Subgroups — #12014397

Order 1280.

410 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	C₂×#65190	C₂	1	1	yes	maximal, characteristic
2	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602228	C₂	1	1	yes	maximal, characteristic
4	#10602192	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600691	C₂	1	1	yes	maximal, characteristic
6	#10598305	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598163	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4047	C₂²	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₅×#405	C₂²	1	1	yes	characteristic
12	#65421	C₄	1	1	yes	characteristic
13	#65390	C₂²	1	1	yes	characteristic
14	C₂×#3944	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₂²	1	1	yes	characteristic
16	#65295	C₄	1	1	yes	characteristic
17	#65190	—	2	1	no	—
18	#65190	—	2	1	no	—
19	#65190	—	2	1	no	—
20	#65190	—	2	1	no	—
21	C₂×#3930	C₄	1	2	yes	complemented
22	C₂×D₅×C₁₆	C₄	1	2	yes	complemented
23	C₅×#346	—	2	1	no	—
24	C₄×C₅×C₁₆	—	2	1	no	—
25	#64779	C₄	1	1	yes	characteristic
26	#64753	C₄	1	1	yes	characteristic
27	#64734	C₄	1	4	yes	complemented
28	#64722	—	2	1	no	—
29	C₁₆×Dic₅	—	2	1	no	—
30	#12613	—	5	1	no	maximal, Sylow
31	C₂×#709	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₅×#117	C₂×C₄	1	1	yes	characteristic
35	#4072	C₂×C₄	1	1	yes	characteristic
36	C₂×#681	C₂×C₄	1	1	yes	characteristic
37	C₂×#681	C₂×C₄	1	1	yes	characteristic
38	#4047	—	2	1	no	—
39	#4047	—	2	1	no	—
40	#4047	—	2	1	no	—
41	#4047	—	2	1	no	—
42	C₂×#676	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
44	C₅×M₃₂	—	2	1	no	—
45	C₅×M₃₂	C₂×C₄	1	2	yes	—
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
50	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
51	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
52	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
53	C₅×#97	—	2	1	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	#3956	C₈	1	2	yes	—
56	#3955	C₈	1	2	yes	—
57	#3947	—	2	1	no	—
58	C₈×Dic₅	—	2	1	no	—
59	#3944	—	2	1	no	—
60	#3944	—	2	1	no	—
61	#3944	—	2	1	no	—
62	#3944	C₂×C₄	1	2	yes	—
63	C₂×#672	—	2	1	no	—
64	C₂×#672	C₂×C₄	1	2	yes	—
65	#3938	C₈	1	4	yes	—
66	#3930	—	2	2	no	—
67	#3930	—	2	2	no	—
68	D₅×C₁₆	—	2	2	no	—
69	D₅×C₁₆	—	2	2	no	—
70	C₂×#504	—	5	1	no	—
71	#2115	—	5	1	no	—
72	#2108	—	5	1	no	—
73	#2108	D₅	1	1	yes	characteristic, complemented
74	#1757	—	5	1	no	—
75	#1378	—	5	1	no	—
76	#1378	—	5	1	no	—
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
79	#709	—	2	1	no	—
80	#709	—	2	1	no	—
81	#709	—	2	1	no	—
82	#709	—	2	1	no	—
83	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₅	C₄²	1	2	yes	—
85	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
86	C₅×M₁₆	—	2	1	no	—
87	C₅×M₁₆	C₂×C₈	1	2	yes	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	#41	1	1	yes	characteristic
94	C₂×C₅×C₈	#41	1	1	yes	characteristic
95	C₅×C₄⋊C₄	—	2	1	no	—
96	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
97	C₄²×C₅	—	2	1	no	—
98	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₅	—	2	1	no	—
100	C₄×Dic₅	—	2	1	no	—
101	#681	—	2	1	no	—
102	#681	—	2	1	no	—
103	#681	—	2	1	no	—
104	#681	C₂×C₈	1	2	yes	—
105	C₂×#168	—	2	1	no	—
106	C₂×#168	C₄²	1	2	yes	—
107	#676	—	2	2	no	—
108	#676	—	2	2	no	—
