Subgroups — #13043470

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×#4053	C₂	1	1	yes	maximal, characteristic
2	#10604742	C₂	1	2	yes	maximal
3	#10604741	C₂	1	2	yes	maximal, complemented
4	#10604150	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
6	#10603871	C₂	1	4	yes	maximal, complemented
7	#10603831	C₂	1	2	yes	maximal, complemented
8	#10603829	C₂	1	2	yes	maximal, complemented
9	C₂×#4053	C₄	1	1	yes	characteristic
10	C₂×#4053	C₂²	1	1	yes	characteristic
11	C₄×#709	C₄	1	1	yes	characteristic
12	C₂×#3998	C₂²	1	2	yes	—
13	C₂²×#674	C₂²	1	2	yes	—
14	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
15	#65411	C₄	1	1	yes	characteristic
16	#65411	C₂²	1	1	yes	characteristic
17	C₂×C₄×#168	C₂²	1	1	yes	characteristic
18	#64997	C₄	1	2	yes	—
19	#64997	C₂²	1	2	yes	—
20	C₄×#676	C₄	1	2	yes	—
21	C₄×C₈×D₅	C₂²	1	2	yes	—
22	#64887	C₂²	1	4	yes	—
23	#64887	C₂²	1	4	yes	—
24	#64886	C₂²	1	4	yes	—
25	#64885	C₂²	1	4	yes	—
26	C₄⋊#674	C₂²	1	1	yes	characteristic
27	C₄⋊#674	C₂²	1	2	yes	—
28	C₄⋊#674	C₂²	1	1	yes	characteristic
29	C₄⋊#674	C₂²	1	2	yes	—
30	C₄⋊#674	C₂²	1	2	yes	—
31	#33423	—	5	1	no	maximal, Sylow
32	C₂×#709	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×M₁₆	C₂×C₄	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	C₂²×#168	C₂×C₄	1	1	yes	characteristic
40	C₂²×#168	C₂³	1	1	yes	characteristic
41	#4053	—	2	4	no	—
42	#4053	—	2	2	no	—
43	#4053	—	2	2	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	#3998	—	2	4	no	—
55	C₂×#674	—	2	4	no	—
56	C₂×#674	C₂³	1	2	yes	—
57	C₂×#674	C₂³	1	4	yes	—
58	C₂×#674	C₂³	1	2	yes	—
59	C₅×#97	C₂×C₄	1	2	yes	—
60	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
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	C₄×#168	C₂×C₄	1	1	yes	characteristic
66	C₄×#168	C₂×C₄	1	1	yes	characteristic
67	C₄×#168	C₂³	1	1	yes	characteristic
68	C₄×#168	C₂³	1	1	yes	characteristic
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	—	5	1	no	—
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	C₅×M₁₆	—	2	2	no	—
89	C₅×M₁₆	M₁₆	1	4	yes	—
90	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
92	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
97	#685	C₂×C₈	1	2	yes	—
98	#685	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
100	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
101	#683	C₂×C₈	1	2	yes	—
102	#683	C₂×C₈	1	2	yes	—
103	C₄×Dic₅	C₂×C₈	1	2	yes	—
104	C₄×Dic₅	C₂×C₈	1	2	yes	—
105	C₄×Dic₅	C₂²×C₄	1	2	yes	—
106	C₂×#168	—	2	2	no	—
107	C₂×#168	C₂²×C₄	1	1	yes	characteristic
108	C₂×#168	C₂²×C₄	1	1	yes	characteristic
109	C₂×#168	C₂²×C₄	1	1	yes	characteristic
110	C₂×#168	#41	1	2	yes	—
111	C₂×#168	#41	1	2	yes	—
112	C₂×#168	C₂⁴	1	1	yes	characteristic
113	#676	—	2	4	no	—
114	#676	—	2	4	no	—
115	C₈×D₅	—	2	4	no	—
116	C₈×D₅	—	2	2	no	—
117	C₈×D₅	—	2	2	no	—
118	#674	—	2	2	no	—
119	#674	—	2	2	no	—
120	#674	—	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	—	5	2	no	—
126	#405	—	5	1	no	—
127	#405	—	5	1	no	—
128	#405	—	5	2	no	—
129	#405	—	5	2	no	—
130	C₄×M₁₆	—	5	2	no	—
131	C₄×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₂×C₄×C₈	—	5	1	no	—
133	C₂×C₄×C₈	—	5	2	no	—
134	C₄⋊C₁₆	—	5	1	no	—
135	C₄⋊C₁₆	—	5	1	no	—
136	C₄⋊C₁₆	—	5	2	no	—
137	C₄⋊C₁₆	—	5	2	no	—
138	C₄⋊C₁₆	—	5	2	no	—
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	2	no	—
166	C₅×C₈	C₂×M₁₆	1	4	yes	—
167	#168	—	2	1	no	—
168	#168	—	2	1	no	—
169	#168	#117	1	2	yes	—
170	#168	C₂×#41	1	1	yes	characteristic
