Subgroups — #12071027

Order 1280.

618 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#10614168	C₂	1	2	yes	maximal
3	#10602416	C₂	1	2	yes	maximal
4	#10601699	C₂	1	2	yes	maximal, complemented
5	#10601696	C₂	1	4	yes	maximal, complemented
6	#10601695	C₂	1	2	yes	maximal, complemented
7	#10600525	C₂	1	1	yes	maximal, characteristic
8	#10600518	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
11	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic
12	C₄×C₅×Q₁₆	C₄	1	2	yes	—
13	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic
14	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
16	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₅×#107	C₄	1	1	yes	characteristic
18	C₂×C₅×#103	C₄	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	#65313	—	2	4	no	—
21	#65313	C₂²	1	4	yes	—
22	Q₈×#168	C₂²	1	4	yes	—
23	#65215	—	2	2	no	—
24	#65215	C₂²	1	2	yes	—
25	#65214	—	2	2	no	—
26	#65214	C₂²	1	1	yes	characteristic
27	C₄×#681	C₂²	1	1	yes	characteristic
28	Q₈⋊#168	C₂²	1	4	yes	—
29	Q₈⋊#168	C₂²	1	4	yes	—
30	#64678	C₂²	1	2	yes	—
31	#64677	C₂²	1	2	yes	—
32	#64675	C₂²	1	2	yes	—
33	#64674	C₂²	1	2	yes	—
34	#18062	—	5	1	no	maximal, Sylow
35	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
36	C₂×C₅×Q₁₆	—	2	4	no	—
37	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₅×Q₈	—	2	2	no	—
43	C₄×C₅×Q₈	—	2	2	no	—
44	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
45	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
46	C₄×C₅×Q₈	C₂³	1	2	yes	—
47	C₄×C₅×Q₈	C₂³	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅	—	2	2	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×#681	—	2	1	no	—
53	C₅×#107	C₂×C₄	1	2	yes	—
54	C₅×#107	C₂×C₄	1	1	yes	characteristic
55	C₅×#107	C₂×C₄	1	1	yes	characteristic
56	C₅×#103	C₂×C₄	1	2	yes	—
57	C₅×#103	C₂×C₄	1	2	yes	—
58	C₅×#103	C₂×C₄	1	2	yes	—
59	C₅×#103	C₂×C₄	1	2	yes	—
60	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
64	C₂²⋊#168	—	2	2	no	—
65	C₂²⋊#168	D₄	1	2	yes	—
66	C₄⋊#168	—	2	4	no	—
67	C₄⋊#168	—	2	4	no	—
68	C₄⋊#168	D₄	1	2	yes	—
69	C₄⋊#168	C₂³	1	4	yes	—
70	#3935	—	2	2	no	—
71	#3935	—	2	1	no	—
72	#3935	—	2	4	no	—
73	C₄×#168	—	2	4	no	—
74	C₄×#168	C₂³	1	2	yes	—
75	C₄×#168	C₂³	1	2	yes	—
76	#2984	—	5	2	no	—
77	C₂×C₄×Q₁₆	D₅	1	1	yes	characteristic
78	#1593	—	5	2	no	—
79	#1584	—	5	4	no	—
80	#1579	—	5	2	no	—
81	#1568	—	5	1	no	—
82	#1491	—	5	2	no	—
83	#1451	—	5	1	no	—
84	C₂×C₅×Q₈	—	2	2	no	—
85	C₂×C₅×Q₈	—	2	4	no	—
86	C₂×C₅×Q₈	—	4	2	no	—
87	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
88	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	—	2	2	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	#41	1	1	yes	characteristic
94	C₅×Q₁₆	—	2	4	no	—
95	C₅×Q₁₆	—	2	2	no	—
96	C₅×Q₁₆	M₁₆	1	4	yes	complemented
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	4	1	no	—
99	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
109	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	2	2	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅	#41	1	1	yes	characteristic
117	C₄²×C₅	C₂⁴	1	1	yes	characteristic
118	#681	—	4	2	no	—
119	C₂×#168	—	2	4	no	—
120	C₂×#168	—	2	2	no	—
121	C₂×#168	C₂×D₄	1	4	yes	—
122	C₂²×Q₁₆	Dic₅	1	1	yes	characteristic
123	C₂×C₄×Q₈	C₂×D₅	1	2	yes	—
124	#446	—	5	4	no	—
125	#446	—	10	4	no	—
126	C₈×Q₈	—	5	4	no	—
127	C₄×Q₁₆	Dic₅	1	2	yes	—
128	C₄×Q₁₆	Dic₅	1	1	yes	characteristic
129	C₄×Q₁₆	Dic₅	1	1	yes	characteristic
130	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
131	C₄×Q₁₆	C₂×D₅	1	2	yes	—
132	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
133	#432	—	5	2	no	—
134	#432	—	10	2	no	—
135	#431	—	5	1	no	—
136	#431	—	10	2	no	—
137	C₂×#107	Dic₅	1	1	yes	characteristic
138	C₂×#103	Dic₅	1	2	yes	—
139	C₄×M₁₆	—	5	1	no	—
140	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
141	#335	—	5	2	no	—
142	#334	—	5	2	no	—
143	Q₈⋊C₈	—	5	4	no	—
144	Q₈⋊C₈	—	5	4	no	—
145	#322	—	5	2	no	—
146	#322	—	5	2	no	—
147	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
148	C₅×Q₈	—	2	2	no	—
149	C₅×Q₈	—	2	2	no	—
150	C₅×Q₈	—	2	2	no	—
151	C₅×Q₈	—	4	4	no	—
152	C₅×Q₈	C₂×M₁₆	1	4	yes	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	4	4	no	—
162	C₂×C₄×C₅	—	4	4	no	—
