Subgroups — C₅×#71788 #14307377

Order 1920.

552 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×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#4936	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic
4	C₅×#4529	C₂	1	4	yes	maximal
5	C₅×#1993	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
8	C₃×C₅×C₄⋊Q₈	C₄	1	4	yes	complemented
9	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₅×C₃⋊#95	—	2	1	no	—
15	C₅×C₃⋊#95	—	2	1	no	—
16	C₅×#803	—	2	2	no	—
17	#71788	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic
19	C₂×C₅×#97	—	3	1	no	—
20	C₅×#382	—	3	1	no	—
21	C₅×#340	—	3	4	no	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	4	4	no	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₅×Dic₃	—	2	2	no	—
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₃×C₄²×C₅	D₄	1	2	yes	—
34	C₅×C₄⋊Dic₃	—	2	4	no	—
35	C₂×C₅×C₃⋊C₈	—	2	2	no	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₂×C₅×C₃⋊C₈	Q₈	1	4	yes	complemented
38	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
39	#4936	C₂×C₅	1	1	yes	characteristic
40	C₂×#803	C₂×C₅	1	1	yes	characteristic
41	#4529	C₂×C₅	1	4	yes	—
42	C₂²×C₅×Q₈	—	2	1	no	—
43	C₂²×C₅×C₈	—	3	2	no	—
44	C₅×C₄⋊Q₈	—	2	2	no	—
45	C₅×C₄⋊Q₈	Dic₃	1	4	yes	complemented
46	C₂×C₅×C₄⋊C₄	—	2	1	no	—
47	C₂×C₅×C₄⋊C₄	—	3	2	no	—
48	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
49	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
50	C₅×#97	—	6	2	no	—
51	C₅×#95	—	6	1	no	—
52	C₅×#95	—	6	1	no	—
53	#1993	—	3	1	no	Sylow
54	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
55	C₃×C₅×Q₈	—	4	8	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
65	C₂×C₃×C₄×C₅	#41	1	2	yes	—
66	C₂×C₅×Dic₃	—	4	2	no	—
67	C₂×C₅×Dic₃	—	4	2	no	—
68	C₂×C₅×Dic₃	—	4	2	no	—
69	C₅×C₃⋊C₈	—	2	4	no	—
70	C₂²×C₃×Q₈	—	2	1	no	—
71	C₃×C₄⋊Q₈	—	2	2	no	—
72	C₃×C₄⋊Q₈	C₄×C₅	1	4	yes	complemented
73	C₂×C₃×C₄⋊C₄	—	2	1	no	—
74	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
75	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
77	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
78	C₃⋊#95	—	2	1	no	—
79	C₃⋊#95	—	2	1	no	—
80	#803	—	2	2	no	—
81	C₂×C₅×Q₈	—	2	4	no	—
82	C₂×C₅×Q₈	—	4	4	no	—
83	C₂²×C₄×C₅	—	2	2	no	—
84	C₂²×C₄×C₅	—	6	2	no	—
85	C₂²×C₄×C₅	#67	1	1	yes	characteristic
86	C₂²×C₄×C₅	#67	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
88	C₂×C₅×C₈	—	3	4	no	—
89	C₂×C₅×C₈	—	6	2	no	—
90	C₂×C₅×C₈	—	6	2	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	—	6	4	no	—
94	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
95	C₄²×C₅	C₂×Dic₃	1	2	yes	—
96	C₄²×C₅	#67	1	2	yes	—
97	C₂×C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic
98	C₂×#97	—	3	1	no	—
99	#382	—	3	1	no	—
100	#340	—	3	4	no	—
101	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₃×C₅	#124	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
105	C₂²×C₃×C₅	#137	1	2	yes	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	2	no	—
108	C₃×C₄×C₅	—	4	4	no	—
109	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
110	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
111	C₅×Dic₃	—	8	2	no	—
112	C₂×C₃×Q₈	—	2	4	no	—
113	C₂×C₃×Q₈	—	4	4	no	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
118	C₂²×Dic₃	—	2	2	no	—
119	C₃×C₄⋊C₄	—	2	4	no	—
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
122	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
123	C₃×C₄²	C₅×D₄	1	2	yes	—
124	C₄⋊Dic₃	—	2	4	no	—
125	C₂×C₃⋊C₈	—	2	2	no	—
126	C₂×C₃⋊C₈	—	2	2	no	—
127	C₂×C₃⋊C₈	C₅×Q₈	1	4	yes	complemented
128	C₂³×C₅	C₂×#67	1	1	yes	characteristic
129	C₅×Q₈	—	4	8	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	12	2	no	—
135	C₂×C₄×C₅	—	12	2	no	—
136	C₂×C₄×C₅	—	12	2	no	—
137	C₂×C₄×C₅	#217	1	2	yes	—
138	C₂×C₄×C₅	#217	1	2	yes	—
139	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
140	C₂×C₄×C₅	#239	1	2	yes	—
141	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
143	C₅×C₈	—	6	4	no	—
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×C₈	—	3	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	C₅×Dic₃	1	4	yes	complemented
