Subgroups — #14313975

Order 1920.

440 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#11752380	C₂	1	1	yes	maximal, characteristic
3	D₅⋊#812	C₂	1	1	yes	maximal, characteristic
4	#11746779	C₂	1	4	yes	maximal, complemented
5	#10604832	—	3	1	no	maximal, Hall
6	C₂×C₃×Q₈×D₅	—	2	1	no	—
7	#91989	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃×#689	—	2	1	no	—
10	C₃×D₅×Q₁₆	—	2	1	no	—
11	C₃×D₅×Q₁₆	—	2	1	no	—
12	C₃×D₅×Q₁₆	C₄	1	2	yes	complemented
13	C₃×D₅×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₃×Dic₂₀	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
16	#91227	—	2	1	no	—
17	#91223	C₂²	1	2	yes	—
18	#91222	—	2	1	no	—
19	#91219	—	2	1	no	—
20	D₅⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
21	#72061	—	5	1	no	maximal, Hall
22	C₂×D₅×Q₁₆	D₃	1	1	yes	characteristic
23	#65721	—	3	1	no	—
24	D₅⋊M₃₂	—	3	1	no	—
25	#64908	—	3	4	no	—
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₃×Q₈×D₅	—	2	2	no	—
28	C₃×Q₈×D₅	—	2	2	no	—
29	C₃×Q₈×D₅	—	4	1	no	—
30	C₃×Q₈×D₅	—	4	1	no	—
31	C₂×C₃×C₄×D₅	—	4	1	no	—
32	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
33	C₂×C₃×Dic₁₀	—	2	1	no	—
34	C₂×C₃×Dic₁₀	—	4	1	no	—
35	#6763	—	4	1	no	—
36	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
37	C₄×ΓL(1,𝔽₁₆)	—	4	1	no	—
38	C₃×C₅×Q₁₆	—	2	1	no	—
39	C₃×C₅×Q₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₃×#689	—	2	2	no	—
42	C₃×#689	—	2	2	no	—
43	C₂×C₃×#168	D₄	1	1	yes	characteristic
44	C₃×Dic₂₀	—	2	1	no	—
45	C₃×Dic₂₀	C₂×C₄	1	2	yes	—
46	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₈×D₅	D₄	1	1	yes	characteristic
48	C₃×C₈×D₅	D₄	1	1	yes	characteristic
49	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
50	C₃⋊#674	—	2	2	no	—
51	C₂²×C₃×Q₁₆	—	5	1	no	—
52	#5106	—	5	1	no	—
53	C₂×#812	—	5	1	no	—
54	#4564	—	5	4	no	—
55	C₂×Q₈×D₅	—	2	1	no	—
56	#4130	—	6	1	no	—
57	C₂×C₅×Q₁₆	Dic₃	1	1	yes	characteristic, complemented
58	C₂×#689	—	2	1	no	—
59	D₅×Q₁₆	—	2	1	no	—
60	D₅×Q₁₆	—	2	1	no	—
61	D₅×Q₁₆	Dic₃	1	2	yes	complemented
62	D₅×Q₁₆	C₂×D₃	1	2	yes	—
63	C₂×Dic₂₀	Dic₃	1	1	yes	characteristic, complemented
64	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
65	#3998	—	6	1	no	—
66	#3994	—	3	2	no	—
67	#3993	—	6	1	no	—
68	#3990	—	6	1	no	—
69	D₅⋊C₁₆	—	3	2	no	—
70	#2141	—	15	1	no	Sylow
71	C₂²×C₃×D₅	D₈	1	1	yes	characteristic
72	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
73	C₃×C₅×Q₈	—	2	2	no	—
74	C₃×C₅×Q₈	—	4	1	no	—
75	C₂×C₃×C₄×C₅	—	4	1	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×Dic₅	—	4	1	no	—
78	C₂×C₃×Dic₅	SD₁₆	1	1	yes	characteristic
79	C₃×C₄×D₅	—	4	2	no	—
80	C₃×C₄×D₅	—	4	2	no	—
81	C₃×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₃×C₄×D₅	D₈	1	1	yes	characteristic
