Subgroups — #11753331

Order 960.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91777	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#4036	C₂	1	1	yes	maximal, characteristic, complemented
4	#91468	C₂	1	1	yes	maximal, characteristic, complemented
5	#91384	C₂	1	1	yes	maximal, characteristic, complemented
6	#91371	C₂	1	1	yes	maximal, characteristic
7	#91365	C₂	1	2	yes	maximal
8	#91352	C₂	1	2	yes	maximal
9	#91346	C₂	1	2	yes	maximal, complemented
10	#91338	C₂	1	1	yes	maximal, characteristic
11	#91328	C₂	1	2	yes	maximal, complemented
12	#65834	—	3	1	no	maximal, Hall
13	C₂×Dic₃₀	C₂²	1	2	yes	—
14	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂×#1204	C₂²	1	2	yes	—
17	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
19	#6712	C₂²	1	2	yes	—
20	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
22	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
25	C₃×#683	C₂²	1	2	yes	—
26	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
27	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic
28	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic
29	Dic₃⋊Dic₅	C₂²	1	2	yes	—
30	#6668	C₂²	1	2	yes	—
31	#6668	C₂²	1	2	yes	—
32	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
33	(C₂×D₃)⋊Dic₅	C₂²	1	1	yes	characteristic
34	(C₂×D₃)⋊Dic₅	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	#5689	—	5	1	no	maximal, Hall
37	C₂×C₅×C₄⋊C₄	—	3	1	no	—
38	#4072	—	3	1	no	—
39	#4072	—	3	1	no	—
40	#4070	—	3	2	no	—
41	#4070	—	3	2	no	—
42	#4036	—	3	2	no	—
43	#4036	—	3	1	no	—
44	#4036	D₃	1	1	yes	characteristic, complemented
45	#4034	—	3	1	no	—
46	#4034	—	3	2	no	—
47	#4034	—	3	1	no	—
48	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₅	C₂³	1	2	yes	—
52	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
54	Dic₃₀	—	4	2	no	—
55	C₂×C₅×Dic₃	C₂³	1	2	yes	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₄×C₅×D₃	—	4	2	no	—
58	C₄×C₅×D₃	Q₈	1	4	yes	complemented
59	C₂×C₃×Dic₅	C₂³	1	2	yes	—
60	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	#1204	—	4	2	no	—
63	C₃×#125	—	5	1	no	—
64	#897	—	5	2	no	—
65	#896	—	5	2	no	—
66	#892	—	5	1	no	—
67	D₃×C₄⋊C₄	D₅	1	1	yes	characteristic
68	#890	—	5	1	no	—
69	#889	—	5	2	no	—
70	#888	—	5	1	no	—
71	#888	—	5	2	no	—
72	#872	—	5	1	no	—
73	#869	—	5	1	no	—
74	C₂²×C₄×C₅	—	3	1	no	—
75	C₂²×C₄×C₅	—	3	2	no	—
76	C₂×Dic₁₀	—	3	2	no	—
77	C₂×Dic₁₀	—	3	2	no	—
78	C₅×C₄⋊C₄	—	3	2	no	—
79	C₅×C₄⋊C₄	—	3	1	no	—
80	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
81	C₂²⋊Dic₅	—	3	1	no	—
82	C₂²⋊Dic₅	—	3	2	no	—
83	C₂²⋊Dic₅	—	3	1	no	—
84	C₄⋊Dic₅	—	3	1	no	—
85	C₄⋊Dic₅	—	3	1	no	—
86	C₄⋊Dic₅	—	3	2	no	—
87	C₄⋊Dic₅	—	3	1	no	—
88	C₄⋊Dic₅	C₂×D₃	1	2	yes	—
89	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
90	#683	—	3	2	no	—
91	#683	—	3	2	no	—
92	#683	—	3	2	no	—
93	#683	C₂×D₃	1	2	yes	—
94	C₄×Dic₅	—	3	1	no	—
