Subgroups — #10611192

Order 640.

388 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#711	C₂	1	1	yes	maximal, characteristic
2	C₅×#435	C₂	1	1	yes	maximal, characteristic, complemented
3	#65417	C₂	1	1	yes	maximal, characteristic, complemented
4	#65407	C₂	1	1	yes	maximal, characteristic, complemented
5	#65400	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#3946	C₂	1	1	yes	maximal, characteristic, complemented
7	#65300	C₂	1	1	yes	maximal, characteristic, complemented
8	#65126	C₂	1	1	yes	maximal, characteristic, complemented
9	#65117	C₂	1	1	yes	maximal, characteristic, complemented
10	#65019	C₂	1	1	yes	maximal, characteristic, complemented
11	#65018	C₂	1	1	yes	maximal, characteristic, complemented
12	#65008	C₂	1	1	yes	maximal, characteristic, complemented
13	#65007	C₂	1	1	yes	maximal, characteristic, complemented
14	#64994	C₂	1	1	yes	maximal, characteristic, complemented
15	#64990	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#711	C₄	1	1	yes	characteristic
17	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
20	D₄×Dic₅	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₅	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₅	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
25	C₂×#681	C₂²	1	1	yes	characteristic
26	C₂²×#168	C₂²	1	1	yes	characteristic
27	C₂×#676	C₂²	1	1	yes	characteristic
28	C₂×#676	C₂²	1	1	yes	characteristic
29	#4038	C₄	1	1	yes	characteristic
30	#4033	C₄	1	1	yes	characteristic
31	#4027	C₄	1	1	yes	characteristic
32	#4027	C₄	1	1	yes	characteristic
33	#4023	C₄	1	1	yes	characteristic
34	#4023	C₄	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×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₅×#97	C₂²	1	1	yes	characteristic
41	C₂²⋊#168	C₂²	1	1	yes	characteristic
42	C₂²⋊#168	C₂²	1	1	yes	characteristic
43	#3952	C₂²	1	1	yes	characteristic
44	#3952	C₂²	1	1	yes	characteristic
45	#3947	C₂²	1	1	yes	characteristic
46	#3947	C₂²	1	1	yes	characteristic
47	#3946	C₂²	1	1	yes	characteristic
48	#3946	C₂²	1	1	yes	characteristic
49	#3946	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	#3946	C₂²	1	1	yes	characteristic
55	C₈×Dic₅	C₂²	1	1	yes	characteristic
56	C₈×Dic₅	C₂²	1	1	yes	characteristic
57	C₄⋊#168	C₂²	1	1	yes	characteristic
58	#3935	C₂²	1	1	yes	characteristic
59	#2989	—	5	1	no	maximal, Sylow
60	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
67	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
68	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
69	#711	—	2	4	no	—
70	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
73	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
74	C₅×M₁₆	—	4	1	no	—
75	C₂×C₅×C₈	—	4	1	no	—
76	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
80	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₅	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
86	#685	C₂×C₄	1	1	yes	characteristic
87	#685	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	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₅	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	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
99	C₄×Dic₅	C₂³	1	1	yes	characteristic
100	C₄×Dic₅	C₂³	1	1	yes	characteristic
101	C₄×Dic₅	C₂³	1	1	yes	characteristic
102	#681	—	4	1	no	—
103	C₂×#168	—	4	1	no	—
104	C₂×#168	C₂³	1	1	yes	characteristic
105	C₂×#168	C₂³	1	1	yes	characteristic
