Subgroups — #10613552

Order 640.

572 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4064	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4061	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#4060	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#4057	C₂	1	1	yes	maximal, characteristic, complemented
5	D₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
6	#66002	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#495	C₂	1	1	yes	maximal, characteristic, complemented
8	#65479	C₂	1	1	yes	maximal, characteristic, complemented
9	#65465	C₂	1	1	yes	maximal, characteristic, complemented
10	#65456	C₂	1	1	yes	maximal, characteristic, complemented
11	#65443	C₂	1	1	yes	maximal, characteristic, complemented
12	#65348	C₂	1	1	yes	maximal, characteristic, complemented
13	#65348	C₂	1	1	yes	maximal, characteristic, complemented
14	#64985	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#676	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#713	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₅	C₂²	1	1	yes	characteristic
18	C₂×#711	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₅	C₂²	1	1	yes	characteristic
20	C₂×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₅×SD₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₅×SD₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₅×D₈	C₂²	1	1	yes	characteristic
24	C₅×#124	C₂²	1	1	yes	characteristic
25	C₅×#124	C₂²	1	1	yes	characteristic, complemented
26	#4093	C₂²	1	1	yes	characteristic
27	#4090	C₂²	1	1	yes	characteristic, complemented
28	C₂×#689	C₂²	1	1	yes	characteristic, complemented
29	C₂×#689	C₂²	1	1	yes	characteristic, complemented
30	C₂×#688	C₂²	1	1	yes	characteristic
31	C₂×#688	C₂²	1	1	yes	characteristic, complemented
32	#4086	C₂²	1	1	yes	characteristic, complemented
33	#4082	C₂²	1	1	yes	characteristic
34	C₂×#687	C₂²	1	1	yes	characteristic
35	C₂×#687	C₂²	1	1	yes	characteristic, complemented
36	C₂×#686	C₂²	1	1	yes	characteristic
37	C₂×#686	C₂²	1	1	yes	characteristic
38	#4064	—	2	4	no	—
39	#4061	—	2	4	no	—
40	#4060	—	2	4	no	—
41	#4057	—	2	4	no	—
42	C₂×Dic₂₀	C₂²	1	1	yes	characteristic, complemented
43	C₂×D₄₀	C₂²	1	1	yes	characteristic
44	C₂×#677	C₂²	1	1	yes	characteristic
45	C₂×#677	C₂²	1	1	yes	characteristic, complemented
46	C₂×#676	C₂²	1	1	yes	characteristic
47	C₂×#676	C₂²	1	1	yes	characteristic
48	#4031	—	2	2	no	—
49	#4030	—	2	2	no	—
50	#4027	—	2	2	no	—
51	D₅×C₂²⋊C₄	—	2	2	no	—
