Subgroups — #14315149

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₈×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#90934	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4549	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745927	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745879	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599897	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₂²⋊D₅	C₄	1	1	yes	characteristic
11	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	#91096	—	2	2	no	—
13	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₃×C₂²⋊#168	C₄	1	1	yes	characteristic
18	C₃×#3952	C₄	1	1	yes	characteristic
19	C₃×#3946	C₄	1	1	yes	characteristic
20	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
21	#90934	—	2	2	no	—
22	#70858	—	5	1	no	maximal, Hall
23	C₈×C₂²⋊D₅	D₃	1	1	yes	characteristic
24	C₂×#3944	—	3	1	no	—
25	C₂×#3930	—	3	1	no	—
26	C₅×C₂²⋊C₁₆	—	3	1	no	—
27	#64727	—	3	1	no	—
28	#64721	—	3	1	no	—
29	C₁₆×Dic₅	—	3	1	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₂²⋊D₅	C₈	1	1	yes	characteristic
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	—	2	1	no	—
34	C₂×C₃×C₅×C₈	—	2	1	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₃×C₂²⋊Dic₅	C₈	1	1	yes	characteristic
38	C₃×#685	C₈	1	1	yes	characteristic
39	C₃×#683	C₈	1	1	yes	characteristic
40	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
43	C₃×C₈×D₅	—	2	1	no	—
44	C₃×C₈×D₅	—	2	1	no	—
45	C₃⋊#672	—	2	1	no	—
46	C₃⋊#672	D₄	1	2	yes	complemented
47	C₅×C₃⋊C₁₆	—	2	1	no	—
48	C₅×C₃⋊C₁₆	—	2	1	no	—
49	C₃×C₈×D₄	—	5	1	no	—
50	C₂×#812	—	5	1	no	—
51	C₂×#812	—	5	1	no	—
52	#4549	—	5	1	no	—
53	#4549	D₅	1	1	yes	characteristic, complemented
54	C₄⋊(C₃⋊C₁₆)	—	5	1	no	—
55	C₄×C₃⋊C₁₆	—	5	1	no	—
56	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
57	C₄×C₂²⋊D₅	Dic₃	1	1	yes	characteristic
58	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
59	C₂×C₅×C₁₆	—	3	1	no	—
60	C₂×C₅×C₁₆	—	3	1	no	—
61	C₂²⋊#168	Dic₃	1	1	yes	characteristic
62	#3952	Dic₃	1	1	yes	characteristic
63	#3946	Dic₃	1	1	yes	characteristic
64	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
65	#3944	—	6	2	no	—
66	C₂×#672	—	3	1	no	—
67	C₂×#672	—	3	1	no	—
68	#3930	—	6	2	no	—
69	#2169	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₂²⋊D₅	—	2	2	no	—
77	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
80	C₃×C₄×D₅	—	2	1	no	—
81	C₃×C₄×D₅	—	2	1	no	—
82	C₃×C₅×C₈	—	2	1	no	—
83	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₃×C₅×C₈	#41	1	1	yes	characteristic
85	C₃×#168	—	2	1	no	—
86	C₃×#168	—	2	1	no	—
87	C₂²×C₃×C₈	—	5	1	no	—
88	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
89	C₃×C₄×D₄	—	5	1	no	—
90	C₃×C₄⋊C₈	—	5	1	no	—
91	C₃×C₂²⋊C₈	—	5	1	no	—
92	C₃×C₂²⋊C₈	—	5	1	no	—
93	C₃×C₄×C₈	—	5	1	no	—
94	#812	—	10	2	no	—
95	#812	—	10	2	no	—
96	C₂×C₃⋊C₁₆	—	5	1	no	—
97	C₂×C₃⋊C₁₆	—	5	1	no	—
98	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
99	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
