Subgroups — #14295641

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6817	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91109	C₂	1	2	yes	maximal
4	#11751033	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746376	C₂	1	2	yes	maximal, complemented
6	#10600659	—	3	1	no	maximal, Hall
7	C₂×#6817	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₄×#1220	—	2	2	no	—
11	#91815	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
13	C₄×D₆₀	C₄	1	2	yes	complemented
14	#91761	—	2	1	no	—
15	C₄²×D₁₅	—	2	1	no	—
16	C₄×Dic₃₀	C₄	1	2	yes	complemented
17	C₃×C₅×#97	—	2	1	no	—
18	C₃×C₄×C₅×C₈	—	2	1	no	—
19	#91115	—	2	2	no	—
20	#91109	C₄	1	8	yes	complemented
21	#69866	—	5	1	no	maximal, Hall
22	C₄×#709	—	3	1	no	—
23	C₄×C₅×M₁₆	D₃	1	1	yes	characteristic, complemented
24	C₂×#3957	—	3	2	no	—
25	#65221	—	3	1	no	—
26	#64775	—	3	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×#1220	—	2	1	no	—
30	C₂²×Dic₁₅	—	2	2	no	—
31	#6817	—	2	2	no	—
32	#6817	C₂×C₄	1	4	yes	—
33	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₅	—	2	1	no	—
35	C₂×C₄×D₁₅	—	2	2	no	—
36	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×M₁₆	—	2	2	no	—
38	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	complemented
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	#6719	—	2	1	no	—
44	#6714	—	2	2	no	—
45	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
46	#6712	—	2	2	no	—
47	C₄×Dic₁₅	—	2	2	no	—
48	C₄×Dic₁₅	—	2	1	no	—
49	C₄×Dic₁₅	—	2	2	no	—
50	C₄×Dic₁₅	C₂×C₄	1	4	yes	complemented
51	C₄×#235	—	5	1	no	—
52	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
53	C₂×#825	—	5	2	no	—
54	#4932	—	5	1	no	—
55	#4555	—	5	2	no	—
56	C₂×#709	—	3	1	no	—
57	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
58	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
59	C₄×C₂²⋊D₅	—	6	2	no	—
60	#4072	—	3	1	no	—
61	C₂×C₄×Dic₅	—	3	2	no	—
62	C₄×D₂₀	—	3	2	no	—
63	#4018	—	6	1	no	—
64	C₄²×D₅	—	6	1	no	—
65	C₄×Dic₁₀	—	3	2	no	—
66	C₅×#97	—	2	1	no	—
67	C₄×C₅×C₈	—	2	1	no	—
68	#3963	—	6	2	no	—
69	#3957	—	3	8	no	—
70	#1759	—	15	1	no	Sylow
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₁₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	4	2	no	—
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	#1220	—	2	4	no	—
81	C₂×Dic₁₅	—	2	1	no	—
82	C₂×Dic₁₅	—	2	2	no	—
83	C₂×Dic₁₅	—	2	4	no	—
84	C₂×Dic₁₅	—	4	4	no	—
85	D₆₀	—	2	1	no	—
86	D₆₀	C₄²	1	2	yes	—
87	C₄×D₁₅	—	2	4	no	—
88	C₄×D₁₅	—	4	4	no	—
89	Dic₃₀	—	2	1	no	—
90	Dic₃₀	C₄²	1	2	yes	—
91	C₃×C₅×C₈	—	2	4	no	—
92	C₂×#235	—	5	1	no	—
93	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₄×#67	—	10	2	no	—
96	#926	—	5	1	no	—
97	C₂×C₄×Dic₃	—	5	2	no	—
98	C₄×D₁₂	—	5	2	no	—
99	#872	—	10	1	no	—
100	C₄²×D₃	—	10	1	no	—
101	C₄×Dic₆	—	5	2	no	—
102	C₃×#97	—	2	1	no	—
103	C₃×C₄×C₈	—	2	1	no	—
104	C₃⋊#95	—	10	2	no	—
105	#825	—	5	8	no	—
106	C₂²×C₄×C₅	D₁₂	1	2	yes	—
107	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
108	C₂×C₂²⋊D₅	—	6	1	no	—
109	C₂²×Dic₅	—	6	2	no	—
110	#709	—	3	4	no	—
111	#709	—	6	2	no	—
112	C₂×D₂₀	—	3	1	no	—
113	C₂×C₄×D₅	—	6	1	no	—
114	C₂×C₄×D₅	—	6	2	no	—
115	C₂×Dic₁₀	—	3	1	no	—
116	C₅×M₁₆	—	2	2	no	—
117	C₅×M₁₆	C₄×D₃	1	4	yes	complemented
118	C₂×C₅×C₈	—	2	2	no	—
119	C₄²×C₅	C₄×D₃	1	2	yes	—
120	C₄²×C₅	#67	1	1	yes	characteristic
121	C₄²×C₅	#67	1	1	yes	characteristic
