Subgroups — #11756327

Order 960.

424 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#91836	C₂	1	1	yes	maximal, characteristic
3	#91833	C₂	1	1	yes	maximal, characteristic, complemented
4	#91787	C₂	1	2	yes	maximal, complemented
5	#91785	C₂	1	2	yes	maximal, complemented
6	#91782	C₂	1	1	yes	maximal, characteristic
7	#91776	C₂	1	1	yes	maximal, characteristic, complemented
8	#91765	C₂	1	2	yes	maximal, complemented
9	#91764	C₂	1	1	yes	maximal, characteristic
10	#91761	C₂	1	2	yes	maximal, complemented
11	C₄×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65902	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×D₆₀	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₅	C₂²	1	2	yes	—
16	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₅	C₂²	1	2	yes	—
22	#6714	C₂²	1	2	yes	—
23	#6714	C₂²	1	2	yes	—
24	#6714	C₂²	1	2	yes	—
25	#6714	C₂²	1	2	yes	—
26	#6714	C₂²	1	2	yes	—
27	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
28	#6712	C₂²	1	1	yes	characteristic
29	#6712	C₂²	1	1	yes	characteristic
30	#6712	C₂²	1	2	yes	—
31	#6712	C₂²	1	2	yes	—
32	#6712	C₂²	1	2	yes	—
33	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₅	C₂²	1	2	yes	—
36	#5568	—	5	1	no	maximal, Hall
37	C₄×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
38	#4093	—	3	1	no	—
39	#4090	—	3	1	no	—
40	#4044	—	3	2	no	—
41	#4042	—	3	2	no	—
42	#4039	—	3	1	no	—
43	#4033	—	3	1	no	—
44	#4022	—	3	2	no	—
45	#4021	—	3	1	no	—
46	#4018	—	3	2	no	—
47	C₄×Dic₁₀	—	3	1	no	—
48	C₂²×D₁₅	C₂³	1	2	yes	—
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×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₅	C₂³	1	2	yes	—
59	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
60	D₆₀	—	2	2	no	—
61	C₄×D₁₅	—	2	4	no	—
62	Dic₃₀	—	2	2	no	—
63	Dic₃₀	—	2	2	no	—
64	C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
65	#947	—	5	1	no	—
66	#944	—	5	1	no	—
67	#898	—	5	2	no	—
68	#896	—	5	2	no	—
69	#893	—	5	1	no	—
70	#887	—	5	1	no	—
71	#876	—	5	2	no	—
72	#875	—	5	1	no	—
73	#872	—	5	2	no	—
74	C₄×Dic₆	—	5	1	no	—
75	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
76	C₂×D₂₀	—	3	1	no	—
77	C₂×C₄×D₅	—	3	2	no	—
78	C₂×Dic₁₀	—	3	1	no	—
79	C₂×Dic₁₀	—	3	1	no	—
80	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
81	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
82	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
83	C₄²×C₅	C₂×D₃	1	2	yes	—
84	#685	—	3	2	no	—
85	#685	—	3	2	no	—
86	#685	—	3	2	no	—
87	#685	—	3	2	no	—
88	#685	—	3	2	no	—
89	C₄⋊Dic₅	—	3	1	no	—
90	#683	—	3	1	no	—
91	#683	—	3	1	no	—
92	#683	—	3	2	no	—
93	#683	—	3	2	no	—
94	#683	—	3	2	no	—
95	C₄×Dic₅	—	3	1	no	—
96	C₄×Dic₅	—	3	1	no	—
97	C₄×Dic₅	—	3	2	no	—