101	C₂×C₂²⋊D₅	C₃⋊C₈	1	1	yes	characteristic
102	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₅×C₈	—	2	1	no	—
104	C₂×C₅×C₈	—	2	1	no	—
105	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
106	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
107	C₂²⋊Dic₅	C₃⋊C₈	1	1	yes	characteristic
108	#685	C₃⋊C₈	1	1	yes	characteristic
109	#683	C₃⋊C₈	1	1	yes	characteristic
110	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
111	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
112	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
113	C₈×D₅	—	2	1	no	—
114	C₈×D₅	—	2	1	no	—
115	C₅×C₁₆	—	6	1	no	—
116	C₅×C₁₆	—	6	1	no	—
117	#672	—	3	2	no	—
118	#672	—	6	1	no	—
119	C₂×M₃₂	—	15	1	no	—
120	C₂×M₃₂	—	15	1	no	—
121	C₈×D₄	—	5	1	no	—
122	C₄⋊C₁₆	—	15	1	no	—
123	C₂²⋊C₁₆	—	3	1	no	—
124	C₂²⋊C₁₆	—	15	1	no	—
125	C₄×C₁₆	—	15	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₃×D₅	—	2	1	no	—
131	C₂×C₃×D₅	—	2	1	no	—
132	C₂×C₃×D₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
135	C₃×C₄×C₅	#131	1	1	yes	characteristic
136	C₃×Dic₅	—	2	1	no	—
137	C₃×Dic₅	—	2	1	no	—
138	C₃×Dic₅	M₃₂	1	2	yes	complemented
139	C₂×C₃×D₄	—	5	1	no	—
140	C₂²×C₃×C₄	—	5	1	no	—
141	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	—	5	1	no	—
145	C₂×C₃×C₈	—	5	1	no	—
146	C₂×C₃×C₈	—	10	1	no	—
147	C₂×C₃×C₈	—	10	1	no	—
148	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	5	1	no	—
151	C₃×C₂²⋊C₄	—	5	1	no	—
152	C₃×C₂²⋊C₄	—	5	1	no	—
153	C₃×C₄²	—	5	1	no	—
154	C₃⋊C₁₆	—	2	1	no	—
155	C₃⋊C₁₆	—	2	1	no	—
156	C₃⋊C₁₆	—	5	2	no	—
157	C₃⋊C₁₆	—	10	1	no	—
158	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
159	C₂²×D₅	C₂×C₃⋊C₈	1	1	yes	characteristic
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
164	C₂²⋊D₅	—	2	2	no	—
165	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
166	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
167	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
168	C₄×D₅	—	2	1	no	—
169	C₄×D₅	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	D₃×D₄	1	1	yes	characteristic
172	C₅×C₈	#237	1	1	yes	characteristic
173	#168	—	2	1	no	—
174	#168	—	2	1	no	—
175	C₂²×C₈	—	5	1	no	—
176	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
177	C₄×D₄	—	5	1	no	—
178	M₃₂	—	30	2	no	—
179	M₃₂	—	30	2	no	—
180	C₂×C₁₆	—	3	1	no	—
181	C₂×C₁₆	—	3	1	no	—
182	C₂×C₁₆	—	15	1	no	—
183	C₂×C₁₆	—	15	1	no	—
184	C₄⋊C₈	—	5	1	no	—
185	C₂²⋊C₈	—	5	1	no	—
186	C₂²⋊C₈	—	5	1	no	—
187	C₄×C₈	—	5	1	no	—
188	C₂×C₃×C₅	—	4	1	no	—
189	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
191	C₂×C₃×C₅	#504	1	1	yes	characteristic
192	C₃×D₅	—	4	1	no	—
193	C₂³×C₃	—	5	1	no	—
194	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
195	C₃×D₄	—	10	2	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	—	5	1	no	—
200	C₂×C₃×C₄	—	5	1	no	—
201	C₂×C₃×C₄	—	10	1	no	—
202	C₂×C₃×C₄	—	10	1	no	—
203	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	10	1	no	—
207	C₃×C₈	—	10	1	no	—
208	C₃×C₈	D₄×D₅	1	1	yes	characteristic
209	C₃×C₈	#711	1	1	yes	characteristic
