Subgroups — #11753397

Order 960.

532 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×Dic₆	C₂	1	1	yes	maximal, characteristic
2	#91776	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4038	C₂	1	1	yes	maximal, characteristic
5	C₄×#1204	C₂	1	2	yes	maximal, complemented
6	#91394	C₂	1	2	yes	maximal
7	#91393	C₂	1	2	yes	maximal
8	#91357	C₂	1	1	yes	maximal, characteristic
9	#91344	C₂	1	1	yes	maximal, characteristic, complemented
10	#91340	C₂	1	1	yes	maximal, characteristic, complemented
11	#91323	C₂	1	2	yes	maximal, complemented
12	#65906	—	3	1	no	maximal, Hall
13	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
14	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
16	C₂×#1204	C₂²	1	2	yes	—
17	C₂×D₅×Dic₃	C₂²	1	2	yes	—
18	D₅×Dic₆	C₄	1	8	yes	complemented
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	#6712	C₂²	1	2	yes	—
21	C₄×Dic₁₅	C₂²	1	2	yes	—
22	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
23	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
24	C₄×C₅×Dic₃	C₂²	1	2	yes	—
25	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
26	C₃×#685	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₃×C₄×Dic₅	C₂²	1	2	yes	—
29	Dic₃⋊Dic₅	C₂²	1	2	yes	—
30	#6669	C₂²	1	2	yes	—
31	#6668	C₂²	1	1	yes	characteristic
32	#6668	C₂²	1	1	yes	characteristic
33	#6665	C₂²	1	1	yes	characteristic
34	#6665	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	Dic₃×Dic₅	C₂²	1	2	yes	—
37	#5516	—	5	1	no	maximal, Hall
38	C₂×Q₈×D₅	—	3	1	no	—
39	C₄×C₅×Q₈	—	3	1	no	—
40	Q₈×Dic₅	—	3	1	no	—
41	#4038	—	3	2	no	—
42	#4038	D₃	1	1	yes	characteristic
43	#4033	—	3	1	no	—
44	#4033	—	3	2	no	—
45	#4018	—	3	1	no	—
46	#4018	—	3	2	no	—
47	C₄×Dic₁₀	—	3	1	no	—
48	C₄×Dic₁₀	—	3	2	no	—
49	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	Dic₃₀	C₂×C₄	1	4	yes	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	2	yes	—
59	C₅×Dic₆	C₂×C₄	1	4	yes	—
60	C₂×C₃×Dic₅	C₂³	1	2	yes	—
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₃×C₄×D₅	C₂×C₄	1	4	yes	—
63	#1204	C₂×C₄	1	8	yes	—
64	D₅×Dic₃	C₂×C₄	1	8	yes	—
65	C₂²×Dic₆	—	5	1	no	—
66	C₃×#117	—	5	1	no	—
67	#926	—	5	1	no	—
68	#887	—	5	1	no	—
69	#887	—	5	1	no	—
70	#887	—	5	1	no	—
71	#887	D₅	1	1	yes	characteristic, complemented
72	#877	—	5	2	no	—
73	#877	—	5	2	no	—
74	C₄×Dic₆	—	5	2	no	—
75	C₄×Dic₆	—	5	2	no	—
76	C₂×C₅×Q₈	—	3	1	no	—
77	Q₈×D₅	—	3	8	no	—
78	C₂×C₄×D₅	—	3	2	no	—
79	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
80	C₂×Dic₁₀	—	3	1	no	—
81	C₂×Dic₁₀	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	2	no	—
83	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
84	C₄²×C₅	—	3	2	no	—
85	C₄²×C₅	—	3	1	no	—
86	#685	—	3	1	no	—
87	#685	—	3	1	no	—
88	#685	C₂×D₃	1	2	yes	—
89	C₄⋊Dic₅	—	3	2	no	—
90	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
91	#683	—	3	2	no	—
92	#683	—	3	1	no	—
93	#683	—	3	1	no	—
94	#683	—	3	2	no	—
95	C₄×Dic₅	—	3	2	no	—
96	C₄×Dic₅	—	3	1	no	—
97	C₄×Dic₅	—	3	2	no	—
