Subgroups — #14439255

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756236	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91321	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×#681	C₂	1	1	yes	maximal, characteristic, complemented
5	#11747097	C₂	1	4	yes	maximal, complemented
6	#11747096	C₂	1	2	yes	maximal, complemented
7	#11747067	C₂	1	2	yes	maximal, complemented
8	#10614934	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	2	yes	—
10	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	#91819	—	2	2	no	—
13	#91764	C₂²	1	2	yes	—
14	#91763	C₂²	1	1	yes	characteristic
15	C₄×D₆₀	—	2	2	no	—
16	#91758	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
18	C₅×#869	—	2	2	no	—
19	C₅×#869	C₂²	1	2	yes	—
20	C₅×#869	C₂²	1	2	yes	—
21	C₂×C₃×#681	C₂²	1	2	yes	—
22	#91321	—	2	16	no	—
23	C₂×#6663	C₂²	1	8	yes	—
24	C₂×#6660	C₂²	1	8	yes	—
25	C₃×#3935	C₂²	1	2	yes	—
26	C₃×C₄×#168	C₂²	1	2	yes	—
27	#78538	—	5	1	no	maximal, Hall
28	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
29	C₂×#4088	—	3	4	no	—
30	#65811	—	3	1	no	—
31	#65381	—	3	2	no	—
32	#65375	—	3	4	no	—
33	#65374	—	3	2	no	—
34	C₄×#681	D₃	1	1	yes	characteristic, complemented
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×#1220	—	2	2	no	—
38	#6817	—	2	8	no	—
39	C₂×D₆₀	—	2	2	no	—
40	C₂×D₆₀	C₂³	1	2	yes	—
41	C₂×C₄×D₁₅	—	2	2	no	—
42	C₂×Dic₃₀	C₂³	1	2	yes	—
43	C₂²×C₅×Dic₃	—	2	2	no	—
44	C₂×C₅×Dic₆	—	2	8	no	—
45	C₂×C₅×Dic₆	—	2	2	no	—
46	C₂×C₅×Dic₆	C₂³	1	4	yes	—
47	C₃×C₄²×C₅	D₄	1	2	yes	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	#6714	—	2	4	no	—
51	C₄⋊Dic₁₅	—	2	2	no	—
52	C₅×C₄⋊Dic₃	—	2	4	no	—
53	C₅×C₄⋊Dic₃	—	2	4	no	—
54	C₃×#681	D₄	1	8	yes	complemented
55	C₂×C₃×#168	C₂³	1	4	yes	—
56	#6663	—	2	16	no	—
57	#6660	—	2	16	no	—
58	C₂×#909	—	5	4	no	—
59	#5477	—	5	1	no	—
60	C₂×#869	D₅	1	1	yes	characteristic, complemented
61	C₃×C₄×M₁₆	—	5	1	no	—
62	#4715	—	5	4	no	—
63	#4699	—	5	2	no	—
64	#4693	—	5	2	no	—
65	C₂²×C₅×Q₈	—	3	2	no	—
66	C₂×#709	—	3	2	no	—
67	C₅×C₄⋊Q₈	—	3	2	no	—
68	C₅×C₄⋊Q₈	—	3	2	no	—
69	C₅×C₄⋊Q₈	—	6	2	no	—
70	C₂×C₅×C₄⋊C₄	—	6	2	no	—
71	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
72	C₂×#689	—	3	8	no	—
73	#4088	—	6	16	no	—
74	C₂×#688	—	3	8	no	—
75	#4076	—	6	2	no	—
76	C₂×#681	C₂×D₃	1	2	yes	—
77	#4021	—	3	2	no	—
78	#4020	—	3	1	no	—
79	C₄×D₂₀	—	6	2	no	—
80	#4015	—	3	1	no	—
81	#3935	C₂×D₃	1	2	yes	—
82	C₄×#168	C₂×D₃	1	2	yes	—
83	#3152	—	15	1	no	Sylow
84	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₁₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	—	2	4	no	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	#1220	—	4	4	no	—
