Subgroups — #14315145

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#798	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#1203	C₂	1	1	yes	maximal, characteristic
3	C₂×#90936	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746328	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64721	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₃)⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602223	—	3	1	no	maximal, Hall
9	C₂×C₈×D₁₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
11	C₄×#1203	C₄	1	1	yes	characteristic
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅×#798	—	2	2	no	—
16	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×C₃×#672	C₂²	1	1	yes	characteristic
18	#90982	C₄	1	1	yes	characteristic
19	#90957	C₄	1	1	yes	characteristic
20	(C₂×D₃)⋊#168	C₄	1	1	yes	characteristic
21	#90936	—	2	2	no	—
22	#69279	—	5	1	no	maximal, Hall
23	C₂×C₅×M₃₂	—	3	1	no	—
24	C₈×C₂²⋊D₅	—	3	1	no	—
25	C₂×#3930	—	3	1	no	—
26	C₂²⋊#672	—	3	1	no	—
27	#64727	—	3	1	no	—
28	#64721	D₃	1	1	yes	characteristic, complemented
29	C₁₆×Dic₅	—	3	1	no	—
30	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#1203	C₈	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
35	C₈×D₁₅	—	2	1	no	—
36	C₈×D₁₅	—	2	1	no	—
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₅×D₃×C₈	—	2	1	no	—
39	C₅×D₃×C₈	—	2	1	no	—
40	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	#6668	C₈	1	1	yes	characteristic
43	#6667	C₈	1	1	yes	characteristic
44	(C₂×D₃)⋊Dic₅	C₈	1	1	yes	characteristic
45	C₃×C₅×C₁₆	—	2	1	no	—
46	C₃⋊#672	—	2	1	no	—
47	C₃×#672	—	2	1	no	—
48	C₅×C₃⋊C₁₆	D₄	1	2	yes	complemented
49	C₂×#798	—	5	1	no	—
50	C₂×#798	D₅	1	1	yes	characteristic, complemented
51	C₈×D₁₂	—	5	1	no	—
52	C₃×C₄⋊C₁₆	—	5	1	no	—
53	(C₂×D₃)⋊C₁₆	—	5	1	no	—
54	(C₂×D₃)⋊C₁₆	—	5	1	no	—
55	C₄×C₃⋊C₁₆	—	5	1	no	—
56	C₂²×C₅×C₈	—	3	1	no	—
57	C₄×C₂²⋊D₅	—	3	1	no	—
58	C₂×C₈×D₅	—	3	1	no	—
59	C₅×M₃₂	—	6	2	no	—
60	C₂×C₅×C₁₆	—	3	1	no	—
61	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
62	C₂²⋊#168	—	3	1	no	—
63	#3952	—	3	1	no	—
64	#3946	—	3	1	no	—
65	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
66	C₂×#672	—	3	1	no	—
67	C₂×#672	C₂×D₃	1	1	yes	characteristic
68	#3930	—	6	2	no	—
69	#2169	—	15	1	no	Sylow
70	C₂²×D₁₅	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
74	C₄×D₁₅	—	2	1	no	—
75	C₄×D₁₅	—	2	1	no	—
76	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
77	C₄×C₅×D₃	—	2	1	no	—
