Subgroups — C₅×#69206 #14315343

Order 1920.

324 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	C₅×Dic₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#828	C₂	1	1	yes	maximal, characteristic
3	C₅×#4872	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#358	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4509	C₂	1	1	yes	maximal, characteristic
6	C₅×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4490	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×SD₃₂	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₅×Q₈×Dic₃	—	2	1	no	—
11	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
12	C₅×#893	—	2	1	no	—
13	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₅×#107	C₂²	1	1	yes	characteristic
15	C₅×#835	—	2	1	no	—
16	C₅×#828	C₄	1	4	yes	complemented
17	C₅×#818	C₂²	1	1	yes	characteristic
18	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₅×#809	—	2	1	no	—
21	#69206	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×SD₃₂	—	3	1	no	—
23	C₄×C₅×Q₁₆	—	3	1	no	—
24	C₄×C₅×D₈	—	3	1	no	—
25	C₅×#367	—	3	1	no	—
26	C₅×#358	D₃	1	1	yes	characteristic, complemented
27	C₅×#357	—	3	1	no	—
28	C₄×C₅×C₁₆	—	3	1	no	—
29	C₂×C₃×C₅×Q₈	—	2	1	no	—
30	C₂×C₅×D₁₂	—	2	1	no	—
31	C₂×C₄×C₅×D₃	—	4	1	no	—
32	C₃×C₅×Q₁₆	—	2	1	no	—
33	C₃×C₅×Q₁₆	C₂×C₄	1	2	yes	complemented
34	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₅×(C₂×D₃)⋊C₄	—	4	1	no	—
37	C₅×C₄⋊Dic₃	—	2	1	no	—
38	C₅×C₄⋊Dic₃	—	4	1	no	—
39	C₄×C₅×Dic₃	—	4	1	no	—
40	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
41	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
42	C₅×D₂₄	—	2	1	no	—
43	C₅×D₂₄	C₂×C₄	1	2	yes	—
44	C₃×C₅×C₁₆	—	2	1	no	—
45	C₅×C₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
46	Dic₃×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂×#828	C₂×C₅	1	1	yes	characteristic
48	#4872	C₂×C₅	1	1	yes	characteristic
49	C₃×#358	C₂×C₅	1	1	yes	characteristic, complemented
50	#4509	C₂×C₅	1	1	yes	characteristic
51	Dic₃×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
52	#4490	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂×C₅×Q₁₆	C₂×D₃	1	1	yes	characteristic
54	C₂×C₅×D₈	—	3	1	no	—
55	C₄×C₅×Q₈	—	6	1	no	—
56	C₄×C₅×D₄	—	6	1	no	—
57	C₅×SD₃₂	—	3	4	no	—
58	C₂×C₅×C₁₆	—	3	1	no	—
59	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
60	C₅×#107	—	3	1	no	—
61	C₅×#107	C₂×D₃	1	1	yes	characteristic
62	C₅×#103	—	6	1	no	—
63	C₅×#102	—	6	1	no	—
64	C₄×C₅×C₈	—	3	1	no	—
65	C₄×SD₃₂	—	3	1	no	Sylow
66	C₂²×C₅×D₃	—	4	1	no	—
67	C₃×C₅×Q₈	—	2	2	no	—
68	C₃×C₅×Q₈	—	4	1	no	—
69	C₂×C₃×C₄×C₅	—	4	1	no	—
70	C₂×C₃×C₄×C₅	—	4	1	no	—
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₅×Dic₃	—	4	1	no	—
73	C₂×C₅×Dic₃	D₈	1	1	yes	characteristic
74	C₂×C₅×Dic₃	D₈	1	1	yes	characteristic
75	C₅×D₁₂	—	2	2	no	—
76	C₅×D₁₂	—	4	1	no	—
77	C₄×C₅×D₃	—	8	2	no	—
78	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₅×C₈	#41	1	1	yes	characteristic
80	C₅×C₃⋊C₈	—	2	1	no	—
81	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
82	Q₈×Dic₃	—	2	1	no	—
83	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
84	#893	—	2	1	no	—
85	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
86	C₃×#107	C₂²×C₅	1	1	yes	characteristic
87	#835	—	2	1	no	—
88	#828	C₄×C₅	1	4	yes	complemented
89	#818	C₂²×C₅	1	1	yes	characteristic
90	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
92	#809	—	2	1	no	—
93	C₂×C₅×Q₈	—	2	1	no	—
94	C₂×C₅×D₄	—	6	1	no	—
95	C₂²×C₄×C₅	—	12	1	no	—
96	C₅×Q₁₆	—	2	1	no	—
97	C₅×Q₁₆	C₄×D₃	1	2	yes	complemented
98	C₅×D₈	—	3	2	no	—
99	C₅×D₈	—	6	1	no	—
100	C₂×C₅×C₈	—	3	1	no	—
101	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₄⋊C₄	—	6	1	no	—
104	C₅×C₄⋊C₄	—	12	1	no	—
105	C₅×C₂²⋊C₄	—	12	1	no	—
106	C₄²×C₅	—	3	1	no	—
107	C₄²×C₅	—	12	1	no	—
108	C₅×C₁₆	—	2	1	no	—
109	C₅×C₁₆	—	3	2	no	—
110	C₂×SD₃₂	—	3	1	no	—
111	C₄×Q₁₆	—	3	1	no	—
112	C₄×D₈	—	3	1	no	—
113	#367	—	3	1	no	—
114	#358	C₅×D₃	1	1	yes	characteristic, complemented
115	#357	—	3	1	no	—
116	C₄×C₁₆	—	3	1	no	—
117	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
118	C₂×C₅×D₃	—	4	2	no	—
119	C₂×C₅×D₃	—	8	1	no	—
120	C₂×C₅×D₃	—	8	1	no	—
121	C₃×C₄×C₅	—	4	2	no	—
122	C₃×C₄×C₅	—	8	1	no	—
123	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
124	C₃×C₄×C₅	#135	1	1	yes	characteristic
125	C₅×Dic₃	—	2	1	no	—
126	C₅×Dic₃	—	8	1	no	—
127	C₅×Dic₃	SD₃₂	1	2	yes	complemented
128	C₂×C₃×Q₈	—	2	1	no	—
129	C₂×D₁₂	—	2	1	no	—
130	C₂×C₄×D₃	—	4	1	no	—
131	C₃×Q₁₆	—	2	1	no	—
132	C₃×Q₁₆	C₂×C₄×C₅	1	2	yes	complemented
133	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	2	1	no	—
