Subgroups — #14439885

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752506	C₂	1	2	yes	maximal
2	#11750783	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4682	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
5	#11747986	C₂	1	4	yes	maximal, complemented
6	#11747672	C₂	1	4	yes	maximal
7	#11746941	C₂	1	2	yes	maximal, complemented
8	#10613669	—	3	1	no	maximal, Hall
9	#91758	C₂²	1	2	yes	—
10	C₅×#869	C₂²	1	2	yes	—
11	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
12	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
13	#91388	C₂²	1	4	yes	—
14	#91360	—	2	2	no	—
15	#91359	—	2	2	no	—
16	#91358	C₂²	1	4	yes	—
17	#91333	—	2	2	no	—
18	Dic₅×Dic₆	—	2	2	no	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	#91102	C₂²	1	4	yes	—
21	C₅×#818	C₂²	1	4	yes	—
22	C₃×C₈×Dic₅	C₂²	1	2	yes	—
23	C₃×C₄×#168	C₂²	1	1	yes	characteristic
24	#90985	C₂²	1	8	yes	—
25	#78442	—	5	1	no	maximal, Hall
26	#66062	—	3	2	no	—
27	C₅×#500	—	3	1	no	—
28	#65368	—	3	2	no	—
29	#65169	—	3	4	no	—
30	#65167	—	3	4	no	—
31	C₄×C₈×D₅	D₃	1	1	yes	characteristic
32	#64971	—	3	1	no	—
33	C₂×Dic₃₀	—	2	2	no	—
34	C₂×C₅×Dic₆	—	2	2	no	—
35	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
37	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₄⋊Dic₁₅	—	2	2	no	—
41	C₄⋊Dic₁₅	C₂³	1	4	yes	—
42	C₅×C₄⋊Dic₃	—	2	2	no	—
43	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
44	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
45	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
46	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
47	C₂×C₃×#168	C₂³	1	2	yes	—
48	C₃×C₈×D₅	Q₈	1	8	yes	complemented
49	Dic₃⋊Dic₅	—	2	4	no	—
50	#6669	—	2	4	no	—
51	#6665	—	2	4	no	—
52	Dic₃×Dic₅	—	2	4	no	—
53	#5470	—	5	2	no	—
54	C₂×C₃×C₄×C₈	—	5	1	no	—
55	#5106	—	5	4	no	—
56	#4683	—	5	4	no	—
57	#4682	—	5	1	no	—
58	#4682	D₅	1	1	yes	characteristic, complemented
59	#4680	—	5	2	no	—
60	C₅×C₄⋊Q₈	—	3	2	no	—
61	#4092	—	6	2	no	—
62	Q₈×Dic₅	—	6	2	no	—
63	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
64	#4043	—	6	2	no	—
65	#4038	—	3	4	no	—
66	#4036	—	3	4	no	—
67	#4033	—	6	2	no	—
68	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
69	#4015	—	3	2	no	—
70	C₅×#107	—	3	4	no	—
71	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	#3950	—	3	4	no	—
73	C₈×Dic₅	C₂×D₃	1	2	yes	—
74	#3940	—	3	8	no	—
75	C₄×#168	C₂×D₃	1	1	yes	characteristic
76	#3160	—	15	1	no	Sylow
77	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₅	—	2	4	no	—
82	Dic₃₀	—	4	4	no	—
83	C₂×C₅×Dic₃	—	2	4	no	—
84	C₅×Dic₆	—	4	4	no	—
85	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
88	C₃×C₄×D₅	—	2	2	no	—
89	C₃×C₄×D₅	—	2	2	no	—
90	C₃×C₄×D₅	C₂×Q₈	1	4	yes	—
91	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
92	C₃×#168	C₂×Q₈	1	4	yes	—
93	C₂²×C₃×C₈	—	5	2	no	—
94	C₂×C₃×C₄²	—	5	1	no	—
95	#926	—	5	4	no	—
96	#924	—	10	2	no	—
97	#924	—	10	2	no	—
98	#870	—	5	4	no	—
99	#869	—	5	2	no	—
100	#869	C₂×D₅	1	2	yes	—
101	C₄×Dic₆	—	10	2	no	—
102	C₄×Dic₆	—	10	2	no	—
103	C₃×C₄×C₈	—	5	2	no	—
104	C₃×C₄×C₈	—	5	1	no	—
105	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
106	#818	—	5	4	no	—
107	#818	C₂×D₅	1	4	yes	—
108	#817	—	5	8	no	—
109	C₂×C₅×Q₈	—	6	2	no	—
