Subgroups — #12252481

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2929	C₂	1	1	yes	maximal, characteristic
2	C₂×#65429	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#3944	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602797	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602756	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601651	C₂	1	4	yes	maximal, complemented
7	#10601650	C₂	1	4	yes	maximal, complemented
8	#10600675	C₂	1	1	yes	maximal, characteristic, complemented
9	#10600674	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#525	C₄	1	1	yes	characteristic
13	C₅×#435	C₄	1	4	yes	—
14	C₅×C₈×D₄	C₂²	1	4	yes	—
15	C₅×#424	C₄	1	1	yes	characteristic
16	C₅×#408	C₄	1	1	yes	characteristic
17	C₅×#407	C₄	1	1	yes	characteristic
18	C₅×#405	C₂²	1	1	yes	characteristic
19	#65429	—	2	4	no	—
20	C₂×#3944	—	2	4	no	—
21	C₂×#3944	C₂²	1	1	yes	characteristic
22	C₂×#3944	C₂²	1	1	yes	characteristic
23	C₂×#3944	C₂²	1	2	yes	—
24	C₂×#3944	C₂²	1	2	yes	—
25	C₂×#3944	C₂²	1	2	yes	complemented
26	C₂²×#672	C₂²	1	1	yes	characteristic
27	C₂²×#672	C₂²	1	1	yes	characteristic
28	C₂²×#672	C₂²	1	2	yes	complemented
29	C₂²⋊#672	C₂²	1	4	yes	—
30	C₂²⋊#672	C₂²	1	4	yes	complemented
31	C₄⋊#672	C₂²	1	4	yes	complemented
32	#64681	C₂²	1	2	yes	complemented
33	C₄×#672	C₂²	1	2	yes	complemented
34	#33433	—	5	1	no	maximal, Sylow
35	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
37	C₅×#131	—	2	2	no	—
38	C₅×#131	—	2	2	no	—
39	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
42	C₂²×C₅×C₈	—	2	2	no	—
43	C₂²×C₅×C₈	—	2	2	no	—
44	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₅×C₈	C₂³	1	2	yes	—
47	C₂²×C₅×C₈	C₂³	1	2	yes	—
48	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₅×#123	C₈	1	2	yes	—
50	C₅×C₂²⋊Q₈	C₈	1	2	yes	—
51	C₅×#121	C₈	1	2	yes	—
52	C₅×#120	C₈	1	2	yes	—
53	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
54	C₅×#117	C₂×C₄	1	1	yes	characteristic
55	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
57	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
58	C₅×#97	C₂×C₄	1	2	yes	—
59	C₄×C₅×C₈	C₂³	1	2	yes	—
60	#3944	—	2	2	no	—
61	#3944	—	2	2	no	—
62	#3944	—	2	4	no	—
63	#3944	—	2	4	no	—
64	#3944	—	2	4	no	—
65	C₂×#672	—	2	2	no	—
66	C₂×#672	—	2	4	no	—
67	C₂×#672	D₄	1	4	yes	complemented
68	C₂×#672	C₂³	1	2	yes	—
69	C₂×#672	C₂³	1	2	yes	—
70	C₂×#672	C₂³	1	2	yes	—
71	C₂×#672	C₂³	1	1	yes	characteristic
72	C₂×#672	C₂³	1	1	yes	characteristic
73	C₂×#504	—	5	1	no	—
74	C₂²×M₃₂	—	5	1	no	—
75	#2929	D₅	1	1	yes	characteristic
76	#2169	—	5	4	no	—
77	#2168	—	5	4	no	—
78	#2151	—	5	1	no	—
79	#2113	—	5	1	no	—
80	#2112	—	5	1	no	—
81	#2108	—	5	1	no	—
82	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
83	C₅×#41	—	2	2	no	—
84	C₅×#41	—	2	2	no	—
85	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
86	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
87	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
88	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
89	C₂²×C₄×C₅	—	2	2	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
92	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₅×M₁₆	—	2	2	no	—
97	C₅×M₁₆	—	2	2	no	—
98	C₅×M₁₆	—	2	4	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	4	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₅×C₈	#41	1	2	yes	—
