Subgroups — C₂×C₃×C₅×#332 #14274573

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 304 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×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#332	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×#332	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
11	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
14	C₂×C₃×C₅×C₄⋊C₄	C₄	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×C₄⋊C₈	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃×#332	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
23	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₅×#332	C₂×C₃	1	4	yes	direct product factor
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	4	2	no	—
28	C₂×C₃×C₅×Q₈	—	4	2	no	—
29	C₂²×C₃×C₄×C₅	—	2	2	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	—	4	2	no	—
36	C₂×C₃×C₅×C₈	—	4	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	C₂×C₄	1	4	yes	—
41	C₃×C₄²×C₅	C₂×C₄	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	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
46	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
47	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
48	C₃×#332	C₂×C₅	1	4	yes	direct product factor
49	C₂²×C₅×Q₈	—	2	1	no	—
50	C₂²×C₅×C₈	—	2	1	no	—
51	C₂²×C₅×C₈	—	2	1	no	—
52	C₅×C₄⋊Q₈	—	2	1	no	—
53	C₅×C₄⋊Q₈	—	2	1	no	—
54	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	—
55	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
56	C₂×C₅×C₄⋊C₄	—	2	1	no	—
57	C₂×C₅×C₄⋊C₄	C₃×C₄	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×C₄⋊C₈	C₂²×C₃	1	2	yes	—
62	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
63	C₂×#332	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	2	no	—
67	C₃×C₅×Q₈	—	4	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	D₈	1	2	yes	—
75	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
76	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
77	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₃×C₅×C₈	—	4	2	no	—
82	C₃×C₅×C₈	—	4	2	no	—
83	C₂²×C₃×Q₈	—	2	1	no	—
84	C₂²×C₃×C₈	—	2	1	no	—
85	C₂²×C₃×C₈	—	2	1	no	—
86	C₃×C₄⋊Q₈	—	2	1	no	—
87	C₃×C₄⋊Q₈	—	2	1	no	—
88	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	—
89	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
90	C₂×C₃×C₄⋊C₄	—	2	1	no	—
91	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄⋊C₈	—	2	1	no	—
94	C₃×C₄⋊C₈	—	2	1	no	—
95	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
96	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
97	C₂×C₅×Q₈	—	2	2	no	—
98	C₂×C₅×Q₈	—	2	2	no	—
99	C₂×C₅×Q₈	—	4	2	no	—
100	C₂×C₅×Q₈	—	4	2	no	—
101	C₂²×C₄×C₅	—	2	2	no	—
102	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	4	2	no	—
108	C₂×C₅×C₈	—	4	2	no	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
113	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
114	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
115	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
116	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
118	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
119	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
120	#332	C₂×C₃×C₅	1	4	yes	direct product factor
121	C₂²×C₃×C₅	#101	1	2	yes	—
122	C₂²×C₃×C₅	#102	1	2	yes	—
123	C₂²×C₃×C₅	#103	1	2	yes	—
124	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₃×C₄×C₅	—	2	1	no	—
126	C₃×C₄×C₅	—	2	1	no	—
127	C₃×C₄×C₅	—	4	4	no	—
128	C₃×C₄×C₅	#102	1	2	yes	—
129	C₃×C₄×C₅	#103	1	2	yes	—
130	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
131	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
132	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
133	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₃×Q₈	—	2	2	no	—
135	C₂×C₃×Q₈	—	2	2	no	—
136	C₂×C₃×Q₈	—	4	2	no	—
137	C₂×C₃×Q₈	—	4	2	no	—
138	C₂²×C₃×C₄	—	2	2	no	—
139	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
142	C₂×C₃×C₈	—	2	2	no	—
143	C₂×C₃×C₈	—	2	2	no	—
144	C₂×C₃×C₈	—	4	2	no	—
145	C₂×C₃×C₈	—	4	2	no	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	—	2	2	no	—
149	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
150	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
151	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
152	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
153	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
154	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
155	C₅×Q₈	—	4	4	no	—
156	C₅×Q₈	—	4	2	no	—
157	C₅×Q₈	—	4	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	4	no	—
160	C₂×C₄×C₅	—	4	4	no	—
161	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
165	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
166	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
167	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
168	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
171	C₅×C₈	—	4	2	no	—