52	#4021	C₂²	1	1	yes	characteristic
53	#4021	C₂²	1	1	yes	characteristic, complemented
54	C₄²×D₅	C₂²	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
56	#3947	C₂²	1	1	yes	characteristic, complemented
57	#3947	C₂²	1	1	yes	characteristic, complemented
58	C₄×#168	C₂²	1	1	yes	characteristic, complemented
59	#3153	—	5	1	no	maximal, Sylow
60	C₂³×D₅	—	2	1	no	—
61	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
63	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
64	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₅	—	2	1	no	—
66	C₂×C₂²⋊D₅	—	2	1	no	—
67	C₂×C₂²⋊D₅	—	2	2	no	—
68	C₂²×Dic₅	—	2	1	no	—
69	#713	—	2	2	no	—
70	#713	—	2	2	no	—
71	Q₈×D₅	—	2	2	no	—
72	Q₈×D₅	—	2	2	no	—
73	#711	—	2	2	no	—
74	#711	—	2	2	no	—
75	D₄×D₅	—	2	2	no	—
76	D₄×D₅	—	2	2	no	—
77	C₂×D₂₀	—	2	1	no	—
78	C₂×D₂₀	C₂³	1	1	yes	characteristic
79	C₂×D₂₀	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₅	—	2	1	no	—
81	C₂×C₄×D₅	—	2	1	no	—
82	C₂×C₄×D₅	D₄	1	2	yes	—
83	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
84	C₂×Dic₁₀	—	2	1	no	—
85	C₂×Dic₁₀	C₂³	1	1	yes	characteristic
86	C₂×Dic₁₀	C₂³	1	1	yes	characteristic
87	C₅×Q₁₆	—	2	2	no	—
88	C₅×SD₁₆	—	2	2	no	—
89	C₅×SD₁₆	—	2	2	no	—
90	C₅×D₈	—	2	2	no	—
91	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
93	C₅×C₂²⋊C₄	—	2	2	no	—
94	C₅×C₂²⋊C₄	—	2	2	no	—
95	C₄²×C₅	C₂³	1	1	yes	characteristic
96	C₂²⋊Dic₅	—	2	2	no	—
97	#689	—	2	2	no	—
98	#689	—	2	2	no	—
99	#688	—	2	2	no	—
100	#688	—	2	2	no	—
101	#687	—	2	2	no	—
102	#687	—	2	2	no	—
103	#686	—	2	2	no	—
104	#686	—	2	2	no	—
105	#685	—	2	2	no	—
106	#685	—	2	2	no	—
107	#685	—	2	2	no	—
108	#683	—	2	2	no	—
109	C₄×Dic₅	D₄	1	2	yes	—
110	C₄×Dic₅	C₂³	1	1	yes	characteristic
111	C₂×#168	C₂³	1	1	yes	characteristic
112	C₂×#168	C₂³	1	1	yes	characteristic
113	Dic₂₀	—	2	2	no	—
114	D₄₀	—	2	2	no	—
115	#677	—	2	2	no	—
116	#677	—	2	2	no	—
117	#676	D₄	1	4	yes	complemented
118	#676	D₄	1	4	yes	complemented
119	C₂×#137	—	5	1	no	—
120	C₂×#137	—	5	1	no	—
121	C₂×AGL(1,ℤ/8)	—	5	1	no	—
122	C₂×AGL(1,ℤ/8)	—	5	1	no	—
123	#532	—	5	1	no	—
124	C₂×#124	—	5	1	no	—
125	#497	—	5	1	no	—
126	#497	—	5	1	no	—
127	#496	—	5	1	no	—
128	#496	—	5	1	no	—
129	#495	—	5	1	no	—
130	#495	—	5	1	no	—
131	#495	—	5	1	no	—
132	#495	D₅	1	1	yes	characteristic, complemented
133	C₄×M₁₆	—	5	1	no	—
134	C₂³×C₅	—	2	1	no	—
135	C₂³×C₅	—	2	1	no	—
136	C₂²×D₅	—	2	1	no	—
137	C₂²×D₅	—	2	1	no	—
138	C₂²×D₅	—	4	4	no	—
139	C₂²×D₅	—	4	2	no	—
