Subgroups — #12014174

Order 1280.

586 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#65390	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3944	C₂	1	1	yes	maximal, characteristic
4	#10598306	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×#4072	C₄	1	1	yes	characteristic
7	C₂²×#681	C₄	1	1	yes	characteristic
8	C₂×C₅×M₃₂	—	2	2	no	—
9	C₂×C₅×M₃₂	—	2	1	no	—
10	C₂×C₅×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₅×C₁₆	—	2	1	no	—
13	#65390	—	2	1	no	—
14	#65390	—	2	1	no	—
15	#65390	C₄	1	2	yes	—
16	#65390	C₂²	1	2	yes	—
17	C₂×#3947	—	2	1	no	—
18	C₂×C₈×Dic₅	—	2	1	no	—
19	C₂×#3944	—	2	1	no	—
20	C₂×#3944	—	2	1	no	—
21	C₂×#3944	—	2	2	no	—
22	C₂×#3944	C₄	1	4	yes	complemented
23	C₂²×#672	C₂²	1	2	yes	—
24	#12554	—	5	1	no	maximal, Sylow
25	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	2	yes	—
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	Q₈	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₂²⋊Dic₅	—	2	1	no	—
37	#4072	—	2	1	no	—
38	#4072	—	2	1	no	—
39	#4072	C₈	1	2	yes	—
40	#4072	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊Dic₅	C₈	1	2	yes	—
42	C₂×C₄×Dic₅	—	2	1	no	—
43	C₂×#681	—	2	1	no	—
44	C₂×#681	—	2	2	no	—
45	C₂×#681	C₈	1	4	yes	—
46	C₂×#681	C₂×C₄	1	2	yes	—
47	C₂²×#168	—	2	1	no	—
48	C₅×M₃₂	—	2	4	no	—
49	C₅×M₃₂	—	2	2	no	—
50	C₅×M₃₂	C₂×C₄	1	4	yes	complemented
51	C₂×C₅×C₁₆	—	2	4	no	—
52	C₂×C₅×C₁₆	—	2	2	no	—
53	#3947	—	2	2	no	—
54	#3947	—	2	2	no	—
55	C₈×Dic₅	—	2	2	no	—
56	C₈×Dic₅	—	2	2	no	—
57	#3944	—	2	4	no	—
58	#3944	—	2	4	no	—
59	C₂×#672	—	2	2	no	—
60	C₂×#672	—	2	2	no	—
61	C₂×#672	C₂×C₄	1	4	yes	complemented
62	C₂²×M₃₂	—	5	1	no	—
63	C₂²×M₃₂	D₅	1	1	yes	characteristic
64	C₂×#405	—	5	1	no	—
65	#1379	—	5	4	no	—
66	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₄²	1	2	yes	—
68	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₅	M₁₆	1	2	yes	—
76	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×Dic₅	—	2	2	no	—
78	C₂×C₅×C₈	C₄²	1	2	yes	—
79	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
82	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
88	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
89	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
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	1	yes	characteristic
93	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₂²⋊Dic₅	—	2	2	no	—
97	C₂²⋊Dic₅	—	2	2	no	—
98	C₄⋊Dic₅	—	2	2	no	—
99	C₄⋊Dic₅	C₂×C₈	1	4	yes	—
100	C₄×Dic₅	—	2	2	no	—
101	C₄×Dic₅	—	2	2	no	—
102	#681	—	2	4	no	—
103	#681	—	2	2	no	—
104	#681	C₂×C₈	1	4	yes	—
105	C₂×#168	—	2	4	no	—
106	C₂×#168	—	2	2	no	—
107	C₅×C₁₆	—	2	4	no	—
108	#672	—	2	4	no	—
109	C₂²×M₁₆	—	5	1	no	—
110	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
111	C₂×#117	—	5	1	no	—
112	C₂×M₃₂	—	2	2	no	—
113	C₂×M₃₂	—	2	1	no	—
114	C₂×M₃₂	—	5	4	no	—
115	C₂×M₃₂	—	10	1	no	—
116	C₂×M₃₂	—	10	1	no	—
117	C₂×M₃₂	—	10	2	no	—
118	C₂×M₃₂	Dic₅	1	4	yes	complemented
119	C₂×M₃₂	C₂×D₅	1	2	yes	—
120	C₂²×C₁₆	—	2	1	no	—
121	C₂²×C₁₆	—	5	2	no	—
122	#405	—	5	2	no	—
123	#405	—	5	2	no	—
124	#405	—	10	1	no	—
125	#405	—	10	1	no	—
126	C₂×#97	—	10	1	no	—
127	C₂×C₄×C₈	—	10	1	no	—
