Subgroups — #11988797

Order 1280.

666 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#65214	C₂	1	1	yes	maximal, characteristic
3	C₂×#65214	C₂	1	1	yes	maximal, characteristic
4	#10598158	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₄	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₄×C₅×M₁₆	—	2	1	no	—
8	C₄×C₅×M₁₆	—	2	1	no	—
9	C₄×C₅×M₁₆	C₄	1	2	yes	—
10	C₄×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₅×#97	—	2	1	no	—
12	C₂×C₄×C₅×C₈	—	2	1	no	—
13	C₂×C₂²⋊#168	—	2	1	no	—
14	C₂×C₂²⋊#168	—	2	1	no	—
15	#65214	—	2	1	no	—
16	#65214	—	2	1	no	—
17	#65214	—	2	1	no	—
18	#65214	—	2	1	no	—
19	#65214	C₄	1	2	yes	—
20	#65214	C₄	1	2	yes	—
21	#65214	C₄	1	2	yes	—
22	#65214	C₂²	1	2	yes	—
23	C₂×C₄⋊#168	—	2	1	no	—
24	C₂×C₄⋊#168	C₄	1	2	yes	—
25	C₂×C₄×#168	—	2	1	no	—
26	C₂×C₄×#168	C₂²	1	2	yes	—
27	#7882	—	5	1	no	maximal, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂×C₅×M₁₆	—	2	4	no	—
33	C₂×C₅×M₁₆	C₈	1	8	yes	complemented
34	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₂²×#168	—	2	2	no	—
49	C₂²×#168	—	2	2	no	—
50	C₅×#97	—	2	2	no	—
51	C₅×#97	—	2	2	no	—
52	C₄×C₅×C₈	—	2	2	no	—
53	C₄×C₅×C₈	—	2	2	no	—
54	C₂²⋊#168	—	2	2	no	—
55	C₂²⋊#168	—	2	2	no	—
56	C₂²⋊#168	—	2	2	no	—
57	C₂²⋊#168	—	2	2	no	—
58	C₄⋊#168	—	2	2	no	—
59	C₄⋊#168	—	2	2	no	—
60	C₄⋊#168	—	2	2	no	—
61	C₄⋊#168	C₂×C₄	1	4	yes	—
62	C₄×#168	—	2	2	no	—
63	C₄×#168	—	2	2	no	—
64	C₄×#168	—	2	2	no	—
65	C₄×#168	C₂×C₄	1	4	yes	—
66	C₂×#431	—	5	1	no	—
67	C₂×#431	—	5	1	no	—
68	C₂×C₄×M₁₆	D₅	1	1	yes	characteristic
69	#1278	—	5	4	no	—
70	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	—	4	2	no	—
72	C₂²×C₄×C₅	—	4	2	no	—
73	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
89	C₂²×C₄×C₅	M₁₆	1	4	yes	—
90	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
93	C₅×M₁₆	—	2	8	no	—
94	C₅×M₁₆	—	2	4	no	—
95	C₅×M₁₆	C₂×C₈	1	8	yes	complemented
96	C₂×C₅×C₈	—	2	8	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₄²×C₅	C₄²	1	1	yes	characteristic
99	C₄²×C₅	C₄²	1	1	yes	characteristic
100	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
106	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
114	C₂×#168	—	2	4	no	—
115	C₂×#168	—	2	4	no	—
116	C₂×#168	—	4	4	no	—
117	C₂×#168	—	4	4	no	—
118	C₂²×M₁₆	Dic₅	1	2	yes	—
119	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
120	#431	—	5	2	no	—
121	#431	—	5	2	no	—
122	#431	—	5	2	no	—
123	#431	—	5	2	no	—
124	#431	—	10	1	no	—
125	#431	—	10	1	no	—
126	#431	—	10	1	no	—
127	#431	—	10	1	no	—
128	C₂×C₄⋊C₈	—	5	2	no	—
129	C₂×C₄⋊C₈	—	10	1	no	—
130	C₂×C₂²⋊C₈	—	10	1	no	—
131	C₂×C₂²⋊C₈	—	10	1	no	—
132	C₄×M₁₆	—	2	1	no	—
133	C₄×M₁₆	—	2	1	no	—
134	C₄×M₁₆	Dic₅	1	2	yes	—
135	C₄×M₁₆	C₂×D₅	1	2	yes	—
136	C₂×#97	—	2	1	no	—
137	C₂×C₄×C₈	—	2	1	no	—
138	C₂×C₄×C₈	—	5	2	no	—
139	C₂×C₄×C₈	—	10	1	no	—
140	C₂³×C₅	—	4	1	no	—
141	C₂³×C₅	—	4	1	no	—
142	C₂³×C₅	#95	1	1	yes	characteristic
143	C₂³×C₅	#95	1	1	yes	characteristic
144	C₂³×C₅	C₄⋊C₈	1	4	yes	—
145	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	—	4	1	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	—	4	1	no	—
158	C₂×C₄×C₅	#95	1	1	yes	characteristic
