Subgroups — C₃×C₅×#2369 #14275403

Order 1920.

992 conjugacy classes of subgroups.

The subgroup lattice has 260 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	C₃×C₅×#391	C₂	1	4	yes	maximal
3	C₃×C₅×#389	C₂	1	4	yes	maximal
4	C₃×C₄×C₅×C₄⋊C₄	C₂	1	2	yes	maximal
5	C₃×C₄×C₅×C₄⋊C₄	C₂	1	4	yes	maximal
6	C₅×#2369	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	C₄	1	8	yes	complemented
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₅×#95	C₂²	1	8	yes	—
17	C₃×C₅×#95	C₂²	1	4	yes	—
18	C₃×#2369	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
20	C₅×#391	C₂×C₃	1	4	yes	—
21	C₅×#389	C₂×C₃	1	4	yes	—
22	C₄×C₅×C₄⋊C₄	C₂×C₃	1	2	yes	—
23	C₄×C₅×C₄⋊C₄	C₂×C₃	1	4	yes	—
24	C₂×C₃×C₅×Q₈	—	4	2	no	—
25	C₂×C₃×C₅×Q₈	—	4	2	no	—
26	C₂×C₃×C₅×Q₈	C₂×C₄	1	8	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₅×C₄⋊C₄	—	4	4	no	—
34	C₃×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
35	C₃×C₅×C₄⋊C₄	Q₈	1	8	yes	complemented
36	C₃×C₄²×C₅	—	2	8	no	—
37	C₃×C₄²×C₅	—	4	4	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
39	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
40	C₃×#391	C₂×C₅	1	4	yes	—
41	C₃×#389	C₂×C₅	1	4	yes	—
42	C₃×C₄×C₄⋊C₄	C₂×C₅	1	4	yes	—
43	C₃×C₄×C₄⋊C₄	C₂×C₅	1	2	yes	—
44	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
45	C₅×C₄⋊Q₈	C₃×C₄	1	8	yes	complemented
46	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
48	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
51	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
52	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₅×#95	C₂²×C₃	1	8	yes	—
54	C₅×#95	C₂²×C₃	1	4	yes	—
55	#2369	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
57	C₃×C₅×Q₈	—	4	8	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	—	2	4	no	—
61	C₂×C₃×C₄×C₅	—	2	4	no	—
62	C₂×C₃×C₄×C₅	—	2	1	no	—
63	C₂×C₃×C₄×C₅	—	2	1	no	—
64	C₂×C₃×C₄×C₅	—	2	4	no	—
65	C₂×C₃×C₄×C₅	—	2	4	no	—
66	C₂×C₃×C₄×C₅	—	2	4	no	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	8	yes	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
70	C₂×C₃×C₄×C₅	C₂×Q₈	1	8	yes	—
71	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
72	C₂×C₃×C₄×C₅	#41	1	4	yes	—
73	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
74	C₃×C₄⋊Q₈	C₄×C₅	1	8	yes	complemented
75	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
76	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
77	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
79	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
80	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
81	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
82	C₃×#95	C₂²×C₅	1	8	yes	—
83	C₃×#95	C₂²×C₅	1	4	yes	—
84	C₂×C₅×Q₈	—	4	2	no	—
85	C₂×C₅×Q₈	—	4	2	no	—
86	C₂×C₅×Q₈	C₂×C₃×C₄	1	8	yes	—
87	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
88	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
89	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
91	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	4	4	no	—
94	C₅×C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
95	C₅×C₄⋊C₄	C₃×Q₈	1	8	yes	complemented
96	C₄²×C₅	—	2	8	no	—
97	C₄²×C₅	—	4	4	no	—
98	C₄²×C₅	C₂×C₃×C₄	1	4	yes	—
99	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
100	#391	C₂×C₃×C₅	1	4	yes	—
101	#389	C₂×C₃×C₅	1	4	yes	—
102	C₄×C₄⋊C₄	C₂×C₃×C₅	1	4	yes	—
103	C₄×C₄⋊C₄	C₂×C₃×C₅	1	2	yes	—
104	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
106	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
107	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
109	C₃×C₄×C₅	—	2	8	no	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	2	8	no	—
112	C₃×C₄×C₅	—	4	4	no	—
113	C₃×C₄×C₅	C₄×Q₈	1	8	yes	—
114	C₂×C₃×Q₈	—	4	2	no	—
115	C₂×C₃×Q₈	—	4	2	no	—
116	C₂×C₃×Q₈	C₂×C₄×C₅	1	8	yes	—
117	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
118	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
120	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
121	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
122	C₃×C₄⋊C₄	—	2	4	no	—
123	C₃×C₄⋊C₄	—	4	4	no	—
124	C₃×C₄⋊C₄	C₂×C₄×C₅	1	16	yes	—
125	C₃×C₄⋊C₄	C₅×Q₈	1	8	yes	complemented
126	C₃×C₄²	—	2	8	no	—
127	C₃×C₄²	—	4	4	no	—
128	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
129	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
130	C₅×Q₈	—	4	8	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
141	C₂×C₄×C₅	C₂²×C₃×C₄	1	8	yes	—
142	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
143	C₂×C₄×C₅	C₂×C₃×Q₈	1	8	yes	—
