Subgroups — C₂×C₃×C₅×#358 #14275171

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#358	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×#358	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	—	2	2	no	—
9	C₂×C₃×C₅×Q₁₆	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
13	C₂×C₃×C₅×C₁₆	—	2	2	no	—
14	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
15	C₃×C₅×#107	—	2	1	no	—
16	C₃×C₅×#107	C₂²	1	2	yes	—
17	C₂×C₃×#358	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic
20	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic
21	C₅×#358	C₂×C₃	1	4	yes	direct product factor
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	—	4	1	no	—
25	C₂×C₃×C₅×Q₈	—	4	2	no	—
26	C₂²×C₃×C₄×C₅	—	4	1	no	—
27	C₂²×C₃×C₄×C₅	—	4	1	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₃×C₅×Q₁₆	—	2	4	no	—
30	C₃×C₅×Q₁₆	—	2	2	no	—
31	C₃×C₅×Q₁₆	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	4	1	no	—
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₂²×C₃×Q₁₆	C₂×C₅	1	1	yes	characteristic
40	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic
41	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic
42	C₃×#358	C₂×C₅	1	4	yes	direct product factor
43	C₂²×C₅×Q₈	—	2	1	no	—
44	C₂×C₅×Q₁₆	—	2	1	no	—
45	C₂×C₅×Q₁₆	—	2	2	no	—
46	C₂×C₅×Q₁₆	C₃×C₄	1	4	yes	complemented
47	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
48	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	—	2	1	no	—
50	C₂×C₅×C₁₆	—	2	2	no	—
51	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
52	C₅×#107	—	2	1	no	—
53	C₅×#107	C₂²×C₃	1	2	yes	—
54	C₂×#358	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₃×C₅×Q₈	—	2	4	no	—
57	C₃×C₅×Q₈	—	4	4	no	—
58	C₃×C₅×Q₈	—	4	2	no	—
59	C₂×C₃×C₄×C₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	—	8	2	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×Q₈	—	2	1	no	—
71	C₂×C₃×Q₁₆	—	2	1	no	—
72	C₂×C₃×Q₁₆	—	2	2	no	—
73	C₂×C₃×Q₁₆	C₄×C₅	1	4	yes	complemented
74	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
75	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₃×C₄⋊C₄	—	2	1	no	—
77	C₂×C₃×C₁₆	—	2	2	no	—
78	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
79	C₃×#107	—	2	1	no	—
80	C₃×#107	C₂²×C₅	1	2	yes	—
81	C₂×C₅×Q₈	—	2	4	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×Q₈	—	4	2	no	—
85	C₂²×C₄×C₅	—	4	1	no	—
86	C₂²×C₄×C₅	—	4	1	no	—
87	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
88	C₅×Q₁₆	—	2	4	no	—
89	C₅×Q₁₆	—	2	2	no	—
90	C₅×Q₁₆	C₂×C₃×C₄	1	4	yes	complemented
91	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
94	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	4	1	no	—
97	C₅×C₁₆	—	2	2	no	—
98	C₂²×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic
99	C₂²×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
100	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic
101	#358	C₂×C₃×C₅	1	4	yes	direct product factor
102	C₂²×C₃×C₅	#102	1	2	yes	—
103	C₂²×C₃×C₅	SD₃₂	1	2	yes	—
104	C₂²×C₃×C₅	Q₃₂	1	2	yes	—
105	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
106	C₃×C₄×C₅	—	4	4	no	—
107	C₃×C₄×C₅	—	8	2	no	—
108	C₃×C₄×C₅	#102	1	2	yes	—
109	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₃×Q₈	—	2	4	no	—
112	C₂×C₃×Q₈	—	2	2	no	—
113	C₂×C₃×Q₈	—	4	1	no	—
114	C₂×C₃×Q₈	—	4	2	no	—
115	C₂²×C₃×C₄	—	4	1	no	—
116	C₂²×C₃×C₄	—	4	1	no	—
117	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
118	C₃×Q₁₆	—	2	4	no	—
119	C₃×Q₁₆	—	2	2	no	—
120	C₃×Q₁₆	C₂×C₄×C₅	1	4	yes	complemented
121	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
122	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
123	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
124	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	—	4	1	no	—
127	C₃×C₁₆	—	2	2	no	—
128	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
129	C₅×Q₈	—	2	4	no	—
130	C₅×Q₈	—	4	4	no	—
131	C₅×Q₈	—	4	2	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	8	2	no	—
136	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
137	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
139	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
140	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
141	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
143	C₂²×Q₈	—	2	1	no	—
144	C₂×Q₁₆	—	2	1	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	C₃×C₄×C₅	1	4	yes	complemented
147	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
148	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
