Subgroups — #14415393

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5477	C₂	1	1	yes	maximal, characteristic
2	C₂×#91288	C₂	1	2	yes	maximal, complemented
3	C₂×#91087	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748347	C₂	1	2	yes	maximal, complemented
6	#11746999	C₂	1	8	yes	maximal, complemented
7	#10615007	—	3	1	no	maximal, Hall
8	C₂×C₅×#235	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
12	C₅×#930	C₄	1	4	yes	—
13	C₅×#875	C₄	1	2	yes	—
14	C₅×#874	C₄	1	1	yes	characteristic
15	C₄×C₅×D₁₂	C₂²	1	4	yes	—
16	C₅×#869	C₄	1	1	yes	characteristic
17	C₂×C₃×#681	C₂²	1	2	yes	—
18	C₂×#6650	C₂²	1	4	yes	—
19	#91288	—	2	8	no	—
20	C₂×D₃×#168	C₂²	1	4	yes	—
21	#91087	C₂²	1	4	yes	—
22	C₃×C₄⋊#168	C₂²	1	4	yes	—
23	(C₂×D₃)⋊#168	C₂²	1	8	yes	—
24	#79042	—	5	1	no	maximal, Hall
25	C₅×#532	—	3	1	no	—
26	C₂×#4094	—	3	2	no	—
27	#65523	—	3	2	no	—
28	#65300	—	3	8	no	—
29	#65213	D₃	1	1	yes	characteristic, complemented
30	C₂×#3935	—	3	1	no	—
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×#67	C₂×C₄	1	4	yes	—
34	C₅×#235	—	2	4	no	—
35	C₅×#235	—	2	4	no	—
36	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
37	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
38	C₂×C₄×C₅×D₃	C₂³	1	4	yes	—
39	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	2	yes	—
43	C₂×C₃⋊#168	—	2	2	no	—
44	C₂×C₃⋊#168	D₄	1	4	yes	complemented
45	C₂×C₃⋊#168	C₂³	1	2	yes	—
46	C₂×C₃⋊#168	C₂³	1	2	yes	—
47	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	—
48	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
49	C₃×#681	—	2	4	no	—
50	C₂×C₃×#168	C₂³	1	4	yes	—
51	#6650	—	2	8	no	—
52	D₃×#168	—	2	8	no	—
53	C₂×#907	—	5	2	no	—
54	#5477	D₅	1	1	yes	characteristic
55	C₃×#423	—	5	1	no	—
56	#5129	—	5	2	no	—
57	C₂×#803	—	5	1	no	—
58	#4836	—	5	8	no	—
59	C₂×C₅×#41	—	3	2	no	—
60	C₅×C₄⋊Q₈	—	3	1	no	—
61	C₅×#127	—	3	1	no	—
62	C₅×#124	—	3	2	no	—
63	C₅×#121	—	3	4	no	—
64	C₄×C₅×D₄	—	3	4	no	—
65	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
66	#4094	—	6	8	no	—
67	C₂×#681	—	3	4	no	—
68	C₂×#681	C₂×D₃	1	2	yes	—
69	C₂²×#168	—	3	1	no	—
70	C₂²×#168	—	3	1	no	—
71	C₂²×#168	—	3	4	no	—
72	C₂²⋊#168	—	3	8	no	—
73	C₄⋊#168	C₂×D₃	1	4	yes	—
74	#3935	—	3	4	no	—
75	#2932	—	15	1	no	Sylow
76	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	—	4	2	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	#41	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₅×#67	—	2	8	no	—
86	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
87	C₅×D₁₂	—	2	4	no	—
88	C₅×D₁₂	—	4	4	no	—
89	C₄×C₅×D₃	—	2	4	no	—
90	C₄×C₅×D₃	—	2	4	no	—
91	C₅×Dic₆	—	2	4	no	—
92	C₃⋊#168	—	2	2	no	—
93	C₃⋊#168	C₂×D₄	1	4	yes	—
94	C₃×#168	—	2	4	no	—
95	C₂×#235	C₂×D₅	1	2	yes	—
96	C₂×C₃×M₁₆	—	5	2	no	—
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	#930	Dic₅	1	4	yes	—
99	C₂²×C₃⋊C₈	—	5	1	no	—
100	C₂²×C₃⋊C₈	—	5	1	no	—
101	#907	—	10	8	no	—
102	C₂×ΓL(1,𝔽₂₅)	—	5	4	no	—
103	C₂×D₃×C₈	—	5	4	no	—
104	#875	Dic₅	1	2	yes	—
105	#874	Dic₅	1	1	yes	characteristic
106	C₄×D₁₂	C₂×D₅	1	4	yes	—
107	#869	Dic₅	1	1	yes	characteristic
108	C₃×C₄⋊C₈	—	5	4	no	—
109	(C₂×D₃)⋊C₈	—	5	8	no	—
110	#803	—	5	4	no	—
111	C₅×#41	—	6	4	no	—
112	C₅×#41	—	6	4	no	—
113	C₂×C₅×Q₈	—	3	2	no	—
114	C₂×C₅×D₄	—	3	4	no	—
115	C₂×C₅×D₄	—	3	2	no	—
116	C₂×C₅×D₄	—	3	4	no	—
117	C₂²×C₄×C₅	—	3	4	no	—
