Subgroups — Dic₃×#711 #11754868

Order 960.

620 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#711	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#91559	C₂	1	2	yes	maximal, complemented
5	Dic₃×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
6	#91528	C₂	1	2	yes	maximal, complemented
7	C₂×Dic₃×Dic₅	C₂	1	2	yes	maximal, complemented
8	C₄×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#91358	C₂	1	1	yes	maximal, characteristic, complemented
10	#91331	C₂	1	1	yes	maximal, characteristic, complemented
11	Dic₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#711	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅	C₂²	1	2	yes	—
15	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
18	C₃×#711	C₄	1	8	yes	complemented
19	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
20	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
21	C₂×D₅×Dic₃	C₂²	1	1	yes	characteristic
22	C₂×D₅×Dic₃	C₂²	1	1	yes	characteristic
23	#6719	C₂²	1	2	yes	—
24	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
26	C₅×#217	C₂²	1	2	yes	—
27	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
29	#6668	C₂²	1	2	yes	—
30	#6668	C₂²	1	2	yes	—
31	#6665	C₂²	1	2	yes	—
32	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
33	Dic₃×Dic₅	C₂²	1	2	yes	—
34	Dic₃×Dic₅	C₂²	1	2	yes	—
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
37	Dic₃×#41	—	5	1	no	maximal, Hall
38	C₂×#711	D₃	1	1	yes	characteristic
39	C₄×C₅×D₄	—	3	1	no	—
40	D₄×Dic₅	—	3	1	no	—
41	C₄×C₂²⋊D₅	—	3	2	no	—
42	C₂×C₄×Dic₅	—	3	2	no	—
43	#4038	—	3	1	no	—
44	#4033	—	3	1	no	—
45	#4027	—	3	2	no	—
46	#4023	—	3	2	no	—
47	C₄²×D₅	—	3	1	no	—
48	C₄×Dic₁₀	—	3	1	no	—
49	C₂³×C₃×C₅	C₂³	1	2	yes	—
50	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
51	C₃×C₅×D₄	C₂×C₄	1	4	yes	complemented
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	—	2	4	no	—
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	—	2	4	no	—
58	C₂×C₅×Dic₃	C₂³	1	2	yes	—
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₃×C₂²⋊D₅	C₂×C₄	1	8	yes	complemented
62	C₂×C₃×Dic₅	C₂×C₄	1	8	yes	complemented
63	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
64	C₂×C₃×Dic₅	C₂³	1	2	yes	—
65	C₃×C₄×D₅	C₂×C₄	1	4	yes	complemented
66	C₃×Dic₁₀	C₂×C₄	1	4	yes	complemented
67	D₅×Dic₃	—	2	2	no	—
68	D₅×Dic₃	—	2	2	no	—
69	C₂×C₃×#41	—	5	1	no	—
70	Q₈×Dic₃	—	5	1	no	—
71	Q₈×Dic₃	—	5	1	no	—
72	D₄×Dic₃	—	5	1	no	—
73	D₄×Dic₃	—	5	2	no	—
74	D₄×Dic₃	—	5	2	no	—
75	D₄×Dic₃	D₅	1	1	yes	characteristic, complemented
76	#926	—	5	2	no	—
77	#926	—	5	1	no	—
78	C₂×C₄×Dic₃	—	5	2	no	—
79	C₂×C₄×Dic₃	—	5	1	no	—
80	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
