Subgroups — #11748874

Order 960.

468 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	C₂²×D₃×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#683	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#6668	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₃)⋊Dic₅	C₂	1	1	yes	maximal, characteristic
6	#91115	C₂	1	1	yes	maximal, characteristic, complemented
7	#90994	C₂	1	1	yes	maximal, characteristic, complemented
8	#65231	—	3	1	no	maximal, Hall
9	C₂³×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₅	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×D₃×Dic₅	—	2	2	no	—
17	C₂×D₃×Dic₅	—	2	2	no	—
18	C₂×D₃×Dic₅	C₄	1	4	yes	complemented
19	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
20	C₃×#683	—	2	2	no	—
21	#6668	—	2	2	no	—
22	(C₂×D₃)⋊Dic₅	—	2	2	no	—
23	#4987	—	5	1	no	maximal, Hall
24	C₂³×Dic₅	—	3	1	no	—
25	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
26	C₂×C₂²⋊Dic₅	—	3	1	no	—
27	C₂×#683	—	3	1	no	—
28	C₂×#683	D₃	1	1	yes	characteristic, complemented
29	#3963	—	3	1	no	—
30	#3963	—	3	1	no	—
31	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×D₃	—	2	1	no	—
33	C₂²×C₅×D₃	—	2	1	no	—
34	C₂²×C₅×D₃	—	2	1	no	—
35	C₂²×C₅×D₃	—	2	1	no	—
36	C₂²×C₅×D₃	C₂×C₄	1	2	yes	—
37	C₂²×C₅×D₃	D₄	1	2	yes	—
38	C₂²×C₅×D₃	Q₈	1	2	yes	—
39	C₂×C₃×C₄×C₅	—	2	1	no	—
40	C₂×C₃×C₄×C₅	—	2	1	no	—
41	C₂×C₃×C₄×C₅	—	2	1	no	—
42	C₂×Dic₁₅	—	2	1	no	—
43	C₂×Dic₁₅	—	2	1	no	—
44	C₂×Dic₁₅	—	2	1	no	—
45	C₂×Dic₁₅	—	2	1	no	—
46	C₂×Dic₁₅	—	2	1	no	—
47	C₂×Dic₁₅	C₂×C₄	1	2	yes	—
48	C₂×C₅×Dic₃	—	2	1	no	—
49	C₂×C₅×Dic₃	—	2	1	no	—
50	C₂×C₅×Dic₃	—	2	1	no	—
51	C₂×C₃×Dic₅	—	2	1	no	—
52	C₂×C₃×Dic₅	—	2	1	no	—
53	C₂×C₃×Dic₅	—	2	1	no	—
54	C₂×C₃×Dic₅	C₂×C₄	1	2	yes	—
55	C₂×C₃×Dic₅	D₄	1	2	yes	—
56	C₂×C₃×Dic₅	D₄	1	2	yes	—
57	D₃×Dic₅	—	2	8	no	—
58	C₂²×C₄×D₃	—	5	1	no	—
59	C₂×C₃×C₄⋊C₄	—	5	1	no	—
60	C₂×(C₂×D₃)⋊C₄	—	5	1	no	—
61	C₂×(C₂×D₃)⋊C₄	D₅	1	1	yes	characteristic
62	C₂×C₄⋊Dic₃	—	5	1	no	—
63	C₃⋊#95	—	5	1	no	—
64	C₃⋊#95	—	5	1	no	—
65	C₂⁴×C₅	—	3	1	no	—
66	C₂²×C₄×C₅	—	3	1	no	—
67	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
68	C₂²×Dic₅	—	3	1	no	—
69	C₂²×Dic₅	—	3	4	no	—
70	C₂²×Dic₅	—	3	1	no	—
71	C₂²×Dic₅	—	6	2	no	—
72	C₂²×Dic₅	—	6	2	no	—
73	C₂²×Dic₅	C₂×D₃	1	1	yes	characteristic
74	C₂²×Dic₅	C₂×D₃	1	1	yes	characteristic
75	C₅×C₂²⋊C₄	—	6	2	no	—
76	C₂²⋊Dic₅	—	6	2	no	—
77	#683	—	2	2	no	—
78	#683	—	6	2	no	—
79	#380	—	15	1	no	Sylow
