Subgroups — #14280725

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₄×#6650	C₂	1	1	yes	maximal, characteristic
3	C₄×#6650	C₂	1	1	yes	maximal, characteristic
4	#11746275	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64674	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745832	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600642	—	3	1	no	maximal, Hall
9	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
10	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
11	C₂×#6650	C₄	1	2	yes	—
12	C₂×#6650	C₄	1	2	yes	—
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
16	C₅×#815	C₄	1	2	yes	—
17	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄×#168	C₂²	1	1	yes	characteristic
19	C₃×C₄×#168	C₂²	1	1	yes	characteristic
20	#90953	C₄	1	2	yes	—
21	#90953	C₄	1	2	yes	—
22	#68075	—	5	1	no	maximal, Hall
23	C₄×C₅×M₁₆	—	3	1	no	—
24	C₄×#681	—	3	1	no	—
25	C₄×#681	—	3	1	no	—
26	#64675	—	3	1	no	—
27	#64674	—	3	1	no	—
28	#64674	D₃	1	1	yes	characteristic, complemented
29	C₈×#168	—	3	1	no	—
30	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
37	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
42	C₂×C₃×#168	C₂×C₄	1	2	yes	—
43	C₂×C₃×#168	C₂×C₄	1	2	yes	—
44	#6650	—	2	4	no	—
45	#6650	—	2	4	no	—
46	C₄×ΓL(1,𝔽₂₅)	—	5	1	no	—
47	C₄×ΓL(1,𝔽₂₅)	—	5	1	no	—
48	C₄×ΓL(1,𝔽₂₅)	D₅	1	1	yes	characteristic
49	C₃×#322	—	5	1	no	—
50	#4454	—	5	1	no	—
51	#4454	—	5	1	no	—
52	C₈×C₃⋊C₈	—	5	1	no	—
53	C₂×C₅×M₁₆	—	3	2	no	—
54	C₂×C₄²×C₅	—	3	1	no	—
55	C₂×#681	—	3	2	no	—
56	C₂×#681	—	3	2	no	—
57	C₅×#97	—	3	2	no	—
58	C₄×C₅×C₈	—	3	1	no	—
59	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
60	#3935	—	3	2	no	—
61	#3935	—	3	2	no	—
62	C₄×#168	—	3	1	no	—
63	C₄×#168	—	3	1	no	—
64	C₄×#168	C₂×D₃	1	1	yes	characteristic
65	C₄×#168	C₂×D₃	1	1	yes	characteristic
66	#1455	—	15	1	no	Sylow
67	C₂²×C₅×D₃	C₄²	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
72	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
73	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
74	C₄×C₅×D₃	—	2	2	no	—
75	C₄×C₅×D₃	—	2	2	no	—
76	C₄×C₅×D₃	—	2	2	no	—
77	C₃×C₅×C₈	—	2	2	no	—
78	C₃⋊#168	—	2	2	no	—
79	C₃⋊#168	M₁₆	1	4	yes	complemented
80	C₃×#168	—	2	2	no	—
81	C₃×#168	M₁₆	1	4	yes	complemented
82	C₅×C₃⋊C₈	M₁₆	1	4	yes	complemented
83	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
84	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
85	C₂×ΓL(1,𝔽₂₅)	Dic₅	1	2	yes	—
86	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
87	C₃×C₄×C₈	—	5	1	no	—
88	C₃×C₄×C₈	—	5	1	no	—
89	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
90	#815	—	5	2	no	—
91	#815	—	5	2	no	—
92	#815	Dic₅	1	2	yes	—
93	C₄×C₃⋊C₈	—	5	1	no	—
94	C₄×C₃⋊C₈	—	5	1	no	—
95	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
96	C₂²×C₄×C₅	—	3	1	no	—
97	C₂²×C₄×C₅	—	3	1	no	—
98	C₂²×C₄×C₅	—	3	1	no	—
99	C₅×M₁₆	—	6	4	no	—
100	C₂×C₅×C₈	—	3	2	no	—