93	C₂×Dic₁₅	—	2	2	no	—
94	D₆₀	—	2	4	no	—
95	D₆₀	—	4	4	no	—
96	C₄×D₁₅	—	4	4	no	—
97	Dic₃₀	—	2	4	no	—
98	C₂×C₅×Dic₃	—	2	4	no	—
99	C₂×C₅×Dic₃	—	4	4	no	—
100	C₅×Dic₆	—	2	8	no	—
101	C₅×Dic₆	—	4	4	no	—
102	C₃×#168	C₂×D₄	1	8	yes	—
103	C₂×#235	—	5	2	no	—
104	C₂²×Dic₆	C₂×D₅	1	2	yes	—
105	C₂×C₃×M₁₆	—	5	2	no	—
106	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
107	#930	—	10	2	no	—
108	C₂×C₄⋊Dic₃	—	2	2	no	—
109	#909	—	10	16	no	—
110	C₂×Dic₁₂	—	5	8	no	—
111	C₂×#204	—	5	8	no	—
112	#875	—	5	2	no	—
113	#874	—	5	1	no	—
114	C₄×D₁₂	—	10	2	no	—
115	#869	—	2	2	no	—
116	#869	—	5	1	no	—
117	#869	C₂×D₅	1	2	yes	—
118	#869	C₂×D₅	1	2	yes	—
119	C₃×#97	—	5	2	no	—
120	C₃×C₄×C₈	—	5	2	no	—
121	C₂×C₅×Q₈	—	3	4	no	—
122	C₂×C₅×Q₈	—	6	2	no	—
123	C₂×C₅×Q₈	—	6	8	no	—
124	C₂²×C₄×C₅	—	6	2	no	—
125	C₂²×C₄×C₅	D₁₂	1	2	yes	—
126	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
127	C₂×C₂²⋊D₅	—	6	2	no	—
128	#709	—	6	8	no	—
129	C₂×D₂₀	—	3	2	no	—
130	C₂×D₂₀	—	6	2	no	—
131	C₂×C₄×D₅	—	6	2	no	—
132	C₂×Dic₁₀	—	3	2	no	—
133	C₅×C₄⋊C₄	—	6	4	no	—
134	C₅×C₄⋊C₄	—	6	4	no	—
135	C₄²×C₅	D₁₂	1	2	yes	—
136	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
137	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
138	#689	—	6	16	no	—
139	#688	—	6	16	no	—
140	#685	—	6	4	no	—
141	C₄⋊Dic₅	—	6	2	no	—
142	#681	D₁₂	1	8	yes	complemented
143	C₂×#168	C₂²×D₃	1	4	yes	—
144	C₂×#137	—	15	4	no	—
145	#532	—	15	1	no	—
146	C₂×C₄⋊Q₈	—	3	1	no	—
147	#497	—	15	4	no	—
148	#494	—	15	2	no	—
149	#492	—	15	2	no	—
150	C₄×M₁₆	—	5	1	no	—
151	C₂²×C₃×C₅	—	2	2	no	—
152	C₂²×C₃×C₅	#127	1	2	yes	—
153	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₅	—	4	2	no	—
155	C₂×D₁₅	—	4	4	no	—
156	C₃×C₄×C₅	—	2	2	no	—
157	C₃×C₄×C₅	#127	1	2	yes	—
158	C₃×C₄×C₅	#137	1	4	yes	complemented
159	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
160	Dic₁₅	—	4	2	no	—
161	C₅×Dic₃	—	4	4	no	—
162	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
163	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
164	C₂×#67	—	10	2	no	—
165	C₂²×Dic₃	—	2	2	no	—
166	#235	—	10	8	no	—
167	C₂×D₁₂	—	5	2	no	—
168	C₂×D₁₂	—	10	2	no	—
169	C₂×C₄×D₃	—	10	2	no	—
170	C₂×Dic₆	—	2	8	no	—
171	C₂×Dic₆	—	2	2	no	—
172	C₂×Dic₆	—	5	2	no	—
173	C₂×Dic₆	C₂²×D₅	1	4	yes	—
174	C₃×M₁₆	—	5	8	no	—
175	C₂×C₃×C₈	—	5	4	no	—
176	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
177	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
178	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
179	(C₂×D₃)⋊C₄	—	10	4	no	—
180	C₄⋊Dic₃	—	2	4	no	—
181	C₄⋊Dic₃	—	2	4	no	—
182	C₄⋊Dic₃	—	10	2	no	—
183	Dic₁₂	—	10	16	no	—