78	C₄×C₅×D₃	—	2	1	no	—
79	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
80	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
81	#1203	—	2	2	no	—
82	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₃×C₅×C₈	#41	1	1	yes	characteristic
84	C₃⋊#168	—	2	1	no	—
85	C₃×#168	—	2	1	no	—
86	C₅×C₃⋊C₈	—	2	1	no	—
87	C₂×D₃×C₈	—	5	1	no	—
88	C₂×D₃×C₈	C₂×D₅	1	1	yes	characteristic
89	C₄×D₁₂	—	5	1	no	—
90	C₂×C₃×C₁₆	—	5	1	no	—
91	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
92	C₃×C₄×C₈	—	5	1	no	—
93	(C₂×D₃)⋊C₈	—	5	1	no	—
94	(C₂×D₃)⋊C₈	—	5	1	no	—
95	C₂×C₃⋊C₁₆	—	5	1	no	—
96	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
97	C₄⋊(C₃⋊C₈)	—	5	1	no	—
98	#798	—	2	2	no	—
99	#798	—	10	2	no	—
100	C₂²×C₄×C₅	—	3	1	no	—
101	C₂×C₂²⋊D₅	—	3	1	no	—
102	C₂×C₄×D₅	—	3	1	no	—
103	C₂×C₅×C₈	—	3	1	no	—
104	C₂×C₅×C₈	—	6	1	no	—
105	C₂×C₅×C₈	—	6	1	no	—
106	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
107	C₂²⋊Dic₅	—	3	1	no	—
108	#685	—	3	1	no	—
109	#683	—	3	1	no	—
110	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
111	C₂×#168	—	3	1	no	—
112	C₂×#168	C₄×D₃	1	1	yes	characteristic
113	C₈×D₅	—	6	1	no	—
114	C₈×D₅	—	6	1	no	—
115	C₅×C₁₆	—	2	1	no	—
116	C₅×C₁₆	—	3	2	no	—
117	#672	—	2	1	no	—
118	#672	—	6	1	no	—
119	C₂×M₃₂	—	3	1	no	—
120	C₂×M₃₂	—	15	1	no	—
121	C₈×D₄	—	15	1	no	—
122	C₄⋊C₁₆	—	5	1	no	—
123	C₂²⋊C₁₆	—	15	1	no	—
124	C₂²⋊C₁₆	—	15	1	no	—
125	C₄×C₁₆	—	15	1	no	—
126	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
127	C₂×D₁₅	—	2	1	no	—
128	C₂×D₁₅	—	2	1	no	—
129	C₂×D₁₅	—	2	1	no	—
130	C₂×C₅×D₃	—	2	1	no	—
131	C₂×C₅×D₃	—	2	1	no	—
132	C₂×C₅×D₃	—	2	1	no	—
133	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
134	C₃×C₄×C₅	#131	1	1	yes	characteristic
135	Dic₁₅	—	2	1	no	—
136	C₃×Dic₅	—	2	1	no	—
137	C₃×Dic₅	M₃₂	1	2	yes	complemented
138	C₅×Dic₃	—	2	1	no	—
139	C₂×D₁₂	—	5	1	no	—
140	C₂×C₄×D₃	—	5	1	no	—
141	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
142	C₂×C₃×C₈	—	5	1	no	—
143	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
144	C₃×C₄²	—	5	1	no	—
145	(C₂×D₃)⋊C₄	—	5	1	no	—
146	(C₂×D₃)⋊C₄	—	5	1	no	—
147	C₄⋊Dic₃	—	5	1	no	—
148	C₂×C₃⋊C₈	—	5	1	no	—
149	C₂×C₃⋊C₈	C₄×D₅	1	1	yes	characteristic
150	D₃×C₈	—	2	1	no	—
151	D₃×C₈	—	2	1	no	—
152	D₃×C₈	—	10	1	no	—
153	D₃×C₈	—	10	1	no	—
154	C₃×C₁₆	—	2	1	no	—
155	C₃×C₁₆	—	10	1	no	—
156	C₃⋊C₁₆	—	10	1	no	—
157	C₃⋊C₁₆	C₂²⋊D₅	1	2	yes	complemented
158	C₂³×C₅	—	3	1	no	—
159	C₂²×D₅	—	3	1	no	—
