Subgroups — #14280731

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6739	C₂	1	1	yes	maximal, characteristic
2	C₄×#6739	C₂	1	1	yes	maximal, characteristic
3	Dic₅×ΓL(1,𝔽₂₅)	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	#11746485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604737	—	3	1	no	maximal, Hall
9	C₂×#6739	C₄	1	2	yes	—
10	C₂×#6739	C₄	1	2	yes	—
11	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
12	C₄×D₃×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×#6650	C₄	1	1	yes	characteristic
14	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
16	C₃×C₄×#170	C₂²	1	1	yes	characteristic
17	C₃×C₄×#170	C₂²	1	1	yes	characteristic
18	#91169	C₄	1	2	yes	—
19	#91169	C₄	1	2	yes	—
20	#91099	C₄	1	1	yes	characteristic
21	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	#90954	C₄	1	1	yes	characteristic
23	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68075	—	5	1	no	maximal, Hall
25	C₄×#704	—	3	1	no	—
26	C₄×#704	—	3	1	no	—
27	M₁₆×Dic₅	—	3	1	no	—
28	#170⋊C₈	—	3	1	no	—
29	#170⋊C₈	D₃	1	1	yes	characteristic, complemented
30	#64879	—	3	1	no	—
31	C₈×#170	—	3	1	no	—
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×D₃×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂×D₃×Dic₅	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	#6739	—	2	4	no	—
40	#6739	—	2	4	no	—
41	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
46	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
47	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
48	Dic₃×Dic₅	C₂×C₄	1	1	yes	characteristic
49	Dic₃×Dic₅	C₂×C₄	1	1	yes	characteristic
50	#6650	—	2	2	no	—
51	C₄×ΓL(1,𝔽₂₅)	—	5	1	no	—
52	C₄×ΓL(1,𝔽₂₅)	—	5	1	no	—
53	C₄×ΓL(1,𝔽₂₅)	—	5	1	no	—
54	C₃×#322	—	5	1	no	—
55	#4454	—	5	1	no	—
56	#4454	—	5	1	no	—
57	C₈×C₃⋊C₈	—	5	1	no	—
58	C₂×#704	—	3	2	no	—
59	C₂×#704	—	3	2	no	—
60	C₂×C₅×M₁₆	—	3	1	no	—
61	C₂×C₄×Dic₅	—	3	1	no	—
62	C₂×#681	—	3	1	no	—
63	#4002	—	3	2	no	—
64	#4002	—	3	2	no	—
65	C₄×#170	—	3	1	no	—
66	C₄×#170	—	3	1	no	—
67	C₄×#170	C₂×D₃	1	1	yes	characteristic
68	C₄×#170	C₂×D₃	1	1	yes	characteristic
69	#3947	—	3	1	no	—
70	#3947	—	3	1	no	—
71	C₈×Dic₅	—	3	1	no	—
72	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
73	#1455	—	15	1	no	Sylow
74	C₂²×C₅×D₃	C₄²	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
77	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
78	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
79	C₄×C₅×D₃	—	2	1	no	—
80	C₄×C₅×D₃	—	2	1	no	—
81	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
83	D₃×Dic₅	—	2	2	no	—
84	D₃×Dic₅	—	2	2	no	—
85	C₃⋊#170	—	2	2	no	—
86	C₃⋊#170	M₁₆	1	4	yes	complemented
87	C₃×#170	—	2	2	no	—
88	C₃×#170	M₁₆	1	4	yes	complemented
89	C₃×C₅×C₈	—	2	1	no	—
90	C₃⋊#168	M₁₆	1	2	yes	complemented
91	C₃×#168	—	2	1	no	—
92	C₅×C₃⋊C₈	M₁₆	1	2	yes	complemented
93	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
94	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
95	C₂×ΓL(1,𝔽₂₅)	—	5	1	no	—
96	C₂×ΓL(1,𝔽₂₅)	AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₄²×D₃	—	5	1	no	—
98	C₃×C₄×C₈	—	5	1	no	—
99	C₃×C₄×C₈	—	5	1	no	—
100	C₃×C₄×C₈	—	5	1	no	—
101	#815	—	5	2	no	—
102	#815	—	5	2	no	—
103	#815	—	5	1	no	—
104	#815	—	5	1	no	—
105	C₄×C₃⋊C₈	—	5	1	no	—
106	C₄×C₃⋊C₈	—	5	1	no	—
