Subgroups — #14306012

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 292 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	C₄²×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90978	C₂	1	4	yes	maximal, complemented
5	#10602608	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅	—	2	2	no	—
9	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
10	C₅×#930	—	2	2	no	—
11	C₅×#875	C₄	1	2	yes	complemented
12	C₅×#874	C₄	1	1	yes	characteristic, complemented
13	C₄×C₅×D₁₂	—	2	2	no	—
14	C₅×#869	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×#681	C₂²	1	2	yes	—
16	C₃×C₄⋊#168	—	2	2	no	—
17	#90978	—	2	8	no	—
18	#70512	—	5	1	no	maximal, Hall
19	C₅×#532	—	3	1	no	—
20	C₂×#3969	—	3	4	no	—
21	C₄²×Dic₅	—	3	1	no	—
22	#65213	D₃	1	1	yes	characteristic, complemented
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₅	—	2	2	no	—
26	C₂×C₅×#67	—	2	2	no	—
27	C₅×#235	—	2	4	no	—
28	C₅×#235	—	2	4	no	—
29	C₂×C₅×D₁₂	—	2	2	no	—
30	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₅×D₃	—	2	2	no	—
32	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	D₄	1	2	yes	—
36	C₄×Dic₁₅	—	2	8	no	—
37	C₄×Dic₁₅	—	2	2	no	—
38	C₄×Dic₁₅	D₄	1	4	yes	complemented
39	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
40	C₅×C₄⋊Dic₃	—	2	2	no	—
41	C₃×#681	—	2	4	no	—
42	C₂×C₃×#168	—	2	2	no	—
43	#5477	D₅	1	1	yes	characteristic
44	C₃×#423	—	5	1	no	—
45	C₂×#825	—	5	4	no	—
46	C₄²×Dic₃	—	5	1	no	—
47	C₂×C₅×#41	—	3	2	no	—
48	C₅×C₄⋊Q₈	—	3	1	no	—
49	C₅×#127	—	3	1	no	—
50	C₅×#124	—	3	2	no	—
51	C₅×#121	—	6	2	no	—
52	C₄×C₅×D₄	—	6	2	no	—
53	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
54	C₂×C₄×Dic₅	—	3	2	no	—
55	C₂×C₄×Dic₅	—	6	2	no	—
56	C₂×#681	C₂×D₃	1	2	yes	—
57	#3969	—	6	8	no	—
58	C₄⋊#168	—	2	2	no	—
59	#1962	—	15	1	no	Sylow
60	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₅×D₃	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	2	yes	—
69	C₂×Dic₁₅	—	2	8	no	—
70	C₂×Dic₁₅	—	2	4	no	—
71	C₅×#67	—	4	4	no	—
72	C₂×C₅×Dic₃	—	2	2	no	—
73	C₅×D₁₂	—	2	4	no	—
74	C₅×D₁₂	—	4	4	no	—
75	C₄×C₅×D₃	—	4	2	no	—
76	C₄×C₅×D₃	—	4	2	no	—
77	C₅×Dic₆	—	2	4	no	—
78	C₃×#168	—	4	2	no	—
79	C₂×#235	C₂×D₅	1	2	yes	—
80	C₂×C₃×M₁₆	—	5	2	no	—
81	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
82	#930	—	2	2	no	—
83	C₂×C₄×Dic₃	—	5	2	no	—
84	C₂×C₄×Dic₃	—	10	2	no	—
85	#875	Dic₅	1	2	yes	complemented
86	#874	Dic₅	1	1	yes	characteristic, complemented
87	C₄×D₁₂	—	2	2	no	—
88	#869	Dic₅	1	1	yes	characteristic, complemented
89	C₃×C₄⋊C₈	—	10	2	no	—
90	#825	—	10	8	no	—
91	C₅×#41	—	6	4	no	—
92	C₅×#41	—	6	4	no	—
93	C₂×C₅×Q₈	—	3	2	no	—
94	C₂×C₅×D₄	—	3	2	no	—
95	C₂×C₅×D₄	—	6	2	no	—
96	C₂×C₅×D₄	—	6	2	no	—
97	C₂²×C₄×C₅	—	6	2	no	—
98	C₂²×C₄×C₅	D₁₂	1	2	yes	—
99	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
100	C₂²×Dic₅	—	6	2	no	—
101	C₅×C₄⋊C₄	—	6	2	no	—
102	C₅×C₂²⋊C₄	—	6	4	no	—
103	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
104	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
105	C₄²×C₅	#67	1	2	yes	—
106	C₄×Dic₅	—	3	4	no	—
107	C₄×Dic₅	—	6	8	no	—
108	C₄×Dic₅	—	6	2	no	—
109	#681	—	2	4	no	—
110	C₂×#168	—	2	2	no	—
111	#532	—	3	1	no	—
112	#423	—	5	1	no	—
113	C₂×#104	—	15	4	no	—
