Subgroups — #12252414

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#4047	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602202	C₂	1	2	yes	maximal
4	#10602192	C₂	1	2	yes	maximal, complemented
5	#10600536	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600069	C₂	1	4	yes	maximal, complemented
7	#10600068	C₂	1	2	yes	maximal, complemented
8	#10600067	C₂	1	2	yes	maximal, complemented
9	C₂×#4047	C₄	1	1	yes	characteristic
10	C₂×#4047	C₂²	1	1	yes	characteristic
11	C₄×#709	C₄	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#65390	C₄	1	1	yes	characteristic
15	#65390	C₂²	1	1	yes	characteristic
16	C₂×#3944	C₂²	1	2	yes	—
17	C₄×#681	C₄	1	1	yes	characteristic
18	#64978	C₄	1	2	yes	—
19	#64978	C₂²	1	2	yes	—
20	C₄×#676	C₄	1	2	yes	—
21	C₄×C₈×D₅	C₂²	1	2	yes	—
22	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
25	#64727	C₂²	1	4	yes	—
26	#64727	C₂²	1	4	yes	—
27	#64722	C₂²	1	4	yes	—
28	C₁₆×Dic₅	C₂²	1	4	yes	—
29	C₄⋊#672	C₂²	1	2	yes	—
30	C₄⋊#672	C₂²	1	2	yes	—
31	#33423	—	5	1	no	maximal, Sylow
32	C₂×#709	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₄×C₂²⋊D₅	C₈	1	2	yes	—
37	C₄×C₂²⋊D₅	C₈	1	2	yes	—
38	#4072	C₂×C₄	1	1	yes	characteristic
39	C₂×#681	C₂×C₄	1	1	yes	characteristic
40	C₂×#681	C₂×C₄	1	1	yes	characteristic
41	#4047	—	2	2	no	—
42	#4047	—	2	2	no	—
43	#4047	—	2	4	no	—
44	C₂×#676	C₂×C₄	1	2	yes	—
45	C₂×#676	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₅	C₂³	1	2	yes	—
48	C₄×D₂₀	C₈	1	1	yes	characteristic
49	C₄×D₂₀	C₈	1	1	yes	characteristic
50	#4018	C₂×C₄	1	2	yes	—
51	C₄²×D₅	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₀	C₈	1	1	yes	characteristic
53	C₄×Dic₁₀	C₈	1	1	yes	characteristic
54	C₂×C₅×C₁₆	—	2	4	no	—
55	C₂×C₅×C₁₆	C₂³	1	2	yes	—
56	C₂×C₅×C₁₆	C₂³	1	2	yes	—
57	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
60	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
61	#3947	C₂×C₄	1	2	yes	—
62	#3947	C₂×C₄	1	2	yes	—
63	C₈×Dic₅	C₂×C₄	1	2	yes	—
64	C₈×Dic₅	C₂³	1	2	yes	—
65	#3944	—	2	4	no	—
66	C₂×#672	C₂³	1	4	yes	—
67	#3935	C₂×C₄	1	2	yes	—
68	C₄×#168	C₂×C₄	1	2	yes	—
69	C₄×#131	—	5	1	no	—
70	#2164	—	5	4	no	—
71	#2163	—	5	2	no	—
72	#2162	—	5	2	no	—
73	#2150	—	5	1	no	—
74	C₂×C₄⋊C₁₆	D₅	1	1	yes	characteristic, complemented
75	#2113	—	5	2	no	—
76	#2108	—	5	2	no	—
77	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
81	#709	—	2	2	no	—
82	#709	—	2	2	no	—
83	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	#41	1	2	yes	—
93	C₂×C₅×C₈	#41	1	2	yes	—
94	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
95	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
100	#685	C₂×C₈	1	2	yes	—
101	#685	C₂×C₈	1	2	yes	—
102	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
103	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
104	#683	C₂×C₈	1	2	yes	—
105	#683	C₂×C₈	1	2	yes	—
106	C₄×Dic₅	C₂×C₈	1	2	yes	—
107	C₄×Dic₅	C₂×C₈	1	2	yes	—
108	C₄×Dic₅	C₂²×C₄	1	2	yes	—
109	#681	—	2	2	no	—
110	#681	M₁₆	1	4	yes	—
111	C₂×#168	C₂²×C₄	1	2	yes	—
112	C₂×#168	C₂²×C₄	1	2	yes	—
113	#676	—	2	4	no	—
114	#676	—	2	4	no	—
