Subgroups — #11754608

Order 960.

476 conjugacy classes of subgroups.

The subgroup lattice has 316 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	C₅×#930	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#91571	C₂	1	2	yes	maximal, complemented
5	#67⋊Dic₅	C₂	1	2	yes	maximal, complemented
6	#91457	C₂	1	2	yes	maximal, complemented
7	#91449	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#91388	C₂	1	1	yes	maximal, characteristic, complemented
10	#91358	C₂	1	1	yes	maximal, characteristic, complemented
11	#91342	C₂	1	2	yes	maximal, complemented
12	#65952	—	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	2	yes	—
16	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₂²⋊D₅	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
19	C₂×#1201	C₂²	1	2	yes	—
20	C₂×D₃×Dic₅	C₂²	1	2	yes	—
21	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₁₅	C₂²	1	2	yes	—
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₃×#685	C₂²	1	1	yes	characteristic
28	C₃×#683	C₂²	1	1	yes	characteristic
29	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
30	Dic₃⋊Dic₅	C₂²	1	2	yes	—
31	#6669	C₂²	1	2	yes	—
32	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
33	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
34	#6665	C₂²	1	2	yes	—
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	#5565	—	5	1	no	maximal, Hall
37	C₅×#121	—	3	1	no	—
38	#4085	—	3	2	no	—
39	#4084	—	3	1	no	—
40	D₄×Dic₅	—	3	2	no	—
41	D₄×Dic₅	—	3	1	no	—
42	C₄×C₂²⋊D₅	D₃	1	1	yes	characteristic, complemented
43	C₂×C₄⋊Dic₅	—	3	1	no	—
44	#4038	—	3	1	no	—
45	#4036	—	3	1	no	—
46	#4032	—	3	2	no	—
47	#4025	—	3	2	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂²×C₅×D₃	C₂³	1	2	yes	—
50	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₅	—	4	2	no	—
55	C₂×Dic₁₅	C₂³	1	2	yes	—
56	C₂×Dic₁₅	C₂³	1	2	yes	—
57	C₅×#67	—	2	4	no	—
58	C₂×C₅×Dic₃	C₂³	1	2	yes	—
59	C₅×D₁₂	—	2	2	no	—
60	C₃×C₂²⋊D₅	—	2	2	no	—
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
64	C₃×C₄×D₅	—	2	2	no	—
65	#1201	—	4	2	no	—
66	D₃×Dic₅	—	4	2	no	—
67	C₃×C₄×D₄	—	5	1	no	—
68	#939	—	5	2	no	—
69	#930	—	5	1	no	—
70	#930	D₅	1	1	yes	characteristic, complemented
71	#926	—	5	1	no	—
72	C₂×C₄⋊Dic₃	—	5	1	no	—
73	#898	—	5	2	no	—
74	#883	—	5	2	no	—
75	#67⋊C₄	—	5	2	no	—
76	C₄×D₁₂	—	5	1	no	—
77	#870	—	5	1	no	—
78	C₂×C₅×D₄	—	3	2	no	—
79	C₂×C₅×D₄	—	3	1	no	—
80	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
81	C₂×C₂²⋊D₅	—	3	2	no	—
82	C₂×C₂²⋊D₅	C₂×D₃	1	1	yes	characteristic
83	C₂²×Dic₅	—	3	2	no	—
84	C₂²×Dic₅	—	3	2	no	—
85	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	3	1	no	—
87	C₅×C₂²⋊C₄	—	3	2	no	—
88	C₂²⋊Dic₅	—	3	2	no	—
89	C₂²⋊Dic₅	—	3	2	no	—
90	C₂²⋊Dic₅	C₂×D₃	1	1	yes	characteristic
91	#685	—	3	2	no	—
92	#685	C₂×D₃	1	1	yes	characteristic
93	C₄⋊Dic₅	—	3	2	no	—
94	C₄⋊Dic₅	—	3	1	no	—
95	C₄⋊Dic₅	—	3	1	no	—
96	C₄⋊Dic₅	—	3	2	no	—
97	#683	—	3	2	no	—
98	#683	C₂×D₃	1	1	yes	characteristic
99	C₄×Dic₅	—	3	2	no	—
100	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
101	#553	—	15	1	no	Sylow
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	—	2	1	no	—
104	C₂²×C₃×C₅	#41	1	2	yes	—
105	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
