Subgroups — #11754514

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₁₅	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	#91576	C₂	1	2	yes	maximal, complemented
5	#67⋊Dic₅	C₂	1	2	yes	maximal
6	#91469	C₂	1	1	yes	maximal, characteristic, complemented
7	D₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#91417	C₂	1	2	yes	maximal, complemented
9	#91369	C₂	1	1	yes	maximal, characteristic, complemented
10	#91343	C₂	1	2	yes	maximal, complemented
11	Dic₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65877	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅	C₂²	1	2	yes	—
15	C₂×C₅×#67	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₂×#1204	C₂²	1	2	yes	—
19	C₂×D₃×Dic₅	C₂²	1	2	yes	—
20	C₄⋊Dic₁₅	C₂²	1	2	yes	—
21	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
23	C₅×#217	C₂²	1	1	yes	characteristic
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
26	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
27	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
28	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
29	C₃×#683	C₂²	1	2	yes	—
30	Dic₃⋊Dic₅	C₂²	1	2	yes	—
31	Dic₃⋊Dic₅	C₂²	1	2	yes	—
32	#6669	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	#5635	—	5	1	no	maximal, Hall
37	C₄×C₅×D₄	—	3	1	no	—
38	D₄×Dic₅	—	3	2	no	—
39	C₂×C₄⋊Dic₅	—	3	1	no	—
40	C₂×C₄⋊Dic₅	—	3	1	no	—
41	#4070	—	3	1	no	—
42	#4070	D₃	1	1	yes	characteristic, complemented
43	#4036	—	3	2	no	—
44	#4024	—	3	2	no	—
45	#4024	—	3	2	no	—
46	#4015	—	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	—	4	2	no	—
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	C₂×Dic₁₅	C₂³	1	2	yes	—
56	C₅×#67	Q₈	1	4	yes	complemented
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₄×C₅×D₃	—	2	2	no	—
61	C₂×C₃×Dic₅	C₂³	1	2	yes	—
62	C₂×C₃×Dic₅	C₂³	1	2	yes	—
63	C₃×Dic₁₀	—	2	2	no	—
64	#1204	—	4	2	no	—
65	D₃×Dic₅	—	4	2	no	—
66	C₃×C₂²⋊Q₈	—	5	1	no	—
67	#946	—	5	1	no	—
68	Q₈×Dic₃	—	5	1	no	—
69	C₄×#67	D₅	1	1	yes	characteristic
70	C₂×C₄⋊Dic₃	—	5	1	no	—
71	#897	—	5	2	no	—
72	D₃×C₄⋊C₄	—	5	1	no	—
73	#889	—	5	2	no	—
74	#888	—	5	1	no	—
75	#67⋊C₄	—	5	2	no	—
76	#878	—	5	2	no	—
77	C₂×C₅×D₄	—	3	1	no	—
78	C₂²×C₄×C₅	—	3	1	no	—
79	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
80	C₂²×Dic₅	—	3	2	no	—
81	C₂²×Dic₅	—	3	2	no	—
82	C₂×Dic₁₀	—	3	2	no	—
83	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic
84	C₅×C₄⋊C₄	—	3	1	no	—
85	C₅×C₂²⋊C₄	—	3	1	no	—
86	C₅×C₂²⋊C₄	—	3	1	no	—
87	C₄²×C₅	—	3	1	no	—
88	C₂²⋊Dic₅	—	3	2	no	—
89	C₂²⋊Dic₅	C₂×D₃	1	2	yes	—
90	C₄⋊Dic₅	—	3	1	no	—
91	C₄⋊Dic₅	—	3	1	no	—
92	C₄⋊Dic₅	—	3	2	no	—
93	C₄⋊Dic₅	—	3	2	no	—
94	C₄⋊Dic₅	—	3	2	no	—
95	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
96	#683	—	3	2	no	—
97	#683	—	3	2	no	—
98	#683	C₂×D₃	1	2	yes	—
99	C₄×Dic₅	—	3	2	no	—
100	#554	—	15	1	no	Sylow
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	—
104	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
105	C₂×C₅×D₃	—	2	1	no	—
