Subgroups — C₃×C₅×#2327 #14275361

Order 1920.

1032 conjugacy classes of subgroups.

The subgroup lattice has 268 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×C₄×C₅×Q₈	C₂	1	2	yes	maximal
3	C₃×C₅×#391	C₂	1	2	yes	maximal
4	C₃×C₅×#387	C₂	1	8	yes	maximal
5	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic
7	C₅×#2327	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	8	yes	complemented
10	C₃×C₄×C₅×Q₈	—	2	8	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₃×C₅×#95	C₂²	1	4	yes	—
19	C₃×C₅×#95	C₂²	1	4	yes	—
20	C₃×C₅×#95	C₂²	1	8	yes	—
21	C₃×#2327	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
23	C₂×C₄×C₅×Q₈	C₂×C₃	1	2	yes	—
24	C₅×#391	C₂×C₃	1	2	yes	—
25	C₅×#387	C₂×C₃	1	8	yes	—
26	C₅×#382	C₂×C₃	1	1	yes	characteristic
27	C₅×#376	C₂×C₃	1	1	yes	characteristic
28	C₂×C₃×C₅×Q₈	—	2	4	no	—
29	C₂×C₃×C₅×Q₈	—	2	4	no	—
30	C₂×C₃×C₅×Q₈	C₂×C₄	1	8	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₅×C₄⋊C₄	—	2	8	no	—
38	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
40	C₃×C₄²×C₅	—	2	8	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
43	C₃×C₄²×C₅	D₄	1	4	yes	—
44	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
45	C₂×C₃×C₄×Q₈	C₂×C₅	1	2	yes	—
46	C₃×#391	C₂×C₅	1	2	yes	—
47	C₃×#387	C₂×C₅	1	8	yes	—
48	C₃×#382	C₂×C₅	1	1	yes	characteristic
49	C₃×#376	C₂×C₅	1	1	yes	characteristic
50	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
51	C₅×C₄⋊Q₈	C₃×C₄	1	8	yes	complemented
52	C₄×C₅×Q₈	—	2	8	no	—
53	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
55	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
56	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₅×#95	C₂²×C₃	1	4	yes	—
61	C₅×#95	C₂²×C₃	1	4	yes	—
62	C₅×#95	C₂²×C₃	1	8	yes	—
63	#2327	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
64	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
65	C₃×C₅×Q₈	—	2	16	no	—
66	C₂×C₃×C₄×C₅	—	2	4	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	2	4	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	—	2	4	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	8	no	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	8	yes	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
81	C₃×C₄⋊Q₈	C₄×C₅	1	8	yes	complemented
82	C₃×C₄×Q₈	—	2	8	no	—
83	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
84	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
85	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
86	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
88	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
89	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
90	C₃×#95	C₂²×C₅	1	4	yes	—
91	C₃×#95	C₂²×C₅	1	4	yes	—
92	C₃×#95	C₂²×C₅	1	8	yes	—
93	C₂×C₅×Q₈	—	2	4	no	—
94	C₂×C₅×Q₈	—	2	4	no	—
95	C₂×C₅×Q₈	C₂×C₃×C₄	1	8	yes	—
96	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
97	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
98	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
99	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
100	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
101	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	8	no	—
103	C₅×C₄⋊C₄	—	2	4	no	—
104	C₅×C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
105	C₄²×C₅	—	2	8	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	C₂×C₃×C₄	1	4	yes	—
108	C₄²×C₅	C₃×D₄	1	4	yes	—
109	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
110	C₂×C₄×Q₈	C₂×C₃×C₅	1	2	yes	—
111	#391	C₂×C₃×C₅	1	2	yes	—
112	#387	C₂×C₃×C₅	1	8	yes	—
113	#382	C₂×C₃×C₅	1	1	yes	characteristic
114	#376	C₂×C₃×C₅	1	1	yes	characteristic
115	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
118	C₂²×C₃×C₅	2₋¹⁺⁴	1	4	yes	—
119	C₃×C₄×C₅	—	2	8	no	—
120	C₃×C₄×C₅	—	2	4	no	—
121	C₃×C₄×C₅	—	2	4	no	—
122	C₃×C₄×C₅	—	4	4	no	—
123	C₃×C₄×C₅	—	4	2	no	—
124	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
125	C₂×C₃×Q₈	—	2	4	no	—
126	C₂×C₃×Q₈	—	2	4	no	—
127	C₂×C₃×Q₈	C₂×C₄×C₅	1	8	yes	—
128	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
129	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
130	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
131	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
132	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
133	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	2	8	no	—
135	C₃×C₄⋊C₄	—	2	4	no	—
136	C₃×C₄⋊C₄	C₂×C₄×C₅	1	16	yes	—
137	C₃×C₄²	—	2	8	no	—
138	C₃×C₄²	—	2	2	no	—
139	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
140	C₃×C₄²	C₅×D₄	1	4	yes	—
141	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