98	#556	—	15	1	no	Sylow
99	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×D₁₅	—	2	2	no	—
101	C₂×D₁₅	—	2	2	no	—
102	C₂×D₁₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	1	no	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	#41	1	2	yes	—
108	Dic₁₅	—	2	2	no	—
109	Dic₁₅	—	2	1	no	—
110	Dic₁₅	—	2	1	no	—
111	Dic₁₅	—	2	1	no	—
112	Dic₁₅	#41	1	2	yes	—
113	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
114	C₂×D₁₂	—	5	1	no	—
115	C₂×C₄×D₃	—	5	2	no	—
116	C₂×Dic₆	—	5	1	no	—
117	C₂×Dic₆	—	5	1	no	—
118	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
119	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
120	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
121	C₃×C₄²	C₂×D₅	1	2	yes	—
122	(C₂×D₃)⋊C₄	—	5	2	no	—
123	(C₂×D₃)⋊C₄	—	5	2	no	—
124	(C₂×D₃)⋊C₄	—	5	2	no	—
125	(C₂×D₃)⋊C₄	—	5	2	no	—
126	(C₂×D₃)⋊C₄	—	5	2	no	—
127	C₄⋊Dic₃	—	5	1	no	—
128	Dic₃⋊C₄	—	5	1	no	—
129	Dic₃⋊C₄	—	5	1	no	—
130	Dic₃⋊C₄	—	5	2	no	—
131	Dic₃⋊C₄	—	5	2	no	—
132	Dic₃⋊C₄	—	5	2	no	—
133	C₄×Dic₃	—	5	1	no	—
134	C₄×Dic₃	—	5	1	no	—
135	C₄×Dic₃	—	5	2	no	—
136	C₂²×D₅	—	3	2	no	—
137	C₅×Q₈	—	2	2	no	—
138	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
139	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
141	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₂×Dic₅	—	3	1	no	—
144	C₂×Dic₅	—	3	1	no	—
145	C₂×Dic₅	—	3	1	no	—
146	C₂×Dic₅	—	3	2	no	—
147	C₂×Dic₅	—	3	1	no	—
148	D₂₀	—	6	2	no	—
149	C₄×D₅	—	6	4	no	—
150	Dic₁₀	—	6	2	no	—
151	Dic₁₀	—	6	2	no	—
152	C₄⋊Q₈	—	15	1	no	—
153	#126	—	15	2	no	—
154	#126	—	15	2	no	—
155	#124	—	15	1	no	—
156	#124	—	15	1	no	—
157	C₂²⋊Q₈	—	15	2	no	—
158	C₄×Q₈	—	15	1	no	—
159	C₄×Q₈	—	15	1	no	—
160	C₄×Q₈	D₁₅	1	1	yes	characteristic, complemented
161	C₄×D₄	—	15	1	no	—
162	#117	—	15	2	no	—
163	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
166	D₁₅	—	4	2	no	—
167	C₂²×D₃	—	5	2	no	—
168	C₃×Q₈	—	2	2	no	—
169	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
170	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
172	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×Dic₃	—	5	1	no	—
175	C₂×Dic₃	—	5	1	no	—
176	C₂×Dic₃	—	5	1	no	—
177	C₂×Dic₃	—	5	2	no	—
178	C₂×Dic₃	—	5	1	no	—
179	D₁₂	—	10	2	no	—
180	C₄×D₃	—	10	4	no	—
181	Dic₆	—	10	2	no	—
182	Dic₆	—	10	2	no	—
183	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
184	C₂×D₅	—	6	2	no	—
185	C₂×D₅	—	6	2	no	—
186	C₂×D₅	—	6	2	no	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	#235	1	2	yes	—
192	Dic₅	—	3	2	no	—
193	Dic₅	—	6	2	no	—
194	Dic₅	—	6	1	no	—
195	Dic₅	—	6	1	no	—
196	Dic₅	—	6	1	no	—
197	C₂×Q₈	—	15	1	no	—
198	C₂×Q₈	—	15	1	no	—
199	C₂×Q₈	C₂×D₁₅	1	1	yes	characteristic
200	C₂×D₄	—	15	1	no	—
