Subgroups — C₃×#10604118 #14314631

Order 1920.

312 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65721	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3994	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₃×#64850	C₂	1	2	yes	maximal, complemented
5	C₃×#64849	C₂	1	2	yes	maximal, complemented
6	#10604118	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×#4130	—	2	1	no	—
8	C₂×C₃×C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₃×#3994	—	2	1	no	—
11	C₃×#3994	C₂²	1	2	yes	—
12	C₃×#3994	C₂²	1	1	yes	characteristic
13	C₃×#3994	C₂²	1	1	yes	characteristic
14	C₃×#3993	—	2	1	no	—
15	C₃×C₅×M₃₂	C₄	1	1	yes	characteristic, complemented
16	C₃×#3944	C₄	1	1	yes	characteristic, complemented
17	C₃×#3930	C₄	1	2	yes	complemented
18	C₃×D₅×C₁₆	C₂²	1	2	yes	—
19	C₃×#2159	—	5	1	no	maximal, Hall
20	#65721	C₂×C₃	1	1	yes	characteristic
21	C₂×#3994	C₂×C₃	1	1	yes	characteristic
22	D₅×M₃₂	C₂×C₃	1	1	yes	characteristic
23	#64850	C₂×C₃	1	2	yes	complemented
24	#64849	C₂×C₃	1	2	yes	complemented
25	C₂²×C₃×AGL(1,𝔽₅)	—	4	1	no	—
26	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
27	C₃×C₂²⋊AGL(1,𝔽₅)	—	4	1	no	—
28	C₃×C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
29	C₃×C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
30	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
31	C₃×C₄⋊AGL(1,𝔽₅)	—	4	1	no	—
32	C₃×C₄×AGL(1,𝔽₅)	—	4	1	no	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×#168	Q₈	1	1	yes	characteristic
35	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₈×D₅	D₄	1	1	yes	characteristic
37	C₃×C₈×D₅	Q₈	1	1	yes	characteristic
38	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
39	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
40	C₃×#672	C₂×C₄	1	2	yes	complemented
41	C₂×C₃×M₃₂	—	5	1	no	—
42	C₃×#427	—	5	1	no	—
43	C₂×C₃×#107	—	5	1	no	—
44	C₃×#367	—	5	2	no	—
45	C₃×#366	—	5	2	no	—
46	#4130	—	2	1	no	—
47	C₂×C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
48	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
49	#3994	—	2	1	no	—
50	#3994	C₂²×C₃	1	1	yes	characteristic
51	#3994	C₂²×C₃	1	2	yes	—
52	#3994	C₂²×C₃	1	1	yes	characteristic
53	#3993	—	2	1	no	—
54	C₅×M₃₂	C₃×C₄	1	1	yes	characteristic, complemented
55	#3944	C₃×C₄	1	1	yes	characteristic, complemented
56	#3930	C₃×C₄	1	2	yes	complemented
57	D₅×C₁₆	C₂²×C₃	1	2	yes	—
58	#2159	—	5	1	no	Sylow
59	C₂²×C₃×D₅	D₈	1	1	yes	characteristic
60	C₂×C₃×AGL(1,𝔽₅)	—	4	1	no	—
61	C₂×C₃×AGL(1,𝔽₅)	—	4	1	no	—
62	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
63	C₂×C₃×AGL(1,𝔽₅)	—	8	2	no	—
64	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₃×Dic₅	Q₁₆	1	1	yes	characteristic
66	C₃×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
67	C₃×C₄×D₅	D₈	1	1	yes	characteristic
68	C₃×C₄×D₅	Q₁₆	1	1	yes	characteristic
69	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
70	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
73	C₃×#168	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₃×C₈	—	5	1	no	—
75	C₃×#117	—	10	1	no	—
76	C₂×C₃×C₄⋊C₄	—	10	1	no	—
77	C₃×M₃₂	—	5	1	no	—
78	C₃×M₃₂	—	5	2	no	—
79	C₃×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
80	C₂×C₃×C₁₆	—	5	2	no	—
81	C₃×#107	—	5	1	no	—
82	C₃×#107	—	5	2	no	—
83	C₃×#107	—	5	1	no	—
84	C₃×#107	—	10	1	no	—
85	C₃×#106	—	10	1	no	—
86	C₂²×AGL(1,𝔽₅)	—	4	1	no	—
87	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
88	C₂²⋊AGL(1,𝔽₅)	—	4	1	no	—
89	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
90	C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
91	C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
92	C₄⋊AGL(1,𝔽₅)	—	4	1	no	—
93	C₄×AGL(1,𝔽₅)	—	4	1	no	—
94	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×#168	C₃×Q₈	1	1	yes	characteristic
96	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₈×D₅	C₃×D₄	1	1	yes	characteristic
98	C₈×D₅	C₃×Q₈	1	1	yes	characteristic
99	C₈×D₅	C₂³×C₃	1	1	yes	characteristic
100	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	complemented
101	#672	C₂×C₃×C₄	1	2	yes	complemented
102	C₂×M₃₂	—	5	1	no	—
103	#427	—	5	1	no	—
104	C₂×#107	—	5	1	no	—
105	#367	—	5	2	no	—
106	#366	—	5	2	no	—
107	C₂²×C₃×C₅	#107	1	1	yes	characteristic
108	C₂×C₃×D₅	—	2	1	no	—
109	C₂×C₃×D₅	—	2	1	no	—
110	C₂×C₃×D₅	#107	1	1	yes	characteristic
111	C₂×C₃×D₅	C₂×D₈	1	1	yes	characteristic
112	C₃×AGL(1,𝔽₅)	—	8	1	no	—
113	C₃×AGL(1,𝔽₅)	—	8	2	no	—
114	C₃×AGL(1,𝔽₅)	—	8	1	no	—
115	C₃×C₄×C₅	#107	1	1	yes	characteristic
116	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₃×Dic₅	#107	1	1	yes	characteristic
118	C₃×Dic₅	C₂×Q₁₆	1	1	yes	characteristic
119	C₂²×C₃×C₄	—	5	1	no	—
120	C₂²×C₃×C₄	—	20	1	no	—
121	C₂×C₃×C₈	—	5	1	no	—
122	C₂×C₃×C₈	—	5	1	no	—
123	C₂×C₃×C₈	—	5	1	no	—
124	C₂×C₃×C₈	—	5	1	no	—
125	C₂×C₃×C₈	—	5	1	no	—
