Subgroups — #14315212

Order 1920.

292 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5144	C₂	1	1	yes	maximal, characteristic
2	C₃×#65429	C₂	1	1	yes	maximal, characteristic, complemented
3	#11747527	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746382	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745932	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745887	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×#672	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601716	—	3	1	no	maximal, Hall
9	C₃×C₅×#131	C₂²	1	1	yes	characteristic
10	C₅×#950	C₄	1	1	yes	characteristic
11	C₅×#901	C₂²	1	1	yes	characteristic
12	#91096	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
14	C₅×#837	C₄	1	1	yes	characteristic
15	C₅×#825	C₄	1	1	yes	characteristic
16	C₅×#822	C₄	1	1	yes	characteristic
17	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
18	C₃×#3944	—	2	1	no	—
19	C₃×#3944	C₂²	1	1	yes	characteristic
20	C₂×C₃×#672	—	2	1	no	—
21	C₂×C₃×#672	C₂²	1	1	yes	characteristic
22	#90935	—	2	1	no	—
23	D₃×#672	—	2	1	no	—
24	#71452	—	5	1	no	maximal, Hall
25	C₅×#443	—	3	1	no	—
26	#65429	—	3	1	no	—
27	#65429	D₃	1	1	yes	characteristic, complemented
28	#64781	—	3	1	no	—
29	#64781	—	3	1	no	—
30	#64683	—	3	1	no	—
31	C₄×#672	—	3	1	no	—
32	C₃×C₅×#41	C₂×C₄	1	1	yes	characteristic
33	C₅×#235	C₂×C₄	1	1	yes	characteristic
34	C₃×C₅×M₁₆	—	2	1	no	—
35	C₃×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₅×#216	C₈	1	1	yes	characteristic
39	C₅×#215	C₈	1	1	yes	characteristic
40	C₅×#214	C₈	1	1	yes	characteristic
41	C₅×#213	C₈	1	1	yes	characteristic
42	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₅×#208	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
46	C₅×D₃×C₈	—	2	1	no	—
47	C₃⋊#672	—	2	1	no	—
48	C₃⋊#672	—	2	1	no	—
49	C₃×#672	—	2	1	no	—
50	C₃×#672	D₄	1	1	yes	characteristic
51	C₃×#672	D₄	1	1	yes	characteristic
52	C₃×#504	—	5	1	no	—
53	#5144	D₅	1	1	yes	characteristic
54	#4901	—	5	1	no	—
55	#4561	—	5	1	no	—
56	#4515	—	5	1	no	—
57	#4511	—	5	1	no	—
58	Dic₃×C₁₆	—	5	1	no	—
59	C₅×#135	—	3	1	no	—
60	C₅×#131	—	3	1	no	—
61	C₅×#131	C₂×D₃	1	1	yes	characteristic
62	C₅×#108	—	3	1	no	—
63	C₅×#104	—	3	1	no	—
64	C₅×#104	—	3	1	no	—
65	C₄×C₅×C₈	—	3	1	no	—
66	#3944	—	2	1	no	—
67	#3944	—	3	1	no	—
68	#3944	—	6	1	no	—
69	#3944	C₂×D₃	1	1	yes	characteristic
70	C₂×#672	—	2	1	no	—
71	C₂×#672	—	3	1	no	—
72	C₂×#672	—	6	1	no	—
73	C₂×#672	C₂×D₃	1	1	yes	characteristic
74	#2170	—	15	1	no	Sylow
75	C₃×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₂×C₈	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₅×#67	—	2	1	no	—
81	C₂×C₅×Dic₃	—	2	1	no	—
82	C₅×D₁₂	C₂×C₈	1	1	yes	characteristic
83	C₄×C₅×D₃	—	2	1	no	—
84	C₅×Dic₆	C₂×C₈	1	1	yes	characteristic
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₃×C₅×C₈	#41	1	1	yes	characteristic
88	C₅×C₃⋊C₈	—	2	1	no	—
89	C₅×C₃⋊C₈	C₂×C₈	1	1	yes	characteristic
90	C₅×C₃⋊C₈	C₂×C₈	1	1	yes	characteristic
91	C₃×#131	C₂×D₅	1	1	yes	characteristic
92	#950	Dic₅	1	1	yes	characteristic
93	#901	C₂×D₅	1	1	yes	characteristic
94	C₃×M₃₂	—	5	1	no	—
95	C₃×M₃₂	—	10	1	no	—
