Subgroups — #14314337

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3930	C₂	1	1	yes	maximal, characteristic
2	C₈×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#90934	C₂	1	1	yes	maximal, characteristic
4	#11746322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745928	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600077	—	3	1	no	maximal, Hall
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₄×D₅×Dic₃	C₄	1	1	yes	characteristic
11	C₂×D₅×C₃⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₅	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
15	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃×#672	C₂²	1	1	yes	characteristic
18	C₃×#3930	—	2	2	no	—
19	#90951	C₄	1	1	yes	characteristic
20	Dic₃×#168	C₄	1	1	yes	characteristic
21	#90934	—	2	2	no	—
22	#71420	—	5	1	no	maximal, Hall
23	C₂×#3930	—	3	1	no	—
24	C₂×#3930	D₃	1	1	yes	characteristic
25	C₄×C₈×D₅	—	3	1	no	—
26	C₅×C₄⋊C₁₆	—	3	1	no	—
27	#64721	—	3	1	no	—
28	#64721	—	3	1	no	—
29	C₄⋊#672	—	3	1	no	—
30	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×D₅×Dic₃	C₈	1	1	yes	characteristic
32	C₂×D₅×Dic₃	C₈	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
36	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
39	C₃×C₈×D₅	D₄	1	2	yes	—
40	C₃×C₈×D₅	Q₈	1	2	yes	—
41	Dic₃×Dic₅	C₈	1	1	yes	characteristic
42	Dic₃×Dic₅	C₈	1	1	yes	characteristic
43	D₅×C₃⋊C₈	—	2	2	no	—
44	C₃×C₅×C₁₆	—	2	1	no	—
45	C₃⋊#672	—	2	1	no	—
46	C₃×#672	—	2	1	no	—
47	C₅×C₃⋊C₁₆	—	2	1	no	—
48	C₂×C₃×M₃₂	—	5	1	no	—
49	C₂×C₈×Dic₃	—	5	1	no	—
50	C₂×#812	—	5	1	no	—
51	Dic₃⋊C₁₆	—	5	1	no	—
52	Dic₃⋊C₁₆	—	5	1	no	—
53	Dic₃⋊C₁₆	—	5	1	no	—
54	Dic₃⋊C₁₆	D₅	1	1	yes	characteristic, complemented
55	C₂×C₈×D₅	—	3	1	no	—
56	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
57	C₄²×D₅	—	3	1	no	—
58	C₂×C₅×C₁₆	—	3	1	no	—
59	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
60	C₄×C₅×C₈	—	3	1	no	—
61	C₈×Dic₅	—	3	1	no	—
62	C₈×Dic₅	—	3	1	no	—
63	C₂×#672	—	3	1	no	—
64	C₂×#672	C₂×D₃	1	1	yes	characteristic
65	C₄×#168	—	3	1	no	—
66	#3930	—	2	2	no	—
67	#3930	—	6	2	no	—
68	#2150	—	15	1	no	Sylow
69	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
72	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
73	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
74	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
76	C₃×C₄×D₅	C₄⋊C₄	1	2	yes	—
77	C₃×C₄×D₅	M₁₆	1	2	yes	—
78	D₅×Dic₃	—	2	2	no	—
79	D₅×Dic₃	—	2	2	no	—
80	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₃⋊#168	—	2	1	no	—
83	C₃×#168	C₄⋊C₄	1	2	yes	—
84	C₅×C₃⋊C₈	—	2	1	no	—
85	C₂²×C₃×C₈	—	5	1	no	—
86	C₂×C₄×Dic₃	—	5	1	no	—
87	C₂²×C₃⋊C₈	—	5	1	no	—
88	C₃×M₃₂	—	10	2	no	—
89	C₂×C₃×C₁₆	—	5	1	no	—
90	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
91	C₈×Dic₃	—	5	1	no	—
92	C₈×Dic₃	—	5	1	no	—
93	C₈×Dic₃	—	5	1	no	—
94	C₈×Dic₃	C₂×D₅	1	1	yes	characteristic
95	#812	—	10	2	no	—
96	C₂×C₃⋊C₁₆	—	5	1	no	—
97	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
98	C₂×C₄×D₅	—	3	1	no	—
99	C₂×C₄×D₅	—	3	1	no	—
100	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
