Subgroups — C₅×#70942 #14291529

Order 1920.

312 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#5020	C₂	1	2	yes	maximal
3	C₅×#4699	C₂	1	1	yes	maximal, characteristic
4	C₅×#4482	C₂	1	2	yes	maximal
5	C₅×#4458	C₂	1	1	yes	maximal, characteristic
6	C₅×#1670	—	3	1	no	maximal, Hall
7	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×Dic₁₂	—	2	1	no	—
9	C₂×C₅×Dic₁₂	—	2	1	no	—
10	C₅×#869	C₂²	1	2	yes	—
11	C₃×C₅×C₄⋊C₈	—	2	1	no	—
12	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	—	2	1	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#835	—	2	2	no	—
16	C₅×#808	—	2	2	no	—
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	#70942	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#494	—	3	1	no	—
20	C₅×#475	—	3	2	no	—
21	C₅×C₈×Q₈	D₃	1	1	yes	characteristic
22	C₅×#335	—	3	1	no	—
23	C₅×#332	—	3	2	no	—
24	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
25	C₂×C₅×Dic₆	—	2	2	no	—
26	C₂×C₃×C₅×C₈	—	2	1	no	—
27	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	1	no	—
30	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	—	2	1	no	—
32	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₅×C₄⋊Dic₃	—	2	2	no	—
34	C₅×C₄⋊Dic₃	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	—	2	2	no	—
36	C₅×Dic₁₂	—	4	2	no	—
37	C₅×Dic₁₂	—	4	2	no	—
38	C₃×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
39	#5020	C₂×C₅	1	2	yes	—
40	#4699	C₂×C₅	1	1	yes	characteristic
41	#4482	C₂×C₅	1	2	yes	—
42	#4458	C₂×C₅	1	1	yes	characteristic
43	C₂×C₅×Q₁₆	—	6	1	no	—
44	C₂×C₅×Q₁₆	—	6	1	no	—
45	C₅×C₄⋊Q₈	—	3	2	no	—
46	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
47	C₅×#106	—	6	2	no	—
48	C₅×C₄⋊C₈	—	2	1	no	—
49	C₅×C₄⋊C₈	—	3	2	no	—
50	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
51	C₅×#103	—	6	2	no	—
52	C₄×C₅×C₈	—	2	1	no	—
53	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
54	#1670	—	3	1	no	Sylow
55	C₃×C₅×Q₈	—	2	1	no	—
56	C₃×C₅×Q₈	Q₁₆	1	2	yes	complemented
57	C₂×C₃×C₄×C₅	—	2	1	no	—
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
62	C₂×C₅×Dic₃	—	4	2	no	—
63	C₅×Dic₆	—	4	2	no	—
64	C₅×Dic₆	—	4	2	no	—
65	C₃×C₅×C₈	—	2	1	no	—
66	C₃×C₅×C₈	—	4	1	no	—
67	C₃×C₅×C₈	SD₁₆	1	2	yes	—
68	C₅×C₃⋊C₈	—	4	2	no	—
69	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
70	C₂×Dic₁₂	—	2	1	no	—
71	C₂×Dic₁₂	—	2	1	no	—
72	#869	C₂²×C₅	1	2	yes	—
73	C₃×C₄⋊C₈	—	2	1	no	—
74	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
75	C₃×C₄×C₈	—	2	1	no	—
76	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
77	#835	—	2	2	no	—
78	#808	—	2	2	no	—
79	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
80	C₂×C₅×Q₈	—	6	2	no	—
81	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
82	C₅×Q₁₆	—	12	2	no	—
83	C₅×Q₁₆	—	12	2	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	6	2	no	—
86	C₂×C₅×C₈	#67	1	1	yes	characteristic
87	C₂×C₅×C₈	#67	1	1	yes	characteristic
88	C₅×C₄⋊C₄	—	2	1	no	—
89	C₅×C₄⋊C₄	—	6	2	no	—
90	C₅×C₄⋊C₄	—	6	2	no	—
91	C₅×C₄⋊C₄	D₁₂	1	1	yes	characteristic
92	C₄²×C₅	—	2	1	no	—
93	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
94	#494	—	3	1	no	—
95	#475	—	3	2	no	—
96	C₈×Q₈	C₅×D₃	1	1	yes	characteristic
97	#335	—	3	1	no	—
98	#332	—	3	2	no	—
99	C₂²×C₃×C₅	#121	1	1	yes	characteristic
100	C₃×C₄×C₅	—	2	2	no	—
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	4	1	no	—
103	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
104	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
105	C₃×C₄×C₅	#135	1	1	yes	characteristic
106	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₅×Dic₃	—	8	2	no	—
108	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
109	C₂×Dic₆	—	2	2	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
112	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	2	1	no	—
114	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
115	C₃×C₄²	—	2	1	no	—
116	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
117	C₄⋊Dic₃	—	2	2	no	—
118	C₄⋊Dic₃	—	2	2	no	—
119	C₂×C₃⋊C₈	—	2	2	no	—
120	Dic₁₂	—	4	2	no	—
121	Dic₁₂	—	4	2	no	—
122	C₅×Q₈	—	2	1	no	—
123	C₅×Q₈	—	12	2	no	—
124	C₅×Q₈	—	12	2	no	—
