Subgroups — #14285853

Order 1920.

336 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	#11750938	C₂	1	1	yes	maximal, characteristic
2	D₁₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#11750786	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746300	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746300	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599365	—	3	1	no	maximal, Hall
9	C₂×#6705	—	2	1	no	—
10	C₂×C₈×D₁₅	—	2	1	no	—
11	C₄²×D₁₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#91104	C₄	1	1	yes	characteristic
16	#91104	C₄	1	1	yes	characteristic
17	#91098	—	2	1	no	—
18	#91098	—	2	1	no	—
19	#91098	C₄	1	1	yes	characteristic
20	#91098	C₄	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
23	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
24	#68572	—	5	1	no	maximal, Hall
25	#65126	—	3	1	no	—
26	D₅×C₄⋊C₈	—	3	1	no	—
27	#64974	—	3	1	no	—
28	C₅×#335	D₃	1	1	yes	characteristic, complemented
29	#64699	—	3	1	no	—
30	#64699	—	3	1	no	—
31	#64678	—	3	1	no	—
32	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
34	C₂×C₄×D₁₅	Q₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₁₅	D₄	1	1	yes	characteristic
42	C₄×Dic₁₅	Q₈	1	1	yes	characteristic
43	C₂×C₃⋊#168	—	2	1	no	—
44	C₂×C₃⋊#168	—	2	1	no	—
45	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
47	#6705	—	4	2	no	—
48	C₈×D₁₅	—	4	2	no	—
49	#4837	—	5	1	no	—
50	D₃×C₄⋊C₈	—	5	1	no	—
51	#4685	—	5	1	no	—
52	C₃×#335	D₅	1	1	yes	characteristic, complemented
53	#4479	—	5	1	no	—
54	#4479	—	5	1	no	—
55	#4458	—	5	1	no	—
56	C₂×#676	—	6	1	no	—
57	C₂×C₈×D₅	—	6	1	no	—
58	C₄²×D₅	—	3	1	no	—
59	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
60	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
62	#3952	—	3	1	no	—
63	#3952	—	3	1	no	—
64	#3946	—	3	1	no	—
65	#3946	—	3	1	no	—
66	#3946	—	6	1	no	—
67	#3946	—	6	1	no	—
68	C₄⋊#168	—	3	1	no	—
69	C₄⋊#168	—	3	1	no	—
70	C₄×#168	—	3	1	no	—
71	#1571	—	15	1	no	Sylow
72	C₂²×D₁₅	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
76	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
77	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
78	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
79	C₄×D₁₅	—	2	1	no	—
80	C₄×D₁₅	—	2	1	no	—
81	C₄×D₁₅	—	4	1	no	—
82	C₄×D₁₅	D₈	1	2	yes	—
83	C₄×D₁₅	Q₁₆	1	2	yes	—
84	C₃×C₅×C₈	—	2	1	no	—
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	—	4	1	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₃⋊#168	—	2	1	no	—
89	C₃⋊#168	—	2	1	no	—
90	C₃⋊#168	—	4	1	no	—
91	C₃⋊#168	—	4	1	no	—
92	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
93	C₂×D₃×C₈	—	10	1	no	—
94	C₄²×D₃	—	5	1	no	—
95	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
96	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
97	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
98	(C₂×D₃)⋊C₈	—	5	1	no	—
99	(C₂×D₃)⋊C₈	—	5	1	no	—
100	Dic₃⋊C₈	—	5	1	no	—
101	Dic₃⋊C₈	—	5	1	no	—
102	Dic₃⋊C₈	—	10	1	no	—
103	Dic₃⋊C₈	—	10	1	no	—
104	C₄⋊(C₃⋊C₈)	—	5	1	no	—
105	C₄⋊(C₃⋊C₈)	—	5	1	no	—
106	C₄×C₃⋊C₈	—	5	1	no	—
107	C₂×C₄×D₅	—	3	1	no	—
108	C₂×C₄×D₅	—	3	1	no	—
109	C₂×C₄×D₅	—	3	1	no	—
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
