Subgroups — C₅×#67813 #14278754

Order 1920.

264 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₅×#5101	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic
4	C₅×#1378	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₅×#926	C₄	1	1	yes	characteristic
7	C₂×C₅×#208	C₄	1	1	yes	characteristic
8	C₃×C₅×M₃₂	—	2	1	no	—
9	C₃×C₅×M₃₂	—	2	1	no	—
10	C₂×C₃×C₅×C₁₆	C₄	1	2	yes	complemented
11	C₅×#815	—	2	1	no	—
12	C₅×C₈×Dic₃	—	2	1	no	—
13	C₅×#812	—	2	1	no	—
14	C₅×#812	C₄	1	2	yes	complemented
15	C₂×C₅×C₃⋊C₁₆	—	2	1	no	—
16	#67813	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×M₃₂	—	3	1	no	—
18	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic
19	C₅×#405	—	3	1	no	—
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
27	C₅×#217	C₈	1	2	yes	—
28	C₅×C₄⋊Dic₃	—	2	1	no	—
29	C₄×C₅×Dic₃	—	2	1	no	—
30	C₅×#208	—	2	1	no	—
31	C₅×#208	C₈	1	2	yes	—
32	C₂×C₅×C₃⋊C₈	—	2	1	no	—
33	C₃×C₅×C₁₆	—	2	2	no	—
34	C₅×C₃⋊C₁₆	—	2	2	no	—
35	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic
36	#5101	C₂×C₅	1	1	yes	characteristic
37	C₂×#812	C₂×C₅	1	1	yes	characteristic
38	C₂×C₅×M₁₆	—	3	1	no	—
39	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
40	C₅×#117	—	3	1	no	—
41	C₅×M₃₂	—	2	1	no	—
42	C₅×M₃₂	—	2	1	no	—
43	C₅×M₃₂	—	3	2	no	—
44	C₅×M₃₂	—	6	1	no	—
45	C₂×C₅×C₁₆	—	6	1	no	—
46	C₂×C₅×C₁₆	Dic₃	1	2	yes	complemented
47	C₅×#97	—	6	1	no	—
48	C₄×C₅×C₈	—	6	1	no	—
49	#1378	—	3	1	no	Sylow
50	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
57	C₂×C₅×Dic₃	—	2	2	no	—
58	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₅×C₃⋊C₈	—	2	2	no	—
63	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
64	#926	C₄×C₅	1	1	yes	characteristic
65	C₂×#208	C₄×C₅	1	1	yes	characteristic
66	C₃×M₃₂	—	2	1	no	—
67	C₃×M₃₂	—	2	1	no	—
68	C₂×C₃×C₁₆	C₄×C₅	1	2	yes	complemented
69	#815	—	2	1	no	—
70	C₈×Dic₃	—	2	1	no	—
71	#812	—	2	1	no	—
72	#812	C₄×C₅	1	2	yes	complemented
73	C₂×C₃⋊C₁₆	—	2	1	no	—
74	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
75	C₅×M₁₆	—	3	2	no	—
76	C₅×M₁₆	—	6	1	no	—
77	C₂×C₅×C₈	—	6	1	no	—
78	C₂×C₅×C₈	Dic₆	1	1	yes	characteristic
79	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
80	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₅×C₈	#67	1	1	yes	characteristic
83	C₂×C₅×C₈	#67	1	1	yes	characteristic
84	C₅×C₄⋊C₄	—	6	1	no	—
85	C₅×C₂²⋊C₄	—	3	2	no	—
86	C₄²×C₅	—	6	1	no	—
87	C₅×C₁₆	—	2	2	no	—
88	C₅×C₁₆	—	6	2	no	—
89	C₂×M₃₂	—	3	1	no	—
90	C₂×M₃₂	C₅×D₃	1	1	yes	characteristic
91	#405	—	3	1	no	—
92	C₂²×C₃×C₅	—	4	1	no	—
93	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
94	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₄⋊C₈	1	1	yes	characteristic
96	C₃×C₄×C₅	#95	1	1	yes	characteristic
97	C₃×C₄×C₅	#97	1	1	yes	characteristic
98	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
99	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
100	C₅×Dic₃	—	4	2	no	—
101	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
104	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
105	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
107	C₂×C₃×C₈	C₅×Q₈	1	1	yes	characteristic
108	#217	C₅×C₈	1	2	yes	—
109	C₄⋊Dic₃	—	2	1	no	—
110	C₄×Dic₃	—	2	1	no	—
111	#208	—	2	1	no	—
112	#208	C₅×C₈	1	2	yes	—
113	C₂×C₃⋊C₈	—	2	1	no	—
114	C₃×C₁₆	—	2	2	no	—
115	C₃⋊C₁₆	—	2	2	no	—
116	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅	—	6	2	no	—
118	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
120	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
121	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
122	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
124	C₅×C₈	—	6	2	no	—
125	C₅×C₈	Dic₃⋊C₄	1	1	yes	characteristic
126	C₅×C₈	C₄⋊Dic₃	1	1	yes	characteristic
127	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
128	C₅×C₈	#217	1	1	yes	characteristic
129	C₂×M₁₆	—	3	1	no	—
130	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
131	#117	—	3	1	no	—
132	M₃₂	—	2	1	no	—
133	M₃₂	—	2	1	no	—
134	M₃₂	—	3	2	no	—
135	M₃₂	—	6	1	no	—
136	C₂×C₁₆	—	6	1	no	—
137	C₂×C₁₆	C₅×Dic₃	1	2	yes	complemented
138	#97	—	6	1	no	—
139	C₄×C₈	—	6	1	no	—
140	C₂×C₃×C₅	—	2	1	no	—
141	C₂×C₃×C₅	—	2	1	no	—
142	C₂×C₃×C₅	—	2	1	no	—
