Subgroups — #11999039

Order 1280.

642 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65374	C₂	1	1	yes	maximal, characteristic
2	C₅×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602580	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600565	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600373	C₂	1	1	yes	maximal, characteristic
7	#10600369	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂²×#688	—	2	2	no	—
10	C₂×#4020	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#384	C₂²	1	1	yes	characteristic
13	C₂×#3953	—	2	2	no	—
14	#65374	—	2	2	no	—
15	#65374	C₄	1	4	yes	complemented
16	C₂×#3941	—	2	2	no	—
17	C₂×#3940	—	2	2	no	—
18	#65224	C₂²	1	1	yes	characteristic
19	C₂×C₄×#168	C₂²	1	1	yes	characteristic
20	C₂×C₄×#168	C₂²	1	1	yes	characteristic
21	C₅×#340	—	2	2	no	—
22	#64749	—	2	2	no	—
23	C₈×#168	—	2	2	no	—
24	#9820	—	5	1	no	maximal, Sylow
25	C₂²×C₅×Q₈	—	2	1	no	—
26	C₂²×D₂₀	—	2	1	no	—
27	C₂²×D₂₀	—	2	1	no	—
28	C₂²×D₂₀	—	2	1	no	—
29	C₂²×C₅×C₈	D₄	1	2	yes	—
30	C₅×C₄⋊Q₈	—	2	1	no	—
31	C₅×C₄⋊Q₈	—	2	1	no	—
32	C₅×C₄⋊Q₈	—	2	1	no	—
33	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	—	2	1	no	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×#688	—	2	8	no	—
41	C₂×#688	—	4	4	no	—
42	C₂×#688	—	4	4	no	—
43	C₂×C₄⋊Dic₅	—	2	1	no	—
44	C₂×C₄⋊Dic₅	—	2	1	no	—
45	C₂×C₄⋊Dic₅	—	2	1	no	—
46	C₂²×#168	D₄	1	2	yes	—
47	C₂²×#168	D₄	1	2	yes	—
48	#4020	—	2	1	no	—
49	#4020	—	2	1	no	—
50	#4020	—	2	1	no	—
51	#4020	C₂×C₄	1	2	yes	—
52	C₄×C₅×C₈	—	2	1	no	—
53	C₄×C₅×C₈	—	2	1	no	—
54	#3953	—	4	4	no	—
55	#3941	—	4	4	no	—
56	#3940	—	4	4	no	—
57	C₄×#168	—	2	1	no	—
58	C₄×#168	—	2	1	no	—
59	C₄×#168	C₂×C₄	1	2	yes	—
60	C₄×#168	D₄	1	2	yes	—
61	C₂×#492	—	5	1	no	—
62	#1990	—	5	1	no	—
63	#1990	D₅	1	1	yes	characteristic, complemented
64	#1957	—	5	1	no	—
65	#1957	—	5	1	no	—
66	#1844	—	5	1	no	—
67	C₂×C₈²	—	5	1	no	—
68	C₂³×D₅	—	4	1	no	—
69	C₂×C₅×Q₈	—	2	2	no	—
70	C₂×C₅×Q₈	—	4	1	no	—
71	C₂×C₅×Q₈	—	4	2	no	—
72	C₂×C₅×Q₈	—	4	2	no	—
73	C₂²×C₄×C₅	—	4	1	no	—
74	C₂²×C₄×C₅	—	4	1	no	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₅	—	4	1	no	—
79	C₂×D₂₀	—	2	2	no	—
80	C₂×D₂₀	—	2	2	no	—
81	C₂×D₂₀	—	2	2	no	—
82	C₂×D₂₀	—	4	2	no	—
83	C₂×D₂₀	—	4	2	no	—
84	C₂×D₂₀	—	4	2	no	—
85	C₂×D₂₀	—	4	2	no	—
86	C₂×D₂₀	—	4	2	no	—
87	C₂×D₂₀	—	4	2	no	—
88	C₂×D₂₀	—	4	1	no	—
89	C₂×D₂₀	—	4	1	no	—
90	C₂×D₂₀	—	4	1	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	SD₁₆	1	4	yes	complemented