152	#107	—	2	1	no	—
153	#107	C₂²×C₃×C₅	1	2	yes	—
154	C₂×C₃×C₅	#358	1	4	yes	direct product factor
155	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
156	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
158	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
159	C₃×Q₈	—	2	4	no	—
160	C₃×Q₈	—	4	2	no	—
161	C₃×Q₈	—	4	4	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	4	4	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	—	8	2	no	—
166	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
167	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
168	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
169	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
170	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
171	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
172	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂²×C₅	C₃×#102	1	2	yes	—
174	C₂²×C₅	C₃×SD₃₂	1	2	yes	—
175	C₂²×C₅	C₃×Q₃₂	1	2	yes	—
176	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
177	C₄×C₅	—	4	4	no	—
178	C₄×C₅	—	8	2	no	—
179	C₄×C₅	C₃×#102	1	2	yes	—
180	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	4	1	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
189	Q₁₆	—	2	4	no	—
190	Q₁₆	—	2	2	no	—
191	Q₁₆	C₂×C₃×C₄×C₅	1	4	yes	complemented
192	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
193	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
194	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
195	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₁₆	—	2	2	no	—
199	C₃×C₅	C₂×#358	1	1	yes	characteristic, direct product factor, Hall
200	C₂²×C₃	C₅×#102	1	2	yes	—
201	C₂²×C₃	C₅×SD₃₂	1	2	yes	—
202	C₂²×C₃	C₅×Q₃₂	1	2	yes	—
203	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
204	C₃×C₄	—	4	4	no	—
205	C₃×C₄	—	8	2	no	—
206	C₃×C₄	C₅×#102	1	2	yes	—
207	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
209	C₂×C₅	C₃×#358	1	4	yes	direct product factor
210	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
211	C₂×C₅	C₂×C₃×SD₃₂	1	1	yes	characteristic
212	C₂×C₅	C₂×C₃×Q₃₂	1	1	yes	characteristic
213	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
214	Q₈	—	2	4	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
222	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
223	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
224	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
225	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
226	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
227	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
228	C₂×C₃	C₅×#358	1	4	yes	direct product factor
229	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
230	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
231	C₂×C₃	C₂×C₅×Q₃₂	1	1	yes	characteristic
232	C₅	C₂×C₃×#358	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²	C₃×C₅×#102	1	2	yes	—
234	C₂²	C₃×C₅×SD₃₂	1	2	yes	—
235	C₂²	C₃×C₅×Q₃₂	1	2	yes	—
236	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
237	C₄	—	4	4	no	—
238	C₄	—	8	2	no	—
239	C₄	C₃×C₅×#102	1	2	yes	—
240	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
241	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
242	C₃	C₂×C₅×#358	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	C₃×C₅×#358	1	4	yes	direct product factor
244	C₂	C₂×C₃×C₅×#102	1	1	yes	characteristic
245	C₂	C₂×C₃×C₅×SD₃₂	1	1	yes	characteristic
246	C₂	C₂×C₃×C₅×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#358	C₂	direct (N × H)
5. C₂×C₅×#358	C₃	direct (N × H)
9. C₂×C₃×C₅×Q₁₆	C₄	semidirect
17. C₂×C₃×#358	C₅	direct (N × H)
21. C₅×#358	C₂×C₃	direct (N × H)
31. C₃×C₅×Q₁₆	C₂×C₄	semidirect
42. C₃×#358	C₂×C₅	direct (N × H)
46. C₂×C₅×Q₁₆	C₃×C₄	semidirect
54. C₂×#358	C₃×C₅	direct (N × H)
73. C₂×C₃×Q₁₆	C₄×C₅	semidirect
90. C₅×Q₁₆	C₂×C₃×C₄	semidirect
101. #358	C₂×C₃×C₅	direct (N × H)
120. C₃×Q₁₆	C₂×C₄×C₅	semidirect
146. C₂×Q₁₆	C₃×C₄×C₅	semidirect
154. C₂×C₃×C₅	#358	direct (N × H)
191. Q₁₆	C₂×C₃×C₄×C₅	semidirect
199. C₃×C₅	C₂×#358	direct (N × H)
209. C₂×C₅	C₃×#358	direct (N × H)
228. C₂×C₃	C₅×#358	direct (N × H)
232. C₅	C₂×C₃×#358	direct (N × H)
242. C₃	C₂×C₅×#358	direct (N × H)
243. C₂	C₃×C₅×#358	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×Q₁₆	2
C₂²×C₃×C₅×C₁₆	2
C₂×C₃×C₅×#107	2
C₃×C₅×#358	2
C₂×C₅×#358	3
C₂×C₃×#358	5

Subgroups — C2×C3×C5×#358 #14275171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#358 #14275171