Subgroups — C₅×#13246 #11935966

Order 1280.

422 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic
5	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1979	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1978	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×SD₃₂	—	2	1	no	—
11	C₂×C₅×SD₃₂	—	2	1	no	—
12	C₂×C₅×SD₃₂	—	2	2	no	—
13	C₂×C₅×SD₃₂	C₄	1	4	yes	complemented
14	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₅×#108	C₂²	1	1	yes	characteristic
17	C₂×C₅×#108	C₂²	1	1	yes	characteristic
18	C₅×#419	—	2	1	no	—
19	C₅×#418	—	2	1	no	—
20	C₂×C₅×#101	—	2	1	no	—
21	C₂×C₅×#100	—	2	1	no	—
22	C₅×#405	C₂²	1	1	yes	characteristic
23	C₅×#368	—	2	2	no	—
24	C₅×#362	—	2	2	no	—
25	C₅×#361	—	2	2	no	—
26	C₅×#346	—	2	1	no	—
27	C₄×C₅×C₁₆	—	2	1	no	—
28	#13246	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂²×C₅×Q₈	—	2	1	no	—
30	C₂²×C₅×D₄	—	2	1	no	—
31	C₂×C₅×Q₁₆	—	2	1	no	—
32	C₂×C₅×Q₁₆	—	2	1	no	—
33	C₂×C₅×Q₁₆	—	2	1	no	—
34	C₂×C₅×Q₁₆	—	2	1	no	—
35	C₂×C₅×Q₁₆	—	2	1	no	—
36	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₅×D₈	—	2	1	no	—
38	C₂×C₅×D₈	—	2	1	no	—
39	C₂×C₅×D₈	—	2	1	no	—
40	C₂×C₅×D₈	—	2	1	no	—
41	C₂×C₅×D₈	—	2	1	no	—
42	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
43	C₂×C₅×M₁₆	—	2	1	no	—
44	C₂×C₅×M₁₆	—	2	1	no	—
45	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₅×#117	D₄	1	1	yes	characteristic
48	C₅×SD₃₂	—	2	4	no	—
49	C₅×SD₃₂	—	2	4	no	—
50	C₅×M₃₂	—	2	1	no	—
51	C₅×M₃₂	D₄	1	2	yes	—
52	C₂×C₅×C₁₆	—	2	1	no	—
53	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
54	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
55	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
56	C₂×C₅×C₁₆	D₄	1	2	yes	—
57	C₅×#108	—	2	1	no	—
58	C₅×#108	—	2	1	no	—
59	C₅×#108	—	2	1	no	—
60	C₅×#108	—	2	1	no	—
61	C₅×#103	—	4	2	no	—
62	C₅×#102	—	4	2	no	—
63	C₅×#101	—	4	2	no	—
64	C₅×#100	—	4	2	no	—
65	C₅×#97	—	2	1	no	—
66	C₄×C₅×C₈	—	2	1	no	—
67	C₂²×SD₃₂	C₂×C₅	1	1	yes	characteristic
68	C₂×#368	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂×#362	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂×#361	C₂×C₅	1	1	yes	characteristic
71	#2108	C₂×C₅	1	1	yes	characteristic, complemented
72	#1979	C₂×C₅	1	1	yes	characteristic, complemented
73	#1978	C₂×C₅	1	1	yes	characteristic
74	C₂⁴×C₅	—	4	1	no	—
75	C₂×C₅×Q₈	—	2	2	no	—
76	C₂×C₅×Q₈	—	4	1	no	—
77	C₂×C₅×Q₈	—	4	1	no	—
78	C₂×C₅×Q₈	—	4	1	no	—
79	C₂×C₅×Q₈	—	4	1	no	—
80	C₂×C₅×Q₈	—	4	1	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂×C₅×D₄	—	4	1	no	—
84	C₂×C₅×D₄	—	4	1	no	—
85	C₂×C₅×D₄	—	4	1	no	—
86	C₂×C₅×D₄	—	4	1	no	—
87	C₂²×C₄×C₅	—	4	1	no	—
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₅×Q₁₆	—	2	2	no	—
90	C₅×Q₁₆	—	2	2	no	—
91	C₅×Q₁₆	—	4	1	no	—
92	C₅×Q₁₆	—	4	1	no	—
93	C₅×D₈	—	2	2	no	—
94	C₅×D₈	—	2	2	no	—
95	C₅×D₈	—	4	1	no	—
96	C₅×D₈	—	4	1	no	—
97	C₅×M₁₆	—	2	1	no	—
98	C₅×M₁₆	—	4	1	no	—
99	C₅×M₁₆	—	4	1	no	—
100	C₅×M₁₆	—	4	1	no	—
101	C₅×M₁₆	—	4	1	no	—
102	C₅×M₁₆	D₈	1	2	yes	—
103	C₂×C₅×C₈	—	2	1	no	—
104	C₂×C₅×C₈	—	4	1	no	—
105	C₂×C₅×C₈	—	4	1	no	—
106	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₅×C₈	#41	1	1	yes	characteristic