128	C₂³×C₅	—	4	2	no	—
129	C₂³×C₅	#97	1	2	yes	—
130	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
131	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
132	C₂³×C₅	C₄⋊C₈	1	2	yes	—
133	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅	#95	1	2	yes	—
135	C₂×C₄×C₅	#95	1	2	yes	—
136	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
137	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
138	C₂×C₄×C₅	#97	1	2	yes	—
139	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
142	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
143	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
144	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
145	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
152	C₂×Dic₅	—	2	4	no	—
153	C₂×Dic₅	—	4	4	no	—
154	C₅×C₈	#95	1	2	yes	—
155	C₅×C₈	#95	1	2	yes	—
156	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
160	#168	—	2	4	no	—
161	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
162	C₂×M₁₆	—	5	4	no	—
163	C₂×M₁₆	—	5	2	no	—
164	C₂×M₁₆	—	10	2	no	—
165	C₂×M₁₆	—	10	1	no	—
166	C₂²×C₈	—	10	1	no	—
167	C₂²×C₈	Dic₁₀	1	2	yes	—
168	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
169	C₂²×C₈	C₄×D₅	1	2	yes	—
170	C₂²×C₈	D₂₀	1	1	yes	characteristic
171	C₂²×C₈	D₂₀	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₅	1	2	yes	—
173	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
174	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
176	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
177	#117	—	5	2	no	—
178	#117	—	5	2	no	—
179	#117	—	10	1	no	—
180	#117	—	10	1	no	—
181	C₂×C₄⋊C₄	—	5	2	no	—
182	C₂×C₂²⋊C₄	—	10	1	no	—
183	C₂×C₄²	—	10	1	no	—
184	M₃₂	—	2	4	no	—
185	M₃₂	—	2	2	no	—
186	M₃₂	—	10	4	no	—
187	M₃₂	—	10	4	no	—
188	M₃₂	C₂×Dic₅	1	4	yes	complemented
189	C₂×C₁₆	—	2	4	no	—
190	C₂×C₁₆	—	2	2	no	—
191	C₂×C₁₆	—	5	4	no	—
192	C₂×C₁₆	—	10	2	no	—
193	C₂×C₁₆	—	10	2	no	—
194	#97	—	10	2	no	—
195	#97	—	10	2	no	—
196	C₄×C₈	—	10	2	no	—
197	C₄×C₈	—	10	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	—	4	2	no	—
202	C₂²×C₅	—	4	2	no	—
203	C₂²×C₅	#336	1	2	yes	—
204	C₂²×C₅	#336	1	2	yes	—
205	C₂²×C₅	C₂×#97	1	1	yes	characteristic
206	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
207	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
208	C₄×C₅	#336	1	2	yes	—
209	C₄×C₅	#336	1	2	yes	—
210	C₄×C₅	C₂×#95	1	1	yes	characteristic
211	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
212	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
213	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
214	Dic₅	—	4	4	no	—
215	C₂⁴	#676	1	1	yes	characteristic
216	C₂²×C₄	—	10	2	no	—
217	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
219	C₂²×C₄	C₈×D₅	1	2	yes	—
220	C₂²×C₄	#676	1	1	yes	characteristic
221	C₂²×C₄	#676	1	2	yes	—
222	C₂²×C₄	C₄×Dic₅	1	2	yes	—
223	C₂²×C₄	#683	1	2	yes	—
224	C₂²×C₄	#685	1	1	yes	characteristic
225	C₂²×C₄	#685	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
227	M₁₆	—	5	4	no	—
228	M₁₆	—	10	4	no	—
229	M₁₆	—	10	2	no	—
230	C₂×C₈	—	10	4	no	—
231	C₂×C₈	—	10	2	no	—
232	C₂×C₈	C₄×Dic₅	1	2	yes	—
233	C₂×C₈	#683	1	2	yes	—
