Subgroups — #11753668

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6771	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#892	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#1202	C₂	1	2	yes	maximal, complemented
6	#91440	C₂	1	2	yes	maximal, complemented
7	D₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#91406	C₂	1	2	yes	maximal, complemented
9	#91358	C₂	1	1	yes	maximal, characteristic, complemented
10	#91347	C₂	1	1	yes	maximal, characteristic, complemented
11	#91324	C₂	1	2	yes	maximal, complemented
12	#65857	—	3	1	no	maximal, Hall
13	C₂×C₄×D₁₅	C₂²	1	2	yes	—
14	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
17	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
18	C₂×#1204	C₂²	1	1	yes	characteristic
19	C₂×#1203	C₂²	1	1	yes	characteristic
20	C₂×#1202	C₂²	1	1	yes	characteristic
21	C₂×#1201	C₂²	1	1	yes	characteristic
22	C₂×D₃×Dic₅	C₂²	1	2	yes	—
23	#6771	C₄	1	8	yes	complemented
24	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
26	#6712	C₂²	1	2	yes	—
27	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₄×C₅×Dic₃	C₂²	1	2	yes	—
30	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
31	C₃×#683	C₂²	1	2	yes	—
32	Dic₃⋊Dic₅	C₂²	1	2	yes	—
33	#6667	C₂²	1	2	yes	—
34	#6665	C₂²	1	2	yes	—
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	Dic₃×Dic₅	C₂²	1	2	yes	—
37	#5504	—	5	1	no	maximal, Hall
38	C₂×#709	—	3	1	no	—
39	C₅×#117	—	3	1	no	—
40	C₂×C₄⋊Dic₅	—	3	1	no	—
41	#4038	—	3	1	no	—
42	#4038	—	3	1	no	—
43	D₅×C₄⋊C₄	—	3	1	no	—
44	D₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
45	#4027	—	3	2	no	—
46	#4027	—	3	2	no	—
47	C₄×D₂₀	—	3	2	no	—
48	C₄×Dic₁₀	—	3	2	no	—
49	C₂²×D₁₅	C₂³	1	1	yes	characteristic
50	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₅	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×Dic₁₅	C₂³	1	2	yes	—
55	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
56	C₄×D₁₅	—	4	2	no	—
57	C₄×D₁₅	C₂×C₄	1	4	yes	complemented
58	C₂×C₅×Dic₃	C₂³	1	2	yes	—
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₄×C₅×D₃	C₂×C₄	1	4	yes	complemented
61	C₂×C₃×Dic₅	C₂³	1	2	yes	—
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₃×C₄×D₅	—	4	2	no	—
64	C₃×C₄×D₅	C₂×C₄	1	4	yes	complemented
65	#1204	C₂×C₄	1	4	yes	complemented
66	#1203	C₂×C₄	1	4	yes	complemented
67	#1202	C₂×C₄	1	4	yes	complemented
68	#1201	C₂×C₄	1	4	yes	complemented
69	D₃×Dic₅	—	4	2	no	—
70	C₂×#235	—	5	1	no	—
71	C₂×C₃×C₄⋊C₄	—	5	1	no	—
72	C₄×#67	—	5	2	no	—
73	C₄×#67	—	5	2	no	—
74	#926	—	5	1	no	—
75	#893	—	5	2	no	—
76	#892	—	5	1	no	—
77	#892	D₅	1	1	yes	characteristic, complemented
78	D₃×C₄⋊C₄	—	5	1	no	—
79	D₃×C₄⋊C₄	—	5	1	no	—
