Subgroups — C₂×C₃×#65375 #14439019

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 332 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, complemented
2	C₂²×C₃×#689	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃×#688	C₂	1	2	yes	maximal
4	C₂×C₃×#4021	C₂	1	1	yes	maximal, characteristic
5	C₃×#65375	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#65375	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
10	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	4	yes	complemented
13	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×#689	—	2	8	no	—
16	C₂×C₃×#689	C₂²	1	8	yes	complemented
17	C₂×C₃×#688	—	2	8	no	—
18	C₂×C₃×#688	C₂²	1	8	yes	—
19	C₂×C₃×#685	—	2	2	no	—
20	C₂²×C₃×#168	C₂²	1	2	yes	—
21	C₃×#4021	—	2	2	no	—
22	C₃×#4021	C₂²	1	4	yes	—
23	C₃×#3935	C₂²	1	4	yes	complemented
24	C₂×C₃×#497	—	5	1	no	maximal, Hall
25	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂²×#689	C₂×C₃	1	2	yes	complemented
27	C₂²×#688	C₂×C₃	1	2	yes	—
28	C₂×#4021	C₂×C₃	1	1	yes	characteristic
29	#65375	C₂×C₃	1	8	yes	direct product factor
30	C₂×#3935	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂³×C₃×D₅	—	2	1	no	—
32	C₂×C₃×C₅×Q₈	—	2	2	no	—
33	C₂×C₃×C₅×Q₈	—	2	4	no	—
34	C₂×C₃×C₅×Q₈	—	2	4	no	—
35	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
36	C₂²×C₃×C₄×C₅	—	2	2	no	—
37	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₃×Dic₅	—	2	1	no	—
40	C₂×C₃×D₂₀	—	2	4	no	—
41	C₂×C₃×D₂₀	—	2	1	no	—
42	C₂×C₃×D₂₀	C₂³	1	2	yes	—
43	C₂×C₃×Dic₁₀	—	2	4	no	—
44	C₂×C₃×Dic₁₀	—	2	1	no	—
45	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
46	C₃×C₅×C₄⋊C₄	—	2	8	no	—
47	C₃×C₄²×C₅	D₄	1	2	yes	—
48	C₃×C₄²×C₅	C₂³	1	2	yes	—
49	C₃×#689	—	2	16	no	—
50	C₃×#688	—	2	16	no	—
51	C₃×#685	—	2	8	no	—
52	C₂×C₃×#168	D₄	1	8	yes	—
53	C₂×C₃×#168	C₂³	1	4	yes	—
54	C₂²×C₃×Q₁₆	—	5	2	no	—
55	C₂²×C₃×SD₁₆	—	5	2	no	—
56	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
57	C₂×C₃×#124	—	5	1	no	—
58	C₃×#497	—	5	8	no	—
59	C₂×C₃×#97	—	5	1	no	—
60	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
61	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
62	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
63	C₅×C₄⋊Q₈	—	2	2	no	—
64	C₅×C₄⋊Q₈	C₂²×C₃	1	4	yes	complemented
65	C₂×C₅×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
67	C₂×#689	—	2	8	no	—
68	C₂×#689	C₂²×C₃	1	8	yes	complemented
69	C₂×#688	—	2	8	no	—
70	C₂×#688	C₂²×C₃	1	8	yes	—
71	C₂×#685	—	2	2	no	—
72	C₂²×#168	C₂²×C₃	1	2	yes	—
73	#4021	—	2	2	no	—
74	#4021	C₂²×C₃	1	4	yes	—
75	#3935	C₂²×C₃	1	4	yes	complemented
76	C₂×#497	—	5	1	no	Sylow
77	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×D₅	—	2	2	no	—
79	C₂²×C₃×D₅	—	4	2	no	—
80	C₂²×C₃×D₅	—	4	4	no	—
81	C₃×C₅×Q₈	—	2	4	no	—
82	C₃×C₅×Q₈	—	2	4	no	—
83	C₃×C₅×Q₈	—	4	4	no	—
84	C₂×C₃×C₄×C₅	—	2	4	no	—
85	C₂×C₃×C₄×C₅	—	2	4	no	—
86	C₂×C₃×C₄×C₅	—	4	4	no	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
