Subgroups — #11754435

Order 960.

428 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91815	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#67	C₂	1	1	yes	maximal, characteristic
3	C₃×#4070	C₂	1	1	yes	maximal, characteristic, complemented
4	#91555	C₂	1	2	yes	maximal
5	#91531	C₂	1	2	yes	maximal, complemented
6	D₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#91400	C₂	1	2	yes	maximal, complemented
8	#91382	C₂	1	2	yes	maximal, complemented
9	#91381	C₂	1	1	yes	maximal, characteristic, complemented
10	#91369	C₂	1	1	yes	maximal, characteristic, complemented
11	Dic₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65875	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×#67	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
17	C₂×D₃×Dic₅	C₂²	1	2	yes	—
18	#6719	C₂²	1	2	yes	—
19	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₅	C₂²	1	2	yes	—
22	C₅×#217	C₂²	1	1	yes	characteristic
23	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
24	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
25	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
26	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₃×#683	C₂²	1	2	yes	—
29	Dic₃⋊Dic₅	C₂²	1	2	yes	—
30	#6669	C₂²	1	2	yes	—
31	#6669	C₂²	1	2	yes	—
32	#6668	C₂²	1	2	yes	—
33	#6668	C₂²	1	2	yes	—
34	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	#5639	—	5	1	no	maximal, Hall
37	C₄×C₅×D₄	—	3	1	no	—
38	#4081	—	3	2	no	—
39	#4072	—	3	1	no	—
40	C₂×C₄⋊Dic₅	—	3	1	no	—
41	#4070	—	3	1	no	—
42	#4070	D₃	1	1	yes	characteristic, complemented
43	#4035	—	3	2	no	—
44	#4025	—	3	2	no	—
45	#4023	—	3	2	no	—
46	#4016	—	3	1	no	—
47	C₄×Dic₁₀	—	3	1	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	2	yes	—
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	C₅×#67	—	2	2	no	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₄×C₅×D₃	—	2	2	no	—
60	C₂×C₃×Dic₅	C₂³	1	2	yes	—
61	C₂×C₃×Dic₅	C₂³	1	2	yes	—
62	C₃×Dic₁₀	—	2	2	no	—
63	D₃×Dic₅	—	2	4	no	—
64	C₃×C₂²⋊Q₈	—	5	1	no	—
65	#946	—	5	1	no	—
66	Q₈×Dic₃	—	5	1	no	—
67	C₄×#67	D₅	1	1	yes	characteristic
68	#926	—	5	1	no	—
69	#892	—	5	2	no	—
70	D₃×C₄⋊C₄	—	5	1	no	—
71	#890	—	5	1	no	—
72	#889	—	5	2	no	—
73	#883	—	5	2	no	—
74	#879	—	5	2	no	—
75	C₂×C₅×D₄	—	3	1	no	—
76	C₂²×C₄×C₅	—	3	1	no	—
77	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
78	C₂²×Dic₅	—	3	2	no	—
79	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	3	1	no	—
81	C₅×C₂²⋊C₄	—	3	1	no	—
82	C₅×C₂²⋊C₄	—	3	1	no	—
83	C₄²×C₅	—	3	1	no	—
84	C₂²⋊Dic₅	—	3	2	no	—
85	C₂²⋊Dic₅	—	3	2	no	—
86	C₂²⋊Dic₅	C₂×D₃	1	2	yes	—
87	C₄⋊Dic₅	—	3	1	no	—
88	C₄⋊Dic₅	—	3	1	no	—
89	C₄⋊Dic₅	—	3	2	no	—
90	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
91	#683	—	3	2	no	—
92	#683	—	3	2	no	—
93	#683	—	3	2	no	—
94	#683	—	3	2	no	—
95	#683	C₂×D₃	1	2	yes	—
96	C₄×Dic₅	—	3	2	no	—
97	C₄×Dic₅	—	3	2	no	—
98	#552	—	15	1	no	Sylow
