Subgroups — #67×Dic₁₀ #11754318

Order 960.

560 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	#91813	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#67	C₂	1	1	yes	maximal, characteristic
5	#91574	C₂	1	2	yes	maximal, complemented
6	Dic₅×#67	C₂	1	2	yes	maximal
7	#91410	C₂	1	2	yes	maximal, complemented
8	#91389	C₂	1	1	yes	maximal, characteristic, complemented
9	#91369	C₂	1	1	yes	maximal, characteristic, complemented
10	Dic₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
11	#91327	C₂	1	2	yes	maximal, complemented
12	D₄×Dic₁₀	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
15	C₂×C₅×#67	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
18	C₂×C₃×Dic₁₀	—	2	4	no	—
19	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
20	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
21	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
22	C₂×#1204	C₂²	1	2	yes	—
23	C₂×D₃×Dic₅	C₂²	1	2	yes	—
24	D₃×Dic₁₀	—	2	4	no	—
25	#6719	C₂²	1	2	yes	—
26	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
27	#6712	C₂²	1	2	yes	—
28	C₅×#217	C₂²	1	1	yes	characteristic
29	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
30	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
32	Dic₃⋊Dic₅	C₂²	1	2	yes	—
33	Dic₃⋊Dic₅	C₂²	1	2	yes	—
34	#6669	C₂²	1	2	yes	—
35	#6668	C₂²	1	2	yes	—
36	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
37	Dic₃×Dic₅	C₂²	1	2	yes	—
38	Q₈×#67	—	5	1	no	maximal, Hall
39	C₂²×Dic₁₀	—	3	1	no	—
40	C₂²×Dic₁₀	D₃	1	1	yes	characteristic, complemented
41	C₄×C₅×D₄	—	3	1	no	—
42	D₄×Dic₅	—	3	2	no	—
43	#4070	—	3	1	no	—
44	#4070	—	3	1	no	—
45	#4034	—	3	2	no	—
46	#4024	—	3	2	no	—
47	#4024	—	3	2	no	—
48	#4015	—	3	1	no	—
49	C₄×Dic₁₀	—	3	1	no	—
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₅	C₂³	1	2	yes	—
56	C₂×Dic₁₅	C₂³	1	2	yes	—
57	Dic₃₀	—	2	2	no	—
58	C₅×#67	Q₈	1	4	yes	complemented
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
62	C₄×C₅×D₃	—	2	2	no	—
63	C₂×C₃×Dic₅	—	2	4	no	—
64	C₂×C₃×Dic₅	C₂³	1	2	yes	—
65	C₂×C₃×Dic₅	C₂³	1	2	yes	—
66	C₃×Dic₁₀	—	2	2	no	—
67	C₃×Dic₁₀	—	2	4	no	—
68	C₃×Dic₁₀	D₄	1	4	yes	complemented
69	#1204	—	2	4	no	—
70	D₃×Dic₅	—	2	4	no	—
71	C₂²×C₃×Q₈	—	5	1	no	—
72	C₂×D₃×Q₈	—	5	1	no	—
73	#946	—	5	1	no	—
74	#946	—	5	2	no	—
75	Q₈×Dic₃	—	5	1	no	—
76	#944	—	5	2	no	—
77	#944	—	5	1	no	—
78	C₄×#67	—	5	2	no	—
79	C₄×#67	D₅	1	1	yes	characteristic
80	#924	—	5	1	no	—
81	#924	—	5	2	no	—
82	C₂×C₅×D₄	—	3	1	no	—
83	C₂²×C₄×C₅	—	3	1	no	—
84	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
85	C₂²×Dic₅	—	3	2	no	—
