Subgroups — #12069138

Order 1280.

662 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611019	C₂	1	1	yes	maximal, characteristic
2	#10610995	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#677	C₂	1	1	yes	maximal, characteristic
4	C₅×#1450	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601560	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601550	C₂	1	1	yes	maximal, characteristic
7	#10600522	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598875	C₂	1	4	yes	maximal, complemented
9	#10598848	C₂	1	4	yes	maximal, complemented
10	C₂²×#677	C₄	1	1	yes	characteristic
11	C₂×C₄×D₂₀	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₀	C₂²	1	1	yes	characteristic
13	C₅×#431	C₂²	1	1	yes	characteristic
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×#3953	C₄	1	1	yes	characteristic
17	C₂×#3949	C₄	1	1	yes	characteristic
18	C₂×#3948	C₄	1	1	yes	characteristic
19	#65214	C₂²	1	1	yes	characteristic
20	#65214	C₂²	1	1	yes	characteristic
21	#65127	—	2	2	no	—
22	#65120	—	2	2	no	—
23	#65015	—	2	2	no	—
24	#65008	—	2	2	no	—
25	C₄×#677	C₄	1	2	yes	—
26	C₄×#677	C₄	1	2	yes	—
27	C₄×#677	C₂²	1	2	yes	—
28	C₄×#677	C₂²	1	2	yes	—
29	#64977	C₂²	1	4	yes	—
30	#64968	C₂²	1	4	yes	complemented
31	C₅×#322	C₂²	1	2	yes	complemented
32	C₅×C₈²	C₂²	1	2	yes	complemented
33	#64679	C₂²	1	4	yes	—
34	#64678	C₂²	1	2	yes	complemented
35	#64677	C₂²	1	2	yes	complemented
36	#64676	C₂²	1	4	yes	complemented
37	#17896	—	5	1	no	maximal, Sylow
38	C₂²×D₂₀	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×D₅	—	2	1	no	—
40	C₂²×Dic₁₀	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×#685	—	2	1	no	—
45	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
47	C₂×#683	—	2	1	no	—
48	C₂×C₄×Dic₅	—	2	1	no	—
49	C₂×#677	—	2	4	no	—
50	C₂×#677	C₂×C₄	1	2	yes	—
51	C₂×#677	C₂×C₄	1	2	yes	—
52	C₂×#676	—	2	4	no	—
53	C₄×D₂₀	—	2	1	no	—
54	C₄×D₂₀	—	2	1	no	—
55	C₄×D₂₀	C₂×C₄	1	2	yes	—
56	C₄×D₂₀	C₂³	1	2	yes	—
57	C₄×Dic₁₀	—	2	1	no	—
58	C₄×Dic₁₀	—	2	1	no	—
59	C₄×Dic₁₀	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₀	C₂³	1	2	yes	—
61	C₅×C₄⋊C₈	—	2	1	no	—
62	C₅×C₄⋊C₈	D₄	1	2	yes	—
63	C₅×C₂²⋊C₈	—	2	1	no	—
64	C₅×C₂²⋊C₈	D₄	1	2	yes	—
65	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₅×C₈	C₂³	1	2	yes	—
68	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	2	yes	—
71	C₂²⋊#168	—	2	1	no	—
72	C₂²⋊#168	—	2	1	no	—
73	#3953	C₂×C₄	1	2	yes	—
74	#3953	C₂×C₄	1	2	yes	—
75	#3952	—	2	4	no	—
76	#3949	C₂×C₄	1	2	yes	—
77	#3949	C₂×C₄	1	2	yes	—
78	#3948	C₂×C₄	1	2	yes	—
79	#3948	C₂×C₄	1	2	yes	—
80	#3947	—	2	4	no	—
81	#3946	—	2	4	no	—
82	C₄⋊#168	C₂³	1	2	yes	—
83	C₄⋊#168	C₂³	1	2	yes	—
84	C₄×#168	—	2	1	no	—
85	C₄×#168	—	2	1	no	—
86	#2983	—	5	1	no	—
87	#2975	—	5	1	no	—
88	C₂×C₄×SD₁₆	—	5	1	no	—
89	#1592	—	5	4	no	—