144	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
145	C₂×C₄×C₅	C₃×#41	1	4	yes	—
146	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
147	C₄⋊Q₈	C₃×C₄×C₅	1	8	yes	complemented
148	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
150	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
151	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
152	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
153	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
154	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
155	#95	C₂²×C₃×C₅	1	8	yes	—
156	#95	C₂²×C₃×C₅	1	4	yes	—
157	C₂×C₃×C₅	C₂×C₄×Q₈	1	2	yes	—
158	C₂×C₃×C₅	#518	1	1	yes	characteristic
159	C₂×C₃×C₅	#554	1	2	yes	—
160	C₂×C₃×C₅	#555	1	1	yes	characteristic
161	C₂×C₃×C₅	Q₈²	1	1	yes	characteristic
162	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
163	C₃×Q₈	—	4	8	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
174	C₂×C₃×C₄	C₂²×C₄×C₅	1	8	yes	—
175	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
176	C₂×C₃×C₄	C₂×C₅×Q₈	1	8	yes	—
177	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
178	C₂×C₃×C₄	C₅×#41	1	4	yes	—
179	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
180	C₂²×C₅	C₂²×C₃×Q₈	1	2	yes	—
181	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
182	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
183	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
184	C₄×C₅	—	2	8	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	8	no	—
187	C₄×C₅	—	4	4	no	—
188	C₄×C₅	C₃×C₄×Q₈	1	8	yes	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	C₂×C₃×C₄×C₅	1	8	yes	—
192	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
193	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
195	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
196	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	C₂×C₃×C₄×C₅	1	16	yes	—
200	C₄⋊C₄	C₃×C₅×Q₈	1	8	yes	complemented
201	C₄²	—	2	8	no	—
202	C₄²	—	4	4	no	—
203	C₄²	C₂×C₃×C₄×C₅	1	4	yes	—
204	C₃×C₅	#2369	1	1	yes	characteristic, direct product factor, Hall
205	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₃	C₂²×C₅×Q₈	1	2	yes	—
207	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
208	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
209	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
210	C₃×C₄	—	2	8	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	8	no	—
213	C₃×C₄	—	4	4	no	—
214	C₃×C₄	C₄×C₅×Q₈	1	8	yes	—
215	C₂×C₅	C₂×C₃×C₄×Q₈	1	2	yes	—
216	C₂×C₅	C₃×#518	1	1	yes	characteristic
217	C₂×C₅	C₃×#554	1	2	yes	—
218	C₂×C₅	C₃×#555	1	1	yes	characteristic
219	C₂×C₅	C₃×Q₈²	1	1	yes	characteristic
220	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
221	Q₈	—	4	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
232	C₂×C₄	C₂²×C₃×C₄×C₅	1	8	yes	—
233	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
234	C₂×C₄	C₂×C₃×C₅×Q₈	1	8	yes	—
235	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
236	C₂×C₄	C₃×C₅×#41	1	4	yes	—
237	C₂×C₃	C₂×C₄×C₅×Q₈	1	2	yes	—
238	C₂×C₃	C₅×#518	1	1	yes	characteristic
239	C₂×C₃	C₅×#554	1	2	yes	—
240	C₂×C₃	C₅×#555	1	1	yes	characteristic
241	C₂×C₃	C₅×Q₈²	1	1	yes	characteristic
242	C₅	C₃×#2369	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
244	C₂²	C₂²×C₃×C₅×Q₈	1	2	yes	—
245	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
246	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
247	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	8	no	—
250	C₄	—	2	8	no	—
251	C₄	—	4	4	no	—
252	C₄	C₃×C₄×C₅×Q₈	1	8	yes	—
253	C₃	C₅×#2369	1	1	yes	characteristic, direct product factor, Sylow
254	C₂	C₂×C₃×C₄×C₅×Q₈	1	2	yes	—
255	C₂	C₃×C₅×#518	1	1	yes	characteristic
256	C₂	C₃×C₅×#554	1	2	yes	—
257	C₂	C₃×C₅×#555	1	1	yes	characteristic
258	C₂	C₃×C₅×Q₈²	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#2369	C₃	direct (N × H)
8. C₃×C₅×C₄⋊Q₈	C₄	semidirect
18. C₃×#2369	C₅	direct (N × H)
35. C₃×C₅×C₄⋊C₄	Q₈	semidirect
45. C₅×C₄⋊Q₈	C₃×C₄	semidirect
55. #2369	C₃×C₅	direct (N × H)
74. C₃×C₄⋊Q₈	C₄×C₅	semidirect
95. C₅×C₄⋊C₄	C₃×Q₈	semidirect
125. C₃×C₄⋊C₄	C₅×Q₈	semidirect
147. C₄⋊Q₈	C₃×C₄×C₅	semidirect
200. C₄⋊C₄	C₃×C₅×Q₈	semidirect
204. C₃×C₅	#2369	direct (N × H)
242. C₅	C₃×#2369	direct (N × H)
253. C₃	C₅×#2369	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
C₃×C₅×#391	2
C₃×C₅×#389	2
C₃×C₄×C₅×C₄⋊C₄	2
C₅×#2369	3
C₃×#2369	5

Subgroups — C3×C5×#2369 #14275403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2369 #14275403