83	C₃×C₄×D₅	SD₁₆	1	1	yes	characteristic
84	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
85	C₃×Dic₁₀	—	2	2	no	—
86	C₃×Dic₁₀	—	4	1	no	—
87	C₃×Dic₁₀	—	4	2	no	—
88	C₃×Dic₁₀	—	4	2	no	—
89	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
92	C₃×#168	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₃×Q₈	—	10	1	no	—
94	C₂×C₃×Q₁₆	—	5	2	no	—
95	C₂×C₃×Q₁₆	—	5	2	no	—
96	C₂×C₃×Q₁₆	—	5	1	no	—
97	C₂×C₃×Q₁₆	—	10	1	no	—
98	C₂×C₃×Q₁₆	—	10	1	no	—
99	C₂×C₃×Q₁₆	—	10	1	no	—
100	C₂×C₃×Q₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
101	C₂²×C₃×C₈	—	5	1	no	—
102	#926	—	10	1	no	—
103	#818	—	5	2	no	—
104	#817	—	10	1	no	—
105	#812	—	10	1	no	—
106	#812	—	10	1	no	—
107	C₂×C₃⋊C₁₆	—	5	2	no	—
108	C₂×C₅×Q₈	—	2	1	no	—
109	Q₈×D₅	—	2	2	no	—
110	Q₈×D₅	—	2	2	no	—
111	Q₈×D₅	—	4	1	no	—
112	Q₈×D₅	—	4	1	no	—
113	C₂×C₄×D₅	—	4	1	no	—
114	C₂×C₄×D₅	#67	1	1	yes	characteristic
115	C₂×Dic₁₀	—	2	1	no	—
116	C₂×Dic₁₀	—	4	1	no	—
117	C₂²⋊AGL(1,𝔽₅)	—	12	1	no	—
118	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
119	C₄×AGL(1,𝔽₅)	—	12	1	no	—
120	C₅×Q₁₆	—	2	1	no	—
121	C₅×Q₁₆	C₂×Dic₃	1	2	yes	—
122	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
123	#689	—	2	2	no	—
124	#689	—	2	2	no	—
125	C₂×#168	#67	1	1	yes	characteristic
126	Dic₂₀	—	2	1	no	—
127	Dic₂₀	C₂×Dic₃	1	2	yes	—
128	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic
129	C₈×D₅	#67	1	1	yes	characteristic
130	C₈×D₅	#67	1	1	yes	characteristic
131	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
132	#674	—	6	2	no	—
133	C₂²×Q₁₆	—	5	1	no	—
134	C₂×M₃₂	—	15	1	no	—
135	#427	—	15	1	no	—
136	#358	—	15	4	no	—
137	C₂²×C₃×C₅	#102	1	1	yes	characteristic
138	C₂×C₃×D₅	—	2	2	no	—
139	C₂×C₃×D₅	#102	1	1	yes	characteristic
140	C₂×C₃×D₅	C₂×D₈	1	1	yes	characteristic
141	ΓL(1,𝔽₁₆)	—	8	2	no	—
142	C₃×C₄×C₅	—	4	2	no	—
143	C₃×C₄×C₅	#102	1	1	yes	characteristic
144	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₃×Dic₅	—	4	2	no	—
146	C₃×Dic₅	#102	1	1	yes	characteristic
147	C₃×Dic₅	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₃×Q₈	—	2	1	no	—
149	C₂×C₃×Q₈	—	10	2	no	—
150	C₂×C₃×Q₈	—	10	2	no	—
151	C₂×C₃×Q₈	—	10	1	no	—
152	C₂×C₃×Q₈	—	20	1	no	—
153	C₂×C₃×Q₈	—	20	1	no	—
154	C₂×C₃×Q₈	—	20	1	no	—
155	C₂²×C₃×C₄	—	5	1	no	—
156	C₂²×C₃×C₄	—	20	1	no	—
157	C₃×Q₁₆	—	2	1	no	—
158	C₃×Q₁₆	—	5	2	no	—
159	C₃×Q₁₆	—	10	2	no	—
160	C₃×Q₁₆	—	10	2	no	—
161	C₃×Q₁₆	—	10	1	no	—
162	C₃×Q₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
163	C₂×C₃×C₈	—	5	1	no	—
164	C₂×C₃×C₈	—	5	1	no	—
165	C₂×C₃×C₈	—	5	1	no	—
166	C₂×C₃×C₈	—	5	1	no	—
167	C₂×C₃×C₈	—	5	1	no	—
168	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
169	#217	—	20	1	no	—
