Subgroups — #14297475

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#91114	C₂	1	2	yes	maximal
3	C₂×C₄⋊(C₃⋊#168)	C₂	1	2	yes	maximal, complemented
4	#11746345	C₂	1	2	yes	maximal, complemented
5	#10602376	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₃×C₅×#117	—	2	4	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
9	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊#168	C₂²	1	4	yes	—
13	#91114	—	2	4	no	—
14	C₄⋊(C₃⋊#168)	—	2	4	no	—
15	#71583	—	5	1	no	maximal, Hall
16	C₂×C₅×#117	D₃	1	1	yes	characteristic
17	C₂×C₂²⋊#168	—	3	2	no	—
18	C₂×C₄⋊#168	—	3	2	no	—
19	#64744	—	3	2	no	—
20	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
31	C₃×C₄²×C₅	—	2	2	no	—
32	C₃×C₄²×C₅	D₄	1	4	yes	—
33	C₂×C₃⋊#168	—	2	4	no	—
34	C₂×C₃⋊#168	—	2	4	no	—
35	C₂×C₃⋊#168	—	2	4	no	—
36	C₂×C₃×#117	D₅	1	1	yes	characteristic
37	C₂×#830	—	5	2	no	—
38	C₂×C₄⋊(C₃⋊C₈)	—	5	2	no	—
39	#4524	—	5	2	no	—
40	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
41	C₅×#117	—	2	4	no	—
42	C₂×C₅×C₄⋊C₄	Dic₃	1	2	yes	—
43	C₂×C₅×C₂²⋊C₄	Dic₃	1	2	yes	—
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂²×#168	—	3	4	no	—
47	C₂²⋊#168	—	6	4	no	—
48	C₄⋊#168	—	6	4	no	—
49	#1798	—	15	1	no	Sylow
50	C₂³×C₃×C₅	—	2	1	no	—
51	C₂³×C₃×C₅	—	2	1	no	—
52	C₂³×C₃×C₅	—	2	4	no	—
53	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
54	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	4	no	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
63	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	2	yes	—
68	C₃⋊#168	—	4	4	no	—
69	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
70	C₃×#117	—	2	4	no	—
71	C₂×C₃×C₄⋊C₄	Dic₅	1	2	yes	—
72	C₂×C₃×C₂²⋊C₄	Dic₅	1	2	yes	—
73	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
74	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
75	C₂²×C₃⋊C₈	—	5	4	no	—
76	#830	—	10	4	no	—
77	C₄⋊(C₃⋊C₈)	—	10	4	no	—
78	C₂⁴×C₅	C₂×Dic₃	1	1	yes	characteristic
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	Dic₆	1	2	yes	—
82	C₂²×C₄×C₅	D₁₂	1	2	yes	—
83	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
84	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
86	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₂²⋊C₄	—	2	4	no	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	#67	1	4	yes	—
91	C₂×#168	—	6	4	no	—
92	C₂×#168	—	6	4	no	—
93	C₂×#168	—	6	4	no	—
94	C₂×#117	D₁₅	1	1	yes	characteristic
95	C₂×C₄⋊C₈	—	15	2	no	—
96	C₂×C₂²⋊C₈	—	15	2	no	—
97	#336	—	15	2	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	—	2	4	no	—
100	C₂²×C₃×C₅	—	4	2	no	—
101	C₂²×C₃×C₅	—	4	2	no	—
102	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	#117	1	1	yes	characteristic
105	C₂²×C₃×C₅	#131	1	4	yes	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	4	no	—
108	C₃×C₄×C₅	—	4	2	no	—
109	C₃×C₄×C₅	#121	1	2	yes	—
110	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
111	C₂⁴×C₃	C₂×Dic₅	1	1	yes	characteristic
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
115	C₂²×C₃×C₄	D₂₀	1	2	yes	—
116	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
117	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
119	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
120	C₃×C₄⋊C₄	—	2	4	no	—
121	C₃×C₂²⋊C₄	—	2	4	no	—
122	C₃×C₄²	—	2	2	no	—
123	C₃×C₄²	C₂²⋊D₅	1	4	yes	—
124	C₂×C₃⋊C₈	—	10	4	no	—
125	C₂×C₃⋊C₈	—	10	4	no	—
126	C₂×C₃⋊C₈	—	10	4	no	—
127	C₂³×C₅	—	2	1	no	—
128	C₂³×C₅	—	2	4	no	—
129	C₂³×C₅	—	2	1	no	—
130	C₂³×C₅	C₄⋊Dic₃	1	2	yes	—
131	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
140	C₂×C₄×C₅	#217	1	4	yes	—
141	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
143	C₂×C₄×C₅	#235	1	2	yes	—
144	C₂×C₄×C₅	C₂×#67	1	2	yes	—
145	#168	—	12	4	no	—
146	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
