Subgroups — #12252522

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65196	C₂	1	2	yes	maximal, complemented
2	#10611153	C₂	1	1	yes	maximal, characteristic
3	C₅×#2169	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602226	C₂	1	2	yes	maximal, complemented
5	D₄×#672	C₂	1	1	yes	maximal, characteristic, complemented
6	D₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#10600072	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599900	C₂	1	2	yes	maximal, complemented
9	#10599895	C₂	1	2	yes	maximal, complemented
10	C₁₆×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#3930	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#4047	C₂²	1	2	yes	—
13	#65882	C₄	1	1	yes	characteristic
14	C₂×C₅×M₃₂	C₂²	1	2	yes	—
15	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
16	C₈×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂²×#672	C₂²	1	2	yes	—
18	#65301	C₄	1	1	yes	characteristic
19	#65196	—	2	8	no	—
20	C₂×#3930	C₂²	1	2	yes	—
21	C₂×#3930	C₂²	1	1	yes	characteristic
22	C₂×#3930	C₂²	1	1	yes	characteristic
23	C₂×D₅×C₁₆	C₂²	1	2	yes	—
24	C₂×D₅×C₁₆	C₂²	1	2	yes	—
25	#65127	C₄	1	1	yes	characteristic
26	#65126	C₄	1	1	yes	characteristic
27	#65018	C₄	1	2	yes	—
28	#65016	C₄	1	2	yes	—
29	C₈×D₂₀	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
31	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
34	C₂²⋊#672	C₂²	1	2	yes	—
35	#64727	C₂²	1	2	yes	—
36	#64727	C₂²	1	2	yes	—
37	#64722	C₂²	1	1	yes	characteristic
38	#64722	C₂²	1	1	yes	characteristic
39	#64721	C₂²	1	2	yes	—
40	C₄⋊#672	C₂²	1	1	yes	characteristic
41	C₄×#672	C₂²	1	1	yes	characteristic
42	#33438	—	5	1	no	maximal, Sylow
43	C₂×#709	C₂×C₄	1	2	yes	—
44	C₂²×C₅×C₈	C₂³	1	2	yes	—
45	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
46	#4086	C₈	1	1	yes	characteristic
47	#4084	C₈	1	1	yes	characteristic
48	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
49	C₂×#681	C₂×C₄	1	2	yes	—
50	#4047	—	2	4	no	—
51	#4047	—	2	4	no	—
52	C₂×#676	C₂×C₄	1	2	yes	—
53	C₂×#676	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅	C₂³	1	2	yes	—
56	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
57	#4041	C₈	1	1	yes	characteristic
58	#4034	C₈	1	1	yes	characteristic
59	#4031	C₈	1	2	yes	—
60	#4030	C₈	1	2	yes	—
61	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
63	C₅×M₃₂	—	2	4	no	—
64	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₅×C₁₆	C₂³	1	2	yes	—
66	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
67	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
69	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
70	C₂²⋊#168	C₂×C₄	1	2	yes	—
71	#3952	C₂×C₄	1	2	yes	—
72	#3952	C₂×C₄	1	2	yes	—
73	#3946	C₂×C₄	1	2	yes	—
74	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₅	C₂³	1	1	yes	characteristic
76	C₂×#672	—	2	4	no	—
77	C₂×#672	C₂³	1	1	yes	characteristic
78	C₂×#672	C₂³	1	2	yes	—
79	C₂×#672	C₂³	1	1	yes	characteristic
80	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
81	C₄×#168	C₂×C₄	1	1	yes	characteristic
82	#3930	—	2	2	no	—
83	#3930	—	2	4	no	—
84	#3930	D₄	1	4	yes	complemented