106	C₂×C₅×D₃	—	2	2	no	—
107	C₂×C₅×D₃	—	2	2	no	—
108	C₂×C₅×D₃	—	2	2	no	—
109	C₂×C₃×D₅	—	2	1	no	—
110	C₂×C₃×D₅	—	2	1	no	—
111	C₂×C₃×D₅	—	2	1	no	—
112	C₃×C₄×C₅	—	2	1	no	—
113	C₃×C₄×C₅	#41	1	2	yes	—
114	Dic₁₅	—	2	2	no	—
115	Dic₁₅	—	2	2	no	—
116	C₃×Dic₅	—	2	1	no	—
117	C₃×Dic₅	—	2	1	no	—
118	C₃×Dic₅	—	2	1	no	—
119	C₅×Dic₃	—	2	2	no	—
120	C₂×C₃×D₄	—	5	1	no	—
121	C₂²×C₃×C₄	—	5	1	no	—
122	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
123	C₂×#67	—	5	2	no	—
124	C₂×#67	C₂×D₅	1	2	yes	—
125	C₂²×Dic₃	—	5	2	no	—
126	C₂×D₁₂	C₂×D₅	1	1	yes	characteristic
127	C₂×C₄×D₃	—	5	2	no	—
128	C₃×C₄⋊C₄	—	5	1	no	—
129	C₃×C₂²⋊C₄	—	5	1	no	—
130	C₃×C₂²⋊C₄	—	5	1	no	—
131	C₃×C₄²	—	5	1	no	—
132	#217	—	5	2	no	—
133	(C₂×D₃)⋊C₄	—	5	2	no	—
134	(C₂×D₃)⋊C₄	—	5	2	no	—
135	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	—
136	C₄⋊Dic₃	—	5	1	no	—
137	C₄⋊Dic₃	—	5	1	no	—
138	C₄⋊Dic₃	—	5	2	no	—
139	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
140	Dic₃⋊C₄	—	5	2	no	—
141	Dic₃⋊C₄	—	5	2	no	—
142	C₄×Dic₃	—	5	2	no	—
143	C₂³×C₅	—	3	2	no	—
144	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
145	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
146	C₅×D₄	—	6	4	no	—
147	C₅×D₄	—	6	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	3	2	no	—
150	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
152	C₂²⋊D₅	—	2	2	no	—
153	C₂²⋊D₅	—	12	2	no	—
154	C₂×Dic₅	—	3	2	no	—
155	C₂×Dic₅	—	3	2	no	—
156	C₂×Dic₅	—	12	2	no	—
157	C₂×Dic₅	—	12	2	no	—
158	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
159	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
160	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
161	C₄×D₅	—	2	2	no	—
162	#126	—	15	2	no	—
163	#125	—	15	1	no	—
164	#123	—	15	2	no	—
165	#121	—	3	1	no	—
166	#121	—	15	1	no	—
167	#121	—	15	2	no	—
168	C₄×D₄	—	5	1	no	—
169	C₄×D₄	—	15	1	no	—
170	C₄×D₄	—	15	2	no	—
171	#117	—	15	1	no	—
172	C₂×C₄⋊C₄	—	15	1	no	—
173	C₂×C₃×C₅	—	2	2	no	—
174	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
175	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
176	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
177	C₃×D₅	—	4	1	no	—
178	C₅×D₃	—	4	2	no	—
179	C₂³×C₃	—	5	1	no	—
180	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
181	C₂²×D₃	C₂²×D₅	1	2	yes	—
182	C₃×D₄	—	10	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	5	1	no	—
185	C₂×C₃×C₄	—	5	1	no	—
186	C₂×C₃×C₄	—	5	1	no	—
187	C₂×C₃×C₄	—	10	2	no	—
188	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
190	#67	—	2	4	no	—
191	#67	—	20	2	no	—
192	C₂×Dic₃	—	5	2	no	—
193	C₂×Dic₃	—	5	2	no	—
194	C₂×Dic₃	—	20	2	no	—
195	C₂×Dic₃	C₂²×D₅	1	2	yes	—
196	D₁₂	—	2	2	no	—
197	C₄×D₃	—	20	2	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	6	2	no	—
201	C₂²×C₅	—	6	2	no	—
202	C₂²×C₅	—	6	2	no	—
203	C₂²×C₅	#235	1	2	yes	—
204	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₂×D₅	—	2	1	no	—
206	C₂×D₅	—	2	1	no	—
207	C₂×D₅	—	2	1	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	6	2	no	—
210	C₄×C₅	#237	1	2	yes	—
211	Dic₅	—	2	1	no	—
212	Dic₅	—	2	1	no	—
213	Dic₅	—	2	1	no	—
214	Dic₅	—	6	2	no	—
215	Dic₅	—	6	2	no	—
216	C₂×D₄	—	3	2	no	—
217	C₂×D₄	—	3	1	no	—
218	C₂×D₄	—	5	1	no	—