81	C₂²×C₄×C₅	—	3	2	no	—
82	C₂×C₂²⋊D₅	C₂×D₃	1	2	yes	—
83	C₂²×Dic₅	—	3	2	no	—
84	C₂²×Dic₅	C₂×D₃	1	2	yes	—
85	#711	Dic₃	1	8	yes	direct product factor
86	C₂×C₄×D₅	—	3	1	no	—
87	C₂×C₄×D₅	—	3	1	no	—
88	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
89	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	3	1	no	—
91	C₅×C₂²⋊C₄	—	3	2	no	—
92	C₄²×C₅	—	3	1	no	—
93	C₂²⋊Dic₅	—	3	2	no	—
94	#685	—	3	2	no	—
95	C₄⋊Dic₅	—	3	1	no	—
96	#683	—	3	2	no	—
97	#683	—	3	2	no	—
98	C₄×Dic₅	—	3	1	no	—
99	C₄×Dic₅	—	3	2	no	—
100	C₄×Dic₅	—	3	1	no	—
101	C₄×Dic₅	—	3	1	no	—
102	C₄×Dic₅	—	3	2	no	—
103	C₄×Dic₅	—	3	2	no	—
104	C₄×#41	—	15	1	no	Sylow
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	—	2	2	no	—
107	C₂²×C₃×C₅	C₂²×C₄	1	4	yes	—
108	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
109	C₂×C₃×D₅	—	2	1	no	—
110	C₂×C₃×D₅	—	2	1	no	—
111	C₂×C₃×D₅	C₂²×C₄	1	2	yes	—
112	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
113	Dic₁₅	—	2	1	no	—
114	Dic₁₅	—	2	2	no	—
115	Dic₁₅	#41	1	2	yes	complemented
116	C₃×Dic₅	C₂²×C₄	1	4	yes	—
117	C₃×Dic₅	C₂²×C₄	1	2	yes	—
118	C₅×Dic₃	—	2	1	no	—
119	C₅×Dic₃	—	2	2	no	—
120	C₅×Dic₃	#41	1	2	yes	complemented
121	C₃×#41	—	5	8	no	—
122	C₂×C₃×Q₈	—	5	1	no	—
123	C₂×C₃×D₄	—	5	2	no	—
124	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	5	2	no	—
126	C₂²×C₃×C₄	—	5	1	no	—
127	C₂²×Dic₃	—	5	2	no	—
128	C₂²×Dic₃	—	5	1	no	—
129	C₂²×Dic₃	—	5	1	no	—
130	C₂²×Dic₃	C₂×D₅	1	2	yes	—
131	#217	—	5	2	no	—
132	#217	—	5	2	no	—
133	#217	C₂×D₅	1	2	yes	—
134	C₄⋊Dic₃	—	5	1	no	—
135	C₄⋊Dic₃	—	5	2	no	—
136	C₄⋊Dic₃	—	5	2	no	—
137	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
138	C₄×Dic₃	—	5	1	no	—
139	C₄×Dic₃	—	5	1	no	—
140	C₄×Dic₃	—	5	2	no	—
141	C₄×Dic₃	—	5	1	no	—
142	C₄×Dic₃	—	5	2	no	—
143	C₄×Dic₃	—	5	2	no	—
144	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
145	C₂³×C₅	C₂²×D₃	1	2	yes	—
146	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
147	C₅×D₄	C₂×Dic₃	1	4	yes	complemented
148	C₂×C₄×C₅	—	3	1	no	—
149	C₂×C₄×C₅	—	3	2	no	—
150	C₂×C₄×C₅	—	3	1	no	—
151	C₂×C₄×C₅	—	6	4	no	—
152	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
153	C₂²⋊D₅	C₂×Dic₃	1	8	yes	complemented
154	C₂×Dic₅	—	3	1	no	—
155	C₂×Dic₅	—	3	1	no	—
156	C₂×Dic₅	—	3	2	no	—
157	C₂×Dic₅	—	6	4	no	—
158	C₂×Dic₅	C₂×Dic₃	1	8	yes	complemented
159	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
160	C₂×Dic₅	C₂²×D₃	1	2	yes	—
161	C₄×D₅	—	6	2	no	—
162	C₄×D₅	—	6	2	no	—
163	C₄×D₅	C₂×Dic₃	1	4	yes	complemented
164	Dic₁₀	C₂×Dic₃	1	4	yes	complemented
165	C₂×#41	—	5	1	no	—
166	C₄×Q₈	—	15	1	no	—
167	C₄×Q₈	—	15	1	no	—
168	C₄×D₄	—	3	1	no	—
169	C₄×D₄	—	15	2	no	—
