Subgroups — #14471559

Order 1920.

740 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	Q₈×D₆₀	C₂	1	1	yes	maximal, characteristic
2	C₅×#5570	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11747087	C₂	1	3	yes	maximal
5	C₄×#6660	C₂	1	3	yes	maximal
6	#11746945	C₂	1	3	yes	maximal, complemented
7	#11746929	C₂	1	3	yes	maximal, complemented
8	#10615419	—	3	1	no	maximal, Hall
9	C₂×Q₈×D₁₅	—	2	1	no	—
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	#91786	—	2	3	no	—
12	C₄×D₆₀	C₂²	1	3	yes	—
13	#91758	C₂²	1	3	yes	—
14	C₅×Q₈×Dic₃	—	2	1	no	—
15	C₅×#890	—	2	3	no	—
16	C₅×#869	C₂²	1	3	yes	—
17	C₄×C₅×Dic₆	C₂²	1	3	yes	—
18	C₂×#6660	C₂²	1	3	yes	—
19	C₃×C₄⋊#168	C₂²	1	3	yes	—
20	C₃×C₄×#168	C₂²	1	3	yes	—
21	#90985	C₂²	1	3	yes	—
22	#90973	C₂²	1	6	yes	—
23	#90973	C₂²	1	3	yes	—
24	#90969	C₂²	1	1	yes	characteristic
25	#79925	—	5	1	no	maximal, Hall
26	C₅×#557	—	3	1	no	—
27	Q₈×D₂₀	—	3	1	no	—
28	#65372	—	3	3	no	—
29	#65370	—	3	3	no	—
30	C₄×#688	—	3	3	no	—
31	Q₈×#168	D₃	1	1	yes	characteristic, complemented
32	#65309	—	3	3	no	—
33	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
34	Q₈×D₁₅	—	4	4	no	—
35	C₂×D₆₀	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₁₅	—	2	3	no	—
37	C₂×Dic₃₀	—	2	3	no	—
38	C₂×C₅×Dic₆	C₂³	1	3	yes	—
39	C₃×C₅×C₄⋊C₄	D₄	1	3	yes	—
40	C₃×C₄²×C₅	C₂³	1	3	yes	—
41	#6714	—	2	3	no	—
42	C₄⋊Dic₁₅	—	2	3	no	—
43	C₄⋊Dic₁₅	C₂³	1	3	yes	—
44	C₅×C₄⋊Dic₃	—	2	3	no	—
45	C₅×C₄⋊Dic₃	—	2	3	no	—
46	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
47	C₅×Dic₃⋊C₄	—	2	3	no	—
48	C₄×C₅×Dic₃	—	2	3	no	—
49	C₂×C₃×#168	C₂³	1	1	yes	characteristic
50	C₂×C₃×#168	C₂³	1	3	yes	—
51	#6660	—	2	6	no	—
52	Q₈×D₁₂	—	5	1	no	—
53	#5570	D₅	1	1	yes	characteristic, complemented
54	C₃×C₈×Q₈	—	5	1	no	—
55	#4851	—	5	3	no	—
56	#4841	—	5	3	no	—
57	C₄×#204	—	5	3	no	—
58	#4681	—	5	3	no	—
59	C₂×Q₈×D₅	—	6	1	no	—
60	C₅×C₄⋊Q₈	—	3	3	no	—
61	C₅×#125	—	6	3	no	—
62	C₄×C₅×Q₈	—	3	3	no	—
63	C₄×C₅×Q₈	—	6	1	no	—
64	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
65	C₂×#688	—	3	3	no	—
66	#4043	—	6	3	no	—
67	C₄×D₂₀	—	3	3	no	—
68	#4015	—	3	3	no	—
69	#3967	—	3	6	no	—
70	#3967	—	3	3	no	—
71	#3941	—	3	1	no	—
72	#3940	—	3	3	no	—
73	C₄⋊#168	C₂×D₃	1	3	yes	—
74	C₄×#168	C₂×D₃	1	3	yes	—
75	#3359	—	15	1	no	Sylow
76	C₂²×D₁₅	—	2	1	no	—
