Subgroups — #14313870

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91793	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65395	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745901	C₂	1	2	yes	maximal, complemented
5	#11745895	C₂	1	2	yes	maximal, complemented
6	#10602205	—	3	1	no	maximal, Hall
7	C₂×#6817	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₆₀	C₂²	1	1	yes	characteristic
10	#91793	—	2	2	no	—
11	#91793	C₄	1	4	yes	complemented
12	C₂×D₁₂₀	C₂²	1	1	yes	characteristic
13	C₂×#6706	—	2	1	no	—
14	C₃×#4072	—	2	1	no	—
15	C₂×C₅×C₃⋊C₁₆	—	2	2	no	—
16	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	—
17	C₃×#3950	C₂²	1	2	yes	—
18	C₃×#3949	—	2	1	no	—
19	#70651	—	5	1	no	maximal, Hall
20	C₂×#4050	—	3	1	no	—
21	C₂²×C₅×C₁₆	—	3	1	no	—
22	#65395	D₃	1	1	yes	characteristic, complemented
23	#64729	—	3	2	no	—
24	#64723	—	3	2	no	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂×#1220	—	4	1	no	—
27	#6817	—	2	4	no	—
28	#6817	—	4	2	no	—
29	C₂×D₆₀	—	2	1	no	—
30	C₂×C₄×D₁₅	—	4	1	no	—
31	C₂×Dic₃₀	—	2	1	no	—
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	Dic₆₀	—	2	1	no	—
37	Dic₆₀	C₂×C₄	1	2	yes	—
38	D₁₂₀	—	2	1	no	—
39	D₁₂₀	C₂×C₄	1	2	yes	—
40	#6706	—	2	4	no	—
41	C₃×C₂²⋊Dic₅	—	4	1	no	—
42	C₃×C₄⋊Dic₅	—	2	2	no	—
43	C₃×C₄×Dic₅	—	4	1	no	—
44	C₅×C₃⋊C₁₆	—	2	2	no	—
45	C₂×#904	—	5	1	no	—
46	C₃×#427	—	5	1	no	—
47	C₂²×C₃⋊C₁₆	D₅	1	1	yes	characteristic, complemented
48	#4492	—	5	2	no	—
49	#4491	—	5	2	no	—
50	C₂×#709	—	6	1	no	—
51	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
52	#4072	—	2	1	no	—
53	C₂×Dic₂₀	—	3	1	no	—
54	#4050	—	3	4	no	—
55	#4050	—	6	2	no	—
56	C₂×D₄₀	—	3	1	no	—
57	C₂×#677	—	6	1	no	—
58	C₂×C₅×C₁₆	—	3	2	no	—
59	C₂×C₅×C₁₆	—	6	2	no	—
60	#3950	C₂×D₃	1	2	yes	—
61	#3949	—	2	1	no	—
62	#2138	—	15	1	no	Sylow
63	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
64	C₂²×D₁₅	—	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	#1220	—	4	4	no	—
70	C₂×Dic₁₅	—	4	1	no	—
71	D₆₀	—	2	2	no	—
72	D₆₀	—	4	1	no	—
73	C₄×D₁₅	—	4	4	no	—
74	Dic₃₀	—	2	2	no	—
75	Dic₃₀	—	4	1	no	—
76	C₂×C₃×Dic₅	—	4	2	no	—
77	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×#235	—	10	1	no	—
81	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
82	C₃×#117	—	10	1	no	—
83	C₂×Dic₁₂	—	5	1	no	—
84	#904	—	5	4	no	—
85	#904	—	10	2	no	—
86	C₂×D₂₄	—	5	1	no	—
87	C₂×#204	—	10	1	no	—
88	C₃×#107	—	5	2	no	—
89	C₃×#106	—	10	1	no	—
90	C₂×C₃⋊C₁₆	—	2	2	no	—
91	C₂×C₃⋊C₁₆	C₂×D₅	1	2	yes	—
92	C₂²×C₄×C₅	#67	1	1	yes	characteristic
93	C₂×C₂²⋊D₅	—	12	1	no	—
94	#709	—	6	4	no	—
95	#709	—	12	2	no	—
96	C₂×D₂₀	—	6	1	no	—
97	C₂×C₄×D₅	—	12	1	no	—
98	C₂×Dic₁₀	—	6	1	no	—
99	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
100	C₂×C₅×C₈	D₁₂	1	2	yes	—
101	C₂×C₅×C₈	#67	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
103	C₂²⋊Dic₅	—	4	1	no	—
104	C₄⋊Dic₅	—	2	2	no	—