219	C₂×D₄	—	15	2	no	—
220	C₂²×C₄	—	5	1	no	—
221	C₂²×C₄	—	15	2	no	—
222	C₂²×C₄	—	15	2	no	—
223	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	5	1	no	—
226	C₄⋊C₄	—	15	2	no	—
227	C₄⋊C₄	—	15	2	no	—
228	C₄⋊C₄	—	15	1	no	—
229	C₄⋊C₄	—	15	1	no	—
230	C₄⋊C₄	—	15	2	no	—
231	C₂²⋊C₄	—	3	2	no	—
232	C₂²⋊C₄	—	5	1	no	—
233	C₂²⋊C₄	—	5	1	no	—
234	C₂²⋊C₄	—	15	2	no	—
235	C₂²⋊C₄	—	15	2	no	—
236	C₂²⋊C₄	—	15	2	no	—
237	C₄²	—	5	1	no	—
238	C₄²	—	15	2	no	—
239	C₃×C₅	#553	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	10	1	no	—
243	C₂²×C₃	—	10	1	no	—
244	C₂²×C₃	—	10	1	no	—
245	C₂²×C₃	#711	1	2	yes	—
246	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
247	C₂×D₃	—	2	2	no	—
248	C₂×D₃	—	2	2	no	—
249	C₂×D₃	—	2	2	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	10	1	no	—
252	C₃×C₄	—	10	1	no	—
253	C₃×C₄	—	10	1	no	—
254	C₃×C₄	#711	1	2	yes	—
255	Dic₃	—	2	2	no	—
256	Dic₃	—	10	2	no	—
257	Dic₃	—	10	2	no	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	12	2	no	—
260	C₂×C₅	C₂×#235	1	1	yes	characteristic
261	C₂×C₅	C₂×#237	1	1	yes	characteristic
262	C₂×C₅	#1009	1	1	yes	characteristic
263	D₅	—	4	1	no	—
264	C₂³	—	3	2	no	—
265	C₂³	—	5	1	no	—
266	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
267	D₄	—	6	4	no	—
268	D₄	—	6	2	no	—
269	D₄	—	10	2	no	—
270	D₄	—	60	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	3	2	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	5	1	no	—
276	C₂×C₄	—	10	2	no	—
277	C₂×C₄	—	15	2	no	—
278	C₂×C₄	—	15	2	no	—
279	C₂×C₄	—	60	2	no	—
280	C₂×C₄	—	60	2	no	—
281	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
282	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
283	C₂×C₃	—	2	2	no	—
284	C₂×C₃	—	20	1	no	—
285	C₂×C₃	C₂×#711	1	1	yes	characteristic
286	C₂×C₃	C₂×#711	1	1	yes	characteristic
287	C₂×C₃	#4149	1	1	yes	characteristic
288	D₃	—	4	2	no	—
289	C₅	#5565	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	6	2	no	—
293	C₂²	—	6	2	no	—
294	C₂²	—	6	2	no	—
295	C₂²	—	10	1	no	—
296	C₂²	—	10	1	no	—
297	C₂²	—	10	1	no	—
298	C₂²	#6772	1	2	yes	—
299	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	6	2	no	—
302	C₄	—	10	1	no	—
303	C₄	—	10	1	no	—
304	C₄	—	10	1	no	—
305	C₄	—	30	2	no	—
306	C₄	—	30	2	no	—
307	C₄	#6780	1	2	yes	—
308	C₃	#65952	1	1	yes	characteristic, complemented, Sylow
309	C₂	—	2	2	no	—
310	C₂	—	12	2	no	—
311	C₂	—	20	1	no	—
312	C₂	C₂×#6772	1	1	yes	characteristic
313	C₂	#92019	1	1	yes	characteristic
314	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
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	#553
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
2. C₅×#930	C₂	semidirect
3. C₃×C₄×C₂²⋊D₅	C₂	semidirect
4. #91571	C₂	semidirect
5. #67⋊Dic₅	C₂	semidirect
6. #91457	C₂	semidirect
7. #91449	C₂	semidirect
8. Dic₅×D₁₂	C₂	semidirect
9. #91388	C₂	semidirect
10. #91358	C₂	semidirect
11. #91342	C₂	semidirect
42. C₄×C₂²⋊D₅	D₃	semidirect
70. #930	D₅	semidirect
239. C₃×C₅	#553	semidirect
289. C₅	#5565	semidirect
308. C₃	#65952	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
C₅×#930	2
C₃×C₄×C₂²⋊D₅	2
#91571	2
#67⋊Dic₅	2
#91457	2
#91449	2
Dic₅×D₁₂	2
#91388	2
#91358	2
#91342	2
#65952	3
#5565	5