80	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₃×C₅	#41	1	1	yes	characteristic
86	C₂²×C₃×C₅	#41	1	1	yes	characteristic
87	C₂×C₅×D₃	—	2	2	no	—
88	C₂×C₅×D₃	—	2	2	no	—
89	C₂×C₅×D₃	—	2	2	no	—
90	C₂×C₅×D₃	—	2	2	no	—
91	C₂×C₅×D₃	—	2	2	no	—
92	C₂×C₅×D₃	—	2	2	no	—
93	C₂×C₅×D₃	C₄⋊C₄	1	4	yes	complemented
94	C₃×C₄×C₅	—	4	1	no	—
95	Dic₁₅	—	2	2	no	—
96	Dic₁₅	—	4	1	no	—
97	C₃×Dic₅	—	4	1	no	—
98	C₃×Dic₅	C₂²⋊C₄	1	4	yes	complemented
99	C₅×Dic₃	—	4	1	no	—
100	C₂³×D₃	C₂×D₅	1	1	yes	characteristic
101	C₂²×C₃×C₄	—	5	1	no	—
102	C₂²×C₃×C₄	—	5	1	no	—
103	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
104	C₂²×Dic₃	—	5	1	no	—
105	C₂²×Dic₃	—	5	1	no	—
106	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
107	C₂×C₄×D₃	—	5	4	no	—
108	C₂×C₄×D₃	—	10	2	no	—
109	C₂×C₄×D₃	—	10	2	no	—
110	C₃×C₄⋊C₄	—	10	2	no	—
111	(C₂×D₃)⋊C₄	—	2	2	no	—
112	(C₂×D₃)⋊C₄	—	10	2	no	—
113	C₄⋊Dic₃	—	10	2	no	—
114	C₂³×C₅	—	3	2	no	—
115	C₂³×C₅	—	3	2	no	—
116	C₂³×C₅	—	3	2	no	—
117	C₂³×C₅	—	6	1	no	—
118	C₂³×C₅	—	6	1	no	—
119	C₂³×C₅	—	6	1	no	—
120	C₂³×C₅	—	6	1	no	—
121	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	6	1	no	—
126	C₂×C₄×C₅	—	6	1	no	—
127	C₂×C₄×C₅	—	6	1	no	—
128	C₂×Dic₅	—	2	1	no	—
129	C₂×Dic₅	—	2	1	no	—
130	C₂×Dic₅	—	2	1	no	—
131	C₂×Dic₅	—	3	2	no	—
132	C₂×Dic₅	—	6	1	no	—
133	C₂×Dic₅	—	6	1	no	—
134	C₂×Dic₅	—	6	8	no	—
135	C₂×Dic₅	—	6	1	no	—
136	C₂×Dic₅	—	6	1	no	—
137	C₂×Dic₅	—	6	1	no	—
138	C₂×Dic₅	C₄×D₃	1	2	yes	—
139	C₂×Dic₅	D₁₂	1	2	yes	—
140	C₂×Dic₅	#67	1	2	yes	—
141	C₂³×C₄	—	15	1	no	—
142	C₂×C₄⋊C₄	—	5	1	no	—
143	C₂×C₄⋊C₄	—	15	1	no	—
144	C₂×C₂²⋊C₄	—	3	1	no	—
145	C₂×C₂²⋊C₄	—	15	1	no	—
146	#95	—	15	1	no	—
147	#95	—	15	1	no	—
148	C₂×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₅	#117	1	1	yes	characteristic
151	C₂×C₃×C₅	#121	1	1	yes	characteristic
152	C₂×C₃×C₅	#121	1	1	yes	characteristic
153	C₂×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
155	C₅×D₃	—	2	4	no	—
156	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
157	C₂²×D₃	—	2	1	no	—
158	C₂²×D₃	—	2	1	no	—
159	C₂²×D₃	—	2	1	no	—
160	C₂²×D₃	—	2	1	no	—
161	C₂²×D₃	Dic₁₀	1	2	yes	—
162	C₂²×D₃	C₄×D₅	1	2	yes	—
163	C₂²×D₃	C₂²⋊D₅	1	2	yes	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	5	2	no	—
168	C₂×C₃×C₄	—	5	2	no	—
169	C₂×C₃×C₄	—	5	2	no	—
170	C₂×C₃×C₄	—	10	1	no	—
171	C₂×C₃×C₄	—	10	1	no	—
172	C₂×C₃×C₄	—	10	1	no	—
173	C₂×Dic₃	—	2	1	no	—