80	#887	—	5	2	no	—
81	C₂²×C₄×C₅	—	3	1	no	—
82	C₂×C₂²⋊D₅	—	3	1	no	—
83	C₂×C₂²⋊D₅	—	3	1	no	—
84	C₂²×Dic₅	—	3	2	no	—
85	#709	—	3	8	no	—
86	C₂×D₂₀	—	3	1	no	—
87	C₂×C₄×D₅	—	3	1	no	—
88	C₂×C₄×D₅	—	3	2	no	—
89	C₂×C₄×D₅	C₂×D₃	1	2	yes	—
90	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
91	C₂×Dic₁₀	—	3	1	no	—
92	C₅×C₄⋊C₄	—	3	1	no	—
93	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
94	C₅×C₂²⋊C₄	—	3	2	no	—
95	C₄²×C₅	—	3	2	no	—
96	#685	—	3	2	no	—
97	#685	—	3	2	no	—
98	C₄⋊Dic₅	—	3	1	no	—
99	C₄⋊Dic₅	—	3	2	no	—
100	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
101	#683	—	3	2	no	—
102	#683	C₂×D₃	1	2	yes	—
103	C₄×Dic₅	—	3	2	no	—
104	C₄×Dic₅	—	3	2	no	—
105	#520	—	15	1	no	Sylow
106	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
107	C₂×D₁₅	—	2	1	no	—
108	C₂×D₁₅	—	2	1	no	—
109	C₂×D₁₅	C₂²×C₄	1	2	yes	—
110	C₂×C₅×D₃	—	2	1	no	—
111	C₂×C₅×D₃	—	2	1	no	—
112	C₂×C₅×D₃	C₂²×C₄	1	2	yes	—
113	C₂×C₃×D₅	—	2	1	no	—
114	C₂×C₃×D₅	—	2	1	no	—
115	C₂×C₃×D₅	C₂²×C₄	1	2	yes	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
118	Dic₁₅	—	2	2	no	—
119	Dic₁₅	C₂²×C₄	1	2	yes	—
120	C₃×Dic₅	—	2	2	no	—
121	C₃×Dic₅	C₂²×C₄	1	2	yes	—
122	C₅×Dic₃	—	2	2	no	—
123	C₅×Dic₃	C₂²×C₄	1	2	yes	—
124	C₂²×C₃×C₄	—	5	1	no	—
125	C₂²×C₃×C₄	—	5	2	no	—
126	C₂×#67	—	5	1	no	—
127	C₂×#67	—	5	1	no	—
128	#235	—	5	8	no	—
129	C₂×D₁₂	—	5	1	no	—
130	C₂×C₄×D₃	—	5	1	no	—
131	C₂×C₄×D₃	—	5	2	no	—
132	C₂×C₄×D₃	—	5	2	no	—
133	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
134	C₂×Dic₆	—	5	1	no	—
135	C₃×C₄⋊C₄	—	5	2	no	—
136	C₃×C₄⋊C₄	—	5	1	no	—
137	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
138	#217	—	5	2	no	—
139	(C₂×D₃)⋊C₄	—	5	2	no	—
140	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	—
141	C₄⋊Dic₃	—	5	1	no	—
142	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
143	Dic₃⋊C₄	—	5	2	no	—
144	Dic₃⋊C₄	—	5	2	no	—
145	C₄×Dic₃	—	5	2	no	—
146	C₄×Dic₃	—	5	2	no	—
147	C₄×Dic₃	C₂×D₅	1	2	yes	—
148	C₂³×C₅	—	3	1	no	—
149	C₂²×D₅	—	3	1	no	—
150	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
151	C₂×C₄×C₅	—	3	1	no	—
152	C₂×C₄×C₅	—	3	4	no	—
153	C₂×C₄×C₅	—	3	2	no	—
154	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
155	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
156	C₂²⋊D₅	—	3	4	no	—
157	C₂²⋊D₅	—	3	4	no	—
158	C₂×Dic₅	—	3	1	no	—
159	C₂×Dic₅	—	3	2	no	—
160	C₂×Dic₅	—	12	2	no	—
161	C₂×Dic₅	C₂²×D₃	1	2	yes	—
162	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
163	D₂₀	—	3	4	no	—
164	C₄×D₅	—	3	4	no	—
165	C₄×D₅	—	4	2	no	—
166	C₄×D₅	—	12	2	no	—
167	C₄×D₅	C₄×D₃	1	4	yes	complemented
168	Dic₁₀	—	3	4	no	—
169	C₂×#41	—	15	1	no	—
170	C₄×Q₈	—	15	2	no	—
171	C₄×D₄	—	15	2	no	—
