Subgroups — #10611113

Order 640.

346 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	#65862	C₂	1	1	yes	maximal, characteristic
2	C₅×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	#65288	C₂	1	1	yes	maximal, characteristic, complemented
4	#65129	C₂	1	1	yes	maximal, characteristic, complemented
5	#65128	C₂	1	1	yes	maximal, characteristic, complemented
6	#65126	C₂	1	1	yes	maximal, characteristic, complemented
7	#65124	C₂	1	1	yes	maximal, characteristic, complemented
8	#65019	C₂	1	1	yes	maximal, characteristic, complemented
9	#65019	C₂	1	1	yes	maximal, characteristic, complemented
10	#65018	C₂	1	1	yes	maximal, characteristic, complemented
11	#65017	C₂	1	1	yes	maximal, characteristic, complemented
12	#64995	C₂	1	1	yes	maximal, characteristic, complemented
13	#64990	C₂	1	1	yes	maximal, characteristic, complemented
14	#64977	C₂	1	1	yes	maximal, characteristic, complemented
15	#64968	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#117	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊D₅	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₅	C₂²	1	1	yes	characteristic
19	C₂×#681	C₂²	1	1	yes	characteristic
20	C₂²×#168	C₂²	1	1	yes	characteristic
21	C₂×#676	C₂²	1	1	yes	characteristic
22	C₂×#676	C₂²	1	1	yes	characteristic
23	C₂×#676	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
25	#4044	C₄	1	1	yes	characteristic
26	#4042	C₄	1	1	yes	characteristic
27	#4040	C₄	1	1	yes	characteristic
28	#4035	C₄	1	1	yes	characteristic
29	#4031	C₄	1	1	yes	characteristic
30	#4030	C₄	1	1	yes	characteristic
31	#4029	C₄	1	1	yes	characteristic
32	#4025	C₄	1	1	yes	characteristic
33	C₄×D₂₀	C₂²	1	1	yes	characteristic
34	#4018	C₂²	1	1	yes	characteristic
35	C₄²×D₅	C₂²	1	1	yes	characteristic
36	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
37	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₅×#97	C₂²	1	1	yes	characteristic
42	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
43	#3952	C₂²	1	1	yes	characteristic
44	#3952	C₂²	1	1	yes	characteristic
45	#3952	C₂²	1	1	yes	characteristic
46	#3952	C₂²	1	1	yes	characteristic
47	#3947	C₂²	1	1	yes	characteristic
48	#3947	C₂²	1	1	yes	characteristic
49	#3947	C₂²	1	1	yes	characteristic
50	#3946	C₂²	1	1	yes	characteristic
51	#3946	C₂²	1	1	yes	characteristic
52	#3946	C₂²	1	1	yes	characteristic
53	#3946	C₂²	1	1	yes	characteristic
54	C₈×Dic₅	C₂²	1	1	yes	characteristic
55	C₄⋊#168	C₂²	1	1	yes	characteristic
56	C₄⋊#168	C₂²	1	1	yes	characteristic
57	C₄⋊#168	C₂²	1	1	yes	characteristic
58	C₄⋊#168	C₂²	1	1	yes	characteristic
59	#2995	—	5	1	no	maximal, Sylow
60	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
63	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
76	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
77	C₄²×C₅	C₂³	1	1	yes	characteristic
78	C₄²×C₅	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
81	#685	C₂×C₄	1	1	yes	characteristic
82	#685	C₂×C₄	1	1	yes	characteristic
83	#685	C₂×C₄	1	1	yes	characteristic
84	#685	C₂×C₄	1	1	yes	characteristic
85	#685	C₂×C₄	1	1	yes	characteristic
86	#685	C₂×C₄	1	1	yes	characteristic
87	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
89	#683	C₂×C₄	1	1	yes	characteristic
90	#683	C₂×C₄	1	1	yes	characteristic
91	#683	C₂×C₄	1	1	yes	characteristic
92	#683	C₂×C₄	1	1	yes	characteristic
