Subgroups — C₂×#10600107 #12254192

Order 1280.

552 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65182	C₂	1	1	yes	maximal, characteristic
2	C₂×#65157	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#64737	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64734	C₂	1	1	yes	maximal, characteristic
6	C₂×#64716	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64707	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600107	C₂	1	8	yes	maximal, direct product factor
9	C₂×D₅×M₁₆	—	2	1	no	—
10	C₂²×Dic₂₀	C₂²	1	1	yes	characteristic
11	C₂²×#676	C₂²	1	1	yes	characteristic
12	C₂×#4043	—	2	1	no	—
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₅×#108	C₂²	1	1	yes	characteristic
15	C₂×C₅×#106	C₂²	1	1	yes	characteristic
16	C₂×#3954	—	2	1	no	—
17	C₂×#3951	C₂²	1	1	yes	characteristic
18	C₂×#3944	C₂²	1	1	yes	characteristic
19	C₂×#3942	—	2	1	no	—
20	#65182	—	2	2	no	—
21	#65182	C₂²	1	4	yes	—
22	#65157	—	2	2	no	—
23	#65157	C₂²	1	4	yes	—
24	C₅×#365	C₂²	1	4	yes	complemented
25	#64737	C₂²	1	4	yes	complemented
26	#64734	C₂²	1	4	yes	—
27	#64716	C₂²	1	4	yes	complemented
28	#64707	C₂²	1	4	yes	complemented
29	C₂×#2239	—	5	1	no	maximal, Sylow
30	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
31	C₂²×Dic₁₀	—	2	1	no	—
32	C₂×C₅×M₁₆	—	2	1	no	—
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×#685	—	4	1	no	—
36	C₂×C₄⋊Dic₅	—	4	1	no	—
37	C₂×#681	—	2	1	no	—
38	C₂²×#168	D₄	1	1	yes	characteristic
39	D₅×M₁₆	—	2	4	no	—
40	D₅×M₁₆	—	4	2	no	—
41	C₂×Dic₂₀	—	2	1	no	—
42	C₂×Dic₂₀	—	2	1	no	—
43	C₂×Dic₂₀	—	2	1	no	—
44	C₂×Dic₂₀	—	2	1	no	—
45	C₂×Dic₂₀	—	2	1	no	—
46	C₂×Dic₂₀	C₂³	1	2	yes	—
47	C₂×#676	—	2	1	no	—
48	C₂×#676	—	2	1	no	—
49	C₂×#676	—	2	1	no	—
50	C₂×#676	—	2	1	no	—
51	C₂×#676	—	4	1	no	—
52	C₂×#676	D₄	1	2	yes	—
53	C₂×#676	C₂³	1	2	yes	—
54	C₂×C₈×D₅	—	4	1	no	—
55	#4043	—	2	4	no	—
56	#4043	—	4	2	no	—
57	C₅×M₃₂	—	2	1	no	—
58	C₅×M₃₂	C₂³	1	2	yes	—
59	C₂×C₅×C₁₆	—	2	1	no	—
60	C₅×#108	—	2	1	no	—
61	C₅×#108	C₂³	1	2	yes	—
62	C₅×#106	—	2	1	no	—
63	C₅×#106	C₂³	1	2	yes	—
64	#3954	—	2	4	no	—
65	#3951	—	2	1	no	—
66	#3951	C₂³	1	2	yes	—
67	#3944	—	2	1	no	—
68	#3944	C₂³	1	2	yes	—
69	C₂×#672	—	2	1	no	—
70	#3942	—	2	4	no	—
71	C₂×#470	—	5	1	no	—
72	C₂×#430	—	5	1	no	—
73	#2239	—	5	8	no	—
74	C₂×#365	—	5	1	no	—
75	C₂×#365	D₅	1	1	yes	characteristic, complemented
76	C₂×#362	—	5	1	no	—
77	C₂×#362	—	5	1	no	—
78	C₂×#349	—	5	1	no	—
79	C₂³×D₅	SD₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₅	—	4	1	no	—
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₅	—	4	1	no	—
83	C₂²×Dic₅	SD₁₆	1	1	yes	characteristic
84	C₂×C₄×D₅	—	2	1	no	—
85	C₂×C₄×D₅	—	2	1	no	—
