Subgroups — #12130179

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2960	C₂	1	1	yes	maximal, characteristic, complemented
2	#10610310	C₂	1	1	yes	maximal, characteristic
3	#10602635	C₂	1	2	yes	maximal, complemented
4	#10601341	C₂	1	2	yes	maximal, complemented
5	#10601340	C₂	1	1	yes	maximal, characteristic
6	#10601332	C₂	1	1	yes	maximal, characteristic, complemented
7	#10601324	C₂	1	1	yes	maximal, characteristic
8	#10601319	C₂	1	1	yes	maximal, characteristic, complemented
9	#10600691	C₂	1	2	yes	maximal, complemented
10	#10600688	C₂	1	1	yes	maximal, characteristic, complemented
11	#10600582	C₂	1	2	yes	maximal, complemented
12	C₅×#533	C₂²	1	1	yes	characteristic
13	#66082	C₂²	1	1	yes	characteristic
14	C₂×#4047	C₂²	1	2	yes	—
15	C₅×#446	—	2	2	no	—
16	C₅×C₈×Q₈	—	2	2	no	—
17	C₅×#424	C₂²	1	1	yes	characteristic
18	C₅×#424	C₂²	1	1	yes	characteristic
19	C₅×#423	C₂²	1	1	yes	characteristic
20	C₅×#405	C₂²	1	1	yes	characteristic
21	C₅×#405	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	#65421	C₂²	1	2	yes	—
24	#65414	C₂²	1	1	yes	characteristic
25	#65402	C₂²	1	2	yes	—
26	#65390	C₂²	1	2	yes	—
27	#65295	C₂²	1	2	yes	—
28	#65294	C₂²	1	2	yes	—
29	C₂×#3937	C₂²	1	2	yes	—
30	#65006	—	2	2	no	—
31	#64989	—	2	2	no	—
32	#64780	C₂²	1	1	yes	characteristic
33	#64779	C₂²	1	2	yes	—
34	#64777	C₂²	1	1	yes	characteristic
35	#64772	C₂²	1	2	yes	—
36	#64753	C₂²	1	2	yes	—
37	#64751	C₂²	1	2	yes	—
38	#64743	C₂²	1	2	yes	—
39	#23833	—	5	1	no	maximal, Sylow
40	C₂×#709	C₂³	1	2	yes	—
41	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₅×C₄⋊Q₈	—	2	1	no	—
46	C₅×#125	—	2	1	no	—
47	C₅×C₂²⋊Q₈	—	2	2	no	—
48	C₄×C₅×Q₈	—	2	2	no	—
49	C₅×#117	C₂³	1	1	yes	characteristic
50	C₅×#117	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	#4072	C₂³	1	2	yes	—
53	C₂×#681	C₂³	1	2	yes	—
54	C₂×#681	C₂³	1	2	yes	—
55	#4055	—	2	2	no	—
56	#4054	—	2	2	no	—
57	#4050	—	2	2	no	—
58	#4047	—	2	2	no	—
59	#4047	—	2	2	no	—
60	#4047	—	2	2	no	—
61	#4047	—	2	2	no	—
62	C₂×#676	—	2	2	no	—
63	C₂×C₈×D₅	—	2	2	no	—
64	C₅×C₄⋊C₈	—	2	2	no	—
65	C₅×C₄⋊C₈	—	2	2	no	—
66	C₅×C₄⋊C₈	—	2	2	no	—
67	C₅×#97	—	2	1	no	—
68	C₅×#97	D₄	1	2	yes	—
69	C₄×C₅×C₈	—	2	1	no	—
70	C₄×C₅×C₈	—	2	1	no	—
71	C₄×C₅×C₈	—	2	1	no	—
72	C₄×C₅×C₈	D₄	1	2	yes	—
73	#3956	—	2	2	no	—
74	#3955	—	2	2	no	—
75	#3953	—	2	2	no	—
76	#3951	—	2	2	no	—
77	#3948	—	2	2	no	—
78	#3947	—	2	2	no	—
79	C₈×Dic₅	—	2	2	no	—
80	#3938	—	2	4	no	—
81	#3937	—	2	4	no	—
