Subgroups — #12069816

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic
2	#10615006	C₂	1	2	yes	maximal, complemented
3	#10602595	C₂	1	2	yes	maximal, complemented
4	#10601674	C₂	1	4	yes	maximal, complemented
5	D₈×#168	C₂	1	4	yes	maximal, complemented
6	#10600524	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#532	C₂²	1	2	yes	—
9	C₂×#4094	—	2	2	no	—
10	C₅×#500	C₄	1	1	yes	characteristic
11	C₅×#493	C₄	1	1	yes	characteristic
12	C₅×#486	C₄	1	2	yes	—
13	C₅×#466	C₄	1	4	yes	—
14	C₄×C₅×D₈	C₂²	1	4	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65523	—	2	2	no	—
17	#65301	—	2	2	no	—
18	#65301	C₂²	1	4	yes	—
19	D₄×#168	—	2	2	no	—
20	D₄×#168	C₂²	1	4	yes	—
21	#65213	C₂²	1	2	yes	—
22	C₂×C₄×#168	C₂²	1	1	yes	characteristic
23	C₂×C₄×#168	C₂²	1	1	yes	characteristic
24	D₄⋊#168	C₂²	1	8	yes	—
25	#64678	C₂²	1	4	yes	—
26	C₈×#168	C₂²	1	2	yes	—
27	C₈×#168	C₂²	1	2	yes	—
28	#17956	—	5	1	no	maximal, Sylow
29	C₂×C₅×#41	—	2	2	no	—
30	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₅×#127	C₂×C₄	1	2	yes	—
34	C₅×#124	—	2	2	no	—
35	C₅×#121	—	2	2	no	—
36	C₅×#121	C₂×C₄	1	4	yes	—
37	C₄×C₅×D₄	—	2	2	no	—
38	C₄×C₅×D₄	C₂³	1	4	yes	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	#4094	—	4	8	no	—
41	C₂×#681	—	2	2	no	—
42	C₂×#681	—	2	4	no	—
43	C₂²×#168	—	2	1	no	—
44	C₂²×#168	—	2	4	no	—
45	C₂²×#168	D₄	1	1	yes	characteristic
46	C₂²×#168	D₄	1	1	yes	characteristic
47	C₅×#107	C₂×C₄	1	4	yes	—
48	C₅×#102	C₂×C₄	1	8	yes	—
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×C₈	C₂³	1	2	yes	—
52	C₂²⋊#168	—	2	4	no	—
53	C₂²⋊#168	—	2	4	no	—
54	C₄⋊#168	—	2	2	no	—
55	C₄⋊#168	C₂³	1	4	yes	—
56	C₄×#168	—	2	1	no	—
57	C₄×#168	D₄	1	1	yes	characteristic
58	C₄×#168	D₄	1	1	yes	characteristic
59	C₄×#168	C₂³	1	2	yes	—
60	C₄×#168	C₂³	1	1	yes	characteristic
61	C₄×#168	C₂³	1	1	yes	characteristic
62	#3169	D₅	1	1	yes	characteristic
63	#2931	—	5	2	no	—
64	#1589	—	5	4	no	—
65	C₈×D₈	—	5	4	no	—
66	#1567	—	5	1	no	—
67	#1492	—	5	2	no	—
68	C₂×C₈²	—	5	1	no	—
69	C₅×#41	—	4	4	no	—
70	C₅×#41	—	4	4	no	—
71	C₂×C₅×Q₈	—	2	2	no	—
72	C₂×C₅×D₄	—	2	4	no	—
73	C₂×C₅×D₄	—	2	2	no	—
74	C₂×C₅×D₄	—	2	2	no	—
75	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₅	—	2	4	no	—
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	#41	1	1	yes	characteristic
80	C₅×D₈	—	2	8	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₅×C₄⋊C₄	—	2	2	no	—
85	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
86	C₅×C₂²⋊C₄	—	2	4	no	—
87	C₅×C₂²⋊C₄	—	2	4	no	—
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅	#41	1	1	yes	characteristic
91	C₄²×C₅	C₂⁴	1	1	yes	characteristic
92	#681	—	4	4	no	—
93	#681	—	4	8	no	—
94	C₂×#168	—	2	4	no	—
95	C₂×#168	—	2	2	no	—