90	#1590	—	5	4	no	—
91	#1564	—	5	1	no	—
92	#1479	—	5	1	no	—
93	#1478	—	5	1	no	—
94	#1450	D₅	1	1	yes	characteristic, complemented
95	C₂³×D₅	—	2	1	no	—
96	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	#41	1	1	yes	characteristic
99	C₂²×Dic₅	—	2	1	no	—
100	C₂²×Dic₅	—	2	1	no	—
101	C₂²×Dic₅	—	2	1	no	—
102	C₂×D₂₀	—	2	2	no	—
103	C₂×D₂₀	—	2	2	no	—
104	C₂×D₂₀	—	2	1	no	—
105	C₂×D₂₀	C₂²×C₄	1	2	yes	—
106	C₂×C₄×D₅	—	2	2	no	—
107	C₂×C₄×D₅	—	2	2	no	—
108	C₂×C₄×D₅	—	8	2	no	—
109	C₂×Dic₁₀	—	2	2	no	—
110	C₂×Dic₁₀	—	2	2	no	—
111	C₂×Dic₁₀	—	2	1	no	—
112	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	M₁₆	1	4	yes	complemented
116	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
121	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₅	#41	1	1	yes	characteristic
124	C₄²×C₅	C₂⁴	1	1	yes	characteristic
125	#685	—	2	2	no	—
126	#685	—	2	2	no	—
127	C₄⋊Dic₅	—	2	1	no	—
128	C₄⋊Dic₅	—	2	1	no	—
129	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
130	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
131	#683	—	2	2	no	—
132	#683	—	2	2	no	—
133	C₄×Dic₅	—	2	2	no	—
134	C₄×Dic₅	—	2	2	no	—
135	C₂×#168	—	2	2	no	—
136	C₂×#168	—	2	2	no	—
137	#677	—	2	4	no	—
138	#677	—	2	4	no	—
139	#676	—	8	4	no	—
140	C₂²×SD₁₆	—	5	1	no	—
141	C₂×C₄×Q₈	—	5	1	no	—
142	C₂×C₄×D₄	—	5	1	no	—
143	#446	—	5	4	no	—
144	#446	—	10	2	no	—
145	C₄×SD₁₆	—	5	2	no	—
146	C₄×SD₁₆	—	5	2	no	—
147	C₄×SD₁₆	—	5	2	no	—
148	C₄×SD₁₆	—	5	2	no	—
149	#436	—	5	4	no	—
150	#436	—	10	2	no	—
151	#432	—	10	2	no	—
152	#431	—	5	1	no	—
153	#431	—	5	1	no	—
154	#431	C₂×D₅	1	1	yes	characteristic
155	#431	C₂×D₅	1	1	yes	characteristic
156	C₂×#106	—	5	1	no	—
157	C₂×#103	—	5	1	no	—
158	C₂×#102	—	5	1	no	—
159	#407	—	10	2	no	—
160	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
161	#335	—	5	2	no	—
162	#334	—	5	2	no	—
163	Q₈⋊C₈	—	5	4	no	—
164	D₄⋊C₈	—	5	4	no	—
165	#322	C₂×D₅	1	2	yes	complemented
166	C₈²	C₂×D₅	1	2	yes	complemented
167	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
168	C₂²×D₅	—	2	2	no	—
169	C₂²×D₅	—	4	2	no	—
170	C₂²×D₅	—	8	1	no	—
171	C₂²×D₅	—	8	1	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	4	1	no	—
174	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
178	C₂×C₄×C₅	#131	1	2	yes	—
179	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
182	C₂×Dic₅	—	2	2	no	—
183	C₂×Dic₅	—	2	2	no	—
184	C₂×Dic₅	—	2	2	no	—
185	C₂×Dic₅	—	4	2	no	—
186	C₂×Dic₅	—	8	1	no	—
187	C₂×Dic₅	—	8	1	no	—
188	D₂₀	—	2	2	no	—
189	D₂₀	—	2	2	no	—
190	D₂₀	—	4	2	no	—
191	C₄×D₅	—	4	4	no	—
192	C₄×D₅	—	8	2	no	—
193	Dic₁₀	—	2	2	no	—
194	Dic₁₀	—	2	2	no	—
195	Dic₁₀	—	4	2	no	—
196	C₅×C₈	—	2	1	no	—
197	C₅×C₈	—	2	1	no	—
198	C₅×C₈	—	2	4	no	—
199	C₅×C₈	—	4	2	no	—
200	C₅×C₈	C₂×M₁₆	1	2	yes	—
201	C₅×C₈	C₂×M₁₆	1	2	yes	—
202	#168	—	4	2	no	—
203	#168	—	4	2	no	—
