Subgroups — #14313916

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4050	C₂	1	1	yes	maximal, characteristic
2	C₂×#91102	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745898	C₂	1	2	yes	maximal, complemented
5	#11745892	C₂	1	2	yes	maximal, complemented
6	#10602232	—	3	1	no	maximal, Hall
7	C₂×C₃×#709	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₅	—	2	1	no	—
10	C₂×C₃×Dic₂₀	C₂²	1	1	yes	characteristic
11	C₃×#4050	—	2	2	no	—
12	C₃×#4050	C₄	1	4	yes	complemented
13	C₂×C₃×D₄₀	C₂²	1	1	yes	characteristic
14	C₂×C₃×#677	—	2	1	no	—
15	#91102	—	2	1	no	—
16	#91102	C₂²	1	2	yes	—
17	C₅×#812	—	2	2	no	—
18	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	—
19	#72078	—	5	1	no	maximal, Hall
20	C₂×#4050	D₃	1	1	yes	characteristic
21	C₂×C₅×M₃₂	—	3	1	no	—
22	C₂×#3950	—	3	1	no	—
23	#64729	—	3	2	no	—
24	#64723	—	3	2	no	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×Dic₁₅	—	4	1	no	—
27	C₂×C₃×C₂²⋊D₅	—	4	1	no	—
28	C₃×#709	—	2	4	no	—
29	C₃×#709	—	4	2	no	—
30	C₂×C₃×D₂₀	—	2	1	no	—
31	C₂×C₃×C₄×D₅	—	4	1	no	—
32	C₂×C₃×Dic₁₀	—	2	1	no	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₁₅	—	2	2	no	—
38	C₄⋊Dic₁₅	—	4	1	no	—
39	C₃×Dic₂₀	—	2	1	no	—
40	C₃×Dic₂₀	C₂×C₄	1	2	yes	complemented
41	C₃×D₄₀	—	2	1	no	—
42	C₃×D₄₀	C₂×C₄	1	2	yes	complemented
43	C₃×#677	—	2	4	no	—
44	C₅×C₃⋊C₁₆	—	2	2	no	—
45	C₂×C₃×#135	—	5	1	no	—
46	C₂×#818	—	5	1	no	—
47	C₂×#812	D₅	1	1	yes	characteristic, complemented
48	#4564	—	5	2	no	—
49	#4562	—	5	2	no	—
50	C₂×#709	—	2	1	no	—
51	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×C₄⋊Dic₅	—	6	1	no	—
53	C₂×Dic₂₀	C₂×D₃	1	1	yes	characteristic
54	#4050	—	2	2	no	—
55	#4050	Dic₃	1	4	yes	complemented
56	C₂×D₄₀	C₂×D₃	1	1	yes	characteristic
57	C₂×#677	—	2	1	no	—
58	C₅×M₃₂	—	6	2	no	—
59	C₂×C₅×C₁₆	—	3	2	no	—
60	#3950	—	3	2	no	—
61	#3950	—	6	1	no	—
62	#2139	—	15	1	no	Sylow
63	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
64	C₂²×C₃×D₅	—	4	1	no	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×Dic₁₅	—	4	2	no	—
70	C₂×Dic₁₅	—	8	2	no	—
71	C₃×C₂²⋊D₅	—	4	4	no	—
72	C₂×C₃×Dic₅	—	4	1	no	—
73	C₃×D₂₀	—	2	2	no	—
74	C₃×D₂₀	—	4	1	no	—
75	C₃×C₄×D₅	—	4	4	no	—
76	C₃×Dic₁₀	—	2	2	no	—
77	C₃×Dic₁₀	—	4	1	no	—
78	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
79	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×#41	—	10	1	no	—
82	C₃×#135	—	5	4	no	—
83	C₃×#135	—	10	2	no	—
84	C₂×C₃×Q₁₆	—	5	1	no	—
85	C₂×C₃×SD₁₆	—	10	1	no	—
86	C₂×C₃×D₈	—	5	1	no	—
87	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
88	C₂×C₄⋊Dic₃	—	10	1	no	—
89	#818	—	5	2	no	—
90	#818	—	10	1	no	—
91	#812	—	2	2	no	—
92	C₂×C₃⋊C₁₆	C₂×D₅	1	2	yes	—
93	C₂²×C₄×C₅	#67	1	1	yes	characteristic
94	C₂×C₂²⋊D₅	—	4	1	no	—
95	C₂²×Dic₅	—	12	1	no	—
96	#709	—	2	4	no	—
97	#709	—	4	2	no	—
98	C₂×D₂₀	—	2	1	no	—
99	C₂×C₄×D₅	—	4	1	no	—
100	C₂×Dic₁₀	—	2	1	no	—
101	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
102	C₂×C₅×C₈	#67	1	2	yes	—
103	C₂×C₅×C₈	#67	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
105	C₄⋊Dic₅	—	6	2	no	—
106	C₄⋊Dic₅	—	12	1	no	—
107	Dic₂₀	—	2	1	no	—
108	Dic₂₀	C₂×Dic₃	1	2	yes	complemented