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	C₂²×C₃⋊C₈	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	D₄×Dic₃	1	1	yes	characteristic
219	C₄×C₅	#948	1	1	yes	characteristic
220	Dic₅	—	2	1	no	—
221	Dic₅	—	2	1	no	—
222	Dic₅	#812	1	2	yes	complemented
223	C₂×D₄	—	5	1	no	—
224	C₂²×C₄	—	5	1	no	—
225	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	5	1	no	—
229	C₂×C₈	—	5	1	no	—
230	C₂×C₈	—	10	1	no	—
231	C₂×C₈	—	10	1	no	—
232	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
233	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
234	C₄⋊C₄	—	5	1	no	—
235	C₂²⋊C₄	—	5	1	no	—
236	C₂²⋊C₄	—	5	1	no	—
237	C₄²	—	5	1	no	—
238	C₁₆	—	6	1	no	—
239	C₁₆	—	6	1	no	—
240	C₁₆	—	15	2	no	—
241	C₁₆	—	30	1	no	—
242	C₃×C₅	#2169	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	10	1	no	—
247	C₂²×C₃	—	10	1	no	—
248	C₂²×C₃	—	10	1	no	—
249	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	5	2	no	—
252	C₃×C₄	—	10	1	no	—
253	C₃×C₄	—	10	1	no	—
254	C₃×C₄	#4027	1	1	yes	characteristic
255	C₃×C₄	#4053	1	1	yes	characteristic
256	C₂×C₅	—	4	1	no	—
257	C₂×C₅	D₄×C₃⋊C₈	1	1	yes	characteristic
258	C₂×C₅	C₂×#812	1	1	yes	characteristic
259	C₂×C₅	#5140	1	1	yes	characteristic
260	D₅	—	4	1	no	—
261	C₂³	—	5	1	no	—
262	C₂³	D₅×C₃⋊C₈	1	1	yes	characteristic
263	D₄	—	10	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	5	1	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	10	1	no	—
270	C₂×C₄	—	10	1	no	—
271	C₂×C₄	D₅×C₃⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	10	1	no	—
275	C₈	—	10	1	no	—
276	C₈	#6780	1	1	yes	characteristic
277	C₈	#6790	1	1	yes	characteristic
278	C₂×C₃	—	4	1	no	—
279	C₂×C₃	—	20	1	no	—
280	C₂×C₃	#65014	1	1	yes	characteristic
281	C₂×C₃	#65190	1	1	yes	characteristic
282	C₂×C₃	D₅×M₃₂	1	1	yes	characteristic
283	C₅	#70858	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	10	1	no	—
288	C₂²	—	10	1	no	—
289	C₂²	—	10	1	no	—
290	C₂²	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	5	2	no	—
293	C₄	—	10	1	no	—
294	C₄	—	10	1	no	—
295	C₄	#91284	1	1	yes	characteristic
296	C₄	#91559	1	1	yes	characteristic
297	C₃	#10599897	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	4	1	no	—
299	C₂	—	20	1	no	—
300	C₂	D₅×#812	1	1	yes	characteristic
301	C₂	#11747524	1	1	yes	characteristic
302	C₂	#11748493	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂²×C₃×C₅×C₈	C₂²
prime radical	C₂×C₃×C₂²⋊D₅	C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2169
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#91096	C₂	semidirect
3. C₂×#90934	C₂	semidirect
4. C₅×#4549	C₂	semidirect
5. #11745927	C₂	semidirect
6. #11745879	C₂	semidirect
7. Dic₅×C₃⋊C₁₆	C₂	semidirect
46. C₃⋊#672	D₄	semidirect
53. #4549	D₅	semidirect
138. C₃×Dic₅	M₃₂	semidirect
222. Dic₅	#812	semidirect
242. C₃×C₅	#2169	semidirect
283. C₅	#70858	semidirect
297. C₃	#10599897	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91096	2
C₃×C₈×C₂²⋊D₅	2
C₂×#90934	2
C₅×#4549	2
#11745927	2
#11745879	2
Dic₅×C₃⋊C₁₆	2
#10599897	3
#70858	5