160	C₂×C₄×C₅	—	3	1	no	—
161	C₂×C₄×C₅	—	6	1	no	—
162	C₂×C₄×C₅	—	6	1	no	—
163	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂²⋊D₅	—	6	2	no	—
165	C₂×Dic₅	—	3	1	no	—
166	C₂×Dic₅	D₃×C₈	1	1	yes	characteristic
167	C₂×Dic₅	D₃×C₈	1	1	yes	characteristic
168	C₄×D₅	—	6	1	no	—
169	C₄×D₅	—	6	1	no	—
170	C₅×C₈	—	6	1	no	—
171	C₅×C₈	D₃×D₄	1	1	yes	characteristic
172	C₅×C₈	#239	1	1	yes	characteristic
173	#168	—	2	1	no	—
174	#168	—	6	1	no	—
175	C₂²×C₈	—	3	1	no	—
176	C₂²×C₈	—	15	1	no	—
177	C₄×D₄	—	15	1	no	—
178	M₃₂	—	6	2	no	—
179	M₃₂	—	30	2	no	—
180	C₂×C₁₆	—	3	1	no	—
181	C₂×C₁₆	—	5	1	no	—
182	C₂×C₁₆	—	15	1	no	—
183	C₂×C₁₆	D₃×D₅	1	1	yes	characteristic
184	C₄⋊C₈	—	15	1	no	—
185	C₂²⋊C₈	—	15	1	no	—
186	C₂²⋊C₈	—	15	1	no	—
187	C₄×C₈	—	5	1	no	—
188	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
189	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
190	C₂×C₃×C₅	#504	1	1	yes	characteristic
191	D₁₅	—	4	1	no	—
192	C₅×D₃	—	4	1	no	—
193	C₂²×D₃	—	5	1	no	—
194	C₂²×D₃	C₈×D₅	1	1	yes	characteristic
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	—	5	1	no	—
197	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
198	C₂×Dic₃	—	5	1	no	—
199	C₂×Dic₃	C₈×D₅	1	1	yes	characteristic
200	D₁₂	—	10	2	no	—
201	C₄×D₃	—	2	1	no	—
202	C₄×D₃	—	2	1	no	—
203	C₄×D₃	—	10	1	no	—
204	C₄×D₃	—	10	1	no	—
205	C₃×C₈	—	10	1	no	—
206	C₃×C₈	#709	1	1	yes	characteristic
207	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	C₃⋊C₈	—	2	1	no	—
209	C₃⋊C₈	—	10	1	no	—
210	C₂²×C₅	—	6	1	no	—
211	C₂²×C₅	—	6	1	no	—
212	C₂²×C₅	—	6	1	no	—
213	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
214	C₂×D₅	—	6	1	no	—
215	C₂×D₅	—	6	1	no	—
216	C₂×D₅	—	6	1	no	—
217	C₄×C₅	—	6	1	no	—
218	C₄×C₅	#893	1	1	yes	characteristic
219	C₄×C₅	#907	1	1	yes	characteristic
220	Dic₅	—	2	1	no	—
221	Dic₅	—	6	1	no	—
222	Dic₅	#798	1	2	yes	complemented
223	C₂×D₄	—	15	1	no	—
224	C₂²×C₄	—	3	1	no	—
225	C₂²×C₄	—	15	1	no	—
226	C₂×C₈	—	3	1	no	—
227	C₂×C₈	—	5	1	no	—
228	C₂×C₈	—	6	1	no	—
229	C₂×C₈	—	6	1	no	—
230	C₂×C₈	—	15	1	no	—
231	C₂×C₈	—	30	1	no	—
232	C₂×C₈	—	30	1	no	—
233	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
234	C₄⋊C₄	—	15	1	no	—
235	C₂²⋊C₄	—	15	1	no	—
236	C₂²⋊C₄	—	15	1	no	—
237	C₄²	—	5	1	no	—
238	C₁₆	—	2	1	no	—
239	C₁₆	—	3	2	no	—
240	C₁₆	—	10	1	no	—
241	C₁₆	—	30	1	no	—
242	C₃×C₅	#2169	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
244	C₂×D₃	—	2	1	no	—
245	C₂×D₃	—	2	1	no	—
