Subgroups — #14304983

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#91098	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#91087	C₂	1	1	yes	maximal, characteristic
5	#11746371	C₂	1	2	yes	maximal
6	#10601262	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₅	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
11	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	—	2	2	no	—
14	#91115	C₄	1	1	yes	characteristic
15	#91115	C₄	1	1	yes	characteristic
16	#91098	—	2	4	no	—
17	#91087	—	2	2	no	—
18	#70469	—	5	1	no	maximal, Hall
19	C₂×C₅×C₄⋊C₈	D₃	1	1	yes	characteristic
20	C₂×#3946	—	3	2	no	—
21	C₄×C₄⋊Dic₅	—	3	1	no	—
22	C₂×#3935	—	3	1	no	—
23	#64770	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₅	C₂×C₄	1	2	yes	—
28	C₂²×Dic₁₅	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₃×C₄²×C₅	D₄	1	2	yes	—
33	C₃×C₄²×C₅	Q₈	1	2	yes	—
34	C₄⋊Dic₁₅	—	2	4	no	—
35	C₄×Dic₁₅	—	2	2	no	—
36	C₄×Dic₁₅	—	2	2	no	—
37	C₂×C₃⋊#168	—	2	1	no	—
38	C₂×C₃⋊#168	—	2	1	no	—
39	C₂×C₃⋊#168	—	2	1	no	—
40	C₂×C₃⋊#168	—	2	1	no	—
41	C₂×C₃⋊#168	D₄	1	2	yes	—
42	C₂×C₃⋊#168	Q₈	1	2	yes	—
43	C₂×C₃×C₄⋊C₈	D₅	1	1	yes	characteristic
44	C₂×Dic₃⋊C₈	—	5	2	no	—
45	C₄×C₄⋊Dic₃	—	5	1	no	—
46	C₂×#803	—	5	1	no	—
47	#4550	—	5	2	no	—
48	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
49	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
50	C₂×C₄⋊Dic₅	—	3	2	no	—
51	C₂×C₄×Dic₅	—	3	2	no	—
52	C₂²×#168	—	3	1	no	—
53	C₂²×#168	—	3	1	no	—
54	C₅×C₄⋊C₈	—	2	2	no	—
55	#3963	—	3	1	no	—
56	#3963	—	3	1	no	—
57	#3946	—	6	4	no	—
58	#3935	—	6	2	no	—
59	#1940	—	15	1	no	Sylow
60	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
71	C₂×Dic₁₅	—	2	2	no	—
72	C₂×Dic₁₅	—	2	2	no	—
73	C₂×Dic₁₅	—	2	2	no	—
74	C₂×Dic₁₅	—	2	2	no	—
75	C₂×Dic₁₅	—	2	2	no	—
76	C₂×Dic₁₅	M₁₆	1	4	yes	—
77	C₃×C₅×C₈	—	4	2	no	—
78	C₃⋊#168	—	2	2	no	—
79	C₃⋊#168	—	2	2	no	—
80	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
81	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
82	C₂×C₄⋊Dic₃	—	5	2	no	—
83	C₂×C₄×Dic₃	—	5	2	no	—
84	C₂²×C₃⋊C₈	—	5	1	no	—
85	C₂²×C₃⋊C₈	—	5	1	no	—
86	C₃×C₄⋊C₈	—	2	2	no	—
87	C₃⋊#95	—	5	1	no	—
88	C₃⋊#95	—	5	1	no	—
89	Dic₃⋊C₈	—	10	4	no	—
90	#803	—	10	2	no	—
91	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₂²×Dic₅	—	3	2	no	—
95	C₂²×Dic₅	—	3	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₄²×C₅	Dic₆	1	2	yes	—
100	C₄²×C₅	#67	1	2	yes	—
101	C₄⋊Dic₅	—	6	4	no	—
102	C₄×Dic₅	—	6	2	no	—
103	C₄×Dic₅	—	6	2	no	—
104	C₂×#168	—	3	2	no	—
105	C₂×#168	—	3	2	no	—
106	C₂×#168	—	6	1	no	—
107	C₂×#168	—	6	1	no	—
108	C₂×#168	—	6	1	no	—
109	C₂×#168	—	6	1	no	—
110	C₂×C₄⋊C₈	—	15	2	no	—
111	C₂×C₄⋊C₈	D₁₅	1	1	yes	characteristic
112	C₂×#97	—	15	1	no	—
113	C₄×C₄⋊C₄	—	15	1	no	—
114	#336	—	15	2	no	—
115	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
118	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
119	C₂²×C₃×C₅	#131	1	2	yes	—
120	C₃×C₄×C₅	—	2	2	no	—