86	C₂×C₄×D₅	—	2	1	no	—
87	C₂×C₄×D₅	—	2	1	no	—
88	C₂×C₄×D₅	SD₁₆	1	2	yes	—
89	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
90	C₂×Dic₁₀	—	2	2	no	—
91	C₂×Dic₁₀	—	4	1	no	—
92	C₂×Dic₁₀	—	4	1	no	—
93	C₂×Dic₁₀	—	4	1	no	—
94	C₂×Dic₁₀	—	4	1	no	—
95	C₂×Dic₁₀	—	4	1	no	—
96	C₅×M₁₆	—	2	2	no	—
97	C₅×M₁₆	—	4	1	no	—
98	C₂×C₅×C₈	—	4	1	no	—
99	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×C₈	#41	1	1	yes	characteristic
101	C₂×C₅×C₈	#41	1	1	yes	characteristic
102	C₂×C₅×C₈	#41	1	1	yes	characteristic
103	C₂×C₅×C₈	#41	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	4	1	no	—
107	#685	—	4	4	no	—
108	C₄⋊Dic₅	—	4	4	no	—
109	#681	—	2	2	no	—
110	#681	—	4	1	no	—
111	C₂×#168	—	2	1	no	—
112	C₂×#168	—	2	1	no	—
113	C₂×#168	—	4	1	no	—
114	C₂×#168	C₂×D₄	1	2	yes	—
115	Dic₂₀	—	2	2	no	—
116	Dic₂₀	—	2	2	no	—
117	Dic₂₀	—	4	1	no	—
118	Dic₂₀	—	4	1	no	—
119	#676	—	2	2	no	—
120	#676	—	2	2	no	—
121	#676	—	2	2	no	—
122	#676	—	2	2	no	—
123	#676	—	4	4	no	—
124	C₈×D₅	—	4	4	no	—
125	C₅×C₁₆	—	2	2	no	—
126	#672	—	2	2	no	—
127	C₂²×Q₁₆	—	5	1	no	—
128	C₂²×M₁₆	—	5	1	no	—
129	C₂²×M₁₆	—	10	1	no	—
130	C₂×C₂²⋊Q₈	—	10	1	no	—
131	C₂×M₃₂	—	5	1	no	—
132	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
133	#470	—	5	4	no	—
134	#470	—	10	2	no	—
135	#430	—	5	4	no	—
136	#430	—	10	2	no	—
137	C₂×#108	—	5	1	no	—
138	C₂×#108	—	10	1	no	—
139	C₂×#108	C₂×D₅	1	1	yes	characteristic
140	C₂×#106	C₂×D₅	1	1	yes	characteristic
141	C₂×#103	—	10	1	no	—
142	#365	—	5	4	no	—
143	#365	C₂×D₅	1	4	yes	complemented
144	#362	—	5	4	no	—
145	#362	—	5	4	no	—
146	#349	—	5	4	no	—
147	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
148	C₂²×D₅	—	2	1	no	—
149	C₂²×D₅	—	2	1	no	—
150	C₂²×D₅	—	4	2	no	—
151	C₂²×D₅	C₂×SD₁₆	1	2	yes	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	8	2	no	—
154	C₂×C₄×C₅	#123	1	1	yes	characteristic
155	C₂×C₄×C₅	#123	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
157	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
160	C₂×Dic₅	—	2	1	no	—
161	C₂×Dic₅	—	2	1	no	—
162	C₂×Dic₅	—	4	2	no	—
163	C₂×Dic₅	—	8	1	no	—
164	C₂×Dic₅	—	8	1	no	—
165	C₂×Dic₅	C₂×SD₁₆	1	2	yes	—
166	C₄×D₅	—	2	2	no	—
167	C₄×D₅	—	2	2	no	—
168	C₄×D₅	—	2	2	no	—
169	C₄×D₅	—	2	2	no	—
170	Dic₁₀	—	4	2	no	—
171	Dic₁₀	—	4	2	no	—
172	Dic₁₀	—	8	1	no	—
173	Dic₁₀	—	8	1	no	—
174	C₅×C₈	—	4	2	no	—
175	C₅×C₈	#123	1	1	yes	characteristic
176	C₅×C₈	#123	1	1	yes	characteristic
177	C₅×C₈	C₂×#41	1	1	yes	characteristic
178	C₅×C₈	C₂×#41	1	1	yes	characteristic
179	#168	—	2	2	no	—
180	#168	—	4	2	no	—
181	C₂²×Q₈	—	10	1	no	—
182	C₂³×C₄	—	5	1	no	—
183	C₂×Q₁₆	—	5	2	no	—
184	C₂×Q₁₆	—	10	1	no	—