170	C₄×D₄	—	15	2	no	—
171	C₄×D₄	—	15	1	no	—
172	#117	—	15	2	no	—
173	#117	—	15	1	no	—
174	C₂×C₄²	—	15	1	no	—
175	C₂×C₄²	—	15	2	no	—
176	C₂×C₃×C₅	—	2	4	no	—
177	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
179	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
180	C₃×D₅	—	2	2	no	—
181	C₂³×C₃	—	5	1	no	—
182	C₂³×C₃	C₂²×D₅	1	2	yes	—
183	C₃×Q₈	—	5	4	no	—
184	C₃×D₄	—	5	8	no	—
185	C₃×D₄	C₄×D₅	1	4	yes	complemented
186	C₂×C₃×C₄	—	5	8	no	—
187	C₂×C₃×C₄	—	5	1	no	—
188	C₂×C₃×C₄	—	5	2	no	—
189	C₂×C₃×C₄	—	5	4	no	—
190	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
191	C₂×Dic₃	—	2	4	no	—
192	C₂×Dic₃	—	5	1	no	—
193	C₂×Dic₃	—	5	1	no	—
194	C₂×Dic₃	—	5	2	no	—
195	C₂×Dic₃	—	10	2	no	—
196	C₂×Dic₃	—	10	4	no	—
197	C₂×Dic₃	—	10	2	no	—
198	C₂×Dic₃	C₂²×D₅	1	2	yes	—
199	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
200	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	C₂²×Dic₃	1	4	yes	—
204	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₂×D₅	—	2	1	no	—
206	C₂×D₅	—	2	1	no	—
207	C₂×D₅	C₂²×Dic₃	1	2	yes	—
208	C₄×C₅	—	3	2	no	—
209	C₄×C₅	—	6	1	no	—
210	C₄×C₅	—	6	2	no	—
211	C₄×C₅	C₂²×Dic₃	1	2	yes	—
212	Dic₅	—	3	2	no	—
213	Dic₅	—	6	1	no	—
214	Dic₅	—	6	2	no	—
215	Dic₅	C₂²×Dic₃	1	4	yes	—
216	Dic₅	C₂²×Dic₃	1	2	yes	—
217	#41	—	5	8	no	—
218	C₂×Q₈	—	5	1	no	—
219	C₂×D₄	—	5	2	no	—
220	C₂×D₄	D₃×D₅	1	1	yes	characteristic
221	C₂²×C₄	—	3	2	no	—
222	C₂²×C₄	—	5	2	no	—
223	C₂²×C₄	—	5	1	no	—
224	C₂²×C₄	—	15	1	no	—
225	C₂²×C₄	—	15	1	no	—
226	C₂²×C₄	—	15	2	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₄⋊C₄	—	15	2	no	—
229	C₄⋊C₄	—	15	2	no	—
230	C₄⋊C₄	—	15	1	no	—
231	C₂²⋊C₄	—	3	2	no	—
232	C₂²⋊C₄	—	15	2	no	—
233	C₂²⋊C₄	—	15	2	no	—
234	C₄²	—	3	1	no	—
235	C₄²	—	15	1	no	—
236	C₄²	—	15	1	no	—
237	C₄²	—	15	2	no	—
238	C₄²	—	15	2	no	—
239	C₄²	—	15	2	no	—
240	C₄²	—	15	1	no	—
241	C₃×C₅	C₄×#41	1	1	yes	characteristic, complemented, Hall
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	5	2	no	—
245	C₂²×C₃	—	10	1	no	—
246	C₂²×C₃	—	10	1	no	—
247	C₂²×C₃	C₂×C₄×D₅	1	4	yes	—
248	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
249	C₃×C₄	—	5	2	no	—
250	C₃×C₄	—	5	4	no	—
251	C₃×C₄	C₂×C₄×D₅	1	2	yes	—
252	Dic₃	—	2	2	no	—
253	Dic₃	—	2	1	no	—
254	Dic₃	—	5	2	no	—
255	Dic₃	—	10	1	no	—
256	Dic₃	—	10	2	no	—
257	Dic₃	#711	1	2	yes	direct product factor
258	C₂×C₅	—	2	4	no	—
259	C₂×C₅	D₃×#41	1	1	yes	characteristic
260	C₂×C₅	D₃×#41	1	1	yes	characteristic
261	C₂×C₅	C₂³×Dic₃	1	1	yes	characteristic
262	D₅	—	2	2	no	—
263	C₂³	—	5	1	no	—
264	C₂³	C₂×D₃×D₅	1	2	yes	—
265	Q₈	—	5	4	no	—
266	D₄	—	5	8	no	—
267	D₄	D₅×Dic₃	1	4	yes	complemented