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	4	1	no	—
97	C₅×C₄⋊C₄	—	4	1	no	—
98	C₅×C₄⋊C₄	—	4	2	no	—
99	C₅×C₄⋊C₄	—	4	2	no	—
100	C₅×C₄⋊C₄	—	4	2	no	—
101	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅	#41	1	1	yes	characteristic
104	C₄²×C₅	#41	1	1	yes	characteristic
105	#688	—	4	16	no	—
106	C₄⋊Dic₅	—	4	2	no	—
107	C₄⋊Dic₅	—	4	2	no	—
108	C₄⋊Dic₅	—	4	2	no	—
109	C₂×#168	—	2	2	no	—
110	C₂×#168	—	2	2	no	—
111	C₂×#168	C₂²⋊C₄	1	4	yes	—
112	C₂×#168	D₈	1	4	yes	—
113	C₂×#168	SD₁₆	1	4	yes	—
114	C₂×#168	SD₁₆	1	4	yes	complemented
115	C₂²×SD₁₆	—	10	2	no	—
116	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
117	C₂×#127	—	5	1	no	—
118	#492	—	5	4	no	—
119	#492	—	10	2	no	—
120	C₂×#106	—	10	2	no	—
121	C₂×#102	—	10	2	no	—
122	C₂×#102	—	10	2	no	—
123	C₂×C₄×C₈	—	5	1	no	—
124	C₂×C₄×C₈	—	5	1	no	—
125	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
126	#384	—	5	1	no	—
127	#384	C₂×D₅	1	1	yes	characteristic
128	#340	—	2	2	no	—
129	#340	—	10	2	no	—
130	C₈²	—	10	2	no	—
131	C₂³×C₅	#127	1	1	yes	characteristic
132	C₂²×D₅	—	4	2	no	—
133	C₂²×D₅	—	8	1	no	—
134	C₂²×D₅	—	8	1	no	—
135	C₂²×D₅	—	8	1	no	—
136	C₂²×D₅	—	8	1	no	—
137	C₂²×D₅	—	8	1	no	—
138	C₂²×D₅	—	8	1	no	—
139	C₅×Q₈	—	4	4	no	—
140	C₅×Q₈	—	8	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	—	8	1	no	—
143	C₂×C₄×C₅	—	8	1	no	—
144	C₂×C₄×C₅	—	8	1	no	—
145	C₂×C₄×C₅	—	8	1	no	—
146	C₂×C₄×C₅	—	8	1	no	—
147	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	#121	1	1	yes	characteristic
151	C₂×C₄×C₅	#121	1	1	yes	characteristic
152	C₂×C₄×C₅	#124	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
155	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
156	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
157	C₂×C₄×C₅	#135	1	2	yes	—
158	C₂×C₄×C₅	#135	1	2	yes	—
159	C₂×Dic₅	—	8	1	no	—
160	C₂×Dic₅	—	8	1	no	—
161	C₂×Dic₅	—	8	1	no	—
162	D₂₀	—	4	4	no	—
163	D₂₀	—	4	4	no	—
164	D₂₀	—	4	4	no	—
165	D₂₀	—	8	2	no	—
166	D₂₀	—	8	2	no	—
167	D₂₀	—	8	2	no	—
168	C₅×C₈	—	2	4	no	—
169	#168	—	2	4	no	—
170	#168	#102	1	8	yes	complemented
171	C₂²×Q₈	—	2	1	no	—
172	C₂²×D₄	—	10	1	no	—
173	C₂²×D₄	—	10	1	no	—
174	C₂²×D₄	—	10	1	no	—
175	C₂×SD₁₆	—	10	8	no	—
176	C₂×SD₁₆	—	20	4	no	—
177	C₂×SD₁₆	—	20	4	no	—
178	C₂²×C₈	—	5	2	no	—
179	C₂²×C₈	—	5	2	no	—
180	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	—	2	1	no	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	C₄×D₅	1	2	yes	—
185	#127	—	5	2	no	—
186	#127	—	10	1	no	—
187	#127	—	10	1	no	—