96	C₂×#168	—	2	2	no	—
97	C₂×#168	—	2	4	no	—
98	C₂×#168	—	4	8	no	—
99	C₂×#168	D₈	1	4	yes	complemented
100	C₂×#168	C₂×D₄	1	1	yes	characteristic
101	C₂×#168	C₂×D₄	1	1	yes	characteristic
102	C₂×#168	C₂×D₄	1	1	yes	characteristic
103	C₂×#168	C₂×D₄	1	1	yes	characteristic
104	C₂×#131	—	10	2	no	—
105	#532	C₂×D₅	1	2	yes	—
106	#500	Dic₅	1	1	yes	characteristic
107	#493	Dic₅	1	1	yes	characteristic
108	#486	Dic₅	1	2	yes	—
109	#466	Dic₅	1	4	yes	—
110	C₄×D₈	C₂×D₅	1	4	yes	—
111	#436	—	5	4	no	—
112	#436	—	10	2	no	—
113	C₈×D₄	—	5	4	no	—
114	C₈×D₄	—	10	2	no	—
115	#423	—	5	2	no	—
116	#408	—	10	2	no	—
117	C₂×C₄×C₈	—	5	1	no	—
118	C₂×C₄×C₈	—	5	1	no	—
119	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
120	#335	—	5	4	no	—
121	D₄⋊C₈	—	5	8	no	—
122	C₈²	—	5	2	no	—
123	C₈²	—	5	2	no	—
124	C₂³×C₅	—	2	4	no	—
125	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
126	C₅×Q₈	—	4	4	no	—
127	C₅×D₄	—	2	8	no	—
128	C₅×D₄	—	4	8	no	—
129	C₅×D₄	—	4	4	no	—
130	C₅×D₄	—	4	4	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	—	4	4	no	—
136	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₅	#135	1	2	yes	—
141	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	#131	1	4	yes	complemented
146	#168	—	2	4	no	—
147	#168	—	2	1	no	—
148	#168	—	2	1	no	—
149	#168	—	4	4	no	—
150	#168	C₂×D₈	1	2	yes	—
151	#168	C₂×D₈	1	2	yes	—
152	C₂×#41	—	2	2	no	—
153	C₂×D₈	C₂×Dic₅	1	4	yes	—
154	#131	—	20	8	no	—
155	C₂×M₁₆	—	10	4	no	—
156	C₂×M₁₆	—	10	2	no	—
157	C₂²×C₈	—	5	1	no	—
158	C₂²×C₈	—	5	1	no	—
159	C₂²×C₈	—	10	4	no	—
160	C₂²×C₈	—	10	1	no	—
161	C₂²×C₈	C₂×Dic₅	1	2	yes	—
162	C₄⋊Q₈	C₂×Dic₅	1	2	yes	—
163	#127	C₂×Dic₅	1	2	yes	—
164	#124	—	2	2	no	—
165	#121	—	2	2	no	—
166	#121	C₂×Dic₅	1	4	yes	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	C₂²×D₅	1	4	yes	—
169	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
170	#107	C₂×Dic₅	1	4	yes	—
171	C₄⋊C₈	—	5	4	no	—
172	C₄⋊C₈	—	10	2	no	—
173	#102	C₂×Dic₅	1	8	yes	—
174	C₂²⋊C₈	—	10	4	no	—
175	C₂²⋊C₈	—	10	4	no	—
176	C₄×C₈	—	5	1	no	—
177	C₄×C₈	—	5	1	no	—
178	C₄×C₈	—	5	1	no	—
179	C₄×C₈	—	5	1	no	—
180	C₄×C₈	—	5	2	no	—
181	C₄×C₈	—	10	1	no	—
182	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
183	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
184	C₄×C₈	C₂²×D₅	1	2	yes	—
185	C₂²×C₅	—	2	1	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	4	4	no	—
188	C₂²×C₅	—	4	4	no	—
189	C₂²×C₅	—	4	4	no	—
190	C₂²×C₅	C₄×D₈	1	2	yes	—
191	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	4	4	no	—
195	C₄×C₅	C₄×D₈	1	2	yes	—
196	C₄×C₅	#443	1	2	yes	—
197	C₄×C₅	C₂×#131	1	2	yes	—
198	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
199	C₄×C₅	C₂×#135	1	1	yes	characteristic
200	#41	—	4	4	no	—