77	C₃×C₅×Q₈	SD₁₆	1	4	yes	complemented
78	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₅	—	2	3	no	—
82	D₆₀	—	2	1	no	—
83	D₆₀	#41	1	2	yes	—
84	C₄×D₁₅	—	4	6	no	—
85	Dic₃₀	—	4	6	no	—
86	C₂×C₅×Dic₃	—	2	1	no	—
87	C₂×C₅×Dic₃	—	2	3	no	—
88	C₅×Dic₆	—	2	3	no	—
89	C₅×Dic₆	—	2	3	no	—
90	C₃×#168	—	2	3	no	—
91	C₃×#168	—	2	1	no	—
92	C₂×D₃×Q₈	—	10	1	no	—
93	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
94	Q₈×Dic₃	—	2	1	no	—
95	C₂×#204	—	5	3	no	—
96	#897	—	10	3	no	—
97	#890	—	2	3	no	—
98	C₄×D₁₂	—	5	3	no	—
99	#869	—	5	3	no	—
100	#869	C₂×D₅	1	3	yes	—
101	C₄×Dic₆	C₂×D₅	1	3	yes	—
102	C₃×C₄⋊C₈	—	5	3	no	—
103	C₃×C₄×C₈	—	5	3	no	—
104	#821	—	5	1	no	—
105	#817	—	5	3	no	—
106	#816	—	5	3	no	—
107	#816	—	5	6	no	—
108	C₂×C₅×Q₈	—	3	3	no	—
109	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
110	Q₈×D₅	—	12	4	no	—
111	C₂×D₂₀	—	3	1	no	—
112	C₂×C₄×D₅	—	6	3	no	—
113	C₂×Dic₁₀	—	6	3	no	—
114	C₅×C₄⋊C₄	—	3	1	no	—
115	C₅×C₄⋊C₄	—	6	3	no	—
116	C₅×C₄⋊C₄	—	6	3	no	—
117	C₅×C₄⋊C₄	—	6	3	no	—
118	C₅×C₄⋊C₄	D₁₂	1	3	yes	—
119	C₄²×C₅	—	6	3	no	—
120	C₄²×C₅	C₂²×D₃	1	3	yes	—
121	#688	—	6	6	no	—
122	#685	—	6	3	no	—
123	C₄⋊Dic₅	—	3	3	no	—
124	C₄⋊Dic₅	—	6	3	no	—
125	C₂×#168	C₂²×D₃	1	1	yes	characteristic
126	C₂×#168	C₂²×D₃	1	3	yes	—
127	#557	—	3	1	no	—
128	D₄×Q₈	—	15	1	no	—
129	#487	—	15	3	no	—
130	#475	—	15	3	no	—
131	#461	—	15	3	no	—
132	C₈×Q₈	—	5	1	no	—
133	C₄×SD₁₆	—	15	3	no	—
134	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₁₅	—	2	2	no	—
136	C₂×D₁₅	—	4	1	no	—
137	C₂×D₁₅	—	4	1	no	—
138	C₃×C₄×C₅	—	2	3	no	—
139	C₃×C₄×C₅	C₂×SD₁₆	1	6	yes	—
140	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
141	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
142	Dic₁₅	—	4	3	no	—
143	C₅×Dic₃	—	4	3	no	—
144	C₅×Dic₃	—	4	1	no	—
145	C₂×C₃×Q₈	C₂²⋊D₅	1	1	yes	characteristic
146	D₃×Q₈	—	20	4	no	—
147	C₂×D₁₂	—	5	1	no	—
148	C₂×C₄×D₃	—	10	3	no	—
149	C₂×Dic₆	—	10	3	no	—
150	C₂×Dic₆	C₂²×D₅	1	3	yes	—
151	C₂×C₃×C₈	—	5	1	no	—
152	C₂×C₃×C₈	—	5	3	no	—
153	C₃×C₄⋊C₄	C₂²⋊D₅	1	3	yes	—
154	C₃×C₄²	C₂²×D₅	1	3	yes	—
155	(C₂×D₃)⋊C₄	—	10	3	no	—
156	C₄⋊Dic₃	—	2	3	no	—
157	C₄⋊Dic₃	—	2	3	no	—
158	C₄⋊Dic₃	—	5	3	no	—
159	C₄⋊Dic₃	—	10	3	no	—
160	C₄⋊Dic₃	C₂²×D₅	1	1	yes	characteristic
161	Dic₃⋊C₄	—	2	3	no	—
162	C₄×Dic₃	—	2	3	no	—
163	#204	—	10	6	no	—
164	C₂²×D₅	—	6	1	no	—
