Subgroups — C₅×#69779 #14296453

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 260 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×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#819	C₂	1	1	yes	maximal, characteristic
3	C₅×#5106	C₂	1	1	yes	maximal, characteristic
4	C₅×#5101	C₂	1	2	yes	maximal
5	C₅×#4523	C₂	1	2	yes	maximal
6	C₅×#1772	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₅×#926	C₂²	1	2	yes	—
11	C₂×C₅×#208	C₂²	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₄	1	4	yes	complemented
13	C₅×#819	C₄	1	4	yes	complemented
14	C₅×#818	C₄	1	2	yes	complemented
15	C₅×#817	C₄	1	2	yes	complemented
16	C₅×#815	—	2	2	no	—
17	C₅×C₈×Dic₃	—	2	2	no	—
18	#69779	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×#108	—	3	1	no	—
20	C₅×#427	—	3	1	no	—
21	C₅×#405	—	3	2	no	—
22	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
23	C₅×#339	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	—	2	1	no	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
34	C₃×C₄²×C₅	—	2	1	no	—
35	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₅×#217	—	2	2	no	—
37	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
38	C₄×C₅×Dic₃	—	2	2	no	—
39	C₅×#208	—	2	2	no	—
40	C₅×#208	C₂×C₄	1	4	yes	—
41	C₂×C₅×C₃⋊C₈	—	2	2	no	—
42	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
43	C₂×#819	C₂×C₅	1	1	yes	characteristic
44	#5106	C₂×C₅	1	1	yes	characteristic
45	#5101	C₂×C₅	1	2	yes	—
46	#4523	C₂×C₅	1	2	yes	—
47	C₂×C₅×M₁₆	—	3	2	no	—
48	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
50	C₅×#117	—	3	2	no	—
51	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
52	C₅×#108	—	3	4	no	—
53	C₅×#107	—	3	2	no	—
54	C₅×#106	—	3	2	no	—
55	C₅×#97	—	6	2	no	—
56	C₄×C₅×C₈	—	6	2	no	—
57	C₄×C₅×C₈	Dic₃	1	4	yes	complemented
58	#1772	—	3	1	no	Sylow
59	C₂³×C₃×C₅	#41	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	—	2	4	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
68	C₂×C₅×Dic₃	—	2	4	no	—
69	C₃×C₅×C₈	C₄²	1	2	yes	—
70	C₃×C₅×C₈	C₄²	1	2	yes	—
71	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
72	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
73	C₅×C₃⋊C₈	—	2	4	no	—
74	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
77	#926	C₂²×C₅	1	2	yes	—
78	C₂×#208	C₂²×C₅	1	2	yes	—
79	C₃×C₄×C₈	C₄×C₅	1	4	yes	complemented
80	#819	C₄×C₅	1	4	yes	complemented
81	#818	C₄×C₅	1	2	yes	complemented
82	#817	C₄×C₅	1	2	yes	complemented
83	#815	—	2	2	no	—
84	C₈×Dic₃	—	2	2	no	—
85	C₂²×C₄×C₅	—	2	1	no	—
86	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₅×M₁₆	—	3	4	no	—
88	C₅×M₁₆	—	6	2	no	—
89	C₂×C₅×C₈	—	6	2	no	—
90	C₂×C₅×C₈	Dic₆	1	2	yes	—
91	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
92	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
93	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
94	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
95	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
97	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
98	C₅×C₄⋊C₄	—	3	4	no	—
99	C₅×C₂²⋊C₄	—	6	2	no	—
100	C₄²×C₅	—	2	1	no	—
101	C₄²×C₅	—	6	2	no	—
102	C₄²×C₅	C₂×Dic₃	1	2	yes	—
103	C₂×#108	—	3	1	no	—
104	#427	—	3	1	no	—
105	#405	—	3	2	no	—
106	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
107	#339	—	3	2	no	—
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	#117	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
111	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
112	C₃×C₄×C₅	—	2	4	no	—
113	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
114	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
116	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
117	C₅×Dic₃	—	4	4	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
123	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
124	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
125	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
126	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
127	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
130	#217	—	2	2	no	—
131	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
132	C₄×Dic₃	—	2	2	no	—
133	#208	—	2	2	no	—
134	#208	C₂×C₄×C₅	1	4	yes	—
135	C₂×C₃⋊C₈	—	2	2	no	—
136	C₂³×C₅	#235	1	1	yes	characteristic
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	6	4	no	—
140	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
144	C₂×C₄×C₅	#235	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
146	C₅×C₈	—	6	4	no	—
147	C₅×C₈	C₄×Dic₃	1	2	yes	—
148	C₅×C₈	C₄×Dic₃	1	2	yes	—
149	C₅×C₈	C₄⋊Dic₃	1	2	yes	—
150	C₅×C₈	C₄⋊Dic₃	1	2	yes	—