163	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
169	C₅×C₈	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	C₂×M₁₆	1	2	yes	—
173	#168	—	2	4	no	—
174	#168	—	4	2	no	—
175	#168	—	4	4	no	—
176	C₂²×Q₈	C₂×Dic₅	1	2	yes	—
177	C₂×Q₁₆	—	2	4	no	—
178	C₂×Q₁₆	C₂×Dic₅	1	2	yes	—
179	C₂×Q₁₆	C₂×Dic₅	1	1	yes	characteristic
180	C₂×Q₁₆	C₂×Dic₅	1	1	yes	characteristic
181	C₂×M₁₆	—	10	1	no	—
182	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
183	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
184	C₄×Q₈	—	2	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	C₂×Dic₅	1	2	yes	—
187	C₄×Q₈	C₂×Dic₅	1	2	yes	—
188	C₄×Q₈	C₂²×D₅	1	2	yes	—
189	C₄×Q₈	C₂²×D₅	1	2	yes	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
194	#107	C₂×Dic₅	1	2	yes	—
195	#107	C₂×Dic₅	1	1	yes	characteristic
196	#107	C₂×Dic₅	1	1	yes	characteristic
197	C₄⋊C₈	—	5	2	no	—
198	C₄⋊C₈	—	5	4	no	—
199	C₄⋊C₈	—	10	4	no	—
200	C₄⋊C₈	—	10	4	no	—
201	#103	C₂×Dic₅	1	2	yes	—
202	#103	C₂×Dic₅	1	2	yes	—
203	#103	C₂×Dic₅	1	2	yes	—
204	#103	C₂×Dic₅	1	2	yes	—
205	C₂²⋊C₈	—	5	2	no	—
206	C₂²⋊C₈	—	10	2	no	—
207	#97	—	10	4	no	—
208	#97	—	10	2	no	—
209	#97	—	10	1	no	—
210	C₄×C₈	—	5	2	no	—
211	C₄×C₈	—	5	2	no	—
212	C₄×C₈	—	10	4	no	—
213	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
214	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	#444	1	2	yes	—
220	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	—	2	4	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	4	2	no	—
227	C₄×C₅	—	4	2	no	—
228	C₄×C₅	—	4	2	no	—
229	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
230	C₄×C₅	#568	1	1	yes	characteristic
231	C₄×C₅	#578	1	2	yes	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	C₂²×Dic₅	1	2	yes	—
236	C₂×Q₈	C₂²×Dic₅	1	2	yes	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
240	C₂²×C₄	#711	1	1	yes	characteristic
241	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
242	Q₁₆	—	2	4	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	#681	1	4	yes	complemented
245	M₁₆	—	20	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	5	4	no	—
249	C₂×C₈	—	10	4	no	—
250	C₂×C₈	—	10	2	no	—
251	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
252	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
253	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
254	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
261	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	D₄×D₅	1	1	yes	characteristic
267	C₄²	#711	1	1	yes	characteristic
268	C₄²	C₂²×Dic₅	1	1	yes	characteristic
269	C₄²	C₂³×D₅	1	1	yes	characteristic
270	C₂×C₅	—	2	2	no	—
271	C₂×C₅	D₄×M₁₆	1	1	yes	characteristic
272	C₂×C₅	#2950	1	1	yes	characteristic
273	C₂×C₅	C₂×#444	1	1	yes	characteristic
274	C₂³	D₄×Dic₅	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	4	4	no	—
279	Q₈	C₂×#681	1	4	yes	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
291	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
292	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
294	C₂×C₄	C₂×#711	1	1	yes	characteristic
295	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	10	4	no	—
300	C₈	—	20	2	no	—
301	C₈	—	20	4	no	—
302	C₈	C₂×#681	1	2	yes	—
303	C₅	#18062	1	1	yes	characteristic, complemented, Sylow
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	#65486	1	2	yes	—
307	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	4	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	#66100	1	2	yes	—
317	C₄	C₂²×#681	1	1	yes	characteristic
318	C₄	#66153	1	1	yes	characteristic
319	C₂	—	2	2	no	—
320	C₂	D₄×#681	1	1	yes	characteristic
321	C₂	#10615605	1	1	yes	characteristic
322	C₂	C₂×#65486	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₄×C₅×Q₁₆	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₄×D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅
hypocenter	C₅	#18062
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #10601699	C₂	semidirect
5. #10601696	C₂	semidirect
6. #10601695	C₂	semidirect
96. C₅×Q₁₆	M₁₆	semidirect
244. Q₁₆	#681	semidirect
303. C₅	#18062	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×Q₁₆	2
#10614168	2
#10602416	2
#10601699	2
#10601696	2
#10601695	2
#10600525	2
#10600518	2
#18062	5