122	C₂²⋊Dic₅	—	6	1	no	—
123	#685	—	6	2	no	—
124	C₄⋊Dic₅	—	3	2	no	—
125	#683	—	6	2	no	—
126	C₄×Dic₅	—	3	4	no	—
127	C₄×Dic₅	—	6	2	no	—
128	C₄×Dic₅	—	6	2	no	—
129	C₄×Dic₅	—	6	1	no	—
130	C₄×#41	—	15	1	no	—
131	C₂×#104	—	15	2	no	—
132	C₄×M₁₆	D₁₅	1	1	yes	characteristic, complemented
133	#376	—	15	1	no	—
134	#339	—	15	2	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
138	C₂×D₁₅	—	2	2	no	—
139	C₂×D₁₅	—	4	2	no	—
140	C₃×C₄×C₅	—	2	2	no	—
141	C₃×C₄×C₅	—	2	2	no	—
142	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
143	C₃×C₄×C₅	#117	1	1	yes	characteristic
144	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
145	Dic₁₅	—	2	2	no	—
146	Dic₁₅	—	4	4	no	—
147	Dic₁₅	—	4	2	no	—
148	C₂²×C₃×C₄	D₂₀	1	2	yes	—
149	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
150	C₂×#67	—	10	1	no	—
151	C₂²×Dic₃	—	10	2	no	—
152	#235	—	5	4	no	—
153	#235	—	10	2	no	—
154	C₂×D₁₂	—	5	1	no	—
155	C₂×C₄×D₃	—	10	1	no	—
156	C₂×C₄×D₃	—	10	2	no	—
157	C₂×Dic₆	—	5	1	no	—
158	C₃×M₁₆	—	2	2	no	—
159	C₃×M₁₆	C₄×D₅	1	4	yes	complemented
160	C₂×C₃×C₈	—	2	2	no	—
161	C₃×C₄²	C₄×D₅	1	2	yes	—
162	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
163	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
164	#217	—	10	1	no	—
165	(C₂×D₃)⋊C₄	—	10	2	no	—
166	C₄⋊Dic₃	—	5	2	no	—
167	Dic₃⋊C₄	—	10	2	no	—
168	C₄×Dic₃	—	5	4	no	—
169	C₄×Dic₃	—	10	2	no	—
170	C₄×Dic₃	—	10	2	no	—
171	C₄×Dic₃	—	10	1	no	—
172	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
173	C₂²×D₅	—	6	1	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
176	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
177	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
179	C₂×C₄×C₅	#235	1	2	yes	—
180	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
181	C₂²⋊D₅	—	6	4	no	—
182	C₂×Dic₅	—	6	1	no	—
183	C₂×Dic₅	—	6	4	no	—
184	C₂×Dic₅	—	6	2	no	—
185	C₂×Dic₅	—	12	4	no	—
186	D₂₀	—	3	2	no	—
187	D₂₀	—	6	1	no	—
188	C₄×D₅	—	6	4	no	—
189	C₄×D₅	—	12	4	no	—
190	Dic₁₀	—	3	2	no	—
191	Dic₁₀	—	6	1	no	—
192	C₅×C₈	—	2	4	no	—
193	C₂×#41	—	15	1	no	—
194	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
195	C₄×Q₈	—	15	2	no	—
196	C₄×D₄	—	15	2	no	—
197	C₄×D₄	—	30	2	no	—
198	#117	—	15	1	no	—
199	#117	—	30	1	no	—
200	C₂×C₄²	—	15	2	no	—
201	C₂×C₄²	—	30	1	no	—
202	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
203	#104	—	15	8	no	—
204	#97	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	#95	—	30	2	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₃×C₅	#421	1	2	yes	—
210	D₁₅	—	4	2	no	—
211	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
212	C₂²×D₃	—	10	1	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	#685	1	2	yes	—
215	C₂×C₃×C₄	#685	1	2	yes	—
216	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
218	C₂×C₃×C₄	#709	1	2	yes	—
219	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
220	#67	—	10	4	no	—
221	C₂×Dic₃	—	10	1	no	—
222	C₂×Dic₃	—	10	4	no	—
223	C₂×Dic₃	—	10	2	no	—
224	C₂×Dic₃	—	20	4	no	—
225	D₁₂	—	5	2	no	—
226	D₁₂	—	10	1	no	—
227	C₄×D₃	—	10	4	no	—
228	C₄×D₃	—	20	4	no	—
229	Dic₆	—	5	2	no	—
230	Dic₆	—	10	1	no	—
231	C₃×C₈	—	2	4	no	—
232	C₂²×C₅	—	2	2	no	—
233	C₂²×C₅	C₄×D₁₂	1	2	yes	—
234	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
235	C₂×D₅	—	6	2	no	—
236	C₂×D₅	—	12	2	no	—
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	—	2	2	no	—
