Subgroups — #11755180

Order 960.

444 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	#91833	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#946	C₂	1	1	yes	maximal, characteristic
3	C₃×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#91470	C₂	1	1	yes	maximal, characteristic, complemented
5	#91436	C₂	1	2	yes	maximal
6	#91398	C₂	1	2	yes	maximal
7	C₄×D₃×Dic₅	C₂	1	1	yes	maximal, characteristic
8	#91365	C₂	1	1	yes	maximal, characteristic
9	#91339	C₂	1	2	yes	maximal, complemented
10	#91333	C₂	1	1	yes	maximal, characteristic, complemented
11	#91323	C₂	1	2	yes	maximal, complemented
12	#65956	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂×#1204	C₂²	1	2	yes	—
17	C₂×D₃×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×D₃×Dic₅	C₂²	1	1	yes	characteristic
19	#6712	C₂²	1	2	yes	—
20	#6712	C₂²	1	2	yes	—
21	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
22	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
28	C₃×C₄×Dic₅	C₂²	1	2	yes	—
29	#6669	C₂²	1	2	yes	—
30	#6669	C₂²	1	2	yes	—
31	#6668	C₂²	1	2	yes	—
32	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
33	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
34	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	#5578	—	5	1	no	maximal, Hall
37	C₅×C₂²⋊Q₈	—	3	1	no	—
38	Q₈×Dic₅	D₃	1	1	yes	characteristic, complemented
39	#4090	—	3	1	no	—
40	#4070	—	3	1	no	—
41	C₂×C₄×Dic₅	—	3	1	no	—
42	#4035	—	3	2	no	—
43	#4034	—	3	1	no	—
44	#4033	—	3	2	no	—
45	#4033	—	3	1	no	—
46	#4024	—	3	2	no	—
47	#4023	—	3	2	no	—
48	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	2	2	no	—
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×Dic₁₅	C₂³	1	2	yes	—
53	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
55	Dic₃₀	—	2	2	no	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	C₂³	1	2	yes	—
58	C₄×C₅×D₃	—	2	2	no	—
59	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₅	C₂³	1	2	yes	—
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	#1204	—	2	4	no	—
63	D₃×Dic₅	—	2	2	no	—
64	D₃×Dic₅	Q₈	1	4	yes	complemented
65	C₃×C₄×Q₈	—	5	1	no	—
66	#946	D₅	1	1	yes	characteristic
67	#944	—	5	1	no	—
68	#897	—	5	1	no	—
69	#896	—	5	2	no	—
70	#889	—	5	2	no	—
71	#888	—	5	1	no	—
72	#872	—	5	2	no	—
73	C₄²×D₃	—	5	1	no	—
74	C₄×Dic₆	—	5	1	no	—
75	C₄×Dic₆	—	5	2	no	—
76	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
77	C₂²×C₄×C₅	—	3	1	no	—
78	C₂²×Dic₅	—	3	1	no	—
79	C₂²×Dic₅	—	3	1	no	—
80	C₂×Dic₁₀	—	3	1	no	—
81	C₂×Dic₁₀	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	2	no	—
83	C₅×C₄⋊C₄	—	3	1	no	—
84	C₅×C₂²⋊C₄	—	3	2	no	—
85	C₂²⋊Dic₅	—	3	2	no	—
86	C₄⋊Dic₅	C₂×D₃	1	2	yes	—
87	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
88	#683	—	3	2	no	—
89	#683	—	3	2	no	—
90	#683	—	3	2	no	—
91	#683	—	3	2	no	—
92	#683	—	3	2	no	—
93	C₄×Dic₅	—	3	1	no	—
94	C₄×Dic₅	—	3	2	no	—
