Subgroups — #11998548

Order 1280.

522 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65377	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#333	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602572	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601258	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601250	C₂	1	1	yes	maximal, characteristic
6	C₂×#64705	C₂	1	1	yes	maximal, characteristic
7	C₂×#64700	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#4020	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#3953	—	2	2	no	—
13	C₂×C₈×Dic₅	—	2	1	no	—
14	#65377	—	2	2	no	—
15	#65377	C₄	1	4	yes	complemented
16	C₂×#3941	—	2	1	no	—
17	C₂×#3941	—	2	1	no	—
18	C₄×C₄⋊Dic₅	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×#333	—	2	2	no	—
22	#64770	—	2	1	no	—
23	#64749	—	2	2	no	—
24	#64705	—	2	2	no	—
25	#64700	—	2	2	no	—
26	#9735	—	5	1	no	maximal, Sylow
27	C₂²×C₅×Q₈	—	2	1	no	—
28	C₂²×D₂₀	—	2	1	no	—
29	C₂²×D₂₀	—	2	1	no	—
30	C₂²×D₂₀	—	2	1	no	—
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₅×C₄⋊Q₈	—	2	1	no	—
34	C₅×C₄⋊Q₈	—	2	1	no	—
35	C₅×C₄⋊Q₈	—	2	1	no	—
36	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
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×C₄⋊Dic₅	D₄	1	2	yes	—
41	C₂×C₄×Dic₅	—	2	1	no	—
42	C₂²×#168	—	2	1	no	—
43	C₂²×#168	D₄	1	1	yes	characteristic
44	C₂²×#168	D₄	1	1	yes	characteristic
45	#4020	—	2	1	no	—
46	#4020	—	2	1	no	—
47	#4020	—	2	1	no	—
48	#4020	C₂×C₄	1	2	yes	—
49	C₅×C₄⋊C₈	—	2	1	no	—
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₄⋊C₈	—	2	1	no	—
52	C₅×C₄⋊C₈	D₄	1	2	yes	—
53	#3963	—	2	1	no	—
54	#3953	—	4	4	no	—
55	C₈×Dic₅	—	4	2	no	—
56	#3941	—	2	4	no	—
57	#3941	—	2	4	no	—
58	C₄⋊#168	—	2	1	no	—
59	C₄⋊#168	—	2	1	no	—
60	C₄×#168	C₂×C₄	1	2	yes	—
61	C₄×#168	D₄	1	2	yes	—
62	C₂×#486	—	5	1	no	—
63	#1987	—	5	1	no	—
64	#1982	—	5	1	no	—
65	C₈×C₄⋊C₄	—	5	1	no	—
66	C₂×#334	—	5	1	no	—
67	C₂×#333	D₅	1	1	yes	characteristic, complemented
68	C₂×#331	—	5	1	no	—
69	C₂³×D₅	—	4	1	no	—
70	C₂×C₅×Q₈	—	2	2	no	—
71	C₂×C₅×Q₈	—	4	1	no	—
72	C₂×C₅×Q₈	—	8	2	no	—
73	C₂²×C₄×C₅	—	4	1	no	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₂²×Dic₅	—	2	2	no	—
78	C₂×D₂₀	—	2	2	no	—
79	C₂×D₂₀	—	2	2	no	—
80	C₂×D₂₀	—	2	2	no	—
81	C₂×D₂₀	—	4	2	no	—
82	C₂×D₂₀	—	4	1	no	—
83	C₂×D₂₀	—	4	1	no	—
84	C₂×D₂₀	—	4	2	no	—
85	C₂×D₂₀	—	4	1	no	—
86	C₂×D₂₀	—	4	1	no	—
87	C₂×D₂₀	—	4	1	no	—
88	C₂×D₂₀	—	4	1	no	—
89	C₂×D₂₀	—	4	1	no	—
90	C₂×D₂₀	—	8	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	4	1	no	—
93	C₂×C₅×C₈	—	4	1	no	—
94	C₂×C₅×C₈	—	4	1	no	—
95	C₂×C₅×C₈	—	4	2	no	—
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₅×C₄⋊C₄	—	4	1	no	—
99	C₅×C₄⋊C₄	—	4	1	no	—
100	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅	#41	1	1	yes	characteristic
104	C₄⋊Dic₅	—	2	2	no	—
105	C₄⋊Dic₅	SD₁₆	1	4	yes	—
106	C₄×Dic₅	—	4	1	no	—
107	C₄×Dic₅	—	4	1	no	—
108	C₂×#168	—	2	1	no	—
109	C₂×#168	—	2	1	no	—
110	C₂×#168	—	2	1	no	—
111	C₂×#168	—	2	1	no	—
112	C₂×#168	—	4	1	no	—
113	C₂×#168	—	4	2	no	—
114	C₂×#168	C₂²⋊C₄	1	2	yes	—
115	C₂×#168	C₂²⋊C₄	1	2	yes	—
116	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
117	C₂×#127	—	5	1	no	—
118	#486	—	5	4	no	—