174	C₂×Dic₃	—	2	1	no	—
175	C₂×Dic₃	—	2	1	no	—
176	C₂×Dic₃	—	5	2	no	—
177	C₂×Dic₃	—	10	1	no	—
178	C₂×Dic₃	—	10	1	no	—
179	C₂×Dic₃	—	10	1	no	—
180	C₂×Dic₃	—	10	1	no	—
181	C₂×Dic₃	—	10	1	no	—
182	C₄×D₃	—	10	8	no	—
183	C₂²×C₅	—	3	4	no	—
184	C₂²×C₅	—	6	2	no	—
185	C₂²×C₅	—	6	2	no	—
186	C₂²×C₅	—	6	2	no	—
187	C₂²×C₅	—	6	2	no	—
188	C₂²×C₅	—	6	2	no	—
189	C₂²×C₅	—	6	2	no	—
190	C₂²×C₅	C₂×C₄×D₃	1	1	yes	characteristic
191	C₂²×C₅	C₂×D₁₂	1	1	yes	characteristic
192	C₂²×C₅	D₃×D₄	1	1	yes	characteristic
193	C₂²×C₅	#237	1	1	yes	characteristic
194	C₂²×C₅	D₃×Q₈	1	1	yes	characteristic
195	C₂²×C₅	#239	1	1	yes	characteristic
196	C₂²×C₅	C₂×#67	1	1	yes	characteristic
197	C₄×C₅	—	4	1	no	—
198	C₄×C₅	—	12	1	no	—
199	Dic₅	—	4	1	no	—
200	Dic₅	—	6	2	no	—
201	Dic₅	—	12	1	no	—
202	Dic₅	(C₂×D₃)⋊C₄	1	4	yes	complemented
203	C₂⁴	—	3	1	no	—
204	C₂²×C₄	—	3	1	no	—
205	C₂²×C₄	—	5	1	no	—
206	C₂²×C₄	—	5	1	no	—
207	C₂²×C₄	—	15	4	no	—
208	C₂²×C₄	—	15	1	no	—
209	C₂²×C₄	—	15	1	no	—
210	C₂²×C₄	—	30	2	no	—
211	C₂²×C₄	—	30	2	no	—
212	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
213	C₄⋊C₄	—	10	2	no	—
214	C₄⋊C₄	—	30	2	no	—
215	C₂²⋊C₄	—	6	2	no	—
216	C₂²⋊C₄	—	30	2	no	—
217	C₃×C₅	#380	1	1	yes	characteristic, complemented, Hall
218	C₂²×C₃	C₂×Dic₁₀	1	1	yes	characteristic
219	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
220	C₂²×C₃	D₄×D₅	1	1	yes	characteristic
221	C₂²×C₃	D₄×D₅	1	1	yes	characteristic
222	C₂²×C₃	#711	1	1	yes	characteristic
223	C₂²×C₃	#711	1	1	yes	characteristic
224	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
225	C₂×D₃	—	2	2	no	—
226	C₂×D₃	—	2	2	no	—
227	C₂×D₃	—	2	2	no	—
228	C₂×D₃	—	2	2	no	—
229	C₂×D₃	—	2	2	no	—
230	C₂×D₃	—	2	2	no	—
231	C₂×D₃	#683	1	4	yes	complemented
232	C₃×C₄	—	4	1	no	—
233	C₃×C₄	—	5	4	no	—
234	C₃×C₄	—	20	1	no	—
235	Dic₃	—	4	1	no	—
236	Dic₃	—	10	2	no	—
237	Dic₃	—	20	1	no	—
238	C₂×C₅	—	6	4	no	—
239	C₂×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₅	#892	1	1	yes	characteristic
241	C₂×C₅	#895	1	1	yes	characteristic
242	C₂×C₅	#897	1	1	yes	characteristic
243	C₂×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
244	C₂×C₅	#938	1	1	yes	characteristic
245	C₂×C₅	#946	1	1	yes	characteristic
246	C₂³	—	3	2	no	—
247	C₂³	—	3	2	no	—
248	C₂³	—	3	2	no	—
249	C₂³	—	6	1	no	—
250	C₂³	—	6	1	no	—
251	C₂³	—	6	1	no	—
252	C₂³	—	6	1	no	—
253	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	5	2	no	—