114	C₄³	—	15	1	no	—
115	C₂²×C₃×C₅	—	2	2	no	—
116	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	#121	1	2	yes	—
118	C₂×C₅×D₃	—	4	2	no	—
119	C₂×C₅×D₃	—	4	4	no	—
120	C₃×C₄×C₅	—	2	2	no	—
121	C₃×C₄×C₅	#104	1	4	yes	complemented
122	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
123	C₃×C₄×C₅	#124	1	1	yes	characteristic
124	C₃×C₄×C₅	#127	1	1	yes	characteristic
125	Dic₁₅	—	2	8	no	—
126	C₅×Dic₃	—	4	2	no	—
127	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
128	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
129	C₂×#67	—	2	2	no	—
130	C₂²×Dic₃	—	10	2	no	—
131	#235	—	2	4	no	—
132	#235	—	2	4	no	—
133	C₂×D₁₂	—	2	2	no	—
134	C₂×D₁₂	C₂×Dic₅	1	2	yes	—
135	C₂×C₄×D₃	—	2	2	no	—
136	C₂×Dic₆	C₂×Dic₅	1	2	yes	—
137	C₃×M₁₆	—	10	4	no	—
138	C₂×C₃×C₈	—	10	2	no	—
139	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
140	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
141	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
142	(C₂×D₃)⋊C₄	—	2	4	no	—
143	C₄⋊Dic₃	—	2	2	no	—
144	C₄×Dic₃	—	5	4	no	—
145	C₄×Dic₃	—	10	8	no	—
146	C₄×Dic₃	—	10	2	no	—
147	C₂³×C₅	—	6	2	no	—
148	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
149	C₅×Q₈	—	6	4	no	—
150	C₅×D₄	—	6	4	no	—
151	C₅×D₄	—	12	4	no	—
152	C₅×D₄	—	12	4	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	6	2	no	—
155	C₂×C₄×C₅	—	12	2	no	—
156	C₂×C₄×C₅	—	12	2	no	—
157	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
158	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
159	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
160	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
161	C₂×C₄×C₅	#239	1	2	yes	—
162	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
163	C₂×Dic₅	—	6	8	no	—
164	C₂×Dic₅	—	6	4	no	—
165	#168	—	4	2	no	—
166	C₂×#41	—	3	2	no	—
167	C₂×M₁₆	—	5	2	no	—
168	C₄⋊Q₈	—	3	1	no	—
169	#127	—	3	1	no	—
170	#124	—	3	2	no	—
171	#121	—	6	2	no	—
172	C₄×D₄	—	6	2	no	—
173	C₂×C₄²	—	15	2	no	—
174	C₂×C₄²	—	30	2	no	—
175	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
176	C₄⋊C₈	—	10	2	no	—
177	#104	—	30	8	no	—
178	C₂×C₃×C₅	—	2	2	no	—
179	C₂×C₃×C₅	#390	1	1	yes	characteristic
180	C₂×C₃×C₅	C₂×#104	1	2	yes	—
181	C₅×D₃	—	8	2	no	—
182	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
183	C₂²×D₃	—	2	2	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
186	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
187	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
188	C₂×C₃×C₄	#711	1	2	yes	—
189	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
191	#67	—	4	4	no	—
192	C₂×Dic₃	—	2	2	no	—
193	C₂×Dic₃	—	10	8	no	—
194	C₂×Dic₃	—	10	4	no	—
195	D₁₂	—	2	4	no	—
196	D₁₂	—	4	4	no	—
197	C₄×D₃	—	4	2	no	—
198	C₄×D₃	—	4	2	no	—
199	Dic₆	—	2	4	no	—
200	C₃×C₈	—	20	2	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	—	12	4	no	—
203	C₂²×C₅	—	12	2	no	—
204	C₂²×C₅	#895	1	2	yes	—
205	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	12	2	no	—
208	C₄×C₅	#825	1	4	yes	complemented
209	C₄×C₅	#893	1	2	yes	—
210	C₄×C₅	#940	1	1	yes	characteristic
211	C₄×C₅	#947	1	1	yes	characteristic
212	Dic₅	—	6	8	no	—
213	#41	—	6	4	no	—
214	#41	—	6	4	no	—
215	C₂×Q₈	—	3	2	no	—