239	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
240	C₄×C₅	#872	1	1	yes	characteristic
241	C₄×C₅	C₄×#67	1	2	yes	—
242	Dic₅	—	6	2	no	—
243	Dic₅	—	12	4	no	—
244	Dic₅	—	12	2	no	—
245	#41	—	15	4	no	—
246	#41	—	30	2	no	—
247	C₂×Q₈	—	15	1	no	—
248	C₂×D₄	—	15	1	no	—
249	C₂×D₄	—	30	1	no	—
250	C₂²×C₄	—	30	1	no	—
251	C₂²×C₄	—	30	2	no	—
252	C₂²×C₄	—	30	2	no	—
253	C₂²×C₄	D₆₀	1	2	yes	—
254	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	C₄×D₁₅	1	4	yes	complemented
257	C₂×C₈	—	2	2	no	—
258	C₄⋊C₄	—	15	2	no	—
259	C₄⋊C₄	—	30	2	no	—
260	C₂²⋊C₄	—	30	1	no	—
261	C₂²⋊C₄	—	30	2	no	—
262	C₄²	—	15	4	no	—
263	C₄²	—	30	2	no	—
264	C₄²	—	30	2	no	—
265	C₄²	—	30	1	no	—
266	C₄²	C₄×D₁₅	1	2	yes	—
267	C₄²	#1220	1	1	yes	characteristic
268	C₄²	#1220	1	1	yes	characteristic
269	C₃×C₅	#1759	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	C₄×D₂₀	1	2	yes	—
272	C₂²×C₃	C₂×#685	1	1	yes	characteristic
273	C₂×D₃	—	10	2	no	—
274	C₂×D₃	—	20	2	no	—
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
278	C₃×C₄	#4018	1	1	yes	characteristic
279	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
280	Dic₃	—	10	2	no	—
281	Dic₃	—	20	4	no	—
282	Dic₃	—	20	2	no	—
283	C₂×C₅	—	2	2	no	—
284	C₂×C₅	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
285	C₂×C₅	#5139	1	2	yes	—
286	D₅	—	12	2	no	—
287	C₂³	—	30	1	no	—
288	C₂³	C₂×D₆₀	1	1	yes	characteristic
289	Q₈	—	15	2	no	—
290	Q₈	—	30	1	no	—
291	D₄	—	15	2	no	—
292	D₄	—	30	4	no	—
293	D₄	—	30	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	30	1	no	—
296	C₂×C₄	—	30	4	no	—
297	C₂×C₄	—	30	4	no	—
298	C₂×C₄	—	30	2	no	—
299	C₂×C₄	—	60	4	no	—
300	C₂×C₄	—	60	4	no	—
301	C₂×C₄	#6714	1	2	yes	—
302	C₂×C₄	#6714	1	2	yes	—
303	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
305	C₂×C₄	#6817	1	2	yes	—
306	C₂×C₄	C₂×#1220	1	1	yes	characteristic
307	C₈	—	2	4	no	—
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	C₄×#685	1	1	yes	characteristic
310	C₂×C₃	#65428	1	2	yes	—
311	D₃	—	20	2	no	—
312	C₅	#69866	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	30	2	no	—
315	C₂²	—	60	2	no	—
316	C₂²	C₄×D₆₀	1	2	yes	—
317	C₂²	C₂×#6714	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	30	2	no	—
321	C₄	—	60	4	no	—
322	C₄	—	60	2	no	—
323	C₄	C₄²×D₁₅	1	1	yes	characteristic
324	C₄	#91761	1	1	yes	characteristic
325	C₄	C₄×#1220	1	2	yes	—
326	C₃	#10600659	1	1	yes	characteristic, complemented, Sylow
327	C₂	—	2	2	no	—
328	C₂	—	60	2	no	—
329	C₂	C₄×#6714	1	1	yes	characteristic
330	C₂	#11751240	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×#6817	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₄×M₁₆	D₁₅
hypocenter	C₃×C₅	#1759
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×M₁₆	C₂	semidirect
4. #11751033	C₂	semidirect
5. #11746376	C₂	semidirect
13. C₄×D₆₀	C₄	semidirect
16. C₄×Dic₃₀	C₄	semidirect
20. #91109	C₄	semidirect
23. C₄×C₅×M₁₆	D₃	semidirect
38. C₃×C₅×M₁₆	C₂×C₄	semidirect
50. C₄×Dic₁₅	C₂×C₄	semidirect
52. C₃×C₄×M₁₆	D₅	semidirect
117. C₅×M₁₆	C₄×D₃	semidirect
132. C₄×M₁₆	D₁₅	semidirect
159. C₃×M₁₆	C₄×D₅	semidirect
256. M₁₆	C₄×D₁₅	semidirect
269. C₃×C₅	#1759	semidirect
312. C₅	#69866	semidirect
326. C₃	#10600659	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6817	2
C₃×C₄×C₅×M₁₆	2
C₂×#91109	2
#11751033	2
#11746376	2
#10600659	3
#69866	5