Subgroups — #13043639

Order 1280.

490 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611689	C₂	1	1	yes	maximal, characteristic
2	#10604448	C₂	1	2	yes	maximal, complemented
3	#10604026	C₂	1	2	yes	maximal, complemented
4	#10604026	C₂	1	2	yes	maximal, complemented
5	#10604025	C₂	1	2	yes	maximal, complemented
6	#10603869	C₂	1	2	yes	maximal
7	#10603868	C₂	1	2	yes	maximal
8	#10603827	C₂	1	1	yes	maximal, characteristic
9	#10603823	C₂	1	1	yes	maximal, characteristic
10	#65912	C₄	1	1	yes	characteristic
11	C₂×#3990	C₂²	1	2	yes	—
12	C₂×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₅×#446	C₄	1	1	yes	characteristic
15	Q₈×#168	C₂²	1	1	yes	characteristic
16	#65128	C₄	1	2	yes	—
17	#65126	C₄	1	1	yes	characteristic
18	#65119	C₂²	1	2	yes	—
19	#65119	C₂²	1	1	yes	characteristic
20	#64997	C₂²	1	2	yes	—
21	#64990	C₄	1	2	yes	—
22	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
23	#64968	C₄	1	1	yes	characteristic
24	#64888	C₂²	1	1	yes	characteristic
25	#64888	C₂²	1	1	yes	characteristic
26	#64888	C₂²	1	2	yes	—
27	#64888	C₂²	1	2	yes	—
28	#64886	C₂²	1	2	yes	—
29	#64886	C₂²	1	2	yes	—
30	#64885	C₂²	1	2	yes	—
31	#64859	C₂²	1	2	yes	—
32	#64859	C₂²	1	2	yes	—
33	C₄⋊#674	C₂²	1	2	yes	—
34	C₄⋊#674	C₂²	1	2	yes	—
35	C₄⋊#674	C₂²	1	2	yes	—
36	C₄×#674	C₂²	1	2	yes	—
37	#33475	—	5	1	no	maximal, Sylow
38	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
39	#4092	C₈	1	1	yes	characteristic
40	#4090	C₈	1	1	yes	characteristic
41	C₂×#676	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
44	#4043	C₈	1	1	yes	characteristic
45	#4042	C₈	1	2	yes	—
46	#4035	C₈	1	2	yes	—
47	#4034	C₈	1	1	yes	characteristic
48	#4018	C₂×C₄	1	2	yes	—
49	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₀	C₂×C₄	1	2	yes	—
52	C₂×#674	C₂³	1	2	yes	—
53	C₂×#674	C₂³	1	2	yes	—
54	C₂×#674	C₂³	1	2	yes	—
55	C₂×#674	C₂³	1	2	yes	—
56	#3990	—	2	4	no	—
57	D₅⋊C₁₆	—	2	2	no	—
58	D₅⋊C₁₆	Q₈	1	4	yes	complemented
59	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₅×#97	C₂×C₄	1	2	yes	—
62	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
63	#3947	C₂×C₄	1	2	yes	—
64	#3946	C₂×C₄	1	2	yes	—
65	#3946	C₂×C₄	1	2	yes	—
66	#3946	C₂×C₄	1	2	yes	—
67	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₅	C₂³	1	2	yes	—
69	C₈×Dic₅	C₂³	1	1	yes	characteristic
70	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
71	C₄⋊#168	C₂×C₄	1	2	yes	—
72	C₄×#168	C₂³	1	2	yes	—
73	C₄×#168	C₂³	1	1	yes	characteristic
74	#2960	—	5	1	no	—
75	#2183	—	5	2	no	—
76	#2183	—	5	2	no	—
77	#2183	—	5	2	no	—
78	#2183	—	5	2	no	—
79	#2151	—	5	2	no	—
80	#2150	—	5	1	no	—
81	#2108	—	5	2	no	—
82	C₂×#346	—	5	1	no	—
83	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₀	C₂×C₈	1	2	yes	—
88	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
93	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
94	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅	C₂²×C₄	1	2	yes	—
96	#685	C₂×C₈	1	2	yes	—
97	#685	C₂×C₈	1	2	yes	—
98	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
99	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