109	D₄₀	—	2	1	no	—
110	D₄₀	C₂×Dic₃	1	2	yes	complemented
111	#677	—	2	4	no	—
112	C₅×C₁₆	—	6	2	no	—
113	C₂×#135	—	5	1	no	—
114	C₂×M₃₂	—	3	1	no	—
115	C₂×#107	—	15	1	no	—
116	#358	—	15	2	no	—
117	#357	—	15	2	no	—
118	C₂²×C₃×C₅	—	2	1	no	—
119	C₂²×C₃×C₅	—	2	1	no	—
120	C₂²×C₃×C₅	#102	1	2	yes	—
121	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
122	C₂×C₃×D₅	—	4	2	no	—
123	C₂×C₃×D₅	—	8	1	no	—
124	C₂×C₃×D₅	—	8	1	no	—
125	C₃×C₄×C₅	#102	1	2	yes	—
126	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
128	Dic₁₅	—	8	2	no	—
129	C₃×Dic₅	—	4	2	no	—
130	C₃×#41	—	10	4	no	—
131	C₃×#41	—	20	2	no	—
132	C₂×C₃×Q₈	—	10	1	no	—
133	C₂×C₃×D₄	—	10	1	no	—
134	C₂×C₃×D₄	—	20	1	no	—
135	C₂²×C₃×C₄	—	20	1	no	—
136	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
137	C₂²×Dic₃	—	20	1	no	—
138	C₃×Q₁₆	—	5	2	no	—
139	C₃×Q₁₆	—	10	1	no	—
140	C₃×SD₁₆	—	10	4	no	—
141	C₃×D₈	—	5	2	no	—
142	C₃×D₈	—	10	1	no	—
143	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
144	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
145	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
146	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
147	C₄⋊Dic₃	—	10	2	no	—
148	C₄⋊Dic₃	—	20	1	no	—
149	C₃⋊C₁₆	—	2	2	no	—
150	C₂³×C₅	#213	1	1	yes	characteristic
151	C₂²×D₅	—	4	1	no	—
152	C₂×C₄×C₅	#213	1	1	yes	characteristic
153	C₂×C₄×C₅	#214	1	2	yes	—
154	C₂×C₄×C₅	#217	1	2	yes	—
155	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
156	C₂²⋊D₅	—	4	4	no	—
157	C₂×Dic₅	—	4	1	no	—
158	C₂×Dic₅	—	12	2	no	—
159	C₂×Dic₅	—	24	2	no	—
160	D₂₀	—	2	2	no	—
161	D₂₀	—	4	1	no	—
162	C₄×D₅	—	4	4	no	—
163	Dic₁₀	—	2	2	no	—
164	Dic₁₀	—	4	1	no	—
165	C₅×C₈	#217	1	2	yes	—
166	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
167	C₅×C₈	C₂×#67	1	1	yes	characteristic
168	C₂×#41	—	10	1	no	—
169	#135	—	5	4	no	—
170	#135	—	10	2	no	—
171	C₂×Q₁₆	—	5	1	no	—
172	C₂×SD₁₆	—	10	1	no	—
173	C₂×D₈	—	5	1	no	—
174	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	30	1	no	—
176	M₃₂	—	6	2	no	—
177	C₂×C₁₆	—	3	2	no	—
178	#107	—	15	2	no	—
179	#107	—	30	1	no	—
180	C₂×C₃×C₅	—	2	2	no	—
181	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
182	C₂×C₃×C₅	#509	1	1	yes	characteristic
183	C₂×C₃×C₅	#510	1	1	yes	characteristic
184	C₃×D₅	—	8	2	no	—
185	C₂³×C₃	—	20	1	no	—
186	C₂³×C₃	D₄₀	1	1	yes	characteristic
187	C₃×Q₈	—	10	2	no	—
188	C₃×Q₈	—	20	1	no	—
189	C₃×D₄	—	10	2	no	—
190	C₃×D₄	—	20	4	no	—
191	C₃×D₄	—	20	1	no	—
192	C₂×C₃×C₄	—	20	1	no	—
193	C₂×C₃×C₄	—	20	4	no	—
194	C₂×C₃×C₄	#677	1	2	yes	—
195	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
196	C₂×C₃×C₄	#685	1	2	yes	—
197	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
198	C₂×Dic₃	—	20	2	no	—
199	C₂×Dic₃	—	40	2	no	—
200	C₃×C₈	#685	1	2	yes	—
201	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
202	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	#832	1	2	yes	—
206	C₂²×C₅	C₂×#213	1	1	yes	characteristic
207	C₂×D₅	—	4	2	no	—
208	C₂×D₅	—	8	1	no	—
209	C₂×D₅	—	8	1	no	—
210	C₄×C₅	#832	1	2	yes	—
211	C₄×C₅	C₂×#214	1	1	yes	characteristic
212	C₄×C₅	C₂×#217	1	1	yes	characteristic