106	C₂×#168	C₂³	1	1	yes	characteristic
107	C₂×#168	C₂³	1	1	yes	characteristic
108	#676	—	4	1	no	—
109	#676	—	4	1	no	—
110	C₄×#41	—	5	1	no	—
111	#446	—	5	1	no	—
112	#446	—	5	1	no	—
113	#436	—	5	1	no	—
114	#436	—	5	1	no	—
115	#435	—	5	1	no	—
116	#435	—	5	1	no	—
117	#435	—	5	1	no	—
118	#435	D₅	1	1	yes	characteristic, complemented
119	#432	—	5	1	no	—
120	#432	—	5	1	no	—
121	#431	—	5	1	no	—
122	#423	—	5	1	no	—
123	#423	—	5	1	no	—
124	C₂×C₄⋊C₈	—	5	1	no	—
125	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
126	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
127	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
128	C₅×D₄	—	2	2	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
138	C₂²⋊D₅	—	2	2	no	—
139	C₂²⋊D₅	—	2	2	no	—
140	C₂×Dic₅	—	2	2	no	—
141	C₂×Dic₅	—	4	1	no	—
142	C₂×Dic₅	—	4	1	no	—
143	C₂×Dic₅	M₁₆	1	4	yes	complemented
144	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
145	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
146	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
147	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
148	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
149	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
150	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
151	C₄×D₅	—	2	1	no	—
152	C₄×D₅	—	2	1	no	—
153	C₄×D₅	—	2	2	no	—
154	C₄×D₅	—	4	1	no	—
155	Dic₁₀	—	2	2	no	—
156	C₅×C₈	—	2	1	no	—
157	C₅×C₈	—	2	1	no	—
158	C₅×C₈	—	2	1	no	—
159	C₅×C₈	—	2	1	no	—
160	#168	—	2	1	no	—
161	#168	—	2	1	no	—
162	#168	—	2	1	no	—
163	#168	—	2	1	no	—
164	C₂×#41	—	5	1	no	—
165	C₂×M₁₆	—	5	1	no	—
166	C₂×M₁₆	—	5	1	no	—
167	C₂×M₁₆	—	5	1	no	—
168	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
169	C₂²×C₈	—	5	1	no	—
170	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
171	C₄×Q₈	—	5	1	no	—
172	C₄×Q₈	—	5	1	no	—
173	C₄×D₄	—	5	1	no	—
174	C₄×D₄	—	5	1	no	—
175	C₄×D₄	—	5	1	no	—
176	C₄×D₄	—	5	1	no	—
177	C₄×D₄	—	5	1	no	—
178	C₄×D₄	C₂×D₅	1	1	yes	characteristic
179	#117	—	5	1	no	—
180	#117	—	5	1	no	—
181	#117	—	5	1	no	—
182	C₂×C₄²	—	5	1	no	—
183	C₂×C₄²	—	5	1	no	—
184	C₂×C₄²	—	5	1	no	—
185	C₄⋊C₈	—	5	1	no	—
186	C₄⋊C₈	—	5	1	no	—
187	C₄⋊C₈	—	5	1	no	—
188	C₄⋊C₈	—	5	1	no	—
189	C₄⋊C₈	—	5	1	no	—
190	C₄⋊C₈	—	5	1	no	—
191	C₄⋊C₈	—	5	1	no	—
192	C₄⋊C₈	—	5	1	no	—
193	C₄⋊C₈	—	5	1	no	—
194	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
195	C₂²⋊C₈	—	5	1	no	—
196	C₂²⋊C₈	—	5	1	no	—
197	C₂²⋊C₈	—	5	1	no	—
198	C₂²⋊C₈	—	5	1	no	—
199	C₂²⋊C₈	C₂×D₅	1	1	yes	characteristic
200	C₂²⋊C₈	C₂×D₅	1	1	yes	characteristic
201	#97	—	5	1	no	—
202	#97	—	5	1	no	—
203	#97	—	5	1	no	—
204	#97	C₂×D₅	1	1	yes	characteristic
205	C₄×C₈	—	5	1	no	—
206	C₄×C₈	—	5	1	no	—
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	C₂×M₁₆	1	2	yes	—
213	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
214	C₂×D₅	—	2	1	no	—
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	—	2	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
222	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
223	Dic₅	—	2	1	no	—
224	Dic₅	—	2	1	no	—
225	Dic₅	—	2	1	no	—
226	Dic₅	—	2	1	no	—
227	Dic₅	—	2	1	no	—
228	Dic₅	C₂×M₁₆	1	2	yes	—
229	Dic₅	C₂×M₁₆	1	2	yes	—
230	#41	—	10	4	no	—
231	C₂×Q₈	—	5	1	no	—
232	C₂×D₄	—	5	1	no	—
233	C₂×D₄	—	5	1	no	—
234	C₂×D₄	C₄×D₅	1	1	yes	characteristic
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	C₂²×C₄	—	5	1	no	—
238	C₂²×C₄	—	5	1	no	—
