Subgroups — C₃×C₅×#1997 #14275031

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 292 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×#423	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#343	C₂	1	4	yes	maximal
5	C₅×#1997	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₄	1	2	yes	—
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
11	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
12	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	—	2	1	no	—
17	C₃×C₅×C₄⋊C₈	—	2	1	no	—
18	C₃×C₅×#97	—	2	1	no	—
19	C₃×C₄×C₅×C₈	—	2	1	no	—
20	C₃×#1997	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₅×#423	C₂×C₃	1	1	yes	characteristic
23	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
24	C₅×#343	C₂×C₃	1	4	yes	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	4	4	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₃×C₅×M₁₆	—	2	2	no	—
33	C₃×C₅×M₁₆	—	2	4	no	—
34	C₃×C₅×M₁₆	Q₈	1	4	yes	complemented
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
46	C₃×#423	C₂×C₅	1	1	yes	characteristic
47	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
48	C₃×#343	C₂×C₅	1	4	yes	—
49	C₂²×C₅×Q₈	C₃×C₄	1	2	yes	—
50	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
51	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
52	C₅×C₄⋊Q₈	—	2	1	no	—
53	C₅×C₄⋊Q₈	—	2	1	no	—
54	C₅×C₄⋊Q₈	—	2	1	no	—
55	C₅×C₄⋊Q₈	—	2	1	no	—
56	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	—
57	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₅×C₄⋊C₈	—	2	1	no	—
60	C₅×C₄⋊C₈	—	2	1	no	—
61	C₅×#97	—	2	1	no	—
62	C₄×C₅×C₈	—	2	1	no	—
63	#1997	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
64	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₃×C₅×Q₈	—	4	4	no	—
66	C₃×C₅×Q₈	—	4	4	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	—	4	4	no	—
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
78	C₂×C₃×C₄×C₅	#41	1	2	yes	—
79	C₃×C₅×C₈	—	2	4	no	—
80	C₃×C₅×C₈	—	4	2	no	—
81	C₂²×C₃×Q₈	C₄×C₅	1	2	yes	—
82	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
83	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
84	C₃×C₄⋊Q₈	—	2	1	no	—
85	C₃×C₄⋊Q₈	—	2	1	no	—
86	C₃×C₄⋊Q₈	—	2	1	no	—
87	C₃×C₄⋊Q₈	—	2	1	no	—
88	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
89	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
90	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
91	C₃×C₄⋊C₈	—	2	1	no	—
92	C₃×C₄⋊C₈	—	2	1	no	—
93	C₃×#97	—	2	1	no	—
94	C₃×C₄×C₈	—	2	1	no	—
95	C₂×C₅×Q₈	—	2	2	no	—
96	C₂×C₅×Q₈	—	2	2	no	—
97	C₂×C₅×Q₈	—	4	4	no	—
98	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
101	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
102	C₅×M₁₆	—	2	2	no	—
103	C₅×M₁₆	—	2	4	no	—
104	C₅×M₁₆	C₃×Q₈	1	4	yes	complemented
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
112	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
113	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
114	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
115	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
116	#423	C₂×C₃×C₅	1	1	yes	characteristic
117	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
118	#343	C₂×C₃×C₅	1	4	yes	—
119	C₂²×C₃×C₅	—	2	1	no	—
120	C₂²×C₃×C₅	—	2	1	no	—
121	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₃×C₅	C₄×Q₈	1	2	yes	—
123	C₃×C₄×C₅	—	2	1	no	—
124	C₃×C₄×C₅	—	2	1	no	—
125	C₃×C₄×C₅	—	4	4	no	—
126	C₃×C₄×C₅	#100	1	2	yes	—
127	C₃×C₄×C₅	#101	1	2	yes	—
128	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
129	C₃×C₄×C₅	#124	1	1	yes	characteristic
130	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₃×Q₈	—	2	2	no	—
132	C₂×C₃×Q₈	—	2	2	no	—
133	C₂×C₃×Q₈	—	4	4	no	—
134	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
136	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
137	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
138	C₃×M₁₆	—	2	2	no	—
139	C₃×M₁₆	—	2	4	no	—
140	C₃×M₁₆	C₅×Q₈	1	4	yes	complemented
141	C₂×C₃×C₈	—	2	2	no	—
142	C₂×C₃×C₈	—	2	2	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
148	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
