Subgroups — C₅×#71635 #14307086

Order 1920.

560 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	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic
4	C₅×#4524	C₂	1	4	yes	maximal
5	C₅×#1985	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	C₄	1	1	yes	characteristic
7	C₃×C₄×C₅×Q₈	—	2	2	no	—
8	C₃×C₄×C₅×Q₈	—	2	2	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
15	C₅×C₄⋊(C₃⋊C₈)	—	2	2	no	—
16	C₅×#803	—	2	2	no	—
17	#71635	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₄×C₅×Q₈	D₃	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₈	—	3	1	no	—
20	C₂×C₅×#97	—	3	1	no	—
21	C₅×#336	—	3	4	no	—
22	C₂×C₃×C₅×Q₈	—	2	2	no	—
23	C₂×C₃×C₅×Q₈	—	2	4	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₄²×C₅	—	2	2	no	—
32	C₃×C₄²×C₅	—	2	2	no	—
33	C₃×C₄²×C₅	D₄	1	2	yes	—
34	C₃×C₄²×C₅	D₄	1	2	yes	—
35	C₂×C₅×C₃⋊C₈	—	2	2	no	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₂×C₅×C₃⋊C₈	—	2	2	no	—
38	C₂×C₅×C₃⋊C₈	—	2	4	no	—
39	C₂×C₅×C₃⋊C₈	Q₈	1	4	yes	complemented
40	C₂×C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic
41	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic
42	C₂×#803	C₂×C₅	1	1	yes	characteristic
43	#4524	C₂×C₅	1	4	yes	—
44	C₂²×C₅×Q₈	Dic₃	1	1	yes	characteristic
45	C₂²×C₅×C₈	—	3	2	no	—
46	C₂²×C₅×C₈	—	3	2	no	—
47	C₄×C₅×Q₈	—	2	2	no	—
48	C₄×C₅×Q₈	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	Dic₃	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₄	Dic₃	1	2	yes	—
51	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
52	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
53	C₅×C₄⋊C₈	—	6	2	no	—
54	C₅×#97	—	6	2	no	—
55	#1985	—	3	1	no	Sylow
56	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₅×Q₈	—	2	8	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
70	C₂×C₃×C₄×C₅	#41	1	2	yes	—
71	C₅×C₃⋊C₈	—	2	4	no	—
72	C₅×C₃⋊C₈	—	4	2	no	—
73	C₂²×C₃×Q₈	C₄×C₅	1	1	yes	characteristic
74	C₃×C₄×Q₈	—	2	2	no	—
75	C₃×C₄×Q₈	—	2	2	no	—
76	C₂×C₃×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
77	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
78	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
79	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
80	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
81	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
82	C₄⋊(C₃⋊C₈)	—	2	2	no	—
83	#803	—	2	2	no	—
84	C₂×C₅×Q₈	—	2	4	no	—
85	C₂×C₅×Q₈	—	2	2	no	—
86	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
87	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
88	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₅×C₈	—	3	4	no	—
92	C₂×C₅×C₈	—	6	2	no	—
93	C₂×C₅×C₈	—	6	2	no	—
94	C₂×C₅×C₈	—	6	4	no	—
95	C₂×C₅×C₈	—	6	2	no	—
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₄²×C₅	—	2	2	no	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	#67	1	2	yes	—
101	C₄²×C₅	#67	1	2	yes	—
102	C₂×C₄×Q₈	C₅×D₃	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	3	1	no	—
104	C₂×#97	—	3	1	no	—
105	#336	—	3	4	no	—
106	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₄×Q₈	1	2	yes	—
108	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
109	C₂²×C₃×C₅	#131	1	2	yes	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	2	4	no	—
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×C₄×C₅	—	4	2	no	—
114	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
115	C₃×C₄×C₅	#124	1	1	yes	characteristic
116	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₃×Q₈	—	2	2	no	—
118	C₂×C₃×Q₈	—	2	4	no	—
119	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
120	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
123	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	—	2	4	no	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄²	—	2	2	no	—
127	C₃×C₄²	—	2	2	no	—
128	C₃×C₄²	C₅×D₄	1	2	yes	—
129	C₃×C₄²	C₅×D₄	1	2	yes	—
130	C₂×C₃⋊C₈	—	2	2	no	—
131	C₂×C₃⋊C₈	—	2	2	no	—
132	C₂×C₃⋊C₈	—	2	2	no	—
133	C₂×C₃⋊C₈	—	2	4	no	—
134	C₂×C₃⋊C₈	C₅×Q₈	1	4	yes	complemented
135	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
136	C₅×Q₈	—	2	8	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	#208	1	4	yes	—
144	C₂×C₄×C₅	#217	1	2	yes	—