151	C₂×M₁₆	—	3	2	no	—
152	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
153	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
154	#117	—	3	2	no	—
155	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
156	#108	—	3	4	no	—
157	#107	—	3	2	no	—
158	#106	—	3	2	no	—
159	#97	—	6	2	no	—
160	C₄×C₈	—	6	2	no	—
161	C₄×C₈	C₅×Dic₃	1	4	yes	complemented
162	C₂×C₃×C₅	—	2	1	no	—
163	C₂×C₃×C₅	—	2	1	no	—
164	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₃×C₅	#443	1	2	yes	—
166	C₂³×C₃	C₅×#41	1	1	yes	characteristic
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
174	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
175	C₂×Dic₃	—	2	4	no	—
176	C₃×C₈	C₄²×C₅	1	2	yes	—
177	C₃×C₈	C₄²×C₅	1	2	yes	—
178	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
179	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
180	C₃⋊C₈	—	2	4	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	C₄×Dic₆	1	1	yes	characteristic
183	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
184	C₂²×C₅	#926	1	1	yes	characteristic
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	—	12	4	no	—
187	C₄×C₅	C₄×Dic₆	1	1	yes	characteristic
188	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
189	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
190	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
193	M₁₆	—	3	4	no	—
194	M₁₆	—	6	2	no	—
195	C₂×C₈	—	6	2	no	—
196	C₂×C₈	C₅×Dic₆	1	2	yes	—
197	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
198	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
199	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
200	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
201	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
202	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
203	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
204	C₄⋊C₄	—	3	4	no	—
205	C₂²⋊C₄	—	6	2	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	6	2	no	—
208	C₄²	C₂×C₅×Dic₃	1	2	yes	—
209	C₃×C₅	#1772	1	1	yes	characteristic, complemented, Hall
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	C₅×#117	1	1	yes	characteristic
212	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
213	C₂²×C₃	C₄×C₅×Q₈	1	1	yes	characteristic
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
216	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
217	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
218	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
219	Dic₃	—	4	4	no	—
220	C₂×C₅	—	2	1	no	—
221	C₂×C₅	—	2	1	no	—
222	C₂×C₅	#4700	1	2	yes	—
223	C₂×C₅	C₄×C₄⋊Dic₃	1	1	yes	characteristic
224	C₂³	C₅×#235	1	1	yes	characteristic
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	6	4	no	—
228	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
231	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
232	C₂×C₄	C₅×#235	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
234	C₈	—	6	4	no	—
235	C₈	C₄×C₅×Dic₃	1	2	yes	—
236	C₈	C₄×C₅×Dic₃	1	2	yes	—
237	C₈	C₅×C₄⋊Dic₃	1	2	yes	—
238	C₈	C₅×C₄⋊Dic₃	1	2	yes	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₃	C₅×#443	1	2	yes	—
243	C₅	#69779	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	—	2	2	no	—
245	C₂²	C₄×C₅×Dic₆	1	1	yes	characteristic
246	C₂²	C₄×C₅×D₁₂	1	1	yes	characteristic
247	C₂²	C₅×#926	1	1	yes	characteristic
248	C₄	—	2	4	no	—
249	C₄	—	12	4	no	—
250	C₄	C₄×C₅×Dic₆	1	1	yes	characteristic
251	C₄	C₄×C₅×D₁₂	1	1	yes	characteristic
252	C₄	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
253	C₄	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
254	C₃	C₅×#1772	1	1	yes	characteristic, complemented, Sylow
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	C₅×#4700	1	2	yes	—
258	C₂	C₄×C₅×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	D₁₂
commutator	C₃×C₄	C₂×C₄²×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×#1772
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
12. C₃×C₄×C₅×C₈	C₄	semidirect
13. C₅×#819	C₄	semidirect
14. C₅×#818	C₄	semidirect
15. C₅×#817	C₄	semidirect
18. #69779	C₅	direct (N × H)
57. C₄×C₅×C₈	Dic₃	semidirect
79. C₃×C₄×C₈	C₄×C₅	semidirect
80. #819	C₄×C₅	semidirect
81. #818	C₄×C₅	semidirect
82. #817	C₄×C₅	semidirect
161. C₄×C₈	C₅×Dic₃	semidirect
209. C₃×C₅	#1772	semidirect
243. C₅	#69779	direct (N × H)
254. C₃	C₅×#1772	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×C₈	2
C₂×C₅×#819	2
C₅×#5106	2
C₅×#5101	2
C₅×#4523	2
C₅×#1772	3
#69779	5

Subgroups — C5×#69779 #14296453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69779 #14296453