105	C₄×Dic₅	—	4	1	no	—
106	Dic₂₀	—	3	2	no	—
107	Dic₂₀	—	6	1	no	—
108	D₄₀	—	3	2	no	—
109	D₄₀	—	6	1	no	—
110	#677	—	6	4	no	—
111	C₅×C₁₆	—	6	2	no	—
112	C₂×#135	—	15	1	no	—
113	C₂²×C₁₆	—	3	1	no	—
114	#427	—	5	1	no	—
115	#358	—	15	2	no	—
116	#357	—	15	2	no	—
117	C₂²×C₃×C₅	—	2	2	no	—
118	C₂²×C₃×C₅	#102	1	2	yes	—
119	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
120	C₂×D₁₅	—	4	2	no	—
121	C₂×D₁₅	—	8	2	no	—
122	C₃×C₄×C₅	#102	1	2	yes	—
123	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
125	Dic₁₅	—	4	2	no	—
126	C₃×Dic₅	—	8	2	no	—
127	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
128	C₂×#67	—	20	1	no	—
129	#235	—	10	4	no	—
130	#235	—	20	2	no	—
131	C₂×D₁₂	—	10	1	no	—
132	C₂×C₄×D₃	—	20	1	no	—
133	C₂×Dic₆	—	10	1	no	—
134	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
135	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
137	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
138	C₃×C₄⋊C₄	—	10	2	no	—
139	C₃×C₂²⋊C₄	—	20	1	no	—
140	C₃×C₄²	—	20	1	no	—
141	Dic₁₂	—	5	2	no	—
142	Dic₁₂	—	10	1	no	—
143	D₂₄	—	5	2	no	—
144	D₂₄	—	10	1	no	—
145	#204	—	10	4	no	—
146	C₃⋊C₁₆	—	2	2	no	—
147	C₂³×C₅	#213	1	1	yes	characteristic
148	C₂²×D₅	—	12	1	no	—
149	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
150	C₂×C₄×C₅	#213	1	1	yes	characteristic
151	C₂×C₄×C₅	#215	1	2	yes	—
152	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
153	C₂²⋊D₅	—	12	4	no	—
154	C₂×Dic₅	—	4	2	no	—
155	C₂×Dic₅	—	12	1	no	—
156	D₂₀	—	6	2	no	—
157	D₂₀	—	12	1	no	—
158	C₄×D₅	—	12	4	no	—
159	Dic₁₀	—	6	2	no	—
160	Dic₁₀	—	12	1	no	—
161	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
162	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
163	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
164	C₂×#41	—	30	1	no	—
165	#135	—	15	4	no	—
166	#135	—	30	2	no	—
167	C₂×Q₁₆	—	15	1	no	—
168	C₂×SD₁₆	—	30	1	no	—
169	C₂×D₈	—	15	1	no	—
170	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
171	#117	—	10	1	no	—
172	C₂×C₁₆	—	3	2	no	—
173	C₂×C₁₆	—	6	2	no	—
174	#107	—	5	2	no	—
175	#106	—	10	1	no	—
176	C₂×C₃×C₅	—	2	2	no	—
177	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
178	C₂×C₃×C₅	#508	1	2	yes	—
179	D₁₅	—	8	2	no	—
180	C₂³×C₃	D₄₀	1	1	yes	characteristic
181	C₂²×D₃	—	20	1	no	—
182	C₂×C₃×C₄	—	20	2	no	—
183	C₂×C₃×C₄	#677	1	2	yes	—
184	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
185	C₂×C₃×C₄	#685	1	2	yes	—
186	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
187	#67	—	20	4	no	—
188	C₂×Dic₃	—	20	1	no	—
189	D₁₂	—	10	2	no	—
190	D₁₂	—	20	1	no	—
191	C₄×D₃	—	20	4	no	—
192	Dic₆	—	10	2	no	—
193	Dic₆	—	20	1	no	—
194	C₃×C₈	#685	1	2	yes	—
195	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
196	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	#809	1	2	yes	—
199	C₂²×C₅	C₂×#213	1	1	yes	characteristic
200	C₂×D₅	—	12	2	no	—