201	#41	—	4	4	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	C₂²×Dic₅	1	4	yes	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
209	C₂²×C₄	#711	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
211	D₈	—	2	8	no	—
212	M₁₆	—	20	8	no	—
213	M₁₆	—	20	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	5	1	no	—
216	C₂×C₈	—	5	1	no	—
217	C₂×C₈	—	5	4	no	—
218	C₂×C₈	—	5	1	no	—
219	C₂×C₈	—	5	1	no	—
220	C₂×C₈	—	10	4	no	—
221	C₂×C₈	—	10	2	no	—
222	C₂×C₈	—	10	2	no	—
223	C₂×C₈	—	10	4	no	—
224	C₂×C₈	—	20	8	no	—
225	C₂×C₈	C₂²×Dic₅	1	2	yes	—
226	C₂×C₈	C₂²×Dic₅	1	2	yes	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	C₂²×Dic₅	1	4	yes	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₄²	D₄×D₅	1	1	yes	characteristic
232	C₄²	#711	1	1	yes	characteristic
233	C₄²	C₂²×Dic₅	1	1	yes	characteristic
234	C₄²	C₂³×D₅	1	1	yes	characteristic
235	C₂×C₅	—	2	2	no	—
236	C₂×C₅	—	8	4	no	—
237	C₂×C₅	#2931	1	1	yes	characteristic
238	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
239	C₂×C₅	C₂×#443	1	1	yes	characteristic
240	C₂³	—	2	4	no	—
241	C₂³	D₄×Dic₅	1	1	yes	characteristic
242	Q₈	—	4	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	8	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	D₅×D₈	1	2	yes	—
253	C₂×C₄	#4058	1	2	yes	—
254	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
258	C₂×C₄	C₂×#711	1	1	yes	characteristic
259	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	5	2	no	—
262	C₈	—	5	2	no	—
263	C₈	—	10	4	no	—
264	C₈	—	10	1	no	—
265	C₈	—	10	1	no	—
266	C₈	—	20	4	no	—
267	C₈	#4094	1	4	yes	complemented
268	C₅	#17956	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	4	4	no	—
274	C₂²	D₈×Dic₅	1	2	yes	—
275	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	4	no	—
279	C₄	D₈×Dic₅	1	2	yes	—
280	C₄	#65485	1	2	yes	—
281	C₄	C₂×D₅×D₈	1	1	yes	characteristic
282	C₄	C₂×#4058	1	1	yes	characteristic
283	C₄	C₂×#4094	1	2	yes	—
284	C₂	—	2	2	no	—
285	C₂	—	8	4	no	—
286	C₂	#10615006	1	1	yes	characteristic
287	C₂	C₂×D₈×Dic₅	1	1	yes	characteristic
288	C₂	C₂×#65485	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting, prime radical	C₅×#3169	C₂
socle	C₂²×C₅	C₂×C₄×D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#3169	D₅
hypocenter	C₅	#17956
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
2. #10615006	C₂	semidirect
3. #10602595	C₂	semidirect
4. #10601674	C₂	semidirect
5. D₈×#168	C₂	semidirect
6. #10600524	C₂	semidirect
7. C₂×C₈×#168	C₂	semidirect
99. C₂×#168	D₈	semidirect
145. C₅×C₈	#131	semidirect
267. C₈	#4094	semidirect
268. C₅	#17956	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3169	2
#10615006	2
#10602595	2
#10601674	2
D₈×#168	2
#10600524	2
C₂×C₈×#168	2
#17956	5