Subgroups — C₅×#71786 #14305929

Order 1920.

616 conjugacy classes of subgroups.

The subgroup lattice has 236 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	C₂×C₅×#835	C₂	1	4	yes	maximal
3	C₅×#4936	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic
5	C₅×#1960	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
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	2	no	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
13	C₅×#835	—	2	8	no	—
14	C₅×C₃⋊#95	—	2	2	no	—
15	C₅×#803	—	2	2	no	—
16	#71786	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic
18	C₂×C₅×#103	—	3	4	no	—
19	C₂×C₅×#97	—	3	1	no	—
20	C₅×#382	—	3	1	no	—
21	C₂×C₃×C₅×Q₈	—	2	2	no	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	2	4	no	—
24	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₅×Dic₃	—	2	2	no	—
29	C₃×C₅×C₄⋊C₄	—	2	8	no	—
30	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₅×C₄⋊Dic₃	—	2	4	no	—
33	C₂×C₅×C₃⋊C₈	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
36	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
37	C₂×#835	C₂×C₅	1	4	yes	—
38	#4936	C₂×C₅	1	1	yes	characteristic
39	C₂×#803	C₂×C₅	1	1	yes	characteristic
40	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
41	C₂²×C₅×C₈	—	3	2	no	—
42	C₅×C₄⋊Q₈	—	2	2	no	—
43	C₅×C₄⋊Q₈	Dic₃	1	4	yes	complemented
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	—	3	2	no	—
46	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
47	C₅×#103	—	6	8	no	—
48	C₅×#97	—	6	2	no	—
49	C₅×#95	—	6	2	no	—
50	#1960	—	3	1	no	Sylow
51	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
52	C₃×C₅×Q₈	—	2	8	no	—
53	C₃×C₅×Q₈	—	4	4	no	—
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	#41	1	2	yes	—
63	C₂×C₅×Dic₃	—	4	2	no	—
64	C₂×C₅×Dic₃	—	4	4	no	—
65	C₅×C₃⋊C₈	—	2	4	no	—
66	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
67	C₃×C₄⋊Q₈	—	2	2	no	—
68	C₃×C₄⋊Q₈	C₄×C₅	1	4	yes	complemented
69	C₂×C₃×C₄⋊C₄	—	2	2	no	—
70	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
72	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
73	#835	—	2	8	no	—
74	C₃⋊#95	—	2	2	no	—
75	#803	—	2	2	no	—
76	C₂×C₅×Q₈	—	2	2	no	—
77	C₂×C₅×Q₈	—	2	4	no	—
78	C₂×C₅×Q₈	—	2	4	no	—
79	C₂×C₅×Q₈	C₂×Dic₃	1	4	yes	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	6	2	no	—
82	C₂²×C₄×C₅	#67	1	2	yes	—
83	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₅×C₈	—	3	4	no	—
85	C₂×C₅×C₈	—	6	2	no	—
86	C₂×C₅×C₈	—	6	2	no	—
87	C₅×C₄⋊C₄	—	2	8	no	—
88	C₅×C₄⋊C₄	—	6	4	no	—
89	C₄²×C₅	C₂×Dic₃	1	2	yes	—
90	C₄²×C₅	#67	1	2	yes	—
91	C₂×C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic
92	C₂×#103	—	3	4	no	—
93	C₂×#97	—	3	1	no	—
94	#382	—	3	1	no	—
95	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₃×C₅	#124	1	1	yes	characteristic
97	C₂²×C₃×C₅	#127	1	1	yes	characteristic
98	C₂²×C₃×C₅	#137	1	4	yes	—
99	C₃×C₄×C₅	—	2	4	no	—
100	C₃×C₄×C₅	—	4	4	no	—
101	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
102	C₃×C₄×C₅	#121	1	2	yes	—
103	C₅×Dic₃	—	8	2	no	—
104	C₂×C₃×Q₈	—	2	2	no	—
105	C₂×C₃×Q₈	—	2	4	no	—
106	C₂×C₃×Q₈	—	2	4	no	—
107	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
110	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
111	C₂²×Dic₃	—	2	2	no	—
112	C₃×C₄⋊C₄	—	2	8	no	—
113	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
114	C₃×C₄²	C₅×D₄	1	2	yes	—
115	C₄⋊Dic₃	—	2	4	no	—
116	C₂×C₃⋊C₈	—	2	2	no	—
117	C₂×C₃⋊C₈	—	2	2	no	—
118	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	—
119	C₂³×C₅	C₂×#67	1	1	yes	characteristic
120	C₅×Q₈	—	2	8	no	—
121	C₅×Q₈	—	4	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	4	2	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	—	12	2	no	—
127	C₂×C₄×C₅	—	12	4	no	—
128	C₂×C₄×C₅	#217	1	4	yes	—