258	C₂×C₄	—	5	2	no	—
259	C₂×C₄	—	5	2	no	—
260	C₂×C₄	—	6	1	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	—	6	1	no	—
263	C₂×C₄	—	10	1	no	—
264	C₂×C₄	—	10	1	no	—
265	C₂×C₄	—	10	1	no	—
266	C₂×C₄	—	15	2	no	—
267	C₂×C₄	—	30	8	no	—
268	C₂×C₄	—	30	1	no	—
269	C₂×C₄	—	30	1	no	—
270	C₂×C₄	—	30	1	no	—
271	C₂×C₄	—	30	1	no	—
272	C₂×C₄	—	30	1	no	—
273	C₂×C₃	#4023	1	1	yes	characteristic
274	C₂×C₃	#4024	1	1	yes	characteristic
275	C₂×C₃	#4024	1	1	yes	characteristic
276	C₂×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₃	C₂×#683	1	1	yes	characteristic
278	C₂×C₃	#4085	1	1	yes	characteristic
279	C₂×C₃	#4085	1	1	yes	characteristic
280	D₃	—	2	4	no	—
281	C₅	#4987	1	1	yes	characteristic, complemented, Sylow
282	C₂²	—	3	4	no	—
283	C₂²	—	6	2	no	—
284	C₂²	—	6	2	no	—
285	C₂²	—	6	2	no	—
286	C₂²	—	6	2	no	—
287	C₂²	—	6	2	no	—
288	C₂²	—	6	2	no	—
289	C₂²	D₃×Dic₁₀	1	1	yes	characteristic
290	C₂²	#6768	1	1	yes	characteristic
291	C₂²	C₄×D₃×D₅	1	1	yes	characteristic
292	C₂²	D₅×D₁₂	1	1	yes	characteristic
293	C₂²	#6780	1	1	yes	characteristic
294	C₂²	D₅×#67	1	1	yes	characteristic
295	C₂²	D₃×C₂²⋊D₅	1	1	yes	characteristic
296	C₄	—	4	1	no	—
297	C₄	—	5	4	no	—
298	C₄	—	12	1	no	—
299	C₄	—	20	1	no	—
300	C₄	—	30	2	no	—
301	C₄	—	60	1	no	—
302	C₃	#65231	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	6	4	no	—
304	C₂	D₃×#683	1	1	yes	characteristic
305	C₂	#91400	1	1	yes	characteristic
306	C₂	#91410	1	1	yes	characteristic
307	C₂	#91416	1	1	yes	characteristic
308	C₂	#91417	1	1	yes	characteristic
309	C₂	D₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
310	C₂	#91565	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×D₃×D₅
commutator	C₂²×C₃×C₅	C₂²×C₄
Fitting	C₂²×C₃×C₄×C₅	C₂²
socle, cosocle	C₂³×C₃×C₅	C₂³
prime radical	C₂³×C₅×D₃	C₂²
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	#380
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#683	C₂	semidirect
4. C₂×#6668	C₂	semidirect
6. #91115	C₂	semidirect
7. #90994	C₂	semidirect
18. C₂×D₃×Dic₅	C₄	semidirect
28. C₂×#683	D₃	semidirect
93. C₂×C₅×D₃	C₄⋊C₄	semidirect
98. C₃×Dic₅	C₂²⋊C₄	semidirect
202. Dic₅	(C₂×D₃)⋊C₄	semidirect
217. C₃×C₅	#380	semidirect
231. C₂×D₃	#683	semidirect
281. C₅	#4987	semidirect
302. C₃	#65231	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₃×Dic₅	2
C₂×C₅×(C₂×D₃)⋊C₄	2
C₂×C₃×#683	2
C₂×#6668	2
C₂×(C₂×D₃)⋊Dic₅	2
#91115	2
#90994	2
#65231	3
#4987	5