Subgroups — #12254731

Order 1280.

918 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3147	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#65482	C₂	1	4	yes	maximal, complemented
3	C₂×#64984	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602114	C₂	1	4	yes	maximal, complemented
5	#10602106	C₂	1	2	yes	maximal, complemented
6	#10602083	C₂	1	2	yes	maximal, complemented
7	C₄×#3944	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	—	2	1	no	—
9	C₂×C₅×#135	C₂²	1	2	yes	—
10	C₅×#532	—	2	1	no	—
11	C₂²×D₄₀	—	2	1	no	—
12	C₂²×D₄₀	C₂²	1	2	yes	—
13	C₂×#4020	—	2	1	no	—
14	C₅×#495	—	2	1	no	—
15	C₅×#495	C₂²	1	2	yes	complemented
16	C₅×#494	C₂²	1	1	yes	characteristic, complemented
17	C₅×#493	C₂²	1	1	yes	characteristic, complemented
18	C₅×#492	—	2	1	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	#65482	—	2	16	no	—
21	C₂×#3960	C₂²	1	8	yes	complemented
22	C₂×#3958	C₂²	1	8	yes	complemented
23	C₂×#3944	C₂²	1	2	yes	—
24	#64984	—	2	1	no	—
25	#64984	—	2	1	no	—
26	#64984	C₂²	1	2	yes	—
27	#64984	C₂²	1	2	yes	—
28	#64681	C₂²	1	2	yes	complemented
29	C₄×#672	C₂²	1	2	yes	complemented
30	#33717	—	5	1	no	maximal, Sylow
31	C₂×C₅×#41	—	2	2	no	—
32	C₂²×D₂₀	—	2	2	no	—
33	C₂²×D₂₀	—	4	2	no	—
34	C₅×#135	—	2	8	no	—
35	C₅×#135	—	4	4	no	—
36	C₂×C₅×Q₁₆	—	2	1	no	—
37	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
38	C₂×C₅×SD₁₆	—	2	2	no	—
39	C₂×C₅×SD₁₆	—	2	2	no	—
40	C₂×C₅×D₈	—	2	1	no	—
41	C₂×C₅×D₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×C₄⋊Q₈	—	2	1	no	—
45	C₅×#127	—	2	1	no	—
46	C₅×#124	—	2	2	no	—
47	C₅×#121	—	4	2	no	—
48	C₄×C₅×D₄	—	4	2	no	—
49	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
50	C₂×D₄₀	—	2	8	no	—
51	C₂×D₄₀	—	2	2	no	—
52	C₂×D₄₀	—	2	2	no	—
53	C₂×D₄₀	—	2	2	no	—
54	C₂×D₄₀	—	4	4	no	—
55	C₂×D₄₀	C₂³	1	4	yes	—
56	#4020	—	2	2	no	—
57	#4020	—	2	2	no	—
58	#4020	—	4	2	no	—
59	C₄×C₅×C₈	D₄	1	1	yes	characteristic
60	C₄×C₅×C₈	D₄	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
63	#3960	—	2	16	no	—
64	#3958	—	2	16	no	—
65	#3944	D₄	1	8	yes	complemented
66	C₂×#672	C₂³	1	4	yes	—
67	C₂×#509	—	5	4	no	—
68	#3147	D₅	1	1	yes	characteristic, complemented
69	C₂×#493	—	5	1	no	—
70	#2250	—	5	4	no	—
71	#2248	—	5	2	no	—
72	#2246	—	5	2	no	—
73	C₄×M₃₂	—	5	1	no	—
74	C₂³×D₅	—	4	2	no	—
75	C₅×#41	—	4	4	no	—
76	C₅×#41	—	4	4	no	—
77	C₂×C₅×Q₈	—	2	2	no	—
78	C₂×C₅×D₄	—	2	2	no	—
79	C₂×C₅×D₄	—	4	2	no	—
80	C₂×C₅×D₄	—	4	2	no	—
81	C₂²×C₄×C₅	—	4	2	no	—
82	C₂²×C₄×C₅	D₈	1	2	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×D₂₀	—	2	4	no	—
85	C₂×D₂₀	—	4	4	no	—
86	C₂×D₂₀	—	4	4	no	—
87	C₂×D₂₀	—	4	2	no	—
88	C₂×D₂₀	—	4	4	no	—
89	C₂×D₂₀	—	4	4	no	—
90	C₂×D₂₀	—	8	8	no	—
91	C₅×Q₁₆	—	2	4	no	—
92	C₅×Q₁₆	—	4	2	no	—
93	C₅×SD₁₆	—	4	4	no	—
94	C₅×SD₁₆	—	4	4	no	—
95	C₅×D₈	—	2	4	no	—
96	C₅×D₈	—	4	2	no	—
97	C₂×C₅×C₈	D₈	1	4	yes	—
98	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
99	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
103	C₅×C₄⋊C₄	—	4	2	no	—
104	C₅×C₂²⋊C₄	—	4	4	no	—
105	C₄²×C₅	D₈	1	2	yes	—
106	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
108	D₄₀	—	2	8	no	—