111	C₂×C₅×C₈	#41	1	1	yes	characteristic
112	C₅×C₄⋊C₄	D₈	1	2	yes	—
113	C₅×C₂²⋊C₄	—	2	1	no	—
114	C₄²×C₅	—	2	1	no	—
115	C₅×C₁₆	—	2	2	no	—
116	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
117	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
118	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
119	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
120	C₂×SD₃₂	—	2	1	no	—
121	C₂×SD₃₂	—	2	1	no	—
122	C₂×SD₃₂	—	2	2	no	—
123	C₂×SD₃₂	C₄×C₅	1	4	yes	complemented
124	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
125	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
126	C₂×#108	C₂²×C₅	1	1	yes	characteristic
127	C₂×#108	C₂²×C₅	1	1	yes	characteristic
128	#419	—	2	1	no	—
129	#418	—	2	1	no	—
130	C₂×#101	—	2	1	no	—
131	C₂×#100	—	2	1	no	—
132	#405	C₂²×C₅	1	1	yes	characteristic
133	#368	—	2	2	no	—
134	#362	—	2	2	no	—
135	#361	—	2	2	no	—
136	#346	—	2	1	no	—
137	C₄×C₁₆	—	2	1	no	—
138	C₂³×C₅	—	4	2	no	—
139	C₂³×C₅	—	8	1	no	—
140	C₂³×C₅	—	8	1	no	—
141	C₂³×C₅	—	16	2	no	—
142	C₂³×C₅	#135	1	1	yes	characteristic
143	C₅×Q₈	—	4	2	no	—
144	C₅×Q₈	—	4	2	no	—
145	C₅×Q₈	—	8	1	no	—
146	C₅×Q₈	—	8	1	no	—
147	C₅×D₄	—	4	2	no	—
148	C₅×D₄	—	4	2	no	—
149	C₅×D₄	—	8	1	no	—
150	C₅×D₄	—	8	1	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	8	1	no	—
154	C₂×C₄×C₅	—	8	1	no	—
155	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅	#121	1	1	yes	characteristic
157	C₂×C₄×C₅	#127	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₅	#135	1	1	yes	characteristic
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	—	8	1	no	—
163	C₅×C₈	—	8	1	no	—
164	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₅×C₈	C₄×D₄	1	1	yes	characteristic
166	C₅×C₈	#121	1	1	yes	characteristic
167	C₅×C₈	#124	1	1	yes	characteristic
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂×Q₁₆	—	2	1	no	—
171	C₂×Q₁₆	—	2	1	no	—
172	C₂×Q₁₆	—	2	1	no	—
173	C₂×Q₁₆	—	2	1	no	—
174	C₂×Q₁₆	—	2	1	no	—
175	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	—	2	1	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
186	#117	C₅×D₄	1	1	yes	characteristic
187	SD₃₂	—	2	4	no	—
188	SD₃₂	—	2	4	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	C₅×D₄	1	2	yes	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
193	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
194	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
195	C₂×C₁₆	C₅×D₄	1	2	yes	—
196	#108	—	2	1	no	—
197	#108	—	2	1	no	—
198	#108	—	2	1	no	—
199	#108	—	2	1	no	—
200	#103	—	4	2	no	—
201	#102	—	4	2	no	—
202	#101	—	4	2	no	—
203	#100	—	4	2	no	—
204	#97	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₂²×C₅	—	2	2	no	—
207	C₂²×C₅	—	8	2	no	—
208	C₂²×C₅	—	16	2	no	—
209	C₂²×C₅	—	16	2	no	—
210	C₂²×C₅	—	16	2	no	—
211	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
212	C₂²×C₅	#466	1	1	yes	characteristic
213	C₂²×C₅	#495	1	1	yes	characteristic
214	C₄×C₅	—	4	2	no	—
215	C₄×C₅	—	8	2	no	—
216	C₄×C₅	#390	1	1	yes	characteristic
217	C₄×C₅	C₄×D₈	1	1	yes	characteristic
218	C₄×C₅	#466	1	1	yes	characteristic