93	#683	C₂×C₄	1	1	yes	characteristic
94	#683	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₅	C₂³	1	1	yes	characteristic
98	C₄×Dic₅	C₂³	1	1	yes	characteristic
99	#681	—	4	1	no	—
100	C₂×#168	—	4	1	no	—
101	C₂×#168	C₂³	1	1	yes	characteristic
102	C₂×#168	C₂³	1	1	yes	characteristic
103	C₂×#168	C₂³	1	1	yes	characteristic
104	C₂×#168	C₂³	1	1	yes	characteristic
105	#676	—	4	1	no	—
106	#676	—	4	1	no	—
107	#676	—	4	1	no	—
108	C₈×D₅	—	4	1	no	—
109	#529	—	5	1	no	—
110	#446	—	5	1	no	—
111	#446	—	5	1	no	—
112	#436	—	5	1	no	—
113	#436	—	5	1	no	—
114	#435	—	5	1	no	—
115	#435	—	5	1	no	—
116	#435	—	5	1	no	—
117	#435	—	5	1	no	—
118	#433	—	5	1	no	—
119	#433	D₅	1	1	yes	characteristic, complemented
120	#432	—	5	1	no	—
121	#424	—	5	1	no	—
122	#424	—	5	1	no	—
123	#423	—	5	1	no	—
124	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
125	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
126	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
135	C₂²⋊D₅	—	4	1	no	—
136	C₂²⋊D₅	—	4	1	no	—
137	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
138	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
139	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
140	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
141	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
142	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
143	D₂₀	—	4	1	no	—
144	C₄×D₅	—	2	1	no	—
145	C₄×D₅	—	2	1	no	—
146	C₄×D₅	—	2	1	no	—
147	C₄×D₅	—	2	1	no	—
148	C₄×D₅	—	4	1	no	—
149	C₄×D₅	—	4	1	no	—
150	Dic₁₀	—	4	1	no	—
151	C₅×C₈	—	2	1	no	—
152	C₅×C₈	—	2	1	no	—
153	C₅×C₈	—	2	1	no	—
154	C₅×C₈	—	2	1	no	—
155	#168	—	2	1	no	—
156	#168	—	2	1	no	—
157	#168	—	2	1	no	—
158	#168	—	2	1	no	—
159	C₂×M₁₆	—	5	1	no	—
160	C₂×M₁₆	—	5	1	no	—
161	C₂×M₁₆	—	5	1	no	—
162	C₂×M₁₆	—	5	1	no	—
163	C₂²×C₈	—	5	1	no	—
164	C₂²×C₈	—	5	1	no	—
165	#126	—	5	1	no	—
166	#126	—	5	1	no	—
167	#125	—	5	1	no	—
168	#124	—	5	1	no	—
169	#123	—	5	1	no	—
170	#123	—	5	1	no	—
171	C₂²⋊Q₈	—	5	1	no	—
172	#121	—	5	1	no	—
173	C₄×Q₈	—	5	1	no	—
174	C₄×D₄	—	5	1	no	—
175	C₄×D₄	—	5	1	no	—
176	C₄×D₄	—	5	1	no	—
177	#117	—	5	1	no	—
178	#117	C₂×D₅	1	1	yes	characteristic
179	C₂×C₄²	—	5	1	no	—
180	C₄⋊C₈	—	5	1	no	—
181	C₄⋊C₈	—	5	1	no	—
182	C₄⋊C₈	—	5	1	no	—
183	C₄⋊C₈	—	5	1	no	—
184	C₄⋊C₈	—	5	1	no	—
185	C₄⋊C₈	—	5	1	no	—
186	C₄⋊C₈	—	5	1	no	—
187	C₄⋊C₈	—	5	1	no	—
188	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
189	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
190	C₂²⋊C₈	—	5	1	no	—
191	C₂²⋊C₈	—	5	1	no	—
192	C₂²⋊C₈	—	5	1	no	—
193	C₂²⋊C₈	—	5	1	no	—
194	C₂²⋊C₈	C₂×D₅	1	1	yes	characteristic
195	C₂²⋊C₈	C₂×D₅	1	1	yes	characteristic
196	#97	—	5	1	no	—
197	#97	—	5	1	no	—
198	#97	—	5	1	no	—
199	#97	C₂×D₅	1	1	yes	characteristic
200	C₄×C₈	—	5	1	no	—
201	C₄×C₈	C₂×D₅	1	1	yes	characteristic
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
206	C₂×D₅	—	2	1	no	—
207	C₂×D₅	—	2	1	no	—
208	C₂×D₅	—	2	1	no	—
209	C₂×D₅	—	2	1	no	—
210	C₂×D₅	—	2	1	no	—
211	C₂×D₅	—	2	1	no	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
218	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
219	Dic₅	—	2	1	no	—
220	Dic₅	—	2	1	no	—
221	Dic₅	—	2	1	no	—
222	Dic₅	—	2	1	no	—
223	Dic₅	—	2	1	no	—
224	Dic₅	—	2	1	no	—
225	C₂×Q₈	—	5	1	no	—
226	C₂×D₄	—	5	1	no	—
227	C₂×D₄	—	5	1	no	—
228	C₂×D₄	—	5	1	no	—
229	C₂²×C₄	—	5	1	no	—