85	D₅×C₁₆	—	2	4	no	—
86	D₅×C₁₆	—	2	4	no	—
87	C₂×#504	—	5	2	no	—
88	#2931	—	5	1	no	—
89	#2169	—	5	1	no	—
90	#2169	D₅	1	1	yes	characteristic, complemented
91	#2168	—	5	2	no	—
92	#2168	—	5	2	no	—
93	D₄×C₁₆	—	5	1	no	—
94	D₄×C₁₆	—	5	1	no	—
95	C₂×C₄⋊C₁₆	—	5	1	no	—
96	#2113	—	5	2	no	—
97	C₄×M₃₂	—	5	1	no	—
98	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
100	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
101	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
102	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
103	#709	—	2	4	no	—
104	#709	—	2	4	no	—
105	C₂×D₂₀	C₂×C₈	1	2	yes	—
106	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
107	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
108	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
109	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
110	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
111	C₂×Dic₁₀	C₂×C₈	1	2	yes	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
117	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
118	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
119	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
120	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
121	#685	C₂×C₈	1	2	yes	—
122	#685	C₂×C₈	1	2	yes	—
123	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
124	#683	C₂×C₈	1	2	yes	—
125	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
126	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
127	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
128	#681	—	2	4	no	—
129	C₂×#168	C₂²×C₄	1	1	yes	characteristic
130	C₂×#168	C₂²×C₄	1	2	yes	—
131	C₂×#168	C₂²×C₄	1	1	yes	characteristic
132	#676	—	2	4	no	—
133	#676	—	2	4	no	—
134	C₈×D₅	—	2	2	no	—
135	C₈×D₅	—	2	2	no	—
136	C₈×D₅	—	2	2	no	—
137	C₈×D₅	C₂×D₄	1	2	yes	—
138	C₈×D₅	#41	1	2	yes	—
139	C₅×C₁₆	—	2	2	no	—
140	C₅×C₁₆	—	2	1	no	—
141	C₅×C₁₆	C₂×D₄	1	2	yes	—
142	#672	—	2	2	no	—
143	#672	—	2	1	no	—
144	#672	C₂×D₄	1	2	yes	—
145	C₂×#131	—	5	2	no	—
146	#532	—	5	1	no	—
147	#504	—	10	8	no	—
148	C₂×M₃₂	—	5	1	no	—
149	C₂×M₃₂	—	5	1	no	—
150	C₂×M₃₂	—	5	2	no	—
151	C₂×M₃₂	C₂×D₅	1	2	yes	—
152	C₂²×C₁₆	—	5	2	no	—
153	C₂²×C₁₆	—	5	2	no	—
154	C₂²×C₁₆	—	5	2	no	—
155	#436	—	5	1	no	—
156	#436	—	5	1	no	—
157	#436	—	5	2	no	—
158	C₈×D₄	—	5	2	no	—
159	C₈×D₄	—	5	1	no	—
160	C₈×D₄	C₂×D₅	1	1	yes	characteristic
161	#423	—	5	1	no	—
162	#408	—	5	2	no	—
163	C₂×C₄×C₈	—	5	1	no	—
164	C₄⋊C₁₆	—	5	2	no	—
165	C₄⋊C₁₆	—	5	1	no	—
166	C₄⋊C₁₆	C₂×D₅	1	1	yes	characteristic
167	C₂²⋊C₁₆	—	5	2	no	—
168	C₂²⋊C₁₆	—	5	2	no	—
169	C₂²⋊C₁₆	—	5	2	no	—
170	C₂²⋊C₁₆	C₂×D₅	1	2	yes	—
171	#346	—	5	1	no	—
172	#346	—	5	1	no	—
173	C₄×C₁₆	—	5	1	no	—
174	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
175	C₂³×C₅	C₂²×C₈	1	2	yes	—
176	C₂²×D₅	C₂²×C₈	1	2	yes	—
177	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
178	C₅×D₄	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
183	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
185	C₂²⋊D₅	—	2	4	no	—
186	C₂²⋊D₅	—	2	4	no	—
187	C₂²⋊D₅	—	4	2	no	—