159	C₂×C₄×C₅	#95	1	1	yes	characteristic
160	C₂×C₄×C₅	#95	1	1	yes	characteristic
161	C₂×C₄×C₅	#95	1	1	yes	characteristic
162	C₂×C₄×C₅	#95	1	1	yes	characteristic
163	C₂×C₄×C₅	#95	1	1	yes	characteristic
164	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
165	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
166	C₂×C₄×C₅	#97	1	2	yes	—
167	C₂×C₄×C₅	#97	1	2	yes	—
168	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
169	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
170	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
171	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
178	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
179	C₅×C₈	—	2	8	no	—
180	#168	—	4	4	no	—
181	#168	—	4	4	no	—
182	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
183	C₂³×C₄	D₂₀	1	1	yes	characteristic
184	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	#168	1	8	yes	complemented
188	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	10	2	no	—
191	C₂²×C₈	—	10	2	no	—
192	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
197	C₂×C₄²	D₂₀	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
203	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
204	C₄⋊C₈	—	5	4	no	—
205	C₄⋊C₈	—	10	2	no	—
206	C₄⋊C₈	—	10	2	no	—
207	C₄⋊C₈	—	10	2	no	—
208	C₂²⋊C₈	—	10	2	no	—
209	C₂²⋊C₈	—	10	2	no	—
210	C₂²⋊C₈	—	10	2	no	—
211	C₂²⋊C₈	—	10	2	no	—
212	#97	—	2	2	no	—
213	#97	—	2	2	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	—	2	2	no	—
216	C₄×C₈	—	5	4	no	—
217	C₄×C₈	—	10	2	no	—
218	C₄×C₈	—	10	2	no	—
219	C₄×C₈	—	10	2	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	4	4	no	—
227	C₂²×C₅	#336	1	2	yes	—
228	C₂²×C₅	#336	1	2	yes	—
229	C₂²×C₅	C₂×#95	1	1	yes	characteristic
230	C₂²×C₅	C₂×C₄⋊C₈	1	2	yes	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	#336	1	2	yes	—
238	C₄×C₅	#336	1	2	yes	—
239	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
240	C₄×C₅	C₂×#97	1	1	yes	characteristic
241	C₄×C₅	C₂×C₂²⋊C₈	1	2	yes	—
242	C₄×C₅	#421	1	2	yes	—
243	C₄×C₅	#430	1	2	yes	—
244	C₂⁴	C₄⋊Dic₅	1	1	yes	characteristic
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×#168	1	4	yes	—
248	C₂²×C₄	#681	1	4	yes	—
249	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
250	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
251	C₂²×C₄	#683	1	1	yes	characteristic
252	C₂²×C₄	#683	1	1	yes	characteristic
253	C₂²×C₄	#683	1	1	yes	characteristic
254	C₂²×C₄	#683	1	1	yes	characteristic
255	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
256	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
257	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
258	C₂²×C₄	#685	1	1	yes	characteristic
259	C₂²×C₄	#685	1	1	yes	characteristic
260	C₂²×C₄	#685	1	1	yes	characteristic
261	C₂²×C₄	#685	1	1	yes	characteristic
262	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
263	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
264	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
266	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
267	M₁₆	—	2	8	no	—
268	M₁₆	—	2	4	no	—
269	M₁₆	C₂×#168	1	8	yes	complemented