230	C₂²×C₄	—	5	1	no	—
231	C₂²×C₄	—	5	1	no	—
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
234	M₁₆	—	20	1	no	—
235	M₁₆	—	20	1	no	—
236	M₁₆	—	20	1	no	—
237	M₁₆	—	20	1	no	—
238	C₂×C₈	—	5	1	no	—
239	C₂×C₈	—	5	1	no	—
240	C₂×C₈	—	5	1	no	—
241	C₂×C₈	—	5	1	no	—
242	C₂×C₈	—	20	1	no	—
243	C₂×C₈	—	20	1	no	—
244	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
248	C₄⋊C₄	—	5	1	no	—
249	C₄⋊C₄	—	5	1	no	—
250	C₄⋊C₄	—	5	1	no	—
251	C₄⋊C₄	—	5	1	no	—
252	C₄⋊C₄	—	5	1	no	—
253	C₄⋊C₄	—	5	1	no	—
254	C₄⋊C₄	—	5	1	no	—
255	C₄⋊C₄	—	5	1	no	—
256	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
257	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
258	C₂²⋊C₄	—	5	1	no	—
259	C₂²⋊C₄	—	5	1	no	—
260	C₂²⋊C₄	—	5	1	no	—
261	C₂²⋊C₄	—	5	1	no	—
262	C₂²⋊C₄	—	5	1	no	—
263	C₂²⋊C₄	—	5	1	no	—
264	C₂²⋊C₄	—	5	1	no	—
265	C₂²⋊C₄	—	5	1	no	—
266	C₂²⋊C₄	C₄×D₅	1	1	yes	characteristic
267	C₂²⋊C₄	C₄×D₅	1	1	yes	characteristic
268	C₄²	—	5	1	no	—
269	C₄²	—	5	1	no	—
270	C₄²	—	5	1	no	—
271	C₄²	—	5	1	no	—
272	C₄²	C₂²×D₅	1	1	yes	characteristic
273	C₄²	C₂²×D₅	1	1	yes	characteristic
274	C₂×C₅	—	4	1	no	—
275	C₂×C₅	#520	1	1	yes	characteristic
276	C₂×C₅	#568	1	1	yes	characteristic
277	C₂×C₅	#568	1	1	yes	characteristic
278	D₅	—	4	1	no	—
279	D₅	—	4	1	no	—
280	C₂³	—	5	1	no	—
281	C₂³	—	5	1	no	—
282	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
283	Q₈	—	20	1	no	—
284	D₄	—	20	1	no	—
285	D₄	—	20	1	no	—
286	D₄	—	20	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	5	1	no	—
290	C₂×C₄	—	5	1	no	—
291	C₂×C₄	—	5	1	no	—
292	C₂×C₄	—	5	1	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	10	1	no	—
296	C₂×C₄	—	10	1	no	—
297	C₂×C₄	—	10	1	no	—
298	C₂×C₄	—	10	1	no	—
299	C₂×C₄	—	20	1	no	—
300	C₂×C₄	—	20	1	no	—
301	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	10	1	no	—
312	C₈	—	10	1	no	—
313	C₈	—	10	1	no	—
314	C₈	—	10	1	no	—
315	C₅	#2995	1	1	yes	characteristic, complemented, Sylow
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	10	1	no	—
320	C₂²	—	10	1	no	—
321	C₂²	—	10	1	no	—
322	C₂²	—	10	1	no	—
323	C₂²	—	10	1	no	—
324	C₂²	—	10	1	no	—
325	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	10	1	no	—
332	C₄	—	10	1	no	—
333	C₄	—	10	1	no	—
334	C₄	—	10	1	no	—
335	C₄	—	10	1	no	—
336	C₄	—	10	1	no	—
337	C₄	#4149	1	1	yes	characteristic
338	C₄	#4150	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	20	1	no	—
341	C₂	—	20	1	no	—
342	C₂	#65857	1	1	yes	characteristic
343	C₂	#66084	1	1	yes	characteristic
344	C₂	#66084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#433	C₂
prime radical	#4030	C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#433	D₅
hypocenter	C₅	#2995
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. C₅×#433	C₂	semidirect
3. #65288	C₂	semidirect
4. #65129	C₂	semidirect
5. #65128	C₂	semidirect
6. #65126	C₂	semidirect
7. #65124	C₂	semidirect
8. #65019	C₂	semidirect
9. #65019	C₂	semidirect
10. #65018	C₂	semidirect
11. #65017	C₂	semidirect
12. #64995	C₂	semidirect
13. #64990	C₂	semidirect
14. #64977	C₂	semidirect
15. #64968	C₂	semidirect
119. #433	D₅	semidirect
315. C₅	#2995	semidirect

Maximal subgroups

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

subgroup	index
#65862	2
C₅×#433	2
#65288	2
#65129	2
#65128	2
#65126	2
#65124	2
#65019	2
#65018	2
#65017	2
#64995	2
#64990	2
#64977	2
#64968	2
#2995	5