201	C₂²×C₄	—	15	2	no	—
202	C₄⋊C₄	—	15	1	no	—
203	C₄⋊C₄	—	15	2	no	—
204	C₄⋊C₄	—	15	2	no	—
205	C₄⋊C₄	—	15	1	no	—
206	C₄⋊C₄	—	15	1	no	—
207	C₄⋊C₄	—	15	2	no	—
208	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
209	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
210	C₂²⋊C₄	—	15	2	no	—
211	C₂²⋊C₄	—	15	2	no	—
212	C₂²⋊C₄	—	15	2	no	—
213	C₂²⋊C₄	—	15	2	no	—
214	C₂²⋊C₄	—	15	2	no	—
215	C₄²	—	15	2	no	—
216	C₄²	—	15	1	no	—
217	C₄²	—	15	1	no	—
218	C₄²	C₂×D₁₅	1	2	yes	—
219	C₄²	C₂×D₁₅	1	1	yes	characteristic
220	C₃×C₅	#556	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
222	C₂×D₃	—	10	2	no	—
223	C₂×D₃	—	10	2	no	—
224	C₂×D₃	—	10	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	#709	1	2	yes	—
230	Dic₃	—	5	2	no	—
231	Dic₃	—	10	2	no	—
232	Dic₃	—	10	1	no	—
233	Dic₃	—	10	1	no	—
234	Dic₃	—	10	1	no	—
235	C₂×C₅	C₂×#235	1	1	yes	characteristic
236	C₂×C₅	#1007	1	1	yes	characteristic
237	C₂×C₅	D₃×#41	1	1	yes	characteristic
238	D₅	—	12	2	no	—
239	C₂³	—	15	2	no	—
240	Q₈	—	2	2	no	—
241	Q₈	—	30	2	no	—
242	Q₈	—	30	2	no	—
243	D₄	—	30	2	no	—
244	C₂×C₄	—	15	1	no	—
245	C₂×C₄	—	15	1	no	—
246	C₂×C₄	—	15	1	no	—
247	C₂×C₄	—	15	2	no	—
248	C₂×C₄	—	15	1	no	—
249	C₂×C₄	—	30	4	no	—
250	C₂×C₄	C₂²×D₁₅	1	2	yes	—
251	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₁₅	1	2	yes	—
253	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
255	C₂×C₃	C₂×#709	1	1	yes	characteristic
256	C₂×C₃	#4147	1	1	yes	characteristic
257	C₂×C₃	D₅×#41	1	1	yes	characteristic
258	D₃	—	20	2	no	—
259	C₅	#5568	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	30	2	no	—
261	C₂²	—	30	2	no	—
262	C₂²	—	30	2	no	—
263	C₂²	C₂³×D₁₅	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	15	2	no	—
269	C₄	—	30	2	no	—
270	C₄	—	30	1	no	—
271	C₄	—	30	1	no	—
272	C₄	—	30	1	no	—
273	C₄	#6817	1	2	yes	—
274	C₃	#65902	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	60	2	no	—
276	C₂	C₂×#6817	1	1	yes	characteristic
277	C₂	#92098	1	1	yes	characteristic
278	C₂	#41×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₄×C₅×Q₈	C₂
prime radical	C₂×D₆₀	C₂²
hypercenter	C₄×Q₈	D₁₅
hypocenter	C₃×C₅	#556
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×Q₈	C₂	semidirect
3. #91833	C₂	semidirect
4. #91787	C₂	semidirect
5. #91785	C₂	semidirect
7. #91776	C₂	semidirect
8. #91765	C₂	semidirect
10. #91761	C₂	semidirect
11. C₄×Dic₃₀	C₂	semidirect
37. C₄×C₅×Q₈	D₃	semidirect
64. C₃×C₄×Q₈	D₅	semidirect
160. C₄×Q₈	D₁₅	semidirect
220. C₃×C₅	#556	semidirect
259. C₅	#5568	semidirect
274. C₃	#65902	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×Q₈	2
#91836	2
#91833	2
#91787	2
#91785	2
#91782	2
#91776	2
#91765	2
#91764	2
#91761	2
C₄×Dic₃₀	2
#65902	3
#5568	5