204	C₂²×Q₈	—	5	1	no	—
205	C₂²×D₄	—	5	1	no	—
206	C₂³×C₄	—	10	1	no	—
207	C₂×SD₁₆	—	5	2	no	—
208	C₂×SD₁₆	—	5	2	no	—
209	C₂×SD₁₆	—	10	4	no	—
210	C₂×M₁₆	—	10	4	no	—
211	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
213	C₄×Q₈	—	5	2	no	—
214	C₄×Q₈	—	5	2	no	—
215	C₄×Q₈	—	10	1	no	—
216	C₄×Q₈	—	10	1	no	—
217	C₄×D₄	—	5	2	no	—
218	C₄×D₄	—	5	2	no	—
219	C₄×D₄	—	10	1	no	—
220	C₄×D₄	—	10	1	no	—
221	C₂×C₄⋊C₄	—	5	1	no	—
222	C₂×C₄⋊C₄	—	5	1	no	—
223	C₂×C₄⋊C₄	—	10	1	no	—
224	C₂×C₂²⋊C₄	—	10	1	no	—
225	C₂×C₄²	—	10	1	no	—
226	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
227	#106	—	5	2	no	—
228	#106	—	5	2	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	5	2	no	—
231	C₄⋊C₈	—	5	2	no	—
232	C₄⋊C₈	—	10	4	no	—
233	C₄⋊C₈	D₂₀	1	2	yes	—
234	#103	—	5	2	no	—
235	#103	—	5	2	no	—
236	#102	—	5	2	no	—
237	#102	—	5	2	no	—
238	C₂²⋊C₈	—	2	1	no	—
239	C₂²⋊C₈	—	10	1	no	—
240	C₂²⋊C₈	—	10	1	no	—
241	C₂²⋊C₈	—	10	4	no	—
242	C₂²⋊C₈	D₂₀	1	2	yes	—
243	#97	—	10	4	no	—
244	C₄×C₈	—	10	1	no	—
245	C₄×C₈	—	10	1	no	—
246	C₄×C₈	C₄×D₅	1	1	yes	characteristic
247	C₄×C₈	C₄×D₅	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₅	1	2	yes	—
249	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₅	1	2	yes	—
252	C₂²×C₅	—	2	1	no	—
253	C₂²×C₅	—	2	1	no	—
254	C₂²×C₅	#436	1	2	yes	—
255	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
256	C₂×D₅	—	4	2	no	—
257	C₂×D₅	—	4	2	no	—
258	C₂×D₅	—	4	2	no	—
259	C₂×D₅	—	8	4	no	—
260	C₄×C₅	—	2	1	no	—
261	C₄×C₅	—	2	1	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	#436	1	2	yes	—
264	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
265	C₄×C₅	C₂×#131	1	1	yes	characteristic
266	C₄×C₅	#577	1	2	yes	—
267	Dic₅	—	4	2	no	—
268	Dic₅	—	4	2	no	—
269	Dic₅	—	4	2	no	—
270	C₂⁴	—	10	1	no	—
271	C₂×Q₈	—	5	2	no	—
272	C₂×Q₈	—	10	1	no	—
273	C₂×Q₈	—	10	2	no	—
274	C₂×Q₈	—	10	2	no	—
275	C₂×D₄	—	5	2	no	—
276	C₂×D₄	—	10	1	no	—
277	C₂×D₄	—	10	2	no	—
278	C₂×D₄	—	10	2	no	—
279	C₂²×C₄	—	10	1	no	—
280	C₂²×C₄	—	10	1	no	—
281	C₂²×C₄	—	10	1	no	—
282	C₂²×C₄	—	10	2	no	—
283	C₂²×C₄	—	10	2	no	—
284	C₂²×C₄	—	40	2	no	—
285	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
287	C₂²×C₄	#709	1	1	yes	characteristic
288	SD₁₆	—	10	4	no	—
289	SD₁₆	—	10	4	no	—
290	M₁₆	—	40	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	10	2	no	—
294	C₂×C₈	—	10	2	no	—
295	C₂×C₈	#676	1	4	yes	complemented
296	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
299	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
300	C₂×C₈	C₂×D₂₀	1	4	yes	—
301	C₄⋊C₄	—	5	2	no	—
302	C₄⋊C₄	—	5	2	no	—
303	C₄⋊C₄	—	10	2	no	—
304	C₄⋊C₄	—	10	1	no	—
305	C₄⋊C₄	—	10	1	no	—
306	C₄⋊C₄	—	10	2	no	—
307	C₂²⋊C₄	—	10	2	no	—
308	C₂²⋊C₄	—	10	2	no	—
309	C₄²	—	10	2	no	—
310	C₄²	—	10	2	no	—
311	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