125	C₅×Q₈	Dic₁₂	1	2	yes	complemented
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	12	2	no	—
129	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
130	C₂×C₄×C₅	#235	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	4	1	no	—
134	C₅×C₈	—	12	2	no	—
135	C₅×C₈	#215	1	2	yes	—
136	C₂×Q₁₆	—	6	1	no	—
137	C₂×Q₁₆	—	6	1	no	—
138	C₄⋊Q₈	—	3	2	no	—
139	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
140	#106	—	6	2	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	3	2	no	—
143	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
144	#103	—	6	2	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
147	C₂×C₃×C₅	#460	1	1	yes	characteristic
148	C₂×C₃×C₅	#467	1	1	yes	characteristic
149	C₂×C₃×C₅	#473	1	1	yes	characteristic
150	C₃×Q₈	—	2	1	no	—
151	C₃×Q₈	C₅×Q₁₆	1	2	yes	complemented
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
157	C₂×Dic₃	—	4	2	no	—
158	Dic₆	—	4	2	no	—
159	Dic₆	—	4	2	no	—
160	C₃×C₈	—	2	1	no	—
161	C₃×C₈	—	4	1	no	—
162	C₃×C₈	C₅×SD₁₆	1	2	yes	—
163	C₃⋊C₈	—	4	2	no	—
164	C₂²×C₅	#930	1	1	yes	characteristic
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	4	1	no	—
168	C₄×C₅	—	24	2	no	—
169	C₄×C₅	#904	1	1	yes	characteristic
170	C₄×C₅	C₂×Dic₁₂	1	1	yes	characteristic
171	C₄×C₅	C₂×#215	1	1	yes	characteristic
172	C₄×C₅	#949	1	1	yes	characteristic
173	C₂×Q₈	—	6	2	no	—
174	C₂×Q₈	C₅×D₁₂	1	1	yes	characteristic
175	Q₁₆	—	12	2	no	—
176	Q₁₆	—	12	2	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	6	2	no	—
179	C₂×C₈	C₅×#67	1	1	yes	characteristic
180	C₂×C₈	C₅×#67	1	1	yes	characteristic
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	6	2	no	—
183	C₄⋊C₄	—	6	2	no	—
184	C₄⋊C₄	C₅×D₁₂	1	1	yes	characteristic
185	C₄²	—	2	1	no	—
186	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
187	C₃×C₅	#1670	1	1	yes	characteristic, complemented, Hall
188	C₂²×C₃	C₅×#121	1	1	yes	characteristic
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	4	1	no	—
192	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
193	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
194	C₃×C₄	C₅×#135	1	1	yes	characteristic
195	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
196	Dic₃	—	8	2	no	—
197	C₂×C₅	#5027	1	1	yes	characteristic
198	C₂×C₅	#5116	1	1	yes	characteristic
199	C₂×C₅	#5143	1	1	yes	characteristic
200	Q₈	—	2	1	no	—
201	Q₈	—	12	2	no	—
202	Q₈	—	12	2	no	—
203	Q₈	C₅×Dic₁₂	1	2	yes	complemented
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	12	2	no	—
207	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
208	C₂×C₄	C₅×#235	1	1	yes	characteristic
209	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	4	1	no	—
212	C₈	—	12	2	no	—
213	C₈	C₅×#215	1	2	yes	—
214	C₂×C₃	C₅×#460	1	1	yes	characteristic
215	C₂×C₃	C₅×#467	1	1	yes	characteristic
216	C₂×C₃	C₅×#473	1	1	yes	characteristic
217	C₅	#70942	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²	C₅×#930	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	24	2	no	—
223	C₄	C₅×#904	1	1	yes	characteristic
224	C₄	C₂×C₅×Dic₁₂	1	1	yes	characteristic
225	C₄	C₂×C₅×#215	1	1	yes	characteristic
226	C₄	C₅×#949	1	1	yes	characteristic
227	C₃	C₅×#1670	1	1	yes	characteristic, complemented, Sylow
228	C₂	C₅×#5027	1	1	yes	characteristic
229	C₂	C₅×#5116	1	1	yes	characteristic
230	C₂	C₅×#5143	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#930
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈×Q₈	C₂
socle, prime radical	C₂²×C₃×C₅	#121
hypercenter	C₅×C₈×Q₈	D₃
hypocenter	C₃	C₅×#1670
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×Q₈

Complements

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

N	H ≅ G/N	type
18. #70942	C₅	direct (N × H)
56. C₃×C₅×Q₈	Q₁₆	semidirect
125. C₅×Q₈	Dic₁₂	semidirect
151. C₃×Q₈	C₅×Q₁₆	semidirect
187. C₃×C₅	#1670	semidirect
203. Q₈	C₅×Dic₁₂	semidirect
217. C₅	#70942	direct (N × H)
227. C₃	C₅×#1670	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₈×Q₈	2
C₅×#5020	2
C₅×#4699	2
C₅×#4482	2
C₅×#4458	2
C₅×#1670	3
#70942	5

Subgroups — C5×#70942 #14291529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70942 #14291529