188	#127	—	10	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	10	1	no	—
195	C₂×C₄⋊C₄	—	10	1	no	—
196	C₂×C₄⋊C₄	—	10	1	no	—
197	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
198	#106	—	20	4	no	—
199	#102	—	20	4	no	—
200	#102	—	20	4	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	5	2	no	—
204	C₄×C₈	—	5	2	no	—
205	C₄×C₈	—	10	1	no	—
206	C₄×C₈	—	10	1	no	—
207	C₂²×C₅	#390	1	1	yes	characteristic
208	C₂²×C₅	#492	1	1	yes	characteristic
209	C₂²×C₅	#492	1	1	yes	characteristic
210	C₂²×C₅	#492	1	1	yes	characteristic
211	C₂²×C₅	#493	1	1	yes	characteristic
212	C₂²×C₅	#495	1	1	yes	characteristic
213	C₂²×C₅	#495	1	1	yes	characteristic
214	C₂×D₅	—	8	2	no	—
215	C₂×D₅	—	8	2	no	—
216	C₂×D₅	—	8	2	no	—
217	C₂×D₅	—	16	1	no	—
218	C₂×D₅	—	16	1	no	—
219	C₂×D₅	—	16	1	no	—
220	C₂×D₅	—	16	1	no	—
221	C₄×C₅	—	8	2	no	—
222	C₄×C₅	—	16	1	no	—
223	C₄×C₅	C₂×#102	1	2	yes	—
224	C₄×C₅	C₄×SD₁₆	1	2	yes	—
225	C₄×C₅	C₄×SD₁₆	1	2	yes	—
226	C₄×C₅	#465	1	2	yes	—
227	C₄×C₅	#465	1	2	yes	—
228	C₄×C₅	#486	1	2	yes	—
229	Dic₅	—	16	1	no	—
230	C₂⁴	—	20	1	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×D₄	—	10	2	no	—
236	C₂×D₄	—	10	2	no	—
237	C₂×D₄	—	10	2	no	—
238	C₂×D₄	—	20	2	no	—
239	C₂×D₄	—	20	1	no	—
240	C₂×D₄	—	20	1	no	—
241	C₂×D₄	—	20	2	no	—
242	C₂×D₄	—	20	2	no	—
243	C₂×D₄	—	20	2	no	—
244	C₂×D₄	—	20	2	no	—
245	C₂×D₄	—	20	1	no	—
246	C₂×D₄	—	20	2	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	20	1	no	—
250	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
253	SD₁₆	—	20	16	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	5	4	no	—
257	C₂×C₈	—	5	4	no	—
258	C₂×C₈	—	5	4	no	—
259	C₂×C₈	—	5	4	no	—
260	C₂×C₈	—	10	2	no	—
261	C₂×C₈	—	10	2	no	—
262	C₂×C₈	#688	1	4	yes	complemented
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	20	2	no	—
271	C₄⋊C₄	—	20	2	no	—
272	C₄⋊C₄	—	20	2	no	—
273	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
274	C₄²	#709	1	1	yes	characteristic
275	C₄²	D₄×D₅	1	1	yes	characteristic
276	C₄²	#711	1	1	yes	characteristic
277	C₂×C₅	#1705	1	1	yes	characteristic
278	C₂×C₅	#1706	1	1	yes	characteristic
279	C₂×C₅	#1706	1	1	yes	characteristic
280	C₂×C₅	#1708	1	1	yes	characteristic
281	C₂×C₅	#1957	1	1	yes	characteristic
282	C₂×C₅	#1968	1	1	yes	characteristic
283	C₂×C₅	#1968	1	1	yes	characteristic
284	D₅	—	16	2	no	—
285	C₂³	—	20	2	no	—
286	C₂³	—	40	1	no	—
287	C₂³	—	40	1	no	—
288	C₂³	—	40	1	no	—
289	C₂³	—	40	1	no	—
290	C₂³	—	40	1	no	—
291	C₂³	—	40	1	no	—
292	C₂³	#4086	1	1	yes	characteristic
