Subgroups — C₅×#21692 #11944412

Order 1280.

1230 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#492	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#440	C₂	1	4	yes	maximal, complemented
3	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#127	C₂	1	1	yes	maximal, characteristic
5	C₅×#1973	C₂	1	4	yes	maximal, complemented
6	C₅×#1957	C₂	1	1	yes	maximal, characteristic
7	C₅×#1956	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1845	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×SD₁₆	C₂²	1	4	yes	—
11	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×#127	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
14	C₂×C₄×C₅×D₄	—	2	2	no	—
15	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
16	C₅×#492	C₄	1	8	yes	complemented
17	C₅×#440	—	2	16	no	—
18	C₂×C₅×#107	C₂²	1	4	yes	—
19	C₂×C₅×#103	C₂²	1	4	yes	—
20	C₂×C₅×#102	C₂²	1	4	yes	—
21	C₂×C₅×#97	C₂²	1	2	yes	—
22	C₂×C₅×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×#391	C₂²	1	2	yes	—
26	C₅×#390	—	2	2	no	—
27	C₅×#390	C₂²	1	2	yes	—
28	C₅×#389	—	2	2	no	—
29	C₅×#384	C₂²	1	1	yes	characteristic
30	C₅×#384	C₂²	1	1	yes	characteristic
31	C₄×C₅×C₄⋊C₄	—	2	2	no	—
32	C₄³×C₅	C₂²	1	1	yes	characteristic
33	#21692	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₅×Q₈	C₂³	1	2	yes	—
35	C₂²×C₅×D₄	—	2	2	no	—
36	C₂²×C₅×D₄	C₂³	1	2	yes	—
37	C₂³×C₄×C₅	—	2	2	no	—
38	C₂×C₅×SD₁₆	—	2	8	no	—
39	C₂×C₅×SD₁₆	—	2	8	no	—
40	C₂×C₅×SD₁₆	C₂×C₄	1	16	yes	complemented
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	2	yes	—
43	C₅×C₄⋊Q₈	—	2	2	no	—
44	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
45	C₅×#127	—	2	2	no	—
46	C₅×#127	C₂×C₄	1	4	yes	—
47	C₄×C₅×Q₈	—	2	8	no	—
48	C₄×C₅×D₄	—	2	8	no	—
49	C₄×C₅×D₄	—	4	8	no	—
50	C₂×C₅×C₄⋊C₄	—	2	2	no	—
51	C₂×C₅×C₄⋊C₄	—	2	2	no	—
52	C₂×C₅×C₄⋊C₄	—	2	2	no	—
53	C₂×C₅×C₄⋊C₄	—	2	2	no	—
54	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
57	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
58	C₂×C₄²×C₅	—	2	2	no	—
59	C₂×C₄²×C₅	D₄	1	2	yes	—
60	C₂×C₄²×C₅	D₄	1	2	yes	—
61	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
64	C₅×#107	—	2	8	no	—
65	C₅×#103	—	2	8	no	—
66	C₅×#102	—	2	8	no	—
67	C₅×#97	—	2	4	no	—
68	C₅×#97	D₄	1	8	yes	complemented
69	C₄×C₅×C₈	—	2	2	no	—
70	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
71	C₅×#95	—	2	4	no	—
72	C₂×#492	C₂×C₅	1	1	yes	characteristic
73	C₂×#440	C₂×C₅	1	4	yes	complemented
74	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
75	C₄×#127	C₂×C₅	1	1	yes	characteristic
76	#1973	C₂×C₅	1	4	yes	complemented
77	#1957	C₂×C₅	1	1	yes	characteristic
78	#1956	C₂×C₅	1	1	yes	characteristic, complemented
79	#1845	C₂×C₅	1	1	yes	characteristic, complemented
80	C₄×#97	C₂×C₅	1	1	yes	characteristic, complemented
81	C₂⁴×C₅	—	2	2	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	2	4	no	—
84	C₂×C₅×Q₈	—	2	4	no	—
85	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
86	C₂×C₅×D₄	—	2	8	no	—
87	C₂×C₅×D₄	—	2	4	no	—
88	C₂×C₅×D₄	—	2	4	no	—
89	C₂×C₅×D₄	—	2	2	no	—
90	C₂×C₅×D₄	—	4	4	no	—
91	C₂×C₅×D₄	—	4	4	no	—
92	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	—	2	2	no	—
95	C₂²×C₄×C₅	—	2	2	no	—
96	C₂²×C₄×C₅	—	4	8	no	—
97	C₂²×C₄×C₅	—	4	4	no	—
98	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
103	C₅×SD₁₆	—	2	32	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
107	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
108	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
109	C₂×C₅×C₈	#41	1	4	yes	—
110	C₅×C₄⋊C₄	—	2	8	no	—
111	C₅×C₄⋊C₄	—	2	4	no	—
112	C₅×C₄⋊C₄	—	2	4	no	—
113	C₅×C₄⋊C₄	—	4	4	no	—
114	C₅×C₄⋊C₄	—	4	4	no	—
