Subgroups — #12117765

Order 1280.

870 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65184	C₂	1	2	yes	maximal
2	#10610780	C₂	1	2	yes	maximal, complemented
3	C₄×#4050	C₂	1	1	yes	maximal, characteristic
4	C₅×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601515	C₂	1	4	yes	maximal, complemented
6	#10601497	C₂	1	2	yes	maximal, complemented
7	#10601322	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#4053	C₂²	1	4	yes	—
10	C₂×#4050	C₂²	1	1	yes	characteristic
11	C₄×#709	C₂²	1	2	yes	—
12	C₂×C₅×#108	C₂²	1	2	yes	—
13	C₂×C₅×#108	C₂²	1	2	yes	—
14	C₅×#423	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×#3957	C₂²	1	4	yes	—
17	#65408	C₂²	1	4	yes	—
18	#65402	C₂²	1	2	yes	—
19	#65395	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₅	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
23	#65184	—	2	8	no	—
24	#65128	—	2	4	no	—
25	#65126	—	2	2	no	—
26	D₅×C₄⋊C₈	—	2	2	no	—
27	C₄×Dic₂₀	C₄	1	1	yes	characteristic, complemented
28	C₄×Dic₂₀	C₄	1	1	yes	characteristic, complemented
29	C₄×D₄₀	C₄	1	1	yes	characteristic, complemented
30	C₄×D₄₀	C₄	1	1	yes	characteristic, complemented
31	C₄×#677	C₄	1	2	yes	complemented
32	C₄×#677	C₄	1	2	yes	complemented
33	#64775	C₂²	1	4	yes	—
34	#64770	C₂²	1	2	yes	—
35	#22688	—	5	1	no	maximal, Sylow
36	C₂×#709	C₂³	1	2	yes	—
37	C₂×C₅×M₁₆	C₂³	1	4	yes	—
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₄×C₂²⋊D₅	—	2	2	no	—
42	C₄×C₂²⋊D₅	—	2	2	no	—
43	#4072	C₂³	1	2	yes	—
44	C₂×C₄⋊Dic₅	—	2	1	no	—
45	C₂×C₄×Dic₅	C₂³	1	2	yes	—
46	C₂²×#168	C₂³	1	2	yes	—
47	#4053	—	2	8	no	—
48	#4053	—	4	4	no	—
49	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
50	#4050	D₄	1	4	yes	—
51	#4050	Q₈	1	4	yes	—
52	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
53	C₂×#677	C₂×C₄	1	2	yes	—
54	C₂×#676	—	2	4	no	—
55	C₂×C₈×D₅	—	2	4	no	—
56	C₄×D₂₀	C₂×C₄	1	2	yes	—
57	C₄×D₂₀	C₂×C₄	1	2	yes	—
58	#4018	—	2	2	no	—
59	C₄²×D₅	—	2	2	no	—
60	C₄×Dic₁₀	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₀	C₂×C₄	1	2	yes	—
62	C₅×#108	—	2	4	no	—
63	C₅×#108	—	4	2	no	—
64	C₅×C₄⋊C₈	—	2	2	no	—
65	C₅×C₄⋊C₈	—	2	2	no	—
66	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
70	#3963	—	2	1	no	—
71	#3957	—	2	8	no	—
72	#3953	C₂×C₄	1	2	yes	—
73	#3953	C₂×C₄	1	2	yes	—
74	#3950	C₂×C₄	1	1	yes	characteristic
75	#3950	C₂×C₄	1	1	yes	characteristic
76	#3949	C₂×C₄	1	2	yes	—
77	#3948	C₂×C₄	1	2	yes	—
78	#3948	C₂×C₄	1	2	yes	—
79	#3946	—	2	4	no	—
80	#3946	—	2	4	no	—
81	C₈×Dic₅	—	2	4	no	—
82	C₄⋊#168	D₄	1	1	yes	characteristic
83	C₄⋊#168	D₄	1	1	yes	characteristic
84	C₄⋊#168	D₄	1	2	yes	—
85	C₂×#443	—	5	2	no	—
86	C₄×#135	—	5	1	no	—
87	#2906	—	5	2	no	—
88	#1945	—	5	2	no	—
89	#1866	—	5	4	no	—
90	#1848	D₅	1	1	yes	characteristic, complemented
91	#1806	—	5	2	no	—
92	C₈×C₄⋊C₄	—	5	1	no	—
93	C₂²×C₄×C₅	#41	1	1	yes	characteristic
94	C₂²×C₄×C₅	#41	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₂×C₂²⋊D₅	—	2	2	no	—