293	Q₈	—	4	4	no	—
294	Q₈	—	8	2	no	—
295	D₄	—	20	4	no	—
296	D₄	—	20	4	no	—
297	D₄	—	20	4	no	—
298	D₄	—	40	2	no	—
299	D₄	—	40	2	no	—
300	D₄	—	40	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	40	1	no	—
308	C₂×C₄	—	40	1	no	—
309	C₂×C₄	—	40	1	no	—
310	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
311	C₂×C₄	#4027	1	1	yes	characteristic
312	C₂×C₄	#4030	1	1	yes	characteristic
313	C₂×C₄	#4031	1	1	yes	characteristic
314	C₂×C₄	D₅×D₈	1	2	yes	—
315	C₂×C₄	D₅×SD₁₆	1	2	yes	—
316	C₂×C₄	D₅×SD₁₆	1	2	yes	—
317	C₂×C₄	#4065	1	2	yes	—
318	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
319	C₂×C₄	#4085	1	1	yes	characteristic
320	C₂×C₄	C₂×#688	1	2	yes	—
321	C₂×C₄	#4096	1	2	yes	—
322	C₈	—	2	4	no	—
323	C₈	—	5	8	no	—
324	C₈	—	10	4	no	—
325	C₅	#9820	1	1	yes	characteristic, complemented, Sylow
326	C₂²	—	40	2	no	—
327	C₂²	—	40	2	no	—
328	C₂²	—	40	2	no	—
329	C₂²	—	80	1	no	—
330	C₂²	—	80	1	no	—
331	C₂²	—	80	1	no	—
332	C₂²	—	80	1	no	—
333	C₂²	#65246	1	1	yes	characteristic
334	C₂²	#65348	1	1	yes	characteristic
335	C₂²	#65374	1	1	yes	characteristic
336	C₂²	#65440	1	1	yes	characteristic
337	C₂²	#65464	1	1	yes	characteristic
338	C₂²	#65464	1	1	yes	characteristic
339	C₂²	#65480	1	1	yes	characteristic
340	C₄	—	8	2	no	—
341	C₄	—	16	1	no	—
342	C₄	—	80	1	no	—
343	C₄	#65049	1	2	yes	—
344	C₄	D₅×#102	1	2	yes	—
345	C₄	#65077	1	2	yes	—
346	C₄	#65096	1	2	yes	—
347	C₄	C₄×#688	1	2	yes	—
348	C₄	#65337	1	2	yes	—
349	C₂	—	80	2	no	—
350	C₂	#10600469	1	1	yes	characteristic
351	C₂	#10600950	1	1	yes	characteristic
352	C₂	#10601050	1	1	yes	characteristic
353	C₂	#10601973	1	1	yes	characteristic
354	C₂	#10602006	1	1	yes	characteristic
355	C₂	#10602125	1	1	yes	characteristic
356	C₂	#10602152	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4086
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1990	C₂
socle	C₂³×C₅	#127
prime radical	C₂×#4020	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1990	D₅
hypocenter	C₅	#9820
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
2. C₅×#1990	C₂	semidirect
3. #10602580	C₂	semidirect
5. C₂×C₈×#168	C₂	semidirect
7. #10600369	C₂	semidirect
15. #65374	C₄	semidirect
63. #1990	D₅	semidirect
93. C₂×C₅×C₈	SD₁₆	semidirect
114. C₂×#168	SD₁₆	semidirect
170. #168	#102	semidirect
262. C₂×C₈	#688	semidirect
325. C₅	#9820	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65374	2
C₅×#1990	2
#10602580	2
#10600565	2
C₂×C₈×#168	2
#10600373	2
#10600369	2
#9820	5