Subgroups — #14285908

Order 1920.

300 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752342	C₂	1	1	yes	maximal, characteristic
2	#11752339	C₂	1	1	yes	maximal, characteristic
3	C₃×#65122	C₂	1	1	yes	maximal, characteristic
4	#11746755	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746754	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746753	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746752	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604017	—	3	1	no	maximal, Hall
9	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
10	#91232	—	2	1	no	—
11	#91232	—	2	1	no	—
12	#91231	—	2	1	no	—
13	C₄⋊(C₃⋊#170)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃⋊#170)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₃⋊#170)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₃⋊#170)	C₂²	1	1	yes	characteristic
17	C₄×C₃⋊#170	—	2	1	no	—
18	C₃×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
19	C₃×#3952	C₄	1	1	yes	characteristic
20	C₃×#3952	C₄	1	1	yes	characteristic
21	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊#168	C₄	1	1	yes	characteristic
24	#70751	—	5	1	no	maximal, Hall
25	#65683	—	3	1	no	—
26	#65680	—	3	1	no	—
27	#65122	D₃	1	1	yes	characteristic
28	#64884	—	3	1	no	—
29	#64883	—	3	1	no	—
30	#64882	—	3	1	no	—
31	#64881	—	3	1	no	—
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
35	C₂×C₃⋊#170	—	2	1	no	—
36	C₂×C₃⋊#170	—	2	1	no	—
37	C₂×C₃⋊#170	—	2	1	no	—
38	C₂×C₃⋊#170	—	2	1	no	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
43	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
44	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₃×#431	—	5	1	no	—
48	#4926	—	5	1	no	—
49	#4925	—	5	1	no	—
50	#4458	—	5	1	no	—
51	#4458	—	5	1	no	—
52	#4457	—	5	1	no	—
53	#4457	—	5	1	no	—
54	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
55	#4003	—	6	1	no	—
56	#4003	—	6	1	no	—
57	#4002	—	6	1	no	—
58	C₄⋊#170	—	3	1	no	—
59	C₄⋊#170	—	3	1	no	—
60	C₄⋊#170	—	3	1	no	—
61	C₄⋊#170	—	3	1	no	—
62	C₄×#170	—	6	1	no	—
63	C₅×C₄⋊C₈	Dic₃	1	1	yes	characteristic
64	#3952	Dic₃	1	1	yes	characteristic
65	#3952	Dic₃	1	1	yes	characteristic
66	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
67	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
68	C₄⋊#168	Dic₃	1	1	yes	characteristic
69	#1572	—	15	1	no	Sylow
70	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
77	C₃×C₄×D₅	—	2	1	no	—
78	C₃×C₄×D₅	—	2	1	no	—
79	C₃×C₄×D₅	—	2	1	no	—
80	C₃×C₄×D₅	—	2	1	no	—
81	C₃×C₄×D₅	—	4	1	no	—
82	C₃⋊#170	—	4	1	no	—
83	C₃⋊#170	—	4	1	no	—
84	C₃⋊#170	—	4	1	no	—
85	C₃⋊#170	—	4	1	no	—
86	C₃×C₅×C₈	—	2	1	no	—
87	C₃×C₅×C₈	—	2	1	no	—
88	C₃×#168	—	2	1	no	—
89	C₃×#168	—	2	1	no	—
90	C₂×C₃×C₄²	—	5	1	no	—
91	C₃×C₄⋊C₈	—	5	1	no	—
92	C₃×C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₃×C₂²⋊C₈	—	5	1	no	—
94	C₃×C₂²⋊C₈	—	5	1	no	—
95	C₃×C₄×C₈	—	5	1	no	—
96	C₃×C₄×C₈	—	5	1	no	—
97	#830	—	10	1	no	—
98	#830	—	10	1	no	—
99	C₄⋊(C₃⋊C₈)	—	5	1	no	—
100	C₄⋊(C₃⋊C₈)	—	5	1	no	—
101	C₄⋊(C₃⋊C₈)	—	5	1	no	—
102	C₄⋊(C₃⋊C₈)	—	5	1	no	—
103	#803	—	10	1	no	—
104	C₄×C₃⋊C₈	—	10	1	no	—
105	C₂×C₄×D₅	Dic₆	1	1	yes	characteristic