170	C₄⋊Dic₃	—	10	2	no	—
171	C₄×Dic₃	—	20	1	no	—
172	C₃⋊C₁₆	—	10	2	no	—
173	C₂²×D₅	#213	1	1	yes	characteristic
174	C₂×AGL(1,𝔽₅)	—	12	2	no	—
175	C₅×Q₈	—	2	2	no	—
176	C₅×Q₈	—	4	1	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	#217	1	1	yes	characteristic
179	C₂×Dic₅	—	4	1	no	—
180	C₂×Dic₅	#214	1	1	yes	characteristic
181	C₄×D₅	—	4	2	no	—
182	C₄×D₅	—	4	2	no	—
183	C₄×D₅	#213	1	1	yes	characteristic
184	C₄×D₅	#214	1	1	yes	characteristic
185	C₄×D₅	#217	1	1	yes	characteristic
186	C₄×D₅	C₂×#67	1	1	yes	characteristic
187	Dic₁₀	—	2	2	no	—
188	Dic₁₀	—	4	1	no	—
189	Dic₁₀	—	4	2	no	—
190	Dic₁₀	—	4	2	no	—
191	C₅×C₈	#217	1	1	yes	characteristic
192	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
193	#168	#217	1	1	yes	characteristic
194	#168	C₂×#67	1	1	yes	characteristic
195	C₂²×Q₈	—	10	1	no	—
196	C₂×Q₁₆	—	5	2	no	—
197	C₂×Q₁₆	—	5	2	no	—
198	C₂×Q₁₆	—	5	1	no	—
199	C₂×Q₁₆	—	10	1	no	—
200	C₂×Q₁₆	—	10	1	no	—
201	C₂×Q₁₆	—	10	1	no	—
202	C₂×Q₁₆	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
203	C₂²×C₈	—	5	1	no	—
204	#117	—	30	1	no	—
205	M₃₂	—	30	1	no	—
206	M₃₂	—	30	1	no	—
207	C₂×C₁₆	—	15	2	no	—
208	#107	—	15	2	no	—
209	#106	—	30	1	no	—
210	C₂×C₃×C₅	—	2	1	no	—
211	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
212	C₃×D₅	—	2	1	no	—
213	C₃×D₅	#510	1	2	yes	—
214	C₂³×C₃	—	5	1	no	—
215	C₃×Q₈	—	2	2	no	—
216	C₃×Q₈	—	4	1	no	—
217	C₃×Q₈	—	10	2	no	—
218	C₃×Q₈	—	20	2	no	—
219	C₃×Q₈	—	20	2	no	—
220	C₃×Q₈	—	20	1	no	—
221	C₂×C₃×C₄	—	4	1	no	—
222	C₂×C₃×C₄	—	5	1	no	—
223	C₂×C₃×C₄	—	5	1	no	—
224	C₂×C₃×C₄	—	5	1	no	—
225	C₂×C₃×C₄	—	5	1	no	—
226	C₂×C₃×C₄	—	5	1	no	—
227	C₂×C₃×C₄	—	20	2	no	—
228	C₂×C₃×C₄	—	20	2	no	—
229	C₂×C₃×C₄	—	20	1	no	—
230	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
231	C₂×Dic₃	—	20	2	no	—
232	C₃×C₈	—	5	1	no	—
233	C₃×C₈	—	5	1	no	—
234	C₃×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
236	C₂²×C₅	#832	1	1	yes	characteristic
237	C₂×D₅	—	2	2	no	—
238	C₂×D₅	#832	1	1	yes	characteristic
239	C₂×D₅	C₂×#213	1	1	yes	characteristic
240	AGL(1,𝔽₅)	—	24	2	no	—
241	C₄×C₅	—	4	2	no	—
242	C₄×C₅	#832	1	1	yes	characteristic
243	C₄×C₅	C₂×#217	1	1	yes	characteristic
244	Dic₅	—	4	2	no	—
245	Dic₅	#832	1	1	yes	characteristic
246	Dic₅	C₂×#214	1	1	yes	characteristic
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	10	2	no	—
249	C₂×Q₈	—	10	2	no	—
250	C₂×Q₈	—	10	1	no	—
251	C₂×Q₈	—	20	1	no	—
252	C₂×Q₈	—	20	1	no	—
253	C₂×Q₈	—	20	1	no	—
254	C₂²×C₄	—	5	1	no	—
255	C₂²×C₄	—	20	1	no	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	5	2	no	—
258	Q₁₆	—	10	2	no	—
259	Q₁₆	—	10	2	no	—
260	Q₁₆	—	10	1	no	—
261	Q₁₆	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
262	C₂×C₈	—	5	1	no	—