113	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
114	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₄×Dic₅	—	3	1	no	—
116	C₄×Dic₅	—	3	1	no	—
117	C₄×Dic₅	—	3	1	no	—
118	C₂×#168	—	3	1	no	—
119	C₂×#168	—	3	1	no	—
120	C₂×#168	—	6	1	no	—
121	C₂×#168	—	6	1	no	—
122	#676	—	12	2	no	—
123	C₈×D₅	—	12	2	no	—
124	#431	—	15	1	no	—
125	#423	—	15	1	no	—
126	C₂×C₄⋊C₈	—	15	1	no	—
127	#335	—	15	1	no	—
128	#335	—	15	1	no	—
129	#335	—	15	1	no	—
130	#335	D₁₅	1	1	yes	characteristic, complemented
131	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×D₁₅	—	2	1	no	—
133	C₂×D₁₅	—	2	1	no	—
134	C₂×D₁₅	#107	1	2	yes	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	#131	1	1	yes	characteristic
137	C₃×C₄×C₅	#131	1	1	yes	characteristic
138	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
140	Dic₁₅	—	2	1	no	—
141	Dic₁₅	—	2	1	no	—
142	Dic₁₅	#107	1	2	yes	—
143	C₂×C₄×D₃	—	5	1	no	—
144	C₂×C₄×D₃	—	5	1	no	—
145	C₂×C₄×D₃	—	5	1	no	—
146	C₂×C₃×C₈	—	2	1	no	—
147	C₂×C₃×C₈	—	2	1	no	—
148	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
149	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
150	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
151	C₄×Dic₃	—	5	1	no	—
152	C₄×Dic₃	—	5	1	no	—
153	C₄×Dic₃	—	5	1	no	—
154	C₂×C₃⋊C₈	—	5	1	no	—
155	C₂×C₃⋊C₈	—	5	1	no	—
156	C₂×C₃⋊C₈	—	10	1	no	—
157	C₂×C₃⋊C₈	—	10	1	no	—
158	ΓL(1,𝔽₂₅)	—	20	2	no	—
159	D₃×C₈	—	20	2	no	—
160	C₂²×D₅	—	3	1	no	—
161	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
164	C₂×Dic₅	—	3	1	no	—
165	C₂×Dic₅	—	3	1	no	—
166	C₂×Dic₅	—	3	1	no	—
167	C₄×D₅	—	3	2	no	—
168	C₄×D₅	—	3	2	no	—
169	C₄×D₅	—	6	1	no	—
170	C₄×D₅	—	6	1	no	—
171	C₄×D₅	—	12	1	no	—
172	C₅×C₈	—	2	1	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	4	1	no	—
175	C₅×C₈	—	4	1	no	—
176	#168	—	6	1	no	—
177	#168	—	6	1	no	—
178	#168	—	12	1	no	—
179	#168	—	12	1	no	—
180	C₂×M₁₆	—	30	1	no	—
181	C₂²×C₈	—	30	1	no	—
182	C₂×C₄²	—	15	1	no	—
183	C₄⋊C₈	—	15	1	no	—
184	C₄⋊C₈	—	15	1	no	—
185	C₄⋊C₈	—	15	1	no	—
186	C₄⋊C₈	—	15	1	no	—
187	C₄⋊C₈	—	30	1	no	—
188	C₄⋊C₈	—	30	1	no	—
189	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
190	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
191	C₂²⋊C₈	—	15	1	no	—
192	C₂²⋊C₈	—	15	1	no	—
193	C₄×C₈	—	15	1	no	—
194	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
195	C₂×C₃×C₅	#424	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
197	C₂×C₃×C₅	#430	1	1	yes	characteristic
198	D₁₅	—	2	2	no	—
199	C₂²×D₃	—	5	1	no	—
200	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
201	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
202	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
203	C₂×Dic₃	—	5	1	no	—
204	C₂×Dic₃	—	5	1	no	—
205	C₂×Dic₃	—	5	1	no	—
206	C₄×D₃	—	5	2	no	—
207	C₄×D₃	—	5	2	no	—
208	C₄×D₃	—	10	1	no	—
209	C₄×D₃	—	10	1	no	—
210	C₄×D₃	—	20	1	no	—
211	C₃×C₈	—	2	1	no	—
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	4	1	no	—
214	C₃×C₈	—	4	1	no	—
215	C₃⋊C₈	—	10	1	no	—
216	C₃⋊C₈	—	10	1	no	—
217	C₃⋊C₈	—	20	1	no	—
218	C₃⋊C₈	—	20	1	no	—
219	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
220	C₂×D₅	—	3	2	no	—
221	C₂×D₅	—	6	1	no	—
