Subgroups — #12016249

Order 1280.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 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×#3950	C₂	1	1	yes	maximal, characteristic
3	C₂×#64784	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊#672	C₂	1	1	yes	maximal, characteristic
5	#10598213	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	C₂²×C₄⋊Dic₅	—	2	1	no	—
10	C₅×#467	—	2	2	no	—
11	C₅×#467	C₄	1	4	yes	complemented
12	C₂×C₅×#107	C₂²	1	1	yes	characteristic
13	C₂×C₅×#103	—	2	1	no	—
14	C₂×#3950	—	2	4	no	—
15	C₂×#3950	—	2	1	no	—
16	C₂×#3950	C₂²	1	2	yes	—
17	C₂²×#672	C₂²	1	2	yes	—
18	#64784	—	2	4	no	—
19	C₂²⋊#672	—	2	2	no	—
20	#12993	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂³×Dic₅	—	4	1	no	—
24	C₂×C₅×Q₁₆	—	2	2	no	—
25	C₂×C₅×Q₁₆	—	2	1	no	—
26	C₂×C₅×Q₁₆	—	2	2	no	—
27	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	complemented
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₅×C₂²⋊Q₈	—	2	4	no	—
35	C₅×C₂²⋊Q₈	—	4	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	1	no	—
37	C₂×C₅×C₄⋊C₄	—	4	1	no	—
38	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
39	C₂×C₄⋊Dic₅	—	2	4	no	—
40	C₂×C₄⋊Dic₅	—	2	2	no	—
41	C₂×C₄⋊Dic₅	—	4	2	no	—
42	C₂×C₄⋊Dic₅	—	4	1	no	—
43	C₅×#107	—	2	1	no	—
44	C₅×#107	C₂×C₄	1	2	yes	complemented
45	C₅×#103	—	2	4	no	—
46	#3950	—	2	4	no	—
47	#3950	—	2	2	no	—
48	#3950	D₄	1	4	yes	—
49	C₂×#672	—	2	2	no	—
50	C₂×#672	—	2	2	no	—
51	C₂×#672	—	2	2	no	—
52	C₂×#467	D₅	1	1	yes	characteristic
53	C₂²×#107	—	5	1	no	—
54	C₂×#358	—	5	2	no	—
55	C₂×C₂²⋊C₁₆	—	5	1	no	—
56	#1380	—	5	2	no	—
57	C₂⁴×C₅	D₈	1	1	yes	characteristic
58	C₂×C₅×Q₈	—	2	2	no	—
59	C₂×C₅×Q₈	—	4	1	no	—
60	C₂×C₅×Q₈	—	4	1	no	—
61	C₂×C₅×Q₈	—	4	1	no	—
62	C₂×C₅×Q₈	—	4	1	no	—
63	C₂×C₅×Q₈	—	4	1	no	—
64	C₂²×C₄×C₅	—	2	1	no	—
65	C₂²×C₄×C₅	—	2	1	no	—
66	C₂²×C₄×C₅	—	2	1	no	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	—	4	1	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₅	—	4	4	no	—
75	C₂²×Dic₅	—	4	2	no	—
76	C₂²×Dic₅	—	8	2	no	—
77	C₂²×Dic₅	—	8	2	no	—
78	C₅×Q₁₆	—	2	4	no	—
79	C₅×Q₁₆	—	4	2	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	#41	1	2	yes	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	—	4	4	no	—
91	C₅×C₄⋊C₄	—	4	1	no	—
92	C₅×C₂²⋊C₄	—	4	4	no	—
93	C₄⋊Dic₅	—	2	4	no	—
94	C₄⋊Dic₅	—	4	4	no	—
95	C₄⋊Dic₅	—	4	2	no	—
96	#672	—	4	2	no	—
97	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic
98	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
99	C₂×C₂²⋊Q₈	—	2	1	no	—
100	C₂²×C₄⋊C₄	—	10	1	no	—
101	C₂²×C₁₆	—	5	2	no	—
102	#467	—	2	2	no	—
103	#467	Dic₅	1	4	yes	complemented
104	C₂×#107	—	5	2	no	—
105	C₂×#107	—	10	4	no	—
106	C₂×#107	—	10	1	no	—
107	C₂×#107	C₂×D₅	1	1	yes	characteristic
108	C₂×#103	—	2	1	no	—
109	#358	—	10	4	no	—
110	C₂²⋊C₁₆	—	10	2	no	—
111	C₂³×C₅	—	2	1	no	—