95	C₄×Dic₅	—	3	1	no	—
96	C₄×Dic₅	—	3	1	no	—
97	C₄×Dic₅	C₂×D₃	1	2	yes	—
98	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
99	#533	—	15	1	no	Sylow
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₂×C₅×D₃	—	2	1	no	—
102	C₂×C₅×D₃	—	2	1	no	—
103	C₂×C₅×D₃	C₂×Q₈	1	2	yes	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	#41	1	2	yes	—
106	Dic₁₅	—	2	2	no	—
107	Dic₁₅	C₂×Q₈	1	2	yes	—
108	Dic₁₅	#41	1	2	yes	—
109	C₃×Dic₅	—	2	1	no	—
110	C₃×Dic₅	—	2	2	no	—
111	C₃×Dic₅	C₂×Q₈	1	2	yes	—
112	C₅×Dic₃	—	2	2	no	—
113	C₅×Dic₃	—	2	1	no	—
114	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
115	C₂×C₄×D₃	—	5	1	no	—
116	C₂×C₄×D₃	—	5	1	no	—
117	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
118	C₂×Dic₆	—	5	1	no	—
119	C₂×Dic₆	—	5	2	no	—
120	C₃×C₄⋊C₄	—	5	1	no	—
121	C₃×C₄⋊C₄	—	5	2	no	—
122	C₃×C₄²	—	5	1	no	—
123	C₃×C₄²	—	5	2	no	—
124	(C₂×D₃)⋊C₄	—	5	2	no	—
125	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	—
126	C₄⋊Dic₃	—	5	2	no	—
127	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
128	Dic₃⋊C₄	—	5	2	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	1	no	—
134	C₄×Dic₃	—	5	1	no	—
135	C₄×Dic₃	—	5	2	no	—
136	C₄×Dic₃	—	5	1	no	—
137	C₂³×C₅	—	3	1	no	—
138	C₅×Q₈	—	2	2	no	—
139	C₂×C₄×C₅	—	3	1	no	—
140	C₂×C₄×C₅	—	3	2	no	—
141	C₂×C₄×C₅	—	6	2	no	—
142	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
143	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂×Dic₅	—	3	1	no	—
145	C₂×Dic₅	—	3	1	no	—
146	C₂×Dic₅	—	3	4	no	—
147	C₂×Dic₅	—	3	2	no	—
148	C₂×Dic₅	—	6	2	no	—
149	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
150	C₂×Dic₅	C₂²×D₃	1	2	yes	—
151	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
152	Dic₁₀	—	6	4	no	—
153	Dic₁₀	—	6	2	no	—
154	C₄⋊Q₈	—	15	1	no	—
155	C₄⋊Q₈	—	15	1	no	—
156	#125	—	15	2	no	—
157	C₂²⋊Q₈	—	3	1	no	—
158	C₂²⋊Q₈	—	15	2	no	—
159	C₂²⋊Q₈	—	15	1	no	—
160	C₄×Q₈	—	5	1	no	—
161	C₄×Q₈	—	15	1	no	—
162	C₄×Q₈	—	15	2	no	—
163	#117	—	15	2	no	—
164	C₂×C₄²	—	15	1	no	—
165	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₅×D₃	—	2	2	no	—
169	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
170	C₃×Q₈	—	2	2	no	—
171	C₂×C₃×C₄	—	5	1	no	—
172	C₂×C₃×C₄	—	5	1	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	1	no	—
177	C₂×Dic₃	—	5	1	no	—
178	C₂×Dic₃	—	5	2	no	—
179	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
180	C₂×Dic₃	C₂²×D₅	1	2	yes	—
181	C₄×D₃	—	2	2	no	—
182	C₄×D₃	—	5	4	no	—
183	C₄×D₃	—	10	2	no	—
184	Dic₆	—	10	4	no	—
185	Dic₆	—	10	2	no	—
186	C₂²×C₅	—	3	2	no	—
187	C₂²×C₅	—	6	1	no	—
188	C₂²×C₅	—	6	1	no	—
189	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	6	1	no	—
192	C₄×C₅	—	6	2	no	—
193	C₄×C₅	#235	1	2	yes	—
194	Dic₅	—	2	2	no	—
195	Dic₅	—	2	1	no	—
196	Dic₅	—	3	2	no	—
197	Dic₅	—	3	2	no	—
198	Dic₅	—	6	2	no	—
199	Dic₅	D₃×Q₈	1	2	yes	—
200	C₂×Q₈	—	15	2	no	—
201	C₂×Q₈	—	15	1	no	—