111	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
112	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
113	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
114	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
115	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
116	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
117	C₄²×C₅	C₂²×C₄	1	2	yes	—
118	#672	—	2	2	no	—
119	#672	—	2	2	no	—
120	#672	—	2	2	no	—
121	#672	C₂×D₄	1	2	yes	—
122	#672	C₂×D₄	1	2	yes	—
123	C₂×#131	C₂×D₅	1	1	yes	characteristic
124	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
125	#525	Dic₅	1	1	yes	characteristic
126	#504	—	10	4	no	—
127	C₂×M₃₂	—	5	1	no	—
128	C₂×M₃₂	—	5	1	no	—
129	C₂×M₃₂	—	5	2	no	—
130	C₂×M₃₂	—	5	2	no	—
131	C₂×M₃₂	—	5	2	no	—
132	C₂×M₃₂	—	10	4	no	—
133	C₂²×C₁₆	—	5	1	no	—
134	C₂²×C₁₆	—	5	1	no	—
135	C₂²×C₁₆	—	5	2	no	—
136	#435	Dic₅	1	4	yes	—
137	C₈×D₄	C₂×D₅	1	4	yes	—
138	#424	Dic₅	1	1	yes	characteristic
139	#408	Dic₅	1	1	yes	characteristic
140	#407	Dic₅	1	1	yes	characteristic
141	#405	C₂×D₅	1	1	yes	characteristic
142	C₄⋊C₁₆	—	5	4	no	—
143	C₂²⋊C₁₆	—	5	4	no	—
144	C₂²⋊C₁₆	—	5	4	no	—
145	#346	—	5	2	no	—
146	C₄×C₁₆	—	5	2	no	—
147	C₂³×C₅	—	2	2	no	—
148	C₂³×C₅	—	4	2	no	—
149	C₂³×C₅	C₂²×C₈	1	2	yes	—
150	C₂³×C₅	C₂²×C₈	1	2	yes	—
151	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
152	C₅×Q₈	—	2	2	no	—
153	C₅×D₄	—	2	4	no	—
154	C₅×D₄	—	2	2	no	—
155	C₅×D₄	—	4	4	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
163	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
164	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
165	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
166	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
167	C₂×C₄×C₅	#131	1	2	yes	—
168	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
169	C₅×C₈	—	2	2	no	—
170	C₅×C₈	—	2	2	no	—
171	C₅×C₈	—	2	2	no	—
172	C₅×C₈	C₄×D₄	1	2	yes	—
173	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
174	C₅×C₈	C₂×#41	1	1	yes	characteristic
175	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
176	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
177	#131	—	2	2	no	—
178	#131	—	2	2	no	—
179	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
180	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
181	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₅	1	2	yes	—
186	C₂²×C₈	C₂²×D₅	1	2	yes	—
187	C₂²×C₈	C₂²×D₅	1	2	yes	—
188	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
189	#123	#168	1	2	yes	—
190	C₂²⋊Q₈	#168	1	2	yes	—
191	#121	#168	1	2	yes	—
192	#120	#168	1	2	yes	—
193	C₄×D₄	C₂×Dic₅	1	4	yes	—
194	#117	C₂×Dic₅	1	1	yes	characteristic
195	M₃₂	—	10	4	no	—
196	M₃₂	—	10	4	no	—
197	M₃₂	—	10	4	no	—
198	M₃₂	—	10	2	no	—
199	M₃₂	—	10	2	no	—
200	C₂×C₁₆	—	5	4	no	—
201	C₂×C₁₆	—	5	1	no	—
202	C₂×C₁₆	—	5	1	no	—
203	C₂×C₁₆	—	5	2	no	—
204	C₂×C₁₆	—	5	2	no	—
205	C₂×C₁₆	—	5	2	no	—
206	C₂×C₁₆	—	10	4	no	—
207	C₂×C₁₆	—	10	2	no	—
208	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
209	C₂²⋊C₈	C₂×Dic₅	1	4	yes	—
210	C₂²⋊C₈	C₂×Dic₅	1	4	yes	—
211	#97	C₂×Dic₅	1	2	yes	—
212	C₄×C₈	C₂²×D₅	1	2	yes	—
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	—	2	2	no	—
