Subgroups — C₃×C₅×#2734 #14275768

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 316 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	2	yes	maximal
2	C₃×C₅×#398	C₂	1	2	yes	maximal
3	C₃×C₅×#395	C₂	1	4	yes	maximal
4	C₃×C₅×#391	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#389	C₂	1	2	yes	maximal
6	C₃×C₅×#387	C₂	1	4	yes	maximal
7	C₅×#2734	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊Q₈	—	2	8	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
18	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₃×C₅×#95	C₂²	1	2	yes	—
20	C₃×C₅×#95	C₂²	1	4	yes	—
21	C₃×C₅×#95	C₂²	1	4	yes	—
22	C₃×C₅×#95	C₂²	1	4	yes	—
23	C₃×#2734	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	2	yes	—
25	C₅×#398	C₂×C₃	1	2	yes	—
26	C₅×#395	C₂×C₃	1	4	yes	—
27	C₅×#391	C₂×C₃	1	1	yes	characteristic
28	C₅×#389	C₂×C₃	1	2	yes	—
29	C₅×#387	C₂×C₃	1	4	yes	—
30	C₂×C₃×C₅×Q₈	—	2	4	no	—
31	C₂×C₃×C₅×Q₈	—	4	1	no	—
32	C₂×C₃×C₅×Q₈	—	4	1	no	—
33	C₂×C₃×C₅×Q₈	—	4	2	no	—
34	C₂×C₃×C₅×Q₈	—	4	2	no	—
35	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
41	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
42	C₃×C₅×C₄⋊C₄	—	2	8	no	—
43	C₃×C₅×C₄⋊C₄	—	2	8	no	—
44	C₃×C₅×C₄⋊C₄	—	4	2	no	—
45	C₃×C₅×C₄⋊C₄	—	4	1	no	—
46	C₃×C₅×C₄⋊C₄	—	4	2	no	—
47	C₃×C₅×C₄⋊C₄	Q₈	1	8	yes	complemented
48	C₃×C₄²×C₅	—	2	4	no	—
49	C₃×C₄²×C₅	—	4	1	no	—
50	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	2	yes	—
51	C₃×#398	C₂×C₅	1	2	yes	—
52	C₃×#395	C₂×C₅	1	4	yes	—
53	C₃×#391	C₂×C₅	1	1	yes	characteristic
54	C₃×#389	C₂×C₅	1	2	yes	—
55	C₃×#387	C₂×C₅	1	4	yes	—
56	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
57	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×C₄⋊Q₈	—	2	8	no	—
59	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
60	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
61	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
66	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
67	C₅×#95	C₂²×C₃	1	2	yes	—
68	C₅×#95	C₂²×C₃	1	4	yes	—
69	C₅×#95	C₂²×C₃	1	4	yes	—
70	C₅×#95	C₂²×C₃	1	4	yes	—
71	#2734	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
73	C₃×C₅×Q₈	—	4	4	no	—
74	C₃×C₅×Q₈	—	4	8	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	—	2	4	no	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	—	2	4	no	—
86	C₂×C₃×C₄×C₅	—	2	4	no	—
87	C₂×C₃×C₄×C₅	—	2	4	no	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
91	C₂×C₃×C₄×C₅	C₂×Q₈	1	8	yes	—
92	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
93	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
94	C₃×C₄⋊Q₈	—	2	8	no	—
95	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
96	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
97	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
98	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
100	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
101	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
102	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
103	C₃×#95	C₂²×C₅	1	2	yes	—
104	C₃×#95	C₂²×C₅	1	4	yes	—
105	C₃×#95	C₂²×C₅	1	4	yes	—
106	C₃×#95	C₂²×C₅	1	4	yes	—
107	C₂×C₅×Q₈	—	2	4	no	—
108	C₂×C₅×Q₈	—	4	1	no	—
109	C₂×C₅×Q₈	—	4	1	no	—
110	C₂×C₅×Q₈	—	4	2	no	—
111	C₂×C₅×Q₈	—	4	2	no	—
112	C₂×C₅×Q₈	C₃×D₄	1	4	yes	—
113	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
114	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
115	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
117	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
118	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
119	C₅×C₄⋊C₄	—	2	8	no	—
120	C₅×C₄⋊C₄	—	2	8	no	—
121	C₅×C₄⋊C₄	—	4	2	no	—
122	C₅×C₄⋊C₄	—	4	1	no	—
123	C₅×C₄⋊C₄	—	4	2	no	—
124	C₅×C₄⋊C₄	C₃×Q₈	1	8	yes	complemented
125	C₄²×C₅	—	2	4	no	—
126	C₄²×C₅	—	4	1	no	—
127	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	2	yes	—
128	#398	C₂×C₃×C₅	1	2	yes	—
129	#395	C₂×C₃×C₅	1	4	yes	—
130	#391	C₂×C₃×C₅	1	1	yes	characteristic
131	#389	C₂×C₃×C₅	1	2	yes	—
132	#387	C₂×C₃×C₅	1	4	yes	—
133	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
135	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
138	C₃×C₄×C₅	—	2	2	no	—
139	C₃×C₄×C₅	—	2	4	no	—
140	C₃×C₄×C₅	—	2	4	no	—
141	C₃×C₄×C₅	—	2	8	no	—
142	C₃×C₄×C₅	—	4	4	no	—
143	C₃×C₄×C₅	—	4	2	no	—
144	C₂×C₃×Q₈	—	2	4	no	—
145	C₂×C₃×Q₈	—	4	1	no	—
146	C₂×C₃×Q₈	—	4	1	no	—
147	C₂×C₃×Q₈	—	4	2	no	—
148	C₂×C₃×Q₈	—	4	2	no	—
149	C₂×C₃×Q₈	C₅×D₄	1	4	yes	—
150	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
151	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
152	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
153	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
154	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
155	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
156	C₃×C₄⋊C₄	—	2	8	no	—
157	C₃×C₄⋊C₄	—	2	8	no	—
158	C₃×C₄⋊C₄	—	4	2	no	—
159	C₃×C₄⋊C₄	—	4	1	no	—
160	C₃×C₄⋊C₄	—	4	2	no	—
161	C₃×C₄⋊C₄	C₅×Q₈	1	8	yes	complemented
162	C₃×C₄²	—	2	4	no	—