184	#204	—	10	16	no	—
185	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
186	C₂²×D₅	—	6	2	no	—
187	C₅×Q₈	—	6	8	no	—
188	C₅×Q₈	—	12	4	no	—
189	C₂×C₄×C₅	—	2	4	no	—
190	C₂×C₄×C₅	—	6	4	no	—
191	C₂×C₄×C₅	—	12	4	no	—
192	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
193	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
194	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
196	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
197	C₂²⋊D₅	—	12	4	no	—
198	C₂×Dic₅	—	6	2	no	—
199	D₂₀	—	6	4	no	—
200	D₂₀	—	12	4	no	—
201	C₄×D₅	—	12	4	no	—
202	Dic₁₀	—	6	4	no	—
203	#168	C₂×D₁₂	1	8	yes	—
204	C₂×#41	—	15	2	no	—
205	C₂²×Q₈	—	3	2	no	—
206	#137	—	30	16	no	—
207	C₂×Q₁₆	—	15	8	no	—
208	C₂×SD₁₆	—	15	8	no	—
209	C₂×M₁₆	—	5	2	no	—
210	C₄⋊Q₈	—	3	2	no	—
211	C₄⋊Q₈	—	3	2	no	—
212	C₄⋊Q₈	—	6	2	no	—
213	C₄⋊Q₈	—	15	1	no	—
214	#127	—	15	1	no	—
215	#124	—	15	2	no	—
216	#121	—	30	2	no	—
217	C₄×D₄	—	30	2	no	—
218	C₂×C₄⋊C₄	—	6	2	no	—
219	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
220	#97	—	5	2	no	—
221	C₄×C₈	—	5	2	no	—
222	C₂×C₃×C₅	—	2	2	no	—
223	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
224	C₂×C₃×C₅	C₂×#137	1	2	yes	—
225	D₁₅	—	8	2	no	—
226	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₂²×D₃	—	10	2	no	—
228	C₂×C₃×C₄	—	2	4	no	—
229	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
230	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
231	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
233	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
234	#67	—	20	4	no	—
235	C₂×Dic₃	—	2	4	no	—
236	C₂×Dic₃	—	4	4	no	—
237	C₂×Dic₃	—	10	2	no	—
238	D₁₂	—	10	4	no	—
239	D₁₂	—	20	4	no	—
240	C₄×D₃	—	20	4	no	—
241	Dic₆	—	2	8	no	—
242	Dic₆	—	4	4	no	—
243	Dic₆	—	10	4	no	—
244	C₃×C₈	—	5	8	no	—
245	C₂²×C₅	—	2	2	no	—
246	C₂²×C₅	#874	1	2	yes	—
247	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
248	C₂×D₅	—	12	2	no	—
249	C₂×D₅	—	12	4	no	—
250	C₄×C₅	—	2	2	no	—
251	C₄×C₅	—	12	4	no	—
252	C₄×C₅	#874	1	2	yes	—
253	C₄×C₅	#909	1	4	yes	complemented
254	C₄×C₅	C₂²×D₁₂	1	2	yes	—
255	Dic₅	—	12	2	no	—
256	#41	—	30	8	no	—
257	C₂×Q₈	—	3	4	no	—
258	C₂×Q₈	—	6	2	no	—
259	C₂×Q₈	—	6	8	no	—
260	C₂×Q₈	—	15	2	no	—
261	C₂×D₄	—	15	2	no	—
262	C₂×D₄	—	30	2	no	—
263	C₂×D₄	—	30	2	no	—
264	C₂²×C₄	—	6	2	no	—
265	C₂²×C₄	—	30	2	no	—
266	C₂²×C₄	#1203	1	2	yes	—
267	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
268	Q₁₆	—	30	16	no	—
269	SD₁₆	—	30	16	no	—
270	M₁₆	—	5	8	no	—
271	C₂×C₈	—	5	4	no	—
272	C₄⋊C₄	—	6	4	no	—
273	C₄⋊C₄	—	6	4	no	—
274	C₄⋊C₄	—	30	2	no	—
275	C₂²⋊C₄	—	30	4	no	—
276	C₄²	#1203	1	2	yes	—