312	C₄²	C₂×D₂₀	1	1	yes	characteristic
313	C₄²	#709	1	1	yes	characteristic
314	C₄²	C₂³×D₅	1	1	yes	characteristic
315	C₂×C₅	—	2	2	no	—
316	C₂×C₅	C₂×#436	1	1	yes	characteristic
317	C₂×C₅	#2949	1	1	yes	characteristic
318	C₂×C₅	#2957	1	1	yes	characteristic
319	D₅	—	8	2	no	—
320	C₂³	—	10	2	no	—
321	C₂³	—	20	2	no	—
322	C₂³	—	40	1	no	—
323	C₂³	—	40	1	no	—
324	C₂³	C₄×D₂₀	1	1	yes	characteristic
325	Q₈	—	10	2	no	—
326	Q₈	—	10	2	no	—
327	Q₈	—	20	2	no	—
328	D₄	—	10	2	no	—
329	D₄	—	10	2	no	—
330	D₄	—	20	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	10	2	no	—
334	C₂×C₄	—	10	2	no	—
335	C₂×C₄	—	10	2	no	—
336	C₂×C₄	—	20	2	no	—
337	C₂×C₄	—	20	4	no	—
338	C₂×C₄	—	40	2	no	—
339	C₂×C₄	—	40	1	no	—
340	C₂×C₄	—	40	1	no	—
341	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
342	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
343	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
344	C₂×C₄	C₂×#676	1	2	yes	—
345	C₂×C₄	#4047	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
347	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
348	C₂×C₄	C₂×#709	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	4	no	—
352	C₈	—	4	2	no	—
353	C₈	—	20	2	no	—
354	C₈	—	20	2	no	—
355	C₈	C₂×#676	1	2	yes	—
356	C₈	C₂×#676	1	2	yes	—
357	C₅	#17896	1	1	yes	characteristic, complemented, Sylow
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	20	2	no	—
361	C₂²	—	20	2	no	—
362	C₂²	—	20	2	no	—
363	C₂²	—	40	4	no	—
364	C₂²	#64977	1	2	yes	—
365	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	20	2	no	—
370	C₄	—	20	2	no	—
371	C₄	—	20	2	no	—
372	C₄	#64977	1	2	yes	—
373	C₄	C₂²×#676	1	1	yes	characteristic
374	C₄	C₂×#4047	1	1	yes	characteristic
375	C₄	#66085	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	40	2	no	—
378	C₂	C₂×#64977	1	1	yes	characteristic
379	C₂	#10610303	1	1	yes	characteristic
380	C₂	#10610311	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂²×D₂₀	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1450	D₅
hypocenter	C₅	#17896
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #10610995	C₂	semidirect
4. C₅×#1450	C₂	semidirect
5. #10601560	C₂	semidirect
7. #10600522	C₂	semidirect
8. #10598875	C₂	semidirect
9. #10598848	C₂	semidirect
30. #64968	C₂²	semidirect
31. C₅×#322	C₂²	semidirect
32. C₅×C₈²	C₂²	semidirect
34. #64678	C₂²	semidirect
35. #64677	C₂²	semidirect
36. #64676	C₂²	semidirect
94. #1450	D₅	semidirect
115. C₂×C₅×C₈	M₁₆	semidirect
165. #322	C₂×D₅	semidirect
166. C₈²	C₂×D₅	semidirect
295. C₂×C₈	#676	semidirect
357. C₅	#17896	semidirect

Maximal subgroups

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

subgroup	index
#10611019	2
#10610995	2
C₂×C₄×#677	2
C₅×#1450	2
#10601560	2
#10601550	2
#10600522	2
#10598875	2
#10598848	2
#17896	5