185	C₂×Q₁₆	—	10	1	no	—
186	C₂×Q₁₆	—	10	1	no	—
187	C₂×Q₁₆	—	10	1	no	—
188	C₂×Q₁₆	—	10	1	no	—
189	C₂×M₁₆	—	2	1	no	—
190	C₂×M₁₆	—	5	2	no	—
191	C₂×M₁₆	—	5	2	no	—
192	C₂×M₁₆	—	10	1	no	—
193	C₂×M₁₆	—	10	1	no	—
194	C₂×M₁₆	—	10	1	no	—
195	C₂×M₁₆	—	10	1	no	—
196	C₂×M₁₆	—	10	4	no	—
197	C₂×M₁₆	—	10	1	no	—
198	C₂×M₁₆	—	20	1	no	—
199	C₂×M₁₆	—	20	2	no	—
200	C₂²×C₈	—	5	1	no	—
201	C₂²×C₈	—	20	1	no	—
202	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
203	C₂²⋊Q₈	—	10	4	no	—
204	C₂²⋊Q₈	—	20	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	20	1	no	—
207	C₂×C₂²⋊C₄	—	20	1	no	—
208	M₃₂	—	2	1	no	—
209	M₃₂	—	5	2	no	—
210	M₃₂	—	10	1	no	—
211	M₃₂	C₂²×D₅	1	2	yes	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	10	1	no	—
214	#108	—	2	1	no	—
215	#108	—	5	2	no	—
216	#108	—	10	4	no	—
217	#108	—	10	1	no	—
218	#108	C₂²×D₅	1	2	yes	—
219	#106	—	2	1	no	—
220	#106	C₂²×D₅	1	2	yes	—
221	#103	—	10	4	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	#481	1	1	yes	characteristic
224	C₂²×C₅	#481	1	1	yes	characteristic
225	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
226	C₂×D₅	—	2	2	no	—
227	C₂×D₅	—	4	2	no	—
228	C₂×D₅	—	4	2	no	—
229	C₂×D₅	—	8	1	no	—
230	C₄×C₅	—	8	2	no	—
231	C₄×C₅	#481	1	1	yes	characteristic
232	C₄×C₅	#481	1	1	yes	characteristic
233	C₄×C₅	C₂×#123	1	1	yes	characteristic
234	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	Dic₅	—	2	2	no	—
236	Dic₅	—	8	2	no	—
237	C₂⁴	—	5	1	no	—
238	C₂×Q₈	—	10	2	no	—
239	C₂×Q₈	—	20	1	no	—
240	C₂×Q₈	—	20	1	no	—
241	C₂×Q₈	—	20	1	no	—
242	C₂×Q₈	—	20	1	no	—
243	C₂×Q₈	—	20	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	5	1	no	—
246	C₂²×C₄	—	5	2	no	—
247	C₂²×C₄	—	5	2	no	—
248	C₂²×C₄	—	10	1	no	—
249	C₂²×C₄	—	10	1	no	—
250	C₂²×C₄	—	10	1	no	—
251	C₂²×C₄	—	10	1	no	—
252	C₂²×C₄	—	20	1	no	—
253	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
254	Q₁₆	—	10	2	no	—
255	Q₁₆	—	10	2	no	—
256	Q₁₆	—	20	1	no	—
257	Q₁₆	—	20	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	10	2	no	—
261	M₁₆	—	10	2	no	—
262	M₁₆	—	10	2	no	—
263	M₁₆	—	10	2	no	—
264	M₁₆	—	10	2	no	—
265	M₁₆	—	20	4	no	—
266	M₁₆	—	20	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	5	2	no	—
269	C₂×C₈	—	10	1	no	—
270	C₂×C₈	—	10	1	no	—
271	C₂×C₈	—	20	4	no	—
272	C₂×C₈	—	20	1	no	—
273	C₂×C₈	#709	1	1	yes	characteristic
274	C₂×C₈	#709	1	1	yes	characteristic
275	C₂×C₈	D₄×D₅	1	1	yes	characteristic
276	C₂×C₈	#713	1	1	yes	characteristic
277	C₂×C₈	#713	1	1	yes	characteristic
278	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	20	4	no	—
282	C₂²⋊C₄	—	20	4	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	10	2	no	—
285	C₂×C₅	—	2	1	no	—