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	3	2	no	—
150	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
151	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
152	#97	—	6	2	no	—
153	#95	—	6	1	no	—
154	#95	—	6	1	no	—
155	C₂×C₃×C₅	#391	1	1	yes	characteristic
156	C₂×C₃×C₅	#418	1	1	yes	characteristic
157	C₂×C₃×C₅	#419	1	1	yes	characteristic
158	C₂×C₃×C₅	#489	1	2	yes	—
159	C₂×C₃×C₅	#501	1	2	yes	—
160	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
161	C₃×Q₈	—	4	8	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	4	4	no	—
166	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
167	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
168	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
171	C₂×C₃×C₄	C₅×#41	1	2	yes	—
172	C₂×Dic₃	—	4	2	no	—
173	C₂×Dic₃	—	4	2	no	—
174	C₂×Dic₃	—	4	2	no	—
175	C₃⋊C₈	—	2	4	no	—
176	C₂²×C₅	#932	1	2	yes	—
177	C₂²×C₅	#942	1	2	yes	—
178	C₂²×C₅	#944	1	1	yes	characteristic
179	C₂²×C₅	#947	1	1	yes	characteristic
180	C₂²×C₅	C₂×#217	1	1	yes	characteristic
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	4	4	no	—
184	C₄×C₅	—	24	2	no	—
185	C₄×C₅	Q₈×Dic₃	1	2	yes	—
186	C₄×C₅	#946	1	2	yes	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	4	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	6	2	no	—
191	C₂²×C₄	C₅×#67	1	1	yes	characteristic
192	C₂²×C₄	C₅×#67	1	1	yes	characteristic
193	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
194	C₂×C₈	—	3	4	no	—
195	C₂×C₈	—	6	2	no	—
196	C₂×C₈	—	6	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	6	4	no	—
200	C₄⋊C₄	C₂×C₅×Dic₃	1	4	yes	—
201	C₄²	C₂×C₅×Dic₃	1	2	yes	—
202	C₄²	C₅×#67	1	2	yes	—
203	C₃×C₅	#1993	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₃	C₅×#124	1	1	yes	characteristic
206	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
207	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
208	C₂²×C₃	C₅×#137	1	2	yes	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	4	4	no	—
212	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
213	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
214	Dic₃	—	8	2	no	—
215	C₂×C₅	#5081	1	2	yes	—
216	C₂×C₅	#5089	1	2	yes	—
217	C₂×C₅	#5215	1	1	yes	characteristic
218	C₂×C₅	#5225	1	1	yes	characteristic
219	C₂×C₅	#5229	1	1	yes	characteristic
220	C₂³	C₂×C₅×#67	1	1	yes	characteristic
221	Q₈	—	4	8	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	12	2	no	—
227	C₂×C₄	—	12	2	no	—
228	C₂×C₄	—	12	2	no	—
229	C₂×C₄	C₅×#217	1	2	yes	—
230	C₂×C₄	C₅×#217	1	2	yes	—
231	C₂×C₄	C₅×D₃×Q₈	1	2	yes	—
232	C₂×C₄	C₅×#239	1	2	yes	—
233	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
235	C₈	—	6	4	no	—
236	C₂×C₃	C₅×#391	1	1	yes	characteristic
237	C₂×C₃	C₅×#418	1	1	yes	characteristic
238	C₂×C₃	C₅×#419	1	1	yes	characteristic
239	C₂×C₃	C₅×#489	1	2	yes	—
240	C₂×C₃	C₅×#501	1	2	yes	—
241	C₅	#71788	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²	C₅×#932	1	2	yes	—
243	C₂²	C₅×#942	1	2	yes	—
244	C₂²	C₅×#944	1	1	yes	characteristic
245	C₂²	C₅×#947	1	1	yes	characteristic
246	C₂²	C₂×C₅×#217	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	4	no	—
250	C₄	—	24	2	no	—
251	C₄	C₅×Q₈×Dic₃	1	2	yes	—
252	C₄	C₅×#946	1	2	yes	—
253	C₃	C₅×#1993	1	1	yes	characteristic, complemented, Sylow
254	C₂	C₅×#5081	1	2	yes	—
255	C₂	C₅×#5089	1	2	yes	—
256	C₂	C₅×#5215	1	1	yes	characteristic
257	C₂	C₅×#5225	1	1	yes	characteristic
258	C₂	C₅×#5229	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#1993
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
8. C₃×C₅×C₄⋊Q₈	C₄	semidirect
17. #71788	C₅	direct (N × H)
37. C₂×C₅×C₃⋊C₈	Q₈	semidirect
45. C₅×C₄⋊Q₈	Dic₃	semidirect
72. C₃×C₄⋊Q₈	C₄×C₅	semidirect
127. C₂×C₃⋊C₈	C₅×Q₈	semidirect
147. C₄⋊Q₈	C₅×Dic₃	semidirect
203. C₃×C₅	#1993	semidirect
241. C₅	#71788	direct (N × H)
253. C₃	C₅×#1993	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₅×#4936	2
C₂×C₅×#803	2
C₅×#4529	2
C₅×#1993	3
#71788	5

Subgroups — C5×#71788 #14307377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71788 #14307377