140	C₂²×D₅	C₂×D₄	1	1	yes	characteristic
141	C₅×Q₈	—	2	1	no	—
142	C₅×Q₈	—	2	1	no	—
143	C₅×Q₈	—	2	1	no	—
144	C₅×Q₈	—	2	1	no	—
145	C₅×D₄	—	2	1	no	—
146	C₅×D₄	—	2	1	no	—
147	C₅×D₄	—	2	1	no	—
148	C₅×D₄	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
152	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
153	C₂²⋊D₅	—	4	4	no	—
154	C₂²⋊D₅	—	4	2	no	—
155	C₂²⋊D₅	—	4	2	no	—
156	C₂×Dic₅	—	2	1	no	—
157	C₂×Dic₅	—	2	1	no	—
158	C₂×Dic₅	—	4	2	no	—
159	C₂×Dic₅	C₂×D₄	1	2	yes	—
160	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
161	D₂₀	—	2	1	no	—
162	D₂₀	—	2	1	no	—
163	D₂₀	—	2	1	no	—
164	D₂₀	—	2	1	no	—
165	D₂₀	—	4	2	no	—
166	C₄×D₅	—	4	2	no	—
167	C₄×D₅	—	4	2	no	—
168	C₄×D₅	—	4	2	no	—
169	C₄×D₅	C₂×D₄	1	2	yes	—
170	C₄×D₅	C₂×D₄	1	2	yes	—
171	Dic₁₀	—	2	1	no	—
172	Dic₁₀	—	2	1	no	—
173	Dic₁₀	—	2	1	no	—
174	Dic₁₀	—	2	1	no	—
175	Dic₁₀	—	4	2	no	—
176	C₅×C₈	C₂×D₄	1	2	yes	—
177	C₅×C₈	C₂×D₄	1	2	yes	—
178	#168	C₂×D₄	1	2	yes	—
179	#168	C₂×D₄	1	2	yes	—
180	C₂×#41	—	5	1	no	—
181	C₂×#41	—	5	1	no	—
182	C₂²×Q₈	—	5	1	no	—
183	C₂²×D₄	—	5	1	no	—
184	#137	—	10	4	no	—
185	#137	—	10	4	no	—
186	AGL(1,ℤ/8)	—	10	4	no	—
187	AGL(1,ℤ/8)	—	10	4	no	—
188	C₂×Q₁₆	—	5	1	no	—
189	C₂×Q₁₆	—	5	1	no	—
190	C₂×Q₁₆	—	5	1	no	—
191	C₂×Q₁₆	C₂×D₅	1	1	yes	characteristic, complemented
192	C₂×SD₁₆	—	5	1	no	—
193	C₂×SD₁₆	—	5	1	no	—
194	C₂×SD₁₆	—	5	1	no	—
195	C₂×SD₁₆	—	5	1	no	—
196	C₂×SD₁₆	—	5	1	no	—
197	C₂×SD₁₆	—	5	1	no	—
198	C₂×SD₁₆	C₂×D₅	1	1	yes	characteristic
199	C₂×SD₁₆	C₂×D₅	1	1	yes	characteristic, complemented
200	C₂×D₈	—	5	1	no	—
201	C₂×D₈	—	5	1	no	—
202	C₂×D₈	—	5	1	no	—
203	C₂×D₈	C₂×D₅	1	1	yes	characteristic
204	C₂×M₁₆	—	5	1	no	—
205	C₂×M₁₆	—	5	1	no	—
206	C₄⋊Q₈	—	5	1	no	—
207	#127	—	5	1	no	—
208	#124	—	5	1	no	—
209	#124	—	5	1	no	—
210	#124	—	5	1	no	—
211	#124	—	5	1	no	—
212	#124	—	10	2	no	—
213	#124	C₂×D₅	1	1	yes	characteristic
214	#124	C₂×D₅	1	1	yes	characteristic, complemented
215	#121	—	10	2	no	—
216	C₄×D₄	—	10	2	no	—
217	C₂×C₂²⋊C₄	—	10	2	no	—
218	C₂×C₄²	—	5	1	no	—
219	#97	—	5	1	no	—
220	#97	—	5	1	no	—
221	C₄×C₈	—	5	1	no	—
222	C₄×C₈	C₂×D₅	1	1	yes	characteristic, complemented
223	C₂²×C₅	—	4	1	no	—
224	C₂²×C₅	—	4	1	no	—
225	C₂²×C₅	—	4	2	no	—
226	C₂²×C₅	—	4	2	no	—
227	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
228	C₂×D₅	—	2	2	no	—
229	C₂×D₅	—	4	1	no	—
230	C₂×D₅	—	4	1	no	—
231	C₂×D₅	—	4	2	no	—
232	C₂×D₅	—	4	2	no	—
233	C₂×D₅	—	8	4	no	—
234	C₂×D₅	#127	1	2	yes	—
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	4	1	no	—
