Subgroups — #14315126

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×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#1220	C₂	1	1	yes	maximal, characteristic
3	C₂×#91082	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746370	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746328	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₆×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602223	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₄×#1220	C₄	1	1	yes	characteristic
11	C₂×C₈×D₁₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×M₃₂	—	2	2	no	—
13	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	#91114	C₄	1	1	yes	characteristic
16	#91104	C₄	1	1	yes	characteristic
17	#91098	C₄	1	1	yes	characteristic
18	C₈×Dic₁₅	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
20	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
21	#91082	—	2	2	no	—
22	#71430	—	5	1	no	maximal, Hall
23	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic, complemented
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	—	3	1	no	—
29	C₁₆×Dic₅	—	3	1	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×#1220	C₈	1	1	yes	characteristic
32	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	—	2	1	no	—
34	C₂×C₃×C₅×C₈	—	2	1	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
37	#6719	C₈	1	1	yes	characteristic
38	#6714	C₈	1	1	yes	characteristic
39	#6712	C₈	1	1	yes	characteristic
40	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
43	C₈×D₁₅	—	2	1	no	—
44	C₈×D₁₅	—	2	1	no	—
45	C₃×C₅×C₁₆	—	2	1	no	—
46	C₃×C₅×C₁₆	D₄	1	2	yes	complemented
47	C₃⋊#672	—	2	1	no	—
48	C₃⋊#672	—	2	1	no	—
49	C₂×C₃×M₃₂	D₅	1	1	yes	characteristic, complemented
50	C₈×#67	—	5	1	no	—
51	C₂×#798	—	5	1	no	—
52	#4549	—	5	1	no	—
53	(C₂×D₃)⋊C₁₆	—	5	1	no	—
54	Dic₃⋊C₁₆	—	5	1	no	—
55	Dic₃×C₁₆	—	5	1	no	—
56	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
57	C₄×C₂²⋊D₅	—	3	1	no	—
58	C₂×C₈×D₅	—	3	1	no	—
59	C₅×M₃₂	—	2	2	no	—
60	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
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	—	3	1	no	—
66	C₂×#672	—	3	1	no	—
67	C₂×#672	—	3	1	no	—
68	#3930	—	6	2	no	—
69	#2169	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
71	C₂²×D₁₅	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	#1220	—	2	2	no	—
77	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
78	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
79	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
80	C₄×D₁₅	—	2	1	no	—
81	C₄×D₁₅	—	2	1	no	—
82	C₃×C₅×C₈	—	2	1	no	—
83	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₃×C₅×C₈	#41	1	1	yes	characteristic
85	C₃⋊#168	—	2	1	no	—
86	C₃⋊#168	—	2	1	no	—
87	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
88	C₄×#67	—	5	1	no	—
89	C₂×D₃×C₈	—	5	1	no	—
90	C₃×M₃₂	—	2	2	no	—
91	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
92	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
93	#830	—	5	1	no	—
94	(C₂×D₃)⋊C₈	—	5	1	no	—