263	C₂×C₈	—	5	1	no	—
264	C₂×C₈	—	5	1	no	—
265	C₂×C₈	—	5	1	no	—
266	C₂×C₈	—	5	1	no	—
267	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
268	C₄⋊C₄	—	30	2	no	—
269	C₂²⋊C₄	—	60	1	no	—
270	C₄²	—	60	1	no	—
271	C₁₆	—	30	2	no	—
272	C₃×C₅	#2141	1	1	yes	characteristic, complemented, Hall
273	C₂²×C₃	—	5	1	no	—
274	C₂²×C₃	—	5	1	no	—
275	C₂²×C₃	—	10	2	no	—
276	C₂²×C₃	#4007	1	1	yes	characteristic
277	C₃×C₄	—	4	2	no	—
278	C₃×C₄	—	5	1	no	—
279	C₃×C₄	—	5	1	no	—
280	C₃×C₄	—	20	2	no	—
281	C₃×C₄	#4007	1	1	yes	characteristic
282	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
283	Dic₃	—	40	2	no	—
284	C₂×C₅	—	2	1	no	—
285	C₂×C₅	C₂×#832	1	1	yes	characteristic
286	D₅	—	2	1	no	—
287	D₅	#5179	1	2	yes	—
288	C₂³	—	5	1	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	10	2	no	—
292	Q₈	—	20	2	no	—
293	Q₈	—	20	2	no	—
294	Q₈	—	20	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	5	1	no	—
297	C₂×C₄	—	5	1	no	—
298	C₂×C₄	—	5	1	no	—
299	C₂×C₄	—	5	1	no	—
300	C₂×C₄	—	5	1	no	—
301	C₂×C₄	—	20	2	no	—
302	C₂×C₄	—	20	2	no	—
303	C₂×C₄	—	20	1	no	—
304	C₂×C₄	—	60	2	no	—
305	C₂×C₄	#6763	1	1	yes	characteristic
306	C₈	—	5	1	no	—
307	C₈	—	5	1	no	—
308	C₈	#6763	1	1	yes	characteristic
309	C₈	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	5	2	no	—
312	C₂×C₃	—	10	1	no	—
313	C₂×C₃	C₂×#4007	1	1	yes	characteristic
314	C₅	#72061	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	5	1	no	—
316	C₂²	—	5	1	no	—
317	C₂²	—	10	2	no	—
318	C₂²	#91236	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	5	1	no	—
321	C₄	—	5	1	no	—
322	C₄	—	20	2	no	—
323	C₄	—	120	2	no	—
324	C₄	#91236	1	1	yes	characteristic
325	C₄	C₂×#6763	1	1	yes	characteristic
326	C₃	#10604832	1	1	yes	characteristic, complemented, Sylow
327	C₂	—	2	1	no	—
328	C₂	—	5	2	no	—
329	C₂	—	10	1	no	—
330	C₂	C₂×#91236	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#91236
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₈	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₅×Q₁₆	C₄
socle	C₂×C₃×C₅	C₂×#102
prime radical	C₂²×C₃×D₅	D₈
cosocle	C₃×C₈×D₅	C₂³
hypercenter	C₂×Q₁₆	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2141
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #11746779	C₂	semidirect
8. C₂×C₃×C₅×Q₁₆	C₄	semidirect
12. C₃×D₅×Q₁₆	C₄	semidirect
14. C₂×C₃×Dic₂₀	C₄	semidirect
57. C₂×C₅×Q₁₆	Dic₃	semidirect
61. D₅×Q₁₆	Dic₃	semidirect
63. C₂×Dic₂₀	Dic₃	semidirect
100. C₂×C₃×Q₁₆	AGL(1,𝔽₅)	semidirect
202. C₂×Q₁₆	ΓL(1,𝔽₁₆)	semidirect
272. C₃×C₅	#2141	semidirect
314. C₅	#72061	semidirect
326. C₃	#10604832	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₅×Q₁₆	2
#11752380	2
D₅⋊#812	2
#11746779	2
#10604832	3
#72061	5