Subgroups — #12016698

Order 1280.

484 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#467	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3944	C₂	1	1	yes	maximal, characteristic
3	C₂×#64785	C₂	1	2	yes	maximal, complemented
4	#10602770	C₂	1	1	yes	maximal, characteristic
5	#10598214	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₂²⋊Q₈	—	2	1	no	—
9	C₅×#467	—	2	1	no	—
10	C₅×#467	—	2	1	no	—
11	C₅×#467	C₄	1	2	yes	—
12	C₅×#467	C₄	1	2	yes	—
13	C₂×C₅×#107	C₂²	1	1	yes	characteristic
14	C₂×C₅×#103	—	2	1	no	—
15	#65496	—	2	1	no	—
16	C₂×#3951	C₂²	1	2	yes	—
17	C₂×#3944	—	2	4	no	—
18	C₂×#3944	—	2	1	no	—
19	C₂×#3944	C₂²	1	2	yes	—
20	C₂²×#672	—	2	1	no	—
21	#64785	—	2	4	no	—
22	#64772	—	2	1	no	—
23	#13066	—	5	1	no	maximal, Sylow
24	C₂²×C₅×Q₈	—	2	1	no	—
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂×C₅×Q₁₆	—	2	2	no	—
27	C₂×C₅×Q₁₆	—	2	1	no	—
28	C₂×C₅×Q₁₆	—	2	2	no	—
29	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₅×C₂²⋊Q₈	—	2	2	no	—
39	C₅×C₂²⋊Q₈	—	2	2	no	—
40	C₅×C₂²⋊Q₈	—	4	1	no	—
41	C₅×C₂²⋊Q₈	—	4	1	no	—
42	C₂×C₅×C₄⋊C₄	—	2	1	no	—
43	C₂×C₅×C₄⋊C₄	—	4	1	no	—
44	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
45	C₂×#681	—	2	2	no	—
46	C₅×#107	—	2	1	no	—
47	C₅×#107	C₂×C₄	1	2	yes	—
48	C₅×#103	—	2	2	no	—
49	C₅×#103	—	2	2	no	—
50	C₂²⋊#168	—	4	1	no	—
51	C₂²⋊#168	—	4	1	no	—
52	#3951	—	2	2	no	—
53	#3951	—	2	2	no	—
54	#3944	—	2	4	no	—
55	#3944	—	2	2	no	—
56	#3944	D₄	1	4	yes	—
57	C₂×#672	—	2	4	no	—
58	C₂×#672	—	2	2	no	—
59	C₂²×M₃₂	—	5	1	no	—
60	C₂×#467	D₅	1	1	yes	characteristic
61	C₂×#362	—	5	2	no	—
62	#1810	—	5	1	no	—
63	#1387	—	5	2	no	—
64	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
65	C₂×C₅×Q₈	—	2	2	no	—
66	C₂×C₅×Q₈	—	4	1	no	—
67	C₂×C₅×Q₈	—	4	2	no	—
68	C₂×C₅×Q₈	—	4	2	no	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	4	1	no	—
72	C₂²×C₄×C₅	—	4	1	no	—
73	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₅×Q₁₆	—	2	4	no	—
80	C₅×Q₁₆	—	4	1	no	—
81	C₅×Q₁₆	—	4	1	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	2	1	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	#41	1	2	yes	—
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₄⋊C₄	—	4	1	no	—
99	C₅×C₂²⋊C₄	—	4	2	no	—
100	C₅×C₂²⋊C₄	—	4	2	no	—
101	#681	—	4	2	no	—
102	#681	—	4	2	no	—
103	C₂×#168	—	4	2	no	—
104	#672	—	2	4	no	—
105	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic
106	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
107	C₂×C₂²⋊Q₈	—	2	1	no	—
108	C₂×M₃₂	—	5	2	no	—
109	C₂×M₃₂	—	10	4	no	—
110	C₂×M₃₂	—	10	1	no	—
111	C₂²×C₁₆	—	10	1	no	—
112	#467	—	2	1	no	—
113	#467	—	2	1	no	—
114	#467	Dic₅	1	2	yes	—
115	#467	Dic₅	1	2	yes	—
116	C₂×#108	—	5	2	no	—
117	C₂×#107	C₂×D₅	1	1	yes	characteristic
118	C₂×#103	—	2	1	no	—
119	#407	—	10	1	no	—
120	#362	—	10	4	no	—
121	#341	—	10	1	no	—
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	—	2	2	no	—
124	C₂³×C₅	—	4	1	no	—
125	C₂³×C₅	#102	1	1	yes	characteristic
126	C₂³×C₅	#102	1	1	yes	characteristic
127	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
128	C₅×Q₈	—	4	2	no	—
129	C₅×Q₈	—	4	2	no	—
130	C₅×Q₈	—	8	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	8	2	no	—
141	C₂×C₄×C₅	—	8	2	no	—
142	C₂×C₄×C₅	#102	1	1	yes	characteristic
143	C₂×C₄×C₅	#102	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	#121	1	2	yes	—
146	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₅	#135	1	2	yes	—
148	C₅×C₈	—	2	1	no	—
149	C₅×C₈	—	2	1	no	—
150	C₅×C₈	C₄×D₄	1	2	yes	—
151	C₅×C₈	#120	1	1	yes	characteristic
152	C₅×C₈	#123	1	1	yes	characteristic
153	#168	—	8	2	no	—
154	C₂²×Q₈	—	2	1	no	—
155	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