115	C₅×C₄⋊C₄	—	4	4	no	—
116	C₅×C₂²⋊C₄	—	4	8	no	—
117	C₅×C₂²⋊C₄	—	4	4	no	—
118	C₄²×C₅	—	2	4	no	—
119	C₄²×C₅	—	2	4	no	—
120	C₄²×C₅	—	2	2	no	—
121	C₄²×C₅	—	4	4	no	—
122	C₄²×C₅	C₂²×C₄	1	2	yes	—
123	C₄²×C₅	C₂×D₄	1	4	yes	—
124	C₄²×C₅	#41	1	2	yes	—
125	C₂²×SD₁₆	C₂²×C₅	1	4	yes	—
126	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
127	C₂×#127	C₂²×C₅	1	1	yes	characteristic
128	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
129	C₂×C₄×D₄	—	2	2	no	—
130	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
131	#492	C₄×C₅	1	8	yes	complemented
132	#440	—	2	16	no	—
133	C₂×#107	C₂²×C₅	1	4	yes	—
134	C₂×#103	C₂²×C₅	1	4	yes	—
135	C₂×#102	C₂²×C₅	1	4	yes	—
136	C₂×#97	C₂²×C₅	1	2	yes	—
137	C₂×#97	C₂²×C₅	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
140	#391	C₂²×C₅	1	2	yes	—
141	#390	—	2	2	no	—
142	#390	C₂²×C₅	1	2	yes	—
143	#389	—	2	2	no	—
144	#384	C₂²×C₅	1	1	yes	characteristic
145	#384	C₂²×C₅	1	1	yes	characteristic
146	C₄×C₄⋊C₄	—	2	2	no	—
147	C₄³	C₂²×C₅	1	1	yes	characteristic
148	C₂³×C₅	—	2	4	no	—
149	C₂³×C₅	—	4	2	no	—
150	C₂³×C₅	—	4	2	no	—
151	C₂³×C₅	—	4	4	no	—
152	C₂³×C₅	—	4	4	no	—
153	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₅×Q₈	—	2	8	no	—
155	C₅×Q₈	—	4	4	no	—
156	C₅×D₄	—	2	8	no	—
157	C₅×D₄	—	4	4	no	—
158	C₅×D₄	—	4	16	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	4	2	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	4	2	no	—
168	C₂×C₄×C₅	—	4	2	no	—
169	C₂×C₄×C₅	—	4	4	no	—
170	C₂×C₄×C₅	—	4	4	no	—
171	C₂×C₄×C₅	—	8	8	no	—
172	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
173	C₂×C₄×C₅	#127	1	2	yes	—
174	C₂×C₄×C₅	#127	1	2	yes	—
175	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
176	C₂×C₄×C₅	#137	1	4	yes	—
177	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
179	C₂×C₄×C₅	C₂×#41	1	2	yes	—
180	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
181	C₅×C₈	—	2	4	no	—
182	C₅×C₈	C₄×D₄	1	8	yes	complemented
183	C₂²×Q₈	C₂³×C₅	1	2	yes	—
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	C₂³×C₅	1	2	yes	—
186	C₂³×C₄	—	2	2	no	—
187	C₂×SD₁₆	—	2	8	no	—
188	C₂×SD₁₆	—	2	8	no	—
189	C₂×SD₁₆	C₂×C₄×C₅	1	16	yes	complemented
190	C₂²×C₈	C₂³×C₅	1	2	yes	—
191	C₂²×C₈	C₂³×C₅	1	2	yes	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	complemented
194	#127	—	2	2	no	—
195	#127	C₂×C₄×C₅	1	4	yes	—
196	C₄×Q₈	—	2	8	no	—
197	C₄×D₄	—	2	8	no	—
198	C₄×D₄	—	4	8	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
205	C₂×C₂²⋊C₄	—	2	4	no	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	C₅×D₄	1	2	yes	—
209	C₂×C₄²	C₅×D₄	1	2	yes	—
210	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
213	#107	—	2	8	no	—
214	#103	—	2	8	no	—
215	#102	—	2	8	no	—
216	#97	—	2	4	no	—
217	#97	C₅×D₄	1	8	yes	complemented
218	C₄×C₈	—	2	2	no	—
219	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
220	#95	—	2	4	no	—
221	C₂²×C₅	—	4	4	no	—
222	C₂²×C₅	—	4	8	no	—
223	C₂²×C₅	—	8	2	no	—
224	C₂²×C₅	—	8	2	no	—
225	C₂²×C₅	—	8	4	no	—
226	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	C₂²×C₅	C₂×#127	1	1	yes	characteristic
228	C₂²×C₅	#532	1	1	yes	characteristic
229	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
230	C₂²×C₅	C₂×#137	1	2	yes	—
231	C₄×C₅	—	2	2	no	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	4	4	no	—
234	C₄×C₅	—	4	2	no	—
235	C₄×C₅	—	8	2	no	—
236	C₄×C₅	—	8	2	no	—
237	C₄×C₅	#440	1	8	yes	complemented
238	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
239	C₄×C₅	#532	1	2	yes	—
240	C₂⁴	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