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	—	2	1	no	—
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
103	C₂×C₅×D₃	—	2	1	no	—
104	C₂×C₅×D₃	—	2	1	no	—
105	C₂×C₅×D₃	#41	1	2	yes	—
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	#41	1	2	yes	—
108	Dic₁₅	—	2	2	no	—
109	Dic₁₅	—	2	2	no	—
110	C₃×Dic₅	—	2	2	no	—
111	C₃×Dic₅	—	2	2	no	—
112	C₅×Dic₃	—	2	1	no	—
113	C₅×Dic₃	—	2	1	no	—
114	C₅×Dic₃	—	2	1	no	—
115	C₂×C₃×Q₈	—	5	1	no	—
116	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
117	C₂×#67	C₂×D₅	1	1	yes	characteristic
118	C₂×C₄×D₃	—	5	2	no	—
119	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
120	C₃×C₄⋊C₄	—	5	2	no	—
121	C₃×C₄⋊C₄	—	5	1	no	—
122	C₃×C₂²⋊C₄	—	5	2	no	—
123	#217	—	5	2	no	—
124	#217	C₂×D₅	1	1	yes	characteristic
125	(C₂×D₃)⋊C₄	—	5	2	no	—
126	(C₂×D₃)⋊C₄	C₂×D₅	1	1	yes	characteristic
127	C₄⋊Dic₃	—	5	1	no	—
128	C₄⋊Dic₃	—	5	1	no	—
129	C₄⋊Dic₃	—	5	2	no	—
130	C₄⋊Dic₃	—	5	2	no	—
131	Dic₃⋊C₄	—	5	2	no	—
132	Dic₃⋊C₄	—	5	2	no	—
133	Dic₃⋊C₄	—	5	2	no	—
134	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic
135	C₄×Dic₃	—	5	2	no	—
136	C₄×Dic₃	—	5	2	no	—
137	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
138	C₂³×C₅	—	3	1	no	—
139	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₅×D₄	—	6	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	—	3	1	no	—
145	C₂×C₄×C₅	—	6	2	no	—
146	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
148	C₂×Dic₅	—	3	2	no	—
149	C₂×Dic₅	—	3	2	no	—
150	C₂×Dic₅	—	6	4	no	—
151	C₂×Dic₅	C₂²×D₃	1	2	yes	—
152	C₂×Dic₅	C₂²×D₃	1	2	yes	—
153	Dic₁₀	—	2	2	no	—
154	#126	—	15	2	no	—
155	#125	—	15	1	no	—
156	#125	—	15	2	no	—
157	#123	—	15	2	no	—
158	C₂²⋊Q₈	—	5	1	no	—
159	C₂²⋊Q₈	—	15	1	no	—
160	C₄×Q₈	—	15	1	no	—
161	C₄×D₄	—	3	1	no	—
162	#117	—	15	2	no	—
163	#117	—	15	1	no	—
164	C₂×C₄⋊C₄	—	15	1	no	—
165	C₂×C₃×C₅	—	4	1	no	—
166	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
169	C₅×D₃	—	2	2	no	—
170	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
171	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
172	C₃×Q₈	—	10	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	5	2	no	—
175	C₂×C₃×C₄	—	5	2	no	—
176	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	#67	—	2	2	no	—
179	C₂×Dic₃	—	5	2	no	—
180	C₂×Dic₃	—	5	2	no	—
181	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
182	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
183	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
184	C₄×D₃	—	2	2	no	—
185	C₄×D₃	—	10	4	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	—	3	2	no	—
190	C₂²×C₅	—	6	1	no	—
191	C₂²×C₅	—	6	1	no	—
192	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	6	1	no	—
195	C₄×C₅	—	6	1	no	—
196	C₄×C₅	—	6	1	no	—
197	C₄×C₅	#237	1	2	yes	—
198	Dic₅	—	2	2	no	—
199	Dic₅	—	2	2	no	—
200	Dic₅	—	6	2	no	—
201	Dic₅	—	6	2	no	—
202	C₂×Q₈	—	5	1	no	—
203	C₂×D₄	—	3	1	no	—
204	C₂²×C₄	—	3	1	no	—