143	C₂×C₃×C₅	#336	1	1	yes	characteristic
144	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
145	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
146	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
149	C₂×C₃×C₄	C₅×M₁₆	1	1	yes	characteristic
150	C₂×C₃×C₄	C₅×M₁₆	1	1	yes	characteristic
151	C₂×Dic₃	—	2	2	no	—
152	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
153	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
154	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
155	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
156	C₃⋊C₈	—	2	2	no	—
157	C₂²×C₅	—	4	1	no	—
158	C₂²×C₅	C₈×Dic₃	1	1	yes	characteristic
159	C₂²×C₅	Dic₃⋊C₈	1	1	yes	characteristic
160	C₂²×C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic
161	C₄×C₅	—	12	2	no	—
162	C₄×C₅	Dic₃⋊C₈	1	1	yes	characteristic
163	C₄×C₅	#815	1	1	yes	characteristic
164	C₄×C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic
165	C₄×C₅	C₃⋊#95	1	1	yes	characteristic
166	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
167	M₁₆	—	3	2	no	—
168	M₁₆	—	6	1	no	—
169	C₂×C₈	—	6	1	no	—
170	C₂×C₈	C₅×Dic₆	1	1	yes	characteristic
171	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
172	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
173	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
174	C₂×C₈	C₅×#67	1	1	yes	characteristic
175	C₂×C₈	C₅×#67	1	1	yes	characteristic
176	C₄⋊C₄	—	6	1	no	—
177	C₂²⋊C₄	—	3	2	no	—
178	C₄²	—	6	1	no	—
179	C₁₆	—	2	2	no	—
180	C₁₆	—	6	2	no	—
181	C₃×C₅	#1378	1	1	yes	characteristic, complemented, Hall
182	C₂²×C₃	—	4	1	no	—
183	C₂²×C₃	C₄×C₅×C₈	1	1	yes	characteristic
184	C₂²×C₃	C₅×C₂²⋊C₈	1	1	yes	characteristic
185	C₂²×C₃	C₅×C₄⋊C₈	1	1	yes	characteristic
186	C₃×C₄	C₅×#95	1	1	yes	characteristic
187	C₃×C₄	C₅×#97	1	1	yes	characteristic
188	C₃×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
189	C₃×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
190	Dic₃	—	4	2	no	—
191	C₂×C₅	—	2	1	no	—
192	C₂×C₅	—	2	1	no	—
193	C₂×C₅	—	2	1	no	—
194	C₂×C₅	#4550	1	1	yes	characteristic
195	C₂³	C₅×D₃×C₈	1	1	yes	characteristic
196	C₂×C₄	—	6	2	no	—
197	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
198	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
199	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
200	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
201	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
202	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₈	—	6	2	no	—
204	C₈	C₅×Dic₃⋊C₄	1	1	yes	characteristic
205	C₈	C₅×C₄⋊Dic₃	1	1	yes	characteristic
206	C₈	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
207	C₈	C₅×#217	1	1	yes	characteristic
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	C₅×#336	1	1	yes	characteristic
212	C₅	#67813	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²	—	4	1	no	—
214	C₂²	C₅×C₈×Dic₃	1	1	yes	characteristic
215	C₂²	C₅×Dic₃⋊C₈	1	1	yes	characteristic
216	C₂²	C₅×(C₂×D₃)⋊C₈	1	1	yes	characteristic
217	C₄	—	12	2	no	—
218	C₄	C₅×Dic₃⋊C₈	1	1	yes	characteristic
219	C₄	C₅×#815	1	1	yes	characteristic
220	C₄	C₅×(C₂×D₃)⋊C₈	1	1	yes	characteristic
221	C₄	C₅×C₃⋊#95	1	1	yes	characteristic
222	C₃	C₅×#1378	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	C₅×#4550	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	#217
commutator	C₂²×C₃	C₄×C₅×C₈
Frattini	C₂²×C₈	C₂×C₅×D₃
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂×C₃×C₅	#336
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂²×C₃×C₈	C₂²×C₅
hypercenter	C₂×C₅×M₃₂	D₃
hypocenter	C₃	C₅×#1378
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂

Complements

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

N	H ≅ G/N	type
10. C₂×C₃×C₅×C₁₆	C₄	semidirect
14. C₅×#812	C₄	semidirect
16. #67813	C₅	direct (N × H)
46. C₂×C₅×C₁₆	Dic₃	semidirect
68. C₂×C₃×C₁₆	C₄×C₅	semidirect
72. #812	C₄×C₅	semidirect
137. C₂×C₁₆	C₅×Dic₃	semidirect
181. C₃×C₅	#1378	semidirect
212. C₅	#67813	direct (N × H)
222. C₃	C₅×#1378	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×M₃₂	2
C₅×#5101	2
C₂×C₅×#812	2
C₅×#1378	3
#67813	5

Subgroups — C5×#67813 #14278754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#67813 #14278754