Subgroups — #14313890

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91793	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746313	C₂	1	2	yes	maximal, complemented
5	#11746312	C₂	1	2	yes	maximal, complemented
6	#10601391	—	3	1	no	maximal, Hall
7	C₂×#6817	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
10	C₂×Dic₆₀	C₂²	1	1	yes	characteristic
11	#91793	—	2	2	no	—
12	#91793	C₄	1	4	yes	complemented
13	C₂×D₁₂₀	C₂²	1	1	yes	characteristic
14	C₂×#6706	—	2	1	no	—
15	C₃×C₅×#107	—	2	1	no	—
16	C₃×C₅×#107	C₂²	1	2	yes	—
17	#91096	—	2	2	no	—
18	C₂×C₃⋊#672	C₂²	1	2	yes	—
19	#70598	—	5	1	no	maximal, Hall
20	C₂×#4050	—	3	1	no	—
21	C₂×C₅×#107	D₃	1	1	yes	characteristic, complemented
22	C₂×#3944	—	3	1	no	—
23	#64712	—	3	2	no	—
24	#64711	—	3	2	no	—
25	C₂²×C₃×C₄×C₅	—	4	1	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂×#1220	—	4	1	no	—
28	#6817	—	2	4	no	—
29	#6817	—	4	2	no	—
30	C₂×D₆₀	—	2	1	no	—
31	C₂×C₄×D₁₅	—	4	1	no	—
32	C₂×Dic₃₀	—	2	1	no	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	4	1	no	—
39	Dic₆₀	—	2	1	no	—
40	Dic₆₀	C₂×C₄	1	2	yes	complemented
41	D₁₂₀	—	2	1	no	—
42	D₁₂₀	C₂×C₄	1	2	yes	complemented
43	#6706	—	2	4	no	—
44	C₃⋊#672	—	2	2	no	—
45	C₂×#904	—	5	1	no	—
46	C₂×C₃×#107	D₅	1	1	yes	characteristic, complemented
47	C₂×#812	—	5	1	no	—
48	#4492	—	5	2	no	—
49	#4491	—	5	2	no	—
50	C₂×#709	—	6	1	no	—
51	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×C₅×C₄⋊C₄	—	2	1	no	—
53	C₂×Dic₂₀	—	3	1	no	—
54	#4050	—	3	4	no	—
55	#4050	—	6	2	no	—
56	C₂×D₄₀	—	3	1	no	—
57	C₂×#677	—	6	1	no	—
58	C₅×#107	—	2	1	no	—
59	C₅×#107	C₂×D₃	1	2	yes	—
60	#3944	—	6	2	no	—
61	C₂×#672	—	3	2	no	—
62	#2139	—	15	1	no	Sylow
63	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
64	C₂²×D₁₅	—	4	1	no	—
65	C₂×C₃×C₄×C₅	—	4	2	no	—
66	C₂×C₃×C₄×C₅	—	8	2	no	—
67	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	#1220	—	4	4	no	—
72	C₂×Dic₁₅	—	4	1	no	—
73	D₆₀	—	2	2	no	—
74	D₆₀	—	4	1	no	—
75	C₄×D₁₅	—	4	4	no	—
76	Dic₃₀	—	2	2	no	—
77	Dic₃₀	—	4	1	no	—
78	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
79	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×#235	—	10	1	no	—
82	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
83	C₂×C₃×C₄⋊C₄	—	2	1	no	—
84	C₂×Dic₁₂	—	5	1	no	—
85	#904	—	5	4	no	—
86	#904	—	10	2	no	—
87	C₂×D₂₄	—	5	1	no	—
88	C₂×#204	—	10	1	no	—
89	C₃×#107	—	2	1	no	—
90	C₃×#107	C₂×D₅	1	2	yes	—
91	#812	—	10	2	no	—
92	C₂×C₃⋊C₁₆	—	5	2	no	—
93	C₂²×C₄×C₅	—	4	1	no	—
94	C₂²×C₄×C₅	#67	1	1	yes	characteristic
95	C₂×C₂²⋊D₅	—	12	1	no	—
96	#709	—	6	4	no	—
97	#709	—	12	2	no	—
98	C₂×D₂₀	—	6	1	no	—
99	C₂×C₄×D₅	—	12	1	no	—
100	C₂×Dic₁₀	—	6	1	no	—
101	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
102	C₂×C₅×C₈	D₁₂	1	2	yes	—
103	C₂×C₅×C₈	#67	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	4	1	no	—
107	Dic₂₀	—	3	2	no	—
108	Dic₂₀	—	6	1	no	—
109	D₄₀	—	3	2	no	—
110	D₄₀	—	6	1	no	—
111	#677	—	6	4	no	—
112	#672	—	6	2	no	—
113	C₂×#135	—	15	1	no	—
114	C₂×M₃₂	—	15	1	no	—
115	C₂×#107	D₁₅	1	1	yes	characteristic, complemented