96	C₂×C₃×C₁₆	—	5	1	no	—
97	C₂×C₃×C₁₆	—	10	1	no	—
98	#837	Dic₅	1	1	yes	characteristic
99	#825	Dic₅	1	1	yes	characteristic
100	#822	Dic₅	1	1	yes	characteristic
101	C₈×Dic₃	C₂×D₅	1	1	yes	characteristic
102	#812	—	5	1	no	—
103	C₂×C₃⋊C₁₆	—	5	1	no	—
104	#798	—	10	1	no	—
105	D₃×C₁₆	—	10	1	no	—
106	C₅×#41	—	3	1	no	—
107	C₅×#41	C₄×D₃	1	1	yes	characteristic
108	C₅×Q₁₆	—	3	1	no	—
109	C₅×SD₁₆	—	3	1	no	—
110	C₅×SD₁₆	—	3	1	no	—
111	C₅×D₈	—	3	1	no	—
112	C₅×M₁₆	—	2	1	no	—
113	C₅×M₁₆	—	3	1	no	—
114	C₅×M₁₆	—	6	1	no	—
115	C₅×M₁₆	C₄×D₃	1	1	yes	characteristic
116	C₂×C₅×C₈	—	2	1	no	—
117	C₂×C₅×C₈	—	3	1	no	—
118	C₂×C₅×C₈	—	6	1	no	—
119	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
120	C₄²×C₅	—	3	1	no	—
121	#672	—	2	1	no	—
122	#672	—	6	1	no	—
123	#672	—	6	1	no	—
124	#672	#67	1	1	yes	characteristic
125	#672	#67	1	1	yes	characteristic
126	#504	—	5	1	no	—
127	#504	—	15	1	no	—
128	#443	—	3	1	no	—
129	#364	—	15	1	no	—
130	#350	—	15	1	no	—
131	#350	—	15	1	no	—
132	C₄×C₁₆	—	15	1	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	#131	1	1	yes	characteristic
135	C₂×C₅×D₃	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
139	C₅×Dic₃	—	2	2	no	—
140	C₅×Dic₃	—	2	1	no	—
141	C₃×#41	C₂×Dic₅	1	1	yes	characteristic
142	#235	C₂×Dic₅	1	1	yes	characteristic
143	C₃×M₁₆	—	2	1	no	—
144	C₃×M₁₆	C₂×Dic₅	1	1	yes	characteristic
145	C₂×C₃×C₈	—	2	1	no	—
146	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
147	#216	#168	1	1	yes	characteristic
148	#215	#168	1	1	yes	characteristic
149	#214	#168	1	1	yes	characteristic
150	#213	#168	1	1	yes	characteristic
151	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
152	#208	C₂×Dic₅	1	1	yes	characteristic
153	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
154	ΓL(1,𝔽₂₅)	—	2	1	no	—
155	D₃×C₈	—	2	1	no	—
156	C₃×C₁₆	—	5	1	no	—
157	C₃×C₁₆	—	5	1	no	—
158	C₃×C₁₆	—	10	1	no	—
159	C₃⋊C₁₆	—	10	1	no	—
160	C₃⋊C₁₆	—	10	1	no	—
161	C₅×Q₈	—	3	1	no	—
162	C₅×Q₈	D₃×C₈	1	1	yes	characteristic
163	C₅×D₄	—	2	1	no	—
164	C₅×D₄	—	3	1	no	—
165	C₅×D₄	—	6	1	no	—
166	C₅×D₄	D₃×C₈	1	1	yes	characteristic
167	C₂×C₄×C₅	—	2	1	no	—
168	C₂×C₄×C₅	—	6	1	no	—
169	C₂×C₄×C₅	—	6	1	no	—
170	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	—	3	1	no	—
173	C₅×C₈	—	3	1	no	—
174	C₅×C₈	—	6	1	no	—
175	C₅×C₈	#235	1	1	yes	characteristic
176	C₅×C₈	C₂×#67	1	1	yes	characteristic
177	#135	—	3	1	no	—
178	#131	—	3	1	no	—
179	#131	D₃×D₅	1	1	yes	characteristic
180	M₃₂	—	5	1	no	—
181	M₃₂	—	10	1	no	—
182	M₃₂	—	15	1	no	—
183	M₃₂	—	30	1	no	—
184	C₂×C₁₆	—	5	1	no	—
185	C₂×C₁₆	—	10	1	no	—
186	C₂×C₁₆	—	15	1	no	—
187	C₂×C₁₆	—	30	1	no	—
188	#108	—	3	1	no	—
189	#104	—	3	1	no	—
190	#104	—	3	1	no	—
191	C₄×C₈	—	3	1	no	—
192	C₂×C₃×C₅	—	2	1	no	—
193	C₂×C₃×C₅	—	4	1	no	—
194	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
195	C₅×D₃	—	4	1	no	—
196	C₃×Q₈	C₂×#168	1	1	yes	characteristic
197	C₃×D₄	—	2	1	no	—
198	C₃×D₄	C₂×#168	1	1	yes	characteristic
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
201	#67	—	2	1	no	—
202	C₂×Dic₃	—	2	1	no	—
203	D₁₂	C₂×#168	1	1	yes	characteristic
204	C₄×D₃	—	2	1	no	—
205	Dic₆	C₂×#168	1	1	yes	characteristic
