Subgroups — C₃×#10603926 #14295522

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4131	C₂	1	1	yes	maximal, characteristic
2	C₃×#65771	C₂	1	1	yes	maximal, characteristic
3	C₃×#65728	C₂	1	2	yes	maximal, complemented
4	C₃×#65034	C₂	1	1	yes	maximal, characteristic
5	C₃×#64902	C₂	1	2	yes	maximal, complemented
6	#10603926	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#711	C₂²	1	1	yes	characteristic
8	C₂×C₃×#704	C₄	1	2	yes	complemented
9	C₂²×C₃×#170	C₄	1	2	yes	complemented
10	C₃×#4131	—	2	2	no	—
11	C₃×#4131	—	2	2	no	—
12	C₃×#4130	C₂²	1	2	yes	—
13	C₂×C₃×#700	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×D₅×M₁₆	C₂²	1	2	yes	—
16	C₃×#4011	C₄	1	4	yes	—
17	C₃×#3992	—	2	2	no	—
18	C₃×C₈×AGL(1,𝔽₅)	—	2	2	no	—
19	C₃×C₅×#100	C₄	1	1	yes	characteristic, complemented
20	C₃×#3965	C₄	1	1	yes	characteristic, complemented
21	C₃×#3956	C₄	1	2	yes	complemented
22	C₃×#1757	—	5	1	no	maximal, Hall
23	C₂×#4131	C₂×C₃	1	1	yes	characteristic
24	#65771	C₂×C₃	1	1	yes	characteristic
25	#65728	C₂×C₃	1	2	yes	complemented
26	#65034	C₂×C₃	1	1	yes	characteristic
27	#64902	C₂×C₃	1	2	yes	complemented
28	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
31	C₃×#711	—	2	2	no	—
32	C₃×#711	—	2	2	no	—
33	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
35	C₃×C₂²⋊AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
36	C₃×#704	—	2	4	no	—
37	C₂×C₃×#170	—	2	2	no	—
38	C₂×C₃×#170	—	2	2	no	—
39	C₂×C₃×#170	C₂×C₄	1	2	yes	—
40	C₂×C₃×#170	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×#170	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
43	C₃×C₄×AGL(1,𝔽₅)	—	2	2	no	—
44	C₃×#700	—	2	2	no	—
45	C₃×D₅⋊C₈	D₄	1	2	yes	—
46	C₃×D₅⋊C₈	D₄	1	2	yes	—
47	C₃×C₅×M₁₆	C₂×C₄	1	2	yes	—
48	C₃×#681	C₂×C₄	1	2	yes	—
49	C₃×#676	—	2	2	no	—
50	C₃×C₈×D₅	—	2	2	no	—
51	C₂×C₃×#131	—	5	1	no	—
52	C₃×#413	—	5	1	no	—
53	C₃×#410	—	5	1	no	—
54	C₃×#405	—	5	2	no	—
55	C₃×#344	—	5	2	no	—
56	C₂×#711	C₂²×C₃	1	1	yes	characteristic
57	C₂×#704	C₃×C₄	1	2	yes	complemented
58	C₂²×#170	C₃×C₄	1	2	yes	complemented
59	#4131	—	2	2	no	—
60	#4131	—	2	2	no	—
61	#4130	C₂²×C₃	1	2	yes	—
62	C₂×#700	C₂²×C₃	1	1	yes	characteristic
63	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	D₅×M₁₆	C₂²×C₃	1	2	yes	—
65	#4011	C₃×C₄	1	4	yes	—
66	#3992	—	2	2	no	—
67	C₈×AGL(1,𝔽₅)	—	2	2	no	—
68	C₅×#100	C₃×C₄	1	1	yes	characteristic, complemented
69	#3965	C₃×C₄	1	1	yes	characteristic, complemented
70	#3956	C₃×C₄	1	2	yes	complemented
71	#1757	—	5	1	no	Sylow
72	C₂³×C₃×C₅	C₄²	1	2	yes	—
73	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
75	C₃×C₅×D₄	—	2	2	no	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₃×C₂²⋊D₅	—	2	4	no	—
78	C₂×C₃×Dic₅	—	2	2	no	—
79	C₂×C₃×Dic₅	—	2	2	no	—
