Subgroups — #12129270

Order 1280.

664 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#430	C₂	1	1	yes	maximal, characteristic
2	#10615481	C₂	1	1	yes	maximal, characteristic
3	C₂×#64776	C₂	1	2	yes	maximal
4	C₂×#64777	C₂	1	1	yes	maximal, characteristic
5	#10602775	C₂	1	2	yes	maximal
6	#10601521	C₂	1	8	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×#4072	C₂²	1	2	yes	—
10	C₂×#4072	C₂²	1	1	yes	characteristic
11	C₂²×#681	C₂²	1	1	yes	characteristic
12	C₅×#430	—	2	2	no	—
13	C₅×#430	C₂²	1	4	yes	—
14	C₂×C₅×#108	—	2	2	no	—
15	#65408	—	2	2	no	—
16	#65407	C₂²	1	4	yes	—
17	#65391	—	2	2	no	—
18	#64777	C₂²	1	4	yes	—
19	#64776	C₂²	1	8	yes	—
20	#64775	C₂²	1	8	yes	—
21	#23751	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	D₄	1	2	yes	—
31	C₂×C₅×M₁₆	C₂³	1	4	yes	—
32	C₂×C₅×M₁₆	C₂³	1	2	yes	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂×C₂²⋊Dic₅	—	2	1	no	—
36	C₂×C₂²⋊Dic₅	—	2	2	no	—
37	#4072	—	2	1	no	—
38	#4072	—	2	1	no	—
39	#4072	—	2	2	no	—
40	#4072	—	2	2	no	—
41	#4072	—	2	2	no	—
42	#4072	Q₈	1	2	yes	—
43	#4072	Q₈	1	2	yes	—
44	#4072	C₂³	1	4	yes	—
45	C₂×C₄⋊Dic₅	—	2	1	no	—
46	C₂×C₄⋊Dic₅	—	2	2	no	—
47	C₂×C₄×Dic₅	—	2	2	no	—
48	C₂×C₄×Dic₅	C₂³	1	2	yes	—
49	C₂×#681	—	2	1	no	—
50	C₂×#681	—	2	2	no	—
51	C₂×#681	—	2	2	no	—
52	C₂×#681	C₂³	1	2	yes	—
53	C₂²×#168	—	2	1	no	—
54	C₅×#108	—	2	8	no	—
55	#3946	—	2	8	no	—
56	C₂×#430	D₅	1	1	yes	characteristic
57	#2905	—	5	1	no	—
58	#2102	—	5	8	no	—
59	C₂×#344	—	5	2	no	—
60	#1733	—	5	2	no	—
61	C₂×H(3,ℤ/4)	—	5	1	no	—
62	C₂⁴×C₅	#41	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	#41	1	2	yes	—
68	C₂²×C₄×C₅	#41	1	2	yes	—
69	C₂²×C₄×C₅	#41	1	2	yes	—
70	C₂²×C₄×C₅	#41	1	2	yes	—
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₅	—	2	2	no	—
74	C₂²×Dic₅	—	4	2	no	—
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	4	2	no	—
80	C₅×M₁₆	—	4	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
87	C₂²⋊Dic₅	—	2	2	no	—
88	C₂²⋊Dic₅	—	2	2	no	—
89	C₂²⋊Dic₅	—	2	4	no	—
90	C₂²⋊Dic₅	—	4	2	no	—
91	C₄⋊Dic₅	—	2	2	no	—
92	C₄⋊Dic₅	—	2	2	no	—
93	C₄⋊Dic₅	—	2	4	no	—
94	C₄⋊Dic₅	—	4	2	no	—
95	C₄×Dic₅	—	2	2	no	—
96	C₄×Dic₅	—	2	4	no	—
97	C₄×Dic₅	—	4	2	no	—
98	C₄×Dic₅	C₂×Q₈	1	4	yes	—
99	#681	—	2	4	no	—
100	#681	—	4	1	no	—
101	#681	—	4	1	no	—
102	C₂×#168	—	2	2	no	—
103	C₂×#168	—	4	2	no	—
104	C₂²×M₁₆	—	5	1	no	—
105	C₂²×M₁₆	C₂×D₅	1	2	yes	—
106	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
107	C₂×#117	—	5	1	no	—
108	C₂×#117	—	5	2	no	—
109	#430	—	2	2	no	—
110	#430	C₂×D₅	1	4	yes	—
111	C₂×#108	—	2	2	no	—
112	#424	—	10	2	no	—
113	#424	—	10	2	no	—
