Subgroups — #12014334

Order 1280.

514 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×#64978	C₂	1	1	yes	maximal, characteristic
2	C₂×#64727	C₂	1	1	yes	maximal, characteristic
3	C₂×#64727	C₂	1	1	yes	maximal, characteristic
4	C₂×#64722	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1374	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598148	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#676	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₂×#4018	C₄	1	1	yes	characteristic
11	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×#3947	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
16	C₂²×#672	C₂²	1	1	yes	characteristic
17	C₂²×#672	C₂²	1	1	yes	characteristic
18	C₂×#3935	C₄	1	1	yes	characteristic
19	#64978	—	2	2	no	—
20	#64978	—	2	2	no	—
21	#64727	C₄	1	4	yes	complemented
22	#64727	C₄	1	4	yes	complemented
23	#64722	—	2	2	no	—
24	C₁₆×Dic₅	—	2	2	no	—
25	#12600	—	5	1	no	maximal, Sylow
26	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×#685	C₈	1	2	yes	—
31	C₂×#683	C₈	1	2	yes	—
32	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×#168	C₂×C₄	1	1	yes	characteristic
34	C₂²×#168	C₂×C₄	1	1	yes	characteristic
35	C₂×#676	—	2	2	no	—
36	C₂×#676	—	2	2	no	—
37	C₂×#676	C₈	1	4	yes	—
38	C₂×C₈×D₅	—	2	1	no	—
39	C₂×C₈×D₅	—	2	1	no	—
40	C₂×C₈×D₅	—	2	1	no	—
41	C₂×C₈×D₅	—	2	1	no	—
42	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
44	#4018	—	2	2	no	—
45	#4018	—	2	2	no	—
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₂×C₅×C₁₆	—	2	1	no	—
48	C₂×C₅×C₁₆	—	2	1	no	—
49	C₂×C₅×C₁₆	—	2	1	no	—
50	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
51	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
52	C₄×C₅×C₈	—	2	1	no	—
53	C₄×C₅×C₈	—	2	1	no	—
54	#3947	—	2	2	no	—
55	C₈×Dic₅	D₄	1	2	yes	—
56	C₈×Dic₅	D₄	1	2	yes	—
57	C₂×#672	—	2	1	no	—
58	C₂×#672	—	2	1	no	—
59	C₂×#672	—	2	1	no	—
60	C₂×#672	—	2	1	no	—
61	C₂×#672	C₂×C₄	1	2	yes	—
62	C₂×#672	C₂×C₄	1	2	yes	—
63	#3935	—	2	2	no	—
64	C₂×#405	—	5	1	no	—
65	C₂×C₂²⋊C₁₆	—	5	1	no	—
66	C₂×C₂²⋊C₁₆	—	5	1	no	—
67	C₂×#346	—	5	1	no	—
68	C₂×C₄×C₁₆	—	5	1	no	—
69	#1374	—	5	1	no	—
70	#1374	D₅	1	1	yes	characteristic, complemented
71	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
75	C₂²×Dic₅	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₅	—	2	1	no	—
77	C₂×C₄×D₅	—	2	1	no	—
78	C₂×C₄×D₅	—	2	1	no	—
79	C₂×C₄×D₅	—	2	1	no	—
80	C₂×C₄×D₅	C₄²	1	2	yes	—
81	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	#41	1	1	yes	characteristic
90	C₂×C₅×C₈	#41	1	1	yes	characteristic
91	C₄²×C₅	—	2	1	no	—
92	C₄²×C₅	—	2	1	no	—
93	#685	—	2	4	no	—
94	#683	—	2	4	no	—
95	C₄×Dic₅	C₂²⋊C₄	1	2	yes	—
96	C₄×Dic₅	M₁₆	1	2	yes	—
97	C₂×#168	—	2	1	no	—
98	C₂×#168	—	2	1	no	—
99	C₂×#168	C₄²	1	2	yes	—
100	C₂×#168	C₂²⋊C₄	1	2	yes	—
101	C₂×#168	C₂×C₈	1	2	yes	—
102	C₂×#168	M₁₆	1	2	yes	—
103	#676	—	2	8	no	—
104	C₈×D₅	—	2	2	no	—
105	C₈×D₅	—	2	2	no	—
106	C₈×D₅	—	2	2	no	—
107	C₈×D₅	—	2	2	no	—
108	C₅×C₁₆	—	2	2	no	—
109	C₅×C₁₆	—	2	2	no	—
110	#672	—	2	2	no	—
111	#672	—	2	2	no	—
112	C₂²×M₁₆	—	5	1	no	—