172	C₄×D₄	—	15	2	no	—
173	C₄×D₄	—	15	2	no	—
174	#117	—	3	1	no	—
175	#117	—	15	1	no	—
176	#117	—	15	1	no	—
177	C₂×C₄⋊C₄	—	5	1	no	—
178	C₂×C₄⋊C₄	—	15	1	no	—
179	C₂×C₄⋊C₄	—	15	1	no	—
180	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
182	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
183	D₁₅	—	2	2	no	—
184	C₃×D₅	—	2	2	no	—
185	C₅×D₃	—	2	2	no	—
186	C₂³×C₃	—	5	1	no	—
187	C₂²×D₃	—	5	1	no	—
188	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
189	C₂×C₃×C₄	—	5	1	no	—
190	C₂×C₃×C₄	—	5	4	no	—
191	C₂×C₃×C₄	—	5	2	no	—
192	C₂×C₃×C₄	—	20	2	no	—
193	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
194	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
195	#67	—	5	4	no	—
196	#67	—	5	4	no	—
197	C₂×Dic₃	—	5	1	no	—
198	C₂×Dic₃	—	5	2	no	—
199	C₂×Dic₃	C₂²×D₅	1	2	yes	—
200	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
201	D₁₂	—	5	4	no	—
202	C₄×D₃	—	5	4	no	—
203	C₄×D₃	—	20	2	no	—
204	C₄×D₃	—	20	2	no	—
205	C₄×D₃	C₄×D₅	1	4	yes	complemented
206	Dic₆	—	5	4	no	—
207	C₂²×C₅	—	3	2	no	—
208	C₂²×C₅	—	6	1	no	—
209	C₂²×C₅	—	6	1	no	—
210	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
211	C₂×D₅	—	2	1	no	—
212	C₂×D₅	—	2	1	no	—
213	C₂×D₅	—	3	2	no	—
214	C₂×D₅	—	6	1	no	—
215	C₂×D₅	—	6	1	no	—
216	C₂×D₅	C₂×C₄×D₃	1	2	yes	—
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	3	2	no	—
219	C₄×C₅	—	6	2	no	—
220	C₄×C₅	C₂×C₄×D₃	1	2	yes	—
221	Dic₅	—	2	2	no	—
222	Dic₅	—	3	2	no	—
223	Dic₅	—	6	2	no	—
224	Dic₅	C₂×C₄×D₃	1	2	yes	—
225	#41	—	15	8	no	—
226	C₂×Q₈	—	15	1	no	—
227	C₂×D₄	—	15	1	no	—
228	C₂×D₄	—	15	1	no	—
229	C₂×D₄	—	15	1	no	—
230	C₂²×C₄	—	3	1	no	—
231	C₂²×C₄	—	5	1	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	—	15	2	no	—
234	C₂²×C₄	—	15	1	no	—
235	C₂²×C₄	—	15	2	no	—
236	C₄⋊C₄	—	3	1	no	—
237	C₄⋊C₄	—	5	1	no	—
238	C₄⋊C₄	—	5	2	no	—
239	C₄⋊C₄	—	15	2	no	—
240	C₄⋊C₄	—	15	1	no	—
241	C₄⋊C₄	—	15	2	no	—
242	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
243	C₂²⋊C₄	—	3	2	no	—
244	C₂²⋊C₄	—	15	2	no	—
245	C₂²⋊C₄	—	15	2	no	—
246	C₄²	—	3	2	no	—
247	C₄²	—	15	2	no	—
248	C₄²	—	15	2	no	—
249	C₃×C₅	#520	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	5	2	no	—
251	C₂²×C₃	—	10	1	no	—
252	C₂²×C₃	—	10	1	no	—
253	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	—	5	2	no	—
257	C₂×D₃	—	10	1	no	—
258	C₂×D₃	—	10	1	no	—
259	C₂×D₃	C₂×C₄×D₅	1	2	yes	—
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	—	5	2	no	—
262	C₃×C₄	—	10	2	no	—
263	C₃×C₄	C₂×C₄×D₅	1	2	yes	—
264	Dic₃	—	2	2	no	—
265	Dic₃	—	5	2	no	—
266	Dic₃	—	10	2	no	—
267	Dic₃	C₂×C₄×D₅	1	2	yes	—
268	C₂×C₅	—	6	2	no	—
269	C₂×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
