Subgroups — #14439218

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	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×#91840	C₂	1	4	yes	maximal, complemented
3	C₂×#91758	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751193	C₂	1	2	yes	maximal, complemented
5	#11751177	C₂	1	2	yes	maximal, complemented
6	#11751161	C₂	1	4	yes	maximal, complemented
7	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
8	#10615019	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
10	C₂²×Dic₃₀	C₂²	1	2	yes	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#127	C₂²	1	1	yes	characteristic
13	C₃×C₅×#124	C₂²	1	2	yes	—
14	C₃×C₅×#121	—	2	2	no	—
15	C₃×C₄×C₅×D₄	—	2	2	no	—
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	#91840	—	2	16	no	—
18	C₂×#6718	C₂²	1	8	yes	—
19	C₂×#6716	C₂²	1	8	yes	—
20	C₂×C₄⋊Dic₁₅	—	2	2	no	—
21	C₂×#6710	C₂²	1	2	yes	—
22	#91758	—	2	2	no	—
23	#91758	C₂²	1	2	yes	—
24	#91758	C₂²	1	2	yes	—
25	#91087	C₂²	1	2	yes	—
26	C₄×C₃⋊#168	C₂²	1	2	yes	—
27	#83288	—	5	1	no	maximal, Hall
28	C₅×#532	D₃	1	1	yes	characteristic
29	C₂×#4097	—	3	4	no	—
30	C₂×#4015	—	3	1	no	—
31	#65381	—	3	2	no	—
32	#65365	—	3	2	no	—
33	#65349	—	3	4	no	—
34	C₄×#681	—	3	1	no	—
35	C₃×C₅×#41	—	2	8	no	—
36	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
37	C₂×C₃×C₅×D₄	—	2	2	no	—
38	C₂×C₃×C₅×D₄	—	2	2	no	—
39	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
40	C₂²×C₃×C₄×C₅	—	2	2	no	—
41	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
42	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₅	—	2	2	no	—
44	C₂×Dic₃₀	—	2	8	no	—
45	C₂×Dic₃₀	—	2	2	no	—
46	C₂×Dic₃₀	C₂³	1	4	yes	—
47	C₃×C₅×C₄⋊C₄	—	2	2	no	—
48	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
49	C₃×C₄²×C₅	D₄	1	2	yes	—
50	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	#6718	—	2	16	no	—
53	#6716	—	2	16	no	—
54	C₄⋊Dic₁₅	—	2	4	no	—
55	C₄⋊Dic₁₅	—	2	4	no	—
56	#6710	D₄	1	8	yes	complemented
57	C₂×C₃⋊#168	C₂³	1	4	yes	—
58	C₃×#532	D₅	1	1	yes	characteristic
59	C₂×#951	—	5	4	no	—
60	C₂×#869	—	5	1	no	—
61	#5092	—	5	2	no	—
62	#5076	—	5	2	no	—
63	#5060	—	5	4	no	—
64	C₄×#208	—	5	1	no	—
65	C₂×C₅×#41	C₂×D₃	1	2	yes	—
66	C₂²×Dic₁₀	—	3	2	no	—
67	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
68	C₅×#127	C₂×D₃	1	1	yes	characteristic
69	C₅×#124	C₂×D₃	1	2	yes	—
70	C₅×#121	—	2	2	no	—
71	C₄×C₅×D₄	—	2	2	no	—
72	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
73	#4097	—	6	16	no	—
74	C₂×#689	—	3	8	no	—
75	C₂×#687	—	3	8	no	—
76	C₂×C₄⋊Dic₅	—	6	2	no	—
77	C₂×#681	—	3	2	no	—
78	#4015	—	3	2	no	—
79	#4015	—	3	2	no	—
80	#4015	—	6	2	no	—
81	#3935	—	3	2	no	—
82	C₄×#168	—	3	2	no	—
83	#3152	—	15	1	no	Sylow
84	C₂³×C₃×C₅	—	2	2	no	—
85	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
86	C₃×C₅×Q₈	—	2	4	no	—
87	C₃×C₅×D₄	—	2	4	no	—
88	C₃×C₅×D₄	—	4	4	no	—
89	C₃×C₅×D₄	—	4	4	no	—
90	C₂×C₃×C₄×C₅	—	2	4	no	—
91	C₂×C₃×C₄×C₅	—	2	2	no	—
92	C₂×C₃×C₄×C₅	—	4	4	no	—
93	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
94	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
96	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
97	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₂×Dic₁₅	—	2	4	no	—
99	C₂×Dic₁₅	—	4	4	no	—
100	Dic₃₀	—	2	8	no	—
101	Dic₃₀	—	4	4	no	—
102	C₃⋊#168	C₂×D₄	1	8	yes	—
103	C₂×C₃×#41	C₂×D₅	1	2	yes	—
104	C₂²×Dic₆	—	5	2	no	—
105	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
106	C₃×#127	C₂×D₅	1	1	yes	characteristic
107	C₃×#124	C₂×D₅	1	2	yes	—
108	C₃×#121	—	2	2	no	—
109	C₃×C₄×D₄	—	2	2	no	—
110	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