119	#486	—	10	2	no	—
120	C₂×C₄⋊C₈	—	5	1	no	—
121	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
123	C₂×#102	—	10	1	no	—
124	C₂×#102	—	10	1	no	—
125	C₂×#102	—	10	2	no	—
126	C₂×C₄×C₈	—	5	1	no	—
127	C₂×C₄×C₈	—	10	1	no	—
128	C₄×C₄⋊C₄	—	5	1	no	—
129	#340	—	10	2	no	—
130	#336	—	10	1	no	—
131	#334	—	10	2	no	—
132	#333	—	2	2	no	—
133	#331	—	10	2	no	—
134	C₂³×C₅	#121	1	1	yes	characteristic
135	C₂²×D₅	—	4	2	no	—
136	C₂²×D₅	—	8	1	no	—
137	C₂²×D₅	—	8	1	no	—
138	C₂²×D₅	—	8	2	no	—
139	C₂²×D₅	—	8	2	no	—
140	C₅×Q₈	—	8	2	no	—
141	C₅×Q₈	—	8	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	4	2	no	—
144	C₂×C₄×C₅	—	8	2	no	—
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	#121	1	1	yes	characteristic
149	C₂×C₄×C₅	#124	1	1	yes	characteristic
150	C₂×C₄×C₅	#127	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
152	C₂×C₄×C₅	#135	1	1	yes	characteristic
153	C₂×C₄×C₅	#135	1	1	yes	characteristic
154	C₂×C₄×C₅	#135	1	2	yes	—
155	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₅	#137	1	1	yes	characteristic
157	C₂×Dic₅	—	2	4	no	—
158	C₂×Dic₅	—	4	2	no	—
159	C₂×Dic₅	—	4	2	no	—
160	D₂₀	—	4	2	no	—
161	D₂₀	—	4	2	no	—
162	D₂₀	—	4	4	no	—
163	D₂₀	—	8	2	no	—
164	D₂₀	—	8	2	no	—
165	D₂₀	—	8	1	no	—
166	D₂₀	—	8	1	no	—
167	D₂₀	—	8	1	no	—
168	D₂₀	—	8	1	no	—
169	C₅×C₈	—	4	2	no	—
170	C₅×C₈	—	8	1	no	—
171	#168	—	2	2	no	—
172	#168	—	2	2	no	—
173	#168	—	8	1	no	—
174	C₂²×Q₈	—	2	1	no	—
175	C₂²×D₄	—	10	1	no	—
176	C₂²×D₄	—	10	1	no	—
177	C₂²×D₄	—	10	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	5	1	no	—
181	C₂²×C₈	—	5	1	no	—
182	C₂²×C₈	—	10	1	no	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	C₄×D₅	1	2	yes	—
187	#127	—	5	2	no	—
188	#127	—	10	1	no	—
189	#127	—	10	1	no	—
190	#127	—	10	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	5	2	no	—
194	C₂×C₄²	—	10	1	no	—
195	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	10	1	no	—
200	C₄⋊C₈	—	10	1	no	—
201	C₄⋊C₈	C₂²⋊D₅	1	2	yes	—
202	#102	—	10	4	no	—
203	#102	—	10	4	no	—
204	#102	—	20	4	no	—
205	C₄×C₈	—	5	2	no	—
206	C₄×C₈	—	5	2	no	—
207	C₄×C₈	—	20	2	no	—
208	#95	—	10	1	no	—
209	C₂²×C₅	#390	1	1	yes	characteristic
210	C₂²×C₅	#463	1	1	yes	characteristic
211	C₂²×C₅	#464	1	1	yes	characteristic
212	C₂²×C₅	#465	1	2	yes	—
213	C₂²×C₅	#472	1	2	yes	—
214	C₂×D₅	—	8	2	no	—
215	C₂×D₅	—	8	2	no	—
216	C₂×D₅	—	8	2	no	—
217	C₂×D₅	—	16	2	no	—
218	C₂×D₅	—	16	2	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	8	2	no	—
222	C₄×C₅	#416	1	1	yes	characteristic
223	C₄×C₅	#417	1	1	yes	characteristic
224	C₄×C₅	C₄×SD₁₆	1	2	yes	—
225	C₄×C₅	#463	1	1	yes	characteristic
226	C₄×C₅	#464	1	1	yes	characteristic
227	C₄×C₅	#492	1	1	yes	characteristic
228	C₄×C₅	#495	1	1	yes	characteristic
229	Dic₅	—	4	4	no	—
230	C₂⁴	—	20	1	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	8	2	no	—
234	C₂×D₄	—	10	2	no	—
235	C₂×D₄	—	10	2	no	—
236	C₂×D₄	—	10	2	no	—
237	C₂×D₄	—	20	1	no	—
238	C₂×D₄	—	20	2	no	—
239	C₂×D₄	—	20	1	no	—
240	C₂×D₄	—	20	1	no	—
241	C₂×D₄	—	20	1	no	—
242	C₂×D₄	—	20	2	no	—
243	C₂×D₄	—	20	1	no	—
244	C₂×D₄	—	20	1	no	—
245	C₂×D₄	—	20	1	no	—
246	C₂×D₄	—	40	2	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	10	2	no	—