95	C₄×Dic₅	—	3	2	no	—
96	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
97	#544	—	15	1	no	Sylow
98	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×C₅×D₃	—	2	1	no	—
100	C₂×C₅×D₃	—	2	1	no	—
101	C₂×C₅×D₃	C₂×Q₈	1	2	yes	—
102	C₃×C₄×C₅	—	2	2	no	—
103	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
104	Dic₁₅	—	2	2	no	—
105	Dic₁₅	—	2	1	no	—
106	Dic₁₅	—	2	1	no	—
107	C₃×Dic₅	—	2	1	no	—
108	C₃×Dic₅	—	2	2	no	—
109	C₃×Dic₅	—	2	1	no	—
110	C₅×Dic₃	—	2	2	no	—
111	C₅×Dic₃	C₂×Q₈	1	2	yes	—
112	C₂×C₄×D₃	C₂×D₅	1	2	yes	—
113	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
114	C₂×Dic₆	—	5	2	no	—
115	C₂×Dic₆	—	5	2	no	—
116	C₃×C₄⋊C₄	—	5	2	no	—
117	C₃×C₄⋊C₄	—	5	2	no	—
118	C₃×C₄⋊C₄	—	5	1	no	—
119	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
120	C₃×C₄²	—	5	1	no	—
121	(C₂×D₃)⋊C₄	—	5	1	no	—
122	(C₂×D₃)⋊C₄	—	5	1	no	—
123	(C₂×D₃)⋊C₄	—	5	2	no	—
124	C₄⋊Dic₃	—	5	1	no	—
125	C₄⋊Dic₃	—	5	2	no	—
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	1	no	—
130	Dic₃⋊C₄	—	5	1	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	2	no	—
135	C₂³×C₅	—	3	1	no	—
136	C₂×C₄×C₅	—	3	1	no	—
137	C₂×C₄×C₅	—	3	4	no	—
138	C₂×C₄×C₅	—	3	2	no	—
139	C₂×C₄×C₅	—	12	2	no	—
140	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
141	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂×Dic₅	—	3	1	no	—
143	C₂×Dic₅	—	3	1	no	—
144	C₂×Dic₅	—	3	2	no	—
145	C₂×Dic₅	C₂²×D₃	1	2	yes	—
146	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
147	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
148	Dic₁₀	—	12	2	no	—
149	Dic₁₀	—	12	2	no	—
150	C₄⋊Q₈	—	15	1	no	—
151	C₄⋊Q₈	—	15	2	no	—
152	C₄⋊Q₈	—	15	1	no	—
153	#125	—	5	1	no	—
154	#125	—	15	2	no	—
155	#125	—	15	1	no	—
156	C₂²⋊Q₈	—	15	2	no	—
157	C₂²⋊Q₈	—	15	2	no	—
158	#117	—	15	1	no	—
159	#117	—	15	1	no	—
160	C₂×C₄⋊C₄	—	3	1	no	—
161	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
164	C₅×D₃	—	2	2	no	—
165	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₃×C₄	—	5	2	no	—
167	C₂×C₃×C₄	—	5	1	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
170	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
171	C₂×Dic₃	—	5	1	no	—
172	C₂×Dic₃	—	5	2	no	—
173	C₂×Dic₃	—	5	1	no	—
174	C₂×Dic₃	C₂²×D₅	1	2	yes	—
175	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
176	C₄×D₃	—	4	2	no	—
177	C₄×D₃	Dic₁₀	1	4	yes	complemented
178	Dic₆	—	20	2	no	—
179	Dic₆	—	20	2	no	—
180	C₂²×C₅	—	3	2	no	—
181	C₂²×C₅	—	6	1	no	—
182	C₂²×C₅	—	6	1	no	—
183	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	3	2	no	—
186	C₄×C₅	—	6	2	no	—
187	C₄×C₅	D₃×Q₈	1	2	yes	—
188	Dic₅	—	2	1	no	—
189	Dic₅	—	2	2	no	—
190	Dic₅	—	2	1	no	—
191	Dic₅	—	6	2	no	—
192	Dic₅	—	6	1	no	—
193	Dic₅	—	6	1	no	—
194	C₂×Q₈	—	15	2	no	—
195	C₂×Q₈	—	15	2	no	—
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	3	2	no	—
198	C₄⋊C₄	—	3	2	no	—
199	C₄⋊C₄	—	3	1	no	—
200	C₄⋊C₄	—	5	1	no	—