111	#951	—	10	16	no	—
112	C₂×#216	—	5	8	no	—
113	C₂×#214	—	5	8	no	—
114	C₂×C₄⋊Dic₃	—	10	2	no	—
115	C₂×#208	—	5	2	no	—
116	#869	—	5	2	no	—
117	#869	—	5	2	no	—
118	#869	—	10	2	no	—
119	#803	—	5	2	no	—
120	C₄×C₃⋊C₈	—	5	2	no	—
121	C₅×#41	—	2	8	no	—
122	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
123	C₂×C₅×D₄	—	2	2	no	—
124	C₂×C₅×D₄	—	2	2	no	—
125	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
126	C₂²×C₄×C₅	—	2	2	no	—
127	C₂²×C₄×C₅	#67	1	2	yes	—
128	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
129	C₂²×Dic₅	—	6	2	no	—
130	C₂×Dic₁₀	—	3	4	no	—
131	C₂×Dic₁₀	—	6	8	no	—
132	C₂×Dic₁₀	—	6	2	no	—
133	C₅×C₄⋊C₄	—	2	2	no	—
134	C₅×C₂²⋊C₄	—	2	4	no	—
135	C₄²×C₅	#67	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	#687	—	6	16	no	—
140	C₄⋊Dic₅	—	6	4	no	—
141	C₄⋊Dic₅	—	6	4	no	—
142	#681	—	3	8	no	—
143	C₂×#168	—	3	4	no	—
144	C₂×#137	—	15	4	no	—
145	#532	D₁₅	1	1	yes	characteristic
146	C₂×C₄⋊Q₈	—	15	1	no	—
147	#497	—	15	4	no	—
148	#494	—	15	2	no	—
149	#492	—	15	2	no	—
150	C₄×M₁₆	—	15	1	no	—
151	C₂²×C₃×C₅	—	2	2	no	—
152	C₂²×C₃×C₅	—	4	2	no	—
153	C₂²×C₃×C₅	—	4	4	no	—
154	C₂²×C₃×C₅	#127	1	2	yes	—
155	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
156	C₃×C₄×C₅	—	2	2	no	—
157	C₃×C₄×C₅	—	4	2	no	—
158	C₃×C₄×C₅	#127	1	2	yes	—
159	C₃×C₄×C₅	#137	1	4	yes	complemented
160	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
161	Dic₁₅	—	4	4	no	—
162	C₃×#41	—	2	8	no	—
163	C₂×C₃×Q₈	C₂²×D₅	1	2	yes	—
164	C₂×C₃×D₄	—	2	2	no	—
165	C₂×C₃×D₄	—	2	2	no	—
166	C₂×C₃×D₄	C₂²×D₅	1	2	yes	—
167	C₂²×C₃×C₄	—	2	2	no	—
168	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
169	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
170	C₂²×Dic₃	—	10	2	no	—
171	C₂×Dic₆	—	5	4	no	—
172	C₂×Dic₆	—	10	8	no	—
173	C₂×Dic₆	—	10	2	no	—
174	C₃×C₄⋊C₄	—	2	2	no	—
175	C₃×C₂²⋊C₄	—	2	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	#216	—	10	16	no	—
180	#214	—	10	16	no	—
181	C₄⋊Dic₃	—	10	4	no	—
182	C₄⋊Dic₃	—	10	4	no	—
183	#208	—	5	8	no	—
184	C₂×C₃⋊C₈	—	5	4	no	—
185	C₂³×C₅	—	2	2	no	—
186	C₂³×C₅	C₂×#67	1	1	yes	characteristic
187	C₅×Q₈	—	2	4	no	—
188	C₅×D₄	—	2	4	no	—
189	C₅×D₄	—	4	4	no	—
190	C₅×D₄	—	4	4	no	—
191	C₂×C₄×C₅	—	2	4	no	—
192	C₂×C₄×C₅	—	2	2	no	—
193	C₂×C₄×C₅	—	4	4	no	—
194	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
195	C₂×C₄×C₅	C₂×#67	1	2	yes	—
196	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂×#67	1	2	yes	—
198	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₂×Dic₅	—	6	4	no	—
200	C₂×Dic₅	—	12	4	no	—
201	Dic₁₀	—	6	8	no	—
202	Dic₁₀	—	12	4	no	—
203	#168	—	3	8	no	—
204	C₂×#41	C₂×D₁₅	1	2	yes	—
205	C₂²×Q₈	—	15	2	no	—
206	#137	—	30	16	no	—
207	C₂×Q₁₆	—	15	8	no	—
208	C₂×SD₁₆	—	15	8	no	—
209	C₂×M₁₆	—	15	2	no	—
210	C₄⋊Q₈	—	15	2	no	—
211	C₄⋊Q₈	—	15	2	no	—
212	C₄⋊Q₈	—	30	2	no	—
213	C₄⋊Q₈	C₂×D₁₅	1	1	yes	characteristic
214	#127	C₂×D₁₅	1	1	yes	characteristic
215	#124	C₂×D₁₅	1	2	yes	—
216	#121	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	30	2	no	—
219	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
220	#97	—	15	2	no	—
221	C₄×C₈	—	15	2	no	—
222	C₂×C₃×C₅	—	2	2	no	—
223	C₂×C₃×C₅	—	8	2	no	—
224	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
225	C₂×C₃×C₅	C₂×#137	1	2	yes	—
226	C₂³×C₃	—	2	2	no	—
227	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
228	C₃×Q₈	—	2	4	no	—
229	C₃×D₄	—	2	4	no	—
230	C₃×D₄	—	4	4	no	—
231	C₃×D₄	—	4	4	no	—
232	C₂×C₃×C₄	—	2	4	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	4	4	no	—
235	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
236	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