101	C₂×C₅×C₈	—	3	1	no	—
102	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
103	C₄²×C₅	—	3	1	no	—
104	C₄×Dic₅	—	3	1	no	—
105	C₄×Dic₅	—	3	1	no	—
106	C₄×Dic₅	—	3	1	no	—
107	C₂×#168	—	3	1	no	—
108	C₂×#168	C₄×D₃	1	1	yes	characteristic
109	C₈×D₅	—	6	2	no	—
110	C₈×D₅	Dic₆	1	2	yes	—
111	C₈×D₅	#67	1	2	yes	—
112	C₅×C₁₆	—	2	1	no	—
113	C₅×C₁₆	—	6	1	no	—
114	#672	—	2	1	no	—
115	#672	—	6	1	no	—
116	C₂×M₃₂	—	5	1	no	—
117	C₂×M₃₂	—	15	1	no	—
118	C₂×C₄×C₈	—	15	1	no	—
119	C₄⋊C₁₆	—	3	1	no	—
120	C₄⋊C₁₆	—	15	1	no	—
121	C₄⋊C₁₆	—	15	1	no	—
122	C₄⋊C₁₆	—	15	1	no	—
123	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₃×D₅	—	2	1	no	—
125	C₂×C₃×D₅	—	2	1	no	—
126	C₂×C₃×D₅	C₄⋊C₈	1	2	yes	—
127	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
129	Dic₁₅	—	2	1	no	—
130	Dic₁₅	M₃₂	1	2	yes	complemented
131	C₃×Dic₅	C₄⋊C₈	1	2	yes	—
132	C₅×Dic₃	—	2	1	no	—
133	C₅×Dic₃	M₃₂	1	2	yes	complemented
134	C₂²×C₃×C₄	—	5	1	no	—
135	C₂²×Dic₃	—	5	1	no	—
136	C₂²×Dic₃	—	5	1	no	—
137	C₂×C₃×C₈	—	5	1	no	—
138	C₂×C₃×C₈	—	5	2	no	—
139	C₂×C₃×C₈	—	5	2	no	—
140	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
141	C₄×Dic₃	—	5	1	no	—
142	C₄×Dic₃	—	5	1	no	—
143	C₄×Dic₃	—	5	1	no	—
144	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
145	C₂×C₃⋊C₈	—	5	1	no	—
146	C₂×C₃⋊C₈	—	10	2	no	—
147	C₂×C₃⋊C₈	C₄×D₅	1	1	yes	characteristic
148	C₃×C₁₆	—	2	1	no	—
149	C₃×C₁₆	—	10	1	no	—
150	C₃⋊C₁₆	—	2	1	no	—
151	C₃⋊C₁₆	—	10	1	no	—
152	C₂²×D₅	D₃×C₈	1	1	yes	characteristic
153	C₂×C₄×C₅	—	3	1	no	—
154	C₂×C₄×C₅	—	3	1	no	—
155	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂×Dic₅	—	3	1	no	—
157	C₂×Dic₅	—	3	1	no	—
158	C₂×Dic₅	D₃×C₈	1	1	yes	characteristic
159	C₄×D₅	—	6	2	no	—
160	C₄×D₅	—	6	2	no	—
161	C₄×D₅	ΓL(1,𝔽₂₅)	1	2	yes	—
162	C₄×D₅	Dic₃⋊C₄	1	2	yes	—
163	C₅×C₈	—	6	1	no	—
164	C₅×C₈	C₂×Dic₆	1	1	yes	characteristic
165	C₅×C₈	C₂×#67	1	1	yes	characteristic
166	#168	—	6	1	no	—
167	#168	Dic₃⋊C₄	1	2	yes	—
168	C₂²×C₈	—	5	1	no	—
169	C₂²×C₈	—	15	1	no	—
170	C₂×C₄²	—	15	1	no	—
171	M₃₂	—	10	2	no	—
172	M₃₂	—	30	2	no	—
173	C₂×C₁₆	—	3	1	no	—
174	C₂×C₁₆	—	5	1	no	—
175	C₂×C₁₆	—	15	1	no	—
176	C₂×C₁₆	D₃×D₅	1	1	yes	characteristic
177	C₄×C₈	—	3	1	no	—
178	C₄×C₈	—	15	1	no	—
179	C₄×C₈	—	15	1	no	—
180	C₄×C₈	—	15	1	no	—
181	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
182	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
183	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
184	C₃×D₅	—	2	2	no	—
185	C₂³×C₃	—	5	1	no	—
186	C₂×C₃×C₄	—	5	1	no	—
187	C₂×C₃×C₄	—	5	2	no	—
188	C₂×C₃×C₄	—	5	2	no	—
189	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
190	C₂×Dic₃	—	5	1	no	—
191	C₂×Dic₃	—	5	1	no	—
192	C₂×Dic₃	—	10	2	no	—
193	C₂×Dic₃	—	10	2	no	—
194	C₂×Dic₃	C₈×D₅	1	1	yes	characteristic
195	C₂×Dic₃	C₈×D₅	1	1	yes	characteristic
196	C₃×C₈	—	5	2	no	—
197	C₃×C₈	D₄×D₅	1	1	yes	characteristic
198	C₃×C₈	Q₈×D₅	1	1	yes	characteristic
199	C₃⋊C₈	—	2	1	no	—
200	C₃⋊C₈	—	10	1	no	—
201	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
202	C₂×D₅	—	2	1	no	—
203	C₂×D₅	—	2	1	no	—
