Subgroups — #13043635

Order 1280.

506 conjugacy classes of subgroups.

The subgroup lattice has 326 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	1	yes	maximal, characteristic, complemented
3	Q₈×#674	C₂	1	1	yes	maximal, characteristic, complemented
4	#10604026	C₂	1	2	yes	maximal, complemented
5	#10604025	C₂	1	1	yes	maximal, characteristic, complemented
6	#10604024	C₂	1	1	yes	maximal, characteristic, complemented
7	#10604024	C₂	1	2	yes	maximal, complemented
8	#10603869	C₂	1	2	yes	maximal
9	#10603868	C₂	1	2	yes	maximal
10	#10603827	C₂	1	1	yes	maximal, characteristic
11	C₄×D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic
12	#65912	C₄	1	1	yes	characteristic
13	C₂×#3990	C₂²	1	2	yes	—
14	C₂×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#446	C₄	1	1	yes	characteristic
17	Q₈×#168	C₂²	1	1	yes	characteristic
18	#65128	C₄	1	2	yes	—
19	#65126	C₄	1	1	yes	characteristic
20	#65119	C₂²	1	1	yes	characteristic
21	#65119	C₂²	1	2	yes	—
22	#64997	C₂²	1	2	yes	—
23	#64990	C₄	1	2	yes	—
24	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
25	#64968	C₄	1	1	yes	characteristic
26	#64888	C₂²	1	2	yes	—
27	#64888	C₂²	1	2	yes	—
28	#64888	C₂²	1	2	yes	—
29	#64886	C₂²	1	2	yes	—
30	#64885	C₂²	1	1	yes	characteristic
31	#64885	C₂²	1	1	yes	characteristic
32	#64885	C₂²	1	2	yes	—
33	#64859	C₂²	1	2	yes	—
34	C₄⋊#674	C₂²	1	1	yes	characteristic
35	C₄⋊#674	C₂²	1	1	yes	characteristic
36	C₄⋊#674	C₂²	1	2	yes	—
37	C₄⋊#674	C₂²	1	2	yes	—
38	C₄×#674	C₂²	1	1	yes	characteristic
39	C₄×#674	C₂²	1	1	yes	characteristic
40	C₄×#674	C₂²	1	2	yes	—
41	#33474	—	5	1	no	maximal, Sylow
42	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	#4092	C₈	1	1	yes	characteristic
44	#4090	C₈	1	1	yes	characteristic
45	C₂×#676	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
48	#4043	C₈	1	1	yes	characteristic
49	#4042	C₈	1	2	yes	—
50	#4035	C₈	1	2	yes	—
51	#4034	C₈	1	1	yes	characteristic
52	#4018	C₂×C₄	1	2	yes	—
53	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₀	C₂×C₄	1	2	yes	—
56	C₂×#674	C₂³	1	1	yes	characteristic
57	C₂×#674	C₂³	1	1	yes	characteristic
58	C₂×#674	C₂³	1	2	yes	—
59	C₂×#674	C₂³	1	2	yes	—
60	C₂×#674	C₂³	1	1	yes	characteristic
61	C₂×#674	C₂³	1	1	yes	characteristic
62	#3990	—	2	4	no	—
63	D₅⋊C₁₆	—	2	2	no	—
64	D₅⋊C₁₆	Q₈	1	4	yes	complemented
65	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
66	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
67	C₅×#97	C₂×C₄	1	2	yes	—
68	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
69	#3947	C₂×C₄	1	2	yes	—
70	#3946	C₂×C₄	1	2	yes	—
71	#3946	C₂×C₄	1	2	yes	—
72	#3946	C₂×C₄	1	2	yes	—
73	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₅	C₂³	1	1	yes	characteristic
75	C₈×Dic₅	C₂³	1	2	yes	—
76	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
77	C₄⋊#168	C₂×C₄	1	2	yes	—
78	C₄×#168	C₂³	1	1	yes	characteristic
79	C₄×#168	C₂³	1	2	yes	—
80	#2960	—	5	1	no	—
81	#2183	—	5	1	no	—
82	#2183	—	5	1	no	—
83	#2183	—	5	2	no	—
84	Q₈×C₁₆	—	5	1	no	—
85	Q₈×C₁₆	—	5	1	no	—
86	Q₈×C₁₆	—	5	2	no	—
87	#2151	—	5	2	no	—
88	#2150	—	5	1	no	—
89	#2108	—	5	2	no	—
90	C₂×C₄×C₁₆	—	5	1	no	—
91	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
93	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
94	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₀	C₂×C₈	1	2	yes	—