101	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
102	C₄²×C₅	—	3	1	no	—
103	C₄²×C₅	—	3	1	no	—
104	C₄²×C₅	—	3	1	no	—
105	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
106	#681	—	6	4	no	—
107	#681	—	6	4	no	—
108	C₂×#168	—	3	2	no	—
109	C₂×#168	—	3	2	no	—
110	C₂×#168	C₄×D₃	1	2	yes	—
111	C₂×#168	C₄×D₃	1	2	yes	—
112	C₄×M₁₆	—	3	1	no	—
113	C₄×M₁₆	—	15	1	no	—
114	C₄×M₁₆	—	15	1	no	—
115	#322	—	5	1	no	—
116	#322	—	15	1	no	—
117	#322	—	15	1	no	—
118	C₈²	—	15	1	no	—
119	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₅×D₃	—	2	1	no	—
121	C₂×C₅×D₃	—	2	1	no	—
122	C₂×C₅×D₃	—	2	1	no	—
123	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
124	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
125	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
126	C₅×Dic₃	—	2	1	no	—
127	C₅×Dic₃	—	2	1	no	—
128	C₅×Dic₃	—	2	1	no	—
129	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
130	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
131	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
132	C₂×C₃×C₈	—	5	2	no	—
133	C₂×C₃×C₈	—	5	2	no	—
134	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
135	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
136	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
137	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
138	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
139	C₂×C₃⋊C₈	—	5	2	no	—
140	C₂×C₃⋊C₈	—	5	2	no	—
141	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
142	ΓL(1,𝔽₂₅)	—	2	4	no	—
143	ΓL(1,𝔽₂₅)	—	10	4	no	—
144	ΓL(1,𝔽₂₅)	—	10	4	no	—
145	C₂³×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	3	1	no	—
148	C₂×C₄×C₅	—	3	1	no	—
149	C₂×C₄×C₅	—	6	2	no	—
150	C₂×C₄×C₅	—	6	2	no	—
151	C₂×C₄×C₅	—	6	2	no	—
152	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
155	C₅×C₈	—	2	2	no	—
156	C₅×C₈	—	3	4	no	—
157	#168	—	2	2	no	—
158	#168	—	3	4	no	—
159	#168	—	6	2	no	—
160	#168	ΓL(1,𝔽₂₅)	1	4	yes	complemented
161	C₂×M₁₆	—	3	2	no	—
162	C₂×M₁₆	—	15	2	no	—
163	C₂×M₁₆	—	15	2	no	—
164	C₂×C₄²	—	3	1	no	—
165	#97	—	3	2	no	—
166	#97	—	15	2	no	—
167	#97	—	15	2	no	—
168	C₄×C₈	—	3	1	no	—
169	C₄×C₈	—	5	1	no	—
170	C₄×C₈	—	5	1	no	—
171	C₄×C₈	—	15	1	no	—
172	C₄×C₈	—	15	1	no	—
173	C₄×C₈	D₃×D₅	1	1	yes	characteristic
174	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
175	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
176	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
177	C₅×D₃	—	4	1	no	—
178	C₂²×D₃	C₄×Dic₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
182	C₂×Dic₃	C₄×Dic₅	1	1	yes	characteristic
183	C₂×Dic₃	C₄×Dic₅	1	1	yes	characteristic
184	C₂×Dic₃	C₄×Dic₅	1	1	yes	characteristic
185	C₄×D₃	—	2	2	no	—
186	C₄×D₃	—	2	2	no	—
187	C₄×D₃	—	2	2	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	5	4	no	—
190	C₃×C₈	—	10	2	no	—
191	C₃⋊C₈	—	5	4	no	—
192	C₃⋊C₈	—	10	2	no	—
193	C₃⋊C₈	#681	1	4	yes	complemented
194	C₂²×C₅	—	6	1	no	—