202	C₂×Q₈	D₃×D₅	1	1	yes	characteristic
203	C₂²×C₄	—	3	1	no	—
204	C₂²×C₄	—	15	1	no	—
205	C₂²×C₄	—	15	1	no	—
206	C₄⋊C₄	—	3	2	no	—
207	C₄⋊C₄	—	3	1	no	—
208	C₄⋊C₄	—	5	2	no	—
209	C₄⋊C₄	—	5	1	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	C₂²⋊C₄	—	3	2	no	—
216	C₂²⋊C₄	—	15	2	no	—
217	C₄²	—	5	1	no	—
218	C₄²	—	5	2	no	—
219	C₄²	—	15	1	no	—
220	C₄²	—	15	1	no	—
221	C₄²	—	15	2	no	—
222	C₄²	—	15	1	no	—
223	C₃×C₅	#533	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
225	C₂×D₃	—	2	1	no	—
226	C₂×D₃	—	2	1	no	—
227	C₂×D₃	Q₈×D₅	1	2	yes	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	5	2	no	—
230	C₃×C₄	—	10	2	no	—
231	C₃×C₄	—	10	1	no	—
232	C₃×C₄	#711	1	2	yes	—
233	Dic₃	—	2	1	no	—
234	Dic₃	—	2	2	no	—
235	Dic₃	—	5	2	no	—
236	Dic₃	—	5	2	no	—
237	Dic₃	—	10	2	no	—
238	C₂×C₅	—	6	2	no	—
239	C₂×C₅	C₂×#235	1	1	yes	characteristic
240	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
241	C₂×C₅	D₃×#41	1	1	yes	characteristic
242	C₂³	—	3	1	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	30	4	no	—
245	Q₈	—	30	2	no	—
246	C₂×C₄	—	3	1	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	5	1	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	5	2	no	—
251	C₂×C₄	—	6	2	no	—
252	C₂×C₄	—	15	4	no	—
253	C₂×C₄	—	15	1	no	—
254	C₂×C₄	—	15	1	no	—
255	C₂×C₄	—	15	2	no	—
256	C₂×C₄	—	30	2	no	—
257	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
258	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
259	C₂×C₃	C₂×#711	1	1	yes	characteristic
260	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
261	C₂×C₃	D₅×#41	1	1	yes	characteristic
262	D₃	—	2	2	no	—
263	C₅	#5578	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	3	2	no	—
265	C₂²	—	6	1	no	—
266	C₂²	—	6	1	no	—
267	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	5	2	no	—
270	C₄	—	6	2	no	—
271	C₄	—	6	1	no	—
272	C₄	—	10	1	no	—
273	C₄	—	10	2	no	—
274	C₄	—	15	2	no	—
275	C₄	—	15	2	no	—
276	C₄	—	30	2	no	—
277	C₄	#6782	1	2	yes	—
278	C₃	#65956	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	6	2	no	—
280	C₂	D₃×Q₈×D₅	1	1	yes	characteristic
281	C₂	#92034	1	1	yes	characteristic
282	C₂	C₂×#6782	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₂×C₃×C₅×Q₈	C₂²
prime radical	C₂²×C₅×D₃	C₂³
hypercenter	C₂×Q₈	D₃×D₅
hypocenter	C₃×C₅	#533
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #91833	C₂	semidirect
3. C₃×Q₈×Dic₅	C₂	semidirect
4. #91470	C₂	semidirect
9. #91339	C₂	semidirect
10. #91333	C₂	semidirect
11. #91323	C₂	semidirect
38. Q₈×Dic₅	D₃	semidirect
64. D₃×Dic₅	Q₈	semidirect
223. C₃×C₅	#533	semidirect
263. C₅	#5578	semidirect
278. C₃	#65956	semidirect

Maximal subgroups

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

subgroup	index
#91833	2
C₅×#946	2
C₃×Q₈×Dic₅	2
#91470	2
#91436	2
#91398	2
C₄×D₃×Dic₅	2
#91365	2
#91339	2
#91333	2
#91323	2
#65956	3
#5578	5