188	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
189	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
190	C₂×Dic₅	C₂²×C₈	1	2	yes	—
191	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
192	D₂₀	—	2	4	no	—
193	D₂₀	—	2	2	no	—
194	C₄×D₅	—	2	2	no	—
195	C₄×D₅	—	2	2	no	—
196	C₄×D₅	—	2	2	no	—
197	C₄×D₅	—	4	1	no	—
198	C₄×D₅	C₄×D₄	1	2	yes	—
199	C₄×D₅	#131	1	2	yes	—
200	Dic₁₀	—	2	4	no	—
201	C₅×C₈	—	2	2	no	—
202	C₅×C₈	—	2	1	no	—
203	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
204	C₅×C₈	C₂×#41	1	1	yes	characteristic
205	#168	—	2	2	no	—
206	#168	—	2	1	no	—
207	#168	C₄×D₄	1	2	yes	—
208	C₂×#41	—	5	2	no	—
209	#131	—	10	4	no	—
210	#131	—	10	4	no	—
211	C₂×M₁₆	—	5	2	no	—
212	C₂×M₁₆	—	5	2	no	—
213	C₂×M₁₆	—	5	2	no	—
214	C₂²×C₈	—	5	1	no	—
215	C₂²×C₈	—	5	1	no	—
216	C₂²×C₈	—	5	2	no	—
217	C₂²×C₈	C₂²×D₅	1	2	yes	—
218	C₄⋊Q₈	—	5	1	no	—
219	#127	—	5	1	no	—
220	#124	—	5	2	no	—
221	#121	—	5	1	no	—
222	#121	—	5	1	no	—
223	#121	—	5	2	no	—
224	C₄×D₄	—	5	2	no	—
225	C₄×D₄	—	5	1	no	—
226	C₄×D₄	C₄×D₅	1	1	yes	characteristic
227	C₂×C₄²	—	5	1	no	—
228	M₃₂	—	2	4	no	—
229	M₃₂	—	5	4	no	—
230	M₃₂	—	10	4	no	—
231	M₃₂	—	10	2	no	—
232	C₂×C₁₆	—	5	1	no	—
233	C₂×C₁₆	—	5	1	no	—
234	C₂×C₁₆	—	5	2	no	—
235	C₂×C₁₆	—	10	4	no	—
236	C₂×C₁₆	—	10	4	no	—
237	C₂×C₁₆	—	10	4	no	—
238	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₅	1	2	yes	—
240	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
241	C₄⋊C₈	—	5	2	no	—
242	C₄⋊C₈	—	5	1	no	—
243	C₄⋊C₈	C₄×D₅	1	1	yes	characteristic
244	C₂²⋊C₈	—	5	2	no	—
245	C₂²⋊C₈	—	5	2	no	—
246	C₂²⋊C₈	—	5	2	no	—
247	C₂²⋊C₈	C₄×D₅	1	2	yes	—
248	C₄×C₈	—	5	1	no	—
249	C₄×C₈	—	5	1	no	—
250	C₄×C₈	—	5	1	no	—
251	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
252	C₂²×C₅	—	2	2	no	—
253	C₂²×C₅	—	2	2	no	—
254	C₂²×C₅	—	2	2	no	—
255	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
256	C₂×D₅	—	2	1	no	—
257	C₂×D₅	—	2	1	no	—
258	C₂×D₅	—	2	2	no	—
259	C₂×D₅	—	2	2	no	—
260	C₂×D₅	—	2	2	no	—
261	C₂×D₅	C₈×D₄	1	2	yes	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	2	2	no	—
264	C₄×C₅	#504	1	2	yes	complemented
265	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
266	C₄×C₅	C₂×#131	1	1	yes	characteristic
267	Dic₅	—	2	1	no	—
268	Dic₅	—	2	1	no	—
269	Dic₅	—	2	2	no	—
270	Dic₅	C₈×D₄	1	2	yes	—
271	#41	—	10	4	no	—
272	#41	—	10	4	no	—
273	C₂×Q₈	—	5	2	no	—
274	C₂×D₄	—	5	2	no	—
275	C₂×D₄	—	5	2	no	—
276	C₂×D₄	—	5	2	no	—
277	C₂×D₄	—	5	1	no	—
278	C₂×D₄	—	5	2	no	—
279	C₂×D₄	C₈×D₅	1	1	yes	characteristic
280	C₂²×C₄	—	5	1	no	—
281	C₂²×C₄	—	5	2	no	—
282	C₂²×C₄	—	5	1	no	—
283	C₂²×C₄	—	5	1	no	—
284	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
285	M₁₆	—	10	4	no	—
286	M₁₆	—	10	4	no	—
287	M₁₆	—	10	4	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	5	1	no	—
291	C₂×C₈	—	5	2	no	—
292	C₂×C₈	—	5	2	no	—
293	C₂×C₈	—	5	1	no	—
294	C₂×C₈	—	5	2	no	—
295	C₂×C₈	—	10	2	no	—
296	C₂×C₈	—	10	2	no	—
297	C₂×C₈	—	10	2	no	—
298	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
299	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