205	C₂²×C₄	—	15	2	no	—
206	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
207	C₄⋊C₄	—	3	1	no	—
208	C₄⋊C₄	—	5	1	no	—
209	C₄⋊C₄	—	5	2	no	—
210	C₄⋊C₄	—	15	2	no	—
211	C₄⋊C₄	—	15	2	no	—
212	C₄⋊C₄	—	15	2	no	—
213	C₄⋊C₄	—	15	2	no	—
214	C₄⋊C₄	—	15	2	no	—
215	C₄⋊C₄	—	15	1	no	—
216	C₄⋊C₄	—	15	1	no	—
217	C₂²⋊C₄	—	3	1	no	—
218	C₂²⋊C₄	—	3	1	no	—
219	C₂²⋊C₄	—	5	2	no	—
220	C₂²⋊C₄	—	15	2	no	—
221	C₂²⋊C₄	—	15	2	no	—
222	C₄²	—	3	1	no	—
223	C₄²	—	15	2	no	—
224	C₄²	—	15	2	no	—
225	C₃×C₅	#552	1	1	yes	characteristic, complemented, Hall
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
230	C₂×D₃	—	2	1	no	—
231	C₂×D₃	—	2	1	no	—
232	C₂×D₃	#709	1	2	yes	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	10	2	no	—
235	C₃×C₄	—	10	2	no	—
236	C₃×C₄	#711	1	2	yes	—
237	Dic₃	—	2	1	no	—
238	Dic₃	—	2	1	no	—
239	Dic₃	—	2	1	no	—
240	Dic₃	—	10	2	no	—
241	Dic₃	—	10	2	no	—
242	C₂×C₅	—	4	1	no	—
243	C₂×C₅	—	6	2	no	—
244	C₂×C₅	C₂×#237	1	1	yes	characteristic
245	C₂×C₅	#1007	1	1	yes	characteristic
246	C₂×C₅	D₃×#41	1	1	yes	characteristic
247	C₂³	—	3	1	no	—
248	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
249	Q₈	—	10	2	no	—
250	D₄	—	6	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	3	1	no	—
253	C₂×C₄	—	3	1	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	5	2	no	—
256	C₂×C₄	—	5	2	no	—
257	C₂×C₄	—	6	2	no	—
258	C₂×C₄	—	15	2	no	—
259	C₂×C₄	—	15	2	no	—
260	C₂×C₄	—	30	4	no	—
261	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
262	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
263	C₂×C₃	—	4	1	no	—
264	C₂×C₃	C₂×#709	1	1	yes	characteristic
265	C₂×C₃	C₂×#711	1	1	yes	characteristic
266	C₂×C₃	#4150	1	1	yes	characteristic
267	D₃	—	2	2	no	—
268	C₅	#5639	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	3	2	no	—
273	C₂²	—	6	1	no	—
274	C₂²	—	6	1	no	—
275	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	6	1	no	—
278	C₄	—	6	1	no	—
279	C₄	—	6	1	no	—
280	C₄	—	10	2	no	—
281	C₄	—	10	2	no	—
282	C₄	—	30	2	no	—
283	C₄	—	30	2	no	—
284	C₄	#6780	1	2	yes	—
285	C₃	#65875	1	1	yes	characteristic, complemented, Sylow
286	C₂	—	4	1	no	—
287	C₂	—	6	2	no	—
288	C₂	#92004	1	1	yes	characteristic
289	C₂	D₃×#709	1	1	yes	characteristic
290	C₂	C₂×#6780	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	#552
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. #91815	C₂	semidirect
3. C₃×#4070	C₂	semidirect
5. #91531	C₂	semidirect
6. D₃×C₄⋊Dic₅	C₂	semidirect
7. #91400	C₂	semidirect
8. #91382	C₂	semidirect
9. #91381	C₂	semidirect
10. #91369	C₂	semidirect
11. Dic₃×Dic₁₀	C₂	semidirect
42. #4070	D₃	semidirect
225. C₃×C₅	#552	semidirect
268. C₅	#5639	semidirect
285. C₃	#65875	semidirect

Maximal subgroups

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

subgroup	index
#91815	2
C₄×C₅×#67	2
C₃×#4070	2
#91555	2
#91531	2
D₃×C₄⋊Dic₅	2
#91400	2
#91382	2
#91381	2
#91369	2
Dic₃×Dic₁₀	2
#65875	3
#5639	5