126	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₃×C₄⋊C₄	—	10	1	no	—
128	C₃×C₄⋊C₄	—	10	1	no	—
129	C₃×C₄⋊C₄	—	10	2	no	—
130	C₃×C₄⋊C₄	—	20	1	no	—
131	C₃×C₂²⋊C₄	—	20	1	no	—
132	C₃×C₄²	—	20	1	no	—
133	C₃×C₁₆	—	5	2	no	—
134	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
135	C₂²×D₅	C₃×D₈	1	1	yes	characteristic
136	C₂×AGL(1,𝔽₅)	—	4	1	no	—
137	C₂×AGL(1,𝔽₅)	—	4	1	no	—
138	C₂×AGL(1,𝔽₅)	—	4	2	no	—
139	C₂×AGL(1,𝔽₅)	—	8	2	no	—
140	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
141	C₂×Dic₅	C₃×Q₁₆	1	1	yes	characteristic
142	C₄×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
143	C₄×D₅	C₃×D₈	1	1	yes	characteristic
144	C₄×D₅	C₃×Q₁₆	1	1	yes	characteristic
145	C₄×D₅	C₂×C₃×D₄	1	1	yes	characteristic
146	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
147	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
148	#168	C₃×C₄⋊C₄	1	1	yes	characteristic
149	#168	C₂×C₃×Q₈	1	1	yes	characteristic
150	C₂²×C₈	—	5	1	no	—
151	#117	—	10	1	no	—
152	C₂×C₄⋊C₄	—	10	1	no	—
153	M₃₂	—	5	1	no	—
154	M₃₂	—	5	2	no	—
155	M₃₂	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
156	C₂×C₁₆	—	5	2	no	—
157	#107	—	5	1	no	—
158	#107	—	5	2	no	—
159	#107	—	5	1	no	—
160	#107	—	10	1	no	—
161	#106	—	10	1	no	—
162	C₂×C₃×C₅	—	2	1	no	—
163	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
164	C₃×D₅	—	2	1	no	—
165	C₃×D₅	#509	1	1	yes	characteristic
166	C₃×D₅	#510	1	1	yes	characteristic
167	C₂³×C₃	—	5	1	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	5	1	no	—
170	C₂×C₃×C₄	—	5	1	no	—
171	C₂×C₃×C₄	—	5	1	no	—
172	C₂×C₃×C₄	—	5	1	no	—
173	C₂×C₃×C₄	—	20	1	no	—
174	C₂×C₃×C₄	—	20	1	no	—
175	C₂×C₃×C₄	—	20	2	no	—
176	C₂×C₃×C₄	—	40	2	no	—
177	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₃×C₈	—	5	1	no	—
179	C₃×C₈	—	5	1	no	—
180	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
181	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₂²×C₅	C₃×#107	1	1	yes	characteristic
183	C₂×D₅	—	2	1	no	—
184	C₂×D₅	—	2	1	no	—
185	C₂×D₅	C₃×#107	1	1	yes	characteristic
186	C₂×D₅	C₂×C₃×D₈	1	1	yes	characteristic
187	AGL(1,𝔽₅)	—	8	1	no	—
188	AGL(1,𝔽₅)	—	8	2	no	—
189	AGL(1,𝔽₅)	—	8	1	no	—
190	C₄×C₅	C₃×#107	1	1	yes	characteristic
191	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
192	Dic₅	C₃×#107	1	1	yes	characteristic
193	Dic₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
194	C₂²×C₄	—	5	1	no	—
195	C₂²×C₄	—	20	1	no	—
196	C₂×C₈	—	5	1	no	—
197	C₂×C₈	—	5	1	no	—
198	C₂×C₈	—	5	1	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	5	1	no	—
201	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₄⋊C₄	—	10	1	no	—
203	C₄⋊C₄	—	10	1	no	—
204	C₄⋊C₄	—	10	2	no	—
205	C₄⋊C₄	—	20	1	no	—
206	C₂²⋊C₄	—	20	1	no	—
207	C₄²	—	20	1	no	—
208	C₁₆	—	5	2	no	—
209	C₁₆	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
210	C₃×C₅	#2159	1	1	yes	characteristic, complemented, Hall
211	C₂²×C₃	—	5	1	no	—
212	C₂²×C₃	—	5	1	no	—
213	C₂²×C₃	—	10	1	no	—
214	C₂²×C₃	—	10	1	no	—
215	C₂²×C₃	#3994	1	1	yes	characteristic
216	C₃×C₄	—	5	1	no	—
217	C₃×C₄	—	5	1	no	—
218	C₃×C₄	—	40	1	no	—
219	C₃×C₄	—	40	1	no	—
220	C₃×C₄	—	40	2	no	—
221	C₃×C₄	#3994	1	1	yes	characteristic
222	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
225	D₅	—	2	1	no	—
226	D₅	C₃×#509	1	1	yes	characteristic
227	D₅	C₃×#510	1	1	yes	characteristic
228	C₂³	—	5	1	no	—
229	C₂×C₄	—	5	1	no	—
230	C₂×C₄	—	5	1	no	—
231	C₂×C₄	—	5	1	no	—
232	C₂×C₄	—	5	1	no	—
233	C₂×C₄	—	5	1	no	—
234	C₂×C₄	—	20	1	no	—
235	C₂×C₄	—	20	1	no	—
236	C₂×C₄	—	20	2	no	—
237	C₂×C₄	—	40	2	no	—
238	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₈	—	5	1	no	—
240	C₈	—	5	1	no	—
241	C₈	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
242	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	5	1	no	—
245	C₂×C₃	—	5	1	no	—
246	C₂×C₃	—	10	1	no	—
247	C₂×C₃	C₂×#3994	1	1	yes	characteristic
248	C₅	C₃×#2159	1	1	yes	characteristic, complemented, Sylow
249	C₂²	—	5	1	no	—
250	C₂²	—	5	1	no	—
251	C₂²	—	10	1	no	—
252	C₂²	—	10	1	no	—
253	C₂²	C₃×#3994	1	1	yes	characteristic
254	C₄	—	5	1	no	—
255	C₄	—	5	1	no	—
256	C₄	—	40	2	no	—
257	C₄	—	40	1	no	—
258	C₄	—	40	1	no	—
259	C₄	C₃×#3994	1	1	yes	characteristic
260	C₄	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
261	C₃	#10604118	1	1	yes	characteristic, direct product factor, Sylow
262	C₂	—	2	1	no	—
263	C₂	—	5	1	no	—
264	C₂	—	5	1	no	—
265	C₂	—	10	1	no	—
266	C₂	C₂×C₃×#3994	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×#3994
commutator	C₅×C₈	C₂²×C₃×C₄
Frattini	C₈	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×M₃₂	C₄
socle	C₂×C₃×C₅	C₂×#107
prime radical	C₂²×C₃×D₅	D₈
cosocle	C₈×D₅	C₂³×C₃
hypercenter	C₃×M₃₂	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2159
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×M₃₂