121	C₃×C₄×C₅	—	2	2	no	—
122	C₃×C₄×C₅	#121	1	1	yes	characteristic
123	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
124	C₃×C₄×C₅	#125	1	1	yes	characteristic
125	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
126	Dic₁₅	—	2	4	no	—
127	Dic₁₅	—	4	2	no	—
128	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
131	C₂²×Dic₃	—	5	2	no	—
132	C₂²×Dic₃	—	5	2	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₃×C₄²	Dic₁₀	1	2	yes	—
137	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
138	C₄⋊Dic₃	—	10	4	no	—
139	C₄×Dic₃	—	10	2	no	—
140	C₄×Dic₃	—	10	2	no	—
141	C₂×C₃⋊C₈	—	5	2	no	—
142	C₂×C₃⋊C₈	—	5	2	no	—
143	C₂×C₃⋊C₈	—	10	1	no	—
144	C₂×C₃⋊C₈	—	10	1	no	—
145	C₂×C₃⋊C₈	—	10	1	no	—
146	C₂×C₃⋊C₈	—	10	1	no	—
147	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
151	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
152	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
153	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	#237	1	1	yes	characteristic
155	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅	#239	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
158	C₂×Dic₅	—	3	4	no	—
159	C₂×Dic₅	—	6	2	no	—
160	C₂×Dic₅	—	6	2	no	—
161	C₂×Dic₅	—	6	2	no	—
162	C₂×Dic₅	—	6	2	no	—
163	C₂×Dic₅	—	6	2	no	—
164	C₅×C₈	—	4	2	no	—
165	#168	—	6	2	no	—
166	#168	—	6	2	no	—
167	C₂²×C₈	—	15	1	no	—
168	C₂²×C₈	—	15	1	no	—
169	C₂²×C₈	C₂×D₁₅	1	2	yes	—
170	C₂×C₄⋊C₄	—	15	2	no	—
171	C₂×C₄²	—	15	2	no	—
172	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	30	4	no	—
175	#97	—	30	2	no	—
176	#95	—	15	1	no	—
177	#95	—	15	1	no	—
178	C₂×C₃×C₅	#389	1	1	yes	characteristic
179	C₂×C₃×C₅	#423	1	1	yes	characteristic
180	C₂×C₃×C₅	#424	1	1	yes	characteristic
181	C₂×C₃×C₅	#435	1	2	yes	—
182	C₂×C₃×C₅	#446	1	2	yes	—
183	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	#683	1	2	yes	—
187	C₂×C₃×C₄	#683	1	2	yes	—
188	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
189	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
190	C₂×C₃×C₄	#711	1	1	yes	characteristic
191	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
192	C₂×C₃×C₄	#713	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
194	C₂×Dic₃	—	5	4	no	—
195	C₂×Dic₃	—	10	2	no	—
196	C₂×Dic₃	—	10	2	no	—
197	C₂×Dic₃	—	10	2	no	—
198	C₂×Dic₃	—	10	2	no	—
199	C₂×Dic₃	—	10	2	no	—
200	C₃×C₈	—	4	2	no	—
201	C₃⋊C₈	—	10	2	no	—
202	C₃⋊C₈	—	10	2	no	—
203	C₂²×C₅	#887	1	1	yes	characteristic
204	C₂²×C₅	#893	1	1	yes	characteristic
205	C₂²×C₅	#901	1	2	yes	—
206	C₂²×C₅	D₃×M₁₆	1	2	yes	—
207	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	#888	1	1	yes	characteristic
211	C₄×C₅	#890	1	1	yes	characteristic
212	C₄×C₅	#938	1	1	yes	characteristic
213	C₄×C₅	#946	1	1	yes	characteristic
214	Dic₅	—	6	4	no	—
215	Dic₅	—	12	2	no	—
216	C₂²×C₄	—	15	2	no	—
217	C₂²×C₄	—	15	2	no	—
218	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
