Subgroups — C₅×#71625 #14281145

Order 1920.

472 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	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#4926	C₂	1	2	yes	maximal
3	C₅×#4528	C₂	1	4	yes	maximal
4	C₅×#1467	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅×Q₈	—	2	2	no	—
7	C₃×C₄×C₅×Q₈	—	2	2	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#830	C₂²	1	4	yes	—
13	C₅×C₄⋊(C₃⋊C₈)	—	2	2	no	—
14	C₅×#803	—	2	2	no	—
15	#71625	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₄×C₅×Q₈	D₃	1	1	yes	characteristic
17	C₅×#432	—	3	2	no	—
18	C₅×#324	—	3	4	no	—
19	C₂×C₃×C₅×Q₈	—	2	2	no	—
20	C₂×C₃×C₅×Q₈	—	2	4	no	—
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₃×C₅×C₄⋊C₄	—	2	2	no	—
27	C₃×C₅×C₄⋊C₄	—	2	4	no	—
28	C₃×C₄²×C₅	—	2	2	no	—
29	C₃×C₄²×C₅	—	2	2	no	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₂×C₅×C₃⋊C₈	—	2	4	no	—
33	C₂×C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic
34	#4926	C₂×C₅	1	2	yes	—
35	#4528	C₂×C₅	1	4	yes	—
36	C₂²×C₅×Q₈	Dic₃	1	1	yes	characteristic
37	C₄×C₅×Q₈	—	2	2	no	—
38	C₄×C₅×Q₈	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	Dic₃	1	1	yes	characteristic
40	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
41	C₂×C₄²×C₅	Dic₃	1	2	yes	—
42	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	6	2	no	—
44	C₅×C₂²⋊C₈	—	3	4	no	—
45	C₅×#97	—	6	2	no	—
46	#1467	—	3	1	no	Sylow
47	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₅×Q₈	—	2	8	no	—
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	4	1	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
60	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₅×C₃⋊C₈	—	4	4	no	—
62	C₂²×C₃×Q₈	C₄×C₅	1	1	yes	characteristic
63	C₃×C₄×Q₈	—	2	2	no	—
64	C₃×C₄×Q₈	—	2	2	no	—
65	C₂×C₃×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
66	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
67	C₂×C₃×C₄²	C₄×C₅	1	2	yes	—
68	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
69	#830	C₂²×C₅	1	4	yes	—
70	C₄⋊(C₃⋊C₈)	—	2	2	no	—
71	#803	—	2	2	no	—
72	C₂×C₅×Q₈	—	2	4	no	—
73	C₂×C₅×Q₈	—	2	2	no	—
74	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
75	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
78	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
79	C₂×C₅×C₈	—	6	4	no	—
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	—	2	4	no	—
82	C₄²×C₅	—	2	2	no	—
83	C₄²×C₅	—	2	2	no	—
84	C₄²×C₅	#67	1	2	yes	—
85	C₄²×C₅	#67	1	2	yes	—
86	C₂×C₄×Q₈	C₅×D₃	1	1	yes	characteristic
87	#432	—	3	2	no	—
88	#324	—	3	4	no	—
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	—	2	1	no	—
91	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	—	2	4	no	—
95	C₃×C₄×C₅	—	2	2	no	—
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	—	4	2	no	—
98	C₃×C₄×C₅	#101	1	2	yes	—
99	C₂×C₃×Q₈	—	2	2	no	—
100	C₂×C₃×Q₈	—	2	4	no	—
101	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
104	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
105	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₄⋊C₄	—	2	4	no	—
108	C₃×C₄²	—	2	2	no	—
109	C₃×C₄²	—	2	2	no	—
110	C₃×C₄²	C₅×D₄	1	2	yes	—
111	C₃×C₄²	C₅×D₄	1	2	yes	—
112	C₂×C₃⋊C₈	—	2	4	no	—
113	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
114	C₅×Q₈	—	2	8	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	4	1	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	#208	1	4	yes	—
123	C₂×C₄×C₅	#217	1	1	yes	characteristic
124	C₂×C₄×C₅	#217	1	2	yes	—
125	C₂×C₄×C₅	#217	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×#67	1	2	yes	—
127	C₅×C₈	—	12	4	no	—
128	C₂²×Q₈	C₅×Dic₃	1	1	yes	characteristic
129	C₄×Q₈	—	2	2	no	—
130	C₄×Q₈	—	2	2	no	—
131	C₂×C₄⋊C₄	C₅×Dic₃	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
133	C₂×C₄²	C₅×Dic₃	1	2	yes	—
134	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
135	C₄⋊C₈	—	6	2	no	—
136	C₂²⋊C₈	—	3	4	no	—
137	#97	—	6	2	no	—
138	C₂×C₃×C₅	—	2	2	no	—
139	C₂×C₃×C₅	#407	1	1	yes	characteristic
140	C₂×C₃×C₅	#410	1	1	yes	characteristic
141	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
142	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
143	C₃×Q₈	—	2	8	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	4	1	no	—
150	C₂×C₃×C₄	—	4	2	no	—
151	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
153	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
155	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
156	C₃⋊C₈	—	4	4	no	—
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	C₂×#208	1	2	yes	—
160	C₂²×C₅	C₂×#217	1	1	yes	characteristic
161	C₄×C₅	—	2	4	no	—
162	C₄×C₅	—	2	2	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	4	2	no	—
166	C₄×C₅	#836	1	2	yes	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
170	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
173	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
174	C₂×C₈	—	6	4	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	C₅×#67	1	2	yes	—
180	C₄²	C₅×#67	1	2	yes	—
181	C₃×C₅	#1467	1	1	yes	characteristic, complemented, Hall
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	2	4	no	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	4	2	no	—
191	C₃×C₄	C₅×#101	1	2	yes	—
192	C₂×C₅	—	2	2	no	—
193	C₂×C₅	#5207	1	1	yes	characteristic
194	C₂×C₅	C₂×#836	1	1	yes	characteristic
195	C₂×C₅	#5240	1	1	yes	characteristic
196	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
197	Q₈	—	2	8	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₅×#208	1	4	yes	—
206	C₂×C₄	C₅×#217	1	1	yes	characteristic
207	C₂×C₄	C₅×#217	1	2	yes	—
208	C₂×C₄	C₅×#217	1	1	yes	characteristic
209	C₂×C₄	C₂×C₅×#67	1	2	yes	—
210	C₈	—	12	4	no	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	C₅×#407	1	1	yes	characteristic
213	C₂×C₃	C₅×#410	1	1	yes	characteristic
214	C₂×C₃	C₂×C₅×#101	1	1	yes	characteristic
215	C₅	#71625	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂×C₅×#208	1	2	yes	—
219	C₂²	C₂×C₅×#217	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	2	2	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	C₅×#836	1	2	yes	—
226	C₃	C₅×#1467	1	1	yes	characteristic, complemented, Sylow
227	C₂	—	2	2	no	—
228	C₂	C₅×#5207	1	1	yes	characteristic
229	C₂	C₂×C₅×#836	1	1	yes	characteristic
230	C₂	C₅×#5240	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×#217
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₄×C₅×Q₈	D₃
hypocenter	C₃	C₅×#1467
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
15. #71625	C₅	direct (N × H)
181. C₃×C₅	#1467	semidirect
215. C₅	#71625	direct (N × H)
226. C₃	C₅×#1467	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
C₅×#4926	2
C₅×#4528	2
C₅×#1467	3
#71625	5

Subgroups — C5×#71625 #14281145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71625 #14281145