Subgroups — C₂×#10604098 #13043103

Order 1280.

854 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65718	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂²×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#10604098	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64844	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₆×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
7	C₂×#4130	C₄	1	1	yes	characteristic
8	C₂×D₅⋊M₁₆	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₂×#3998	C₄	1	2	yes	complemented
11	#65718	C₄	1	4	yes	—
12	#65718	C₂²	1	4	yes	—
13	C₂×#3992	C₄	1	2	yes	—
14	C₂×C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	—
15	D₅⋊M₃₂	C₄	1	4	yes	complemented
16	D₅⋊M₃₂	C₂²	1	4	yes	—
17	C₂×#3990	C₄	1	2	yes	complemented
18	C₂×D₅⋊C₁₆	C₂²	1	2	yes	—
19	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic, complemented
20	C₂²×#672	C₄	1	1	yes	characteristic, complemented
21	C₂×D₅×C₁₆	C₄	1	4	yes	complemented
22	C₂×D₅×C₁₆	C₄	1	1	yes	characteristic, complemented
23	C₂×D₅×C₁₆	C₄	1	1	yes	characteristic, complemented
24	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
25	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
26	C₂×D₅×C₁₆	C₂²	1	4	yes	—
27	#64844	C₂²	1	8	yes	complemented
28	C₁₆×AGL(1,𝔽₅)	C₂²	1	8	yes	complemented
29	C₂×#2108	—	5	1	no	maximal, Sylow
30	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊AGL(1,𝔽₅)	C₈	1	2	yes	—
32	C₂×#704	C₈	1	2	yes	—
33	#4130	C₈	1	4	yes	—
34	#4130	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊AGL(1,𝔽₅)	C₈	1	2	yes	—
36	C₂×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
37	D₅⋊M₁₆	C₈	1	4	yes	—
38	D₅⋊M₁₆	C₂×C₄	1	4	yes	—
39	C₂×#700	C₈	1	2	yes	—
40	C₂×D₅⋊C₈	C₂×C₄	1	2	yes	—
41	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×#168	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
50	C₂×C₈×D₅	C₂³	1	2	yes	—
51	#3998	C₂×C₄	1	8	yes	complemented
52	C₂×#674	C₂×C₄	1	4	yes	—
53	#3992	C₂×C₄	1	8	yes	—
54	C₈×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
55	C₈×AGL(1,𝔽₅)	C₂³	1	4	yes	—
56	#3990	C₂×C₄	1	8	yes	complemented
57	D₅⋊C₁₆	C₂×C₄	1	4	yes	—
58	D₅⋊C₁₆	C₂³	1	4	yes	—
59	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
60	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
61	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
62	C₂×#672	C₂×C₄	1	1	yes	characteristic
63	C₂×#672	C₂×C₄	1	1	yes	characteristic
64	C₂×#672	C₂×C₄	1	4	yes	complemented
65	D₅×C₁₆	C₂×C₄	1	2	yes	—
66	D₅×C₁₆	C₂×C₄	1	2	yes	—
67	D₅×C₁₆	C₂×C₄	1	4	yes	complemented
68	D₅×C₁₆	C₂×C₄	1	4	yes	complemented
69	D₅×C₁₆	C₂³	1	2	yes	—
70	D₅×C₁₆	C₂³	1	2	yes	—
71	C₂²×M₃₂	—	5	1	no	—
72	C₂³×C₁₆	—	5	1	no	—
73	C₂×#405	—	5	1	no	—
74	#2108	—	5	8	no	—
75	C₂×#346	—	5	2	no	—
76	C₂×C₄×C₁₆	—	5	2	no	—
77	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
78	C₂²×AGL(1,𝔽₅)	C₂×C₈	1	4	yes	—
79	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