286	C₂×C₅	—	2	1	no	—
287	C₂×C₅	#2239	1	2	yes	complemented
288	C₂×C₅	C₂×#481	1	1	yes	characteristic
289	D₅	—	4	2	no	—
290	C₂³	—	5	2	no	—
291	C₂³	—	10	1	no	—
292	C₂³	—	10	1	no	—
293	C₂³	—	20	2	no	—
294	C₂³	D₅×SD₁₆	1	1	yes	characteristic
295	Q₈	—	20	2	no	—
296	Q₈	—	20	2	no	—
297	Q₈	—	40	1	no	—
298	Q₈	—	40	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	5	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	10	1	no	—
303	C₂×C₄	—	10	1	no	—
304	C₂×C₄	—	10	2	no	—
305	C₂×C₄	—	10	2	no	—
306	C₂×C₄	—	10	2	no	—
307	C₂×C₄	—	10	2	no	—
308	C₂×C₄	—	20	2	no	—
309	C₂×C₄	—	40	1	no	—
310	C₂×C₄	—	40	1	no	—
311	C₂×C₄	#4040	1	1	yes	characteristic
312	C₂×C₄	#4040	1	1	yes	characteristic
313	C₂×C₄	D₅×SD₁₆	1	1	yes	characteristic
314	C₂×C₄	#4060	1	1	yes	characteristic
315	C₂×C₄	#4060	1	1	yes	characteristic
316	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
317	C₈	—	4	2	no	—
318	C₈	—	10	2	no	—
319	C₈	—	20	2	no	—
320	C₈	#4040	1	1	yes	characteristic
321	C₈	#4040	1	1	yes	characteristic
322	C₈	C₂×#709	1	1	yes	characteristic
323	C₈	C₂×#713	1	1	yes	characteristic
324	C₅	C₂×#2239	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	2	2	no	—
326	C₂²	—	10	2	no	—
327	C₂²	—	20	2	no	—
328	C₂²	—	20	2	no	—
329	C₂²	—	40	1	no	—
330	C₂²	#65152	1	1	yes	characteristic
331	C₂²	#65152	1	1	yes	characteristic
332	C₂²	C₂×D₅×SD₁₆	1	1	yes	characteristic
333	C₄	—	8	2	no	—
334	C₄	—	10	2	no	—
335	C₄	—	40	2	no	—
336	C₄	#65152	1	1	yes	characteristic
337	C₄	#65152	1	1	yes	characteristic
338	C₄	C₂×#4040	1	1	yes	characteristic
339	C₄	C₂×#4060	1	1	yes	characteristic
340	C₂	—	2	1	no	—
341	C₂	—	2	1	no	—
342	C₂	—	20	2	no	—
343	C₂	#10600107	1	2	yes	direct product factor
344	C₂	C₂×#65152	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65152
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#365	C₂
socle	C₂²×C₅	#481
prime radical	C₂³×D₅	SD₁₆
hypercenter	C₂×#365	D₅
hypocenter	C₅	C₂×#2239
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#365	C₂	semidirect
4. C₂×#64737	C₂	semidirect
6. C₂×#64716	C₂	semidirect
7. C₂×#64707	C₂	semidirect
8. #10600107	C₂	direct (N × H)
24. C₅×#365	C₂²	semidirect
25. #64737	C₂²	semidirect
27. #64716	C₂²	semidirect
28. #64707	C₂²	semidirect
75. C₂×#365	D₅	semidirect
143. #365	C₂×D₅	semidirect
287. C₂×C₅	#2239	semidirect
324. C₅	C₂×#2239	semidirect
343. C₂	#10600107	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×#65182	2
C₂×#65157	2
C₂×C₅×#365	2
C₂×#64737	2
C₂×#64734	2
C₂×#64716	2
C₂×#64707	2
#10600107	2
C₂×#2239	5

Subgroups — C2×#10600107 #12254192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600107 #12254192