129	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
130	C₂×C₄×C₅	#237	1	2	yes	—
131	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
133	C₅×C₈	—	6	4	no	—
134	C₂²×Q₈	C₂×C₅×D₃	1	2	yes	—
135	C₂²×C₈	—	3	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	C₅×Dic₃	1	4	yes	complemented
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	3	2	no	—
140	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
141	#103	—	6	8	no	—
142	#97	—	6	2	no	—
143	#95	—	6	2	no	—
144	C₂×C₃×C₅	#390	1	1	yes	characteristic
145	C₂×C₃×C₅	#419	1	2	yes	—
146	C₂×C₃×C₅	#491	1	2	yes	—
147	C₂×C₃×C₅	#497	1	2	yes	—
148	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
149	C₃×Q₈	—	2	8	no	—
150	C₃×Q₈	—	4	4	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
156	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
158	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₅×#41	1	2	yes	—
160	C₂×Dic₃	—	4	2	no	—
161	C₂×Dic₃	—	4	4	no	—
162	C₃⋊C₈	—	2	4	no	—
163	C₂²×C₅	#936	1	1	yes	characteristic
164	C₂²×C₅	#940	1	1	yes	characteristic
165	C₂²×C₅	#942	1	4	yes	—
166	C₂²×C₅	C₂×#217	1	1	yes	characteristic
167	C₄×C₅	—	2	4	no	—
168	C₄×C₅	—	4	4	no	—
169	C₄×C₅	—	24	2	no	—
170	C₄×C₅	D₄×Dic₃	1	2	yes	—
171	C₄×C₅	#938	1	2	yes	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	C₂×C₅×Dic₃	1	4	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	6	2	no	—
178	C₂²×C₄	C₅×#67	1	2	yes	—
179	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
180	C₂×C₈	—	3	4	no	—
181	C₂×C₈	—	6	2	no	—
182	C₂×C₈	—	6	2	no	—
183	C₄⋊C₄	—	2	8	no	—
184	C₄⋊C₄	—	6	4	no	—
185	C₄²	C₂×C₅×Dic₃	1	2	yes	—
186	C₄²	C₅×#67	1	2	yes	—
187	C₃×C₅	#1960	1	1	yes	characteristic, complemented, Hall
188	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₃	C₅×#124	1	1	yes	characteristic
190	C₂²×C₃	C₅×#127	1	1	yes	characteristic
191	C₂²×C₃	C₅×#137	1	4	yes	—
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	—	4	4	no	—
194	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
195	C₃×C₄	C₅×#121	1	2	yes	—
196	Dic₃	—	8	2	no	—
197	C₂×C₅	#5082	1	2	yes	—
198	C₂×C₅	#5086	1	2	yes	—
199	C₂×C₅	#5221	1	1	yes	characteristic
200	C₂×C₅	#5225	1	2	yes	—
201	C₂³	C₂×C₅×#67	1	1	yes	characteristic
202	Q₈	—	2	8	no	—
203	Q₈	—	4	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	12	2	no	—
209	C₂×C₄	—	12	4	no	—
210	C₂×C₄	C₅×#217	1	4	yes	—
211	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
212	C₂×C₄	C₅×#237	1	2	yes	—
213	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
214	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
215	C₈	—	6	4	no	—
216	C₂×C₃	C₅×#390	1	1	yes	characteristic
217	C₂×C₃	C₅×#419	1	2	yes	—
218	C₂×C₃	C₅×#491	1	2	yes	—
219	C₂×C₃	C₅×#497	1	2	yes	—
220	C₅	#71786	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²	C₅×#936	1	1	yes	characteristic
222	C₂²	C₅×#940	1	1	yes	characteristic
223	C₂²	C₅×#942	1	4	yes	—
224	C₂²	C₂×C₅×#217	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	4	4	no	—
227	C₄	—	24	2	no	—
228	C₄	C₅×D₄×Dic₃	1	2	yes	—
229	C₄	C₅×#938	1	2	yes	—
230	C₃	C₅×#1960	1	1	yes	characteristic, complemented, Sylow
231	C₂	C₅×#5082	1	2	yes	—
232	C₂	C₅×#5086	1	2	yes	—
233	C₂	C₅×#5221	1	1	yes	characteristic
234	C₂	C₅×#5225	1	2	yes	—

	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×#1960
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
16. #71786	C₅	direct (N × H)
43. C₅×C₄⋊Q₈	Dic₃	semidirect
68. C₃×C₄⋊Q₈	C₄×C₅	semidirect
137. C₄⋊Q₈	C₅×Dic₃	semidirect
187. C₃×C₅	#1960	semidirect
220. C₅	#71786	direct (N × H)
230. C₃	C₅×#1960	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
C₂×C₅×#835	2
C₅×#4936	2
C₂×C₅×#803	2
C₅×#1960	3
#71786	5

Subgroups — C5×#71786 #14305929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71786 #14305929