201	C₄⋊C₄	—	5	2	no	—
202	C₄⋊C₄	—	5	2	no	—
203	C₄⋊C₄	—	15	1	no	—
204	C₄⋊C₄	—	15	1	no	—
205	C₄⋊C₄	—	15	2	no	—
206	C₄⋊C₄	—	15	2	no	—
207	C₄⋊C₄	—	15	2	no	—
208	C₄⋊C₄	—	15	1	no	—
209	C₄⋊C₄	—	15	2	no	—
210	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
211	C₂²⋊C₄	—	15	1	no	—
212	C₂²⋊C₄	—	15	1	no	—
213	C₂²⋊C₄	—	15	2	no	—
214	C₄²	—	5	1	no	—
215	C₄²	—	15	2	no	—
216	C₄²	—	15	1	no	—
217	C₃×C₅	#544	1	1	yes	characteristic, complemented, Hall
218	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
219	C₂×D₃	—	2	1	no	—
220	C₂×D₃	—	2	1	no	—
221	C₂×D₃	C₂×Dic₁₀	1	2	yes	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	10	1	no	—
224	C₃×C₄	—	10	2	no	—
225	C₃×C₄	—	10	1	no	—
226	C₃×C₄	C₂×Dic₁₀	1	2	yes	—
227	Dic₃	—	2	2	no	—
228	Dic₃	—	10	2	no	—
229	Dic₃	—	10	1	no	—
230	Dic₃	—	10	1	no	—
231	Dic₃	C₂×Dic₁₀	1	2	yes	—
232	C₂×C₅	—	6	2	no	—
233	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
234	C₂×C₅	#1009	1	1	yes	characteristic
235	C₂×C₅	#1010	1	1	yes	characteristic
236	C₂³	—	3	1	no	—
237	Q₈	—	60	2	no	—
238	Q₈	—	60	2	no	—
239	C₂×C₄	—	3	1	no	—
240	C₂×C₄	—	3	4	no	—
241	C₂×C₄	—	3	2	no	—
242	C₂×C₄	—	5	1	no	—
243	C₂×C₄	—	5	1	no	—
244	C₂×C₄	—	5	2	no	—
245	C₂×C₄	—	12	2	no	—
246	C₂×C₄	—	15	1	no	—
247	C₂×C₄	—	15	2	no	—
248	C₂×C₄	—	15	1	no	—
249	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
250	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
251	C₂×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
252	C₂×C₃	#4144	1	1	yes	characteristic
253	C₂×C₃	#4147	1	1	yes	characteristic
254	D₃	—	2	2	no	—
255	C₅	#5689	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	3	2	no	—
257	C₂²	—	6	1	no	—
258	C₂²	—	6	1	no	—
259	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	3	2	no	—
262	C₄	—	6	2	no	—
263	C₄	—	10	1	no	—
264	C₄	—	10	1	no	—
265	C₄	—	10	2	no	—
266	C₄	—	30	1	no	—
267	C₄	—	30	2	no	—
268	C₄	—	30	1	no	—
269	C₄	D₃×Dic₁₀	1	2	yes	—
270	C₃	#65834	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	6	2	no	—
272	C₂	C₂×D₃×Dic₁₀	1	1	yes	characteristic
273	C₂	#92027	1	1	yes	characteristic
274	C₂	#92028	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₄⋊C₄	C₂²
prime radical	C₂²×C₅×D₃	C₂³
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	#544
LCS	G ▷ C₂²×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
1. #91777	C₂	semidirect
3. C₃×#4036	C₂	semidirect
4. #91468	C₂	semidirect
5. #91384	C₂	semidirect
9. #91346	C₂	semidirect
11. #91328	C₂	semidirect
44. #4036	D₃	semidirect
58. C₄×C₅×D₃	Q₈	semidirect
177. C₄×D₃	Dic₁₀	semidirect
217. C₃×C₅	#544	semidirect
255. C₅	#5689	semidirect
270. C₃	#65834	semidirect

Maximal subgroups

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

subgroup	index
#91777	2
C₅×D₃×C₄⋊C₄	2
C₃×#4036	2
#91468	2
#91384	2
#91371	2
#91365	2
#91352	2
#91346	2
#91338	2
#91328	2
#65834	3
#5689	5