107	C₄×C₃⋊C₈	—	5	1	no	—
108	C₂²×C₄×C₅	—	3	1	no	—
109	C₂²×Dic₅	—	3	1	no	—
110	C₂²×Dic₅	—	3	1	no	—
111	#704	—	6	4	no	—
112	#704	—	6	4	no	—
113	C₂×#170	—	3	2	no	—
114	C₂×#170	—	3	2	no	—
115	C₂×#170	C₄×D₃	1	2	yes	—
116	C₂×#170	C₄×D₃	1	2	yes	—
117	C₅×M₁₆	—	6	2	no	—
118	C₂×C₅×C₈	—	3	1	no	—
119	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
120	C₄×Dic₅	—	3	1	no	—
121	C₄×Dic₅	—	3	1	no	—
122	C₄×Dic₅	—	3	1	no	—
123	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
124	#681	—	6	2	no	—
125	C₂×#168	—	3	1	no	—
126	C₂×#168	C₄×D₃	1	1	yes	characteristic
127	C₄×M₁₆	—	15	1	no	—
128	C₄×M₁₆	—	15	1	no	—
129	C₄×M₁₆	—	15	1	no	—
130	#322	—	5	1	no	—
131	#322	—	15	1	no	—
132	#322	—	15	1	no	—
133	C₈²	—	15	1	no	—
134	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₅×D₃	—	2	1	no	—
136	C₂×C₅×D₃	—	2	1	no	—
137	C₂×C₅×D₃	—	2	1	no	—
138	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
140	Dic₁₅	—	2	1	no	—
141	Dic₁₅	—	2	1	no	—
142	C₃×Dic₅	C₂×M₁₆	1	2	yes	—
143	C₃×Dic₅	C₂×M₁₆	1	2	yes	—
144	C₅×Dic₃	—	2	1	no	—
145	C₂×C₄×D₃	—	5	1	no	—
146	C₂×C₄×D₃	—	5	1	no	—
147	C₂×C₄×D₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₂×C₃×C₈	—	5	2	no	—
149	C₂×C₃×C₈	—	5	2	no	—
150	C₂×C₃×C₈	—	5	1	no	—
151	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₃×C₄²	—	5	1	no	—
153	C₄×Dic₃	—	5	1	no	—
154	C₄×Dic₃	—	5	1	no	—
155	C₄×Dic₃	—	5	1	no	—
156	C₂×C₃⋊C₈	—	5	2	no	—
157	C₂×C₃⋊C₈	—	5	1	no	—
158	C₂×C₃⋊C₈	—	5	2	no	—
159	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
160	ΓL(1,𝔽₂₅)	—	2	2	no	—
161	ΓL(1,𝔽₂₅)	—	10	2	no	—
162	ΓL(1,𝔽₂₅)	—	10	4	no	—
163	ΓL(1,𝔽₂₅)	—	10	4	no	—
164	C₂³×C₅	—	3	1	no	—
165	C₂×C₄×C₅	—	3	1	no	—
166	C₂×C₄×C₅	—	6	1	no	—
167	C₂×C₄×C₅	—	6	1	no	—
168	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×Dic₅	—	3	1	no	—
170	C₂×Dic₅	—	3	1	no	—
171	C₂×Dic₅	—	6	2	no	—
172	C₂×Dic₅	—	6	2	no	—
173	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
175	#170	—	2	2	no	—
176	#170	—	3	4	no	—
177	#170	—	6	2	no	—
178	#170	ΓL(1,𝔽₂₅)	1	4	yes	complemented
179	C₅×C₈	—	2	1	no	—
180	C₅×C₈	—	3	2	no	—
181	#168	—	2	1	no	—
182	#168	—	3	2	no	—
183	C₂×M₁₆	—	3	1	no	—
184	C₂×M₁₆	—	15	1	no	—
185	C₂×M₁₆	—	15	2	no	—
186	C₂×M₁₆	—	15	2	no	—
187	C₂×C₄²	—	15	1	no	—
188	#97	—	15	1	no	—
189	#97	—	15	2	no	—
190	#97	—	15	2	no	—
191	#97	—	15	1	no	—
192	C₄×C₈	—	5	1	no	—
193	C₄×C₈	—	5	1	no	—
194	C₄×C₈	—	5	1	no	—
195	C₄×C₈	—	15	1	no	—
196	C₄×C₈	—	15	1	no	—
197	C₄×C₈	—	15	1	no	—
198	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
199	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
200	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
201	C₅×D₃	—	4	1	no	—
202	C₂²×D₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₂×C₃×C₄	—	5	1	no	—
204	C₂×C₃×C₄	—	5	1	no	—
205	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₂×Dic₃	—	5	1	no	—
207	C₂×Dic₃	—	5	1	no	—
208	C₂×Dic₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₄×D₃	—	2	1	no	—
210	C₄×D₃	—	2	1	no	—
211	C₄×D₃	—	10	2	no	—
212	C₄×D₃	—	10	2	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	5	4	no	—
215	C₃×C₈	—	10	2	no	—
216	C₃×C₈	—	10	1	no	—
217	C₃⋊C₈	—	5	4	no	—
218	C₃⋊C₈	—	5	2	no	—
219	C₃⋊C₈	—	10	2	no	—
220	C₃⋊C₈	#704	1	2	yes	complemented
221	C₂²×C₅	—	6	1	no	—
222	C₂²×C₅	—	6	1	no	—
223	C₂²×C₅	—	6	1	no	—