216	C₂×D₄	—	3	2	no	—
217	C₂×D₄	—	6	2	no	—
218	C₂×D₄	—	6	2	no	—
219	C₂²×C₄	—	6	2	no	—
220	C₂²×C₄	—	30	2	no	—
221	C₂²×C₄	#1203	1	2	yes	—
222	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
223	M₁₆	—	10	4	no	—
224	C₂×C₈	—	10	2	no	—
225	C₄⋊C₄	—	6	2	no	—
226	C₂²⋊C₄	—	6	4	no	—
227	C₄²	—	15	4	no	—
228	C₄²	—	30	8	no	—
229	C₄²	—	30	2	no	—
230	C₄²	D₃×Dic₅	1	1	yes	characteristic
231	C₄²	D₃×Dic₅	1	1	yes	characteristic
232	C₄²	#1201	1	2	yes	—
233	C₃×C₅	#1962	1	1	yes	characteristic, complemented, Hall
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	#4084	1	2	yes	—
236	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
237	C₂×D₃	—	4	2	no	—
238	C₂×D₃	—	4	4	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	#3969	1	4	yes	complemented
241	C₃×C₄	D₄×Dic₅	1	2	yes	—
242	C₃×C₄	#4082	1	1	yes	characteristic
243	C₃×C₄	#4086	1	1	yes	characteristic
244	Dic₃	—	4	2	no	—
245	Dic₃	—	10	8	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	24	2	no	—
248	C₂×C₅	#4988	1	1	yes	characteristic
249	C₂×C₅	C₂×#825	1	2	yes	—
250	C₂³	—	6	2	no	—
251	C₂³	C₂×#1203	1	1	yes	characteristic
252	Q₈	—	6	4	no	—
253	D₄	—	6	4	no	—
254	D₄	—	12	4	no	—
255	D₄	—	12	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	6	2	no	—
258	C₂×C₄	—	12	2	no	—
259	C₂×C₄	—	12	2	no	—
260	C₂×C₄	—	30	8	no	—
261	C₂×C₄	—	30	4	no	—
262	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
263	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
264	C₂×C₄	#6768	1	2	yes	—
265	C₂×C₄	#6777	1	2	yes	—
266	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
267	C₂×C₄	C₂×#1201	1	1	yes	characteristic
268	C₈	—	20	2	no	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	#65510	1	1	yes	characteristic
271	C₂×C₃	C₂×#3969	1	2	yes	—
272	D₃	—	8	2	no	—
273	C₅	#70512	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	2	2	no	—
275	C₂²	—	12	4	no	—
276	C₂²	—	12	2	no	—
277	C₂²	#91466	1	2	yes	—
278	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	12	2	no	—
281	C₄	—	30	8	no	—
282	C₄	#90978	1	4	yes	complemented
283	C₄	#91370	1	1	yes	characteristic
284	C₄	#91435	1	2	yes	—
285	C₄	#91463	1	1	yes	characteristic
286	C₃	#10602608	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	2	2	no	—
288	C₂	—	24	2	no	—
289	C₂	C₂×#90978	1	2	yes	—
290	C₂	#11748797	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1201
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5477	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1962
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×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. C₄²×Dic₁₅	C₂	semidirect
3. C₃×#65213	C₂	semidirect
4. C₂×#90978	C₂	semidirect
11. C₅×#875	C₄	semidirect
12. C₅×#874	C₄	semidirect
14. C₅×#869	C₄	semidirect
22. #65213	D₃	semidirect
38. C₄×Dic₁₅	D₄	semidirect
85. #875	Dic₅	semidirect
86. #874	Dic₅	semidirect
88. #869	Dic₅	semidirect
121. C₃×C₄×C₅	#104	semidirect
208. C₄×C₅	#825	semidirect
233. C₃×C₅	#1962	semidirect
240. C₃×C₄	#3969	semidirect
273. C₅	#70512	semidirect
282. C₄	#90978	semidirect
286. C₃	#10602608	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
C₄²×Dic₁₅	2
C₃×#65213	2
C₂×#90978	2
#10602608	3
#70512	5