109	C₈×D₅	—	2	2	no	—
110	C₈×D₅	—	2	2	no	—
111	C₅×C₁₆	—	2	2	no	—
112	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
113	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
114	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
115	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
116	#672	—	2	2	no	—
117	#672	—	2	2	no	—
118	C₂×#131	—	5	1	no	—
119	#504	—	10	1	no	—
120	#504	—	10	1	no	—
121	#504	—	10	1	no	—
122	#504	—	10	1	no	—
123	C₂×M₃₂	—	5	1	no	—
124	C₂×M₃₂	—	5	2	no	—
125	C₂×M₃₂	—	5	1	no	—
126	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
127	C₂²×C₁₆	—	5	2	no	—
128	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
129	#413	—	5	1	no	—
130	#410	—	5	1	no	—
131	#405	—	5	1	no	—
132	#405	C₂×D₅	1	1	yes	characteristic
133	#349	—	5	4	no	—
134	#346	—	2	1	no	—
135	#346	—	10	1	no	—
136	C₄×C₁₆	—	2	1	no	—
137	C₄×C₁₆	—	10	1	no	—
138	#344	—	5	1	no	—
139	#344	—	5	1	no	—
140	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
141	C₂²×D₅	C₄×C₈	1	2	yes	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₄×C₅	#131	1	1	yes	characteristic
148	C₂×C₄×C₅	#131	1	1	yes	characteristic
149	C₂²⋊D₅	—	2	2	no	—
150	C₂²⋊D₅	—	2	2	no	—
151	C₂×Dic₅	—	2	2	no	—
152	C₂×Dic₅	C₄×C₈	1	2	yes	—
153	D₂₀	—	2	1	no	—
154	D₂₀	—	2	1	no	—
155	C₄×D₅	—	2	2	no	—
156	C₄×D₅	—	2	2	no	—
157	Dic₁₀	—	2	1	no	—
158	Dic₁₀	—	2	1	no	—
159	C₅×C₈	—	2	2	no	—
160	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₅×C₈	#117	1	1	yes	characteristic
162	C₅×C₈	C₄×D₄	1	1	yes	characteristic
163	C₅×C₈	C₄×D₄	1	1	yes	characteristic
164	#168	—	2	2	no	—
165	#168	—	2	2	no	—
166	C₂×#41	—	5	1	no	—
167	#131	—	10	1	no	—
168	#131	—	10	1	no	—
169	#131	—	10	1	no	—
170	#131	—	10	1	no	—
171	C₂×M₁₆	—	5	1	no	—
172	C₂×M₁₆	—	5	2	no	—
173	C₂×M₁₆	—	5	1	no	—
174	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
175	C₂²×C₈	—	5	2	no	—
176	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
177	#117	—	5	1	no	—
178	#117	C₄×D₅	1	1	yes	characteristic
179	M₃₂	—	2	1	no	—
180	M₃₂	—	5	2	no	—
181	M₃₂	—	10	1	no	—
182	M₃₂	—	10	1	no	—
183	M₃₂	—	10	1	no	—
184	M₃₂	—	10	2	no	—
185	M₃₂	—	10	2	no	—
186	M₃₂	C₄×D₅	1	2	yes	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	5	2	no	—
189	C₂×C₁₆	—	10	1	no	—
190	C₂×C₁₆	—	10	2	no	—
191	C₂×C₁₆	—	10	2	no	—
192	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
193	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
194	C₂×C₁₆	D₂₀	1	1	yes	characteristic
195	C₂×C₁₆	D₂₀	1	1	yes	characteristic
196	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
197	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
198	#101	—	5	2	no	—
199	#100	—	5	2	no	—
200	#99	—	5	4	no	—
201	#97	—	2	1	no	—
202	#97	—	10	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	10	1	no	—
205	C₂²×C₅	—	4	1	no	—
206	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
207	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
208	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
209	C₂×D₅	—	2	2	no	—
210	C₂×D₅	—	4	2	no	—
211	C₄×C₅	—	4	2	no	—
212	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
213	C₄×C₅	#405	1	1	yes	characteristic
214	C₄×C₅	C₈×D₄	1	1	yes	characteristic
215	C₄×C₅	C₈×D₄	1	1	yes	characteristic
216	Dic₅	—	2	2	no	—
217	Dic₅	—	4	2	no	—
