Subgroups — #12016188

Order 1280.

474 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10613738	C₂	1	1	yes	maximal, characteristic, complemented
2	#10610189	C₂	1	1	yes	maximal, characteristic
3	#10602231	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602230	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602228	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1376	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598149	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4047	C₂²	1	1	yes	characteristic
9	C₂×#4047	C₂²	1	1	yes	characteristic
10	#65811	C₄	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#65399	C₄	1	1	yes	characteristic
15	#65399	C₄	1	1	yes	characteristic
16	C₂×#3944	C₂²	1	1	yes	characteristic
17	C₂×#3944	C₂²	1	1	yes	characteristic
18	#65213	C₄	1	1	yes	characteristic
19	#65196	—	2	2	no	—
20	D₅×M₃₂	—	2	2	no	—
21	#65190	—	2	2	no	—
22	#64977	—	2	2	no	—
23	C₈×D₂₀	—	2	2	no	—
24	#64735	—	2	2	no	—
25	#64734	—	2	2	no	—
26	#64728	—	2	2	no	—
27	#12982	—	5	1	no	maximal, Sylow
28	C₂×#709	C₂×C₄	1	1	yes	characteristic
29	C₂×#709	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	#4076	—	2	2	no	—
34	C₂×#681	C₂×C₄	1	1	yes	characteristic
35	C₂×#681	C₂×C₄	1	1	yes	characteristic
36	#4047	—	2	1	no	—
37	#4047	—	2	1	no	—
38	#4047	—	2	1	no	—
39	#4047	—	2	1	no	—
40	#4047	—	2	1	no	—
41	#4047	—	2	1	no	—
42	#4047	—	2	1	no	—
43	#4047	—	2	1	no	—
44	C₂×#676	—	2	1	no	—
45	C₂×#676	D₄	1	2	yes	—
46	C₂×C₈×D₅	—	2	1	no	—
47	C₂×C₈×D₅	D₄	1	2	yes	—
48	#4021	C₈	1	1	yes	characteristic
49	#4021	C₈	1	1	yes	characteristic
50	#4020	C₈	1	1	yes	characteristic
51	C₄×D₂₀	—	2	2	no	—
52	#4015	C₈	1	1	yes	characteristic
53	C₅×M₃₂	—	2	1	no	—
54	C₅×M₃₂	—	2	1	no	—
55	C₅×M₃₂	—	2	1	no	—
56	C₅×M₃₂	D₄	1	2	yes	complemented
57	C₂×C₅×C₁₆	—	2	1	no	—
58	C₂×C₅×C₁₆	—	2	1	no	—
59	C₄×C₅×C₈	—	2	1	no	—
60	C₄×C₅×C₈	—	2	1	no	—
61	#3952	—	2	2	no	—
62	#3952	—	2	2	no	—
63	#3944	—	2	1	no	—
64	#3944	—	2	1	no	—
65	#3944	—	2	1	no	—
66	#3944	D₄	1	2	yes	complemented
67	C₂×#672	—	2	1	no	—
68	C₂×#672	—	2	1	no	—
69	C₄⋊#168	—	2	2	no	—
70	#3930	—	2	4	no	—
71	D₅×C₁₆	—	2	4	no	—
72	#3407	—	5	1	no	—
73	#2931	—	5	1	no	—
74	#2117	—	5	1	no	—
75	#2117	—	5	1	no	—
76	#2115	—	5	1	no	—
77	#1376	—	5	1	no	—
78	#1376	D₅	1	1	yes	characteristic, complemented
79	C₂²×C₄×C₅	—	2	1	no	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₅	—	2	1	no	—
82	C₂×C₂²⋊D₅	M₁₆	1	2	yes	—
83	#709	—	2	1	no	—
84	#709	—	2	1	no	—
85	#709	—	2	1	no	—
86	#709	—	2	1	no	—
87	#709	—	2	1	no	—
88	#709	—	2	1	no	—
89	#709	—	2	1	no	—
90	#709	—	2	1	no	—
91	C₂×D₂₀	—	2	2	no	—