86	C₂²×Dic₅	C₂×D₃	1	2	yes	—
87	C₂×Dic₁₀	—	2	4	no	—
88	C₂×Dic₁₀	—	3	1	no	—
89	C₂×Dic₁₀	—	3	2	no	—
90	C₂×Dic₁₀	—	6	4	no	—
91	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic
92	C₂×Dic₁₀	C₂×D₃	1	2	yes	—
93	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	3	1	no	—
95	C₅×C₂²⋊C₄	—	3	1	no	—
96	C₅×C₂²⋊C₄	—	3	1	no	—
97	C₄²×C₅	—	3	1	no	—
98	C₂²⋊Dic₅	—	3	2	no	—
99	C₂²⋊Dic₅	—	3	2	no	—
100	C₄⋊Dic₅	—	3	1	no	—
101	C₄⋊Dic₅	—	3	2	no	—
102	C₄⋊Dic₅	—	3	2	no	—
103	#683	—	3	2	no	—
104	#683	—	3	2	no	—
105	#683	—	3	2	no	—
106	C₄×Dic₅	—	3	2	no	—
107	D₄×Q₈	—	15	1	no	Sylow
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	complemented
111	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
112	C₂×C₅×D₃	—	2	1	no	—
113	C₂×C₅×D₃	—	2	1	no	—
114	C₂×C₅×D₃	C₂×Q₈	1	2	yes	complemented
115	C₃×C₄×C₅	—	2	1	no	—
116	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
117	Dic₁₅	—	2	2	no	—
118	Dic₁₅	—	2	2	no	—
119	C₃×Dic₅	—	2	2	no	—
120	C₃×Dic₅	C₂×D₄	1	4	yes	—
121	C₅×Dic₃	—	2	1	no	—
122	C₅×Dic₃	—	2	1	no	—
123	C₅×Dic₃	C₂×Q₈	1	2	yes	complemented
124	C₂×C₃×Q₈	—	5	1	no	—
125	C₂×C₃×Q₈	—	5	2	no	—
126	C₂×C₃×Q₈	—	5	1	no	—
127	C₂×C₃×Q₈	—	10	4	no	—
128	C₂²×C₃×C₄	—	5	2	no	—
129	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
130	C₂×#67	C₂×D₅	1	1	yes	characteristic
131	D₃×Q₈	—	10	4	no	—
132	C₂×C₄×D₃	—	5	2	no	—
133	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
134	C₂×Dic₆	—	5	1	no	—
135	C₂×Dic₆	—	5	2	no	—
136	#217	—	5	2	no	—
137	#217	C₂×D₅	1	1	yes	characteristic
138	(C₂×D₃)⋊C₄	—	5	2	no	—
139	(C₂×D₃)⋊C₄	C₂×D₅	1	1	yes	characteristic
140	C₄⋊Dic₃	—	5	1	no	—
141	C₄⋊Dic₃	—	5	2	no	—
142	Dic₃⋊C₄	—	5	2	no	—
143	Dic₃⋊C₄	—	5	2	no	—
144	Dic₃⋊C₄	—	5	2	no	—
145	Dic₃⋊C₄	—	5	2	no	—
146	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic
147	C₄×Dic₃	—	5	2	no	—
148	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
149	C₂³×C₅	—	3	1	no	—
150	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₅×D₄	—	3	4	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	3	1	no	—
154	C₂×C₄×C₅	—	3	1	no	—
155	C₂×C₄×C₅	—	3	1	no	—
156	C₂×C₄×C₅	—	6	2	no	—
157	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
159	C₂×Dic₅	—	2	4	no	—
160	C₂×Dic₅	—	3	2	no	—
161	C₂×Dic₅	—	3	2	no	—
162	C₂×Dic₅	—	6	4	no	—
163	C₂×Dic₅	C₂²×D₃	1	2	yes	—
164	C₂×Dic₅	C₂²×D₃	1	2	yes	—
165	Dic₁₀	—	2	2	no	—
166	Dic₁₀	—	2	4	no	—
167	Dic₁₀	—	6	2	no	—
168	Dic₁₀	—	6	4	no	—
169	Dic₁₀	#67	1	4	yes	direct product factor
170	C₂²×Q₈	—	5	1	no	—
171	C₂²×Q₈	—	15	1	no	—
172	C₄⋊Q₈	—	15	1	no	—
173	C₄⋊Q₈	—	15	2	no	—
174	C₂²⋊Q₈	—	15	2	no	—