135	(C₂×D₃)⋊C₄	—	4	1	no	—
136	C₄⋊Dic₃	—	2	1	no	—
137	C₄⋊Dic₃	—	4	1	no	—
138	C₄×Dic₃	—	4	1	no	—
139	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
140	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
141	D₂₄	—	2	1	no	—
142	D₂₄	C₂×C₄×C₅	1	2	yes	—
143	C₃×C₁₆	—	2	1	no	—
144	C₃⋊C₁₆	C₂×C₄×C₅	1	2	yes	complemented
145	C₂³×C₅	—	12	1	no	—
146	C₅×Q₈	—	2	2	no	—
147	C₅×Q₈	—	4	1	no	—
148	C₅×D₄	—	6	2	no	—
149	C₅×D₄	—	12	1	no	—
150	C₂×C₄×C₅	—	3	1	no	—
151	C₂×C₄×C₅	—	3	1	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	—	4	1	no	—
154	C₂×C₄×C₅	—	12	1	no	—
155	C₂×C₄×C₅	—	24	2	no	—
156	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
157	C₅×C₈	—	6	1	no	—
158	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
159	C₅×C₈	#237	1	1	yes	characteristic
160	C₂×Q₁₆	C₂×C₅×D₃	1	1	yes	characteristic
161	C₂×D₈	—	3	1	no	—
162	C₄×Q₈	—	6	1	no	—
163	C₄×D₄	—	6	1	no	—
164	SD₃₂	—	3	4	no	—
165	C₂×C₁₆	—	3	1	no	—
166	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
167	#107	—	3	1	no	—
168	#107	C₂×C₅×D₃	1	1	yes	characteristic
169	#103	—	6	1	no	—
170	#102	—	6	1	no	—
171	C₄×C₈	—	3	1	no	—
172	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
173	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
174	C₂×C₃×C₅	#508	1	1	yes	characteristic
175	C₅×D₃	—	8	2	no	—
176	C₂²×D₃	—	4	1	no	—
177	C₃×Q₈	—	2	2	no	—
178	C₃×Q₈	—	4	1	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	—	4	1	no	—
181	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
182	C₂×Dic₃	—	4	1	no	—
183	C₂×Dic₃	C₅×D₈	1	1	yes	characteristic
184	C₂×Dic₃	C₅×D₈	1	1	yes	characteristic
185	D₁₂	—	2	2	no	—
186	D₁₂	—	4	1	no	—
187	C₄×D₃	—	8	2	no	—
188	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
189	C₃×C₈	C₅×#41	1	1	yes	characteristic
190	C₃⋊C₈	—	2	1	no	—
191	C₂²×C₅	—	12	2	no	—
192	C₂²×C₅	—	24	1	no	—
193	C₂²×C₅	—	24	1	no	—
194	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
195	C₄×C₅	—	3	2	no	—
196	C₄×C₅	—	4	2	no	—
197	C₄×C₅	—	6	1	no	—
198	C₄×C₅	—	8	1	no	—
199	C₄×C₅	—	24	1	no	—
200	C₄×C₅	#67⋊C₄	1	1	yes	characteristic
201	C₄×C₅	#916	1	1	yes	characteristic
202	C₂×Q₈	—	2	1	no	—
203	C₂×D₄	—	6	1	no	—
204	C₂²×C₄	—	12	1	no	—
205	Q₁₆	—	2	1	no	—
206	Q₁₆	C₄×C₅×D₃	1	2	yes	complemented
207	D₈	—	3	2	no	—
208	D₈	—	6	1	no	—
209	C₂×C₈	—	3	1	no	—
210	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	6	1	no	—
213	C₄⋊C₄	—	12	1	no	—
214	C₂²⋊C₄	—	12	1	no	—
215	C₄²	—	3	1	no	—
216	C₄²	—	12	1	no	—
217	C₁₆	—	2	1	no	—
218	C₁₆	—	3	2	no	—
219	C₃×C₅	C₄×SD₃₂	1	1	yes	characteristic, complemented, Hall
220	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
221	C₂×D₃	—	4	2	no	—
222	C₂×D₃	—	8	1	no	—
223	C₂×D₃	—	8	1	no	—
224	C₃×C₄	—	4	2	no	—
225	C₃×C₄	—	8	1	no	—
226	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
227	C₃×C₄	C₅×#135	1	1	yes	characteristic
228	Dic₃	—	2	1	no	—
229	Dic₃	—	8	1	no	—
230	Dic₃	C₅×SD₃₂	1	2	yes	complemented
231	C₂×C₅	—	24	2	no	—
232	C₂×C₅	#4756	1	1	yes	characteristic
233	C₂×C₅	D₃×SD₃₂	1	1	yes	characteristic
234	C₂×C₅	#4919	1	1	yes	characteristic
235	C₂³	—	12	1	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	4	1	no	—
238	D₄	—	6	2	no	—
239	D₄	—	12	1	no	—
240	C₂×C₄	—	3	1	no	—
241	C₂×C₄	—	3	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	12	1	no	—
245	C₂×C₄	—	24	2	no	—
246	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
247	C₈	—	6	1	no	—
248	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
249	C₈	C₅×#237	1	1	yes	characteristic
250	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
251	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
252	C₂×C₃	C₅×#508	1	1	yes	characteristic
253	D₃	—	8	2	no	—
254	C₅	#69206	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²	—	12	2	no	—
256	C₂²	—	24	1	no	—
257	C₂²	—	24	1	no	—
258	C₂²	C₅×D₃×D₈	1	1	yes	characteristic
259	C₄	—	3	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	6	1	no	—
262	C₄	—	8	1	no	—
263	C₄	—	24	1	no	—
264	C₄	C₅×#67⋊C₄	1	1	yes	characteristic
265	C₄	C₅×#916	1	1	yes	characteristic
266	C₃	C₄×C₅×SD₃₂	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	24	2	no	—
268	C₂	C₅×#4756	1	1	yes	characteristic
269	C₂	C₅×D₃×SD₃₂	1	1	yes	characteristic
270	C₂	C₅×#4919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×D₈
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#358	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₅×D₂₄	C₂²
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#358	D₃
hypocenter	C₃	C₄×C₅×SD₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#358