109	D₄₀	—	4	4	no	—
110	D₄₀	—	4	4	no	—
111	D₄₀	—	4	4	no	—
112	#672	C₂×D₄	1	8	yes	complemented
113	C₂×#135	—	2	1	no	—
114	C₂×#135	C₂×D₅	1	2	yes	—
115	C₂²×D₈	—	5	2	no	—
116	C₂²×D₈	—	10	1	no	—
117	#532	—	2	1	no	—
118	C₂×#127	—	10	1	no	—
119	#509	—	10	16	no	—
120	C₂×SD₃₂	—	5	8	no	—
121	C₂×D₁₆	—	5	8	no	—
122	C₂×M₃₂	—	5	2	no	—
123	#495	—	2	1	no	—
124	#495	C₂×D₅	1	2	yes	complemented
125	#494	C₂×D₅	1	1	yes	characteristic, complemented
126	#493	—	5	2	no	—
127	#493	—	5	2	no	—
128	#493	—	10	1	no	—
129	#493	—	10	1	no	—
130	#493	C₂×D₅	1	1	yes	characteristic, complemented
131	#492	—	2	1	no	—
132	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
133	#346	—	5	2	no	—
134	C₄×C₁₆	—	5	2	no	—
135	C₂³×C₅	—	4	2	no	—
136	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
137	C₂²×D₅	—	4	4	no	—
138	C₂²×D₅	—	8	8	no	—
139	C₂²×D₅	—	8	4	no	—
140	C₅×Q₈	—	4	2	no	—
141	C₅×Q₈	—	4	2	no	—
142	C₅×D₄	—	4	2	no	—
143	C₅×D₄	—	4	2	no	—
144	C₅×D₄	—	8	4	no	—
145	C₅×D₄	—	8	4	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	8	4	no	—
149	C₂×C₄×C₅	#127	1	2	yes	—
150	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
151	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
152	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
155	D₂₀	—	4	4	no	—
156	D₂₀	—	4	4	no	—
157	D₂₀	—	8	8	no	—
158	D₂₀	—	8	4	no	—
159	D₂₀	—	8	8	no	—
160	C₅×C₈	#127	1	2	yes	—
161	C₅×C₈	C₂×D₈	1	2	yes	—
162	C₅×C₈	C₂×D₈	1	2	yes	—
163	C₅×C₈	C₂²×D₄	1	2	yes	—
164	C₂×#41	—	2	2	no	—
165	C₂²×D₄	—	10	2	no	—
166	C₂²×D₄	—	20	2	no	—
167	#135	—	2	8	no	—
168	#135	—	4	4	no	—
169	C₂×Q₁₆	—	2	1	no	—
170	C₂×Q₁₆	C₂²×D₅	1	2	yes	—
171	C₂×SD₁₆	—	2	2	no	—
172	C₂×SD₁₆	—	2	2	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	—	5	4	no	—
175	C₂×D₈	—	10	8	no	—
176	C₂×D₈	—	10	2	no	—
177	C₂×D₈	—	10	2	no	—
178	C₂×D₈	—	10	2	no	—
179	C₂×D₈	—	20	4	no	—
180	C₂×D₈	C₂²×D₅	1	2	yes	—
181	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	1	no	—
184	#127	—	2	1	no	—
185	#127	—	10	2	no	—
186	#127	—	10	2	no	—
187	#127	—	20	2	no	—
188	#124	—	2	2	no	—
189	#121	—	4	2	no	—
190	C₄×D₄	—	4	2	no	—
191	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
192	SD₃₂	—	10	16	no	—
193	D₁₆	—	10	16	no	—
194	M₃₂	—	5	8	no	—
195	C₂×C₁₆	—	5	4	no	—
196	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
197	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	—	8	2	no	—
202	C₂²×C₅	—	8	4	no	—
203	C₂²×C₅	#493	1	2	yes	—
204	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
205	C₂×D₅	—	8	4	no	—
206	C₂×D₅	—	8	8	no	—
207	C₂×D₅	—	16	8	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	8	2	no	—
210	C₄×C₅	#493	1	2	yes	—
211	C₄×C₅	#509	1	4	yes	complemented
212	C₄×C₅	C₂×#127	1	1	yes	characteristic
213	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
214	C₂⁴	—	20	2	no	—
215	#41	—	4	4	no	—
216	#41	—	4	4	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	10	4	no	—
222	C₂×D₄	—	20	4	no	—
223	C₂×D₄	—	20	4	no	—
224	C₂×D₄	—	20	4	no	—
225	C₂×D₄	—	20	2	no	—
226	C₂×D₄	—	20	4	no	—
227	C₂×D₄	—	40	8	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	#686	1	2	yes	—