90	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₂×C₃×Dic₅	—	2	2	no	—
92	C₂×C₃×Dic₅	—	4	2	no	—
93	C₃×D₂₀	—	2	4	no	—
94	C₃×D₂₀	—	4	2	no	—
95	C₃×Dic₁₀	—	2	4	no	—
96	C₃×Dic₁₀	—	4	2	no	—
97	C₃×#168	C₂×D₄	1	8	yes	—
98	C₂²×C₃×Q₈	—	5	1	no	—
99	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
100	C₂²×C₃×D₄	—	5	1	no	—
101	C₂×C₃×Q₁₆	—	5	8	no	—
102	C₂×C₃×Q₁₆	—	10	8	no	—
103	C₂×C₃×SD₁₆	—	5	8	no	—
104	C₂×C₃×SD₁₆	—	10	8	no	—
105	C₂²×C₃×C₈	—	5	2	no	—
106	C₃×C₄⋊Q₈	—	2	2	no	—
107	C₃×C₄⋊Q₈	C₂×D₅	1	4	yes	complemented
108	C₃×#124	—	5	4	no	—
109	C₃×#124	—	10	2	no	—
110	C₂×C₃×C₄⋊C₄	—	2	2	no	—
111	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
112	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
113	C₃×#97	—	5	4	no	—
114	C₂³×D₅	—	2	1	no	—
115	C₂×C₅×Q₈	—	2	2	no	—
116	C₂×C₅×Q₈	—	2	4	no	—
117	C₂×C₅×Q₈	—	2	4	no	—
118	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
119	C₂²×C₄×C₅	—	2	2	no	—
120	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
121	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
122	C₂²×Dic₅	—	2	1	no	—
123	C₂×D₂₀	—	2	4	no	—
124	C₂×D₂₀	—	2	1	no	—
125	C₂×D₂₀	C₂³×C₃	1	2	yes	—
126	C₂×Dic₁₀	—	2	4	no	—
127	C₂×Dic₁₀	—	2	1	no	—
128	C₂×Dic₁₀	C₂³×C₃	1	2	yes	—
129	C₅×C₄⋊C₄	—	2	8	no	—
130	C₄²×C₅	C₃×D₄	1	2	yes	—
131	C₄²×C₅	C₂³×C₃	1	2	yes	—
132	#689	—	2	16	no	—
133	#688	—	2	16	no	—
134	#685	—	2	8	no	—
135	C₂×#168	C₃×D₄	1	8	yes	—
136	C₂×#168	C₂³×C₃	1	4	yes	—
137	C₂²×Q₁₆	—	5	2	no	—
138	C₂²×SD₁₆	—	5	2	no	—
139	C₂×C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
140	C₂×#124	—	5	1	no	—
141	#497	—	5	8	no	—
142	C₂×#97	—	5	1	no	—
143	C₂²×C₃×C₅	#127	1	2	yes	—
144	C₂²×C₃×C₅	#137	1	4	yes	—
145	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₃×D₅	—	4	2	no	—
147	C₂×C₃×D₅	—	4	4	no	—
148	C₂×C₃×D₅	—	8	4	no	—
149	C₃×C₄×C₅	—	2	4	no	—
150	C₃×C₄×C₅	—	4	4	no	—
151	C₃×C₄×C₅	#127	1	2	yes	—
152	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
153	C₃×Dic₅	—	4	2	no	—
154	C₂⁴×C₃	—	10	1	no	—
155	C₂×C₃×Q₈	—	2	2	no	—
156	C₂×C₃×Q₈	—	2	4	no	—
157	C₂×C₃×Q₈	—	2	4	no	—
158	C₂×C₃×Q₈	—	5	2	no	—
159	C₂×C₃×Q₈	—	10	1	no	—
160	C₂×C₃×Q₈	—	10	4	no	—
161	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
162	C₂×C₃×D₄	—	5	2	no	—
163	C₂×C₃×D₄	—	10	4	no	—
164	C₂×C₃×D₄	—	10	1	no	—
165	C₂²×C₃×C₄	—	2	2	no	—
166	C₂²×C₃×C₄	—	10	1	no	—
167	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
168	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
169	C₃×Q₁₆	—	10	16	no	—
170	C₃×SD₁₆	—	10	16	no	—
171	C₂×C₃×C₈	—	5	8	no	—
172	C₂×C₃×C₈	—	5	4	no	—
173	C₃×C₄⋊C₄	—	2	8	no	—
174	C₃×C₂²⋊C₄	—	10	8	no	—
175	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
176	C₃×C₄²	C₂²×D₅	1	2	yes	—
177	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂²×D₅	—	2	2	no	—
179	C₂²×D₅	—	4	2	no	—
180	C₂²×D₅	—	4	4	no	—
181	C₅×Q₈	—	2	4	no	—