98	C₄×Dic₅	—	3	2	no	—
99	C₄×Dic₅	C₂×D₃	1	2	yes	—
100	#519	—	15	1	no	Sylow
101	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
102	C₂×C₃×D₅	—	2	1	no	—
103	C₂×C₃×D₅	—	2	1	no	—
104	C₂×C₃×D₅	C₂²×C₄	1	2	yes	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
107	Dic₁₅	—	2	1	no	—
108	Dic₁₅	—	2	1	no	—
109	Dic₁₅	C₂²×C₄	1	4	yes	—
110	C₃×Dic₅	—	2	2	no	—
111	C₃×Dic₅	C₂²×C₄	1	2	yes	—
112	C₅×Dic₃	—	2	1	no	—
113	C₅×Dic₃	—	2	1	no	—
114	C₅×Dic₃	C₂²×C₄	1	4	yes	—
115	C₂²×C₃×C₄	—	5	1	no	—
116	C₂²×Dic₃	—	5	2	no	—
117	C₂×Dic₆	—	5	1	no	—
118	C₂×Dic₆	—	5	2	no	—
119	C₂×Dic₆	—	5	8	no	—
120	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic
121	C₃×C₄⋊C₄	—	5	1	no	—
122	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
123	C₃×C₂²⋊C₄	—	5	2	no	—
124	C₃×C₄²	—	5	2	no	—
125	#217	—	5	1	no	—
126	#217	—	5	1	no	—
127	C₄⋊Dic₃	—	5	1	no	—
128	C₄⋊Dic₃	—	5	1	no	—
129	Dic₃⋊C₄	—	5	2	no	—
130	Dic₃⋊C₄	—	5	2	no	—
131	Dic₃⋊C₄	—	5	2	no	—
132	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
133	C₄×Dic₃	—	5	2	no	—
134	C₄×Dic₃	—	5	1	no	—
135	C₄×Dic₃	—	5	2	no	—
136	C₄×Dic₃	—	5	2	no	—
137	C₄×Dic₃	C₂×D₅	1	2	yes	—
138	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
139	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
140	C₅×Q₈	—	3	4	no	—
141	C₂×C₄×C₅	—	3	2	no	—
142	C₂×C₄×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
145	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
146	C₂×Dic₅	—	3	2	no	—
147	C₂×Dic₅	—	3	1	no	—
148	C₂×Dic₅	—	3	1	no	—
149	C₂×Dic₅	C₂²×D₃	1	2	yes	—
150	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
151	C₄×D₅	—	3	8	no	—
152	C₄×D₅	C₄×D₃	1	4	yes	—
153	Dic₁₀	—	3	4	no	—
154	Dic₁₀	—	3	8	no	—
155	C₂²×Q₈	—	15	1	no	—
156	C₄×Q₈	—	3	1	no	—
157	C₄×Q₈	—	15	2	no	—
158	C₄×Q₈	—	15	1	no	—
159	C₄×Q₈	—	15	2	no	—
160	C₄×Q₈	—	15	1	no	—
161	C₄×Q₈	—	15	1	no	—
162	#117	—	5	1	no	—
163	#117	—	15	2	no	—
164	#117	—	15	1	no	—
165	#117	—	15	2	no	—
166	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
168	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
169	C₃×D₅	—	2	2	no	—
170	C₂³×C₃	—	5	1	no	—
171	C₂×C₃×C₄	—	5	1	no	—
172	C₂×C₃×C₄	—	5	4	no	—
173	C₂×C₃×C₄	—	5	2	no	—
174	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
175	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
176	C₂×Dic₃	—	5	2	no	—
177	C₂×Dic₃	—	5	8	no	—
178	C₂×Dic₃	—	5	1	no	—
179	C₂×Dic₃	—	5	1	no	—
180	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
181	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
182	C₂×Dic₃	C₂²×D₅	1	2	yes	—
183	Dic₆	—	5	4	no	—
184	Dic₆	—	5	8	no	—
185	Dic₆	C₄×D₅	1	4	yes	—
186	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
187	C₂×D₅	—	2	1	no	—
188	C₂×D₅	—	2	1	no	—
189	C₂×D₅	C₂×C₄×D₃	1	2	yes	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	3	4	no	—
192	C₄×C₅	—	6	1	no	—
193	C₄×C₅	—	6	1	no	—
194	C₄×C₅	C₂×C₄×D₃	1	2	yes	—
195	Dic₅	—	2	2	no	—
196	Dic₅	—	3	4	no	—
197	Dic₅	—	6	1	no	—
198	Dic₅	—	6	1	no	—