118	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
119	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
120	C₅×C₄⋊C₄	—	3	4	no	—
121	C₅×C₂²⋊C₄	—	3	8	no	—
122	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
123	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
124	C₄²×C₅	C₂²×D₃	1	2	yes	—
125	#681	—	2	4	no	—
126	#681	—	6	8	no	—
127	C₂×#168	—	3	4	no	—
128	C₂×#168	—	3	2	no	—
129	C₂×#168	—	3	2	no	—
130	C₂×#168	—	6	8	no	—
131	C₂×#168	—	6	2	no	—
132	C₂×#168	C₂²×D₃	1	4	yes	—
133	C₂×#131	—	15	2	no	—
134	#532	—	3	1	no	—
135	#435	—	15	8	no	—
136	#423	—	5	1	no	—
137	#408	—	15	2	no	—
138	C₂×#97	—	15	1	no	—
139	C₂²×C₃×C₅	—	2	2	no	—
140	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
141	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₅×D₃	—	2	4	no	—
143	C₂×C₅×D₃	—	2	8	no	—
144	C₃×C₄×C₅	—	2	2	no	—
145	C₃×C₄×C₅	—	2	2	no	—
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
149	C₅×Dic₃	—	2	4	no	—
150	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
151	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
152	C₂×#67	C₂×Dic₅	1	4	yes	—
153	#235	—	2	4	no	—
154	#235	—	2	4	no	—
155	C₂×D₁₂	C₂×Dic₅	1	2	yes	—
156	C₂×D₁₂	C₂×Dic₅	1	4	yes	—
157	C₂×C₄×D₃	C₂²×D₅	1	4	yes	—
158	C₂×Dic₆	C₂×Dic₅	1	2	yes	—
159	C₃×M₁₆	—	10	4	no	—
160	C₂×C₃×C₈	—	5	4	no	—
161	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
162	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
163	C₃×C₄²	C₂²×D₅	1	2	yes	—
164	(C₂×D₃)⋊C₄	C₂×Dic₅	1	8	yes	—
165	C₄⋊Dic₃	C₂×Dic₅	1	4	yes	—
166	C₂×C₃⋊C₈	—	5	4	no	—
167	C₂×C₃⋊C₈	—	5	2	no	—
168	C₂×C₃⋊C₈	—	5	2	no	—
169	C₂×C₃⋊C₈	—	10	2	no	—
170	ΓL(1,𝔽₂₅)	—	10	8	no	—
171	D₃×C₈	—	10	8	no	—
172	C₂³×C₅	—	3	4	no	—
173	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₅×Q₈	—	6	4	no	—
175	C₅×D₄	—	6	8	no	—
176	C₅×D₄	—	6	4	no	—
177	C₅×D₄	—	12	4	no	—
178	C₂×C₄×C₅	—	3	4	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	—	6	4	no	—
181	C₂×C₄×C₅	—	6	4	no	—
182	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
183	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
184	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
185	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
186	C₂×C₄×C₅	#239	1	2	yes	—
187	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
188	#168	—	2	4	no	—
189	#168	—	3	4	no	—
190	#168	—	6	2	no	—
191	C₂×#41	—	3	2	no	—
192	#131	—	30	8	no	—
193	C₂×M₁₆	—	5	2	no	—
194	C₂×M₁₆	—	15	4	no	—
195	C₂²×C₈	—	15	4	no	—
196	C₂²×C₈	—	15	1	no	—
197	C₂²×C₈	—	15	1	no	—
198	C₄⋊Q₈	—	3	1	no	—
199	#127	—	3	1	no	—
200	#124	—	3	2	no	—
201	#121	—	3	4	no	—
202	C₄×D₄	—	3	4	no	—
203	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
204	C₄⋊C₈	—	5	4	no	—
205	C₂²⋊C₈	—	15	8	no	—
206	#97	—	15	4	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
209	C₂×C₃×C₅	#568	1	2	yes	—
210	C₅×D₃	—	4	4	no	—
211	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
212	C₂²×D₃	C₂²×Dic₅	1	4	yes	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
215	C₂×C₃×C₄	#711	1	2	yes	—
216	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
217	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
218	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
220	#67	—	2	8	no	—
221	C₂×Dic₃	C₂²×Dic₅	1	4	yes	—
222	D₁₂	—	2	4	no	—
223	D₁₂	—	4	4	no	—
224	C₄×D₃	—	2	4	no	—