115	C₈×D₅	—	2	2	no	—
116	C₈×D₅	—	2	2	no	—
117	C₈×D₅	—	2	4	no	—
118	C₅×C₁₆	—	2	2	no	—
119	C₅×C₁₆	—	2	2	no	—
120	#672	—	2	4	no	—
121	C₂×#131	—	5	1	no	—
122	C₂×#131	—	5	1	no	—
123	C₄×#41	—	5	1	no	—
124	C₂×M₃₂	—	5	2	no	—
125	C₂²×C₁₆	C₂×D₅	1	2	yes	—
126	#405	—	5	1	no	—
127	#405	—	5	1	no	—
128	#405	—	5	2	no	—
129	#405	—	5	2	no	—
130	C₄×M₁₆	—	5	1	no	—
131	C₄×M₁₆	—	5	2	no	—
132	C₂×C₄×C₈	—	5	2	no	—
133	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
134	C₄⋊C₁₆	—	5	2	no	—
135	C₄⋊C₁₆	—	5	2	no	—
136	C₄⋊C₁₆	C₂×D₅	1	1	yes	characteristic
137	C₄⋊C₁₆	C₂×D₅	1	1	yes	characteristic
138	C₄⋊C₁₆	C₂×D₅	1	2	yes	—
139	C₂²⋊C₁₆	—	5	4	no	—
140	C₂²⋊C₁₆	—	5	4	no	—
141	#346	—	5	4	no	—
142	C₄×C₁₆	—	5	4	no	—
143	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
144	C₂²×D₅	C₂²×C₈	1	2	yes	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	4	1	no	—
147	C₂×C₄×C₅	#117	1	2	yes	—
148	C₂×C₄×C₅	C₂²×C₈	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	C₂²×C₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
154	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₅	—	2	4	no	—
156	C₂×Dic₅	C₂²×C₈	1	2	yes	—
157	C₂×Dic₅	C₂²×C₈	1	2	yes	—
158	C₂×Dic₅	C₂²×C₈	1	2	yes	—
159	D₂₀	—	2	2	no	—
160	C₄×D₅	—	2	2	no	—
161	C₄×D₅	—	2	2	no	—
162	C₄×D₅	—	2	4	no	—
163	C₄×D₅	—	4	2	no	—
164	Dic₁₀	—	2	2	no	—
165	C₅×C₈	—	2	1	no	—
166	C₅×C₈	—	2	1	no	—
167	C₅×C₈	#117	1	2	yes	—
168	C₅×C₈	C₂×#41	1	1	yes	characteristic
169	C₅×C₈	C₂×#41	1	1	yes	characteristic
170	#168	—	2	2	no	—
171	#168	C₂×M₁₆	1	4	yes	—
172	C₂×#41	—	5	1	no	—
173	#131	—	10	4	no	—
174	#131	—	10	2	no	—
175	#131	—	10	2	no	—
176	C₂×M₁₆	—	5	1	no	—
177	C₂×M₁₆	—	5	1	no	—
178	C₂×M₁₆	—	5	2	no	—
179	C₂×M₁₆	—	5	2	no	—
180	C₂²×C₈	—	5	2	no	—
181	C₂²×C₈	—	5	2	no	—
182	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
184	C₄×Q₈	—	5	1	no	—
185	C₄×Q₈	—	5	1	no	—
186	C₄×D₄	—	5	2	no	—
187	C₄×D₄	—	5	2	no	—
188	C₄×D₄	—	5	1	no	—
189	C₄×D₄	—	5	1	no	—
190	#117	—	5	1	no	—
191	#117	—	5	2	no	—
192	C₂×C₄²	—	5	2	no	—
193	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
194	M₃₂	—	10	4	no	—
195	C₂×C₁₆	—	2	4	no	—
196	C₂×C₁₆	—	5	4	no	—
197	C₂×C₁₆	C₂²×D₅	1	2	yes	—
198	C₂×C₁₆	C₂²×D₅	1	2	yes	—
199	#97	—	5	2	no	—
200	#97	—	5	2	no	—
201	#97	—	5	2	no	—
202	C₄×C₈	—	5	2	no	—
203	C₄×C₈	—	5	2	no	—
204	C₄×C₈	—	5	2	no	—
205	C₄×C₈	C₄×D₅	1	1	yes	characteristic
206	C₄×C₈	C₄×D₅	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	#431	1	2	yes	—
212	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
213	C₂×D₅	—	2	2	no	—
214	C₂×D₅	—	2	2	no	—
215	C₂×D₅	—	2	2	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	#431	1	2	yes	—
220	C₄×C₅	#504	1	2	yes	—
221	C₄×C₅	C₂×#117	1	1	yes	characteristic
222	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
223	Dic₅	—	2	2	no	—
224	Dic₅	—	2	2	no	—
225	Dic₅	—	2	2	no	—
226	#41	—	10	2	no	—
227	#41	—	10	2	no	—
228	C₂×Q₈	—	5	1	no	—
229	C₂×D₄	—	5	2	no	—
230	C₂×D₄	—	5	1	no	—
231	C₂²×C₄	—	5	2	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	—	5	2	no	—
234	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