165	C₅×Q₈	—	6	3	no	—
166	C₅×Q₈	—	6	3	no	—
167	C₅×Q₈	#204	1	4	yes	complemented
168	C₂×C₄×C₅	—	6	1	no	—
169	C₂×C₄×C₅	—	6	3	no	—
170	C₂×C₄×C₅	C₂×D₁₂	1	3	yes	—
171	C₂×C₄×C₅	C₂×D₁₂	1	3	yes	—
172	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
173	C₂×Dic₅	—	6	3	no	—
174	D₂₀	—	3	2	no	—
175	D₂₀	—	6	1	no	—
176	C₄×D₅	—	12	6	no	—
177	Dic₁₀	—	12	6	no	—
178	#168	—	2	1	no	—
179	#168	—	2	3	no	—
180	C₂²×Q₈	—	30	1	no	—
181	C₂×SD₁₆	—	15	3	no	—
182	C₄⋊Q₈	—	3	3	no	—
183	C₄⋊Q₈	—	15	3	no	—
184	#125	—	6	3	no	—
185	C₂²⋊Q₈	—	30	3	no	—
186	C₄×Q₈	—	3	3	no	—
187	C₄×Q₈	—	6	1	no	—
188	C₄×Q₈	D₃×D₅	1	1	yes	characteristic
189	C₄×D₄	—	15	3	no	—
190	#106	—	15	3	no	—
191	C₄⋊C₈	—	5	3	no	—
192	#103	—	15	3	no	—
193	#103	—	15	6	no	—
194	#102	—	15	1	no	—
195	C₄×C₈	—	5	3	no	—
196	C₂×C₃×C₅	#548	1	1	yes	characteristic
197	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
198	C₂×C₃×C₅	#578	1	1	yes	characteristic
199	D₁₅	—	4	2	no	—
200	C₂²×D₃	—	10	1	no	—
201	C₃×Q₈	#688	1	4	yes	complemented
202	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	3	yes	—
203	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	3	yes	—
204	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
205	C₂×Dic₃	—	2	1	no	—
206	C₂×Dic₃	—	2	3	no	—
207	C₂×Dic₃	—	10	3	no	—
208	D₁₂	—	5	2	no	—
209	D₁₂	—	10	1	no	—
210	C₄×D₃	—	20	6	no	—
211	Dic₆	—	2	3	no	—
212	Dic₆	—	2	3	no	—
213	Dic₆	—	20	6	no	—
214	C₃×C₈	—	10	3	no	—
215	C₃×C₈	—	10	1	no	—
216	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
217	C₂×D₅	—	6	2	no	—
218	C₂×D₅	—	12	1	no	—
219	C₂×D₅	—	12	1	no	—
220	C₄×C₅	—	2	3	no	—
221	C₄×C₅	—	12	1	no	—
222	C₄×C₅	—	12	3	no	—
223	C₄×C₅	C₂×#204	1	6	yes	—
224	C₄×C₅	#1007	1	1	yes	characteristic
225	C₄×C₅	D₃×#41	1	1	yes	characteristic
226	Dic₅	—	12	3	no	—
227	C₂×Q₈	—	3	3	no	—
228	C₂×Q₈	—	30	3	no	—
229	C₂×Q₈	—	60	4	no	—
230	C₂×Q₈	#1203	1	1	yes	characteristic
231	C₂×D₄	—	15	1	no	—
232	C₂²×C₄	—	30	3	no	—
233	SD₁₆	—	30	6	no	—
234	C₂×C₈	—	5	1	no	—
235	C₂×C₈	—	5	3	no	—
236	C₄⋊C₄	—	3	1	no	—
237	C₄⋊C₄	—	6	3	no	—
238	C₄⋊C₄	—	6	3	no	—
239	C₄⋊C₄	—	6	3	no	—
240	C₄⋊C₄	—	15	3	no	—
241	C₄⋊C₄	—	30	3	no	—
242	C₄⋊C₄	#1203	1	3	yes	—
243	C₂²⋊C₄	—	30	3	no	—
244	C₄²	—	6	3	no	—
245	C₄²	C₂×D₃×D₅	1	3	yes	—
246	C₃×C₅	#3359	1	1	yes	characteristic, complemented, Hall