147	C₂²×C₈	—	15	4	no	—
148	#117	—	2	4	no	—
149	C₂×C₄⋊C₄	Dic₁₅	1	2	yes	—
150	C₂×C₂²⋊C₄	Dic₁₅	1	2	yes	—
151	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
152	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
153	C₄⋊C₈	—	30	4	no	—
154	C₂²⋊C₈	—	30	4	no	—
155	C₂×C₃×C₅	—	4	2	no	—
156	C₂×C₃×C₅	#380	1	1	yes	characteristic
157	C₂×C₃×C₅	#408	1	2	yes	—
158	C₂×C₃×C₅	#424	1	2	yes	—
159	C₂×C₃×C₅	#433	1	2	yes	—
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	—	2	1	no	—
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	C₄⋊Dic₅	1	2	yes	—
164	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
173	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
174	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
176	C₂×C₃×C₄	#709	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
178	C₃⋊C₈	—	20	4	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	—	4	2	no	—
183	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
184	C₂²×C₅	#926	1	1	yes	characteristic
185	C₂²×C₅	#948	1	4	yes	—
186	C₂²×C₅	C₂×#217	1	1	yes	characteristic
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	4	no	—
189	C₄×C₅	—	4	2	no	—
190	C₄×C₅	#924	1	2	yes	—
191	C₄×C₅	#930	1	2	yes	—
192	C₂⁴	C₂×Dic₁₅	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	Dic₃₀	1	2	yes	—
196	C₂²×C₄	D₆₀	1	2	yes	—
197	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
198	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
199	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
200	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
201	C₂×C₈	—	30	4	no	—
202	C₂×C₈	—	30	4	no	—
203	C₂×C₈	—	30	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	#1220	1	4	yes	—
208	C₃×C₅	#1798	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	2	no	—
213	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
214	C₂²×C₃	#4072	1	1	yes	characteristic
215	C₂²×C₃	#4094	1	4	yes	—
216	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	4	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	#4070	1	2	yes	—
221	C₃×C₄	#4076	1	2	yes	—
222	C₂×C₅	—	4	2	no	—
223	C₂×C₅	#4999	1	2	yes	—
224	C₂×C₅	#5000	1	2	yes	—
225	C₂×C₅	#5212	1	1	yes	characteristic
226	C₂×C₅	#5234	1	2	yes	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	4	no	—
229	C₂³	—	2	1	no	—
230	C₂³	C₄⋊Dic₁₅	1	2	yes	—
231	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	C₄⋊Dic₁₅	1	2	yes	—
240	C₂×C₄	#6719	1	4	yes	—
241	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
242	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
243	C₂×C₄	#6817	1	2	yes	—
244	C₂×C₄	C₂×#1220	1	2	yes	—
245	C₈	—	60	4	no	—
246	C₂×C₃	—	4	2	no	—
247	C₂×C₃	#65288	1	2	yes	—
248	C₂×C₃	#65289	1	2	yes	—
249	C₂×C₃	#65501	1	1	yes	characteristic
250	C₂×C₃	#65523	1	2	yes	—
251	C₅	#71583	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
257	C₂²	#91815	1	1	yes	characteristic
258	C₂²	#91837	1	4	yes	—
259	C₂²	C₂×#6719	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	#91813	1	2	yes	—
264	C₄	#91819	1	2	yes	—
265	C₃	#10602376	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	4	2	no	—
267	C₂	#11751100	1	2	yes	—
268	C₂	#11751101	1	2	yes	—
269	C₂	#11751313	1	1	yes	characteristic
270	C₂	#11751335	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅
commutator, socle	C₂³×C₃×C₅	C₂²×C₄
Fitting	C₂×C₃×C₅×#117	C₂
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×#117	D₁₅
hypocenter	C₃×C₅	#1798
LCS	G ▷ C₂³×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊(C₃⋊#168)	C₂	semidirect
4. #11746345	C₂	semidirect
208. C₃×C₅	#1798	semidirect
251. C₅	#71583	semidirect
265. C₃	#10602376	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#117	2
C₂×#91114	2
C₂×C₄⋊(C₃⋊#168)	2
#11746345	2
#10602376	3
#71583	5