97	C₂²×Dic₅	—	2	2	no	—
98	#709	—	2	2	no	—
99	#709	—	2	2	no	—
100	#709	C₂×D₄	1	4	yes	—
101	#709	C₂×Q₈	1	4	yes	—
102	C₂×D₂₀	C₂²×C₄	1	2	yes	—
103	C₂×C₄×D₅	—	2	2	no	—
104	C₂×C₄×D₅	—	2	2	no	—
105	C₂×C₄×D₅	—	2	2	no	—
106	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
107	C₅×M₁₆	—	2	4	no	—
108	C₅×M₁₆	—	2	4	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	—	2	4	no	—
112	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
117	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
118	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
122	C₂²⋊Dic₅	—	2	2	no	—
123	#685	—	2	2	no	—
124	#685	—	2	2	no	—
125	C₄⋊Dic₅	—	2	4	no	—
126	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
127	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
128	#683	—	2	2	no	—
129	#683	—	2	2	no	—
130	C₄×Dic₅	—	2	2	no	—
131	C₄×Dic₅	—	2	2	no	—
132	C₄×Dic₅	—	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	C₂×#168	C₂×D₄	1	2	yes	—
138	C₂×#168	C₂×D₄	1	2	yes	—
139	Dic₂₀	C₄⋊C₄	1	4	yes	complemented
140	D₄₀	C₄⋊C₄	1	4	yes	complemented
141	#677	C₄⋊C₄	1	8	yes	complemented
142	#676	—	4	8	no	—
143	C₈×D₅	—	4	8	no	—
144	C₂×#135	—	5	1	no	—
145	C₂×#131	—	5	4	no	—
146	C₄×#41	—	5	2	no	—
147	#443	—	10	8	no	—
148	C₄×Q₁₆	—	5	1	no	—
149	C₄×Q₁₆	—	5	1	no	—
150	C₄×SD₁₆	—	5	2	no	—
151	C₄×SD₁₆	—	5	2	no	—
152	C₄×D₈	—	5	1	no	—
153	C₄×D₈	—	5	1	no	—
154	C₂×#108	C₂×D₅	1	2	yes	—
155	C₂×#108	C₂×D₅	1	2	yes	—
156	#427	—	5	1	no	—
157	#424	—	5	4	no	—
158	#424	—	10	4	no	—
159	#423	—	10	2	no	—
160	#423	C₂×D₅	1	2	yes	—
161	C₂×C₄⋊C₈	—	5	1	no	—
162	C₂×C₄⋊C₈	—	10	2	no	—
163	C₂×#104	—	5	4	no	—
164	#416	—	5	2	no	—
165	C₂×C₄×C₈	—	5	2	no	—
166	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
167	C₄×C₄⋊C₄	—	5	1	no	—
168	#339	—	5	4	no	—
169	#336	—	5	2	no	—
170	C₂³×C₅	C₂×#41	1	1	yes	characteristic
171	C₂²×D₅	—	2	2	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	C₄×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
182	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
184	C₂²⋊D₅	—	2	4	no	—
185	C₂²⋊D₅	—	2	4	no	—
186	C₂×Dic₅	—	2	2	no	—
187	C₂×Dic₅	—	2	4	no	—
188	C₂×Dic₅	—	2	2	no	—
189	C₂×Dic₅	—	2	2	no	—
190	C₂×Dic₅	—	4	2	no	—
191	C₂×Dic₅	—	4	2	no	—
192	D₂₀	—	2	2	no	—
193	D₂₀	C₂×C₄⋊C₄	1	4	yes	—
194	C₄×D₅	—	2	4	no	—
195	C₄×D₅	—	2	4	no	—
196	C₄×D₅	—	4	4	no	—
197	C₄×D₅	—	8	2	no	—
198	Dic₁₀	—	2	2	no	—
199	Dic₁₀	C₂×C₄⋊C₄	1	4	yes	—
200	C₅×C₈	—	2	1	no	—
201	C₅×C₈	—	2	1	no	—
202	C₅×C₈	—	4	4	no	—
203	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
204	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
205	#168	—	2	2	no	—
206	#168	—	2	2	no	—
207	C₂×#41	—	5	2	no	—
208	#135	—	5	4	no	—
209	#135	—	5	4	no	—
210	C₂×Q₁₆	—	5	1	no	—
211	C₂×SD₁₆	—	5	2	no	—
212	C₂×D₈	—	5	1	no	—
213	#131	—	10	8	no	—
214	#131	—	20	4	no	—
215	C₂×M₁₆	—	10	4	no	—
216	C₂×M₁₆	C₂²×D₅	1	4	yes	—
217	C₂²×C₈	—	5	2	no	—
218	C₂²×C₈	—	10	4	no	—