239	C₂²×C₄	—	5	1	no	—
240	C₂²×C₄	—	5	1	no	—
241	C₂²×C₄	—	5	1	no	—
242	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
244	M₁₆	—	4	1	no	—
245	M₁₆	—	20	1	no	—
246	M₁₆	—	20	1	no	—
247	M₁₆	—	20	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	5	1	no	—
250	C₂×C₈	—	5	1	no	—
251	C₂×C₈	—	5	1	no	—
252	C₂×C₈	—	5	1	no	—
253	C₂×C₈	—	20	1	no	—
254	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
258	C₄⋊C₄	—	5	1	no	—
259	C₄⋊C₄	—	5	1	no	—
260	C₄⋊C₄	—	5	1	no	—
261	C₄⋊C₄	—	5	1	no	—
262	C₄⋊C₄	—	5	1	no	—
263	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
264	C₂²⋊C₄	—	5	1	no	—
265	C₂²⋊C₄	—	5	1	no	—
266	C₂²⋊C₄	—	5	1	no	—
267	C₂²⋊C₄	—	5	1	no	—
268	C₂²⋊C₄	C₄×D₅	1	1	yes	characteristic
269	C₂²⋊C₄	C₄×D₅	1	1	yes	characteristic
270	C₄²	—	5	1	no	—
271	C₄²	—	5	1	no	—
272	C₄²	—	5	1	no	—
273	C₄²	—	5	1	no	—
274	C₄²	—	5	1	no	—
275	C₄²	—	5	1	no	—
276	C₄²	—	5	1	no	—
277	C₄²	—	5	1	no	—
278	C₄²	—	5	1	no	—
279	C₄²	C₂²×D₅	1	1	yes	characteristic
280	C₂×C₅	—	2	2	no	—
281	C₂×C₅	—	4	1	no	—
282	C₂×C₅	#520	1	1	yes	characteristic
283	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
284	C₂×C₅	#568	1	1	yes	characteristic
285	D₅	—	4	1	no	—
286	C₂³	—	5	1	no	—
287	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
288	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
289	Q₈	—	10	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	10	2	no	—
292	D₄	—	10	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	5	1	no	—
298	C₂×C₄	—	5	1	no	—
299	C₂×C₄	—	5	4	no	—
300	C₂×C₄	—	5	1	no	—
301	C₂×C₄	—	5	1	no	—
302	C₂×C₄	—	5	1	no	—
303	C₂×C₄	—	5	1	no	—
304	C₂×C₄	—	5	1	no	—
305	C₂×C₄	—	10	2	no	—
306	C₂×C₄	—	10	2	no	—
307	C₂×C₄	—	10	1	no	—
308	C₂×C₄	—	10	1	no	—
309	C₂×C₄	—	20	1	no	—
310	C₂×C₄	—	20	1	no	—
311	C₂×C₄	—	20	1	no	—
312	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	10	1	no	—
322	C₈	—	10	1	no	—
323	C₈	—	10	1	no	—
324	C₈	—	10	1	no	—
325	C₅	#2989	1	1	yes	characteristic, complemented, Sylow
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	10	1	no	—
332	C₂²	—	10	1	no	—
333	C₂²	—	10	1	no	—
334	C₂²	D₅×M₁₆	1	2	yes	—
335	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	5	2	no	—
341	C₄	—	5	2	no	—
342	C₄	—	10	1	no	—
343	C₄	—	10	1	no	—
344	C₄	—	10	1	no	—
345	C₄	—	10	1	no	—
346	C₄	—	10	1	no	—
347	C₄	#4144	1	1	yes	characteristic
348	C₄	#4150	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	20	1	no	—
352	C₂	#65880	1	1	yes	characteristic
353	C₂	C₂×D₅×M₁₆	1	1	yes	characteristic
354	C₂	#66084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#435	C₂
prime radical	C₂×#711	C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#435	D₅
hypocenter	C₅	#2989
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. C₅×#435	C₂	semidirect
3. #65417	C₂	semidirect
4. #65407	C₂	semidirect
5. #65400	C₂	semidirect
6. C₂×#3946	C₂	semidirect
7. #65300	C₂	semidirect
8. #65126	C₂	semidirect
9. #65117	C₂	semidirect
10. #65019	C₂	semidirect
11. #65018	C₂	semidirect
12. #65008	C₂	semidirect
13. #65007	C₂	semidirect
14. #64994	C₂	semidirect
15. #64990	C₂	semidirect
118. #435	D₅	semidirect
143. C₂×Dic₅	M₁₆	semidirect
325. C₅	#2989	semidirect

Maximal subgroups

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

subgroup	index
C₄×#711	2
C₅×#435	2
#65417	2
#65407	2
#65400	2
C₂×#3946	2
#65300	2
#65126	2
#65117	2
#65019	2
#65018	2
#65008	2
#65007	2
#64994	2
#64990	2
#2989	5