142	C₅×Q₈	—	2	16	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	8	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
152	C₂×C₄×C₅	C₂²×C₃×C₄	1	8	yes	—
153	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
154	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
155	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
156	C₂×C₄×C₅	C₃×#41	1	2	yes	—
157	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
158	C₄⋊Q₈	C₃×C₄×C₅	1	8	yes	complemented
159	C₄×Q₈	—	2	8	no	—
160	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
162	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
163	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
164	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
165	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
166	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
167	#95	C₂²×C₃×C₅	1	4	yes	—
168	#95	C₂²×C₃×C₅	1	4	yes	—
169	#95	C₂²×C₃×C₅	1	8	yes	—
170	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×C₃×C₅	#519	1	2	yes	—
172	C₂×C₃×C₅	#536	1	2	yes	—
173	C₂×C₃×C₅	#541	1	2	yes	—
174	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
175	C₃×Q₈	—	2	16	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	8	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
185	C₂×C₃×C₄	C₂²×C₄×C₅	1	8	yes	—
186	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
187	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
188	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
189	C₂×C₃×C₄	C₅×#41	1	2	yes	—
190	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
191	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
192	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
193	C₂²×C₅	C₃×2₋¹⁺⁴	1	4	yes	—
194	C₄×C₅	—	2	8	no	—
195	C₄×C₅	—	2	4	no	—
196	C₄×C₅	—	2	4	no	—
197	C₄×C₅	—	4	4	no	—
198	C₄×C₅	—	4	2	no	—
199	C₄×C₅	C₃×C₄×D₄	1	4	yes	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	C₂×C₃×C₄×C₅	1	8	yes	—
203	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
204	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
205	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
206	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
207	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
208	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	C₂×C₃×C₄×C₅	1	16	yes	—
212	C₄²	—	2	8	no	—
213	C₄²	—	2	2	no	—
214	C₄²	C₂×C₃×C₄×C₅	1	4	yes	—
215	C₄²	C₃×C₅×D₄	1	4	yes	—
216	C₃×C₅	#2327	1	1	yes	characteristic, direct product factor, Hall
217	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
218	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
219	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
220	C₂²×C₃	C₅×2₋¹⁺⁴	1	4	yes	—
221	C₃×C₄	—	2	8	no	—
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	4	4	no	—
225	C₃×C₄	—	4	2	no	—
226	C₃×C₄	C₄×C₅×D₄	1	4	yes	—
227	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
228	C₂×C₅	C₃×#519	1	2	yes	—
229	C₂×C₅	C₃×#536	1	2	yes	—
230	C₂×C₅	C₃×#541	1	2	yes	—
231	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
232	Q₈	—	2	16	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
242	C₂×C₄	C₂²×C₃×C₄×C₅	1	8	yes	—
243	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
244	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
245	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
246	C₂×C₄	C₃×C₅×#41	1	2	yes	—
247	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
248	C₂×C₃	C₅×#519	1	2	yes	—
249	C₂×C₃	C₅×#536	1	2	yes	—
250	C₂×C₃	C₅×#541	1	2	yes	—
251	C₅	C₃×#2327	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
253	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
254	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
255	C₂²	C₃×C₅×2₋¹⁺⁴	1	4	yes	—
256	C₄	—	2	8	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	4	no	—
259	C₄	—	4	4	no	—
260	C₄	—	4	2	no	—
261	C₄	C₃×C₄×C₅×D₄	1	4	yes	—
262	C₃	C₅×#2327	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	C₂×C₃×C₄×C₅×D₄	1	1	yes	characteristic
264	C₂	C₃×C₅×#519	1	2	yes	—
265	C₂	C₃×C₅×#536	1	2	yes	—
266	C₂	C₃×C₅×#541	1	2	yes	—

	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
7. C₅×#2327	C₃	direct (N × H)
9. C₃×C₅×C₄⋊Q₈	C₄	semidirect
21. C₃×#2327	C₅	direct (N × H)
51. C₅×C₄⋊Q₈	C₃×C₄	semidirect
63. #2327	C₃×C₅	direct (N × H)
81. C₃×C₄⋊Q₈	C₄×C₅	semidirect
158. C₄⋊Q₈	C₃×C₄×C₅	semidirect
216. C₃×C₅	#2327	direct (N × H)
251. C₅	C₃×#2327	direct (N × H)
262. C₃	C₅×#2327	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×C₄×C₅×Q₈	2
C₃×C₅×#391	2
C₃×C₅×#387	2
C₃×C₅×#382	2
C₃×C₅×#376	2
C₅×#2327	3
C₃×#2327	5

Subgroups — C3×C5×#2327 #14275361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2327 #14275361