234	C₂×C₈	#683	1	2	yes	—
235	C₂×C₈	#683	1	2	yes	—
236	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
237	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
238	C₂×C₈	#685	1	1	yes	characteristic
239	C₂×C₈	#685	1	1	yes	characteristic
240	C₂×C₈	#685	1	2	yes	—
241	C₂×C₈	#685	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
243	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
244	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
246	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
250	C₄⋊C₄	—	5	4	no	—
251	C₄⋊C₄	—	10	2	no	—
252	C₂²⋊C₄	—	10	2	no	—
253	C₂²⋊C₄	—	10	2	no	—
254	C₄²	—	10	2	no	—
255	C₄²	—	10	2	no	—
256	C₁₆	—	2	4	no	—
257	C₁₆	—	10	4	no	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	2	2	no	—
260	C₂×C₅	—	2	1	no	—
261	C₂×C₅	—	2	1	no	—
262	C₂×C₅	C₂×#336	1	1	yes	characteristic
263	C₂×C₅	#2117	1	2	yes	—
264	C₂³	—	4	2	no	—
265	C₂³	#3946	1	2	yes	—
266	C₂³	#3947	1	2	yes	—
267	C₂³	#3952	1	1	yes	characteristic
268	C₂³	#3952	1	1	yes	characteristic
269	C₂³	C₂×#676	1	1	yes	characteristic
270	C₂×C₄	—	10	4	no	—
271	C₂×C₄	—	20	4	no	—
272	C₂×C₄	C₈×Dic₅	1	2	yes	—
273	C₂×C₄	C₈×Dic₅	1	2	yes	—
274	C₂×C₄	#3946	1	2	yes	—
275	C₂×C₄	#3946	1	2	yes	—
276	C₂×C₄	#3946	1	2	yes	—
277	C₂×C₄	#3947	1	2	yes	—
278	C₂×C₄	#3952	1	1	yes	characteristic
279	C₂×C₄	#3952	1	1	yes	characteristic
280	C₂×C₄	#3952	1	2	yes	—
281	C₂×C₄	#3952	1	2	yes	—
282	C₂×C₄	#3963	1	2	yes	—
283	C₂×C₄	#3963	1	2	yes	—
284	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
285	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
286	C₂×C₄	C₂×#676	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
288	C₂×C₄	C₂×#683	1	1	yes	characteristic
289	C₂×C₄	C₂×#685	1	1	yes	characteristic
290	C₈	—	10	4	no	—
291	C₈	#3963	1	2	yes	—
292	C₈	#3963	1	2	yes	—
293	C₈	C₂×#683	1	1	yes	characteristic
294	C₈	C₂×C₄⋊Dic₅	1	1	yes	characteristic
295	C₈	C₂×#685	1	1	yes	characteristic
296	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
297	C₅	#12554	1	1	yes	characteristic, complemented, Sylow
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#64770	1	2	yes	—
304	C₂²	#64770	1	2	yes	—
305	C₂²	C₂×#3946	1	1	yes	characteristic
306	C₂²	C₂×#3947	1	1	yes	characteristic
307	C₂²	C₂×#3952	1	1	yes	characteristic
308	C₄	—	20	4	no	—
309	C₄	#64770	1	2	yes	—
310	C₄	#64770	1	2	yes	—
311	C₄	C₂×C₈×Dic₅	1	1	yes	characteristic
312	C₄	C₂×#3946	1	1	yes	characteristic
313	C₄	C₂×#3952	1	1	yes	characteristic
314	C₄	C₂×#3963	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	#10602231	1	2	yes	—
320	C₂	C₂×#64770	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂²×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅	M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂²×M₃₂	D₅
hypocenter	C₅	#12554
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #10598306	C₂	semidirect
10. C₂×C₅×M₃₂	C₄	semidirect
22. C₂×#3944	C₄	semidirect
50. C₅×M₃₂	C₂×C₄	semidirect
61. C₂×#672	C₂×C₄	semidirect
118. C₂×M₃₂	Dic₅	semidirect
188. M₃₂	C₂×Dic₅	semidirect
297. C₅	#12554	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
C₂×#65390	2
C₂²×#3944	2
#10598306	2
#12554	5