106	C₂×C₅×D₃	—	2	1	no	—
107	C₂×C₅×D₃	C₂×Q₈	1	2	yes	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	#41	1	2	yes	—
110	Dic₁₅	—	2	2	no	—
111	Dic₁₅	—	2	2	no	—
112	C₃×Dic₅	—	2	2	no	—
113	C₃×Dic₅	—	2	2	no	—
114	C₅×Dic₃	—	2	1	no	—
115	C₅×Dic₃	—	2	1	no	—
116	C₅×Dic₃	C₂×Q₈	1	2	yes	—
117	C₂×C₃×Q₈	—	5	1	no	—
118	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
119	C₂×#67	C₂×D₅	1	1	yes	characteristic
120	C₂²×Dic₃	—	5	2	no	—
121	C₂×C₄×D₃	—	5	2	no	—
122	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
123	C₂×Dic₆	—	5	2	no	—
124	C₃×C₄⋊C₄	—	5	2	no	—
125	C₃×C₄⋊C₄	—	5	1	no	—
126	C₃×C₂²⋊C₄	—	5	2	no	—
127	#217	C₂×D₅	1	1	yes	characteristic
128	(C₂×D₃)⋊C₄	—	5	2	no	—
129	(C₂×D₃)⋊C₄	C₂×D₅	1	1	yes	characteristic
130	C₄⋊Dic₃	—	5	1	no	—
131	C₄⋊Dic₃	—	5	2	no	—
132	C₄⋊Dic₃	—	5	1	no	—
133	C₄⋊Dic₃	—	5	2	no	—
134	Dic₃⋊C₄	—	5	2	no	—
135	Dic₃⋊C₄	—	5	2	no	—
136	Dic₃⋊C₄	—	5	2	no	—
137	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic
138	C₄×Dic₃	—	5	2	no	—
139	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
140	C₂³×C₅	—	3	1	no	—
141	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₅×D₄	—	3	4	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	3	1	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	6	2	no	—
148	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
150	C₂×Dic₅	—	3	2	no	—
151	C₂×Dic₅	—	3	2	no	—
152	C₂×Dic₅	—	12	2	no	—
153	C₂×Dic₅	—	12	2	no	—
154	C₂×Dic₅	C₂²×D₃	1	2	yes	—
155	C₂×Dic₅	C₂²×D₃	1	2	yes	—
156	Dic₁₀	—	2	2	no	—
157	Dic₁₀	—	12	2	no	—
158	C₄⋊Q₈	—	15	1	no	—
159	#125	—	15	2	no	—
160	C₂²⋊Q₈	—	5	1	no	—
161	C₂²⋊Q₈	—	15	2	no	—
162	C₂²⋊Q₈	—	15	2	no	—
163	C₂²⋊Q₈	—	15	1	no	—
164	C₄×Q₈	—	15	1	no	—
165	C₄×D₄	—	3	1	no	—
166	C₄×D₄	—	15	2	no	—
167	C₂×C₄⋊C₄	—	15	1	no	—
168	C₂×C₄⋊C₄	—	15	1	no	—
169	C₂×C₃×C₅	—	2	2	no	—
170	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
172	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
173	C₅×D₃	—	2	2	no	—
174	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
175	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
176	C₃×Q₈	—	10	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	5	2	no	—
179	C₂×C₃×C₄	—	5	2	no	—
180	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	#67	Dic₁₀	1	4	yes	complemented
183	C₂×Dic₃	—	5	2	no	—
184	C₂×Dic₃	—	5	2	no	—
185	C₂×Dic₃	—	20	2	no	—
186	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
187	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
188	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
189	C₄×D₃	—	2	2	no	—
190	C₄×D₃	—	20	2	no	—
191	Dic₆	—	20	2	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	—	2	1	no	—
194	C₂²×C₅	—	3	2	no	—
195	C₂²×C₅	—	6	1	no	—
196	C₂²×C₅	—	6	1	no	—
197	C₂²×C₅	D₃×Q₈	1	2	yes	—
198	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	3	2	no	—
201	C₄×C₅	—	6	1	no	—
202	C₄×C₅	—	6	1	no	—
203	C₄×C₅	#237	1	2	yes	—
204	Dic₅	—	2	2	no	—
205	Dic₅	—	2	2	no	—
206	Dic₅	—	6	2	no	—
207	Dic₅	—	6	2	no	—