219	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	15	2	no	—
225	C₂×C₈	—	15	2	no	—
226	C₂×C₈	—	30	1	no	—
227	C₂×C₈	—	30	1	no	—
228	C₂×C₈	—	30	1	no	—
229	C₂×C₈	—	30	1	no	—
230	C₄⋊C₄	—	30	4	no	—
231	C₄²	—	30	2	no	—
232	C₄²	—	30	2	no	—
233	C₄²	Dic₃₀	1	2	yes	—
234	C₄²	#1220	1	2	yes	—
235	C₃×C₅	#1940	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	#4033	1	1	yes	characteristic
237	C₂²×C₃	#4039	1	1	yes	characteristic
238	C₂²×C₃	#4047	1	2	yes	—
239	C₂²×C₃	D₅×M₁₆	1	2	yes	—
240	C₂²×C₃	C₂×#683	1	1	yes	characteristic
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	#4034	1	1	yes	characteristic
244	C₃×C₄	#4036	1	1	yes	characteristic
245	C₃×C₄	#4084	1	1	yes	characteristic
246	C₃×C₄	#4092	1	1	yes	characteristic
247	Dic₃	—	10	4	no	—
248	Dic₃	—	20	2	no	—
249	C₂×C₅	#4828	1	2	yes	—
250	C₂×C₅	#4836	1	2	yes	—
251	C₂×C₅	#4972	1	1	yes	characteristic
252	C₂×C₅	#5102	1	1	yes	characteristic
253	C₂×C₅	#5118	1	1	yes	characteristic
254	C₂³	C₂×C₄×D₁₅	1	1	yes	characteristic
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	15	4	no	—
258	C₂×C₄	—	30	2	no	—
259	C₂×C₄	—	30	2	no	—
260	C₂×C₄	—	30	2	no	—
261	C₂×C₄	—	30	2	no	—
262	C₂×C₄	—	30	2	no	—
263	C₂×C₄	#6712	1	2	yes	—
264	C₂×C₄	#6712	1	2	yes	—
265	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
266	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
267	C₂×C₄	#6819	1	1	yes	characteristic
268	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
269	C₂×C₄	#6821	1	1	yes	characteristic
270	C₂×C₄	C₂×#1220	1	1	yes	characteristic
271	C₈	—	4	2	no	—
272	C₈	—	30	2	no	—
273	C₈	—	30	2	no	—
274	C₂×C₃	#65117	1	2	yes	—
275	C₂×C₃	#65125	1	2	yes	—
276	C₂×C₃	#65262	1	1	yes	characteristic
277	C₂×C₃	#65391	1	1	yes	characteristic
278	C₂×C₃	#65407	1	1	yes	characteristic
279	C₅	#70469	1	1	yes	characteristic, complemented, Sylow
280	C₂²	#91776	1	1	yes	characteristic
281	C₂²	#91782	1	1	yes	characteristic
282	C₂²	#91790	1	2	yes	—
283	C₂²	M₁₆×D₁₅	1	2	yes	—
284	C₂²	C₂×#6712	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	30	4	no	—
288	C₄	—	60	2	no	—
289	C₄	#91777	1	1	yes	characteristic
290	C₄	#91779	1	1	yes	characteristic
291	C₄	#91827	1	1	yes	characteristic
292	C₄	#91835	1	1	yes	characteristic
293	C₃	#10601262	1	1	yes	characteristic, complemented, Sylow
294	C₂	#11750929	1	2	yes	—
295	C₂	#11750937	1	2	yes	—
296	C₂	#11751073	1	1	yes	characteristic
297	C₂	#11751203	1	1	yes	characteristic
298	C₂	#11751219	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅
commutator, socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
Fitting	C₂×C₃×C₅×C₄⋊C₈	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅
hypocenter	C₃×C₅	#1940
LCS	G ▷ C₂³×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
235. C₃×C₅	#1940	semidirect
279. C₅	#70469	semidirect
293. C₃	#10601262	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊C₈	2
C₂×#91098	2
C₄×C₄⋊Dic₁₅	2
C₂×#91087	2
#11746371	2
#10601262	3
#70469	5