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	3	2	no	—
271	C₂×C₄	—	5	8	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	5	2	no	—
274	C₂×C₄	—	5	4	no	—
275	C₂×C₄	—	6	4	no	—
276	C₂×C₄	—	15	1	no	—
277	C₂×C₄	—	15	1	no	—
278	C₂×C₄	—	15	2	no	—
279	C₂×C₄	—	30	2	no	—
280	C₂×C₄	—	30	2	no	—
281	C₂×C₄	—	30	4	no	—
282	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
283	C₂×C₃	—	2	4	no	—
284	C₂×C₃	—	10	2	no	—
285	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
286	C₂×C₃	C₂×#711	1	1	yes	characteristic
287	C₂×C₃	D₅×#41	1	1	yes	characteristic
288	C₅	Dic₃×#41	1	1	yes	characteristic, complemented, Sylow
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	5	2	no	—
292	C₂²	—	10	1	no	—
293	C₂²	—	10	1	no	—
294	C₂²	C₂×D₅×Dic₃	1	4	yes	—
295	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
296	C₄	—	3	2	no	—
297	C₄	—	5	2	no	—
298	C₄	—	5	4	no	—
299	C₄	—	6	2	no	—
300	C₄	—	6	1	no	—
301	C₄	—	15	2	no	—
302	C₄	—	30	1	no	—
303	C₄	—	30	2	no	—
304	C₄	C₂×D₅×Dic₃	1	2	yes	—
305	C₃	C₄×#711	1	1	yes	characteristic, complemented, Sylow
306	C₂	—	2	4	no	—
307	C₂	—	10	2	no	—
308	C₂	D₃×#711	1	1	yes	characteristic
309	C₂	#92024	1	1	yes	characteristic
310	C₂	C₂²×D₅×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₄
Fitting	C₂×C₃×C₅×D₄	C₂²
socle, cosocle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×#711	C₂
hypercenter	C₂×D₄	D₃×D₅
hypocenter	C₃×C₅	C₄×#41
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. D₄×Dic₁₅	C₂	semidirect
3. C₅×D₄×Dic₃	C₂	semidirect
4. #91559	C₂	semidirect
5. Dic₃×C₂²⋊D₅	C₂	semidirect
6. #91528	C₂	semidirect
7. C₂×Dic₃×Dic₅	C₂	semidirect
8. C₄×D₅×Dic₃	C₂	semidirect
9. #91358	C₂	semidirect
10. #91331	C₂	semidirect
11. Dic₃×Dic₁₀	C₂	semidirect
18. C₃×#711	C₄	semidirect
51. C₃×C₅×D₄	C₂×C₄	semidirect
61. C₃×C₂²⋊D₅	C₂×C₄	semidirect
62. C₂×C₃×Dic₅	C₂×C₄	semidirect
65. C₃×C₄×D₅	C₂×C₄	semidirect
66. C₃×Dic₁₀	C₂×C₄	semidirect
75. D₄×Dic₃	D₅	semidirect
85. #711	Dic₃	direct (N × H)
115. Dic₁₅	#41	semidirect
120. C₅×Dic₃	#41	semidirect
147. C₅×D₄	C₂×Dic₃	semidirect
153. C₂²⋊D₅	C₂×Dic₃	semidirect
158. C₂×Dic₅	C₂×Dic₃	semidirect
163. C₄×D₅	C₂×Dic₃	semidirect
164. Dic₁₀	C₂×Dic₃	semidirect
185. C₃×D₄	C₄×D₅	semidirect
241. C₃×C₅	C₄×#41	semidirect
257. Dic₃	#711	direct (N × H)
267. D₄	D₅×Dic₃	semidirect
288. C₅	Dic₃×#41	semidirect
305. C₃	C₄×#711	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#711	2
D₄×Dic₁₅	2
C₅×D₄×Dic₃	2
#91559	2
Dic₃×C₂²⋊D₅	2
#91528	2
C₂×Dic₃×Dic₅	2
C₄×D₅×Dic₃	2
#91358	2
#91331	2
Dic₃×Dic₁₀	2
C₄×#711	3
Dic₃×#41	5

Subgroups — Dic3×#711 #11754868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×#711 #11754868