92	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
93	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
94	C₂×C₄×D₅	—	2	1	no	—
95	C₂×C₄×D₅	C₂²⋊C₄	1	2	yes	—
96	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
97	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
98	C₂×C₅×C₈	—	2	1	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	2	1	no	—
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	#41	1	1	yes	characteristic
106	C₂×C₅×C₈	#41	1	1	yes	characteristic
107	C₄²×C₅	—	2	1	no	—
108	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
109	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
110	#685	—	2	2	no	—
111	#685	—	2	2	no	—
112	C₄⋊Dic₅	—	2	2	no	—
113	#681	—	2	1	no	—
114	#681	—	2	1	no	—
115	#681	—	2	1	no	—
116	#681	—	2	1	no	—
117	C₂×#168	—	2	1	no	—
118	C₂×#168	C₂²⋊C₄	1	2	yes	—
119	#676	—	2	2	no	—
120	#676	—	2	2	no	—
121	#676	—	4	1	no	—
122	#676	—	4	1	no	—
123	C₈×D₅	—	2	2	no	—
124	C₈×D₅	—	2	2	no	—
125	C₈×D₅	—	4	1	no	—
126	C₈×D₅	—	4	1	no	—
127	C₅×C₁₆	—	2	2	no	—
128	C₅×C₁₆	—	4	1	no	—
129	#672	—	2	2	no	—
130	#672	—	4	1	no	—
131	C₂×#131	—	5	1	no	—
132	C₂×#131	—	5	1	no	—
133	#532	—	5	1	no	—
134	#504	—	10	2	no	—
135	#504	—	10	2	no	—
136	C₂×M₃₂	—	5	1	no	—
137	C₂×M₃₂	—	5	1	no	—
138	C₂×M₃₂	—	10	2	no	—
139	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
140	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
141	#436	—	10	2	no	—
142	C₈×D₄	—	10	2	no	—
143	#423	—	5	1	no	—
144	#408	—	5	1	no	—
145	#408	—	5	1	no	—
146	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
147	#350	—	10	2	no	—
148	#350	—	10	2	no	—
149	#349	—	10	2	no	—
150	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
151	C₂²×D₅	—	2	1	no	—
152	C₂²×D₅	C₂²⋊C₈	1	2	yes	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₄×C₅	#131	1	1	yes	characteristic
161	C₂×C₄×C₅	#131	1	1	yes	characteristic
162	C₂²⋊D₅	—	2	2	no	—
163	C₂²⋊D₅	—	2	2	no	—
164	C₂²⋊D₅	—	4	1	no	—
165	C₂²⋊D₅	—	4	1	no	—
166	C₂×Dic₅	—	2	1	no	—
167	C₂×Dic₅	C₂²⋊C₈	1	2	yes	—
168	D₂₀	—	2	1	no	—
169	D₂₀	—	2	1	no	—
170	D₂₀	—	2	1	no	—
171	D₂₀	—	2	1	no	—
172	D₂₀	—	4	4	no	—
173	C₄×D₅	—	2	2	no	—
174	C₄×D₅	—	2	2	no	—
175	C₄×D₅	—	4	1	no	—
176	C₄×D₅	—	4	1	no	—
177	Dic₁₀	—	2	1	no	—
178	Dic₁₀	—	2	1	no	—
179	Dic₁₀	—	2	1	no	—
180	Dic₁₀	—	2	1	no	—
181	C₅×C₈	—	4	1	no	—
182	C₅×C₈	#120	1	1	yes	characteristic
183	C₅×C₈	#121	1	1	yes	characteristic
184	C₅×C₈	#121	1	1	yes	characteristic
185	C₅×C₈	#123	1	1	yes	characteristic
186	#168	—	2	2	no	—
187	#168	—	4	1	no	—
188	C₂×#41	—	5	1	no	—
189	C₂×#41	—	5	1	no	—
190	#131	—	10	1	no	—
191	#131	—	10	1	no	—
192	#131	—	10	1	no	—
193	#131	—	10	1	no	—
194	#131	—	10	1	no	—