277	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
278	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
279	C₃×C₅	#3152	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	#4086	1	2	yes	—
282	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
283	C₂×D₃	—	20	2	no	—
284	C₂×D₃	—	20	4	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	#4086	1	2	yes	—
287	C₃×C₄	#4088	1	4	yes	complemented
288	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
289	Dic₃	—	4	4	no	—
290	Dic₃	—	20	2	no	—
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	C₂×#874	1	1	yes	characteristic
293	C₂×C₅	C₂×#909	1	2	yes	—
294	D₅	—	24	2	no	—
295	C₂³	—	30	2	no	—
296	C₂³	C₂×#1203	1	1	yes	characteristic
297	Q₈	—	6	8	no	—
298	Q₈	—	12	4	no	—
299	Q₈	—	30	4	no	—
300	D₄	—	30	4	no	—
301	D₄	—	60	4	no	—
302	D₄	—	60	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	6	4	no	—
305	C₂×C₄	—	12	4	no	—
306	C₂×C₄	—	30	2	no	—
307	C₂×C₄	—	60	4	no	—
308	C₂×C₄	D₅×D₁₂	1	4	yes	—
309	C₂×C₄	C₂×#1203	1	2	yes	—
310	C₂×C₄	C₂×#1203	1	1	yes	characteristic
311	C₂×C₄	C₂×#1203	1	2	yes	—
312	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
313	C₈	—	5	8	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	C₂×#4086	1	1	yes	characteristic
316	C₂×C₃	C₂×#4088	1	2	yes	—
317	D₃	—	40	2	no	—
318	C₅	#78538	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	60	2	no	—
321	C₂²	—	60	4	no	—
322	C₂²	#91451	1	2	yes	—
323	C₂²	C₂²×#1203	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	12	4	no	—
326	C₄	—	60	2	no	—
327	C₄	#91321	1	4	yes	complemented
328	C₄	#91451	1	2	yes	—
329	C₄	C₂×D₅×D₁₂	1	2	yes	—
330	C₃	#10614934	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	120	2	no	—
333	C₂	C₂×#91321	1	2	yes	—
334	C₂	C₂×#91451	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1203
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	#11756236	C₂
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#3152
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#869	C₂	semidirect
3. C₂×#91321	C₂	semidirect
4. C₃×C₄×#681	C₂	semidirect
5. #11747097	C₂	semidirect
6. #11747096	C₂	semidirect
7. #11747067	C₂	semidirect
34. C₄×#681	D₃	semidirect
54. C₃×#681	D₄	semidirect
60. C₂×#869	D₅	semidirect
142. #681	D₁₂	semidirect
158. C₃×C₄×C₅	#137	semidirect
253. C₄×C₅	#909	semidirect
279. C₃×C₅	#3152	semidirect
287. C₃×C₄	#4088	semidirect
318. C₅	#78538	semidirect
327. C₄	#91321	semidirect
330. C₃	#10614934	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
C₂×C₅×#869	2
C₂×#91321	2
C₃×C₄×#681	2
#11747097	2
#11747096	2
#11747067	2
#10614934	3
#78538	5