—
315	C₄	C₂×#4033	1	1	yes	characteristic
316	C₄	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
317	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
318	C₄	C₂×D₄×Dic₅	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	2	no	—
322	C₂	#10601495	1	4	yes	direct product factor
323	C₂	C₂×#65184	1	2	yes	—
324	C₂	C₂×C₄⋊C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂²×C₅×#108	C₂
socle	C₂³×C₅	C₄×D₄
prime radical	C₂⁴×C₅	#41
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂²×#108	D₅
hypocenter	C₅	C₂×#1772
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
6. #10601495	C₂	direct (N × H)
12. C₂×C₅×#108	C₄	semidirect
16. #65395	C₄	semidirect
18. C₂×#3950	C₄	semidirect
19. C₂×#3949	C₄	semidirect
22. C₂×C₈×Dic₅	C₄	semidirect
56. C₅×#108	C₂×C₄	semidirect
57. #3950	C₂×C₄	semidirect
58. #3949	C₂×C₄	semidirect
61. C₈×Dic₅	C₂×C₄	semidirect
116. C₂×#108	Dic₅	semidirect
193. #108	C₂×Dic₅	semidirect
263. C₂×C₅	#1772	semidirect
300. C₅	C₂×#1772	semidirect
322. C₂	#10601495	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×#108	2
C₂×#65411	2
C₂×#65395	2
C₂²×C₈×Dic₅	2
C₂×#64775	2
#10601495	2
C₂×#1772	5

Subgroups — C2×#10601495 #12097309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601495 #12097309