247	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
248	C₂×D₃	—	10	2	no	—
249	C₂×D₃	—	20	1	no	—
250	C₂×D₃	—	20	1	no	—
251	C₃×C₄	—	2	3	no	—
252	C₃×C₄	C₂×#688	1	6	yes	—
253	C₃×C₄	D₅×#41	1	1	yes	characteristic
254	C₃×C₄	#4150	1	1	yes	characteristic
255	Dic₃	—	4	3	no	—
256	Dic₃	—	4	1	no	—
257	Dic₃	—	20	3	no	—
258	C₂×C₅	#5576	1	1	yes	characteristic
259	C₂×C₅	C₂²×#204	1	1	yes	characteristic
260	C₂×C₅	#5753	1	1	yes	characteristic
261	D₅	—	12	2	no	—
262	C₂³	—	30	1	no	—
263	Q₈	—	6	3	no	—
264	Q₈	—	6	3	no	—
265	Q₈	—	60	6	no	—
266	Q₈	#6660	1	4	yes	complemented
267	D₄	—	15	2	no	—
268	D₄	—	30	1	no	—
269	C₂×C₄	—	6	1	no	—
270	C₂×C₄	—	6	3	no	—
271	C₂×C₄	—	30	3	no	—
272	C₂×C₄	—	60	6	no	—
273	C₂×C₄	C₂×#1203	1	3	yes	—
274	C₂×C₄	C₂×#1203	1	3	yes	—
275	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
276	C₈	—	10	1	no	—
277	C₈	—	10	3	no	—
278	C₂×C₃	C₂²×#688	1	1	yes	characteristic
279	C₂×C₃	#66165	1	1	yes	characteristic
280	C₂×C₃	#66172	1	1	yes	characteristic
281	D₃	—	20	2	no	—
282	C₅	#79925	1	1	yes	characteristic, complemented, Sylow
283	C₂²	—	30	2	no	—
284	C₂²	—	60	1	no	—
285	C₂²	—	60	1	no	—
286	C₂²	C₂²×#1203	1	1	yes	characteristic
287	C₄	—	2	3	no	—
288	C₄	—	12	1	no	—
289	C₄	—	12	3	no	—
290	C₄	—	60	3	no	—
291	C₄	C₂×#6660	1	6	yes	—
292	C₄	#92030	1	1	yes	characteristic
293	C₄	#92034	1	1	yes	characteristic
294	C₃	#10615419	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	60	2	no	—
296	C₂	C₂²×#6660	1	1	yes	characteristic
297	C₂	#11755134	1	1	yes	characteristic
298	C₂	#11755306	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1203
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₃×C₄×C₅×Q₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×D₆₀	C₂³
hypercenter	C₄×Q₈	D₃×D₅
hypocenter	C₃×C₅	#3359
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₅×#5570	C₂	semidirect
3. C₃×Q₈×#168	C₂	semidirect
6. #11746945	C₂	semidirect
7. #11746929	C₂	semidirect
31. Q₈×#168	D₃	semidirect
53. #5570	D₅	semidirect
77. C₃×C₅×Q₈	SD₁₆	semidirect
167. C₅×Q₈	#204	semidirect
201. C₃×Q₈	#688	semidirect
246. C₃×C₅	#3359	semidirect
266. Q₈	#6660	semidirect
282. C₅	#79925	semidirect
294. C₃	#10615419	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₆₀	2
C₅×#5570	2
C₃×Q₈×#168	2
#11747087	2
C₄×#6660	2
#11746945	2
#11746929	2
#10615419	3
#79925	5