Subgroups — #14280421

Order 1920.

448 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	C₄×#6740	C₂	1	1	yes	maximal, characteristic
2	#11751797	C₂	1	1	yes	maximal, characteristic
3	C₈×#1200	C₂	1	1	yes	maximal, characteristic
4	#11746751	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#170⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃⋊C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#700	—	3	1	no	maximal, Hall
9	C₂×#6740	C₄	1	2	yes	—
10	C₂×C₈×D₁₅	C₄	1	1	yes	characteristic
11	C₄×#1200	C₂²	1	1	yes	characteristic
12	C₂×#6648	C₄	1	1	yes	characteristic
13	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
15	C₃×C₄×#170	C₂²	1	1	yes	characteristic
16	C₃×C₄×#170	C₂²	1	1	yes	characteristic
17	Dic₃⋊#170	C₄	1	2	yes	—
18	#91171	C₄	1	2	yes	—
19	#91169	C₄	1	2	yes	—
20	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic
21	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	Dic₃×#168	C₄	1	1	yes	characteristic
23	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68067	—	5	1	no	maximal, Hall
25	#65680	—	3	1	no	—
26	C₄×#700	—	3	1	no	—
27	C₄×C₈×D₅	—	3	1	no	—
28	#170⋊C₈	—	3	1	no	—
29	#170⋊C₈	D₃	1	1	yes	characteristic, complemented
30	C₈×#170	—	3	1	no	—
31	C₈×#170	—	3	1	no	—
32	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂×#1200	C₂×C₄	1	1	yes	characteristic
34	C₂×#1200	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
37	C₂×C₃×#170	C₂×C₄	1	2	yes	—
38	C₂×C₃×#170	C₂×C₄	1	2	yes	—
39	#6740	C₈	1	8	yes	complemented
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
42	C₈×D₁₅	—	2	2	no	—
43	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	Dic₃×Dic₅	C₂×C₄	1	1	yes	characteristic
48	Dic₃×Dic₅	C₂×C₄	1	1	yes	characteristic
49	#6648	—	2	2	no	—
50	C₄×ΓL(1,𝔽₂₅)	—	5	1	no	—
51	#4685	—	5	1	no	—
52	C₄×D₃×C₈	—	5	1	no	—
53	C₃×#322	—	5	1	no	—
54	#4454	—	5	1	no	—
55	C₈×C₃⋊C₈	—	5	1	no	—
56	C₈×C₃⋊C₈	—	5	1	no	—
57	C₂×#700	—	3	2	no	—
58	C₂×C₈×D₅	—	3	1	no	—
59	C₂×C₈×D₅	—	3	1	no	—
60	C₄²×D₅	—	3	1	no	—
61	#4003	—	3	2	no	—
62	#4002	—	3	2	no	—
63	C₄⋊#170	—	3	2	no	—
64	C₄×#170	—	3	1	no	—
65	C₄×#170	—	3	1	no	—
66	C₄×#170	C₂×D₃	1	1	yes	characteristic
67	C₄×#170	C₂×D₃	1	1	yes	characteristic
68	C₄×C₅×C₈	—	3	1	no	—
69	C₈×Dic₅	—	3	1	no	—
70	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
71	C₄×#168	—	3	1	no	—
72	C₈×M₁₆	—	15	1	no	Sylow
73	C₂²×D₁₅	C₄²	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
76	C₄×D₁₅	—	2	1	no	—
77	C₄×D₁₅	—	2	1	no	—
78	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
79	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
80	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
82	#1200	—	2	2	no	—
83	#1200	C₂×C₈	1	4	yes	—
84	C₃⋊#170	—	2	2	no	—
85	C₃⋊#170	C₂×C₈	1	4	yes	complemented
86	C₃×#170	—	2	2	no	—
87	C₃×#170	C₂×C₈	1	4	yes	complemented
88	C₃×C₅×C₈	—	2	1	no	—
89	C₃⋊#168	M₁₆	1	2	yes	complemented
90	C₃×#168	—	2	1	no	—
91	C₅×C₃⋊C₈	M₁₆	1	2	yes	complemented
92	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
93	C₂×D₃×C₈	—	5	1	no	—
94	C₂×D₃×C₈	—	5	1	no	—
95	C₄²×D₃	—	5	1	no	—
96	C₃×C₄×C₈	—	5	1	no	—
97	C₃×C₄×C₈	—	5	1	no	—
98	C₃×C₄×C₈	—	5	1	no	—
99	(C₂×D₃)⋊C₈	—	5	2	no	—
100	#815	—	5	2	no	—
101	Dic₃⋊C₈	—	5	2	no	—
102	C₈×Dic₃	—	5	1	no	—
103	C₈×Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic
104	C₄×C₃⋊C₈	—	5	1	no	—
105	C₄×C₃⋊C₈	—	5	1	no	—
106	C₄×C₃⋊C₈	—	5	1	no	—
107	C₂×C₄×D₅	—	3	1	no	—
108	C₂×C₄×D₅	—	3	1	no	—
109	C₂×C₄×D₅	—	3	1	no	—
110	C₂×#170	—	3	2	no	—
111	C₂×#170	—	3	2	no	—
112	C₂×#170	C₄×D₃	1	2	yes	—
113	C₂×#170	C₄×D₃	1	2	yes	—
114	#700	—	3	8	no	—
115	C₂×C₅×C₈	—	3	1	no	—
116	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
117	C₄²×C₅	—	3	1	no	—
118	C₄×Dic₅	—	3	1	no	—
119	C₄×Dic₅	—	3	1	no	—
120	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
121	C₂×#168	—	3	1	no	—
122	C₂×#168	C₄×D₃	1	1	yes	characteristic
123	C₈×D₅	—	6	2	no	—
124	C₈×D₅	—	6	2	no	—
125	#431	—	15	1	no	—
126	C₄×M₁₆	—	15	1	no	—
127	C₂×C₄×C₈	—	15	1	no	—
128	#322	—	5	1	no	—
129	#322	—	15	1	no	—
130	C₈²	—	15	1	no	—
131	C₈²	—	15	1	no	—
132	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
133	C₂×D₁₅	—	2	1	no	—
134	C₂×D₁₅	—	2	1	no	—
135	C₂×D₁₅	C₄×C₈	1	2	yes	—
136	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
137	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
138	Dic₁₅	—	2	1	no	—
139	C₃×Dic₅	C₂²×C₈	1	2	yes	—
140	C₃×Dic₅	#131	1	2	yes	—
141	C₅×Dic₃	—	2	1	no	—
142	C₅×Dic₃	C₄×C₈	1	2	yes	—
143	C₂×C₄×D₃	—	5	1	no	—
144	C₂×C₄×D₃	—	5	1	no	—
145	C₂×C₄×D₃	—	5	1	no	—
146	C₂×C₃×C₈	—	5	2	no	—
147	C₂×C₃×C₈	—	5	2	no	—
148	C₂×C₃×C₈	—	5	1	no	—
149	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₃×C₄²	—	5	1	no	—
151	C₄×Dic₃	—	5	1	no	—
152	C₄×Dic₃	—	5	1	no	—
153	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
154	C₂×C₃⋊C₈	—	5	2	no	—
155	C₂×C₃⋊C₈	—	5	1	no	—
156	C₂×C₃⋊C₈	—	5	2	no	—
157	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
158	ΓL(1,𝔽₂₅)	—	5	8	no	—
159	D₃×C₈	—	10	2	no	—
160	D₃×C₈	—	10	2	no	—
161	C₂²×D₅	—	3	1	no	—
162	C₂×C₄×C₅	—	3	1	no	—
163	C₂×C₄×C₅	—	3	1	no	—
164	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
165	C₂×Dic₅	—	3	1	no	—
166	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
168	C₄×D₅	—	3	4	no	—
169	C₄×D₅	—	6	1	no	—
170	C₄×D₅	—	6	1	no	—
171	C₄×D₅	—	6	2	no	—
172	#170	—	2	2	no	—
173	#170	—	3	4	no	—
174	#170	—	6	2	no	—
175	#170	D₃×C₈	1	4	yes	complemented
176	C₅×C₈	—	2	1	no	—
177	C₅×C₈	—	3	2	no	—
178	#168	—	2	1	no	—
179	#168	—	3	2	no	—
180	C₂×M₁₆	—	15	2	no	—
181	C₂²×C₈	—	15	1	no	—
182	C₂²×C₈	—	15	1	no	—
183	C₂×C₄²	—	15	1	no	—
184	C₄⋊C₈	—	15	2	no	—
185	C₂²⋊C₈	—	15	2	no	—
186	#97	—	15	2	no	—
187	C₄×C₈	—	3	1	no	—
188	C₄×C₈	—	5	1	no	—
189	C₄×C₈	—	5	1	no	—
190	C₄×C₈	—	5	1	no	—
191	C₄×C₈	—	15	1	no	—
192	C₄×C₈	—	15	1	no	—
193	C₄×C₈	—	15	1	no	—
194	C₄×C₈	—	15	1	no	—
195	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
196	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
197	C₂×C₃×C₅	#405	1	1	yes	characteristic
198	D₁₅	—	2	2	no	—
199	C₂²×D₃	—	5	1	no	—
200	C₂×C₃×C₄	—	5	1	no	—
201	C₂×C₃×C₄	—	5	1	no	—
202	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₂×Dic₃	—	5	1	no	—
204	C₂×Dic₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₂×Dic₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₄×D₃	—	5	4	no	—
207	C₄×D₃	—	10	2	no	—
208	C₄×D₃	—	10	1	no	—
209	C₄×D₃	—	10	1	no	—
210	C₃×C₈	—	2	1	no	—
211	C₃×C₈	—	5	4	no	—
212	C₃×C₈	—	10	2	no	—
213	C₃×C₈	—	10	1	no	—
214	C₃⋊C₈	—	5	4	no	—
215	C₃⋊C₈	—	5	2	no	—
216	C₃⋊C₈	—	10	2	no	—
217	C₃⋊C₈	#700	1	2	yes	complemented
218	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