112	C₂³×C₅	—	2	1	no	—
113	C₂³×C₅	—	2	1	no	—
114	C₂³×C₅	—	2	1	no	—
115	C₂³×C₅	#102	1	2	yes	—
116	C₂³×C₅	SD₃₂	1	2	yes	—
117	C₂³×C₅	Q₃₂	1	2	yes	—
118	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
119	C₅×Q₈	—	4	2	no	—
120	C₅×Q₈	—	4	2	no	—
121	C₅×Q₈	—	8	1	no	—
122	C₅×Q₈	—	8	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	8	1	no	—
132	C₂×C₄×C₅	—	8	1	no	—
133	C₂×C₄×C₅	—	8	1	no	—
134	C₂×C₄×C₅	—	8	1	no	—
135	C₂×C₄×C₅	#102	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	#120	1	1	yes	characteristic
138	C₂×C₄×C₅	#123	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₅	#137	1	1	yes	characteristic
142	C₂×Dic₅	—	4	4	no	—
143	C₂×Dic₅	—	8	8	no	—
144	C₂×Dic₅	—	8	2	no	—
145	C₂×Dic₅	—	8	2	no	—
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	C₄×D₄	1	2	yes	—
148	C₅×C₈	#121	1	2	yes	—
149	C₂²×Q₈	—	2	1	no	—
150	C₂³×C₄	—	20	1	no	—
151	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	1	no	—
155	C₂×Q₁₆	C₂×Dic₅	1	2	yes	complemented
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	C₂×Dic₅	1	2	yes	—
159	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
160	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
161	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
162	C₂²⋊Q₈	—	2	4	no	—
163	C₂²⋊Q₈	—	4	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	4	1	no	—
166	C₂×C₄⋊C₄	—	10	4	no	—
167	C₂×C₄⋊C₄	—	10	2	no	—
168	C₂×C₄⋊C₄	—	20	2	no	—
169	C₂×C₄⋊C₄	—	20	1	no	—
170	C₂×C₂²⋊C₄	—	4	1	no	—
171	C₂×C₁₆	—	10	2	no	—
172	C₂×C₁₆	—	10	2	no	—
173	C₂×C₁₆	—	10	2	no	—
174	#107	—	2	1	no	—
175	#107	—	5	4	no	—
176	#107	—	10	4	no	—
177	#107	—	10	2	no	—
178	#107	C₂×Dic₅	1	2	yes	complemented
179	#103	—	2	4	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	#358	1	4	yes	complemented
187	C₂²×C₅	C₂×#102	1	1	yes	characteristic
188	C₂²×C₅	#447	1	1	yes	characteristic
189	C₂²×C₅	#480	1	1	yes	characteristic
190	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
191	C₂²×C₅	C₂×Q₃₂	1	1	yes	characteristic
192	C₂²×C₅	#509	1	1	yes	characteristic
193	C₂²×C₅	#510	1	1	yes	characteristic
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	8	2	no	—
196	C₄×C₅	—	8	2	no	—
197	C₄×C₅	#386	1	1	yes	characteristic
198	C₄×C₅	#417	1	1	yes	characteristic
199	C₄×C₅	#448	1	1	yes	characteristic
200	C₄×C₅	#481	1	1	yes	characteristic
201	Dic₅	—	8	4	no	—
202	C₂⁴	#686	1	1	yes	characteristic
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	20	4	no	—
216	C₂²×C₄	—	20	2	no	—
217	C₂²×C₄	—	40	2	no	—
218	C₂²×C₄	—	40	2	no	—
219	C₂²×C₄	#686	1	1	yes	characteristic
220	C₂²×C₄	#687	1	2	yes	—
221	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
223	Q₁₆	—	2	4	no	—
224	Q₁₆	—	4	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
230	C₂×C₈	D₄×D₅	1	2	yes	—
231	C₂×C₈	#711	1	2	yes	—
232	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