204	C₂×D₅	Dic₃⋊C₈	1	2	yes	—
205	C₄×C₅	—	3	2	no	—
206	C₄×C₅	—	6	1	no	—
207	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
208	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
209	Dic₅	—	3	2	no	—
210	Dic₅	—	6	1	no	—
211	Dic₅	Dic₃⋊C₈	1	2	yes	—
212	C₂²×C₄	—	5	1	no	—
213	C₂²×C₄	—	15	1	no	—
214	C₂²×C₄	—	15	1	no	—
215	C₂×C₈	—	3	1	no	—
216	C₂×C₈	—	5	1	no	—
217	C₂×C₈	—	5	2	no	—
218	C₂×C₈	—	5	2	no	—
219	C₂×C₈	—	15	1	no	—
220	C₂×C₈	—	30	2	no	—
221	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
222	C₄²	—	3	1	no	—
223	C₄²	—	15	1	no	—
224	C₄²	—	15	1	no	—
225	C₄²	—	15	1	no	—
226	C₁₆	—	2	1	no	—
227	C₁₆	—	6	1	no	—
228	C₁₆	—	10	1	no	—
229	C₁₆	—	30	1	no	—
230	C₃×C₅	#2150	1	1	yes	characteristic, complemented, Hall
231	C₂²×C₃	—	5	2	no	—
232	C₂²×C₃	—	10	1	no	—
233	C₂²×C₃	—	10	1	no	—
234	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
235	C₃×C₄	—	5	2	no	—
236	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
237	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
238	Dic₃	—	2	1	no	—
239	Dic₃	—	5	2	no	—
240	Dic₃	—	10	1	no	—
241	Dic₃	#3930	1	2	yes	complemented
242	C₂×C₅	D₃×M₃₂	1	1	yes	characteristic
243	C₂×C₅	D₃×M₃₂	1	1	yes	characteristic
244	C₂×C₅	C₂×Dic₃⋊C₈	1	1	yes	characteristic
245	D₅	—	2	2	no	—
246	C₂³	—	5	1	no	—
247	C₂×C₄	—	3	1	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	5	2	no	—
251	C₂×C₄	—	5	2	no	—
252	C₂×C₄	—	15	1	no	—
253	C₂×C₄	—	15	1	no	—
254	C₂×C₄	—	30	2	no	—
255	C₂×C₄	—	30	2	no	—
256	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
257	C₈	—	5	2	no	—
258	C₈	—	6	1	no	—
259	C₈	—	30	1	no	—
260	C₈	D₅×Dic₆	1	1	yes	characteristic
261	C₈	D₅×#67	1	1	yes	characteristic
262	C₂×C₃	—	10	2	no	—
263	C₂×C₃	D₅×C₄⋊C₈	1	1	yes	characteristic
264	C₂×C₃	C₂×#3930	1	1	yes	characteristic
265	C₂×C₃	D₅×M₃₂	1	1	yes	characteristic
266	C₅	#71420	1	1	yes	characteristic, complemented, Sylow
267	C₂²	—	5	2	no	—
268	C₂²	—	10	1	no	—
269	C₂²	—	10	1	no	—
270	C₂²	D₃×C₈×D₅	1	1	yes	characteristic
271	C₄	—	3	2	no	—
272	C₄	—	5	2	no	—
273	C₄	—	6	1	no	—
274	C₄	—	15	2	no	—
275	C₄	—	30	1	no	—
276	C₄	D₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
277	C₄	D₅×Dic₃⋊C₄	1	1	yes	characteristic
278	C₃	#10600077	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	10	2	no	—
280	C₂	D₃×#3930	1	1	yes	characteristic
281	C₂	#11746809	1	1	yes	characteristic
282	C₂	D₅×Dic₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂×C₃×C₅×C₁₆	C₂²
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×C₁₆	D₃×D₅
hypocenter	C₃×C₅	#2150
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
4. #11746322	C₂	semidirect
5. C₅×Dic₃⋊C₁₆	C₂	semidirect
6. Dic₃⋊#672	C₂	semidirect
7. #11745928	C₂	semidirect
54. Dic₃⋊C₁₆	D₅	semidirect
130. Dic₁₅	M₃₂	semidirect
133. C₅×Dic₃	M₃₂	semidirect
230. C₃×C₅	#2150	semidirect
241. Dic₃	#3930	semidirect
266. C₅	#71420	semidirect
278. C₃	#10600077	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#3930	2
C₈×D₅×Dic₃	2
C₂×#90934	2
#11746322	2
C₅×Dic₃⋊C₁₆	2
Dic₃⋊#672	2
#11745928	2
#10600077	3
#71420	5