237	C₄×C₅	—	4	1	no	—
238	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
239	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
240	Dic₅	—	2	2	no	—
241	Dic₅	—	4	1	no	—
242	Dic₅	—	4	1	no	—
243	Dic₅	#127	1	2	yes	—
244	C₂⁴	—	10	1	no	—
245	#41	—	10	2	no	—
246	#41	—	10	2	no	—
247	#41	—	10	2	no	—
248	#41	—	10	2	no	—
249	C₂×Q₈	—	5	1	no	—
250	C₂×Q₈	—	5	1	no	—
251	C₂×Q₈	—	10	1	no	—
252	C₂×Q₈	—	10	2	no	—
253	C₂×Q₈	—	10	2	no	—
254	C₂×Q₈	C₂²×D₅	1	1	yes	characteristic
255	C₂×Q₈	C₂²×D₅	1	1	yes	characteristic
256	C₂×D₄	—	5	1	no	—
257	C₂×D₄	—	5	1	no	—
258	C₂×D₄	—	10	1	no	—
259	C₂×D₄	—	10	2	no	—
260	C₂×D₄	—	10	2	no	—
261	C₂×D₄	—	10	2	no	—
262	C₂×D₄	—	10	1	no	—
263	C₂×D₄	—	10	1	no	—
264	C₂×D₄	C₂²×D₅	1	1	yes	characteristic
265	C₂×D₄	C₂²×D₅	1	1	yes	characteristic
266	C₂²×C₄	—	5	1	no	—
267	C₂²×C₄	—	5	2	no	—
268	C₂²×C₄	—	10	1	no	—
269	C₂²×C₄	—	10	1	no	—
270	C₂²×C₄	—	10	1	no	—
271	Q₁₆	—	2	2	no	—
272	Q₁₆	—	10	2	no	—
273	Q₁₆	—	10	2	no	—
274	Q₁₆	—	10	2	no	—
275	SD₁₆	—	2	2	no	—
276	SD₁₆	—	2	2	no	—
277	SD₁₆	—	10	2	no	—
278	SD₁₆	—	10	2	no	—
279	SD₁₆	—	10	2	no	—
280	SD₁₆	—	10	2	no	—
281	SD₁₆	—	10	2	no	—
282	SD₁₆	—	10	2	no	—
283	D₈	—	2	2	no	—
284	D₈	—	10	2	no	—
285	D₈	—	10	2	no	—
286	D₈	—	10	2	no	—
287	M₁₆	—	5	4	no	—
288	M₁₆	—	5	4	no	—
289	C₂×C₈	—	5	1	no	—
290	C₂×C₈	—	5	1	no	—
291	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
292	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
293	C₄⋊C₄	—	10	2	no	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	10	2	no	—
297	C₂²⋊C₄	—	10	2	no	—
298	C₂²⋊C₄	—	10	2	no	—
299	C₂²⋊C₄	—	10	2	no	—
300	C₄²	—	5	2	no	—
301	C₄²	—	5	1	no	—
302	C₄²	C₂²×D₅	1	1	yes	characteristic
303	C₂×C₅	—	8	1	no	—
304	C₂×C₅	—	8	1	no	—
305	C₂×C₅	C₂×#127	1	1	yes	characteristic
306	C₂×C₅	#575	1	2	yes	—
307	D₅	—	2	2	no	—
308	D₅	—	8	1	no	—
309	D₅	—	8	1	no	—
310	C₂³	—	2	1	no	—
311	C₂³	—	2	1	no	—
312	C₂³	—	5	1	no	—
313	C₂³	—	10	1	no	—
314	C₂³	—	10	1	no	—
315	C₂³	—	20	2	no	—
316	C₂³	—	20	4	no	—
317	Q₈	—	2	1	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	10	1	no	—
322	Q₈	—	10	1	no	—
323	Q₈	—	10	1	no	—
324	Q₈	—	10	1	no	—
325	Q₈	—	20	2	no	—
326	D₄	—	2	1	no	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	10	1	no	—
331	D₄	—	10	1	no	—
332	D₄	—	10	1	no	—
333	D₄	—	10	1	no	—
334	D₄	—	20	2	no	—
335	D₄	—	20	2	no	—
336	D₄	—	20	4	no	—
337	D₄	—	20	2	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	5	1	no	—
341	C₂×C₄	—	5	2	no	—
342	C₂×C₄	—	5	2	no	—