80	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₅	C₄²	1	1	yes	characteristic
82	C₂×C₄×D₅	C₄²	1	2	yes	—
83	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
87	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
89	C₂²⋊AGL(1,𝔽₅)	C₂×C₈	1	8	yes	—
90	#704	C₂×C₈	1	8	yes	—
91	C₂×#170	C₂×C₈	1	4	yes	—
92	C₄⋊AGL(1,𝔽₅)	C₂×C₈	1	8	yes	—
93	C₄×AGL(1,𝔽₅)	C₂×C₈	1	4	yes	—
94	C₄×AGL(1,𝔽₅)	C₂²×C₄	1	4	yes	—
95	#700	C₂×C₈	1	8	yes	—
96	D₅⋊C₈	C₂×C₈	1	4	yes	—
97	D₅⋊C₈	C₂²×C₄	1	4	yes	—
98	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
99	C₂×C₅×C₈	C₄²	1	2	yes	—
100	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
102	C₂×#168	C₄²	1	1	yes	characteristic
103	C₂×#168	C₄²	1	2	yes	—
104	C₂×#168	C₂²×C₄	1	1	yes	characteristic
105	C₂×#168	C₂²×C₄	1	2	yes	—
106	C₈×D₅	C₄²	1	2	yes	—
107	C₈×D₅	C₄²	1	2	yes	—
108	C₈×D₅	C₄²	1	2	yes	—
109	C₈×D₅	C₂²×C₄	1	1	yes	characteristic
110	C₈×D₅	C₂²×C₄	1	2	yes	—
111	C₈×D₅	C₂²×C₄	1	1	yes	characteristic
112	C₈×D₅	C₂²×C₄	1	2	yes	—
113	C₈×D₅	C₂²×C₄	1	1	yes	characteristic
114	C₈×D₅	C₂²×C₄	1	2	yes	—
115	C₈×D₅	C₂⁴	1	1	yes	characteristic
116	#674	C₂²×C₄	1	8	yes	—
117	C₅×C₁₆	C₂²×C₄	1	2	yes	—
118	C₅×C₁₆	C₂²×C₄	1	2	yes	—
119	#672	C₂²×C₄	1	2	yes	—
120	#672	C₂²×C₄	1	2	yes	—
121	C₂²×M₁₆	—	5	1	no	—
122	C₂³×C₈	—	5	1	no	—
123	C₂×#117	—	5	1	no	—
124	C₂×M₃₂	—	5	4	no	—
125	C₂×M₃₂	—	5	2	no	—
126	C₂×M₃₂	—	5	4	no	—
127	C₂×M₃₂	—	5	2	no	—
128	C₂²×C₁₆	—	5	2	no	—
129	C₂²×C₁₆	—	5	1	no	—
130	C₂²×C₁₆	—	5	1	no	—
131	C₂²×C₁₆	—	5	1	no	—
132	C₂²×C₁₆	—	5	1	no	—
133	C₂²×C₁₆	—	5	1	no	—
134	C₂²×C₁₆	—	5	4	no	—
135	C₂²×C₁₆	—	5	4	no	—
136	C₂²×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
137	#405	—	5	4	no	—
138	#405	—	5	4	no	—
139	C₂×#97	—	5	2	no	—
140	C₂×C₄×C₈	—	5	2	no	—
141	#346	—	5	8	no	—
142	C₄×C₁₆	—	5	8	no	—
143	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
144	C₂²×D₅	—	2	4	no	—
145	C₂²×D₅	C₄×C₈	1	1	yes	characteristic
146	C₂²×D₅	C₄×C₈	1	2	yes	—
147	C₂²×D₅	C₂²×C₈	1	2	yes	—
148	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
149	C₂×AGL(1,𝔽₅)	—	2	8	no	—
150	C₂×AGL(1,𝔽₅)	C₂²×C₈	1	8	yes	—
151	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
152	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
154	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
155	C₂×Dic₅	C₄×C₈	1	2	yes	—
156	C₂×Dic₅	C₄×C₈	1	1	yes	characteristic
157	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
158	C₂×Dic₅	C₂²×C₈	1	2	yes	—
159	C₄×D₅	C₄×C₈	1	2	yes	—
160	C₄×D₅	C₄×C₈	1	2	yes	—
161	C₄×D₅	C₄×C₈	1	2	yes	—
162	C₄×D₅	C₂×C₄²	1	2	yes	—
163	C₄×D₅	C₂×C₄²	1	1	yes	characteristic
164	C₄×D₅	C₂²×C₈	1	1	yes	characteristic
165	C₄×D₅	C₂²×C₈	1	2	yes	—
166	C₄×D₅	C₂²×C₈	1	1	yes	characteristic
167	C₄×D₅	C₂²×C₈	1	2	yes	—
168	C₄×D₅	C₂³×C₄	1	1	yes	characteristic
169	#170	C₂²×C₈	1	8	yes	—
170	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
171	C₅×C₈	C₂×C₄²	1	2	yes	—
172	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
173	#168	C₂×C₄²	1	1	yes	characteristic