300	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
301	C₄⋊C₄	—	5	2	no	—
302	C₄⋊C₄	—	5	1	no	—
303	C₄⋊C₄	C₈×D₅	1	1	yes	characteristic
304	C₂²⋊C₄	—	5	2	no	—
305	C₂²⋊C₄	—	5	2	no	—
306	C₂²⋊C₄	—	5	2	no	—
307	C₂²⋊C₄	C₈×D₅	1	2	yes	—
308	C₄²	—	5	1	no	—
309	C₄²	—	5	1	no	—
310	C₄²	—	5	1	no	—
311	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
312	C₁₆	—	2	2	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	—	5	2	no	—
315	C₁₆	—	10	2	no	—
316	C₁₆	—	10	1	no	—
317	C₁₆	D₄×D₅	1	2	yes	—
318	C₂×C₅	—	4	2	no	—
319	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
320	C₂×C₅	C₂×#504	1	1	yes	characteristic
321	C₂×C₅	#3407	1	1	yes	characteristic
322	D₅	—	2	2	no	—
323	D₅	—	4	2	no	—
324	C₂³	—	5	1	no	—
325	C₂³	—	5	2	no	—
326	C₂³	C₂×C₈×D₅	1	2	yes	—
327	Q₈	—	10	4	no	—
328	D₄	—	2	2	no	—
329	D₄	—	10	4	no	—
330	D₄	—	10	4	no	—
331	D₄	—	10	4	no	—
332	D₄	—	10	2	no	—
333	D₄	—	20	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	5	1	no	—
337	C₂×C₄	—	5	2	no	—
338	C₂×C₄	—	5	2	no	—
339	C₂×C₄	—	5	1	no	—
340	C₂×C₄	—	5	1	no	—
341	C₂×C₄	—	5	2	no	—
342	C₂×C₄	—	10	2	no	—
343	C₂×C₄	—	10	2	no	—
344	C₂×C₄	—	10	2	no	—
345	C₂×C₄	—	20	1	no	—
346	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
347	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
348	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
349	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	2	no	—
352	C₈	—	5	2	no	—
353	C₈	—	10	1	no	—
354	C₈	—	10	2	no	—
355	C₈	C₂×D₄×D₅	1	1	yes	characteristic
356	C₈	D₅×#41	1	1	yes	characteristic
357	C₅	#33438	1	1	yes	characteristic, complemented, Sylow
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	5	2	no	—
362	C₂²	—	10	1	no	—
363	C₂²	—	10	1	no	—
364	C₂²	—	10	2	no	—
365	C₂²	—	10	2	no	—
366	C₂²	—	10	2	no	—
367	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	2	1	no	—
370	C₄	—	5	2	no	—
371	C₄	—	10	2	no	—
372	C₄	—	10	1	no	—
373	C₄	—	10	1	no	—
374	C₄	#65196	1	2	yes	complemented
375	C₄	C₄×D₄×D₅	1	1	yes	characteristic
376	C₄	D₅×#131	1	1	yes	characteristic
377	C₂	—	4	2	no	—
378	C₂	—	10	2	no	—
379	C₂	—	20	2	no	—
380	C₂	C₈×D₄×D₅	1	1	yes	characteristic
381	C₂	C₂×#65196	1	1	yes	characteristic
382	C₂	#10613743	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2169	C₂
prime radical	#65882	C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2169	D₅
hypocenter	C₅	#33438
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2169

Complements

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

N	H ≅ G/N	type
1. C₂×#65196	C₂	semidirect
3. C₅×#2169	C₂	semidirect
4. #10602226	C₂	semidirect
5. D₄×#672	C₂	semidirect
6. D₅×C₄⋊C₁₆	C₂	semidirect
7. #10600072	C₂	semidirect
8. #10599900	C₂	semidirect
9. #10599895	C₂	semidirect
10. C₁₆×D₂₀	C₂	semidirect
11. C₄×#3930	C₂	semidirect
84. #3930	D₄	semidirect
90. #2169	D₅	semidirect
264. C₄×C₅	#504	semidirect
357. C₅	#33438	semidirect
374. C₄	#65196	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65196	2
#10611153	2
C₅×#2169	2
#10602226	2
D₄×#672	2
D₅×C₄⋊C₁₆	2
#10600072	2
#10599900	2
#10599895	2
C₁₆×D₂₀	2
C₄×#3930	2
#33438	5