Complements

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

N	H ≅ G/N	type
4. C₃×#64850	C₂	semidirect
5. C₃×#64849	C₂	semidirect
6. #10604118	C₃	direct (N × H)
15. C₃×C₅×M₃₂	C₄	semidirect
16. C₃×#3944	C₄	semidirect
17. C₃×#3930	C₄	semidirect
23. #64850	C₂×C₃	semidirect
24. #64849	C₂×C₃	semidirect
39. C₃×C₅×C₁₆	C₂×C₄	semidirect
40. C₃×#672	C₂×C₄	semidirect
54. C₅×M₃₂	C₃×C₄	semidirect
55. #3944	C₃×C₄	semidirect
56. #3930	C₃×C₄	semidirect
79. C₃×M₃₂	AGL(1,𝔽₅)	semidirect
100. C₅×C₁₆	C₂×C₃×C₄	semidirect
101. #672	C₂×C₃×C₄	semidirect
134. C₃×C₁₆	C₂×AGL(1,𝔽₅)	semidirect
155. M₃₂	C₃×AGL(1,𝔽₅)	semidirect
209. C₁₆	C₂×C₃×AGL(1,𝔽₅)	semidirect
210. C₃×C₅	#2159	semidirect
248. C₅	C₃×#2159	semidirect
261. C₃	#10604118	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#65721	2
C₂×C₃×#3994	2
C₃×D₅×M₃₂	2
C₃×#64850	2
C₃×#64849	2
#10604118	3
C₃×#2159	5

Subgroups — C3×#10604118 #14314631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604118 #14314631