219	C₄×C₅	#493	1	1	yes	characteristic
220	C₂⁴	—	4	1	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×Q₈	—	4	1	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	Q₁₆	—	4	1	no	—
238	Q₁₆	—	4	1	no	—
239	D₈	—	2	2	no	—
240	D₈	—	2	2	no	—
241	D₈	—	4	1	no	—
242	D₈	—	4	1	no	—
243	M₁₆	—	2	1	no	—
244	M₁₆	—	4	1	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	4	1	no	—
248	M₁₆	C₅×D₈	1	2	yes	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
253	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
254	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
255	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
256	C₂×C₈	C₅×#41	1	1	yes	characteristic
257	C₂×C₈	C₅×#41	1	1	yes	characteristic
258	C₄⋊C₄	C₅×D₈	1	2	yes	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₁₆	—	2	2	no	—
262	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
263	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
264	C₂×C₅	—	2	1	no	—
265	C₂×C₅	—	2	1	no	—
266	C₂×C₅	—	16	2	no	—
267	C₂×C₅	#1970	1	1	yes	characteristic
268	C₂×C₅	#2223	1	2	yes	—
269	C₂³	—	4	2	no	—
270	C₂³	—	8	1	no	—
271	C₂³	—	8	1	no	—
272	C₂³	—	16	2	no	—
273	C₂³	C₅×#135	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	8	1	no	—
277	Q₈	—	8	1	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	8	1	no	—
281	D₄	—	8	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
287	C₂×C₄	C₅×#121	1	1	yes	characteristic
288	C₂×C₄	C₅×#127	1	1	yes	characteristic
289	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
290	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
291	C₂×C₄	C₅×#135	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	8	1	no	—
294	C₈	—	8	1	no	—
295	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
296	C₈	C₄×C₅×D₄	1	1	yes	characteristic
297	C₈	C₅×#121	1	1	yes	characteristic
298	C₈	C₅×#124	1	1	yes	characteristic
299	C₅	#13246	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²	—	2	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	16	2	no	—
303	C₂²	—	16	2	no	—
304	C₂²	—	16	2	no	—
305	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
306	C₂²	C₅×#466	1	1	yes	characteristic
307	C₂²	C₅×#495	1	1	yes	characteristic
308	C₄	—	4	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₅×#390	1	1	yes	characteristic
311	C₄	C₄×C₅×D₈	1	1	yes	characteristic
312	C₄	C₅×#466	1	1	yes	characteristic
313	C₄	C₅×#493	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	16	2	no	—
317	C₂	C₅×#1970	1	1	yes	characteristic
318	C₂	C₅×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#368	C₂	semidirect
3. C₂×C₅×#362	C₂	semidirect
5. C₅×#2108	C₂	semidirect
6. C₅×#1979	C₂	semidirect
13. C₂×C₅×SD₃₂	C₄	semidirect
28. #13246	C₅	direct (N × H)
68. C₂×#368	C₂×C₅	semidirect
69. C₂×#362	C₂×C₅	semidirect
71. #2108	C₂×C₅	semidirect
72. #1979	C₂×C₅	semidirect
123. C₂×SD₃₂	C₄×C₅	semidirect
299. C₅	#13246	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×SD₃₂	2
C₂×C₅×#368	2
C₂×C₅×#362	2
C₂×C₅×#361	2
C₅×#2108	2
C₅×#1979	2
C₅×#1978	2
#13246	5

Subgroups — C5×#13246 #11935966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13246 #11935966