222	C₂×D₅	—	6	1	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	#901	1	1	yes	characteristic
225	C₄×C₅	#907	1	1	yes	characteristic
226	C₄×C₅	D₃×D₈	1	1	yes	characteristic
227	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
228	Dic₅	—	3	2	no	—
229	Dic₅	—	6	1	no	—
230	Dic₅	—	6	1	no	—
231	C₂²×C₄	—	15	1	no	—
232	C₂²×C₄	—	15	1	no	—
233	C₂²×C₄	—	15	1	no	—
234	M₁₆	—	60	2	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	15	1	no	—
238	C₂×C₈	—	15	1	no	—
239	C₂×C₈	—	30	1	no	—
240	C₂×C₈	—	30	1	no	—
241	C₂×C₈	—	60	2	no	—
242	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
243	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
244	C₄²	—	15	1	no	—
245	C₄²	—	15	1	no	—
246	C₄²	—	15	1	no	—
247	C₄²	C₂²×D₁₅	1	1	yes	characteristic
248	C₃×C₅	#1571	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
250	C₂×D₃	—	5	2	no	—
251	C₂×D₃	—	10	1	no	—
252	C₂×D₃	—	10	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	#4047	1	1	yes	characteristic
255	C₃×C₄	#4053	1	1	yes	characteristic
256	C₃×C₄	D₅×D₈	1	1	yes	characteristic
257	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
258	Dic₃	—	5	2	no	—
259	Dic₃	—	10	1	no	—
260	Dic₃	—	10	1	no	—
261	C₂×C₅	#4839	1	1	yes	characteristic
262	C₂×C₅	D₃×#107	1	1	yes	characteristic
263	C₂×C₅	#4893	1	1	yes	characteristic
264	D₅	—	6	2	no	—
265	C₂³	—	15	1	no	—
266	C₂×C₄	—	15	1	no	—
267	C₂×C₄	—	15	2	no	—
268	C₂×C₄	—	15	2	no	—
269	C₂×C₄	—	15	1	no	—
270	C₂×C₄	—	15	1	no	—
271	C₂×C₄	—	30	1	no	—
272	C₂×C₄	—	30	1	no	—
273	C₂×C₄	—	60	1	no	—
274	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
275	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
276	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	—	30	1	no	—
282	C₈	—	30	1	no	—
283	C₈	—	60	1	no	—
284	C₈	—	60	1	no	—
285	C₂×C₃	#65128	1	1	yes	characteristic
286	C₂×C₃	D₅×#107	1	1	yes	characteristic
287	C₂×C₃	#65182	1	1	yes	characteristic
288	D₃	—	10	2	no	—
289	C₅	#68572	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	15	2	no	—
291	C₂²	—	30	1	no	—
292	C₂²	—	30	1	no	—
293	C₂²	C₄⋊C₄×D₁₅	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	15	2	no	—
296	C₄	—	30	1	no	—
297	C₄	—	30	1	no	—
298	C₄	#91790	1	1	yes	characteristic
299	C₄	#91796	1	1	yes	characteristic
300	C₄	D₈×D₁₅	1	1	yes	characteristic
301	C₄	Q₁₆×D₁₅	1	1	yes	characteristic
302	C₃	#10599365	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	30	2	no	—
304	C₂	#11750940	1	1	yes	characteristic
305	C₂	D₁₅×#107	1	1	yes	characteristic
306	C₂	#11750994	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₁₅	C₄⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#335	D₁₅
hypocenter	C₃×C₅	#1571
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#335	C₂	semidirect
5. #11746300	C₂	semidirect
6. #11746300	C₂	semidirect
7. #11746279	C₂	semidirect
28. C₅×#335	D₃	semidirect
52. C₃×#335	D₅	semidirect
130. #335	D₁₅	semidirect
248. C₃×C₅	#1571	semidirect
289. C₅	#68572	semidirect
302. C₃	#10599365	semidirect

Maximal subgroups

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

subgroup	index
#11750938	2
D₁₅×C₄⋊C₈	2
#11750786	2
C₃×C₅×#335	2
#11746300	2
#11746279	2
#10599365	3
#68572	5