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	D₄×D₅	1	1	yes	characteristic
208	C₃×C₈	#711	1	1	yes	characteristic
209	C₃⋊C₈	—	2	1	no	—
210	C₃⋊C₈	C₂×#168	1	1	yes	characteristic
211	C₃⋊C₈	C₂×#168	1	1	yes	characteristic
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	6	1	no	—
214	C₂²×C₅	#901	1	1	yes	characteristic
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	6	2	no	—
217	C₄×C₅	—	6	1	no	—
218	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
219	C₄×C₅	C₄×#67	1	1	yes	characteristic
220	#41	—	3	1	no	—
221	#41	D₃×Dic₅	1	1	yes	characteristic
222	Q₁₆	—	3	1	no	—
223	SD₁₆	—	3	1	no	—
224	SD₁₆	—	3	1	no	—
225	D₈	—	3	1	no	—
226	M₁₆	—	2	1	no	—
227	M₁₆	—	3	1	no	—
228	M₁₆	—	6	1	no	—
229	M₁₆	D₃×Dic₅	1	1	yes	characteristic
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	3	1	no	—
232	C₂×C₈	—	6	1	no	—
233	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
234	C₄²	—	3	1	no	—
235	C₁₆	—	5	1	no	—
236	C₁₆	—	5	1	no	—
237	C₁₆	—	10	1	no	—
238	C₁₆	—	30	1	no	—
239	C₁₆	—	30	1	no	—
240	C₃×C₅	#2170	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	#4094	1	1	yes	characteristic
243	C₂×D₃	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	C₂²×#168	1	1	yes	characteristic
246	C₃×C₄	D₄×Dic₅	1	1	yes	characteristic
247	Dic₃	—	2	2	no	—
248	Dic₃	—	2	1	no	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	4	1	no	—
251	C₂×C₅	—	12	1	no	—
252	C₂×C₅	C₈×#67	1	1	yes	characteristic
253	Q₈	—	3	1	no	—
254	Q₈	D₃×#168	1	1	yes	characteristic
255	D₄	—	2	1	no	—
256	D₄	—	3	1	no	—
257	D₄	—	6	1	no	—
258	D₄	D₃×#168	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	6	1	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
263	C₈	—	2	1	no	—
264	C₈	—	3	1	no	—
265	C₈	—	3	1	no	—
266	C₈	—	6	1	no	—
267	C₈	#6782	1	1	yes	characteristic
268	C₈	D₅×#67	1	1	yes	characteristic
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	4	1	no	—
271	C₂×C₃	D₄×#168	1	1	yes	characteristic
272	D₃	—	4	1	no	—
273	C₅	#71452	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	2	1	no	—
275	C₂²	—	6	1	no	—
276	C₂²	#91286	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	6	2	no	—
279	C₄	—	6	1	no	—
280	C₄	C₂×D₃×#168	1	1	yes	characteristic
281	C₄	Dic₅×#67	1	1	yes	characteristic
282	C₃	#10601716	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	2	1	no	—
284	C₂	—	4	1	no	—
285	C₂	—	12	1	no	—
286	C₂	#67×#168	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₅×#67
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₃×C₅×#131	C₂²
socle	C₂×C₃×C₅	C₈×D₄
prime radical	C₅×#950	C₄
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	#131	D₃×D₅
hypocenter	C₃×C₅	#2170
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
2. C₃×#65429	C₂	semidirect
3. #11747527	C₂	semidirect
4. #11746382	C₂	semidirect
5. #11745932	C₂	semidirect
6. #11745887	C₂	semidirect
7. Dic₃×#672	C₂	semidirect
27. #65429	D₃	semidirect
240. C₃×C₅	#2170	semidirect
273. C₅	#71452	semidirect
282. C₃	#10601716	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5144	2
C₃×#65429	2
#11747527	2
#11746382	2
#11745932	2
#11745887	2
Dic₃×#672	2
#10601716	3
#71452	5