208	C₂×Q₈	—	5	1	no	—
209	C₂×Q₈	—	15	2	no	—
210	C₂×D₄	—	3	1	no	—
211	C₂²×C₄	—	3	1	no	—
212	C₂²×C₄	—	15	2	no	—
213	C₂²×C₄	—	15	2	no	—
214	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
215	C₄⋊C₄	—	3	1	no	—
216	C₄⋊C₄	—	5	1	no	—
217	C₄⋊C₄	—	5	2	no	—
218	C₄⋊C₄	—	15	2	no	—
219	C₄⋊C₄	—	15	2	no	—
220	C₄⋊C₄	—	15	2	no	—
221	C₄⋊C₄	—	15	2	no	—
222	C₄⋊C₄	—	15	1	no	—
223	C₄⋊C₄	—	15	2	no	—
224	C₄⋊C₄	—	15	1	no	—
225	C₂²⋊C₄	—	3	1	no	—
226	C₂²⋊C₄	—	3	1	no	—
227	C₂²⋊C₄	—	5	2	no	—
228	C₂²⋊C₄	—	15	2	no	—
229	C₄²	—	3	1	no	—
230	C₄²	—	15	2	no	—
231	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	C₂×Dic₁₀	1	2	yes	—
235	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
236	C₂×D₃	—	2	1	no	—
237	C₂×D₃	—	2	1	no	—
238	C₂×D₃	C₂×Dic₁₀	1	2	yes	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	10	2	no	—
241	C₃×C₄	—	10	2	no	—
242	C₃×C₄	#711	1	2	yes	—
243	Dic₃	—	2	1	no	—
244	Dic₃	—	2	1	no	—
245	Dic₃	—	10	2	no	—
246	Dic₃	—	10	2	no	—
247	Dic₃	C₂×Dic₁₀	1	2	yes	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	—	6	2	no	—
250	C₂×C₅	C₂×#237	1	1	yes	characteristic
251	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
252	C₂×C₅	#1009	1	1	yes	characteristic
253	C₂³	—	3	1	no	—
254	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
255	Q₈	—	10	2	no	—
256	Q₈	—	60	2	no	—
257	D₄	—	3	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	5	2	no	—
264	C₂×C₄	—	6	2	no	—
265	C₂×C₄	—	15	2	no	—
266	C₂×C₄	—	15	2	no	—
267	C₂×C₄	—	60	2	no	—
268	C₂×C₄	—	60	2	no	—
269	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
270	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
273	C₂×C₃	C₂×#711	1	1	yes	characteristic
274	C₂×C₃	#4149	1	1	yes	characteristic
275	D₃	—	2	2	no	—
276	C₅	#5635	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	3	2	no	—
280	C₂²	—	6	1	no	—
281	C₂²	—	6	1	no	—
282	C₂²	D₃×Dic₁₀	1	2	yes	—
283	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	3	2	no	—
286	C₄	—	6	1	no	—
287	C₄	—	6	1	no	—
288	C₄	—	10	2	no	—
289	C₄	—	10	2	no	—
290	C₄	—	30	2	no	—
291	C₄	—	30	2	no	—
292	C₄	#6780	1	2	yes	—
293	C₃	#65877	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	2	2	no	—
295	C₂	—	6	2	no	—
296	C₂	C₂×D₃×Dic₁₀	1	1	yes	characteristic
297	C₂	#92019	1	1	yes	characteristic
298	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	#554
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. C₂×C₄⋊Dic₁₅	C₂	semidirect
3. C₃×#4070	C₂	semidirect
4. #91576	C₂	semidirect
6. #91469	C₂	semidirect
7. D₃×C₄⋊Dic₅	C₂	semidirect
8. #91417	C₂	semidirect
9. #91369	C₂	semidirect
10. #91343	C₂	semidirect
11. Dic₃×Dic₁₀	C₂	semidirect
42. #4070	D₃	semidirect
56. C₅×#67	Q₈	semidirect
182. #67	Dic₁₀	semidirect
231. C₃×C₅	#554	semidirect
276. C₅	#5635	semidirect
293. C₃	#65877	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₁₅	2
C₄×C₅×#67	2
C₃×#4070	2
#91576	2
#67⋊Dic₅	2
#91469	2
D₃×C₄⋊Dic₅	2
#91417	2
#91369	2
#91343	2
Dic₃×Dic₁₀	2
#65877	3
#5635	5