195	#131	—	10	1	no	—
196	#131	—	10	1	no	—
197	#131	—	10	1	no	—
198	C₂×M₁₆	—	5	1	no	—
199	C₂×M₁₆	—	5	2	no	—
200	C₂×M₁₆	—	5	1	no	—
201	C₂×M₁₆	—	10	1	no	—
202	C₂²×C₈	—	5	2	no	—
203	C₂²×C₈	—	10	1	no	—
204	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
206	C₄⋊Q₈	—	5	1	no	—
207	#127	—	5	1	no	—
208	#124	—	5	1	no	—
209	#124	—	5	1	no	—
210	#121	—	10	2	no	—
211	C₄×D₄	—	10	2	no	—
212	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
213	M₃₂	—	2	1	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	—	2	1	no	—
216	M₃₂	—	5	2	no	—
217	M₃₂	—	10	4	no	—
218	M₃₂	—	10	1	no	—
219	M₃₂	—	10	1	no	—
220	M₃₂	—	10	1	no	—
221	M₃₂	D₂₀	1	2	yes	complemented
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	10	4	no	—
225	C₂×C₁₆	—	10	1	no	—
226	C₂×C₁₆	—	10	1	no	—
227	C₄⋊C₈	—	10	2	no	—
228	C₂²⋊C₈	—	10	2	no	—
229	C₂²⋊C₈	—	10	2	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₂²×C₅	—	4	1	no	—
233	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
234	C₂²×C₅	#435	1	1	yes	characteristic
235	C₂²×C₅	#435	1	1	yes	characteristic
236	C₂×D₅	—	2	2	no	—
237	C₂×D₅	—	4	2	no	—
238	C₂×D₅	—	4	2	no	—
239	C₂×D₅	—	4	1	no	—
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	—	4	1	no	—
242	C₄×C₅	#386	1	1	yes	characteristic
243	C₄×C₅	#408	1	1	yes	characteristic
244	C₄×C₅	C₈×D₄	1	1	yes	characteristic
245	C₄×C₅	C₈×D₄	1	1	yes	characteristic
246	Dic₅	—	2	2	no	—
247	Dic₅	—	4	1	no	—
248	#41	—	10	1	no	—
249	#41	—	10	1	no	—
250	#41	—	10	1	no	—
251	#41	—	10	1	no	—
252	#41	—	10	1	no	—
253	#41	—	10	1	no	—
254	#41	—	10	1	no	—
255	#41	—	10	1	no	—
256	C₂×Q₈	—	5	1	no	—
257	C₂×Q₈	—	5	1	no	—
258	C₂×D₄	—	5	2	no	—
259	C₂×D₄	—	5	1	no	—
260	C₂×D₄	—	5	1	no	—
261	C₂×D₄	—	10	2	no	—
262	C₂×D₄	—	10	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	5	2	no	—
265	C₂²×C₄	—	10	1	no	—
266	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
267	M₁₆	—	10	1	no	—
268	M₁₆	—	10	1	no	—
269	M₁₆	—	10	2	no	—
270	M₁₆	—	10	2	no	—
271	M₁₆	—	10	1	no	—
272	M₁₆	—	10	1	no	—
273	M₁₆	—	20	1	no	—
274	M₁₆	—	20	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	5	2	no	—
279	C₂×C₈	—	10	2	no	—
280	C₂×C₈	—	10	2	no	—
281	C₂×C₈	—	10	1	no	—
282	C₂×C₈	—	20	1	no	—
283	C₂×C₈	—	20	1	no	—
284	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
285	C₂×C₈	#709	1	1	yes	characteristic
286	C₂×C₈	D₄×D₅	1	1	yes	characteristic
287	C₂×C₈	D₄×D₅	1	1	yes	characteristic
288	C₂×C₈	D₄×D₅	1	1	yes	characteristic
289	C₂×C₈	#713	1	1	yes	characteristic
290	C₄⋊C₄	—	10	2	no	—
291	C₂²⋊C₄	—	10	2	no	—
292	C₂²⋊C₄	—	10	2	no	—
293	C₄²	—	2	1	no	—
294	C₄²	C₈×D₅	1	1	yes	characteristic
295	C₄²	C₈×D₅	1	1	yes	characteristic