95	Dic₃⋊C₈	—	5	1	no	—
96	C₈×Dic₃	—	5	1	no	—
97	C₂×C₃⋊C₁₆	—	5	1	no	—
98	C₂×C₃⋊C₁₆	—	5	1	no	—
99	#798	—	10	2	no	—
100	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
101	C₂×C₂²⋊D₅	—	3	1	no	—
102	C₂×C₄×D₅	—	3	1	no	—
103	C₂×C₅×C₈	—	2	1	no	—
104	C₂×C₅×C₈	—	2	1	no	—
105	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
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	—	3	1	no	—
111	C₂×#168	—	3	1	no	—
112	C₂×#168	—	3	1	no	—
113	C₈×D₅	—	6	1	no	—
114	C₈×D₅	—	6	1	no	—
115	C₅×C₁₆	—	2	1	no	—
116	C₅×C₁₆	#67	1	2	yes	complemented
117	#672	—	6	1	no	—
118	#672	—	6	1	no	—
119	C₂×M₃₂	—	15	1	no	—
120	C₂×M₃₂	D₁₅	1	1	yes	characteristic, complemented
121	C₈×D₄	—	15	1	no	—
122	C₄⋊C₁₆	—	15	1	no	—
123	C₂²⋊C₁₆	—	15	1	no	—
124	C₂²⋊C₁₆	—	15	1	no	—
125	C₄×C₁₆	—	15	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
130	C₂×D₁₅	—	2	1	no	—
131	C₂×D₁₅	—	2	1	no	—
132	C₂×D₁₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
135	C₃×C₄×C₅	#131	1	1	yes	characteristic
136	Dic₁₅	—	2	1	no	—
137	Dic₁₅	—	2	1	no	—
138	Dic₁₅	M₃₂	1	2	yes	complemented
139	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
140	C₂×#67	—	5	1	no	—
141	C₂×C₄×D₃	—	5	1	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
145	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
146	#217	—	5	1	no	—
147	(C₂×D₃)⋊C₄	—	5	1	no	—
148	Dic₃⋊C₄	—	5	1	no	—
149	C₄×Dic₃	—	5	1	no	—
150	C₂×C₃⋊C₈	—	5	1	no	—
151	C₂×C₃⋊C₈	—	5	1	no	—
152	D₃×C₈	—	10	1	no	—
153	D₃×C₈	—	10	1	no	—
154	C₃×C₁₆	—	2	1	no	—
155	C₃×C₁₆	C₂²⋊D₅	1	2	yes	complemented
156	C₃⋊C₁₆	—	10	1	no	—
157	C₃⋊C₁₆	—	10	1	no	—
158	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
159	C₂²×D₅	—	3	1	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
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	—	3	1	no	—
167	C₂×Dic₅	—	3	1	no	—
168	C₄×D₅	—	6	1	no	—
169	C₄×D₅	—	6	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	#235	1	1	yes	characteristic
172	C₅×C₈	C₂×#67	1	1	yes	characteristic
173	#168	—	6	1	no	—
174	#168	—	6	1	no	—
175	C₂²×C₈	—	15	1	no	—
176	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
177	C₄×D₄	—	15	1	no	—
178	M₃₂	—	2	2	no	—
179	M₃₂	—	30	2	no	—
180	C₂×C₁₆	—	15	1	no	—
181	C₂×C₁₆	—	15	1	no	—
182	C₂×C₁₆	C₂×D₁₅	1	1	yes	characteristic
183	C₂×C₁₆	C₂×D₁₅	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	—	15	1	no	—
188	C₂×C₃×C₅	—	4	1	no	—
189	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
191	C₂×C₃×C₅	#504	1	1	yes	characteristic
192	D₁₅	—	4	1	no	—
193	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
194	C₂²×D₃	—	5	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
199	#67	—	10	2	no	—
200	C₂×Dic₃	—	5	1	no	—
201	C₂×Dic₃	—	5	1	no	—
202	C₂×Dic₃	—	5	1	no	—
203	C₄×D₃	—	10	1	no	—
204	C₄×D₃	—	10	1	no	—
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	#709	1	1	yes	characteristic
207	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	C₃⋊C₈	—	10	1	no	—
209	C₃⋊C₈	—	10	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	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	—	2	1	no	—