106	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
107	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
108	C₂×#170	—	6	1	no	—
109	C₂×#170	—	6	1	no	—
110	C₂×#170	—	6	1	no	—
111	C₂×#170	—	6	1	no	—
112	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
114	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
115	C₄×Dic₅	Dic₆	1	1	yes	characteristic
116	C₄×Dic₅	D₁₂	1	1	yes	characteristic
117	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
118	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
119	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
120	#432	—	15	1	no	—
121	#431	—	5	1	no	—
122	#431	—	15	1	no	—
123	#335	—	15	1	no	—
124	#335	—	15	1	no	—
125	#334	—	15	1	no	—
126	#334	—	15	1	no	—
127	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₃×D₅	—	2	1	no	—
129	C₂×C₃×D₅	—	2	1	no	—
130	C₂×C₃×D₅	—	2	1	no	—
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	#131	1	1	yes	characteristic
134	C₃×C₄×C₅	#131	1	1	yes	characteristic
135	C₃×Dic₅	—	2	1	no	—
136	C₃×Dic₅	—	2	1	no	—
137	C₃×Dic₅	#135	1	1	yes	characteristic
138	C₃×Dic₅	#135	1	1	yes	characteristic
139	C₂²×C₃×C₄	—	5	1	no	—
140	C₂²×C₃×C₄	—	5	1	no	—
141	C₂²×C₃×C₄	—	5	1	no	—
142	C₂×C₃×C₈	—	5	1	no	—
143	C₂×C₃×C₈	—	5	1	no	—
144	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
145	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₃×C₄²	—	5	1	no	—
147	C₃×C₄²	—	5	1	no	—
148	C₃×C₄²	—	5	1	no	—
149	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂×C₃⋊C₈	—	10	1	no	—
151	C₂×C₃⋊C₈	—	10	1	no	—
152	C₂×C₃⋊C₈	—	10	1	no	—
153	C₂×C₃⋊C₈	—	10	1	no	—
154	C₂²×D₅	C₄⋊Dic₃	1	1	yes	characteristic
155	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
156	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
158	C₂×Dic₅	C₄⋊Dic₃	1	1	yes	characteristic
159	C₂×Dic₅	C₂×Dic₆	1	1	yes	characteristic
160	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
161	C₄×D₅	—	2	1	no	—
162	C₄×D₅	—	2	1	no	—
163	C₄×D₅	—	2	1	no	—
164	C₄×D₅	—	2	1	no	—
165	C₄×D₅	—	4	1	no	—
166	#170	—	12	1	no	—
167	#170	—	12	1	no	—
168	#170	—	12	1	no	—
169	#170	—	12	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	2	1	no	—
172	#168	—	2	1	no	—
173	#168	—	2	1	no	—
174	C₂×C₄²	—	5	1	no	—
175	C₄⋊C₈	—	5	1	no	—
176	C₄⋊C₈	—	15	1	no	—
177	C₄⋊C₈	—	15	1	no	—
178	C₄⋊C₈	—	15	1	no	—
179	C₄⋊C₈	—	15	1	no	—
180	C₄⋊C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
181	C₂²⋊C₈	—	5	1	no	—
182	C₂²⋊C₈	—	5	1	no	—
183	C₂²⋊C₈	—	30	1	no	—
184	C₂²⋊C₈	—	30	1	no	—
185	#97	—	30	1	no	—
186	C₄×C₈	—	5	1	no	—
187	C₄×C₈	—	5	1	no	—
188	C₄×C₈	—	30	1	no	—
189	C₂×C₃×C₅	#424	1	1	yes	characteristic
190	C₂×C₃×C₅	#427	1	1	yes	characteristic
191	C₂×C₃×C₅	#430	1	1	yes	characteristic
192	C₃×D₅	—	4	1	no	—
193	C₂³×C₃	—	5	1	no	—
194	C₂×C₃×C₄	—	5	1	no	—
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	—	5	1	no	—
197	C₂×C₃×C₄	—	10	1	no	—
198	C₂×C₃×C₄	—	10	1	no	—
199	C₂×C₃×C₄	—	10	1	no	—
200	C₂×C₃×C₄	—	10	1	no	—
201	C₂×C₃×C₄	—	20	1	no	—
202	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	—	10	1	no	—
208	C₃×C₈	—	10	1	no	—
209	C₃⋊C₈	—	20	1	no	—
210	C₃⋊C₈	—	20	1	no	—
211	C₃⋊C₈	—	20	1	no	—
212	C₃⋊C₈	—	20	1	no	—