116	#358	—	15	2	no	—
117	#357	—	15	2	no	—
118	C₂²×C₃×C₅	—	2	1	no	—
119	C₂²×C₃×C₅	—	2	1	no	—
120	C₂²×C₃×C₅	#102	1	2	yes	—
121	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
122	C₂×D₁₅	—	4	2	no	—
123	C₂×D₁₅	—	8	1	no	—
124	C₂×D₁₅	—	8	1	no	—
125	C₃×C₄×C₅	—	8	2	no	—
126	C₃×C₄×C₅	#102	1	2	yes	—
127	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
129	Dic₁₅	—	4	2	no	—
130	C₂²×C₃×C₄	—	4	1	no	—
131	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
132	C₂×#67	—	20	1	no	—
133	#235	—	10	4	no	—
134	#235	—	20	2	no	—
135	C₂×D₁₂	—	10	1	no	—
136	C₂×C₄×D₃	—	20	1	no	—
137	C₂×Dic₆	—	10	1	no	—
138	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
139	C₂×C₃×C₈	D₂₀	1	2	yes	—
140	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
141	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	—	4	1	no	—
144	Dic₁₂	—	5	2	no	—
145	Dic₁₂	—	10	1	no	—
146	D₂₄	—	5	2	no	—
147	D₂₄	—	10	1	no	—
148	#204	—	10	4	no	—
149	C₃⋊C₁₆	—	10	2	no	—
150	C₂³×C₅	#213	1	1	yes	characteristic
151	C₂²×D₅	—	12	1	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	8	2	no	—
154	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
155	C₂×C₄×C₅	#213	1	1	yes	characteristic
156	C₂×C₄×C₅	#215	1	2	yes	—
157	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
158	C₂²⋊D₅	—	12	4	no	—
159	C₂×Dic₅	—	12	1	no	—
160	D₂₀	—	6	2	no	—
161	D₂₀	—	12	1	no	—
162	C₄×D₅	—	12	4	no	—
163	Dic₁₀	—	6	2	no	—
164	Dic₁₀	—	12	1	no	—
165	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
166	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
167	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
168	C₂×#41	—	30	1	no	—
169	#135	—	15	4	no	—
170	#135	—	30	2	no	—
171	C₂×Q₁₆	—	15	1	no	—
172	C₂×SD₁₆	—	30	1	no	—
173	C₂×D₈	—	15	1	no	—
174	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	2	1	no	—
176	M₃₂	—	30	2	no	—
177	C₂×C₁₆	—	15	2	no	—
178	#107	—	2	1	no	—
179	#107	C₂×D₁₅	1	2	yes	—
180	C₂×C₃×C₅	—	2	2	no	—
181	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
182	C₂×C₃×C₅	#509	1	1	yes	characteristic
183	C₂×C₃×C₅	#510	1	1	yes	characteristic
184	D₁₅	—	8	2	no	—
185	C₂³×C₃	#686	1	1	yes	characteristic
186	C₂²×D₃	—	20	1	no	—
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	—	8	2	no	—
189	C₂×C₃×C₄	#685	1	2	yes	—
190	C₂×C₃×C₄	#686	1	1	yes	characteristic
191	C₂×C₃×C₄	#688	1	2	yes	—
192	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
193	#67	—	20	4	no	—
194	C₂×Dic₃	—	20	1	no	—
195	D₁₂	—	10	2	no	—
196	D₁₂	—	20	1	no	—
197	C₄×D₃	—	20	4	no	—
198	Dic₆	—	10	2	no	—
199	Dic₆	—	20	1	no	—
200	C₃×C₈	#685	1	2	yes	—
201	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
202	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	#809	1	2	yes	—
206	C₂²×C₅	C₂×#213	1	1	yes	characteristic
207	C₂×D₅	—	12	2	no	—
208	C₂×D₅	—	24	1	no	—
209	C₂×D₅	—	24	1	no	—
210	C₄×C₅	—	8	2	no	—
211	C₄×C₅	#809	1	2	yes	—
212	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
213	C₄×C₅	C₂×#215	1	1	yes	characteristic
214	Dic₅	—	12	2	no	—
215	#41	—	30	4	no	—