110	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
111	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
112	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
113	C₂×Dic₁₀	—	6	2	no	—
114	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
115	C₅×C₄⋊C₄	—	3	4	no	—
116	C₅×C₄⋊C₄	—	6	2	no	—
117	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
118	#685	—	6	4	no	—
119	C₄⋊Dic₅	—	3	4	no	—
120	C₄⋊Dic₅	—	6	2	no	—
121	C₄⋊Dic₅	—	6	4	no	—
122	#683	—	6	4	no	—
123	C₄×Dic₅	—	6	4	no	—
124	C₄×Dic₅	D₁₂	1	1	yes	characteristic
125	C₄×Dic₅	D₁₂	1	1	yes	characteristic
126	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
127	C₂×#168	C₂²×D₃	1	2	yes	—
128	C₈×D₅	Dic₆	1	8	yes	complemented
129	#533	—	15	2	no	—
130	#500	—	3	1	no	—
131	#500	—	15	1	no	—
132	#499	—	15	4	no	—
133	#498	—	15	2	no	—
134	#427	—	15	4	no	—
135	C₂×C₄×C₈	—	5	1	no	—
136	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₃×D₅	—	2	1	no	—
138	C₂×C₃×D₅	—	2	1	no	—
139	C₂×C₃×D₅	C₄⋊Q₈	1	2	yes	—
140	C₃×C₄×C₅	#135	1	2	yes	—
141	C₃×C₄×C₅	#135	1	2	yes	—
142	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
143	Dic₁₅	—	4	4	no	—
144	C₃×Dic₅	—	2	1	no	—
145	C₃×Dic₅	—	2	1	no	—
146	C₃×Dic₅	C₄⋊Q₈	1	2	yes	—
147	C₅×Dic₃	—	4	4	no	—
148	C₂²×C₃×C₄	—	5	1	no	—
149	C₂²×C₃×C₄	—	5	1	no	—
150	C₂²×C₃×C₄	—	5	1	no	—
151	C₂×Dic₆	—	2	2	no	—
152	C₂×Dic₆	—	10	2	no	—
153	C₂×C₃×C₈	—	5	2	no	—
154	C₂×C₃×C₈	—	5	8	no	—
155	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
156	C₃×C₄²	—	5	1	no	—
157	C₃×C₄²	—	5	1	no	—
158	C₃×C₄²	—	5	1	no	—
159	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
160	#217	—	10	4	no	—
161	C₄⋊Dic₃	—	2	2	no	—
162	C₄⋊Dic₃	—	5	4	no	—
163	C₄⋊Dic₃	—	10	2	no	—
164	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
165	Dic₃⋊C₄	—	10	4	no	—
166	Dic₃⋊C₄	—	10	4	no	—
167	C₄×Dic₃	—	10	4	no	—
168	C₂²×D₅	C₂×D₁₂	1	1	yes	characteristic
169	C₅×Q₈	—	12	4	no	—
170	C₂×C₄×C₅	—	6	4	no	—
171	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
174	C₂×Dic₅	—	6	4	no	—
175	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
176	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
177	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
178	C₄×D₅	—	2	2	no	—
179	C₄×D₅	—	2	2	no	—
180	C₄×D₅	C₂×Dic₆	1	4	yes	—
181	Dic₁₀	—	12	4	no	—
182	C₅×C₈	C₂×Dic₆	1	4	yes	—
183	#168	C₂×Dic₆	1	4	yes	—
184	C₂²×C₈	—	5	2	no	—
185	C₄⋊Q₈	—	3	2	no	—
186	C₄⋊Q₈	—	15	2	no	—
187	#125	—	15	4	no	—
188	C₂²⋊Q₈	—	30	2	no	—
189	C₂²⋊Q₈	—	30	2	no	—
190	C₄×Q₈	—	30	2	no	—
191	C₄×Q₈	—	30	2	no	—
192	#117	—	15	4	no	—
193	C₂×C₄²	—	5	1	no	—
194	#107	—	3	4	no	—
195	#107	—	15	4	no	—
196	#106	—	15	8	no	—
197	C₄×C₈	—	5	2	no	—
198	C₄×C₈	—	5	1	no	—
199	C₄×C₈	D₃×D₅	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
201	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
202	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
203	C₃×D₅	—	2	2	no	—
204	C₂³×C₃	—	5	1	no	—
205	C₂×C₃×C₄	—	5	1	no	—
206	C₂×C₃×C₄	—	5	4	no	—
207	C₂×C₃×C₄	—	5	1	no	—
208	C₂×C₃×C₄	—	5	1	no	—
209	C₂×C₃×C₄	—	10	2	no	—
210	C₂×C₃×C₄	—	10	2	no	—
211	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
212	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
214	C₂×Dic₃	—	2	4	no	—
215	C₂×Dic₃	—	10	4	no	—