201	C₂×D₅	—	24	2	no	—
202	C₄×C₅	#809	1	2	yes	—
203	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
204	C₄×C₅	C₂×#215	1	1	yes	characteristic
205	Dic₅	—	8	2	no	—
206	Dic₅	—	12	2	no	—
207	#41	—	30	4	no	—
208	#41	—	60	2	no	—
209	C₂×Q₈	—	30	1	no	—
210	C₂×D₄	—	30	1	no	—
211	C₂×D₄	—	60	1	no	—
212	C₂²×C₄	—	60	1	no	—
213	C₂²×C₄	#1202	1	1	yes	characteristic
214	Q₁₆	—	15	2	no	—
215	Q₁₆	—	30	1	no	—
216	SD₁₆	—	30	4	no	—
217	D₈	—	15	2	no	—
218	D₈	—	30	1	no	—
219	C₂×C₈	#1200	1	2	yes	—
220	C₂×C₈	#1202	1	1	yes	characteristic
221	C₂×C₈	#1203	1	2	yes	—
222	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
223	C₄⋊C₄	—	10	2	no	—
224	C₂²⋊C₄	—	20	1	no	—
225	C₄²	—	20	1	no	—
226	C₁₆	—	6	2	no	—
227	C₃×C₅	#2138	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	#3953	1	2	yes	—
230	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
231	C₂×D₃	—	20	2	no	—
232	C₂×D₃	—	40	2	no	—
233	C₃×C₄	—	40	2	no	—
234	C₃×C₄	#3953	1	2	yes	—
235	C₃×C₄	C₂×#677	1	1	yes	characteristic
236	C₃×C₄	C₂×#685	1	1	yes	characteristic
237	Dic₃	—	20	2	no	—
238	C₂×C₅	—	2	2	no	—
239	C₂×C₅	C₂×#809	1	1	yes	characteristic
240	C₂×C₅	#5194	1	2	yes	—
241	D₅	—	24	2	no	—
242	C₂³	—	60	1	no	—
243	C₂³	#6653	1	1	yes	characteristic
244	Q₈	—	30	2	no	—
245	Q₈	—	60	1	no	—
246	D₄	—	30	2	no	—
247	D₄	—	60	4	no	—
248	D₄	—	60	1	no	—
249	C₂×C₄	—	20	2	no	—
250	C₂×C₄	—	60	1	no	—
251	C₂×C₄	—	60	4	no	—
252	C₂×C₄	#6653	1	1	yes	characteristic
253	C₂×C₄	#6658	1	2	yes	—
254	C₂×C₄	#6667	1	2	yes	—
255	C₂×C₄	C₂×#1202	1	1	yes	characteristic
256	C₈	#6667	1	2	yes	—
257	C₈	C₂×#1200	1	1	yes	characteristic
258	C₈	C₂×#1203	1	1	yes	characteristic
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	#65193	1	2	yes	—
261	C₂×C₃	C₂×#3953	1	1	yes	characteristic
262	D₃	—	40	2	no	—
263	C₅	#70651	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	60	2	no	—
266	C₂²	—	120	2	no	—
267	C₂²	#90968	1	2	yes	—
268	C₂²	C₂×#6653	1	1	yes	characteristic
269	C₄	—	40	2	no	—
270	C₄	—	60	2	no	—
271	C₄	#90968	1	2	yes	—
272	C₄	C₂×#6658	1	1	yes	characteristic
273	C₄	C₂×#6667	1	1	yes	characteristic
274	C₃	#10602205	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	120	2	no	—
277	C₂	#11747542	1	2	yes	—
278	C₂	C₂×#90968	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#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×#91793	C₂
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2138
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×C₃⋊C₁₆	C₂	semidirect
3. C₃×#65395	C₂	semidirect
4. #11745901	C₂	semidirect
5. #11745895	C₂	semidirect
11. #91793	C₄	semidirect
22. #65395	D₃	semidirect
47. C₂²×C₃⋊C₁₆	D₅	semidirect
227. C₃×C₅	#2138	semidirect
263. C₅	#70651	semidirect
274. C₃	#10602205	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91793	2
C₂²×C₅×C₃⋊C₁₆	2
C₃×#65395	2
#11745901	2
#11745895	2
#10602205	3
#70651	5