230	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	Q₁₆	—	2	4	no	—
232	Q₁₆	—	4	2	no	—
233	SD₁₆	—	4	4	no	—
234	SD₁₆	—	4	4	no	—
235	D₈	—	2	4	no	—
236	D₈	—	4	2	no	—
237	D₈	—	10	8	no	—
238	D₈	—	20	4	no	—
239	D₈	—	20	4	no	—
240	D₈	—	20	4	no	—
241	C₂×C₈	#686	1	4	yes	—
242	C₂×C₈	D₄×D₅	1	4	yes	—
243	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
244	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
246	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
247	C₄⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₄²	#686	1	2	yes	—
250	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
251	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
252	C₁₆	—	5	8	no	—
253	C₂×C₅	—	2	2	no	—
254	C₂×C₅	—	16	2	no	—
255	C₂×C₅	C₂×#493	1	1	yes	characteristic
256	C₂×C₅	C₂×#509	1	2	yes	—
257	D₅	—	16	4	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	20	4	no	—
260	C₂³	—	40	8	no	—
261	C₂³	—	40	4	no	—
262	C₂³	C₂×#686	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	D₄	—	4	2	no	—
266	D₄	—	4	2	no	—
267	D₄	—	8	4	no	—
268	D₄	—	8	4	no	—
269	D₄	—	20	4	no	—
270	D₄	—	20	4	no	—
271	D₄	—	40	8	no	—
272	D₄	—	40	4	no	—
273	D₄	—	40	8	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	C₂×#686	1	2	yes	—
278	C₂×C₄	C₂×#686	1	2	yes	—
279	C₂×C₄	C₂×#686	1	1	yes	characteristic
280	C₂×C₄	C₂×#686	1	2	yes	—
281	C₂×C₄	#4086	1	2	yes	—
282	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
283	C₈	C₂×#686	1	2	yes	—
284	C₈	C₂×#686	1	2	yes	—
285	C₈	#4086	1	2	yes	—
286	C₈	C₂×D₄×D₅	1	2	yes	—
287	C₅	#33717	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	2	2	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	40	8	no	—
292	C₂²	—	40	4	no	—
293	C₂²	—	80	8	no	—
294	C₂²	#65362	1	2	yes	—
295	C₂²	C₂²×#686	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	8	2	no	—
298	C₄	#65362	1	2	yes	—
299	C₄	#65482	1	4	yes	complemented
300	C₄	C₂²×#686	1	1	yes	characteristic
301	C₄	C₂×#4086	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	16	2	no	—
304	C₂	—	80	4	no	—
305	C₂	C₂×#65362	1	1	yes	characteristic
306	C₂	C₂×#65482	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#686
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#3147	C₂
socle	C₂²×C₅	C₂²×D₈
hypercenter	#3147	D₅
hypocenter	C₅	#33717
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₅×#3147	C₂	semidirect
2. C₂×#65482	C₂	semidirect
3. C₂×#64984	C₂	semidirect
4. #10602114	C₂	semidirect
5. #10602106	C₂	semidirect
6. #10602083	C₂	semidirect
7. C₄×#3944	C₂	semidirect
15. C₅×#495	C₂²	semidirect
16. C₅×#494	C₂²	semidirect
17. C₅×#493	C₂²	semidirect
21. C₂×#3960	C₂²	semidirect
22. C₂×#3958	C₂²	semidirect
28. #64681	C₂²	semidirect
29. C₄×#672	C₂²	semidirect
65. #3944	D₄	semidirect
68. #3147	D₅	semidirect
112. #672	C₂×D₄	semidirect
124. #495	C₂×D₅	semidirect
125. #494	C₂×D₅	semidirect
130. #493	C₂×D₅	semidirect
211. C₄×C₅	#509	semidirect
287. C₅	#33717	semidirect
299. C₄	#65482	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3147	2
C₂×#65482	2
C₂×#64984	2
#10602114	2
#10602106	2
#10602083	2
C₄×#3944	2
#33717	5