218	C₄×C₅	#901	1	1	yes	characteristic
219	C₄×C₅	C₄×#67	1	1	yes	characteristic
220	Dic₅	—	3	2	no	—
221	Dic₅	—	6	1	no	—
222	Dic₅	—	6	1	no	—
223	C₂×D₄	—	15	1	no	—
224	C₂²×C₄	—	15	1	no	—
225	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	15	1	no	—
229	C₂×C₈	—	15	1	no	—
230	C₂×C₈	—	30	1	no	—
231	C₂×C₈	—	30	1	no	—
232	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
234	C₄⋊C₄	—	15	1	no	—
235	C₂²⋊C₄	—	15	1	no	—
236	C₂²⋊C₄	—	15	1	no	—
237	C₄²	—	15	1	no	—
238	C₁₆	—	2	1	no	—
239	C₁₆	—	30	1	no	—
240	C₁₆	—	30	1	no	—
241	C₁₆	#1220	1	2	yes	complemented
242	C₃×C₅	#2169	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
247	C₂×D₃	—	10	1	no	—
248	C₂×D₃	—	10	1	no	—
249	C₂×D₃	—	10	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	#4047	1	1	yes	characteristic
252	C₃×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
253	Dic₃	—	5	2	no	—
254	Dic₃	—	10	1	no	—
255	Dic₃	—	10	1	no	—
256	C₂×C₅	—	4	1	no	—
257	C₂×C₅	D₃×M₃₂	1	1	yes	characteristic
258	C₂×C₅	#4907	1	1	yes	characteristic
259	C₂×C₅	C₈×#67	1	1	yes	characteristic
260	D₅	—	12	1	no	—
261	C₂³	—	15	1	no	—
262	C₂³	C₈×D₁₅	1	1	yes	characteristic
263	D₄	—	30	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	15	1	no	—
267	C₂×C₄	—	15	1	no	—
268	C₂×C₄	—	15	1	no	—
269	C₂×C₄	—	30	1	no	—
270	C₂×C₄	—	30	1	no	—
271	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	30	1	no	—
275	C₈	—	30	1	no	—
276	C₈	#6817	1	1	yes	characteristic
277	C₈	C₂×#1220	1	1	yes	characteristic
278	C₂×C₃	—	4	1	no	—
279	C₂×C₃	D₅×M₃₂	1	1	yes	characteristic
280	C₂×C₃	#65196	1	1	yes	characteristic
281	C₂×C₃	C₈×C₂²⋊D₅	1	1	yes	characteristic
282	D₃	—	20	1	no	—
283	C₅	#71430	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	30	1	no	—
288	C₂²	—	30	1	no	—
289	C₂²	—	30	1	no	—
290	C₂²	C₂×C₈×D₁₅	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	15	2	no	—
293	C₄	—	30	1	no	—
294	C₄	—	30	1	no	—
295	C₄	#91790	1	1	yes	characteristic
296	C₄	C₄×#1220	1	1	yes	characteristic
297	C₃	#10602223	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	4	1	no	—
299	C₂	—	60	1	no	—
300	C₂	D₁₅×M₃₂	1	1	yes	characteristic
301	C₂	#11751008	1	1	yes	characteristic
302	C₂	C₈×#1220	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂×C₃×C₅×M₃₂	C₂
prime radical	C₂×#1220	C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×M₃₂	D₁₅
hypocenter	C₃×C₅	#2169
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₂×#91082	C₂	semidirect
4. #11746370	C₂	semidirect
5. #11746328	C₂	semidirect
6. #11746322	C₂	semidirect
7. C₁₆×Dic₁₅	C₂	semidirect
23. C₂×C₅×M₃₂	D₃	semidirect
46. C₃×C₅×C₁₆	D₄	semidirect
49. C₂×C₃×M₃₂	D₅	semidirect
116. C₅×C₁₆	#67	semidirect
120. C₂×M₃₂	D₁₅	semidirect
138. Dic₁₅	M₃₂	semidirect
155. C₃×C₁₆	C₂²⋊D₅	semidirect
241. C₁₆	#1220	semidirect
242. C₃×C₅	#2169	semidirect
283. C₅	#71430	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×C₃×C₅×M₃₂	2
C₈×#1220	2
C₂×#91082	2
#11746370	2
#11746328	2
#11746322	2
C₁₆×Dic₁₅	2
#10602223	3
#71430	5