175	C₂²⋊Q₈	—	15	2	no	—
176	C₂²⋊Q₈	—	15	1	no	—
177	C₂²⋊Q₈	—	15	1	no	—
178	C₄×Q₈	—	15	1	no	—
179	C₄×D₄	—	3	1	no	—
180	C₄×D₄	—	15	2	no	—
181	C₂×C₃×C₅	—	2	2	no	—
182	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
184	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
185	C₅×D₃	—	2	2	no	—
186	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
187	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
188	C₃×Q₈	—	5	4	no	—
189	C₃×Q₈	—	10	4	no	—
190	C₃×Q₈	—	10	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	5	2	no	—
193	C₂×C₃×C₄	—	5	2	no	—
194	C₂×C₃×C₄	—	10	4	no	—
195	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
197	#67	Dic₁₀	1	4	yes	direct product factor
198	C₂×Dic₃	—	5	2	no	—
199	C₂×Dic₃	—	5	2	no	—
200	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
201	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
202	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
203	C₄×D₃	—	2	2	no	—
204	C₄×D₃	—	10	4	no	—
205	Dic₆	—	10	4	no	—
206	Dic₆	—	10	2	no	—
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	3	2	no	—
210	C₂²×C₅	—	6	1	no	—
211	C₂²×C₅	—	6	1	no	—
212	C₂²×C₅	D₃×Q₈	1	2	yes	complemented
213	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	3	2	no	—
216	C₄×C₅	—	6	1	no	—
217	C₄×C₅	—	6	1	no	—
218	C₄×C₅	C₂×#67	1	2	yes	—
219	Dic₅	—	2	2	no	—
220	Dic₅	—	6	2	no	—
221	Dic₅	—	6	2	no	—
222	Dic₅	C₂×#67	1	4	yes	—
223	C₂×Q₈	—	5	1	no	—
224	C₂×Q₈	—	5	2	no	—
225	C₂×Q₈	—	5	1	no	—
226	C₂×Q₈	—	10	4	no	—
227	C₂×Q₈	—	15	2	no	—
228	C₂×Q₈	—	15	1	no	—
229	C₂×Q₈	—	30	4	no	—
230	C₂×D₄	—	3	1	no	—
231	C₂²×C₄	—	3	1	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	—	15	2	no	—
234	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
235	C₄⋊C₄	—	3	1	no	—
236	C₄⋊C₄	—	15	2	no	—
237	C₄⋊C₄	—	15	2	no	—
238	C₄⋊C₄	—	15	2	no	—
239	C₄⋊C₄	—	15	2	no	—
240	C₄⋊C₄	—	15	1	no	—
241	C₄⋊C₄	—	15	2	no	—
242	C₂²⋊C₄	—	3	1	no	—
243	C₂²⋊C₄	—	3	1	no	—
244	C₂²⋊C₄	—	15	2	no	—
245	C₂²⋊C₄	—	15	2	no	—
246	C₄²	—	3	1	no	—
247	C₄²	—	15	2	no	—
248	C₃×C₅	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	C₂×Dic₁₀	1	2	yes	complemented
252	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
253	C₂×D₃	—	2	1	no	—
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	C₂×Dic₁₀	1	2	yes	complemented
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	5	4	no	—
258	C₃×C₄	—	10	2	no	—
259	C₃×C₄	D₄×D₅	1	2	yes	—
260	Dic₃	—	2	1	no	—
261	Dic₃	—	2	1	no	—
262	Dic₃	—	10	2	no	—
263	Dic₃	—	10	2	no	—
264	Dic₃	C₂×Dic₁₀	1	2	yes	complemented
265	C₂×C₅	—	2	2	no	—
266	C₂×C₅	—	6	2	no	—