114	#423	—	5	4	no	—
115	#344	—	5	8	no	—
116	#339	—	5	8	no	—
117	H(3,ℤ/4)	—	5	4	no	—
118	C₂³×C₅	—	8	1	no	—
119	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂³×C₅	#126	1	2	yes	—
122	C₂³×C₅	#126	1	2	yes	—
123	C₂³×C₅	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅	#123	1	2	yes	—
127	C₂×C₄×C₅	#123	1	1	yes	characteristic
128	C₂×C₄×C₅	#123	1	1	yes	characteristic
129	C₂×C₄×C₅	#124	1	1	yes	characteristic
130	C₂×C₄×C₅	#124	1	2	yes	—
131	C₂×C₄×C₅	#124	1	1	yes	characteristic
132	C₂×C₄×C₅	#125	1	2	yes	—
133	C₂×C₄×C₅	#125	1	2	yes	—
134	C₂×C₄×C₅	#125	1	2	yes	—
135	C₂×C₄×C₅	#125	1	2	yes	—
136	C₂×C₄×C₅	#126	1	2	yes	—
137	C₂×C₄×C₅	#126	1	2	yes	—
138	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×#41	1	2	yes	—
141	C₂×C₄×C₅	C₂×#41	1	2	yes	—
142	C₂×Dic₅	—	2	4	no	—
143	C₂×Dic₅	—	4	4	no	—
144	C₂×Dic₅	—	8	2	no	—
145	C₂×Dic₅	—	8	4	no	—
146	C₅×C₈	—	2	4	no	—
147	C₅×C₈	—	4	4	no	—
148	#168	—	4	2	no	—
149	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	5	2	no	—
158	C₂×M₁₆	—	10	1	no	—
159	C₂×M₁₆	—	10	2	no	—
160	C₂×M₁₆	—	10	2	no	—
161	C₂×M₁₆	C₂²⋊D₅	1	2	yes	—
162	C₂×M₁₆	C₂²×D₅	1	4	yes	—
163	C₂×M₁₆	C₂²×D₅	1	2	yes	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	10	1	no	—
166	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
167	#117	—	5	2	no	—
168	#117	—	5	2	no	—
169	#117	—	5	4	no	—
170	#117	—	10	1	no	—
171	#117	—	10	1	no	—
172	#117	—	10	2	no	—
173	#117	—	10	2	no	—
174	#117	—	10	2	no	—
175	C₂×C₄⋊C₄	—	10	1	no	—
176	C₂×C₄⋊C₄	—	10	2	no	—
177	C₂×C₂²⋊C₄	—	10	1	no	—
178	C₂×C₂²⋊C₄	—	10	2	no	—
179	C₂×C₄²	—	5	2	no	—
180	C₂×C₄²	—	10	2	no	—
181	#108	—	2	8	no	—
182	C₄⋊C₈	—	10	8	no	—
183	C₂²×C₅	—	4	2	no	—
184	C₂²×C₅	—	4	1	no	—
185	C₂²×C₅	—	4	1	no	—
186	C₂²×C₅	—	4	1	no	—
187	C₂²×C₅	—	4	2	no	—
188	C₂²×C₅	#400	1	2	yes	—
189	C₂²×C₅	#400	1	1	yes	characteristic
190	C₂²×C₅	#400	1	1	yes	characteristic
191	C₂²×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
192	C₂²×C₅	C₂×#126	1	2	yes	—
193	C₄×C₅	#400	1	2	yes	—
194	C₄×C₅	#400	1	1	yes	characteristic
195	C₄×C₅	#400	1	1	yes	characteristic
196	C₄×C₅	C₂×#123	1	1	yes	characteristic
197	C₄×C₅	C₂×#124	1	1	yes	characteristic
198	C₄×C₅	C₂×#125	1	2	yes	—
199	Dic₅	—	4	4	no	—
200	Dic₅	—	8	4	no	—
201	C₂⁴	#713	1	1	yes	characteristic
202	C₂²×C₄	—	10	2	no	—
203	C₂²×C₄	—	20	2	no	—
204	C₂²×C₄	#709	1	2	yes	—
205	C₂²×C₄	#709	1	2	yes	—
206	C₂²×C₄	#711	1	2	yes	—
207	C₂²×C₄	#711	1	2	yes	—
208	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
209	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
210	C₂²×C₄	#713	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	10	4	no	—
