Subgroups — C₂×C₃×C₅×#491 #14276168

Order 1920.

768 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×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#491	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₅×#103	C₂	1	4	yes	maximal
5	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#491	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
11	C₃×C₅×#125	—	2	2	no	—
12	C₃×C₅×#125	C₂²	1	4	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
14	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₃×C₅×#103	C₂²	1	16	yes	—
18	C₃×C₅×#97	C₂²	1	4	yes	—
19	C₂×C₃×#491	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic
22	C₅×#491	C₂×C₃	1	8	yes	direct product factor
23	C₂×C₅×#103	C₂×C₃	1	4	yes	—
24	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	2	8	no	—
27	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	—	2	2	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	—	2	4	no	—
33	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
34	C₃×C₅×C₄⋊C₄	—	2	8	no	—
35	C₃×C₅×C₄⋊C₄	—	2	8	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	2	yes	—
40	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
41	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic
42	C₃×#491	C₂×C₅	1	8	yes	direct product factor
43	C₂×C₃×#103	C₂×C₅	1	4	yes	—
44	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
45	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
46	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
47	C₅×C₄⋊Q₈	—	2	2	no	—
48	C₅×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
49	C₅×#125	—	2	2	no	—
50	C₅×#125	C₂²×C₃	1	4	yes	—
51	C₂×C₅×C₄⋊C₄	—	2	2	no	—
52	C₂×C₅×C₄⋊C₄	—	2	2	no	—
53	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
55	C₅×#103	C₂²×C₃	1	16	yes	—
56	C₅×#97	C₂²×C₃	1	4	yes	—
57	C₂×#491	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
59	C₃×C₅×Q₈	—	2	8	no	—
60	C₃×C₅×Q₈	—	4	4	no	—
61	C₂×C₃×C₄×C₅	—	2	4	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	4	4	no	—
65	C₂×C₃×C₄×C₅	—	4	4	no	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	4	yes	—
69	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
70	C₃×C₅×C₈	—	2	4	no	—
71	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
72	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
73	C₃×C₄⋊Q₈	—	2	2	no	—
74	C₃×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
75	C₃×#125	—	2	2	no	—
76	C₃×#125	C₂²×C₅	1	4	yes	—
77	C₂×C₃×C₄⋊C₄	—	2	2	no	—
78	C₂×C₃×C₄⋊C₄	—	2	2	no	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
80	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
81	C₃×#103	C₂²×C₅	1	16	yes	—
82	C₃×#97	C₂²×C₅	1	4	yes	—
83	C₂×C₅×Q₈	—	2	2	no	—
84	C₂×C₅×Q₈	—	2	8	no	—
85	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	—	2	2	no	—
88	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₅×C₈	—	2	4	no	—
91	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
92	C₅×C₄⋊C₄	—	2	8	no	—
93	C₅×C₄⋊C₄	—	2	8	no	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
96	C₄²×C₅	C₃×D₄	1	2	yes	—
97	C₄²×C₅	C₂³×C₃	1	2	yes	—
98	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
99	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic
100	#491	C₂×C₃×C₅	1	8	yes	direct product factor
101	C₂×#103	C₂×C₃×C₅	1	4	yes	—
102	C₂×#97	C₂×C₃×C₅	1	1	yes	characteristic
103	C₂²×C₃×C₅	#124	1	2	yes	—
104	C₂²×C₃×C₅	#137	1	4	yes	—
105	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
106	C₃×C₄×C₅	—	2	4	no	—
107	C₃×C₄×C₅	—	4	4	no	—
108	C₃×C₄×C₅	—	4	4	no	—
109	C₃×C₄×C₅	#124	1	2	yes	—
110	C₃×C₄×C₅	C₂×#41	1	2	yes	—
111	C₂×C₃×Q₈	—	2	2	no	—
112	C₂×C₃×Q₈	—	2	8	no	—
113	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	—	2	2	no	—
116	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
117	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
118	C₂×C₃×C₈	—	2	4	no	—
119	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
120	C₃×C₄⋊C₄	—	2	8	no	—
121	C₃×C₄⋊C₄	—	2	8	no	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