218	#41	—	10	1	no	—
219	#41	—	10	1	no	—
220	#41	—	10	1	no	—
221	#41	—	10	1	no	—
222	C₂×Q₈	—	5	1	no	—
223	C₂×D₄	—	5	2	no	—
224	C₂×D₄	—	5	1	no	—
225	C₂²×C₄	—	5	2	no	—
226	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
227	M₁₆	—	2	1	no	—
228	M₁₆	—	5	2	no	—
229	M₁₆	—	10	1	no	—
230	M₁₆	—	10	1	no	—
231	M₁₆	—	10	1	no	—
232	M₁₆	—	10	2	no	—
233	M₁₆	—	10	2	no	—
234	M₁₆	C₈×D₅	1	2	yes	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	5	2	no	—
237	C₂×C₈	—	10	1	no	—
238	C₂×C₈	—	10	2	no	—
239	C₂×C₈	—	10	2	no	—
240	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
242	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
243	C₂×C₈	#709	1	1	yes	characteristic
244	C₂×C₈	#709	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	10	1	no	—
248	C₂²⋊C₄	—	5	2	no	—
249	C₂²⋊C₄	C₈×D₅	1	2	yes	—
250	C₄²	—	2	1	no	—
251	C₄²	—	10	1	no	—
252	C₁₆	—	2	2	no	—
253	C₁₆	—	10	2	no	—
254	C₁₆	—	10	2	no	—
255	C₁₆	#685	1	1	yes	characteristic
256	C₁₆	#685	1	1	yes	characteristic
257	C₁₆	#685	1	1	yes	characteristic
258	C₁₆	#685	1	1	yes	characteristic
259	C₂×C₅	—	2	1	no	—
260	C₂×C₅	—	2	1	no	—
261	C₂×C₅	—	2	1	no	—
262	C₂×C₅	C₈×C₂²⋊C₄	1	1	yes	characteristic
263	D₅	—	4	2	no	—
264	C₂³	—	5	2	no	—
265	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
266	Q₈	—	10	1	no	—
267	Q₈	—	10	1	no	—
268	D₄	—	10	2	no	—
269	D₄	—	10	2	no	—
270	D₄	—	10	1	no	—
271	D₄	—	10	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	5	2	no	—
274	C₂×C₄	—	10	2	no	—
275	C₂×C₄	—	10	2	no	—
276	C₂×C₄	—	10	2	no	—
277	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
278	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
279	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
280	C₂×C₄	#4047	1	1	yes	characteristic
281	C₂×C₄	#4047	1	1	yes	characteristic
282	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	10	2	no	—
285	C₈	—	10	2	no	—
286	C₈	#4018	1	1	yes	characteristic
287	C₈	C₄×D₂₀	1	1	yes	characteristic
288	C₈	C₂×#685	1	1	yes	characteristic
289	C₈	C₄×C₂²⋊D₅	1	1	yes	characteristic
290	C₅	#12613	1	1	yes	characteristic, complemented, Sylow
291	C₂²	—	4	1	no	—
292	C₂²	—	10	2	no	—
293	C₂²	—	20	2	no	—
294	C₂²	C₄×C₈×D₅	1	1	yes	characteristic
295	C₂²	C₈×D₂₀	1	1	yes	characteristic
296	C₂²	C₈×C₂²⋊D₅	1	1	yes	characteristic
297	C₄	—	4	2	no	—
298	C₄	—	10	2	no	—
299	C₄	—	20	2	no	—
300	C₄	C₈×D₂₀	1	1	yes	characteristic
301	C₄	#64978	1	1	yes	characteristic
302	C₄	C₄×#685	1	1	yes	characteristic
303	C₄	C₈×C₂²⋊D₅	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	20	2	no	—
308	C₂	C₈×#685	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#685
commutator	C₂²×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2108	C₂
socle	C₂×C₅	C₈×C₂²⋊C₄
prime radical	C₂×#709	C₂×C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2108	D₅
hypocenter	C₅	#12613
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
2. C₅×#2108	C₂	semidirect
4. #10602192	C₂	semidirect
6. #10598305	C₂	semidirect
7. #10598163	C₂	semidirect
21. C₂×#3930	C₄	semidirect
22. C₂×D₅×C₁₆	C₄	semidirect
27. #64734	C₄	semidirect
73. #2108	D₅	semidirect
290. C₅	#12613	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65190	2
C₅×#2108	2
#10602228	2
#10602192	2
#10600691	2
#10598305	2
#10598163	2
#12613	5