171	#168	C₂×#41	1	1	yes	characteristic
172	C₂×#41	—	5	1	no	—
173	#131	—	10	2	no	—
174	#131	—	10	2	no	—
175	#131	—	10	4	no	—
176	C₂×M₁₆	—	5	2	no	—
177	C₂×M₁₆	—	5	2	no	—
178	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
179	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
180	C₂²×C₈	—	5	1	no	—
181	C₂²×C₈	—	5	1	no	—
182	C₂²×C₈	—	5	2	no	—
183	C₂²×C₈	—	5	2	no	—
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	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
194	M₃₂	—	10	4	no	—
195	C₂×C₁₆	—	5	2	no	—
196	C₂×C₁₆	—	5	4	no	—
197	C₂×C₁₆	—	5	2	no	—
198	C₂×C₁₆	—	10	4	no	—
199	#97	—	5	2	no	—
200	#97	—	5	2	no	—
201	#97	C₂×AGL(1,𝔽₅)	1	2	yes	—
202	C₄×C₈	—	5	2	no	—
203	C₄×C₈	—	5	2	no	—
204	C₄×C₈	—	5	1	no	—
205	C₄×C₈	—	5	1	no	—
206	C₄×C₈	—	5	1	no	—
207	C₄×C₈	—	5	1	no	—
208	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
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	D₅⋊C₈	1	1	yes	characteristic
235	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
236	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
237	M₁₆	—	2	2	no	—
238	M₁₆	—	10	4	no	—
239	M₁₆	—	10	4	no	—
240	M₁₆	#700	1	4	yes	—
241	C₂×C₈	—	5	2	no	—
242	C₂×C₈	—	5	2	no	—
243	C₂×C₈	—	5	1	no	—
244	C₂×C₈	—	5	1	no	—
245	C₂×C₈	—	5	1	no	—
246	C₂×C₈	—	5	1	no	—
247	C₂×C₈	—	10	2	no	—
248	C₂×C₈	—	10	2	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	—	10	4	no	—
251	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
252	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
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	D₅⋊C₈	1	1	yes	characteristic
264	C₄²	D₅⋊C₈	1	1	yes	characteristic
265	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
267	C₁₆	—	10	2	no	—
268	C₁₆	—	10	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×D₅⋊C₈	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	4	no	—
286	C₂×C₄	—	10	2	no	—
287	C₂×C₄	—	10	2	no	—
288	C₂×C₄	—	20	2	no	—
289	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
290	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
291	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
292	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
293	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
294	C₂×C₄	C₂×#700	1	2	yes	—
295	C₂×C₄	#4130	1	2	yes	—
296	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	5	1	no	—
299	C₈	—	5	1	no	—
300	C₈	—	5	2	no	—
301	C₈	—	10	1	no	—
302	C₈	—	10	1	no	—
303	C₈	C₂×#700	1	4	yes	—
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	#65680	1	2	yes	—
311	C₂²	C₂²×D₅⋊C₈	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	#65680	1	2	yes	—
319	C₄	#65725	1	2	yes	—
320	C₄	C₂²×#700	1	1	yes	characteristic
321	C₄	C₂×#4130	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	20	2	no	—
324	C₂	C₂×#65680	1	1	yes	characteristic
325	C₂	C₂×#65725	1	1	yes	characteristic
326	C₂	#10617404	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₄×C₅×M₁₆	C₄
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×#168	C₂⁴
hypercenter	C₄×M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#33423
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #10604741	C₂	semidirect
4. #10604150	C₂	semidirect
5. C₂×C₄⋊#674	C₂	semidirect
6. #10603871	C₂	semidirect
7. #10603831	C₂	semidirect
8. #10603829	C₂	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×#4053	2
#10604742	2
#10604741	2
#10604150	2
C₂×C₄⋊#674	2
#10603871	2
#10603831	2
#10603829	2
#33423	5