113	C₂³×C₈	—	5	1	no	—
114	C₂×#117	—	5	1	no	—
115	C₂²×C₁₆	—	5	1	no	—
116	C₂²×C₁₆	—	5	1	no	—
117	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
118	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
119	#405	—	10	2	no	—
120	#405	—	10	2	no	—
121	C₂×#97	—	5	1	no	—
122	C₂×#97	—	5	1	no	—
123	C₂×C₄×C₈	—	5	1	no	—
124	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
125	C₂²⋊C₁₆	—	5	4	no	—
126	C₂²⋊C₁₆	—	5	4	no	—
127	#346	—	10	2	no	—
128	C₄×C₁₆	—	10	2	no	—
129	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
130	C₂²×D₅	—	2	1	no	—
131	C₂²×D₅	—	2	1	no	—
132	C₂²×D₅	—	2	2	no	—
133	C₂²×D₅	—	2	2	no	—
134	C₂²×D₅	C₄×C₈	1	2	yes	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	#117	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
144	C₂×Dic₅	—	2	1	no	—
145	C₂×Dic₅	—	2	1	no	—
146	C₂×Dic₅	—	2	2	no	—
147	C₂×Dic₅	—	2	2	no	—
148	C₂×Dic₅	C₄×C₈	1	2	yes	—
149	C₂×Dic₅	C₂²⋊C₈	1	4	yes	—
150	C₄×D₅	—	2	2	no	—
151	C₄×D₅	—	2	2	no	—
152	C₄×D₅	—	2	2	no	—
153	C₄×D₅	—	2	2	no	—
154	C₅×C₈	—	2	2	no	—
155	C₅×C₈	C₄×D₄	1	1	yes	characteristic
156	C₅×C₈	C₄×D₄	1	1	yes	characteristic
157	C₅×C₈	C₄×D₄	1	1	yes	characteristic
158	C₅×C₈	C₄×D₄	1	1	yes	characteristic
159	#168	—	2	2	no	—
160	#168	C₂²⋊C₈	1	4	yes	—
161	C₂³×C₄	—	5	1	no	—
162	C₂×M₁₆	—	5	4	no	—
163	C₂×M₁₆	—	10	2	no	—
164	C₂×M₁₆	—	10	2	no	—
165	C₂²×C₈	—	5	1	no	—
166	C₂²×C₈	—	5	1	no	—
167	C₂²×C₈	—	5	2	no	—
168	C₂²×C₈	—	5	2	no	—
169	C₂²×C₈	—	10	1	no	—
170	C₂²×C₈	—	10	1	no	—
171	C₂²×C₈	—	10	1	no	—
172	C₂²×C₈	—	10	1	no	—
173	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
175	#117	—	10	2	no	—
176	#117	—	10	2	no	—
177	C₂×C₄⋊C₄	—	5	2	no	—
178	C₂×C₂²⋊C₄	—	5	2	no	—
179	C₂×C₄²	—	5	1	no	—
180	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	5	2	no	—
186	C₂×C₁₆	—	5	2	no	—
187	C₂×C₁₆	—	10	1	no	—
188	C₂×C₁₆	—	10	1	no	—
189	C₂×C₁₆	—	10	1	no	—
190	C₂×C₁₆	—	10	1	no	—
191	C₂×C₁₆	C₄×D₅	1	2	yes	—
192	C₂×C₁₆	C₄×D₅	1	2	yes	—
193	#97	—	10	2	no	—
194	#97	—	10	2	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	5	2	no	—
198	C₄×C₈	—	5	2	no	—
199	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
200	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
201	C₂²×C₅	#431	1	1	yes	characteristic
202	C₂²×C₅	#504	1	2	yes	—
203	C₂²×C₅	#504	1	2	yes	—
204	C₂×D₅	—	2	2	no	—
205	C₂×D₅	—	2	2	no	—
206	C₂×D₅	—	2	2	no	—
207	C₂×D₅	—	4	2	no	—
208	C₂×D₅	—	4	2	no	—
209	C₄×C₅	—	4	2	no	—
210	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
211	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
212	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
213	C₄×C₅	#431	1	1	yes	characteristic
214	Dic₅	—	2	4	no	—
215	Dic₅	—	2	2	no	—
216	C₂⁴	—	5	1	no	—
217	C₂²×C₄	—	5	2	no	—
218	C₂²×C₄	—	5	1	no	—
219	C₂²×C₄	—	5	2	no	—
220	C₂²×C₄	—	5	2	no	—
221	C₂²×C₄	—	10	1	no	—
222	C₂²×C₄	—	10	1	no	—
223	C₂²×C₄	—	10	1	no	—
224	C₂²×C₄	—	10	1	no	—
225	C₂²×C₄	C₈×D₅	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
227	M₁₆	—	10	8	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	5	2	no	—
231	C₂×C₈	—	5	2	no	—