163	C₃×C₄²	—	4	1	no	—
164	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
165	C₅×Q₈	—	4	4	no	—
166	C₅×Q₈	—	4	8	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	4	no	—
175	C₂×C₄×C₅	—	2	4	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	2	4	no	—
179	C₂×C₄×C₅	—	2	4	no	—
180	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
181	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
182	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
183	C₂×C₄×C₅	C₂×C₃×Q₈	1	8	yes	—
184	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
185	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	8	no	—
187	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
188	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
189	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
190	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
192	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
193	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
194	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
195	#95	C₂²×C₃×C₅	1	2	yes	—
196	#95	C₂²×C₃×C₅	1	4	yes	—
197	#95	C₂²×C₃×C₅	1	4	yes	—
198	#95	C₂²×C₃×C₅	1	4	yes	—
199	C₂×C₃×C₅	#534	1	1	yes	characteristic
200	C₂×C₃×C₅	#544	1	2	yes	—
201	C₂×C₃×C₅	D₄×Q₈	1	2	yes	—
202	C₂×C₃×C₅	Q₈²	1	1	yes	characteristic
203	C₂×C₃×C₅	#563	1	1	yes	characteristic
204	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
205	C₃×Q₈	—	4	4	no	—
206	C₃×Q₈	—	4	8	no	—
207	C₂×C₃×C₄	—	2	4	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	2	4	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
221	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
222	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
223	C₂×C₃×C₄	C₂×C₅×Q₈	1	8	yes	—
224	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
225	C₂²×C₅	C₂²×C₃×Q₈	1	2	yes	—
226	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
227	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
228	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
229	C₄×C₅	—	2	2	no	—
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	—	2	4	no	—
232	C₄×C₅	—	2	8	no	—
233	C₄×C₅	—	4	2	no	—
234	C₄×C₅	—	4	4	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	C₃×C₅×D₄	1	4	yes	—
241	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
242	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
243	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
244	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
245	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
246	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	C₃×C₅×Q₈	1	8	yes	complemented
253	C₄²	—	2	4	no	—
254	C₄²	—	4	1	no	—
255	C₃×C₅	#2734	1	1	yes	characteristic, direct product factor, Hall
256	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
257	C₂²×C₃	C₂²×C₅×Q₈	1	2	yes	—
258	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
260	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	2	4	no	—
263	C₃×C₄	—	2	4	no	—
264	C₃×C₄	—	2	8	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	—	4	2	no	—
267	C₂×C₅	C₃×#534	1	1	yes	characteristic
268	C₂×C₅	C₃×#544	1	2	yes	—
269	C₂×C₅	C₃×D₄×Q₈	1	2	yes	—
270	C₂×C₅	C₃×Q₈²	1	1	yes	characteristic
271	C₂×C₅	C₃×#563	1	1	yes	characteristic
272	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
273	Q₈	—	4	4	no	—
274	Q₈	—	4	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
289	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
290	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
291	C₂×C₄	C₂×C₃×C₅×Q₈	1	8	yes	—
292	C₂×C₃	C₅×#534	1	1	yes	characteristic
293	C₂×C₃	C₅×#544	1	2	yes	—
294	C₂×C₃	C₅×D₄×Q₈	1	2	yes	—
295	C₂×C₃	C₅×Q₈²	1	1	yes	characteristic
296	C₂×C₃	C₅×#563	1	1	yes	characteristic
297	C₅	C₃×#2734	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
299	C₂²	C₂²×C₃×C₅×Q₈	1	2	yes	—
300	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
301	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
302	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	4	no	—
306	C₄	—	2	8	no	—
307	C₄	—	4	4	no	—
308	C₄	—	4	2	no	—
309	C₃	C₅×#2734	1	1	yes	characteristic, direct product factor, Sylow
310	C₂	C₃×C₅×#534	1	1	yes	characteristic
311	C₂	C₃×C₅×#544	1	2	yes	—
312	C₂	C₃×C₅×D₄×Q₈	1	2	yes	—
313	C₂	C₃×C₅×Q₈²	1	1	yes	characteristic
314	C₂	C₃×C₅×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
LCS	G ▷ C₂³ ▷ 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×#2734	C₃	direct (N × H)
23. C₃×#2734	C₅	direct (N × H)
47. C₃×C₅×C₄⋊C₄	Q₈	semidirect
71. #2734	C₃×C₅	direct (N × H)
124. C₅×C₄⋊C₄	C₃×Q₈	semidirect
161. C₃×C₄⋊C₄	C₅×Q₈	semidirect
252. C₄⋊C₄	C₃×C₅×Q₈	semidirect
255. C₃×C₅	#2734	direct (N × H)
297. C₅	C₃×#2734	direct (N × H)
309. C₃	C₅×#2734	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×#398	2
C₃×C₅×#395	2
C₃×C₅×#391	2
C₃×C₅×#389	2
C₃×C₅×#387	2
C₅×#2734	3
C₃×#2734	5

Subgroups — C3×C5×#2734 #14275768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2734 #14275768