96	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
101	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
102	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂²×C₄	1	2	yes	—
104	#685	C₂×C₈	1	2	yes	—
105	#685	C₂×C₈	1	2	yes	—
106	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
107	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
108	#683	C₂×C₈	1	2	yes	—
109	#683	C₂×C₈	1	2	yes	—
110	#683	C₂×C₈	1	2	yes	—
111	#683	C₂×C₈	1	2	yes	—
112	#683	C₂×C₈	1	2	yes	—
113	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
114	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
115	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₅	C₂²×C₄	1	2	yes	—
117	C₂×#168	C₂²×C₄	1	1	yes	characteristic
118	C₂×#168	C₂²×C₄	1	2	yes	—
119	C₂×#168	C₂⁴	1	1	yes	characteristic
120	#676	—	2	4	no	—
121	C₈×D₅	—	2	2	no	—
122	C₈×D₅	C₂×Q₈	1	2	yes	—
123	C₈×D₅	#41	1	2	yes	—
124	#674	—	2	1	no	—
125	#674	—	2	1	no	—
126	#674	—	2	2	no	—
127	#674	—	2	2	no	—
128	#674	C₂×Q₈	1	2	yes	—
129	#674	C₂×Q₈	1	2	yes	—
130	#533	—	5	1	no	—
131	C₂×M₃₂	—	5	2	no	—
132	C₂²×C₁₆	—	5	1	no	—
133	C₂²×C₁₆	—	5	1	no	—
134	#446	—	5	2	no	—
135	#446	—	5	1	no	—
136	#446	AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₈×Q₈	—	5	1	no	—
138	C₈×Q₈	—	5	2	no	—
139	C₈×Q₈	—	5	1	no	—
140	#424	—	5	2	no	—
141	#423	—	5	1	no	—
142	#405	—	5	2	no	—
143	C₂×C₄×C₈	—	5	1	no	—
144	C₄⋊C₁₆	—	5	1	no	—
145	C₄⋊C₁₆	—	5	1	no	—
146	C₄⋊C₁₆	—	5	2	no	—
147	C₄⋊C₁₆	—	5	2	no	—
148	C₄⋊C₁₆	—	5	2	no	—
149	C₄⋊C₁₆	—	5	2	no	—
150	C₄⋊C₁₆	—	5	2	no	—
151	#346	—	5	2	no	—
152	#346	—	5	2	no	—
153	C₄×C₁₆	—	5	1	no	—
154	C₄×C₁₆	—	5	1	no	—
155	C₄×C₁₆	—	5	1	no	—
156	C₄×C₁₆	—	5	1	no	—
157	C₄×C₁₆	—	5	2	no	—
158	C₄×C₁₆	—	5	2	no	—
159	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
160	C₅×Q₈	—	2	2	no	—
161	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
163	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
164	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
166	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
167	C₂×Dic₅	C₂²×C₈	1	2	yes	—
168	C₂×Dic₅	C₂²×C₈	1	2	yes	—
169	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
170	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
171	C₄×D₅	—	2	2	no	—
172	C₄×D₅	—	2	2	no	—
173	C₄×D₅	C₄×Q₈	1	2	yes	—
174	C₄×D₅	#131	1	2	yes	—
175	Dic₁₀	—	2	2	no	—
176	Dic₁₀	—	2	4	no	—
177	C₅×C₈	—	2	1	no	—
178	C₅×C₈	—	2	2	no	—
179	C₅×C₈	C₄×Q₈	1	2	yes	—
180	#168	—	2	1	no	—
181	#168	—	2	2	no	—
182	#168	C₂²×Q₈	1	1	yes	characteristic
183	#168	C₂×#41	1	1	yes	characteristic
184	C₂×M₁₆	—	5	2	no	—
185	C₂²×C₈	—	5	1	no	—
186	C₂²×C₈	—	5	1	no	—
187	C₄⋊Q₈	—	5	1	no	—
188	C₄⋊Q₈	—	5	1	no	—
189	#125	—	5	2	no	—
190	C₂²⋊Q₈	—	5	1	no	—
191	C₂²⋊Q₈	—	5	2	no	—
192	C₂²⋊Q₈	—	5	1	no	—
193	C₄×Q₈	—	5	2	no	—
194	C₄×Q₈	—	5	1	no	—
195	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
196	#117	—	5	2	no	—
197	C₂×C₄²	—	5	1	no	—
198	M₃₂	—	10	4	no	—
199	C₂×C₁₆	—	5	1	no	—
200	C₂×C₁₆	—	5	1	no	—
201	C₂×C₁₆	—	5	4	no	—
202	C₂×C₁₆	—	5	1	no	—
203	C₂×C₁₆	—	5	1	no	—
204	C₂×C₁₆	—	5	2	no	—
205	C₂×C₁₆	—	5	2	no	—
206	C₂×C₁₆	—	10	2	no	—
207	C₄⋊C₈	—	5	2	no	—
208	C₄⋊C₈	—	5	2	no	—
209	C₄⋊C₈	—	5	2	no	—
210	C₄⋊C₈	—	5	1	no	—
211	C₄⋊C₈	—	5	2	no	—
212	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
214	#97	—	5	2	no	—
215	#97	C₂×AGL(1,𝔽₅)	1	2	yes	—
216	C₄×C₈	—	5	1	no	—