213	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
214	C₂×D₅	—	2	1	no	—
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	—	2	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	#948	1	1	yes	characteristic
220	C₄×C₅	#948	1	1	yes	characteristic
221	Dic₅	—	2	1	no	—
222	Dic₅	—	2	1	no	—
223	Dic₅	#904	1	1	yes	characteristic
224	Dic₅	#904	1	1	yes	characteristic
225	C₂²×C₄	—	5	1	no	—
226	C₂²×C₄	—	5	1	no	—
227	C₂²×C₄	—	5	1	no	—
228	C₂×C₈	—	5	1	no	—
229	C₂×C₈	—	5	1	no	—
230	C₂×C₈	—	30	1	no	—
231	C₂×C₈	—	30	1	no	—
232	C₂×C₈	—	30	1	no	—
233	C₂×C₈	—	30	1	no	—
234	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
235	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
236	C₄²	—	5	1	no	—
237	C₄²	—	5	1	no	—
238	C₄²	—	5	1	no	—
239	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
240	C₃×C₅	#1572	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	10	1	no	—
242	C₂²×C₃	—	10	1	no	—
243	C₂²×C₃	—	10	1	no	—
244	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	5	1	no	—
248	C₃×C₄	—	5	1	no	—
249	C₃×C₄	—	10	1	no	—
250	C₃×C₄	—	10	1	no	—
251	C₃×C₄	#4131	1	1	yes	characteristic
252	C₃×C₄	#4133	1	1	yes	characteristic
253	C₂×C₅	#4999	1	1	yes	characteristic
254	C₂×C₅	#5106	1	1	yes	characteristic
255	C₂×C₅	#5125	1	1	yes	characteristic
256	D₅	—	4	1	no	—
257	C₂³	—	5	1	no	—
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	5	1	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	10	1	no	—
262	C₂×C₄	—	10	1	no	—
263	C₂×C₄	—	10	1	no	—
264	C₂×C₄	—	10	1	no	—
265	C₂×C₄	—	20	1	no	—
266	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
267	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
268	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	10	1	no	—
272	C₈	—	10	1	no	—
273	C₈	—	60	1	no	—
274	C₈	—	60	1	no	—
275	C₈	—	60	1	no	—
276	C₈	—	60	1	no	—
277	C₂×C₃	—	20	1	no	—
278	C₂×C₃	#65708	1	1	yes	characteristic
279	C₂×C₃	#65721	1	1	yes	characteristic
280	C₂×C₃	#65729	1	1	yes	characteristic
281	C₅	#70751	1	1	yes	characteristic, complemented, Sylow
282	C₂²	—	10	1	no	—
283	C₂²	—	10	1	no	—
284	C₂²	—	10	1	no	—
285	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	5	1	no	—
289	C₄	—	5	1	no	—
290	C₄	—	10	1	no	—
291	C₄	—	10	1	no	—
292	C₄	#91990	1	1	yes	characteristic
293	C₄	#91992	1	1	yes	characteristic
294	C₃	#10604017	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	20	1	no	—
296	C₂	#11752367	1	1	yes	characteristic
297	C₂	#11752380	1	1	yes	characteristic
298	C₂	#11752388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×C₄⋊C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₄⋊C₄
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₄⋊C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1572
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #11746755	C₂	semidirect
5. #11746754	C₂	semidirect
6. #11746753	C₂	semidirect
7. #11746752	C₂	semidirect
240. C₃×C₅	#1572	semidirect
281. C₅	#70751	semidirect
294. C₃	#10604017	semidirect

Maximal subgroups

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

subgroup	index
#11752342	2
#11752339	2
C₃×#65122	2
#11746755	2
#11746754	2
#11746753	2
#11746752	2
#10604017	3
#70751	5