174	#168	C₂×C₄²	1	2	yes	—
175	#168	C₂³×C₄	1	1	yes	characteristic
176	C₂³×C₄	—	5	1	no	—
177	C₂×M₁₆	—	5	2	no	—
178	C₂×M₁₆	—	5	4	no	—
179	C₂×M₁₆	—	5	4	no	—
180	C₂×M₁₆	—	5	2	no	—
181	C₂²×C₈	—	5	2	no	—
182	C₂²×C₈	—	5	1	no	—
183	C₂²×C₈	—	5	1	no	—
184	C₂²×C₈	—	5	2	no	—
185	C₂²×C₈	—	5	1	no	—
186	C₂²×C₈	—	5	2	no	—
187	C₂²×C₈	—	5	1	no	—
188	C₂²×C₈	—	5	2	no	—
189	C₂²×C₈	—	5	1	no	—
190	C₂²×C₈	—	5	2	no	—
191	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
192	#117	—	5	4	no	—
193	#117	—	5	4	no	—
194	C₂×C₄⋊C₄	—	5	2	no	—
195	C₂×C₂²⋊C₄	—	5	2	no	—
196	C₂×C₄²	—	5	2	no	—
197	M₃₂	—	5	8	no	—
198	M₃₂	—	5	8	no	—
199	C₂×C₁₆	—	5	4	no	—
200	C₂×C₁₆	—	5	4	no	—
201	C₂×C₁₆	—	5	4	no	—
202	C₂×C₁₆	—	5	1	no	—
203	C₂×C₁₆	—	5	1	no	—
204	C₂×C₁₆	—	5	4	no	—
205	C₂×C₁₆	—	5	2	no	—
206	C₂×C₁₆	—	5	2	no	—
207	C₂×C₁₆	—	5	2	no	—
208	C₂×C₁₆	—	5	4	no	—
209	C₂×C₁₆	—	5	4	no	—
210	C₂×C₁₆	—	5	2	no	—
211	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	4	yes	complemented
214	#97	—	5	8	no	—
215	C₄×C₈	—	5	4	no	—
216	C₄×C₈	—	5	4	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
219	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
220	C₂×D₅	—	2	2	no	—
221	C₂×D₅	—	2	4	no	—
222	C₂×D₅	—	2	4	no	—
223	C₂×D₅	C₂×C₄×C₈	1	2	yes	—
224	C₂×D₅	C₂×C₄×C₈	1	1	yes	characteristic
225	C₂×D₅	#504	1	4	yes	—
226	C₂×D₅	C₂³×C₈	1	1	yes	characteristic
227	AGL(1,𝔽₅)	—	2	8	no	—
228	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
229	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
230	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
231	Dic₅	C₂×C₄×C₈	1	1	yes	characteristic
232	Dic₅	C₂×C₄×C₈	1	2	yes	—
233	Dic₅	C₂³×C₈	1	1	yes	characteristic
234	C₂⁴	—	5	1	no	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	C₂²×C₄	—	5	2	no	—
238	C₂²×C₄	—	5	2	no	—
239	C₂²×C₄	—	5	1	no	—
240	C₂²×C₄	—	5	1	no	—
241	C₂²×C₄	—	5	2	no	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	5	2	no	—
244	C₂²×C₄	—	5	4	no	—
245	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
246	M₁₆	—	5	8	no	—
247	M₁₆	—	5	8	no	—
248	C₂×C₈	—	5	4	no	—
249	C₂×C₈	—	5	4	no	—
250	C₂×C₈	—	5	4	no	—
251	C₂×C₈	—	5	1	no	—
252	C₂×C₈	—	5	2	no	—
253	C₂×C₈	—	5	1	no	—
254	C₂×C₈	—	5	2	no	—
255	C₂×C₈	—	5	1	no	—
256	C₂×C₈	—	5	2	no	—
257	C₂×C₈	—	5	1	no	—
258	C₂×C₈	—	5	2	no	—
259	C₂×C₈	—	5	2	no	—
260	C₂×C₈	—	5	2	no	—
261	C₂×C₈	—	5	2	no	—
262	C₂×C₈	—	5	1	no	—
263	C₂×C₈	—	5	1	no	—
264	C₂×C₈	—	5	2	no	—
265	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
267	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
268	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
269	C₄⋊C₄	—	5	8	no	—
270	C₂²⋊C₄	—	5	8	no	—
271	C₄²	—	5	4	no	—
272	C₄²	—	5	4	no	—
273	C₁₆	—	5	8	no	—
274	C₁₆	—	5	2	no	—
275	C₁₆	—	5	2	no	—
276	C₁₆	C₂²×AGL(1,𝔽₅)	1	2	yes	—
277	C₁₆	C₂²×AGL(1,𝔽₅)	1	2	yes	—
278	C₂×C₅	—	2	2	no	—
279	C₂×C₅	#2108	1	2	yes	complemented
280	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
281	D₅	—	2	2	no	—