195	C₂²×C₅	—	6	1	no	—
196	C₂²×C₅	—	6	1	no	—
197	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
198	C₄×C₅	—	6	1	no	—
199	C₄×C₅	—	6	1	no	—
200	C₄×C₅	—	6	1	no	—
201	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
202	C₄×C₅	D₃×M₁₆	1	2	yes	—
203	C₄×C₅	D₃×M₁₆	1	2	yes	—
204	C₂²×C₄	—	3	1	no	—
205	C₂²×C₄	—	3	1	no	—
206	C₂²×C₄	—	3	1	no	—
207	M₁₆	—	6	4	no	—
208	M₁₆	—	30	4	no	—
209	M₁₆	—	30	4	no	—
210	C₂×C₈	—	3	2	no	—
211	C₂×C₈	—	5	2	no	—
212	C₂×C₈	—	5	2	no	—
213	C₂×C₈	—	15	2	no	—
214	C₂×C₈	—	15	2	no	—
215	C₂×C₈	D₃×Dic₅	1	2	yes	—
216	C₄²	—	3	1	no	—
217	C₄²	—	3	1	no	—
218	C₄²	—	3	1	no	—
219	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
220	C₃×C₅	#1455	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
222	C₂×D₃	—	2	1	no	—
223	C₂×D₃	—	2	1	no	—
224	C₂×D₃	—	2	1	no	—
225	C₃×C₄	D₅×M₁₆	1	2	yes	—
226	C₃×C₄	D₅×M₁₆	1	2	yes	—
227	C₃×C₄	C₂×#681	1	2	yes	—
228	Dic₃	—	2	1	no	—
229	Dic₃	—	2	1	no	—
230	Dic₃	—	2	1	no	—
231	C₂×C₅	—	12	1	no	—
232	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
233	C₂×C₅	#4707	1	1	yes	characteristic
234	C₂×C₅	#4707	1	1	yes	characteristic
235	C₂³	—	3	1	no	—
236	C₂×C₄	—	3	1	no	—
237	C₂×C₄	—	3	1	no	—
238	C₂×C₄	—	3	1	no	—
239	C₂×C₄	—	6	2	no	—
240	C₂×C₄	—	6	2	no	—
241	C₂×C₄	—	6	2	no	—
242	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
243	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
244	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	3	4	no	—
247	C₈	—	5	4	no	—
248	C₈	—	10	2	no	—
249	C₈	—	15	4	no	—
250	C₈	—	30	2	no	—
251	C₂×C₃	C₄×#681	1	1	yes	characteristic
252	C₂×C₃	M₁₆×Dic₅	1	1	yes	characteristic
253	C₂×C₃	M₁₆×Dic₅	1	1	yes	characteristic
254	D₃	—	4	1	no	—
255	C₅	#68075	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	6	1	no	—
257	C₂²	—	6	1	no	—
258	C₂²	—	6	1	no	—
259	C₂²	C₄×D₃×Dic₅	1	1	yes	characteristic
260	C₄	—	6	1	no	—
261	C₄	—	6	1	no	—
262	C₄	—	6	1	no	—
263	C₄	D₅×ΓL(1,𝔽₂₅)	1	2	yes	—
264	C₄	#91246	1	2	yes	—
265	C₄	D₃×#681	1	2	yes	—
266	C₃	#10600642	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	12	1	no	—
268	C₂	#11747005	1	1	yes	characteristic
269	C₂	Dic₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
270	C₂	#11747602	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₃×C₄×C₅×C₈	C₂²
prime radical	C₂²×C₅×D₃	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1455
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #11746275	C₂	semidirect
5. C₃×#64674	C₂	semidirect
6. #11745832	C₂	semidirect
7. C₃⋊C₈×#168	C₂	semidirect
28. #64674	D₃	semidirect
79. C₃⋊#168	M₁₆	semidirect
81. C₃×#168	M₁₆	semidirect
82. C₅×C₃⋊C₈	M₁₆	semidirect
160. #168	ΓL(1,𝔽₂₅)	semidirect
193. C₃⋊C₈	#681	semidirect
220. C₃×C₅	#1455	semidirect
255. C₅	#68075	semidirect
266. C₃	#10600642	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×ΓL(1,𝔽₂₅)	2
C₄×#6650	2
#11746275	2
C₃×#64674	2
#11745832	2
C₃⋊C₈×#168	2
#10600642	3
#68075	5