219	C₂×D₅	—	3	2	no	—
220	C₂×D₅	—	6	1	no	—
221	C₂×D₅	—	6	1	no	—
222	C₄×C₅	—	3	2	no	—
223	C₄×C₅	—	6	1	no	—
224	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
225	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
226	Dic₅	—	6	1	no	—
227	Dic₅	C₂×D₃×C₈	1	2	yes	—
228	Dic₅	#907	1	2	yes	—
229	C₂²×C₄	—	15	1	no	—
230	C₂²×C₄	—	15	1	no	—
231	C₂²×C₄	—	15	1	no	—
232	M₁₆	—	15	8	no	—
233	C₂×C₈	—	3	1	no	—
234	C₂×C₈	—	5	2	no	—
235	C₂×C₈	—	5	2	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	15	2	no	—
238	C₂×C₈	—	15	1	no	—
239	C₂×C₈	—	15	2	no	—
240	C₂×C₈	—	30	2	no	—
241	C₂×C₈	—	30	2	no	—
242	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₄²	—	3	1	no	—
244	C₄²	—	5	1	no	—
245	C₄²	—	15	1	no	—
246	C₄²	—	15	1	no	—
247	C₃×C₅	C₈×M₁₆	1	1	yes	characteristic, complemented, Hall
248	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
249	C₂×D₃	—	5	2	no	—
250	C₂×D₃	—	10	1	no	—
251	C₂×D₃	—	10	1	no	—
252	C₃×C₄	—	5	2	no	—
253	C₃×C₄	—	5	2	no	—
254	C₃×C₄	C₂×#700	1	1	yes	characteristic
255	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
256	Dic₃	—	2	1	no	—
257	Dic₃	—	10	1	no	—
258	Dic₃	C₈×AGL(1,𝔽₅)	1	2	yes	—
259	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
260	C₂×C₅	#4707	1	1	yes	characteristic
261	C₂×C₅	#4708	1	1	yes	characteristic
262	D₅	—	6	2	no	—
263	C₂³	—	15	1	no	—
264	C₂×C₄	—	3	1	no	—
265	C₂×C₄	—	3	1	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	5	1	no	—
268	C₂×C₄	—	15	4	no	—
269	C₂×C₄	—	15	1	no	—
270	C₂×C₄	—	30	2	no	—
271	C₂×C₄	—	30	1	no	—
272	C₂×C₄	—	30	1	no	—
273	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	3	2	no	—
276	C₈	—	5	4	no	—
277	C₈	—	10	2	no	—
278	C₈	—	10	1	no	—
279	C₈	—	15	4	no	—
280	C₈	—	15	2	no	—
281	C₈	—	30	2	no	—
282	C₂×C₃	C₄×#700	1	1	yes	characteristic
283	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
284	C₂×C₃	#65718	1	1	yes	characteristic
285	D₃	—	10	2	no	—
286	C₅	#68067	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	15	2	no	—
288	C₂²	—	30	1	no	—
289	C₂²	—	30	1	no	—
290	C₂²	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
291	C₄	—	3	2	no	—
292	C₄	—	5	2	no	—
293	C₄	—	5	2	no	—
294	C₄	—	6	1	no	—
295	C₄	—	30	1	no	—
296	C₄	D₃×#700	1	1	yes	characteristic
297	C₄	#91931	1	1	yes	characteristic
298	C₃	C₈×#700	1	1	yes	characteristic, complemented, Sylow
299	C₂	—	30	2	no	—
300	C₂	D₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
301	C₂	#11751656	1	1	yes	characteristic
302	C₂	#11751756	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₂×C₃×C₅×C₈	C₂×C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×D₁₅	C₄²
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₈×M₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. #11746751	C₂	semidirect
5. C₃×#170⋊C₈	C₂	semidirect
6. C₃⋊C₈×#170	C₂	semidirect
7. C₃⋊C₈×#170	C₂	semidirect
29. #170⋊C₈	D₃	semidirect
39. #6740	C₈	semidirect
85. C₃⋊#170	C₂×C₈	semidirect
87. C₃×#170	C₂×C₈	semidirect
89. C₃⋊#168	M₁₆	semidirect
91. C₅×C₃⋊C₈	M₁₆	semidirect
175. #170	D₃×C₈	semidirect
217. C₃⋊C₈	#700	semidirect
247. C₃×C₅	C₈×M₁₆	semidirect
286. C₅	#68067	semidirect
298. C₃	C₈×#700	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6740	2
#11751797	2
C₈×#1200	2
#11746751	2
C₃×#170⋊C₈	2
C₃⋊C₈×#170	2
C₈×#700	3
#68067	5