Subgroups — #14298738

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91098	C₂	1	2	yes	maximal
2	#11751033	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic
4	#11746371	C₂	1	2	yes	maximal
5	#11746345	C₂	1	1	yes	maximal, characteristic
6	#10599399	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊#168	C₂²	1	2	yes	—
12	#91115	C₄	1	2	yes	—
13	#91115	C₄	1	2	yes	—
14	#91098	—	2	4	no	—
15	#68606	—	5	1	no	maximal, Hall
16	C₂×#3946	—	3	2	no	—
17	#65221	—	3	1	no	—
18	C₅×#336	D₃	1	1	yes	characteristic
19	#64770	—	3	2	no	—
20	#64744	—	3	1	no	—
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₁₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₁₅	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₃×C₄²×C₅	—	2	2	no	—
30	C₄×Dic₁₅	—	2	2	no	—
31	C₄×Dic₁₅	D₄	1	2	yes	—
32	C₄×Dic₁₅	Q₈	1	2	yes	—
33	C₂×C₃⋊#168	—	2	2	no	—
34	C₂×C₃⋊#168	—	2	2	no	—
35	C₂×C₃⋊#168	—	2	2	no	—
36	C₂×Dic₃⋊C₈	—	5	2	no	—
37	#4932	—	5	1	no	—
38	C₃×#336	D₅	1	1	yes	characteristic
39	#4550	—	5	2	no	—
40	#4524	—	5	1	no	—
41	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
42	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄×Dic₅	—	3	1	no	—
44	C₂×C₄×Dic₅	—	3	1	no	—
45	C₂²×#168	—	3	2	no	—
46	#3963	—	3	2	no	—
47	#3963	—	3	2	no	—
48	#3946	—	6	4	no	—
49	#1832	—	15	1	no	Sylow
50	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	#41	1	2	yes	—
57	C₂×C₃×C₄×C₅	#41	1	2	yes	—
58	C₂×Dic₁₅	—	2	2	no	—
59	C₂×Dic₁₅	—	2	2	no	—
60	C₂×Dic₁₅	—	2	1	no	—
61	C₂×Dic₁₅	—	2	1	no	—
62	C₂×Dic₁₅	—	2	2	no	—
63	C₂×Dic₁₅	—	2	2	no	—
64	C₂×Dic₁₅	C₄⋊C₄	1	4	yes	—
65	C₃×C₅×C₈	—	4	2	no	—
66	C₃⋊#168	—	4	2	no	—
67	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
68	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
69	C₂×C₄×Dic₃	—	5	1	no	—
70	C₂×C₄×Dic₃	—	5	1	no	—
71	C₂²×C₃⋊C₈	—	5	2	no	—
72	C₃⋊#95	—	5	2	no	—
73	C₃⋊#95	—	5	2	no	—
74	Dic₃⋊C₈	—	10	4	no	—
75	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
76	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
77	C₂²×Dic₅	—	3	2	no	—
78	C₂²×Dic₅	—	3	1	no	—
79	C₂²×Dic₅	—	3	1	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₄²×C₅	—	2	2	no	—
84	C₄×Dic₅	—	3	2	no	—
85	C₄×Dic₅	—	3	2	no	—
86	C₄×Dic₅	—	6	2	no	—
87	C₂×#168	—	6	2	no	—
88	C₂×#168	—	6	2	no	—
89	C₂×#168	—	6	2	no	—
90	C₂×C₄⋊C₈	—	15	2	no	—
91	#376	—	15	1	no	—
92	#336	—	15	2	no	—
93	#336	—	15	1	no	—
94	#336	D₁₅	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₃×C₅	#117	1	1	yes	characteristic
97	C₂²×C₃×C₅	#117	1	1	yes	characteristic
98	C₂²×C₃×C₅	#131	1	2	yes	—
99	C₂²×C₃×C₅	#131	1	2	yes	—
100	C₃×C₄×C₅	—	4	2	no	—
101	C₃×C₄×C₅	#124	1	2	yes	—
102	C₃×C₄×C₅	#125	1	2	yes	—
103	Dic₁₅	—	2	4	no	—
104	Dic₁₅	—	4	1	no	—
105	Dic₁₅	—	4	1	no	—
106	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
107	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
108	C₂²×Dic₃	—	5	2	no	—
109	C₂²×Dic₃	—	5	1	no	—
110	C₂²×Dic₃	—	5	1	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₃×C₄²	—	2	2	no	—
115	C₄×Dic₃	—	5	2	no	—
116	C₄×Dic₃	—	5	2	no	—
117	C₄×Dic₃	—	10	2	no	—
118	C₂×C₃⋊C₈	—	10	2	no	—
119	C₂×C₃⋊C₈	—	10	2	no	—
120	C₂×C₃⋊C₈	—	10	2	no	—
121	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	#235	1	2	yes	—
126	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	#237	1	2	yes	—
128	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
129	C₂×Dic₅	—	3	4	no	—
130	C₂×Dic₅	—	6	2	no	—
131	C₂×Dic₅	—	6	2	no	—
132	C₂×Dic₅	—	6	1	no	—
133	C₂×Dic₅	—	6	1	no	—
134	C₂×Dic₅	—	6	2	no	—
135	C₂×Dic₅	—	6	2	no	—
136	C₅×C₈	—	4	2	no	—
137	#168	—	12	2	no	—
138	C₂²×C₈	—	15	2	no	—
139	C₂²×C₈	C₂×D₁₅	1	2	yes	—
140	C₂×C₄²	—	15	1	no	—
141	C₂×C₄²	—	15	1	no	—