199	Dic₅	C₂×C₄×D₃	1	2	yes	—
200	C₂×Q₈	—	3	1	no	—
201	C₂×Q₈	—	15	2	no	—
202	C₂×Q₈	—	15	8	no	—
203	C₂×Q₈	—	15	1	no	—
204	C₂²×C₄	—	5	1	no	—
205	C₂²×C₄	—	15	2	no	—
206	C₄⋊C₄	—	3	2	no	—
207	C₄⋊C₄	—	5	1	no	—
208	C₄⋊C₄	—	15	2	no	—
209	C₄⋊C₄	—	15	1	no	—
210	C₄⋊C₄	—	15	1	no	—
211	C₄⋊C₄	—	15	2	no	—
212	C₄⋊C₄	—	15	2	no	—
213	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
214	C₂²⋊C₄	—	5	2	no	—
215	C₂²⋊C₄	—	15	1	no	—
216	C₂²⋊C₄	—	15	1	no	—
217	C₄²	—	3	2	no	—
218	C₄²	—	3	1	no	—
219	C₄²	—	5	2	no	—
220	C₄²	—	15	2	no	—
221	C₄²	—	15	2	no	—
222	C₄²	—	15	2	no	—
223	C₄²	—	15	1	no	—
224	C₃×C₅	#519	1	1	yes	characteristic, complemented, Hall
225	C₂²×C₃	—	5	2	no	—
226	C₂²×C₃	—	10	1	no	—
227	C₂²×C₃	—	10	1	no	—
228	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	5	2	no	—
231	C₃×C₄	—	10	2	no	—
232	C₃×C₄	C₂×C₄×D₅	1	2	yes	—
233	Dic₃	—	2	1	no	—
234	Dic₃	—	2	1	no	—
235	Dic₃	—	5	4	no	—
236	Dic₃	—	10	1	no	—
237	Dic₃	—	10	1	no	—
238	Dic₃	C₂×C₄×D₅	1	4	yes	—
239	C₂×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
240	C₂×C₅	#1010	1	1	yes	characteristic
241	C₂×C₅	#1010	1	1	yes	characteristic
242	D₅	—	2	2	no	—
243	C₂³	—	5	1	no	—
244	Q₈	—	3	4	no	—
245	Q₈	—	15	8	no	—
246	Q₈	—	15	4	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	5	1	no	—
251	C₂×C₄	—	5	4	no	—
252	C₂×C₄	—	5	2	no	—
253	C₂×C₄	—	15	8	no	—
254	C₂×C₄	—	15	2	no	—
255	C₂×C₄	—	15	1	no	—
256	C₂×C₄	—	15	1	no	—
257	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
258	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
259	C₂×C₃	—	10	2	no	—
260	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
261	C₂×C₃	#4147	1	1	yes	characteristic
262	C₂×C₃	#4150	1	1	yes	characteristic
263	C₅	#5516	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	5	2	no	—
265	C₂²	—	10	1	no	—
266	C₂²	—	10	1	no	—
267	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	3	4	no	—
270	C₄	—	5	2	no	—
271	C₄	—	6	1	no	—
272	C₄	—	6	1	no	—
273	C₄	—	10	2	no	—
274	C₄	—	15	4	no	—
275	C₄	—	30	1	no	—
276	C₄	—	30	1	no	—
277	C₄	C₄×D₃×D₅	1	2	yes	—
278	C₃	#65906	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	10	2	no	—
280	C₂	C₂×C₄×D₃×D₅	1	1	yes	characteristic
281	C₂	#92020	1	1	yes	characteristic
282	C₂	#92028	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₄
Fitting	C₃×C₅×C₄⋊C₄	C₂²
socle, cosocle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂²×C₃×D₅	C₂³
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	#519
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #91776	C₂	semidirect
3. C₅×#887	C₂	semidirect
5. C₄×#1204	C₂	semidirect
9. #91344	C₂	semidirect
10. #91340	C₂	semidirect
11. #91323	C₂	semidirect
18. D₅×Dic₆	C₄	semidirect
71. #887	D₅	semidirect
224. C₃×C₅	#519	semidirect
263. C₅	#5516	semidirect
278. C₃	#65906	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅×Dic₆	2
#91776	2
C₅×#887	2
C₃×#4038	2
C₄×#1204	2
#91394	2
#91393	2
#91357	2
#91344	2
#91340	2
#91323	2
#65906	3
#5516	5