246	C₂×D₃	—	2	1	no	—
247	C₂×D₃	—	10	1	no	—
248	C₂×D₃	—	10	1	no	—
249	C₂×D₃	—	10	1	no	—
250	C₃×C₄	—	5	2	no	—
251	C₃×C₄	—	10	1	no	—
252	C₃×C₄	#4047	1	1	yes	characteristic
253	C₃×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
254	Dic₃	—	2	1	no	—
255	Dic₃	—	10	1	no	—
256	C₂×C₅	—	12	1	no	—
257	C₂×C₅	#4834	1	1	yes	characteristic
258	C₂×C₅	C₂×#798	1	1	yes	characteristic
259	C₂×C₅	#4907	1	1	yes	characteristic
260	D₅	—	12	1	no	—
261	C₂³	—	3	1	no	—
262	C₂³	—	15	1	no	—
263	D₄	—	30	2	no	—
264	C₂×C₄	—	3	1	no	—
265	C₂×C₄	—	5	1	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	6	1	no	—
268	C₂×C₄	—	6	1	no	—
269	C₂×C₄	—	15	1	no	—
270	C₂×C₄	—	30	1	no	—
271	C₂×C₄	—	30	1	no	—
272	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
273	C₈	—	6	1	no	—
274	C₈	—	10	1	no	—
275	C₈	—	30	1	no	—
276	C₈	#6769	1	1	yes	characteristic
277	C₈	D₃×C₂²⋊D₅	1	1	yes	characteristic
278	C₂×C₃	D₅×M₃₂	1	1	yes	characteristic
279	C₂×C₃	#65196	1	1	yes	characteristic
280	C₂×C₃	C₈×C₂²⋊D₅	1	1	yes	characteristic
281	D₃	—	4	1	no	—
282	D₃	—	20	1	no	—
283	C₅	#69279	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	6	1	no	—
285	C₂²	—	6	1	no	—
286	C₂²	—	6	1	no	—
287	C₂²	—	30	1	no	—
288	C₂²	—	30	1	no	—
289	C₂²	—	30	1	no	—
290	C₂²	D₃×C₈×D₅	1	1	yes	characteristic
291	C₄	—	5	2	no	—
292	C₄	—	6	1	no	—
293	C₄	—	10	1	no	—
294	C₄	—	30	1	no	—
295	C₄	#91267	1	1	yes	characteristic
296	C₄	#91446	1	1	yes	characteristic
297	C₃	#10602223	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	12	1	no	—
299	C₂	—	60	1	no	—
300	C₂	D₅×#798	1	1	yes	characteristic
301	C₂	#11746813	1	1	yes	characteristic
302	C₂	#11747700	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂×C₃×C₅×C₁₆	C₂²
prime radical	C₂×#1203	C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×C₁₆	D₃×D₅
hypocenter	C₃×C₅	#2169
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#798	C₂	semidirect
3. C₂×#90936	C₂	semidirect
4. #11746328	C₂	semidirect
5. C₃×#64721	C₂	semidirect
6. (C₂×D₃)⋊#672	C₂	semidirect
7. Dic₅×C₃⋊C₁₆	C₂	semidirect
28. #64721	D₃	semidirect
48. C₅×C₃⋊C₁₆	D₄	semidirect
50. C₂×#798	D₅	semidirect
137. C₃×Dic₅	M₃₂	semidirect
157. C₃⋊C₁₆	C₂²⋊D₅	semidirect
222. Dic₅	#798	semidirect
242. C₃×C₅	#2169	semidirect
283. C₅	#69279	semidirect
297. C₃	#10602223	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#798	2
C₈×#1203	2
C₂×#90936	2
#11746328	2
C₃×#64721	2
(C₂×D₃)⋊#672	2
Dic₅×C₃⋊C₁₆	2
#10602223	3
#69279	5