232	C₂×C₈	—	5	2	no	—
233	C₂×C₈	—	5	2	no	—
234	C₂×C₈	—	10	1	no	—
235	C₂×C₈	—	10	1	no	—
236	C₂×C₈	—	10	2	no	—
237	C₂×C₈	—	10	2	no	—
238	C₂×C₈	—	10	2	no	—
239	C₂×C₈	—	10	2	no	—
240	C₂×C₈	C₈×D₅	1	2	yes	—
241	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
243	C₂×C₈	D₄×D₅	1	1	yes	characteristic
244	C₂×C₈	D₄×D₅	1	1	yes	characteristic
245	C₂×C₈	#711	1	1	yes	characteristic
246	C₂×C₈	#713	1	1	yes	characteristic
247	C₄⋊C₄	—	10	4	no	—
248	C₂²⋊C₄	—	10	4	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	5	2	no	—
252	C₄²	—	5	2	no	—
253	C₁₆	—	2	2	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	10	2	no	—
256	C₁₆	—	10	2	no	—
257	C₂×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic
258	C₂×C₅	#2108	1	1	yes	characteristic
259	C₂×C₅	#2108	1	1	yes	characteristic
260	C₂×C₅	#2113	1	2	yes	—
261	C₂×C₅	#2164	1	2	yes	—
262	D₅	—	4	2	no	—
263	C₂³	—	5	2	no	—
264	C₂³	—	10	1	no	—
265	C₂³	—	10	2	no	—
266	C₂³	—	10	2	no	—
267	C₂³	—	10	1	no	—
268	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	5	2	no	—
273	C₂×C₄	—	5	4	no	—
274	C₂×C₄	—	10	1	no	—
275	C₂×C₄	—	10	2	no	—
276	C₂×C₄	—	10	2	no	—
277	C₂×C₄	—	10	1	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	—	10	2	no	—
280	C₂×C₄	—	10	2	no	—
281	C₂×C₄	—	10	2	no	—
282	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
283	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#4038	1	1	yes	characteristic
285	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
286	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
287	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	5	4	no	—
290	C₈	—	10	2	no	—
291	C₈	#4027	1	1	yes	characteristic
292	C₈	#4027	1	1	yes	characteristic
293	C₈	#4039	1	1	yes	characteristic
294	C₈	#4039	1	1	yes	characteristic
295	C₅	#12600	1	1	yes	characteristic, complemented, Sylow
296	C₂²	—	10	2	no	—
297	C₂²	—	10	2	no	—
298	C₂²	—	10	2	no	—
299	C₂²	—	20	2	no	—
300	C₂²	—	20	2	no	—
301	C₂²	C₄×C₈×D₅	1	1	yes	characteristic
302	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
303	C₂²	#65122	1	1	yes	characteristic
304	C₂²	#65190	1	2	yes	—
305	C₂²	#65196	1	2	yes	—
306	C₄	—	4	2	no	—
307	C₄	—	10	4	no	—
308	C₄	—	10	2	no	—
309	C₄	#64955	1	1	yes	characteristic
310	C₄	#64996	1	1	yes	characteristic
311	C₄	D₅×C₂²⋊C₈	1	1	yes	characteristic
312	C₄	#65122	1	1	yes	characteristic
313	C₂	—	20	2	no	—
314	C₂	#10599392	1	1	yes	characteristic
315	C₂	#10599844	1	1	yes	characteristic
316	C₂	#10599865	1	1	yes	characteristic
317	C₂	#10599900	1	2	yes	—
318	C₂	#10600069	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₅×#1374	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1374	D₅
hypocenter	C₅	#12600
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
4. C₂×#64722	C₂	semidirect
5. C₂×C₁₆×Dic₅	C₂	semidirect
6. C₅×#1374	C₂	semidirect
7. #10598148	C₂	semidirect
21. #64727	C₄	semidirect
22. #64727	C₄	semidirect
70. #1374	D₅	semidirect
295. C₅	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64978	2
C₂×#64727	2
C₂×#64722	2
C₂×C₁₆×Dic₅	2
C₅×#1374	2
#10598148	2
#12600	5