145	C₂×C₄×C₅	#217	1	2	yes	—
146	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
147	C₂×C₄×C₅	#239	1	2	yes	—
148	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
150	C₅×C₈	—	6	4	no	—
151	C₅×C₈	—	12	2	no	—
152	C₂²×Q₈	C₅×Dic₃	1	1	yes	characteristic
153	C₂²×C₈	—	3	2	no	—
154	C₂²×C₈	—	3	2	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₄×Q₈	—	2	2	no	—
157	C₂×C₄⋊C₄	C₅×Dic₃	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₅×Dic₃	1	2	yes	—
159	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
160	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
161	C₄⋊C₈	—	6	2	no	—
162	#97	—	6	2	no	—
163	C₂×C₃×C₅	#391	1	1	yes	characteristic
164	C₂×C₃×C₅	#407	1	1	yes	characteristic
165	C₂×C₃×C₅	#408	1	1	yes	characteristic
166	C₂×C₃×C₅	#446	1	4	yes	—
167	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
168	C₃×Q₈	—	2	8	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
176	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
177	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
178	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
181	C₂×C₃×C₄	C₅×#41	1	2	yes	—
182	C₃⋊C₈	—	2	4	no	—
183	C₃⋊C₈	—	4	2	no	—
184	C₂²×C₅	C₂×#208	1	2	yes	—
185	C₂²×C₅	Q₈×Dic₃	1	2	yes	—
186	C₂²×C₅	#948	1	2	yes	—
187	C₂²×C₅	C₂×#217	1	1	yes	characteristic
188	C₄×C₅	—	2	4	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	4	2	no	—
192	C₄×C₅	#944	1	1	yes	characteristic
193	C₄×C₅	#946	1	2	yes	—
194	C₄×C₅	#947	1	1	yes	characteristic
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
198	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
199	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
202	C₂×C₈	—	3	4	no	—
203	C₂×C₈	—	6	2	no	—
204	C₂×C₈	—	6	2	no	—
205	C₂×C₈	—	6	4	no	—
206	C₂×C₈	—	6	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₅×#67	1	2	yes	—
212	C₄²	C₅×#67	1	2	yes	—
213	C₃×C₅	#1985	1	1	yes	characteristic, complemented, Hall
214	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×C₃	C₄×C₅×Q₈	1	2	yes	—
216	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
217	C₂²×C₃	C₅×#131	1	2	yes	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	4	2	no	—
222	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
223	C₃×C₄	C₅×#124	1	1	yes	characteristic
224	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
225	C₂×C₅	#5024	1	4	yes	—
226	C₂×C₅	#5207	1	1	yes	characteristic
227	C₂×C₅	#5229	1	1	yes	characteristic
228	C₂×C₅	#5234	1	1	yes	characteristic
229	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
230	Q₈	—	2	8	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	C₅×#208	1	4	yes	—
238	C₂×C₄	C₅×#217	1	2	yes	—
239	C₂×C₄	C₅×#217	1	2	yes	—
240	C₂×C₄	C₅×D₃×Q₈	1	2	yes	—
241	C₂×C₄	C₅×#239	1	2	yes	—
242	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
243	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
244	C₈	—	6	4	no	—
245	C₈	—	12	2	no	—
246	C₂×C₃	C₅×#391	1	1	yes	characteristic
247	C₂×C₃	C₅×#407	1	1	yes	characteristic
248	C₂×C₃	C₅×#408	1	1	yes	characteristic
249	C₂×C₃	C₅×#446	1	4	yes	—
250	C₅	#71635	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	C₂×C₅×#208	1	2	yes	—
252	C₂²	C₅×Q₈×Dic₃	1	2	yes	—
253	C₂²	C₅×#948	1	2	yes	—
254	C₂²	C₂×C₅×#217	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	C₅×#944	1	1	yes	characteristic
260	C₄	C₅×#946	1	2	yes	—
261	C₄	C₅×#947	1	1	yes	characteristic
262	C₃	C₅×#1985	1	1	yes	characteristic, complemented, Sylow
263	C₂	C₅×#5024	1	4	yes	—
264	C₂	C₅×#5207	1	1	yes	characteristic
265	C₂	C₅×#5229	1	1	yes	characteristic
266	C₂	C₅×#5234	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₄×C₅×Q₈	D₃
hypocenter	C₃	C₅×#1985
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
17. #71635	C₅	direct (N × H)
39. C₂×C₅×C₃⋊C₈	Q₈	semidirect
134. C₂×C₃⋊C₈	C₅×Q₈	semidirect
213. C₃×C₅	#1985	semidirect
250. C₅	#71635	direct (N × H)
262. C₃	C₅×#1985	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×Q₈	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₂×C₅×#803	2
C₅×#4524	2
C₅×#1985	3
#71635	5

Subgroups — C5×#71635 #14307086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71635 #14307086