213	Dic₅	—	4	2	no	—
214	Dic₅	—	24	2	no	—
215	#41	—	10	4	no	—
216	#41	—	20	2	no	—
217	C₂×Q₈	—	10	1	no	—
218	C₂×D₄	—	10	1	no	—
219	C₂×D₄	—	20	1	no	—
220	C₂²×C₄	—	20	1	no	—
221	C₂²×C₄	—	60	1	no	—
222	C₂²×C₄	#1202	1	1	yes	characteristic
223	Q₁₆	—	5	2	no	—
224	Q₁₆	—	10	1	no	—
225	SD₁₆	—	10	4	no	—
226	D₈	—	5	2	no	—
227	D₈	—	10	1	no	—
228	C₂×C₈	D₅×Dic₃	1	2	yes	—
229	C₂×C₈	#1201	1	2	yes	—
230	C₂×C₈	#1202	1	1	yes	characteristic
231	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
232	C₄⋊C₄	—	30	2	no	—
233	C₄⋊C₄	—	60	1	no	—
234	C₁₆	—	6	2	no	—
235	C₃×C₅	#2139	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	20	2	no	—
239	C₂²×C₃	—	40	1	no	—
240	C₂²×C₃	—	40	1	no	—
241	C₂²×C₃	#3953	1	2	yes	—
242	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
243	C₃×C₄	—	20	2	no	—
244	C₃×C₄	#3953	1	2	yes	—
245	C₃×C₄	C₂×#677	1	1	yes	characteristic
246	C₃×C₄	C₂×#685	1	1	yes	characteristic
247	Dic₃	—	40	2	no	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	#5193	1	1	yes	characteristic
250	C₂×C₅	#5195	1	1	yes	characteristic
251	C₂×C₅	C₂×#832	1	1	yes	characteristic
252	D₅	—	8	2	no	—
253	C₂³	—	20	1	no	—
254	C₂³	#6653	1	1	yes	characteristic
255	Q₈	—	10	2	no	—
256	Q₈	—	20	1	no	—
257	D₄	—	10	2	no	—
258	D₄	—	20	4	no	—
259	D₄	—	20	1	no	—
260	C₂×C₄	—	20	1	no	—
261	C₂×C₄	—	20	4	no	—
262	C₂×C₄	—	60	2	no	—
263	C₂×C₄	—	120	2	no	—
264	C₂×C₄	#6653	1	1	yes	characteristic
265	C₂×C₄	#6657	1	2	yes	—
266	C₂×C₄	#6665	1	2	yes	—
267	C₂×C₄	C₂×#1202	1	1	yes	characteristic
268	C₈	#6665	1	2	yes	—
269	C₈	C₂×D₅×Dic₃	1	1	yes	characteristic
270	C₈	C₂×#1201	1	1	yes	characteristic
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	40	2	no	—
273	C₂×C₃	#65197	1	1	yes	characteristic
274	C₂×C₃	#65198	1	1	yes	characteristic
275	C₂×C₃	C₂×#3953	1	1	yes	characteristic
276	C₅	#72078	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	20	2	no	—
280	C₂²	—	40	1	no	—
281	C₂²	—	40	1	no	—
282	C₂²	#90965	1	2	yes	—
283	C₂²	C₂×#6653	1	1	yes	characteristic
284	C₄	—	20	2	no	—
285	C₄	—	120	2	no	—
286	C₄	#90965	1	2	yes	—
287	C₄	C₂×#6657	1	1	yes	characteristic
288	C₄	C₂×#6665	1	1	yes	characteristic
289	C₃	#10602232	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	2	2	no	—
291	C₂	—	40	2	no	—
292	C₂	#11747544	1	1	yes	characteristic
293	C₂	#11747554	1	1	yes	characteristic
294	C₂	C₂×#90965	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6653
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₃×#4050	C₂
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2139
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#91102	C₂	semidirect
3. C₂×C₅×#812	C₂	semidirect
4. #11745898	C₂	semidirect
5. #11745892	C₂	semidirect
12. C₃×#4050	C₄	semidirect
40. C₃×Dic₂₀	C₂×C₄	semidirect
42. C₃×D₄₀	C₂×C₄	semidirect
47. C₂×#812	D₅	semidirect
55. #4050	Dic₃	semidirect
108. Dic₂₀	C₂×Dic₃	semidirect
110. D₄₀	C₂×Dic₃	semidirect
235. C₃×C₅	#2139	semidirect
276. C₅	#72078	semidirect
289. C₃	#10602232	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4050	2
C₂×#91102	2
C₂×C₅×#812	2
#11745898	2
#11745892	2
#10602232	3
#72078	5