270	C₂×C₅	#1004	1	1	yes	characteristic
271	C₂×C₅	#1007	1	1	yes	characteristic
272	D₅	—	2	2	no	—
273	D₅	—	6	2	no	—
274	C₂³	—	3	1	no	—
275	C₂³	—	5	1	no	—
276	C₂³	—	15	1	no	—
277	Q₈	—	15	4	no	—
278	D₄	—	15	4	no	—
279	D₄	—	15	4	no	—
280	D₄	—	15	4	no	—
281	C₂×C₄	—	3	1	no	—
282	C₂×C₄	—	3	4	no	—
283	C₂×C₄	—	3	2	no	—
284	C₂×C₄	—	5	1	no	—
285	C₂×C₄	—	5	4	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	15	1	no	—
288	C₂×C₄	—	15	4	no	—
289	C₂×C₄	—	15	2	no	—
290	C₂×C₄	—	20	2	no	—
291	C₂×C₄	—	60	2	no	—
292	C₂×C₄	—	60	2	no	—
293	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
294	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
295	C₂×C₃	—	10	2	no	—
296	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
297	C₂×C₃	#4149	1	1	yes	characteristic
298	C₂×C₃	#4150	1	1	yes	characteristic
299	D₃	—	2	2	no	—
300	D₃	—	10	2	no	—
301	C₅	#5504	1	1	yes	characteristic, complemented, Sylow
302	C₂²	—	3	2	no	—
303	C₂²	—	5	2	no	—
304	C₂²	—	6	1	no	—
305	C₂²	—	6	1	no	—
306	C₂²	—	10	1	no	—
307	C₂²	—	10	1	no	—
308	C₂²	—	15	2	no	—
309	C₂²	—	30	1	no	—
310	C₂²	—	30	1	no	—
311	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	3	2	no	—
314	C₄	—	5	2	no	—
315	C₄	—	6	2	no	—
316	C₄	—	10	2	no	—
317	C₄	—	15	2	no	—
318	C₄	—	30	2	no	—
319	C₄	C₄×D₃×D₅	1	2	yes	—
320	C₃	#65857	1	1	yes	characteristic, complemented, Sylow
321	C₂	—	6	2	no	—
322	C₂	—	10	2	no	—
323	C₂	—	30	2	no	—
324	C₂	C₂×C₄×D₃×D₅	1	1	yes	characteristic
325	C₂	#92026	1	1	yes	characteristic
326	C₂	#92030	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₄
Fitting	C₃×C₅×C₄⋊C₄	C₂²
socle, cosocle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×#6771	C₂
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	#520
LCS	G ▷ C₂×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
2. C₄⋊C₄×D₁₅	C₂	semidirect
3. C₅×#892	C₂	semidirect
4. C₃×D₅×C₄⋊C₄	C₂	semidirect
5. C₄×#1202	C₂	semidirect
6. #91440	C₂	semidirect
7. D₃×C₄⋊Dic₅	C₂	semidirect
8. #91406	C₂	semidirect
9. #91358	C₂	semidirect
10. #91347	C₂	semidirect
11. #91324	C₂	semidirect
23. #6771	C₄	semidirect
44. D₅×C₄⋊C₄	D₃	semidirect
57. C₄×D₁₅	C₂×C₄	semidirect
60. C₄×C₅×D₃	C₂×C₄	semidirect
64. C₃×C₄×D₅	C₂×C₄	semidirect
65. #1204	C₂×C₄	semidirect
66. #1203	C₂×C₄	semidirect
67. #1202	C₂×C₄	semidirect
68. #1201	C₂×C₄	semidirect
77. #892	D₅	semidirect
167. C₄×D₅	C₄×D₃	semidirect
205. C₄×D₃	C₄×D₅	semidirect
249. C₃×C₅	#520	semidirect
301. C₅	#5504	semidirect
320. C₃	#65857	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6771	2
C₄⋊C₄×D₁₅	2
C₅×#892	2
C₃×D₅×C₄⋊C₄	2
C₄×#1202	2
#91440	2
D₃×C₄⋊Dic₅	2
#91406	2
#91358	2
#91347	2
#91324	2
#65857	3
#5504	5