217	C₄×C₈	—	5	2	no	—
218	C₄×C₈	—	5	1	no	—
219	C₄×C₈	—	5	1	no	—
220	C₄×C₈	—	5	2	no	—
221	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
223	C₂×D₅	—	2	1	no	—
224	C₂×D₅	—	2	1	no	—
225	C₂×D₅	C₈×Q₈	1	2	yes	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	#504	1	2	yes	—
230	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
231	C₄×C₅	C₂×#131	1	1	yes	characteristic
232	Dic₅	—	2	2	no	—
233	Dic₅	—	2	2	no	—
234	Dic₅	—	2	1	no	—
235	Dic₅	C₈×Q₈	1	2	yes	—
236	Dic₅	#504	1	2	yes	—
237	C₂×Q₈	—	5	2	no	—
238	C₂×Q₈	—	5	1	no	—
239	C₂×Q₈	D₅⋊C₈	1	1	yes	characteristic
240	C₂²×C₄	—	5	1	no	—
241	C₂²×C₄	—	5	1	no	—
242	C₂²×C₄	—	5	1	no	—
243	M₁₆	—	10	4	no	—
244	C₂×C₈	—	5	1	no	—
245	C₂×C₈	—	5	2	no	—
246	C₂×C₈	—	5	2	no	—
247	C₂×C₈	—	5	1	no	—
248	C₂×C₈	—	5	2	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
251	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
252	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
253	C₄⋊C₄	—	5	2	no	—
254	C₄⋊C₄	—	5	2	no	—
255	C₄⋊C₄	—	5	2	no	—
256	C₄⋊C₄	—	5	2	no	—
257	C₄⋊C₄	—	5	2	no	—
258	C₄⋊C₄	—	5	2	no	—
259	C₄⋊C₄	—	5	1	no	—
260	C₄⋊C₄	D₅⋊C₈	1	2	yes	—
261	C₄⋊C₄	D₅⋊C₈	1	1	yes	characteristic
262	C₂²⋊C₄	—	5	2	no	—
263	C₂²⋊C₄	—	5	2	no	—
264	C₄²	—	5	1	no	—
265	C₄²	—	5	1	no	—
266	C₄²	—	5	1	no	—
267	C₄²	—	5	2	no	—
268	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
270	C₁₆	—	5	2	no	—
271	C₁₆	—	5	2	no	—
272	C₁₆	—	10	1	no	—
273	C₁₆	—	10	1	no	—
274	C₁₆	—	10	2	no	—
275	C₁₆	—	10	2	no	—
276	C₂×C₅	C₂×C₈×Q₈	1	1	yes	characteristic
277	C₂×C₅	C₂×#504	1	1	yes	characteristic
278	C₂×C₅	C₂×#504	1	1	yes	characteristic
279	D₅	—	2	2	no	—
280	C₂³	—	5	1	no	—
281	Q₈	—	2	2	no	—
282	Q₈	—	10	4	no	—
283	Q₈	—	10	2	no	—
284	C₂×C₄	—	5	1	no	—
285	C₂×C₄	—	5	1	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	5	2	no	—
288	C₂×C₄	—	5	1	no	—
289	C₂×C₄	—	5	2	no	—
290	C₂×C₄	—	5	2	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	—	10	2	no	—
293	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
294	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
295	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
296	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
297	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	2	no	—
300	C₈	—	5	1	no	—
301	C₈	—	5	1	no	—
302	C₈	—	10	1	no	—
303	C₈	—	10	2	no	—
304	C₈	Q₈×AGL(1,𝔽₅)	1	2	yes	—
305	C₅	#33474	1	1	yes	characteristic, complemented, Sylow
306	C₂²	—	5	2	no	—
307	C₂²	—	10	1	no	—
308	C₂²	—	10	1	no	—
309	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	5	2	no	—
314	C₄	—	5	2	no	—
315	C₄	—	10	1	no	—
316	C₄	—	10	2	no	—
317	C₄	—	10	2	no	—
318	C₄	#65725	1	2	yes	—
319	C₄	C₂×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
320	C₄	D₅⋊#131	1	1	yes	characteristic
321	C₂	—	10	2	no	—
322	C₂	Q₈×D₅⋊C₈	1	1	yes	characteristic
323	C₂	C₂×#65725	1	1	yes	characteristic
324	C₂	D₅⋊#504	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	#33474
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. Q₈×#674	C₂	semidirect
4. #10604026	C₂	semidirect
5. #10604025	C₂	semidirect
6. #10604024	C₂	semidirect
7. #10604024	C₂	semidirect
64. D₅⋊C₁₆	Q₈	semidirect
305. C₅	#33474	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
Q₈×#674	2
#10604026	2
#10604025	2
#10604024	2
#10603869	2
#10603868	2
#10603827	2
C₄×D₅⋊C₁₆	2
#33474	5