219	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
221	C₄×Q₈	—	5	2	no	—
222	C₄×Q₈	—	5	2	no	—
223	C₄×D₄	—	5	2	no	—
224	C₄×D₄	—	5	2	no	—
225	C₄×D₄	—	10	2	no	—
226	C₄×D₄	—	10	2	no	—
227	#117	—	5	2	no	—
228	#117	—	10	2	no	—
229	C₂×C₄⋊C₄	—	10	1	no	—
230	C₂×C₄²	—	5	2	no	—
231	C₂×C₄²	—	10	2	no	—
232	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
233	#108	—	2	4	no	—
234	#108	—	4	2	no	—
235	#107	—	5	1	no	—
236	#107	—	5	1	no	—
237	#106	—	5	2	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	5	1	no	—
241	C₄⋊C₈	—	5	1	no	—
242	C₄⋊C₈	—	5	2	no	—
243	C₄⋊C₈	—	10	4	no	—
244	C₄⋊C₈	—	10	4	no	—
245	#104	—	10	8	no	—
246	#103	—	5	2	no	—
247	#103	—	5	2	no	—
248	#102	—	5	2	no	—
249	#102	—	5	2	no	—
250	C₄×C₈	—	10	4	no	—
251	C₄×C₈	C₄×D₅	1	1	yes	characteristic
252	C₄×C₈	C₄×D₅	1	1	yes	characteristic
253	C₄×C₈	C₄×D₅	1	1	yes	characteristic
254	C₄×C₈	C₄×D₅	1	1	yes	characteristic
255	#95	—	10	1	no	—
256	C₂²×C₅	—	2	1	no	—
257	C₂²×C₅	—	2	1	no	—
258	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
259	C₂²×C₅	#547	1	1	yes	characteristic
260	C₂²×C₅	#554	1	1	yes	characteristic
261	C₂×D₅	—	2	4	no	—
262	C₂×D₅	—	4	2	no	—
263	C₂×D₅	—	4	2	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	#443	1	2	yes	complemented
268	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
269	C₄×C₅	#520	1	1	yes	characteristic
270	C₄×C₅	D₄²	1	1	yes	characteristic
271	C₄×C₅	D₄×Q₈	1	1	yes	characteristic
272	Dic₅	—	2	4	no	—
273	Dic₅	—	4	4	no	—
274	Dic₅	—	4	2	no	—
275	Dic₅	—	4	2	no	—
276	#41	—	5	4	no	—
277	#41	—	5	4	no	—
278	#41	—	10	2	no	—
279	#41	—	10	2	no	—
280	C₂×Q₈	—	5	2	no	—
281	C₂×D₄	—	5	2	no	—
282	C₂×D₄	—	10	2	no	—
283	C₂²×C₄	—	10	2	no	—
284	C₂²×C₄	—	10	2	no	—
285	C₂²×C₄	—	10	2	no	—
286	C₂²×C₄	—	10	2	no	—
287	C₂²×C₄	#713	1	1	yes	characteristic
288	C₂²×C₄	#713	1	1	yes	characteristic
289	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
290	Q₁₆	—	5	4	no	—
291	SD₁₆	—	5	8	no	—
292	D₈	—	5	4	no	—
293	M₁₆	—	2	4	no	—
294	M₁₆	—	2	4	no	—
295	M₁₆	—	20	8	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	4	no	—
299	C₂×C₈	—	5	2	no	—
300	C₂×C₈	—	5	2	no	—
301	C₂×C₈	—	10	2	no	—
302	C₂×C₈	—	10	2	no	—
303	C₂×C₈	—	20	8	no	—
304	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
306	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
307	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
308	C₂×C₈	D₄×D₅	1	2	yes	—
309	C₂×C₈	Q₈×D₅	1	2	yes	—
310	C₄⋊C₄	—	5	2	no	—
311	C₄⋊C₄	—	5	2	no	—
312	C₄⋊C₄	—	10	4	no	—
313	C₄⋊C₄	—	10	2	no	—
314	C₄⋊C₄	—	10	2	no	—
315	C₂²⋊C₄	—	10	2	no	—
316	C₂²⋊C₄	—	10	2	no	—
317	C₂²⋊C₄	—	10	2	no	—
318	C₄²	—	10	2	no	—
319	C₄²	—	10	2	no	—
320	C₄²	—	10	2	no	—
321	C₄²	—	10	2	no	—
322	C₄²	—	10	2	no	—
323	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
324	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
325	C₄²	D₄×D₅	1	1	yes	characteristic
326	C₄²	D₄×D₅	1	1	yes	characteristic