225	C₄×D₃	—	2	4	no	—
226	Dic₆	—	2	4	no	—
227	C₃×C₈	—	10	4	no	—
228	C₃⋊C₈	—	5	4	no	—
229	C₃⋊C₈	—	10	2	no	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	—	6	8	no	—
232	C₂²×C₅	—	6	4	no	—
233	C₂²×C₅	#893	1	2	yes	—
234	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	—	6	4	no	—
238	C₄×C₅	#893	1	2	yes	—
239	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
240	C₄×C₅	C₂×#239	1	1	yes	characteristic
241	#41	—	6	4	no	—
242	#41	—	6	4	no	—
243	C₂×Q₈	—	3	2	no	—
244	C₂×D₄	—	3	4	no	—
245	C₂×D₄	—	3	2	no	—
246	C₂×D₄	—	3	4	no	—
247	C₂²×C₄	—	3	4	no	—
248	C₂²×C₄	D₃×Dic₅	1	2	yes	—
249	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
250	M₁₆	—	10	4	no	—
251	M₁₆	—	30	8	no	—
252	C₂×C₈	—	5	4	no	—
253	C₂×C₈	—	15	4	no	—
254	C₂×C₈	—	15	2	no	—
255	C₂×C₈	—	15	2	no	—
256	C₂×C₈	—	30	8	no	—
257	C₂×C₈	—	30	2	no	—
258	C₄⋊C₄	—	3	4	no	—
259	C₂²⋊C₄	—	3	8	no	—
260	C₄²	D₃×Dic₅	1	1	yes	characteristic
261	C₄²	D₃×Dic₅	1	1	yes	characteristic
262	C₄²	C₂×D₃×D₅	1	2	yes	—
263	C₃×C₅	#2932	1	1	yes	characteristic, complemented, Hall
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	D₄×Dic₅	1	2	yes	—
266	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
267	C₂×D₃	—	2	4	no	—
268	C₂×D₃	—	2	8	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	D₄×Dic₅	1	2	yes	—
272	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
273	C₃×C₄	C₂×#711	1	1	yes	characteristic
274	Dic₃	—	2	4	no	—
275	C₂×C₅	—	2	2	no	—
276	C₂×C₅	—	12	4	no	—
277	C₂×C₅	C₂×#893	1	1	yes	characteristic
278	C₂×C₅	#5745	1	2	yes	—
279	C₂³	—	3	4	no	—
280	C₂³	C₂×D₃×Dic₅	1	1	yes	characteristic
281	Q₈	—	6	4	no	—
282	D₄	—	6	8	no	—
283	D₄	—	6	4	no	—
284	D₄	—	12	4	no	—
285	C₂×C₄	—	3	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	6	4	no	—
288	C₂×C₄	—	6	4	no	—
289	C₂×C₄	#6768	1	2	yes	—
290	C₂×C₄	#6777	1	2	yes	—
291	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
292	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
293	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
295	C₈	—	10	4	no	—
296	C₈	—	15	4	no	—
297	C₈	—	30	2	no	—
298	C₂×C₃	—	2	2	no	—
299	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
300	C₂×C₃	#66153	1	2	yes	—
301	D₃	—	4	4	no	—
302	C₅	#79042	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	2	2	no	—
304	C₂²	—	6	8	no	—
305	C₂²	—	6	4	no	—
306	C₂²	#91435	1	2	yes	—
307	C₂²	C₂²×D₃×Dic₅	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	6	4	no	—
311	C₄	#91435	1	2	yes	—
312	C₄	C₂×#6768	1	1	yes	characteristic
313	C₄	C₂×#6777	1	1	yes	characteristic
314	C₃	#10615007	1	1	yes	characteristic, complemented, Sylow
315	C₂	—	2	2	no	—
316	C₂	—	12	4	no	—
317	C₂	#11754125	1	2	yes	—
318	C₂	C₂×#91435	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₂×C₃×C₄²×C₅	C₂²
prime radical	C₅×#5477	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#2932
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#91288	C₂	semidirect
3. C₂×#91087	C₂	semidirect
4. C₃×#65213	C₂	semidirect
5. #11748347	C₂	semidirect
6. #11746999	C₂	semidirect
29. #65213	D₃	semidirect
44. C₂×C₃⋊#168	D₄	semidirect
263. C₃×C₅	#2932	semidirect
302. C₅	#79042	semidirect
314. C₃	#10615007	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5477	2
C₂×#91288	2
C₂×#91087	2
C₃×#65213	2
#11748347	2
#11746999	2
#10615007	3
#79042	5