82	#3937	—	2	2	no	—
83	#3937	—	2	2	no	—
84	#2960	D₅	1	1	yes	characteristic, complemented
85	#2955	—	5	1	no	—
86	#2103	—	5	1	no	—
87	#2102	—	5	1	no	—
88	#2080	—	5	2	no	—
89	#2053	—	5	1	no	—
90	#2050	—	5	1	no	—
91	#2031	—	5	2	no	—
92	#1764	—	5	2	no	—
93	#1757	—	5	2	no	—
94	#1747	—	5	1	no	—
95	C₂×C₅×Q₈	—	2	2	no	—
96	C₂²×C₄×C₅	—	2	1	no	—
97	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₂×C₂²⋊D₅	—	2	2	no	—
99	#709	—	2	2	no	—
100	#709	—	2	2	no	—
101	#709	—	2	2	no	—
102	#709	—	2	2	no	—
103	C₂×D₂₀	#41	1	2	yes	—
104	C₂×C₄×D₅	—	2	2	no	—
105	C₂×Dic₁₀	#41	1	2	yes	—
106	C₅×M₁₆	—	2	2	no	—
107	C₅×M₁₆	—	2	2	no	—
108	C₂×C₅×C₈	—	2	1	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	#41	1	2	yes	—
115	C₂×C₅×C₈	#41	1	2	yes	—
116	C₅×C₄⋊C₄	—	2	4	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	2	1	no	—
119	C₅×C₄⋊C₄	—	2	1	no	—
120	C₅×C₂²⋊C₄	—	2	1	no	—
121	C₅×C₂²⋊C₄	#41	1	2	yes	—
122	C₄²×C₅	—	2	1	no	—
123	C₄²×C₅	—	2	1	no	—
124	C₄²×C₅	C₂×D₄	1	2	yes	—
125	C₄²×C₅	#41	1	1	yes	characteristic
126	C₄²×C₅	#41	1	1	yes	characteristic
127	C₂²⋊Dic₅	—	2	2	no	—
128	C₄⋊Dic₅	—	2	2	no	—
129	C₄×Dic₅	—	2	2	no	—
130	#681	—	2	2	no	—
131	#681	—	2	2	no	—
132	#681	—	2	2	no	—
133	#681	—	2	2	no	—
134	C₂×#168	—	2	2	no	—
135	C₂×#168	—	2	2	no	—
136	Dic₂₀	—	2	2	no	—
137	D₄₀	—	2	2	no	—
138	#677	—	2	4	no	—
139	#676	—	4	2	no	—
140	#676	—	4	2	no	—
141	C₈×D₅	—	4	2	no	—
142	C₈×D₅	—	4	2	no	—
143	#575	—	5	1	no	—
144	C₂×#131	—	5	2	no	—
145	#533	C₂×D₅	1	1	yes	characteristic
146	#446	—	2	2	no	—
147	C₈×Q₈	—	2	2	no	—
148	#444	—	10	2	no	—
149	#443	—	10	2	no	—
150	#430	—	5	1	no	—
151	#424	C₂×D₅	1	1	yes	characteristic
152	#424	C₂×D₅	1	1	yes	characteristic
153	#423	C₂×D₅	1	1	yes	characteristic
154	#421	—	5	2	no	—
155	C₂×#104	—	5	2	no	—
156	#416	—	5	2	no	—
157	#413	—	5	2	no	—
158	#410	—	5	2	no	—
159	#405	—	5	2	no	—
160	#405	C₂×D₅	1	1	yes	characteristic
161	#405	C₂×D₅	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
163	#344	—	5	2	no	—
164	#344	—	5	2	no	—
165	#341	—	5	2	no	—
166	#339	—	5	2	no	—
167	#338	—	5	1	no	—
168	H(3,ℤ/4)	—	5	1	no	—
169	H(3,ℤ/4)	—	5	2	no	—
170	C₂³×C₅	C₂×#41	1	1	yes	characteristic
171	C₂²×D₅	—	2	2	no	—
172	C₅×Q₈	—	4	4	no	—
173	C₂×C₄×C₅	—	2	1	no	—
174	C₂×C₄×C₅	—	2	1	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	4	2	no	—
178	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×#41	1	2	yes	—
181	C₂×C₄×C₅	C₂×#41	1	2	yes	—
182	C₂²⋊D₅	—	4	2	no	—