80	C₂×C₃×Dic₅	C₄²	1	2	yes	—
81	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
82	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
83	C₃×C₄×D₅	#41	1	2	yes	—
84	C₃×Dic₁₀	—	2	2	no	—
85	C₃×#170	—	2	2	no	—
86	C₃×#170	C₂²⋊C₄	1	1	yes	characteristic
87	C₃×#170	C₂²⋊C₄	1	1	yes	characteristic
88	C₃×#170	C₂²⋊C₄	1	2	yes	—
89	C₃×C₅×C₈	—	2	2	no	—
90	C₃×#168	—	2	2	no	—
91	C₂×C₃×#41	—	5	1	no	—
92	C₃×#131	—	10	2	no	—
93	C₃×#131	—	10	2	no	—
94	C₂×C₃×M₁₆	—	5	1	no	—
95	C₂×C₃×M₁₆	—	5	2	no	—
96	C₂×C₃×M₁₆	—	5	2	no	—
97	C₂²×C₃×C₈	—	5	1	no	—
98	C₂²×C₃×C₈	—	5	2	no	—
99	C₃×#117	—	5	2	no	—
100	C₃×#101	—	5	2	no	—
101	C₃×#100	—	5	1	no	—
102	C₃×#100	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
103	C₃×#99	—	5	4	no	—
104	C₃×#97	—	10	2	no	—
105	C₃×C₄×C₈	—	10	2	no	—
106	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×C₂²⋊D₅	C₂×C₃×C₄	1	2	yes	—
108	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
109	#711	—	2	2	no	—
110	#711	—	2	2	no	—
111	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
112	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂²⋊AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
114	#704	—	2	4	no	—
115	C₂×#170	—	2	2	no	—
116	C₂×#170	—	2	2	no	—
117	C₂×#170	C₂×C₃×C₄	1	2	yes	—
118	C₂×#170	C₂×C₃×C₄	1	1	yes	characteristic
119	C₂×#170	C₂×C₃×C₄	1	1	yes	characteristic
120	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
121	C₄×AGL(1,𝔽₅)	—	2	2	no	—
122	#700	—	2	2	no	—
123	D₅⋊C₈	C₃×D₄	1	2	yes	—
124	D₅⋊C₈	C₃×D₄	1	2	yes	—
125	C₅×M₁₆	C₂×C₃×C₄	1	2	yes	—
126	#681	C₂×C₃×C₄	1	2	yes	—
127	#676	—	2	2	no	—
128	C₈×D₅	—	2	2	no	—
129	C₂×#131	—	5	1	no	—
130	#413	—	5	1	no	—
131	#410	—	5	1	no	—
132	#405	—	5	2	no	—
133	#344	—	5	2	no	—
134	C₂²×C₃×C₅	—	2	2	no	—
135	C₂²×C₃×C₅	—	4	2	no	—
136	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₃×D₅	—	4	1	no	—
138	C₂×C₃×D₅	C₄×D₄	1	2	yes	—
139	C₃×AGL(1,𝔽₅)	—	4	4	no	—
140	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
141	C₃×Dic₅	—	2	2	no	—
142	C₃×Dic₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₃×Dic₅	#117	1	1	yes	characteristic
144	C₃×#41	—	10	2	no	—
145	C₃×#41	—	10	2	no	—
146	C₂×C₃×Q₈	—	5	1	no	—
147	C₂×C₃×D₄	—	5	2	no	—
148	C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₂²×C₃×C₄	—	5	1	no	—
150	C₂²×C₃×C₄	—	5	2	no	—
151	C₃×M₁₆	—	5	2	no	—
152	C₃×M₁₆	—	10	2	no	—
153	C₃×M₁₆	—	10	4	no	—
154	C₃×M₁₆	—	10	2	no	—
155	C₃×M₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
156	C₂×C₃×C₈	—	5	2	no	—
157	C₂×C₃×C₈	—	5	2	no	—
158	C₂×C₃×C₈	—	5	1	no	—
159	C₂×C₃×C₈	—	5	1	no	—
160	C₂×C₃×C₈	—	5	2	no	—
161	C₂×C₃×C₈	—	10	2	no	—
162	C₂×C₃×C₈	—	10	2	no	—
163	C₂×C₃×C₈	—	10	2	no	—
164	C₃×C₄⋊C₄	—	10	2	no	—
165	C₃×C₂²⋊C₄	—	5	4	no	—