237	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
238	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
240	C₂×Dic₃	—	10	4	no	—
241	C₂×Dic₃	—	20	4	no	—
242	Dic₆	—	10	8	no	—
243	Dic₆	—	20	4	no	—
244	C₃⋊C₈	—	5	8	no	—
245	C₂²×C₅	—	2	2	no	—
246	C₂²×C₅	—	4	2	no	—
247	C₂²×C₅	—	4	4	no	—
248	C₂²×C₅	#940	1	2	yes	—
249	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
250	C₄×C₅	—	2	2	no	—
251	C₄×C₅	—	4	2	no	—
252	C₄×C₅	#940	1	2	yes	—
253	C₄×C₅	#951	1	4	yes	complemented
254	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
255	Dic₅	—	12	4	no	—
256	#41	—	2	8	no	—
257	C₂×Q₈	—	15	4	no	—
258	C₂×Q₈	—	30	2	no	—
259	C₂×Q₈	—	30	8	no	—
260	C₂×Q₈	C₂²×D₁₅	1	2	yes	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	C₂²×D₁₅	1	2	yes	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	30	2	no	—
266	C₂²×C₄	#1220	1	2	yes	—
267	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
268	Q₁₆	—	30	16	no	—
269	SD₁₆	—	30	16	no	—
270	M₁₆	—	15	8	no	—
271	C₂×C₈	—	15	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	30	4	no	—
274	C₄⋊C₄	—	30	4	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₄²	#1220	1	2	yes	—
277	C₄²	C₂²×D₁₅	1	1	yes	characteristic
278	C₄²	C₂²×D₁₅	1	1	yes	characteristic
279	C₃×C₅	#3152	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	—	4	2	no	—
282	C₂²×C₃	—	4	4	no	—
283	C₂²×C₃	#4086	1	2	yes	—
284	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	4	2	no	—
287	C₃×C₄	#4086	1	2	yes	—
288	C₃×C₄	#4097	1	4	yes	complemented
289	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
290	Dic₃	—	20	4	no	—
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	—	8	2	no	—
293	C₂×C₅	C₂×#940	1	1	yes	characteristic
294	C₂×C₅	C₂×#951	1	2	yes	—
295	C₂³	—	2	2	no	—
296	C₂³	C₂×#1220	1	1	yes	characteristic
297	Q₈	—	2	4	no	—
298	Q₈	—	30	8	no	—
299	Q₈	—	60	4	no	—
300	D₄	—	2	4	no	—
301	D₄	—	4	4	no	—
302	D₄	—	4	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	30	4	no	—
307	C₂×C₄	—	60	4	no	—
308	C₂×C₄	D₄×D₁₅	1	4	yes	—
309	C₂×C₄	C₂×#1220	1	2	yes	—
310	C₂×C₄	C₂×#1220	1	1	yes	characteristic
311	C₂×C₄	C₂×#1220	1	2	yes	—
312	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
313	C₈	—	15	8	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	—	8	2	no	—
316	C₂×C₃	C₂×#4086	1	1	yes	characteristic
317	C₂×C₃	C₂×#4097	1	2	yes	—
318	C₅	#83288	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	4	no	—
322	C₂²	#91829	1	2	yes	—
323	C₂²	C₂²×#1220	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	60	4	no	—
327	C₄	#91829	1	2	yes	—
328	C₄	#91840	1	4	yes	complemented
329	C₄	C₂×D₄×D₁₅	1	2	yes	—
330	C₃	#10615019	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	8	2	no	—
333	C₂	C₂×#91829	1	1	yes	characteristic
334	C₂	C₂×#91840	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#1220
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	#532	D₁₅
hypocenter	C₃×C₅	#3152
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

Complements

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

N	H ≅ G/N	type
2. C₂×#91840	C₂	semidirect
3. C₂×#91758	C₂	semidirect
4. #11751193	C₂	semidirect
5. #11751177	C₂	semidirect
6. #11751161	C₂	semidirect
7. C₄×#6710	C₂	semidirect
56. #6710	D₄	semidirect
159. C₃×C₄×C₅	#137	semidirect
253. C₄×C₅	#951	semidirect
279. C₃×C₅	#3152	semidirect
288. C₃×C₄	#4097	semidirect
318. C₅	#83288	semidirect
328. C₄	#91840	semidirect
330. C₃	#10615019	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#532	2
C₂×#91840	2
C₂×#91758	2
#11751193	2
#11751177	2
#11751161	2
C₄×#6710	2
#10615019	3
#83288	5