282	D₅	#2108	1	2	yes	complemented
283	D₅	C₂×#504	1	2	yes	—
284	C₂³	—	5	1	no	—
285	C₂³	—	5	2	no	—
286	C₂³	—	5	2	no	—
287	C₂³	—	5	1	no	—
288	C₂³	—	10	4	no	—
289	C₂³	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
290	C₂×C₄	—	5	1	no	—
291	C₂×C₄	—	5	2	no	—
292	C₂×C₄	—	5	2	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	5	2	no	—
296	C₂×C₄	—	5	1	no	—
297	C₂×C₄	—	5	2	no	—
298	C₂×C₄	—	5	2	no	—
299	C₂×C₄	—	5	2	no	—
300	C₂×C₄	—	5	2	no	—
301	C₂×C₄	—	5	1	no	—
302	C₂×C₄	—	5	1	no	—
303	C₂×C₄	—	5	2	no	—
304	C₂×C₄	—	5	8	no	—
305	C₂×C₄	—	10	8	no	—
306	C₂×C₄	C₈×AGL(1,𝔽₅)	1	2	yes	—
307	C₂×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
309	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
310	C₈	—	5	8	no	—
311	C₈	—	5	1	no	—
312	C₈	—	5	1	no	—
313	C₈	—	5	2	no	—
314	C₈	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
315	C₈	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
316	C₈	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
317	C₅	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	2	2	no	—
319	C₂²	—	5	1	no	—
320	C₂²	—	5	1	no	—
321	C₂²	—	5	4	no	—
322	C₂²	—	5	2	no	—
323	C₂²	—	10	4	no	—
324	C₂²	—	10	4	no	—
325	C₂²	—	10	2	no	—
326	C₂²	C₂×C₈×AGL(1,𝔽₅)	1	2	yes	—
327	C₂²	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
328	C₄	—	5	1	no	—
329	C₄	—	5	1	no	—
330	C₄	—	5	2	no	—
331	C₄	—	10	8	no	—
332	C₄	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
333	C₄	C₂×C₈×AGL(1,𝔽₅)	1	2	yes	—
334	C₄	C₂²×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	5	2	no	—
337	C₂	—	5	2	no	—
338	C₂	—	10	2	no	—
339	C₂	#10604098	1	2	yes	direct product factor
340	C₂	C₂²×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₄×C₈
Frattini	C₈	C₂³×AGL(1,𝔽₅)
Fitting	C₂²×C₅×C₁₆	C₄
socle	C₂²×C₅	C₂×C₄×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₈×D₅	C₂⁴
hypercenter	C₂²×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#2108
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
4. #10604098	C₂	direct (N × H)
5. C₂×#64844	C₂	semidirect
6. C₂×C₁₆×AGL(1,𝔽₅)	C₂	semidirect
10. C₂×#3998	C₄	semidirect
15. D₅⋊M₃₂	C₄	semidirect
17. C₂×#3990	C₄	semidirect
19. C₂²×C₅×C₁₆	C₄	semidirect
20. C₂²×#672	C₄	semidirect
21. C₂×D₅×C₁₆	C₄	semidirect
22. C₂×D₅×C₁₆	C₄	semidirect
23. C₂×D₅×C₁₆	C₄	semidirect
27. #64844	C₂²	semidirect
28. C₁₆×AGL(1,𝔽₅)	C₂²	semidirect
51. #3998	C₂×C₄	semidirect
56. #3990	C₂×C₄	semidirect
61. C₂×C₅×C₁₆	C₂×C₄	semidirect
64. C₂×#672	C₂×C₄	semidirect
67. D₅×C₁₆	C₂×C₄	semidirect
68. D₅×C₁₆	C₂×C₄	semidirect
136. C₂²×C₁₆	AGL(1,𝔽₅)	semidirect
213. C₂×C₁₆	C₂×AGL(1,𝔽₅)	semidirect
279. C₂×C₅	#2108	semidirect
282. D₅	#2108	semidirect
317. C₅	C₂×#2108	semidirect
339. C₂	#10604098	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×#65718	2
C₂×D₅⋊M₃₂	2
C₂²×D₅×C₁₆	2
#10604098	2
C₂×#64844	2
C₂×C₁₆×AGL(1,𝔽₅)	2
C₂×#2108	5

Subgroups — C2×#10604098 #13043103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10604098 #13043103