233	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	10	4	no	—
238	C₄⋊C₄	—	20	4	no	—
239	C₄⋊C₄	—	20	2	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₁₆	—	20	2	no	—
242	C₂×C₅	—	2	4	no	—
243	C₂×C₅	#1874	1	1	yes	characteristic
244	C₂×C₅	C₂×#358	1	1	yes	characteristic
245	C₂×C₅	#2139	1	1	yes	characteristic
246	C₂×C₅	#2187	1	1	yes	characteristic
247	C₂×C₅	#2188	1	1	yes	characteristic
248	C₂×C₅	#2235	1	1	yes	characteristic
249	C₂×C₅	#2236	1	1	yes	characteristic
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	#3960	1	2	yes	—
255	C₂³	#3961	1	2	yes	—
256	C₂³	#3964	1	2	yes	—
257	C₂³	C₂×#686	1	1	yes	characteristic
258	Q₈	—	4	2	no	—
259	Q₈	—	4	2	no	—
260	Q₈	—	8	1	no	—
261	Q₈	—	8	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	20	4	no	—
275	C₂×C₄	—	40	8	no	—
276	C₂×C₄	—	40	2	no	—
277	C₂×C₄	—	40	2	no	—
278	C₂×C₄	#3964	1	2	yes	—
279	C₂×C₄	C₂×#687	1	1	yes	characteristic
280	C₂×C₄	#4081	1	1	yes	characteristic
281	C₂×C₄	#4083	1	1	yes	characteristic
282	C₂×C₄	#4095	1	1	yes	characteristic
283	C₂×C₄	#4097	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₈	D₄×Dic₅	1	2	yes	—
287	C₈	#4085	1	2	yes	—
288	C₅	#12993	1	1	yes	characteristic, complemented, Sylow
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#64784	1	4	yes	complemented
296	C₂²	#65322	1	1	yes	characteristic
297	C₂²	#65325	1	1	yes	characteristic
298	C₂²	C₂×#3960	1	1	yes	characteristic
299	C₂²	C₂×#3961	1	1	yes	characteristic
300	C₂²	#65482	1	1	yes	characteristic
301	C₂²	#65484	1	1	yes	characteristic
302	C₂²	C₂×#3964	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	—	40	4	no	—
307	C₄	#65323	1	1	yes	characteristic
308	C₄	#65329	1	1	yes	characteristic
309	C₄	#65509	1	1	yes	characteristic
310	C₄	#65521	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	#10601871	1	1	yes	characteristic
313	C₂	#10601872	1	1	yes	characteristic
314	C₂	#10601878	1	1	yes	characteristic
315	C₂	#10601880	1	1	yes	characteristic
316	C₂	#10602470	1	1	yes	characteristic
317	C₂	C₂×#64784	1	1	yes	characteristic
318	C₂	#10602871	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#686
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#467	C₂
socle	C₂³×C₅	C₂×D₈
prime radical	C₂⁴×C₅	D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#467	D₅
hypocenter	C₅	#12993
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
3. C₂×#64784	C₂	semidirect
5. #10598213	C₂	semidirect
11. C₅×#467	C₄	semidirect
27. C₂×C₅×Q₁₆	C₂×C₄	semidirect
44. C₅×#107	C₂×C₄	semidirect
103. #467	Dic₅	semidirect
155. C₂×Q₁₆	C₂×Dic₅	semidirect
178. #107	C₂×Dic₅	semidirect
186. C₂²×C₅	#358	semidirect
288. C₅	#12993	semidirect
295. C₂²	#64784	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×#3950	2
C₂×#64784	2
C₂×C₂²⋊#672	2
#10598213	2
#12993	5