215	C₂²×C₅	—	2	4	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	4	4	no	—
220	C₂²×C₅	C₈×D₄	1	2	yes	—
221	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
222	C₄×C₅	—	2	4	no	—
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	—	2	2	no	—
225	C₄×C₅	C₈×D₄	1	2	yes	—
226	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	C₄×C₅	C₂×#131	1	1	yes	characteristic
228	C₂⁴	C₂×#168	1	1	yes	characteristic
229	#41	—	2	2	no	—
230	#41	—	2	2	no	—
231	C₂×Q₈	C₂×#168	1	1	yes	characteristic
232	C₂×D₄	C₂×#168	1	2	yes	—
233	C₂×D₄	C₂×#168	1	1	yes	characteristic
234	C₂×D₄	C₂×#168	1	4	yes	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂×#168	1	2	yes	—
238	C₂²×C₄	C₂×#168	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
240	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	D₄×D₅	1	2	yes	—
252	C₂×C₈	#711	1	2	yes	—
253	C₂×C₈	C₂²×Dic₅	1	2	yes	—
254	C₂×C₈	C₂²×Dic₅	1	2	yes	—
255	C₂×C₈	C₂²×Dic₅	1	2	yes	—
256	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
258	C₄⋊C₄	C₂×#168	1	2	yes	—
259	C₄⋊C₄	C₂×#168	1	4	yes	—
260	C₂²⋊C₄	C₂×#168	1	2	yes	—
261	C₂²⋊C₄	C₂×#168	1	4	yes	—
262	C₂²⋊C₄	C₂×#168	1	4	yes	—
263	C₄²	C₂²×Dic₅	1	2	yes	—
264	C₁₆	—	5	2	no	—
265	C₁₆	—	5	2	no	—
266	C₁₆	—	10	2	no	—
267	C₁₆	—	10	2	no	—
268	C₁₆	—	10	2	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	4	2	no	—
271	C₂×C₅	—	4	2	no	—
272	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
273	C₂×C₅	#3407	1	2	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	C₂²×#168	1	2	yes	—
277	C₂³	C₂²×#168	1	2	yes	—
278	C₂³	C₂²×#168	1	1	yes	characteristic
279	Q₈	—	2	2	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	2	no	—
282	D₄	—	4	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	C₂²×#168	1	4	yes	—
290	C₂×C₄	C₂²×#168	1	2	yes	—
291	C₂×C₄	C₂²×#168	1	2	yes	—
292	C₂×C₄	C₂²×#168	1	1	yes	characteristic
293	C₂×C₄	D₄×Dic₅	1	2	yes	—
294	C₂×C₄	#4094	1	2	yes	—
295	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	D₄×Dic₅	1	2	yes	—
300	C₈	C₂×D₄×D₅	1	1	yes	characteristic
301	C₈	C₂×#711	1	1	yes	characteristic
302	C₅	#33433	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	4	4	no	—
310	C₂²	D₄×#168	1	2	yes	—
311	C₂²	C₂³×#168	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	4	no	—
315	C₄	D₄×#168	1	2	yes	—
316	C₄	C₂×D₄×Dic₅	1	1	yes	characteristic
317	C₄	C₂×#4094	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	4	2	no	—
321	C₂	C₂×D₄×#168	1	1	yes	characteristic
322	C₂	#10615511	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2929	C₂
prime radical	C₅×#525	C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2929	D₅
hypocenter	C₅	#33433
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
2. C₂×#65429	C₂	semidirect
3. C₂²×#3944	C₂	semidirect
4. #10602797	C₂	semidirect
5. #10602756	C₂	semidirect
6. #10601651	C₂	semidirect
7. #10601650	C₂	semidirect
8. #10600675	C₂	semidirect
9. #10600674	C₂	semidirect
25. C₂×#3944	C₂²	semidirect
28. C₂²×#672	C₂²	semidirect
30. C₂²⋊#672	C₂²	semidirect
31. C₄⋊#672	C₂²	semidirect
32. #64681	C₂²	semidirect
33. C₄×#672	C₂²	semidirect
67. C₂×#672	D₄	semidirect
302. C₅	#33433	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2929	2
C₂×#65429	2
C₂²×#3944	2
#10602797	2
#10602756	2
#10601651	2
#10601650	2
#10600675	2
#10600674	2
#33433	5