183	C₂²⋊D₅	—	4	2	no	—
184	C₂×Dic₅	—	2	2	no	—
185	C₂×Dic₅	—	4	2	no	—
186	D₂₀	—	2	2	no	—
187	D₂₀	—	2	2	no	—
188	C₄×D₅	—	4	2	no	—
189	C₄×D₅	—	4	2	no	—
190	Dic₁₀	—	2	2	no	—
191	Dic₁₀	—	2	2	no	—
192	C₅×C₈	—	2	2	no	—
193	C₅×C₈	—	4	1	no	—
194	C₅×C₈	—	4	1	no	—
195	C₅×C₈	#124	1	2	yes	—
196	C₅×C₈	#124	1	1	yes	characteristic
197	C₅×C₈	#124	1	1	yes	characteristic
198	#168	—	4	2	no	—
199	#168	—	4	2	no	—
200	C₂×#41	—	5	2	no	—
201	#137	—	10	2	no	—
202	AGL(1,ℤ/8)	—	10	2	no	—
203	#135	—	10	2	no	—
204	#131	—	10	2	no	—
205	#131	—	10	2	no	—
206	#131	—	10	2	no	—
207	#131	—	10	2	no	—
208	C₂×M₁₆	—	5	2	no	—
209	C₂×M₁₆	—	5	2	no	—
210	C₂×M₁₆	—	10	2	no	—
211	C₂×M₁₆	C₂²×D₅	1	1	yes	characteristic
212	C₂×M₁₆	C₂²×D₅	1	1	yes	characteristic
213	C₂²×C₈	—	10	2	no	—
214	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
216	C₄⋊Q₈	—	2	1	no	—
217	#125	—	2	1	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	#117	—	5	2	no	—
221	#117	C₂²×D₅	1	1	yes	characteristic
222	#117	C₂²×D₅	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
224	#108	—	10	2	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	2	2	no	—
228	#104	—	10	2	no	—
229	#104	—	10	2	no	—
230	#104	—	10	4	no	—
231	#103	—	10	2	no	—
232	#102	—	10	2	no	—
233	#101	—	10	2	no	—
234	#100	—	10	2	no	—
235	#99	—	10	4	no	—
236	#97	—	2	1	no	—
237	#97	—	10	2	no	—
238	#97	C₂²⋊D₅	1	2	yes	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	—	10	2	no	—
243	C₄×C₈	C₂²⋊D₅	1	2	yes	—
244	C₂²×C₅	—	4	1	no	—
245	C₂²×C₅	#525	1	1	yes	characteristic
246	C₂²×C₅	#529	1	2	yes	—
247	C₂×D₅	—	4	2	no	—
248	C₂×D₅	—	8	2	no	—
249	C₄×C₅	—	2	2	no	—
250	C₄×C₅	—	4	1	no	—
251	C₄×C₅	—	4	2	no	—
252	C₄×C₅	—	4	2	no	—
253	C₄×C₅	#525	1	1	yes	characteristic
254	C₄×C₅	C₂×#124	1	1	yes	characteristic
255	C₄×C₅	#529	1	2	yes	—
256	Dic₅	—	4	2	no	—
257	Dic₅	—	8	2	no	—
258	#41	—	10	2	no	—
259	#41	—	10	2	no	—
260	#41	—	10	2	no	—
261	#41	—	10	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	5	2	no	—
264	C₂×D₄	—	5	2	no	—
265	C₂×D₄	—	10	2	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	10	2	no	—
268	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
269	Q₁₆	—	10	2	no	—
270	SD₁₆	—	10	4	no	—
271	D₈	—	10	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	10	2	no	—
275	M₁₆	—	10	2	no	—
276	M₁₆	—	10	2	no	—
277	M₁₆	—	10	2	no	—
278	M₁₆	—	20	2	no	—
279	M₁₆	—	20	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	10	2	no	—