296	C₁₆	—	2	2	no	—
297	C₁₆	—	4	1	no	—
298	C₁₆	—	10	2	no	—
299	C₁₆	—	20	1	no	—
300	C₂×C₅	—	2	1	no	—
301	C₂×C₅	—	2	1	no	—
302	C₂×C₅	—	2	1	no	—
303	C₂×C₅	#1869	1	1	yes	characteristic
304	D₅	—	4	2	no	—
305	D₅	—	8	1	no	—
306	C₂³	—	5	2	no	—
307	C₂³	—	10	1	no	—
308	C₂³	D₅×M₁₆	1	1	yes	characteristic
309	Q₈	—	10	1	no	—
310	Q₈	—	10	1	no	—
311	Q₈	—	10	1	no	—
312	Q₈	—	10	1	no	—
313	D₄	—	10	2	no	—
314	D₄	—	10	2	no	—
315	D₄	—	10	1	no	—
316	D₄	—	10	1	no	—
317	D₄	—	10	1	no	—
318	D₄	—	10	1	no	—
319	D₄	—	20	4	no	—
320	D₄	—	20	1	no	—
321	D₄	—	20	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	5	2	no	—
326	C₂×C₄	—	10	2	no	—
327	C₂×C₄	—	10	2	no	—
328	C₂×C₄	—	10	1	no	—
329	C₂×C₄	—	20	1	no	—
330	C₂×C₄	—	20	1	no	—
331	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
332	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
333	C₂×C₄	#4039	1	1	yes	characteristic
334	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
335	C₂×C₄	#4047	1	1	yes	characteristic
336	C₂×C₄	#4053	1	1	yes	characteristic
337	C₈	—	4	1	no	—
338	C₈	—	10	2	no	—
339	C₈	—	20	1	no	—
340	C₈	#4028	1	1	yes	characteristic
341	C₈	#4030	1	1	yes	characteristic
342	C₈	#4040	1	1	yes	characteristic
343	C₈	#4041	1	1	yes	characteristic
344	C₅	#12982	1	1	yes	characteristic, complemented, Sylow
345	C₂²	—	4	1	no	—
346	C₂²	—	10	2	no	—
347	C₂²	—	20	2	no	—
348	C₂²	—	20	2	no	—
349	C₂²	—	20	1	no	—
350	C₂²	#64995	1	1	yes	characteristic
351	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
352	C₂²	#65125	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	10	2	no	—
356	C₄	—	20	1	no	—
357	C₄	#64954	1	1	yes	characteristic
358	C₄	C₈×D₂₀	1	1	yes	characteristic
359	C₄	#65016	1	1	yes	characteristic
360	C₄	#65123	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	20	2	no	—
365	C₂	—	40	1	no	—
366	C₂	#10599461	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#4028
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1376	C₂
socle	C₂×C₅	#1869
prime radical	#65811	C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1376	D₅
hypocenter	C₅	#12982
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
1. #10613738	C₂	semidirect
3. #10602231	C₂	semidirect
4. #10602230	C₂	semidirect
5. #10602228	C₂	semidirect
6. C₅×#1376	C₂	semidirect
7. #10598149	C₂	semidirect
56. C₅×M₃₂	D₄	semidirect
66. #3944	D₄	semidirect
78. #1376	D₅	semidirect
221. M₃₂	D₂₀	semidirect
344. C₅	#12982	semidirect

Maximal subgroups

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

subgroup	index
#10613738	2
#10610189	2
#10602231	2
#10602230	2
#10602228	2
C₅×#1376	2
#10598149	2
#12982	5