343	C₂×C₄	—	5	2	no	—
344	C₂×C₄	—	10	1	no	—
345	C₂×C₄	—	10	1	no	—
346	C₂×C₄	—	20	2	no	—
347	C₂×C₄	—	20	2	no	—
348	C₂×C₄	—	20	2	no	—
349	C₂×C₄	—	20	2	no	—
350	C₂×C₄	D₄×D₅	1	2	yes	—
351	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
352	C₈	—	5	2	no	—
353	C₈	—	5	2	no	—
354	C₈	D₄×D₅	1	2	yes	—
355	C₈	D₄×D₅	1	2	yes	—
356	C₅	#3153	1	1	yes	characteristic, complemented, Sylow
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	5	2	no	—
362	C₂²	—	10	2	no	—
363	C₂²	—	20	1	no	—
364	C₂²	—	20	1	no	—
365	C₂²	—	20	2	no	—
366	C₂²	—	20	2	no	—
367	C₂²	—	40	4	no	—
368	C₂²	C₂×D₄×D₅	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	5	2	no	—
373	C₄	—	10	2	no	—
374	C₄	—	20	1	no	—
375	C₄	—	20	1	no	—
376	C₄	C₂×D₄×D₅	1	1	yes	characteristic
377	C₄	C₂×D₄×D₅	1	1	yes	characteristic
378	C₂	—	8	1	no	—
379	C₂	—	8	1	no	—
380	C₂	—	10	2	no	—
381	C₂	—	40	1	no	—
382	C₂	—	40	1	no	—
383	C₂	D₅×#127	1	1	yes	characteristic
384	C₂	#66102	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₅
Fitting	C₅×#495	C₂
socle	C₂²×C₅	C₂²×D₄
hypercenter	#495	D₅
hypocenter	C₅	#3153
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

Complements

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

N	H ≅ G/N	type
1. C₂×#4064	C₂	semidirect
2. C₂×#4061	C₂	semidirect
3. C₂×#4060	C₂	semidirect
4. C₂×#4057	C₂	semidirect
5. D₅×#124	C₂	semidirect
6. #66002	C₂	semidirect
7. C₅×#495	C₂	semidirect
8. #65479	C₂	semidirect
9. #65465	C₂	semidirect
10. #65456	C₂	semidirect
11. #65443	C₂	semidirect
12. #65348	C₂	semidirect
13. #65348	C₂	semidirect
14. #64985	C₂	semidirect
15. C₄×#676	C₂	semidirect
20. C₂×C₅×Q₁₆	C₂²	semidirect
22. C₂×C₅×SD₁₆	C₂²	semidirect
25. C₅×#124	C₂²	semidirect
27. #4090	C₂²	semidirect
28. C₂×#689	C₂²	semidirect
29. C₂×#689	C₂²	semidirect
31. C₂×#688	C₂²	semidirect
32. #4086	C₂²	semidirect
35. C₂×#687	C₂²	semidirect
42. C₂×Dic₂₀	C₂²	semidirect
45. C₂×#677	C₂²	semidirect
53. #4021	C₂²	semidirect
55. C₄×C₅×C₈	C₂²	semidirect
56. #3947	C₂²	semidirect
57. #3947	C₂²	semidirect
58. C₄×#168	C₂²	semidirect
117. #676	D₄	semidirect
118. #676	D₄	semidirect
132. #495	D₅	semidirect
191. C₂×Q₁₆	C₂×D₅	semidirect
199. C₂×SD₁₆	C₂×D₅	semidirect
214. #124	C₂×D₅	semidirect
222. C₄×C₈	C₂×D₅	semidirect
356. C₅	#3153	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4064	2
C₂×#4061	2
C₂×#4060	2
C₂×#4057	2
D₅×#124	2
#66002	2
C₅×#495	2
#65479	2
#65465	2
#65456	2
#65443	2
#65348	2
#64985	2
C₄×#676	2
#3153	5