172	C₅×C₈	—	4	2	no	—
173	C₂²×Q₈	—	2	1	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₄⋊Q₈	—	2	1	no	—
177	C₄⋊Q₈	—	2	1	no	—
178	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	—
179	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	C₃×C₄×C₅	1	2	yes	—
182	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
186	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
187	C₂×C₃×C₅	#332	1	4	yes	direct product factor
188	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
189	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
191	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
192	C₃×Q₈	—	4	2	no	—
193	C₃×Q₈	—	4	2	no	—
194	C₃×Q₈	—	4	4	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	4	4	no	—
198	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
202	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
203	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
204	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
205	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
208	C₃×C₈	—	4	2	no	—
209	C₃×C₈	—	4	2	no	—
210	C₂²×C₅	C₃×#101	1	2	yes	—
211	C₂²×C₅	C₃×#102	1	2	yes	—
212	C₂²×C₅	C₃×#103	1	2	yes	—
213	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	4	4	no	—
217	C₄×C₅	C₃×#102	1	2	yes	—
218	C₄×C₅	C₃×#103	1	2	yes	—
219	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
220	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
221	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
222	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	4	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
229	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
230	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
239	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
240	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
241	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
242	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
243	C₃×C₅	C₂×#332	1	1	yes	characteristic, direct product factor, Hall
244	C₂²×C₃	C₅×#101	1	2	yes	—
245	C₂²×C₃	C₅×#102	1	2	yes	—
246	C₂²×C₃	C₅×#103	1	2	yes	—
247	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	4	4	no	—
251	C₃×C₄	C₅×#102	1	2	yes	—
252	C₃×C₄	C₅×#103	1	2	yes	—
253	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
254	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
255	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
256	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
257	C₂×C₅	C₃×#332	1	4	yes	direct product factor
258	C₂×C₅	C₂×C₃×#101	1	1	yes	characteristic
259	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
260	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
261	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	4	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
272	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
273	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
274	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
275	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
276	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
277	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
278	C₈	—	4	2	no	—
279	C₈	—	4	2	no	—
280	C₂×C₃	C₅×#332	1	4	yes	direct product factor
281	C₂×C₃	C₂×C₅×#101	1	1	yes	characteristic
282	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
283	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
284	C₅	C₂×C₃×#332	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	C₃×C₅×#101	1	2	yes	—
286	C₂²	C₃×C₅×#102	1	2	yes	—
287	C₂²	C₃×C₅×#103	1	2	yes	—
288	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	4	4	no	—
292	C₄	C₃×C₅×#102	1	2	yes	—
293	C₄	C₃×C₅×#103	1	2	yes	—
294	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
295	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
296	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
297	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
298	C₃	C₂×C₅×#332	1	1	yes	characteristic, direct product factor, Sylow
299	C₂	C₃×C₅×#332	1	4	yes	direct product factor
300	C₂	C₂×C₃×C₅×#101	1	1	yes	characteristic
301	C₂	C₂×C₃×C₅×#102	1	1	yes	characteristic
302	C₂	C₂×C₃×C₅×#103	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#332	C₂	direct (N × H)
5. C₂×C₅×#332	C₃	direct (N × H)
20. C₂×C₃×#332	C₅	direct (N × H)
24. C₅×#332	C₂×C₃	direct (N × H)
48. C₃×#332	C₂×C₅	direct (N × H)
63. C₂×#332	C₃×C₅	direct (N × H)
120. #332	C₂×C₃×C₅	direct (N × H)
187. C₂×C₃×C₅	#332	direct (N × H)
243. C₃×C₅	C₂×#332	direct (N × H)
257. C₂×C₅	C₃×#332	direct (N × H)
280. C₂×C₃	C₅×#332	direct (N × H)
284. C₅	C₂×C₃×#332	direct (N × H)
298. C₃	C₂×C₅×#332	direct (N × H)
299. C₂	C₃×C₅×#332	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×C₅×C₄⋊C₈	2
C₃×C₅×#332	2
C₂×C₅×#332	3
C₂×C₃×#332	5

Subgroups — C2×C3×C5×#332 #14274573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#332 #14274573