166	C₃×C₄²	—	10	2	no	—
167	C₂³×C₅	C₃×C₄²	1	2	yes	—
168	C₂²×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₂×AGL(1,𝔽₅)	—	2	4	no	—
170	C₅×D₄	—	2	2	no	—
171	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
172	C₂²⋊D₅	—	2	4	no	—
173	C₂×Dic₅	—	2	2	no	—
174	C₂×Dic₅	—	2	2	no	—
175	C₂×Dic₅	C₃×C₄²	1	2	yes	—
176	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
178	C₄×D₅	C₃×#41	1	2	yes	—
179	Dic₁₀	—	2	2	no	—
180	#170	—	2	2	no	—
181	#170	C₃×C₂²⋊C₄	1	1	yes	characteristic
182	#170	C₃×C₂²⋊C₄	1	1	yes	characteristic
183	#170	C₃×C₂²⋊C₄	1	2	yes	—
184	C₅×C₈	—	2	2	no	—
185	#168	—	2	2	no	—
186	C₂×#41	—	5	1	no	—
187	#131	—	10	2	no	—
188	#131	—	10	2	no	—
189	C₂×M₁₆	—	5	1	no	—
190	C₂×M₁₆	—	5	2	no	—
191	C₂×M₁₆	—	5	2	no	—
192	C₂²×C₈	—	5	1	no	—
193	C₂²×C₈	—	5	2	no	—
194	#117	—	5	2	no	—
195	#101	—	5	2	no	—
196	#100	—	5	1	no	—
197	#100	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
198	#99	—	5	4	no	—
199	#97	—	10	2	no	—
200	C₄×C₈	—	10	2	no	—
201	C₂×C₃×C₅	—	2	1	no	—
202	C₂×C₃×C₅	—	4	2	no	—
203	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
204	C₃×D₅	—	2	2	no	—
205	C₂³×C₃	—	5	1	no	—
206	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	2	yes	—
207	C₃×Q₈	—	10	2	no	—
208	C₃×D₄	—	2	2	no	—
209	C₃×D₄	—	10	4	no	—
210	C₂×C₃×C₄	—	5	1	no	—
211	C₂×C₃×C₄	—	5	2	no	—
212	C₂×C₃×C₄	—	5	2	no	—
213	C₂×C₃×C₄	—	5	2	no	—
214	C₂×C₃×C₄	—	10	2	no	—
215	C₂×C₃×C₄	—	10	2	no	—
216	C₂×C₃×C₄	—	10	4	no	—
217	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₃×C₈	—	2	2	no	—
219	C₃×C₈	—	5	1	no	—
220	C₃×C₈	—	5	1	no	—
221	C₃×C₈	—	5	2	no	—
222	C₃×C₈	—	10	2	no	—
223	C₃×C₈	—	10	2	no	—
224	C₂²×C₅	—	2	2	no	—
225	C₂²×C₅	—	4	2	no	—
226	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
227	C₂×D₅	—	4	1	no	—
228	C₂×D₅	C₃×C₄×D₄	1	2	yes	—
229	AGL(1,𝔽₅)	—	4	4	no	—
230	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
231	Dic₅	—	2	2	no	—
232	Dic₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
233	Dic₅	C₃×#117	1	1	yes	characteristic
234	#41	—	10	2	no	—
235	#41	—	10	2	no	—
236	C₂×Q₈	—	5	1	no	—
237	C₂×D₄	—	5	2	no	—
238	C₂×D₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₂²×C₄	—	5	1	no	—
240	C₂²×C₄	—	5	2	no	—
241	M₁₆	—	5	2	no	—
242	M₁₆	—	10	2	no	—
243	M₁₆	—	10	4	no	—
244	M₁₆	—	10	2	no	—
245	M₁₆	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
246	C₂×C₈	—	5	2	no	—
247	C₂×C₈	—	5	2	no	—
248	C₂×C₈	—	5	1	no	—
249	C₂×C₈	—	5	1	no	—
250	C₂×C₈	—	5	2	no	—
251	C₂×C₈	—	10	2	no	—
252	C₂×C₈	—	10	2	no	—
253	C₂×C₈	—	10	2	no	—
254	C₄⋊C₄	—	10	2	no	—
255	C₂²⋊C₄	—	5	4	no	—
256	C₄²	—	10	2	no	—
257	C₃×C₅	#1757	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	4	2	no	—
260	C₂²×C₃	—	5	2	no	—
261	C₂²×C₃	—	20	1	no	—