156	C₂×Q₁₆	—	2	2	no	—
157	C₂×Q₁₆	—	2	2	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	C₂×Dic₅	1	2	yes	—
160	C₂×M₁₆	—	10	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
169	C₂²⋊Q₈	—	2	2	no	—
170	C₂²⋊Q₈	—	2	2	no	—
171	C₂²⋊Q₈	—	4	1	no	—
172	C₂²⋊Q₈	—	4	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	4	1	no	—
175	C₂×C₂²⋊C₄	—	4	1	no	—
176	M₃₂	—	5	4	no	—
177	M₃₂	—	10	4	no	—
178	M₃₂	—	10	2	no	—
179	C₂×C₁₆	—	10	4	no	—
180	C₂×C₁₆	—	10	2	no	—
181	#108	—	10	2	no	—
182	#108	—	10	2	no	—
183	#107	—	2	1	no	—
184	#107	C₂×Dic₅	1	2	yes	—
185	#103	—	2	2	no	—
186	#103	—	2	2	no	—
187	C₂²⋊C₈	—	20	1	no	—
188	C₂²⋊C₈	—	20	1	no	—
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	—	2	1	no	—
194	C₂²×C₅	—	2	1	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	4	4	no	—
197	C₂²×C₅	#362	1	2	yes	complemented
198	C₂²×C₅	C₂×#102	1	1	yes	characteristic
199	C₂²×C₅	#465	1	2	yes	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	—	2	1	no	—
202	C₄×C₅	—	8	2	no	—
203	C₄×C₅	—	8	2	no	—
204	C₄×C₅	#386	1	1	yes	characteristic
205	C₄×C₅	#416	1	1	yes	characteristic
206	C₄×C₅	#466	1	2	yes	—
207	C₂⁴	#687	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	4	1	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	#686	1	1	yes	characteristic
217	C₂²×C₄	#686	1	1	yes	characteristic
218	C₂²×C₄	#687	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
222	Q₁₆	—	2	4	no	—
223	Q₁₆	—	4	1	no	—
224	Q₁₆	—	4	1	no	—
225	M₁₆	—	20	2	no	—
226	M₁₆	—	20	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	20	2	no	—
235	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
237	C₂×C₈	D₄×D₅	1	2	yes	—
238	C₂×C₈	#711	1	2	yes	—
239	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₁₆	—	10	4	no	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	—	2	2	no	—
250	C₂×C₅	—	4	1	no	—
251	C₂×C₅	#1893	1	1	yes	characteristic
252	C₂×C₅	C₂×#362	1	1	yes	characteristic
253	C₂×C₅	#2148	1	1	yes	characteristic
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	4	1	no	—
257	C₂³	#3964	1	1	yes	characteristic
258	C₂³	#3964	1	1	yes	characteristic
259	C₂³	C₂×#687	1	1	yes	characteristic
260	Q₈	—	4	2	no	—
261	Q₈	—	4	2	no	—
262	Q₈	—	8	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	#3964	1	1	yes	characteristic
275	C₂×C₄	#3964	1	1	yes	characteristic
276	C₂×C₄	C₂×#686	1	1	yes	characteristic
277	C₂×C₄	#4085	1	2	yes	—
278	C₂×C₄	#4096	1	2	yes	—
279	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	40	2	no	—
283	C₈	D₄×Dic₅	1	2	yes	—
284	C₈	#4081	1	1	yes	characteristic
285	C₈	#4083	1	1	yes	characteristic
286	C₅	#13066	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	4	4	no	—
295	C₂²	#64785	1	2	yes	complemented
296	C₂²	#65330	1	2	yes	—
297	C₂²	C₂×#3964	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	#65327	1	2	yes	—
303	C₄	#65509	1	1	yes	characteristic
304	C₄	#65522	1	1	yes	characteristic
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	1	no	—
308	C₂	#10602474	1	1	yes	characteristic
309	C₂	C₂×#64785	1	1	yes	characteristic
310	C₂	#10602874	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#3964
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#467	C₂
socle	C₂²×C₅	C₂×#102
prime radical	C₂⁴×C₅	SD₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#467	D₅
hypocenter	C₅	#13066
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
3. C₂×#64785	C₂	semidirect
5. #10598214	C₂	semidirect
197. C₂²×C₅	#362	semidirect
286. C₅	#13066	semidirect
295. C₂²	#64785	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#467	2
C₂²×#3944	2
C₂×#64785	2
#10602770	2
#10598214	2
#13066	5