235	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
237	M₁₆	—	5	4	no	—
238	M₁₆	—	10	4	no	—
239	M₁₆	—	10	2	no	—
240	M₁₆	—	10	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	5	2	no	—
243	C₂×C₈	—	5	2	no	—
244	C₂×C₈	—	10	4	no	—
245	C₂×C₈	—	10	2	no	—
246	C₂×C₈	—	10	2	no	—
247	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
250	C₂×C₈	#711	1	2	yes	—
251	C₂×C₈	#713	1	2	yes	—
252	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
253	C₄⋊C₄	—	5	2	no	—
254	C₄⋊C₄	—	5	2	no	—
255	C₄⋊C₄	—	5	1	no	—
256	C₄⋊C₄	—	5	1	no	—
257	C₂²⋊C₄	—	5	2	no	—
258	C₂²⋊C₄	—	5	2	no	—
259	C₂²⋊C₄	—	5	2	no	—
260	C₄²	—	5	2	no	—
261	C₄²	—	5	2	no	—
262	C₄²	—	5	2	no	—
263	C₄²	C₈×D₅	1	1	yes	characteristic
264	C₄²	C₈×D₅	1	1	yes	characteristic
265	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
266	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
267	C₁₆	—	2	2	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	10	4	no	—
270	C₂×C₅	—	2	2	no	—
271	C₂×C₅	C₂×#431	1	1	yes	characteristic
272	C₂×C₅	C₂×#504	1	1	yes	characteristic
273	C₂×C₅	#3407	1	1	yes	characteristic
274	D₅	—	4	2	no	—
275	C₂³	—	5	2	no	—
276	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
277	Q₈	—	10	2	no	—
278	D₄	—	10	4	no	—
279	D₄	—	10	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	5	2	no	—
283	C₂×C₄	—	5	2	no	—
284	C₂×C₄	—	5	2	no	—
285	C₂×C₄	—	10	2	no	—
286	C₂×C₄	—	10	2	no	—
287	C₂×C₄	—	10	4	no	—
288	C₂×C₄	—	20	2	no	—
289	C₂×C₄	#4038	1	2	yes	—
290	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
291	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
292	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
293	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
295	C₂×C₄	D₅×M₁₆	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	5	4	no	—
300	C₈	—	10	2	no	—
301	C₈	#4038	1	2	yes	—
302	C₈	C₂×#711	1	1	yes	characteristic
303	C₈	C₂×#713	1	1	yes	characteristic
304	C₅	#33423	1	1	yes	characteristic, complemented, Sylow
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	10	2	no	—
308	C₂²	—	10	2	no	—
309	C₂²	—	10	2	no	—
310	C₂²	#65122	1	2	yes	—
311	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	10	2	no	—
316	C₄	—	10	2	no	—
317	C₄	—	10	2	no	—
318	C₄	#65122	1	2	yes	—
319	C₄	#65190	1	2	yes	—
320	C₄	C₂×#4038	1	1	yes	characteristic
321	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	20	2	no	—
324	C₂	C₂×#65122	1	1	yes	characteristic
325	C₂	C₂×#65190	1	1	yes	characteristic
326	C₂	#10613738	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₂×C₅×C₄⋊C₁₆	C₂
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₅
hypocenter	C₅	#33423
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
4. #10602192	C₂	semidirect
5. #10600536	C₂	semidirect
6. #10600069	C₂	semidirect
7. #10600068	C₂	semidirect
8. #10600067	C₂	semidirect
74. C₂×C₄⋊C₁₆	D₅	semidirect
304. C₅	#33423	semidirect

Maximal subgroups

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

subgroup	index
C₄×#4047	2
C₂×C₅×C₄⋊C₁₆	2
#10602202	2
#10602192	2
#10600536	2
#10600069	2
#10600068	2
#10600067	2
#33423	5