182	C₅×Q₈	—	2	4	no	—
183	C₅×Q₈	—	4	4	no	—
184	C₂×C₄×C₅	—	2	4	no	—
185	C₂×C₄×C₅	—	2	4	no	—
186	C₂×C₄×C₅	—	4	4	no	—
187	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
188	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
189	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
191	C₂×Dic₅	—	2	2	no	—
192	C₂×Dic₅	—	4	2	no	—
193	D₂₀	—	2	4	no	—
194	D₂₀	—	4	2	no	—
195	Dic₁₀	—	2	4	no	—
196	Dic₁₀	—	4	2	no	—
197	#168	C₂×C₃×D₄	1	8	yes	—
198	C₂²×Q₈	—	5	1	no	—
199	C₂²×Q₈	C₂×C₃×D₅	1	2	yes	—
200	C₂²×D₄	—	5	1	no	—
201	C₂×Q₁₆	—	5	8	no	—
202	C₂×Q₁₆	—	10	8	no	—
203	C₂×SD₁₆	—	5	8	no	—
204	C₂×SD₁₆	—	10	8	no	—
205	C₂²×C₈	—	5	2	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	C₂×C₃×D₅	1	4	yes	complemented
208	#124	—	5	4	no	—
209	#124	—	10	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	—	10	2	no	—
212	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
213	#97	—	5	4	no	—
214	C₂×C₃×C₅	#497	1	4	yes	complemented
215	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
216	C₂×C₃×C₅	C₂×#137	1	2	yes	—
217	C₃×D₅	—	8	2	no	—
218	C₂³×C₃	—	10	2	no	—
219	C₂³×C₃	—	20	2	no	—
220	C₂³×C₃	—	20	4	no	—
221	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
222	C₃×Q₈	—	2	4	no	—
223	C₃×Q₈	—	2	4	no	—
224	C₃×Q₈	—	4	4	no	—
225	C₃×Q₈	—	10	4	no	—
226	C₃×Q₈	—	20	2	no	—
227	C₃×D₄	—	10	4	no	—
228	C₃×D₄	—	20	2	no	—
229	C₂×C₃×C₄	—	2	4	no	—
230	C₂×C₃×C₄	—	2	4	no	—
231	C₂×C₃×C₄	—	4	4	no	—
232	C₂×C₃×C₄	—	10	2	no	—
233	C₂×C₃×C₄	—	20	2	no	—
234	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
235	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
236	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
238	C₃×C₈	—	5	8	no	—
239	C₂²×C₅	C₃×#127	1	2	yes	—
240	C₂²×C₅	C₃×#137	1	4	yes	—
241	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
242	C₂×D₅	—	4	4	no	—
243	C₂×D₅	—	4	2	no	—
244	C₂×D₅	—	8	4	no	—
245	C₄×C₅	—	2	4	no	—
246	C₄×C₅	—	4	4	no	—
247	C₄×C₅	C₃×#127	1	2	yes	—
248	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
249	Dic₅	—	4	2	no	—
250	C₂⁴	—	10	1	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	5	2	no	—
255	C₂×Q₈	—	10	1	no	—
256	C₂×Q₈	—	10	4	no	—
257	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	—
258	C₂×D₄	—	5	2	no	—
259	C₂×D₄	—	10	4	no	—
260	C₂×D₄	—	10	1	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	10	1	no	—
263	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
264	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
265	Q₁₆	—	10	16	no	—
266	SD₁₆	—	10	16	no	—
267	C₂×C₈	—	5	8	no	—
268	C₂×C₈	—	5	4	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₂²⋊C₄	—	10	8	no	—
271	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
272	C₄²	C₂²×C₃×D₅	1	2	yes	—
273	C₃×C₅	C₂×#497	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	20	2	no	—
275	C₂²×C₃	—	20	4	no	—
276	C₂²×C₃	—	40	4	no	—
277	C₂²×C₃	#4086	1	2	yes	—
278	C₂²×C₃	#4088	1	4	yes	—