267	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
268	C₂×C₅	#1007	1	1	yes	characteristic
269	C₂×C₅	C₂²×#67	1	1	yes	characteristic
270	C₂³	—	3	1	no	—
271	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
272	Q₈	—	5	4	no	—
273	Q₈	—	10	4	no	—
274	Q₈	—	10	2	no	—
275	Q₈	—	30	4	no	—
276	Q₈	—	30	2	no	—
277	D₄	—	3	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	3	1	no	—
281	C₂×C₄	—	3	1	no	—
282	C₂×C₄	—	5	2	no	—
283	C₂×C₄	—	5	2	no	—
284	C₂×C₄	—	6	2	no	—
285	C₂×C₄	—	10	4	no	—
286	C₂×C₄	—	15	2	no	—
287	C₂×C₄	—	15	2	no	—
288	C₂×C₄	—	30	4	no	—
289	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
290	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
293	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
294	C₂×C₃	#4150	1	1	yes	characteristic
295	D₃	—	2	2	no	—
296	C₅	Q₈×#67	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	3	2	no	—
300	C₂²	—	6	1	no	—
301	C₂²	—	6	1	no	—
302	C₂²	D₃×Dic₁₀	1	2	yes	complemented
303	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	3	2	no	—
306	C₄	—	5	4	no	—
307	C₄	—	6	1	no	—
308	C₄	—	6	1	no	—
309	C₄	—	10	2	no	—
310	C₄	—	30	2	no	—
311	C₄	—	30	2	no	—
312	C₄	D₅×#67	1	2	yes	—
313	C₃	D₄×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	2	2	no	—
315	C₂	—	6	2	no	—
316	C₂	C₂×D₃×Dic₁₀	1	1	yes	characteristic
317	C₂	#92004	1	1	yes	characteristic
318	C₂	C₂×D₅×#67	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₂²×C₃×C₄×C₅	C₂²
prime radical	C₂×C₅×#67	C₂²
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	D₄×Q₈
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Dic₁₀	C₂	semidirect
2. C₂×D₃×Dic₁₀	C₂	semidirect
3. #91813	C₂	semidirect
5. #91574	C₂	semidirect
7. #91410	C₂	semidirect
8. #91389	C₂	semidirect
9. #91369	C₂	semidirect
10. Dic₃×Dic₁₀	C₂	semidirect
11. #91327	C₂	semidirect
40. C₂²×Dic₁₀	D₃	semidirect
58. C₅×#67	Q₈	semidirect
68. C₃×Dic₁₀	D₄	semidirect
110. C₂²×C₃×C₅	C₂×Q₈	semidirect
114. C₂×C₅×D₃	C₂×Q₈	semidirect
123. C₅×Dic₃	C₂×Q₈	semidirect
169. Dic₁₀	#67	direct (N × H)
197. #67	Dic₁₀	direct (N × H)
212. C₂²×C₅	D₃×Q₈	semidirect
248. C₃×C₅	D₄×Q₈	semidirect
251. C₂²×C₃	C₂×Dic₁₀	semidirect
255. C₂×D₃	C₂×Dic₁₀	semidirect
264. Dic₃	C₂×Dic₁₀	semidirect
296. C₅	Q₈×#67	semidirect
302. C₂²	D₃×Dic₁₀	semidirect
313. C₃	D₄×Dic₁₀	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Dic₁₀	2
C₂×D₃×Dic₁₀	2
#91813	2
C₄×C₅×#67	2
#91574	2
Dic₅×#67	2
#91410	2
#91389	2
#91369	2
Dic₃×Dic₁₀	2
#91327	2
D₄×Dic₁₀	3
Q₈×#67	5

Subgroups — #67×Dic10 #11754318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #67×Dic₁₀ #11754318