Complements

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

N	H ≅ G/N	type
1. C₅×Dic₃×Q₁₆	C₂	semidirect
4. C₃×C₅×#358	C₂	semidirect
6. C₅×Dic₃×C₁₆	C₂	semidirect
7. C₅×#4490	C₂	semidirect
16. C₅×#828	C₄	semidirect
21. #69206	C₅	direct (N × H)
26. C₅×#358	D₃	semidirect
33. C₃×C₅×Q₁₆	C₂×C₄	semidirect
45. C₅×C₃⋊C₁₆	C₂×C₄	semidirect
46. Dic₃×Q₁₆	C₂×C₅	semidirect
49. C₃×#358	C₂×C₅	semidirect
51. Dic₃×C₁₆	C₂×C₅	semidirect
52. #4490	C₂×C₅	semidirect
88. #828	C₄×C₅	semidirect
97. C₅×Q₁₆	C₄×D₃	semidirect
114. #358	C₅×D₃	semidirect
127. C₅×Dic₃	SD₃₂	semidirect
132. C₃×Q₁₆	C₂×C₄×C₅	semidirect
144. C₃⋊C₁₆	C₂×C₄×C₅	semidirect
206. Q₁₆	C₄×C₅×D₃	semidirect
219. C₃×C₅	C₄×SD₃₂	semidirect
230. Dic₃	C₅×SD₃₂	semidirect
254. C₅	#69206	direct (N × H)
266. C₃	C₄×C₅×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₅×Dic₃×Q₁₆	2
C₂×C₅×#828	2
C₅×#4872	2
C₃×C₅×#358	2
C₅×#4509	2
C₅×Dic₃×C₁₆	2
C₅×#4490	2
C₄×C₅×SD₃₂	3
#69206	5

Subgroups — C5×#69206 #14315343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69206 #14315343