221	M₁₆	—	20	1	no	—
222	M₁₆	—	20	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	10	2	no	—
229	C₂×C₈	—	20	2	no	—
230	C₂×C₈	C₂×C₂²⋊D₅	1	4	yes	—
231	C₄⋊C₄	—	10	2	no	—
232	C₄⋊C₄	—	10	2	no	—
233	C₄⋊C₄	—	10	4	no	—
234	C₄⋊C₄	—	20	2	no	—
235	C₂²⋊C₄	—	10	2	no	—
236	C₂²⋊C₄	—	10	2	no	—
237	C₂²⋊C₄	—	10	4	no	—
238	C₂²⋊C₄	—	20	2	no	—
239	C₄²	—	5	4	no	—
240	C₄²	—	10	2	no	—
241	C₄²	—	10	4	no	—
242	C₄²	—	20	2	no	—
243	C₂×C₅	—	2	1	no	—
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	—	2	1	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	2	1	no	—
248	C₂×C₅	C₂×#400	1	1	yes	characteristic
249	C₂³	—	8	1	no	—
250	C₂³	#4044	1	2	yes	—
251	C₂³	#4044	1	2	yes	—
252	C₂³	#4092	1	1	yes	characteristic
253	C₂³	#4092	1	1	yes	characteristic
254	C₂³	C₂×#713	1	1	yes	characteristic
255	C₂×C₄	—	10	4	no	—
256	C₂×C₄	—	20	4	no	—
257	C₂×C₄	—	40	2	no	—
258	C₂×C₄	—	40	4	no	—
259	C₂×C₄	#4035	1	2	yes	—
260	C₂×C₄	#4035	1	2	yes	—
261	C₂×C₄	#4035	1	2	yes	—
262	C₂×C₄	#4035	1	2	yes	—
263	C₂×C₄	#4044	1	2	yes	—
264	C₂×C₄	#4044	1	2	yes	—
265	C₂×C₄	#4075	1	1	yes	characteristic
266	C₂×C₄	#4075	1	1	yes	characteristic
267	C₂×C₄	#4075	1	2	yes	—
268	C₂×C₄	#4082	1	1	yes	characteristic
269	C₂×C₄	#4082	1	2	yes	—
270	C₂×C₄	#4082	1	1	yes	characteristic
271	C₂×C₄	#4092	1	1	yes	characteristic
272	C₂×C₄	#4092	1	1	yes	characteristic
273	C₂×C₄	C₂×#709	1	2	yes	—
274	C₂×C₄	C₂×#711	1	2	yes	—
275	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
277	C₈	—	2	4	no	—
278	C₈	—	4	4	no	—
279	C₈	—	20	2	no	—
280	C₅	#23751	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	#65271	1	1	yes	characteristic
287	C₂²	#65271	1	1	yes	characteristic
288	C₂²	#65271	1	2	yes	—
289	C₂²	C₂×#4044	1	2	yes	—
290	C₂²	C₂×#4092	1	1	yes	characteristic
291	C₄	—	20	4	no	—
292	C₄	—	40	4	no	—
293	C₄	#65271	1	1	yes	characteristic
294	C₄	#65271	1	2	yes	—
295	C₄	#65271	1	1	yes	characteristic
296	C₄	C₂×#4035	1	2	yes	—
297	C₄	C₂×#4075	1	1	yes	characteristic
298	C₄	C₂×#4082	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	1	no	—
304	C₂	C₂×#65271	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#4082
commutator, cosocle	C₂²×C₄×C₅	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₅×#430	C₂
socle	C₂×C₅	C₂×#400
prime radical	C₂⁴×C₅	#41
hypercenter	C₂×#430	D₅
hypocenter	C₅	#23751
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
6. #10601521	C₂	semidirect
280. C₅	#23751	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#430	2
#10615481	2
C₂×#64776	2
C₂×#64777	2
#10602775	2
#10601521	2
#23751	5