142	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
143	C₄⋊C₈	—	30	4	no	—
144	#95	—	15	2	no	—
145	#95	—	15	2	no	—
146	C₂×C₃×C₅	#382	1	1	yes	characteristic
147	C₂×C₃×C₅	#424	1	2	yes	—
148	C₂×C₃×C₅	#433	1	1	yes	characteristic
149	C₂×C₃×C₅	#433	1	1	yes	characteristic
150	C₂×C₃×C₅	#433	1	2	yes	—
151	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	#709	1	2	yes	—
156	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
157	C₂×C₃×C₄	#711	1	2	yes	—
158	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
159	C₂×Dic₃	—	5	4	no	—
160	C₂×Dic₃	—	10	2	no	—
161	C₂×Dic₃	—	10	2	no	—
162	C₂×Dic₃	—	10	1	no	—
163	C₂×Dic₃	—	10	1	no	—
164	C₂×Dic₃	—	10	2	no	—
165	C₂×Dic₃	—	10	2	no	—
166	C₃×C₈	—	4	2	no	—
167	C₃⋊C₈	—	20	2	no	—
168	C₂²×C₅	#872	1	1	yes	characteristic
169	C₂²×C₅	#877	1	1	yes	characteristic
170	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×C₅	#901	1	2	yes	—
172	C₂²×C₅	#907	1	2	yes	—
173	C₄×C₅	—	4	2	no	—
174	C₄×C₅	#885	1	2	yes	—
175	C₄×C₅	#889	1	2	yes	—
176	Dic₅	—	6	4	no	—
177	Dic₅	—	12	1	no	—
178	Dic₅	—	12	1	no	—
179	C₂²×C₄	—	15	2	no	—
180	C₂²×C₄	—	15	1	no	—
181	C₂²×C₄	—	15	1	no	—
182	C₂²×C₄	C₄×D₁₅	1	2	yes	—
183	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	30	2	no	—
188	C₂×C₈	—	30	2	no	—
189	C₂×C₈	—	30	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	15	2	no	—
192	C₄²	—	15	2	no	—
193	C₄²	—	30	2	no	—
194	C₃×C₅	#1832	1	1	yes	characteristic, complemented, Hall
195	C₂²×C₃	#4018	1	1	yes	characteristic
196	C₂²×C₃	#4023	1	1	yes	characteristic
197	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₃	#4047	1	2	yes	—
199	C₂²×C₃	#4053	1	2	yes	—
200	C₃×C₄	—	4	2	no	—
201	C₃×C₄	#4031	1	2	yes	—
202	C₃×C₄	#4035	1	2	yes	—
203	Dic₃	—	10	4	no	—
204	Dic₃	—	20	1	no	—
205	Dic₃	—	20	1	no	—
206	C₂×C₅	#4650	1	1	yes	characteristic
207	C₂×C₅	#4690	1	1	yes	characteristic
208	C₂×C₅	#4709	1	1	yes	characteristic
209	C₂×C₅	#4720	1	2	yes	—
210	C₂×C₅	#4839	1	2	yes	—
211	C₂³	C₂×C₄×D₁₅	1	1	yes	characteristic
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	15	4	no	—
216	C₂×C₄	—	30	2	no	—
217	C₂×C₄	—	30	2	no	—
218	C₂×C₄	—	30	1	no	—
219	C₂×C₄	—	30	1	no	—
220	C₂×C₄	—	30	2	no	—
221	C₂×C₄	—	30	2	no	—
222	C₂×C₄	#6817	1	2	yes	—
223	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
224	C₂×C₄	#6819	1	2	yes	—
225	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
226	C₈	—	4	2	no	—
227	C₈	—	60	2	no	—
228	C₂×C₃	#64939	1	1	yes	characteristic
229	C₂×C₃	#64979	1	1	yes	characteristic
230	C₂×C₃	#64998	1	1	yes	characteristic
231	C₂×C₃	#65009	1	2	yes	—
232	C₂×C₃	#65128	1	2	yes	—
233	C₅	#68606	1	1	yes	characteristic, complemented, Sylow
234	C₂²	#91761	1	1	yes	characteristic
235	C₂²	#91766	1	1	yes	characteristic
236	C₂²	C₄⋊C₄×D₁₅	1	1	yes	characteristic
237	C₂²	#91790	1	2	yes	—
238	C₂²	#91796	1	2	yes	—
239	C₄	—	4	2	no	—
240	C₄	—	30	4	no	—
241	C₄	—	60	1	no	—
242	C₄	—	60	1	no	—
243	C₄	#91774	1	2	yes	—
244	C₄	#91778	1	2	yes	—
245	C₃	#10599399	1	1	yes	characteristic, complemented, Sylow
246	C₂	#11750751	1	1	yes	characteristic
247	C₂	#11750791	1	1	yes	characteristic
248	C₂	#11750810	1	1	yes	characteristic
249	C₂	#11750821	1	2	yes	—
250	C₂	#11750940	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅
commutator, socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
Fitting	C₃×C₅×#336	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#336	D₁₅
hypocenter	C₃×C₅	#1832
LCS	G ▷ C₂³×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
194. C₃×C₅	#1832	semidirect
233. C₅	#68606	semidirect
245. C₃	#10599399	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91098	2
#11751033	2
C₃×C₅×#336	2
#11746371	2
#11746345	2
#10599399	3
#68606	5