149	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
150	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
151	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₅×Q₈	—	4	4	no	—
153	C₅×Q₈	—	4	4	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	4	4	no	—
158	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
165	C₂×C₄×C₅	C₃×#41	1	2	yes	—
166	C₅×C₈	—	2	4	no	—
167	C₅×C₈	—	4	2	no	—
168	C₂²×Q₈	C₃×C₄×C₅	1	2	yes	—
169	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
170	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
171	C₄⋊Q₈	—	2	1	no	—
172	C₄⋊Q₈	—	2	1	no	—
173	C₄⋊Q₈	—	2	1	no	—
174	C₄⋊Q₈	—	2	1	no	—
175	C₂×C₄⋊C₄	C₃×C₄×C₅	1	2	yes	—
176	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
177	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	2	1	no	—
180	#97	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₂×C₃×C₅	—	2	2	no	—
183	C₂×C₃×C₅	#391	1	1	yes	characteristic
184	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
186	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
187	C₃×Q₈	—	4	4	no	—
188	C₃×Q₈	—	4	4	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	4	4	no	—
193	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
200	C₂×C₃×C₄	C₅×#41	1	2	yes	—
201	C₃×C₈	—	2	4	no	—
202	C₃×C₈	—	4	2	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×C₅	C₃×C₄×Q₈	1	2	yes	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	4	4	no	—
210	C₄×C₅	C₃×#100	1	2	yes	—
211	C₄×C₅	C₃×#101	1	2	yes	—
212	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
213	C₄×C₅	C₃×#124	1	1	yes	characteristic
214	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
221	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	C₃×C₅×Q₈	1	4	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
232	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
233	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
234	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
235	C₃×C₅	#1997	1	1	yes	characteristic, direct product factor, Hall
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
239	C₂²×C₃	C₄×C₅×Q₈	1	2	yes	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	4	4	no	—
243	C₃×C₄	C₅×#100	1	2	yes	—
244	C₃×C₄	C₅×#101	1	2	yes	—
245	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
246	C₃×C₄	C₅×#124	1	1	yes	characteristic
247	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	C₃×#391	1	1	yes	characteristic
250	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
251	C₂×C₅	C₂×C₃×#101	1	1	yes	characteristic
252	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
253	Q₈	—	4	4	no	—
254	Q₈	—	4	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
264	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
266	C₂×C₄	C₃×C₅×#41	1	2	yes	—
267	C₈	—	2	4	no	—
268	C₈	—	4	2	no	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	C₅×#391	1	1	yes	characteristic
271	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
272	C₂×C₃	C₂×C₅×#101	1	1	yes	characteristic
273	C₅	C₃×#1997	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₂²	C₃×C₄×C₅×Q₈	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	4	4	no	—
281	C₄	C₃×C₅×#100	1	2	yes	—
282	C₄	C₃×C₅×#101	1	2	yes	—
283	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
284	C₄	C₃×C₅×#124	1	1	yes	characteristic
285	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
286	C₃	C₅×#1997	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	2	2	no	—
288	C₂	C₃×C₅×#391	1	1	yes	characteristic
289	C₂	C₂×C₃×C₅×#100	1	1	yes	characteristic
290	C₂	C₂×C₃×C₅×#101	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1997	C₃	direct (N × H)
20. C₃×#1997	C₅	direct (N × H)
34. C₃×C₅×M₁₆	Q₈	semidirect
63. #1997	C₃×C₅	direct (N × H)
104. C₅×M₁₆	C₃×Q₈	semidirect
140. C₃×M₁₆	C₅×Q₈	semidirect
224. M₁₆	C₃×C₅×Q₈	semidirect
235. C₃×C₅	#1997	direct (N × H)
273. C₅	C₃×#1997	direct (N × H)
286. C₃	C₅×#1997	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×#423	2
C₃×C₄×C₅×M₁₆	2
C₃×C₅×#343	2
C₅×#1997	3
C₃×#1997	5

Subgroups — C3×C5×#1997 #14275031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1997 #14275031