216	#41	—	60	2	no	—
217	C₂×Q₈	—	30	1	no	—
218	C₂×D₄	—	30	1	no	—
219	C₂×D₄	—	60	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	60	1	no	—
222	C₂²×C₄	#1220	1	1	yes	characteristic
223	Q₁₆	—	15	2	no	—
224	Q₁₆	—	30	1	no	—
225	SD₁₆	—	30	4	no	—
226	D₈	—	15	2	no	—
227	D₈	—	30	1	no	—
228	C₂×C₈	C₄×D₁₅	1	2	yes	—
229	C₂×C₈	D₆₀	1	2	yes	—
230	C₂×C₈	#1220	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₁₆	—	30	2	no	—
235	C₃×C₅	#2139	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	#3941	1	2	yes	—
239	C₂²×C₃	C₂×#686	1	1	yes	characteristic
240	C₂×D₃	—	20	2	no	—
241	C₂×D₃	—	40	1	no	—
242	C₂×D₃	—	40	1	no	—
243	C₃×C₄	—	8	2	no	—
244	C₃×C₄	#3941	1	2	yes	—
245	C₃×C₄	C₂×#685	1	1	yes	characteristic
246	C₃×C₄	C₂×#688	1	1	yes	characteristic
247	Dic₃	—	20	2	no	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	C₂×#809	1	1	yes	characteristic
250	C₂×C₅	#5147	1	1	yes	characteristic
251	C₂×C₅	#5179	1	1	yes	characteristic
252	D₅	—	24	2	no	—
253	C₂³	—	60	1	no	—
254	C₂³	#6715	1	1	yes	characteristic
255	Q₈	—	30	2	no	—
256	Q₈	—	60	1	no	—
257	D₄	—	30	2	no	—
258	D₄	—	60	4	no	—
259	D₄	—	60	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	8	2	no	—
262	C₂×C₄	—	60	1	no	—
263	C₂×C₄	—	60	4	no	—
264	C₂×C₄	#6714	1	2	yes	—
265	C₂×C₄	#6715	1	1	yes	characteristic
266	C₂×C₄	#6717	1	2	yes	—
267	C₂×C₄	C₂×#1220	1	1	yes	characteristic
268	C₈	#6714	1	2	yes	—
269	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
270	C₈	C₂×D₆₀	1	1	yes	characteristic
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	C₂×#3941	1	1	yes	characteristic
273	C₂×C₃	#65436	1	1	yes	characteristic
274	C₂×C₃	#65468	1	1	yes	characteristic
275	D₃	—	40	2	no	—
276	C₅	#70598	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	60	2	no	—
280	C₂²	—	120	1	no	—
281	C₂²	—	120	1	no	—
282	C₂²	#91093	1	2	yes	—
283	C₂²	C₂×#6715	1	1	yes	characteristic
284	C₄	—	8	2	no	—
285	C₄	—	60	2	no	—
286	C₄	#91093	1	2	yes	—
287	C₄	C₂×#6714	1	1	yes	characteristic
288	C₄	C₂×#6717	1	1	yes	characteristic
289	C₃	#10601391	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	2	2	no	—
291	C₂	—	120	2	no	—
292	C₂	C₂×#91093	1	1	yes	characteristic
293	C₂	#11751248	1	1	yes	characteristic
294	C₂	#11751280	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6715
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂×C₃×C₅×#107	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×#91793	C₂
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×#107	D₁₅
hypocenter	C₃×C₅	#2139
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#107	C₂	semidirect
3. C₂×#91096	C₂	semidirect
4. #11746313	C₂	semidirect
5. #11746312	C₂	semidirect
12. #91793	C₄	semidirect
21. C₂×C₅×#107	D₃	semidirect
40. Dic₆₀	C₂×C₄	semidirect
42. D₁₂₀	C₂×C₄	semidirect
46. C₂×C₃×#107	D₅	semidirect
115. C₂×#107	D₁₅	semidirect
235. C₃×C₅	#2139	semidirect
276. C₅	#70598	semidirect
289. C₃	#10601391	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91793	2
C₂×C₃×C₅×#107	2
C₂×#91096	2
#11746313	2
#11746312	2
#10601391	3
#70598	5