100	#683	C₂×C₈	1	2	yes	—
101	#683	C₂×C₈	1	2	yes	—
102	#683	C₂×C₈	1	2	yes	—
103	#683	C₂×C₈	1	2	yes	—
104	#683	C₂×C₈	1	2	yes	—
105	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
108	C₄×Dic₅	C₂²×C₄	1	2	yes	—
109	C₂×#168	C₂²×C₄	1	1	yes	characteristic
110	C₂×#168	C₂²×C₄	1	2	yes	—
111	C₂×#168	C₂⁴	1	1	yes	characteristic
112	#676	—	2	4	no	—
113	C₈×D₅	—	2	2	no	—
114	C₈×D₅	C₂×Q₈	1	2	yes	—
115	C₈×D₅	#41	1	2	yes	—
116	#674	—	2	2	no	—
117	#674	—	2	2	no	—
118	#674	—	2	2	no	—
119	#674	C₂×Q₈	1	4	yes	—
120	#533	—	5	1	no	—
121	C₂×M₃₂	—	5	2	no	—
122	C₂²×C₁₆	—	5	1	no	—
123	C₂²×C₁₆	—	5	1	no	—
124	#446	—	5	2	no	—
125	#446	—	5	1	no	—
126	#446	AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₈×Q₈	—	5	1	no	—
128	C₈×Q₈	—	5	1	no	—
129	C₈×Q₈	—	5	2	no	—
130	#424	—	5	2	no	—
131	#423	—	5	1	no	—
132	#405	—	5	2	no	—
133	C₂×C₄×C₈	—	5	1	no	—
134	C₄⋊C₁₆	—	5	2	no	—
135	C₄⋊C₁₆	—	5	2	no	—
136	C₄⋊C₁₆	—	5	2	no	—
137	C₄⋊C₁₆	—	5	1	no	—
138	C₄⋊C₁₆	—	5	1	no	—
139	C₄⋊C₁₆	—	5	2	no	—
140	C₄⋊C₁₆	—	5	2	no	—
141	#346	—	5	2	no	—
142	#346	—	5	2	no	—
143	#346	—	5	2	no	—
144	#346	—	5	2	no	—
145	C₄×C₁₆	—	5	2	no	—
146	C₄×C₁₆	—	5	2	no	—
147	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
148	C₅×Q₈	—	4	1	no	—
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	2	yes	—
152	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
153	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
154	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
155	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
156	C₂×Dic₅	C₂²×C₈	1	2	yes	—
157	C₂×Dic₅	C₂²×C₈	1	2	yes	—
158	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
159	C₄×D₅	—	4	1	no	—
160	C₄×D₅	—	4	1	no	—
161	C₄×D₅	C₄×Q₈	1	2	yes	—
162	C₄×D₅	#131	1	2	yes	—
163	Dic₁₀	—	4	2	no	—
164	Dic₁₀	—	4	1	no	—
165	C₅×C₈	—	2	1	no	—
166	C₅×C₈	—	2	2	no	—
167	C₅×C₈	C₄×Q₈	1	2	yes	—
168	#168	—	2	1	no	—
169	#168	—	2	2	no	—
170	#168	C₂²×Q₈	1	1	yes	characteristic
171	#168	C₂×#41	1	1	yes	characteristic
172	C₂×M₁₆	—	5	2	no	—
173	C₂²×C₈	—	5	1	no	—
174	C₂²×C₈	—	5	1	no	—
175	C₄⋊Q₈	—	5	1	no	—
176	C₄⋊Q₈	—	5	1	no	—
177	#125	—	5	2	no	—
178	C₂²⋊Q₈	—	5	1	no	—
179	C₂²⋊Q₈	—	5	2	no	—
180	C₂²⋊Q₈	—	5	1	no	—
181	C₄×Q₈	—	5	2	no	—
182	C₄×Q₈	—	5	1	no	—
183	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
184	#117	—	5	2	no	—
185	C₂×C₄²	—	5	1	no	—
186	M₃₂	—	10	4	no	—
187	C₂×C₁₆	—	5	2	no	—
188	C₂×C₁₆	—	5	4	no	—
189	C₂×C₁₆	—	5	2	no	—
190	C₂×C₁₆	—	5	2	no	—
191	C₂×C₁₆	—	5	2	no	—
192	C₂×C₁₆	—	10	2	no	—
193	C₄⋊C₈	—	5	2	no	—
194	C₄⋊C₈	—	5	2	no	—
195	C₄⋊C₈	—	5	2	no	—
196	C₄⋊C₈	—	5	1	no	—
197	C₄⋊C₈	—	5	2	no	—
198	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
200	#97	—	5	2	no	—
201	#97	C₂×AGL(1,𝔽₅)	1	2	yes	—
202	C₄×C₈	—	5	2	no	—
203	C₄×C₈	—	5	1	no	—
204	C₄×C₈	—	5	1	no	—
205	C₄×C₈	—	5	2	no	—
206	C₄×C₈	—	5	1	no	—
207	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
208	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
209	C₂×D₅	—	2	1	no	—
210	C₂×D₅	—	2	1	no	—
211	C₂×D₅	C₈×Q₈	1	2	yes	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
217	C₄×C₅	C₂×#131	1	1	yes	characteristic