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	2	8	no	—
272	C₂×C₈	—	10	4	no	—
273	C₂×C₈	—	10	4	no	—
274	C₂×C₈	—	20	4	no	—
275	C₂×C₈	—	20	4	no	—
276	C₄²	C₄×Dic₅	1	1	yes	characteristic
277	C₄²	C₄×Dic₅	1	1	yes	characteristic
278	C₄²	#683	1	1	yes	characteristic
279	C₄²	#683	1	1	yes	characteristic
280	C₄²	#683	1	1	yes	characteristic
281	C₄²	#683	1	1	yes	characteristic
282	C₄²	#685	1	1	yes	characteristic
283	C₄²	#685	1	1	yes	characteristic
284	C₄²	#685	1	1	yes	characteristic
285	C₄²	#685	1	1	yes	characteristic
286	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
287	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
288	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
289	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
290	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
291	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	—	2	2	no	—
294	C₂×C₅	—	2	2	no	—
295	C₂×C₅	C₂×#336	1	1	yes	characteristic
296	C₂×C₅	#1733	1	1	yes	characteristic
297	C₂×C₅	#1738	1	1	yes	characteristic
298	C₂³	—	4	1	no	—
299	C₂³	—	4	1	no	—
300	C₂³	C₄⋊#168	1	4	yes	—
301	C₂³	#3963	1	1	yes	characteristic
302	C₂³	#3963	1	1	yes	characteristic
303	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	C₄×#168	1	2	yes	—
317	C₂×C₄	C₄×#168	1	2	yes	—
318	C₂×C₄	#3935	1	2	yes	—
319	C₂×C₄	#3935	1	2	yes	—
320	C₂×C₄	C₄⋊#168	1	4	yes	—
321	C₂×C₄	C₂²⋊#168	1	4	yes	—
322	C₂×C₄	C₂²⋊#168	1	4	yes	—
323	C₂×C₄	#3963	1	1	yes	characteristic
324	C₂×C₄	#3963	1	1	yes	characteristic
325	C₂×C₄	#3963	1	1	yes	characteristic
326	C₂×C₄	#3963	1	1	yes	characteristic
327	C₂×C₄	#3963	1	1	yes	characteristic
328	C₂×C₄	#3963	1	1	yes	characteristic
329	C₂×C₄	C₂²×#168	1	2	yes	—
330	C₂×C₄	C₂×#681	1	2	yes	—
331	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
332	C₂×C₄	C₂×#683	1	1	yes	characteristic
333	C₂×C₄	C₂×#683	1	1	yes	characteristic
334	C₂×C₄	C₂×#685	1	1	yes	characteristic
335	C₂×C₄	C₂×#685	1	1	yes	characteristic
336	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
337	C₈	—	2	8	no	—
338	C₈	—	20	4	no	—
339	C₈	—	20	4	no	—
340	C₅	#7882	1	1	yes	characteristic, complemented, Sylow
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	#64744	1	2	yes	—
349	C₂²	#64744	1	2	yes	—
350	C₂²	C₂×C₄⋊#168	1	2	yes	—
351	C₂²	C₂×#3963	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	#64744	1	2	yes	—
359	C₄	#64744	1	2	yes	—
360	C₄	C₂×C₄×#168	1	1	yes	characteristic
361	C₄	C₂×#3935	1	1	yes	characteristic
362	C₄	#65413	1	2	yes	—
363	C₄	#65428	1	2	yes	—
364	C₄	C₂×C₂²⋊#168	1	2	yes	—
365	C₂	—	2	2	no	—
366	C₂	—	2	2	no	—
367	C₂	—	2	2	no	—
368	C₂	C₂×#64744	1	1	yes	characteristic
369	C₂	#10602775	1	1	yes	characteristic
370	C₂	#10602787	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₄×C₅×M₁₆	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×C₄×M₁₆	D₅
hypocenter	C₅	#7882
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10598158	C₂	semidirect
33. C₂×C₅×M₁₆	C₈	semidirect
95. C₅×M₁₆	C₂×C₈	semidirect
187. C₂×M₁₆	#168	semidirect
269. M₁₆	C₂×#168	semidirect
340. C₅	#7882	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×M₁₆	2
C₂×#65214	2
#10598158	2
#7882	5