249	C₂²×C₄	#709	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	5	2	no	—
258	C₂×C₈	—	5	2	no	—
259	C₂×C₈	—	10	1	no	—
260	C₂×C₈	—	10	1	no	—
261	C₂×C₈	—	10	1	no	—
262	C₂×C₈	—	10	1	no	—
263	C₂×C₈	—	20	2	no	—
264	C₂×C₈	—	20	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	5	4	no	—
270	C₄⋊C₄	—	10	2	no	—
271	C₄²	—	20	1	no	—
272	C₄²	—	20	1	no	—
273	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
274	C₄²	D₄×D₅	1	1	yes	characteristic
275	C₄²	#711	1	1	yes	characteristic
276	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
277	C₂×C₅	#1677	1	2	yes	—
278	C₂×C₅	#1678	1	2	yes	—
279	C₂×C₅	#1968	1	1	yes	characteristic
280	C₂×C₅	#1978	1	1	yes	characteristic
281	C₂×C₅	#1983	1	1	yes	characteristic
282	D₅	—	16	2	no	—
283	C₂³	—	20	2	no	—
284	C₂³	—	40	2	no	—
285	C₂³	—	40	1	no	—
286	C₂³	—	40	2	no	—
287	C₂³	—	40	1	no	—
288	C₂³	#4030	1	1	yes	characteristic
289	Q₈	—	8	2	no	—
290	Q₈	—	8	2	no	—
291	D₄	—	20	4	no	—
292	D₄	—	20	2	no	—
293	D₄	—	20	2	no	—
294	D₄	—	40	2	no	—
295	D₄	—	40	2	no	—
296	D₄	—	40	1	no	—
297	D₄	—	40	1	no	—
298	D₄	—	40	1	no	—
299	D₄	—	40	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	10	4	no	—
304	C₂×C₄	—	20	2	no	—
305	C₂×C₄	—	20	2	no	—
306	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
307	C₂×C₄	#4027	1	1	yes	characteristic
308	C₂×C₄	#4031	1	1	yes	characteristic
309	C₂×C₄	D₅×SD₁₆	1	2	yes	—
310	C₂×C₄	#4060	1	1	yes	characteristic
311	C₂×C₄	#4062	1	1	yes	characteristic
312	C₂×C₄	#4064	1	1	yes	characteristic
313	C₂×C₄	#4065	1	2	yes	—
314	C₂×C₄	#4065	1	1	yes	characteristic
315	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
316	C₂×C₄	#4085	1	1	yes	characteristic
317	C₂×C₄	#4086	1	1	yes	characteristic
318	C₈	—	4	2	no	—
319	C₈	—	8	1	no	—
320	C₈	—	10	2	no	—
321	C₈	—	10	2	no	—
322	C₈	—	40	1	no	—
323	C₅	#9735	1	1	yes	characteristic, complemented, Sylow
324	C₂²	—	40	2	no	—
325	C₂²	—	40	2	no	—
326	C₂²	—	40	2	no	—
327	C₂²	—	80	2	no	—
328	C₂²	—	80	2	no	—
329	C₂²	#65038	1	2	yes	—
330	C₂²	#65087	1	1	yes	characteristic
331	C₂²	#65096	1	2	yes	—
332	C₂²	#65108	1	1	yes	characteristic
333	C₂²	#65246	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	20	4	no	—
338	C₄	#65077	1	2	yes	—
339	C₄	#65092	1	1	yes	characteristic
340	C₄	#65093	1	1	yes	characteristic
341	C₄	#65454	1	1	yes	characteristic
342	C₄	#65464	1	1	yes	characteristic
343	C₄	#65474	1	1	yes	characteristic
344	C₄	#65480	1	1	yes	characteristic
345	C₂	—	80	2	no	—
346	C₂	#10599291	1	2	yes	—
347	C₂	#10599303	1	2	yes	—
348	C₂	#10600867	1	1	yes	characteristic
349	C₂	#10601023	1	1	yes	characteristic
350	C₂	#10601050	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4030
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#333	C₂
socle	C₂³×C₅	#121
prime radical	C₂×#4020	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#333	D₅
hypocenter	C₅	#9735
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#333	C₂	semidirect
3. #10602572	C₂	semidirect
4. #10601258	C₂	semidirect
7. C₂×#64700	C₂	semidirect
15. #65377	C₄	semidirect
67. C₂×#333	D₅	semidirect
323. C₅	#9735	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65377	2
C₂×C₅×#333	2
#10602572	2
#10601258	2
#10601250	2
C₂×#64705	2
C₂×#64700	2
#9735	5