327	C₂×C₅	—	2	1	no	—
328	C₂×C₅	—	2	1	no	—
329	C₂×C₅	D₄×C₄⋊C₄	1	1	yes	characteristic
330	C₂×C₅	C₂×#443	1	1	yes	characteristic
331	C₂×C₅	#2957	1	1	yes	characteristic
332	D₅	—	4	4	no	—
333	C₂³	—	10	2	no	—
334	C₂³	C₂×#713	1	1	yes	characteristic
335	Q₈	—	5	4	no	—
336	Q₈	—	10	2	no	—
337	D₄	—	5	4	no	—
338	D₄	—	10	4	no	—
339	D₄	—	10	4	no	—
340	D₄	—	10	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	10	2	no	—
344	C₂×C₄	—	10	4	no	—
345	C₂×C₄	—	10	4	no	—
346	C₂×C₄	—	10	4	no	—
347	C₂×C₄	—	10	2	no	—
348	C₂×C₄	—	10	2	no	—
349	C₂×C₄	—	20	2	no	—
350	C₂×C₄	—	20	2	no	—
351	C₂×C₄	—	20	4	no	—
352	C₂×C₄	—	40	2	no	—
353	C₂×C₄	#4039	1	1	yes	characteristic
354	C₂×C₄	#4039	1	1	yes	characteristic
355	C₂×C₄	#4039	1	1	yes	characteristic
356	C₂×C₄	#4039	1	1	yes	characteristic
357	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
359	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
360	C₂×C₄	#4144	1	1	yes	characteristic
361	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
362	C₂×C₄	#4147	1	1	yes	characteristic
363	C₈	—	2	1	no	—
364	C₈	—	2	1	no	—
365	C₈	—	4	4	no	—
366	C₈	—	10	2	no	—
367	C₈	—	10	2	no	—
368	C₈	D₅×C₄⋊C₄	1	2	yes	—
369	C₈	D₅×C₄⋊C₄	1	2	yes	—
370	C₅	#22688	1	1	yes	characteristic, complemented, Sylow
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	10	4	no	—
374	C₂²	—	20	2	no	—
375	C₂²	—	20	2	no	—
376	C₂²	C₂×#4039	1	1	yes	characteristic
377	C₂²	#66060	1	1	yes	characteristic
378	C₂²	#66064	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	10	4	no	—
383	C₄	—	20	4	no	—
384	C₄	—	20	2	no	—
385	C₄	—	20	2	no	—
386	C₄	#65184	1	2	yes	complemented
387	C₄	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
388	C₄	#65838	1	1	yes	characteristic
389	C₄	#66051	1	1	yes	characteristic
390	C₄	#66061	1	1	yes	characteristic
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	—	20	4	no	—
394	C₂	#10606091	1	1	yes	characteristic
395	C₂	C₂×#65184	1	1	yes	characteristic
396	C₂	#10610980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₅×#1848	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×#4050	C₂²
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	#1848	D₅
hypocenter	C₅	#22688
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. #10610780	C₂	semidirect
4. C₅×#1848	C₂	semidirect
5. #10601515	C₂	semidirect
6. #10601497	C₂	semidirect
7. #10601322	C₂	semidirect
8. C₈×C₄⋊Dic₅	C₂	semidirect
27. C₄×Dic₂₀	C₄	semidirect
28. C₄×Dic₂₀	C₄	semidirect
29. C₄×D₄₀	C₄	semidirect
30. C₄×D₄₀	C₄	semidirect
31. C₄×#677	C₄	semidirect
32. C₄×#677	C₄	semidirect
90. #1848	D₅	semidirect
139. Dic₂₀	C₄⋊C₄	semidirect
140. D₄₀	C₄⋊C₄	semidirect
141. #677	C₄⋊C₄	semidirect
267. C₄×C₅	#443	semidirect
370. C₅	#22688	semidirect
386. C₄	#65184	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65184	2
#10610780	2
C₄×#4050	2
C₅×#1848	2
#10601515	2
#10601497	2
#10601322	2
C₈×C₄⋊Dic₅	2
#22688	5