218	Dic₅	—	2	1	no	—
219	Dic₅	—	2	2	no	—
220	Dic₅	—	2	2	no	—
221	Dic₅	—	2	1	no	—
222	Dic₅	C₈×Q₈	1	2	yes	—
223	C₂×Q₈	—	5	2	no	—
224	C₂×Q₈	—	5	1	no	—
225	C₂×Q₈	D₅⋊C₈	1	1	yes	characteristic
226	C₂²×C₄	—	5	1	no	—
227	C₂²×C₄	—	5	1	no	—
228	C₂²×C₄	—	5	1	no	—
229	M₁₆	—	10	4	no	—
230	C₂×C₈	—	5	1	no	—
231	C₂×C₈	—	5	2	no	—
232	C₂×C₈	—	5	2	no	—
233	C₂×C₈	—	5	1	no	—
234	C₂×C₈	—	5	2	no	—
235	C₂×C₈	—	10	2	no	—
236	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
237	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
238	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₄⋊C₄	—	5	2	no	—
240	C₄⋊C₄	—	5	2	no	—
241	C₄⋊C₄	—	5	2	no	—
242	C₄⋊C₄	—	5	2	no	—
243	C₄⋊C₄	—	5	2	no	—
244	C₄⋊C₄	—	5	2	no	—
245	C₄⋊C₄	—	5	1	no	—
246	C₄⋊C₄	D₅⋊C₈	1	2	yes	—
247	C₄⋊C₄	D₅⋊C₈	1	1	yes	characteristic
248	C₂²⋊C₄	—	5	2	no	—
249	C₂²⋊C₄	—	5	2	no	—
250	C₄²	—	5	1	no	—
251	C₄²	—	5	1	no	—
252	C₄²	—	5	1	no	—
253	C₄²	—	5	2	no	—
254	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
255	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
256	C₁₆	—	5	4	no	—
257	C₁₆	—	10	2	no	—
258	C₁₆	—	10	2	no	—
259	C₁₆	—	10	2	no	—
260	C₂×C₅	C₂×C₈×Q₈	1	1	yes	characteristic
261	C₂×C₅	#3407	1	1	yes	characteristic
262	C₂×C₅	#3407	1	1	yes	characteristic
263	D₅	—	2	2	no	—
264	C₂³	—	5	1	no	—
265	Q₈	—	4	1	no	—
266	Q₈	—	20	1	no	—
267	Q₈	—	20	2	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	5	2	no	—
270	C₂×C₄	—	5	2	no	—
271	C₂×C₄	—	5	1	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	5	2	no	—
274	C₂×C₄	—	5	2	no	—
275	C₂×C₄	—	20	1	no	—
276	C₂×C₄	—	20	1	no	—
277	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
278	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
279	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
280	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
281	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	2	no	—
284	C₈	—	5	1	no	—
285	C₈	—	5	1	no	—
286	C₈	—	10	1	no	—
287	C₈	—	10	2	no	—
288	C₈	Q₈×AGL(1,𝔽₅)	1	2	yes	—
289	C₅	#33475	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	5	2	no	—
291	C₂²	—	10	1	no	—
292	C₂²	—	10	1	no	—
293	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	5	2	no	—
299	C₄	—	10	1	no	—
300	C₄	—	10	2	no	—
301	C₄	—	10	1	no	—
302	C₄	—	10	2	no	—
303	C₄	C₂×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
304	C₄	D₅⋊#131	1	1	yes	characteristic
305	C₂	—	10	2	no	—
306	C₂	Q₈×D₅⋊C₈	1	1	yes	characteristic
307	C₂	#10617404	1	1	yes	characteristic
308	C₂	#10617413	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#446	C₄
prime radical	C₂²×D₅	C₂²×C₈
cosocle	C₂×#168	C₂⁴
hypercenter	#446	AGL(1,𝔽₅)
hypocenter	C₅	#33475
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #10604448	C₂	semidirect
3. #10604026	C₂	semidirect
4. #10604026	C₂	semidirect
5. #10604025	C₂	semidirect
58. D₅⋊C₁₆	Q₈	semidirect
289. C₅	#33475	semidirect

Maximal subgroups

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

subgroup	index
#10611689	2
#10604448	2
#10604026	2
#10604025	2
#10603869	2
#10603868	2
#10603827	2
#10603823	2
#33475	5