216	Dic₆	—	4	4	no	—
217	Dic₆	—	20	4	no	—
218	C₃×C₈	—	5	4	no	—
219	C₃×C₈	Q₈×D₅	1	4	yes	—
220	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
221	C₂×D₅	—	2	1	no	—
222	C₂×D₅	—	2	1	no	—
223	C₂×D₅	#869	1	2	yes	—
224	C₄×C₅	—	12	4	no	—
225	C₄×C₅	#904	1	2	yes	—
226	C₄×C₅	#904	1	2	yes	—
227	C₄×C₅	C₂²×Dic₆	1	2	yes	—
228	Dic₅	—	2	1	no	—
229	Dic₅	—	2	1	no	—
230	Dic₅	—	12	4	no	—
231	Dic₅	#869	1	2	yes	—
232	C₂×Q₈	—	6	2	no	—
233	C₂×Q₈	—	30	2	no	—
234	C₂²×C₄	—	5	1	no	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	C₂×C₈	—	5	2	no	—
238	C₂×C₈	—	5	8	no	—
239	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
240	C₄⋊C₄	—	3	4	no	—
241	C₄⋊C₄	—	6	2	no	—
242	C₄⋊C₄	—	15	4	no	—
243	C₄⋊C₄	—	30	4	no	—
244	C₄⋊C₄	—	30	4	no	—
245	C₄⋊C₄	—	30	2	no	—
246	C₂²⋊C₄	—	30	4	no	—
247	C₄²	—	5	1	no	—
248	C₄²	—	5	1	no	—
249	C₄²	—	5	1	no	—
250	C₄²	—	30	4	no	—
251	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
252	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	5	2	no	—
254	C₂²×C₃	—	10	1	no	—
255	C₂²×C₃	—	10	1	no	—
256	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
257	C₃×C₄	—	5	2	no	—
258	C₃×C₄	—	10	1	no	—
259	C₃×C₄	—	10	1	no	—
260	C₃×C₄	#4058	1	2	yes	—
261	C₃×C₄	#4065	1	2	yes	—
262	C₃×C₄	C₂×Q₈×D₅	1	2	yes	—
263	Dic₃	—	4	4	no	—
264	Dic₃	—	20	4	no	—
265	C₂×C₅	C₂×#869	1	1	yes	characteristic
266	C₂×C₅	C₂×#904	1	1	yes	characteristic
267	C₂×C₅	C₂×#904	1	1	yes	characteristic
268	D₅	—	2	2	no	—
269	C₂³	—	5	1	no	—
270	Q₈	—	12	4	no	—
271	Q₈	—	60	4	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	5	4	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	5	1	no	—
276	C₂×C₄	—	6	4	no	—
277	C₂×C₄	—	10	2	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	—	30	4	no	—
280	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
281	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
283	C₈	—	5	4	no	—
284	C₈	D₅×Dic₆	1	4	yes	—
285	C₂×C₃	—	10	2	no	—
286	C₂×C₃	D₅×C₄⋊Q₈	1	1	yes	characteristic
287	C₂×C₃	C₂×#4058	1	1	yes	characteristic
288	C₂×C₃	C₂×#4065	1	1	yes	characteristic
289	C₅	#78442	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	5	2	no	—
291	C₂²	—	10	1	no	—
292	C₂²	—	10	1	no	—
293	C₂²	C₂×D₅×D₁₂	1	1	yes	characteristic
294	C₄	—	5	2	no	—
295	C₄	—	10	1	no	—
296	C₄	—	10	1	no	—
297	C₄	—	12	4	no	—
298	C₄	—	60	4	no	—
299	C₄	#91270	1	2	yes	—
300	C₄	#91271	1	2	yes	—
301	C₄	C₂×D₅×Dic₆	1	2	yes	—
302	C₃	#10613669	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	10	2	no	—
304	C₂	D₅×#869	1	1	yes	characteristic
305	C₂	C₂×#91270	1	1	yes	characteristic
306	C₂	C₂×#91271	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₅×D₁₂
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#3160
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #11750783	C₂	semidirect
3. C₅×#4682	C₂	semidirect
5. #11747986	C₂	semidirect
7. #11746941	C₂	semidirect
48. C₃×C₈×D₅	Q₈	semidirect
58. #4682	D₅	semidirect
128. C₈×D₅	Dic₆	semidirect
252. C₃×C₅	#3160	semidirect
289. C₅	#78442	semidirect
302. C₃	#10613669	semidirect

Maximal subgroups

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

subgroup	index
#11752506	2
#11750783	2
C₅×#4682	2
C₃×C₄×C₈×D₅	2
#11747986	2
#11747672	2
#11746941	2
#10613669	3
#78442	5