124	C₃×C₄²	C₅×D₄	1	2	yes	—
125	C₃×C₄²	C₂³×C₅	1	2	yes	—
126	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
127	C₅×Q₈	—	2	8	no	—
128	C₅×Q₈	—	4	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
135	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₃×#41	1	4	yes	—
137	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
138	C₅×C₈	—	2	4	no	—
139	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
140	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
143	#125	—	2	2	no	—
144	#125	C₂²×C₃×C₅	1	4	yes	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
148	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
149	#103	C₂²×C₃×C₅	1	16	yes	—
150	#97	C₂²×C₃×C₅	1	4	yes	—
151	C₂×C₃×C₅	#491	1	4	yes	direct product factor
152	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
153	C₂×C₃×C₅	C₂×#137	1	2	yes	—
154	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
155	C₃×Q₈	—	2	8	no	—
156	C₃×Q₈	—	4	4	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	4	4	no	—
161	C₂×C₃×C₄	—	4	4	no	—
162	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
163	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₅×#41	1	4	yes	—
165	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
166	C₃×C₈	—	2	4	no	—
167	C₂²×C₅	C₃×#124	1	2	yes	—
168	C₂²×C₅	C₃×#137	1	4	yes	—
169	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
170	C₄×C₅	—	2	4	no	—
171	C₄×C₅	—	4	4	no	—
172	C₄×C₅	—	4	4	no	—
173	C₄×C₅	C₃×#124	1	2	yes	—
174	C₄×C₅	C₂×C₃×#41	1	2	yes	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	8	no	—
177	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
181	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
184	C₄⋊C₄	—	2	8	no	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
188	C₄²	C₃×C₅×D₄	1	2	yes	—
189	C₄²	C₂³×C₃×C₅	1	2	yes	—
190	C₃×C₅	C₂×#491	1	1	yes	characteristic, direct product factor, Hall
191	C₂²×C₃	C₅×#124	1	2	yes	—
192	C₂²×C₃	C₅×#137	1	4	yes	—
193	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
194	C₃×C₄	—	2	4	no	—
195	C₃×C₄	—	4	4	no	—
196	C₃×C₄	—	4	4	no	—
197	C₃×C₄	C₅×#124	1	2	yes	—
198	C₃×C₄	C₂×C₅×#41	1	2	yes	—
199	C₂×C₅	C₃×#491	1	4	yes	direct product factor
200	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
201	C₂×C₅	C₂×C₃×#137	1	2	yes	—
202	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
203	Q₈	—	2	8	no	—
204	Q₈	—	4	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
211	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
212	C₂×C₄	C₃×C₅×#41	1	4	yes	—
213	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
214	C₈	—	2	4	no	—
215	C₂×C₃	C₅×#491	1	4	yes	direct product factor
216	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
217	C₂×C₃	C₂×C₅×#137	1	2	yes	—
218	C₅	C₂×C₃×#491	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	C₃×C₅×#124	1	2	yes	—
220	C₂²	C₃×C₅×#137	1	4	yes	—
221	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
222	C₄	—	2	4	no	—
223	C₄	—	4	4	no	—
224	C₄	—	4	4	no	—
225	C₄	C₃×C₅×#124	1	2	yes	—
226	C₄	C₂×C₃×C₅×#41	1	2	yes	—
227	C₃	C₂×C₅×#491	1	1	yes	characteristic, direct product factor, Sylow
228	C₂	C₃×C₅×#491	1	4	yes	direct product factor
229	C₂	C₂×C₃×C₅×#124	1	1	yes	characteristic
230	C₂	C₂×C₃×C₅×#137	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#491	C₂	direct (N × H)
6. C₂×C₅×#491	C₃	direct (N × H)
19. C₂×C₃×#491	C₅	direct (N × H)
22. C₅×#491	C₂×C₃	direct (N × H)
42. C₃×#491	C₂×C₅	direct (N × H)
57. C₂×#491	C₃×C₅	direct (N × H)
100. #491	C₂×C₃×C₅	direct (N × H)
151. C₂×C₃×C₅	#491	direct (N × H)
190. C₃×C₅	C₂×#491	direct (N × H)
199. C₂×C₅	C₃×#491	direct (N × H)
215. C₂×C₃	C₅×#491	direct (N × H)
218. C₅	C₂×C₃×#491	direct (N × H)
227. C₃	C₂×C₅×#491	direct (N × H)
228. C₂	C₃×C₅×#491	direct (N × H)

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×C₃×C₅×#125	2
C₃×C₅×#491	2
C₂×C₃×C₅×#103	2
C₂×C₃×C₅×#97	2
C₂×C₅×#491	3
C₂×C₃×#491	5

Subgroups — C2×C3×C5×#491 #14276168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#491 #14276168