245	C₂×D₄	—	2	8	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	2	4	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	4	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	8	no	—
256	C₂²×C₄	—	4	4	no	—
257	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
258	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
261	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
262	SD₁₆	—	2	32	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
266	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
267	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
268	C₂×C₈	C₅×#41	1	4	yes	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₂²⋊C₄	—	4	8	no	—
276	C₂²⋊C₄	—	4	4	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	4	4	no	—
281	C₄²	C₂²×C₄×C₅	1	2	yes	—
282	C₄²	C₂×C₅×D₄	1	4	yes	—
283	C₄²	C₅×#41	1	2	yes	—
284	C₂×C₅	—	8	4	no	—
285	C₂×C₅	C₄×#127	1	1	yes	characteristic
286	C₂×C₅	C₂×#440	1	2	yes	—
287	C₂×C₅	#3151	1	1	yes	characteristic
288	C₂×C₅	#3152	1	1	yes	characteristic
289	C₂×C₅	#3182	1	2	yes	—
290	C₂³	—	2	4	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	4	4	no	—
295	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
296	Q₈	—	2	8	no	—
297	Q₈	—	4	4	no	—
298	D₄	—	2	8	no	—
299	D₄	—	4	16	no	—
300	D₄	—	4	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	8	8	no	—
314	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
315	C₂×C₄	C₅×#127	1	2	yes	—
316	C₂×C₄	C₅×#127	1	2	yes	—
317	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
318	C₂×C₄	C₅×#137	1	4	yes	—
319	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
321	C₂×C₄	C₂×C₅×#41	1	2	yes	—
322	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
323	C₈	—	2	4	no	—
324	C₈	C₄×C₅×D₄	1	8	yes	complemented
325	C₅	#21692	1	1	yes	characteristic, direct product factor, Sylow
326	C₂²	—	4	4	no	—
327	C₂²	—	4	8	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	4	no	—
331	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
332	C₂²	C₂×C₅×#127	1	1	yes	characteristic
333	C₂²	C₅×#532	1	1	yes	characteristic
334	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
335	C₂²	C₂×C₅×#137	1	2	yes	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	4	4	no	—
339	C₄	—	4	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	C₅×#440	1	8	yes	complemented
343	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
344	C₄	C₅×#532	1	2	yes	—
345	C₂	—	8	4	no	—
346	C₂	C₄×C₅×#127	1	1	yes	characteristic
347	C₂	C₂×C₅×#440	1	2	yes	—
348	C₂	C₅×#3151	1	1	yes	characteristic
349	C₂	C₅×#3152	1	1	yes	characteristic
350	C₂	C₅×#3182	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#440	C₂	semidirect
3. C₄×C₅×C₄⋊Q₈	C₂	semidirect
5. C₅×#1973	C₂	semidirect
7. C₅×#1956	C₂	semidirect
8. C₅×#1845	C₂	semidirect
9. C₄×C₅×#97	C₂	semidirect
16. C₅×#492	C₄	semidirect
33. #21692	C₅	direct (N × H)
40. C₂×C₅×SD₁₆	C₂×C₄	semidirect
44. C₅×C₄⋊Q₈	C₂×C₄	semidirect
68. C₅×#97	D₄	semidirect
70. C₄×C₅×C₈	C₂×C₄	semidirect
73. C₂×#440	C₂×C₅	semidirect
74. C₄×C₄⋊Q₈	C₂×C₅	semidirect
76. #1973	C₂×C₅	semidirect
78. #1956	C₂×C₅	semidirect
79. #1845	C₂×C₅	semidirect
80. C₄×#97	C₂×C₅	semidirect
131. #492	C₄×C₅	semidirect
182. C₅×C₈	C₄×D₄	semidirect
189. C₂×SD₁₆	C₂×C₄×C₅	semidirect
193. C₄⋊Q₈	C₂×C₄×C₅	semidirect
217. #97	C₅×D₄	semidirect
219. C₄×C₈	C₂×C₄×C₅	semidirect
237. C₄×C₅	#440	semidirect
324. C₈	C₄×C₅×D₄	semidirect
325. C₅	#21692	direct (N × H)
342. C₄	C₅×#440	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#492	2
C₂×C₅×#440	2
C₄×C₅×C₄⋊Q₈	2
C₄×C₅×#127	2
C₅×#1973	2
C₅×#1957	2
C₅×#1956	2
C₅×#1845	2
C₄×C₅×#97	2
#21692	5

Subgroups — C5×#21692 #11944412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21692 #11944412