279	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
280	C₃×C₄	—	2	4	no	—
281	C₃×C₄	—	4	4	no	—
282	C₃×C₄	—	20	2	no	—
283	C₃×C₄	#4086	1	2	yes	—
284	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
285	C₂×C₅	C₃×#497	1	4	yes	complemented
286	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
287	C₂×C₅	C₂×C₃×#137	1	2	yes	—
288	D₅	—	8	2	no	—
289	C₂³	—	10	2	no	—
290	C₂³	—	20	2	no	—
291	C₂³	—	20	4	no	—
292	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
293	Q₈	—	2	4	no	—
294	Q₈	—	2	4	no	—
295	Q₈	—	4	4	no	—
296	Q₈	—	10	4	no	—
297	Q₈	—	20	2	no	—
298	D₄	—	10	4	no	—
299	D₄	—	20	2	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	20	2	no	—
305	C₂×C₄	C₃×D₄×D₅	1	4	yes	—
306	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	4	yes	—
307	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
308	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
309	C₈	—	5	8	no	—
310	C₂×C₃	—	40	2	no	—
311	C₂×C₃	#65375	1	4	yes	direct product factor
312	C₂×C₃	C₂×#4086	1	1	yes	characteristic
313	C₂×C₃	C₂×#4088	1	2	yes	—
314	C₅	C₂×C₃×#497	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	20	2	no	—
316	C₂²	—	20	4	no	—
317	C₂²	—	40	4	no	—
318	C₂²	C₃×#4086	1	2	yes	—
319	C₂²	C₃×#4088	1	4	yes	—
320	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	4	4	no	—
323	C₄	—	20	2	no	—
324	C₄	C₃×#4086	1	2	yes	—
325	C₄	C₂×C₃×D₄×D₅	1	2	yes	—
326	C₃	C₂×#65375	1	1	yes	characteristic, direct product factor, Sylow
327	C₂	—	40	2	no	—
328	C₂	C₃×#65375	1	4	yes	direct product factor
329	C₂	C₂×C₃×#4086	1	1	yes	characteristic
330	C₂	C₂×C₃×#4088	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×D₂₀	C₂²
hypercenter	C₂×C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₂×C₃×#497
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. C₂²×C₃×#689	C₂	semidirect
5. C₃×#65375	C₂	direct (N × H)
6. C₂×C₃×#3935	C₂	semidirect
7. C₂×#65375	C₃	direct (N × H)
12. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
16. C₂×C₃×#689	C₂²	semidirect
23. C₃×#3935	C₂²	semidirect
25. C₂×C₅×C₄⋊Q₈	C₂×C₃	semidirect
26. C₂²×#689	C₂×C₃	semidirect
29. #65375	C₂×C₃	direct (N × H)
30. C₂×#3935	C₂×C₃	semidirect
56. C₂×C₃×C₄⋊Q₈	D₅	semidirect
64. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
68. C₂×#689	C₂²×C₃	semidirect
75. #3935	C₂²×C₃	semidirect
107. C₃×C₄⋊Q₈	C₂×D₅	semidirect
139. C₂×C₄⋊Q₈	C₃×D₅	semidirect
207. C₄⋊Q₈	C₂×C₃×D₅	semidirect
214. C₂×C₃×C₅	#497	semidirect
273. C₃×C₅	C₂×#497	semidirect
285. C₂×C₅	C₃×#497	semidirect
311. C₂×C₃	#65375	direct (N × H)
314. C₅	C₂×C₃×#497	semidirect
326. C₃	C₂×#65375	direct (N × H)
328. C₂	C₃×#65375	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
C₂²×C₃×#689	2
C₂²×C₃×#688	2
C₂×C₃×#4021	2
C₃×#65375	2
C₂×C₃×#3935	2
C₂×#65375	3
C₂×C₃×#497	5

Subgroups — C2×C3×#65375 #14439019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65375 #14439019