284	C₂×C₈	—	10	2	no	—
285	C₂×C₈	—	20	2	no	—
286	C₂×C₈	—	20	2	no	—
287	C₂×C₈	#713	1	2	yes	—
288	C₂×C₈	#713	1	2	yes	—
289	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
290	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	10	2	no	—
297	C₂²⋊C₄	—	2	1	no	—
298	C₂²⋊C₄	—	10	2	no	—
299	C₂²⋊C₄	#709	1	2	yes	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	10	2	no	—
303	C₄²	#709	1	1	yes	characteristic
304	C₄²	#709	1	1	yes	characteristic
305	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
306	C₂×C₅	—	2	2	no	—
307	C₂×C₅	—	2	1	no	—
308	C₂×C₅	#2398	1	1	yes	characteristic
309	D₅	—	8	2	no	—
310	C₂³	—	10	2	no	—
311	C₂³	C₂×#709	1	1	yes	characteristic
312	Q₈	—	4	4	no	—
313	Q₈	—	10	2	no	—
314	Q₈	—	10	2	no	—
315	D₄	—	10	2	no	—
316	D₄	—	10	2	no	—
317	D₄	—	20	2	no	—
318	D₄	—	20	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	10	2	no	—
325	C₂×C₄	—	20	2	no	—
326	C₂×C₄	—	20	2	no	—
327	C₂×C₄	—	20	2	no	—
328	C₂×C₄	C₂×#709	1	1	yes	characteristic
329	C₂×C₄	C₂×#713	1	2	yes	—
330	C₂×C₄	D₅×#41	1	2	yes	—
331	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	4	1	no	—
334	C₈	—	4	1	no	—
335	C₈	—	20	2	no	—
336	C₈	—	20	2	no	—
337	C₈	#4093	1	2	yes	—
338	C₈	#4093	1	1	yes	characteristic
339	C₈	#4093	1	1	yes	characteristic
340	C₅	#23833	1	1	yes	characteristic, complemented, Sylow
341	C₂²	—	4	1	no	—
342	C₂²	—	20	2	no	—
343	C₂²	—	40	2	no	—
344	C₂²	#65914	1	2	yes	—
345	C₂²	#66125	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	20	2	no	—
351	C₄	—	40	2	no	—
352	C₄	#65914	1	2	yes	—
353	C₄	C₂×#4093	1	1	yes	characteristic
354	C₄	#66162	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	40	2	no	—
358	C₂	#10614223	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#4093
commutator, cosocle	C₂²×C₄×C₅	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₅×#2960	C₂
socle	C₂×C₅	#2398
prime radical	#66082	C₂²
hypercenter	#2960	D₅
hypocenter	C₅	#23833
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. C₅×#2960	C₂	semidirect
3. #10602635	C₂	semidirect
4. #10601341	C₂	semidirect
6. #10601332	C₂	semidirect
8. #10601319	C₂	semidirect
9. #10600691	C₂	semidirect
10. #10600688	C₂	semidirect
11. #10600582	C₂	semidirect
84. #2960	D₅	semidirect
340. C₅	#23833	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2960	2
#10610310	2
#10602635	2
#10601341	2
#10601340	2
#10601332	2
#10601324	2
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