224	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
225	C₄×C₅	—	6	1	no	—
226	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
227	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
228	Dic₅	—	6	1	no	—
229	Dic₅	—	6	1	no	—
230	Dic₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
231	Dic₅	D₃×M₁₆	1	2	yes	—
232	C₂²×C₄	—	3	1	no	—
233	C₂²×C₄	—	15	1	no	—
234	C₂²×C₄	—	15	1	no	—
235	M₁₆	—	6	2	no	—
236	M₁₆	—	30	4	no	—
237	M₁₆	—	30	2	no	—
238	M₁₆	—	30	4	no	—
239	C₂×C₈	—	3	1	no	—
240	C₂×C₈	—	5	2	no	—
241	C₂×C₈	—	5	2	no	—
242	C₂×C₈	—	5	1	no	—
243	C₂×C₈	—	15	2	no	—
244	C₂×C₈	—	15	1	no	—
245	C₂×C₈	—	15	2	no	—
246	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
247	C₄²	—	5	1	no	—
248	C₄²	—	15	1	no	—
249	C₄²	—	15	1	no	—
250	C₄²	—	15	1	no	—
251	C₃×C₅	#1455	1	1	yes	characteristic, complemented, Hall
252	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
253	C₂×D₃	—	2	1	no	—
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₃×C₄	—	5	2	no	—
257	C₃×C₄	—	5	2	no	—
258	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
259	C₃×C₄	C₂×#704	1	1	yes	characteristic
260	Dic₃	—	2	1	no	—
261	Dic₃	—	10	1	no	—
262	Dic₃	—	10	1	no	—
263	C₂×C₅	—	12	1	no	—
264	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
265	C₂×C₅	#4707	1	1	yes	characteristic
266	C₂×C₅	#4707	1	1	yes	characteristic
267	C₂³	—	3	1	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	5	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	6	1	no	—
272	C₂×C₄	—	6	1	no	—
273	C₂×C₄	—	15	1	no	—
274	C₂×C₄	—	15	1	no	—
275	C₂×C₄	—	30	2	no	—
276	C₂×C₄	—	30	2	no	—
277	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
278	C₈	—	2	1	no	—
279	C₈	—	3	2	no	—
280	C₈	—	5	4	no	—
281	C₈	—	10	2	no	—
282	C₈	—	10	1	no	—
283	C₈	—	15	4	no	—
284	C₈	—	15	2	no	—
285	C₈	—	30	2	no	—
286	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
287	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
288	C₂×C₃	C₄×#704	1	1	yes	characteristic
289	D₃	—	4	1	no	—
290	C₅	#68075	1	1	yes	characteristic, complemented, Sylow
291	C₂²	—	6	1	no	—
292	C₂²	—	6	1	no	—
293	C₂²	—	6	1	no	—
294	C₂²	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
295	C₄	—	5	2	no	—
296	C₄	—	5	2	no	—
297	C₄	—	6	1	no	—
298	C₄	—	30	1	no	—
299	C₄	—	30	1	no	—
300	C₄	#91915	1	1	yes	characteristic
301	C₄	D₃×#704	1	1	yes	characteristic
302	C₃	#10604737	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	12	1	no	—
304	C₂	AGL(1,𝔽₅)×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
305	C₂	#11751650	1	1	yes	characteristic
306	C₂	#11751907	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₂×C₃×C₅×C₈	C₂×C₄
prime radical	C₂²×C₅×D₃	C₄²
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1455
LCS	G ▷ C₂²×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. #11746485	C₂	semidirect
7. C₃⋊C₈×#170	C₂	semidirect
29. #170⋊C₈	D₃	semidirect
86. C₃⋊#170	M₁₆	semidirect
88. C₃×#170	M₁₆	semidirect
90. C₃⋊#168	M₁₆	semidirect
92. C₅×C₃⋊C₈	M₁₆	semidirect
178. #170	ΓL(1,𝔽₂₅)	semidirect
220. C₃⋊C₈	#704	semidirect
251. C₃×C₅	#1455	semidirect
290. C₅	#68075	semidirect
302. C₃	#10604737	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6739	2
Dic₅×ΓL(1,𝔽₂₅)	2
#11746751	2
C₃×#170⋊C₈	2
#11746485	2
C₃⋊C₈×#170	2
#10604737	3
#68075	5