262	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
263	C₃×C₄	—	5	1	no	—
264	C₃×C₄	—	5	1	no	—
265	C₃×C₄	—	10	2	no	—
266	C₃×C₄	—	20	4	no	—
267	C₃×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
268	C₂×C₅	—	2	1	no	—
269	C₂×C₅	—	4	2	no	—
270	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
271	D₅	—	2	2	no	—
272	C₂³	—	5	1	no	—
273	C₂³	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
274	Q₈	—	10	2	no	—
275	D₄	—	2	2	no	—
276	D₄	—	10	4	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	5	2	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	5	2	no	—
281	C₂×C₄	—	10	2	no	—
282	C₂×C₄	—	10	2	no	—
283	C₂×C₄	—	10	4	no	—
284	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₈	—	5	1	no	—
287	C₈	—	5	1	no	—
288	C₈	—	5	2	no	—
289	C₈	—	10	2	no	—
290	C₈	—	10	2	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	4	2	no	—
293	C₂×C₃	—	10	2	no	—
294	C₂×C₃	C₂²⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
295	C₅	C₃×#1757	1	1	yes	characteristic, complemented, Sylow
296	C₂²	—	2	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	5	2	no	—
299	C₂²	—	20	1	no	—
300	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
301	C₄	—	5	1	no	—
302	C₄	—	5	1	no	—
303	C₄	—	10	2	no	—
304	C₄	—	20	4	no	—
305	C₄	C₃×D₄×AGL(1,𝔽₅)	1	2	yes	—
306	C₃	#10603926	1	1	yes	characteristic, direct product factor, Sylow
307	C₂	—	2	1	no	—
308	C₂	—	4	2	no	—
309	C₂	—	10	2	no	—
310	C₂	C₃×C₂²⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²⋊C₄×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#100	C₄
socle	C₂×C₃×C₅	C₄×C₂²⋊C₄
prime radical	C₂×C₃×#711	C₂²
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₃×#100	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1757
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#100

Complements

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

N	H ≅ G/N	type
3. C₃×#65728	C₂	semidirect
5. C₃×#64902	C₂	semidirect
6. #10603926	C₃	direct (N × H)
8. C₂×C₃×#704	C₄	semidirect
9. C₂²×C₃×#170	C₄	semidirect
19. C₃×C₅×#100	C₄	semidirect
20. C₃×#3965	C₄	semidirect
21. C₃×#3956	C₄	semidirect
25. #65728	C₂×C₃	semidirect
27. #64902	C₂×C₃	semidirect
57. C₂×#704	C₃×C₄	semidirect
58. C₂²×#170	C₃×C₄	semidirect
68. C₅×#100	C₃×C₄	semidirect
69. #3965	C₃×C₄	semidirect
70. #3956	C₃×C₄	semidirect
102. C₃×#100	AGL(1,𝔽₅)	semidirect
197. #100	C₃×AGL(1,𝔽₅)	semidirect
257. C₃×C₅	#1757	semidirect
295. C₅	C₃×#1757	semidirect
306. C₃	#10603926	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#4131	2
C₃×#65771	2
C₃×#65728	2
C₃×#65034	2
C₃×#64902	2
#10603926	3
C₃×#1757	5

Subgroups — C3×#10603926 #14295522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10603926 #14295522