Subgroups — C₄×#90955 #14295022

Order 1920.

872 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90955	C₂	1	2	yes	maximal
5	#11745966	C₂	1	2	yes	maximal, complemented
6	C₄×#3952	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₄×D₅	C₄	1	2	yes	—
8	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊#168	C₂²	1	2	yes	—
11	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄×Dic₅	C₄	1	2	yes	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₃×C₄²×D₅	—	2	2	no	—
15	C₃×C₄²×D₅	—	2	2	no	—
16	C₄×C₃⋊#168	—	2	2	no	—
17	C₄×C₅×C₃⋊C₈	—	2	2	no	—
18	#90955	C₄	1	8	yes	direct product factor
19	C₄×#830	—	5	1	no	maximal, Hall
20	C₂×C₄²×D₅	D₃	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	—	3	1	no	—
22	C₂×#3952	—	3	2	no	—
23	C₂×C₄×#168	—	3	1	no	—
24	#64770	—	3	2	no	—
25	C₂³×C₃×D₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×D₅	—	2	4	no	—
31	C₂×C₃×C₄×D₅	—	2	4	no	—
32	C₂×C₃×C₄×D₅	—	2	1	no	—
33	C₂×C₃×C₄×D₅	—	2	1	no	—
34	C₂×C₃×C₄×D₅	—	2	2	no	—
35	C₂×C₃×C₄×D₅	C₈	1	8	yes	complemented
36	C₂×C₃×C₄×D₅	C₂×C₄	1	4	yes	—
37	C₃×C₄²×C₅	D₄	1	2	yes	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₂×C₃⋊#168	—	2	2	no	—
40	C₂×C₃⋊#168	—	2	2	no	—
41	C₂×C₃⋊#168	C₂×C₄	1	4	yes	complemented
42	C₂×C₅×C₃⋊C₈	—	2	2	no	—
43	C₂×C₅×C₃⋊C₈	—	2	2	no	—
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
45	C₃×C₄×Dic₅	—	2	4	no	—
46	C₃×C₄×Dic₅	—	2	1	no	—
47	C₃×C₄×Dic₅	—	2	1	no	—
48	C₂²×C₃×C₄²	—	5	1	no	—
49	C₂×#830	—	5	2	no	—
50	C₂×C₄×C₃⋊C₈	—	5	1	no	—
51	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
52	#4524	—	5	2	no	—
53	C₂²×C₄×D₅	Dic₃	1	2	yes	—
54	C₂²×C₄×D₅	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₅×C₈	—	3	2	no	—
56	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄×Dic₅	Dic₃	1	2	yes	—
58	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
59	C₂²×#168	—	3	2	no	—
60	C₄²×D₅	—	2	2	no	—
61	C₄²×D₅	—	2	2	no	—
62	C₄×C₅×C₈	—	6	2	no	—
63	#3952	—	3	8	no	—
64	C₄×#168	—	6	2	no	—
65	C₄×C₂²⋊C₈	—	15	1	no	Sylow
66	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₃×D₅	—	2	1	no	—
68	C₂²×C₃×D₅	—	2	1	no	—
69	C₂²×C₃×D₅	—	2	2	no	—
70	C₂²×C₃×D₅	C₄²	1	2	yes	—
71	C₂²×C₃×D₅	C₂×C₈	1	4	yes	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
73	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
74	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	#41	1	2	yes	—
79	C₂×C₃×Dic₅	—	2	1	no	—
80	C₂×C₃×Dic₅	—	2	2	no	—
81	C₂×C₃×Dic₅	—	2	1	no	—
82	C₂×C₃×Dic₅	—	2	2	no	—
83	C₂×C₃×Dic₅	C₄²	1	2	yes	—
84	C₂×C₃×Dic₅	C₂×C₈	1	4	yes	—
85	C₃×C₄×D₅	—	2	16	no	—
86	C₃×C₄×D₅	—	2	4	no	—
87	C₃×C₄×D₅	—	2	4	no	—
88	C₃⋊#168	—	2	4	no	—
89	C₅×C₃⋊C₈	—	2	4	no	—
90	C₂³×C₃×C₄	—	5	2	no	—
91	C₂³×C₃×C₄	—	5	1	no	—
92	C₂×C₃×C₄²	—	5	2	no	—
93	C₂×C₃×C₄²	—	5	1	no	—
94	C₂×C₃×C₄²	—	10	2	no	—
95	C₂×C₃×C₄²	—	10	2	no	—
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	C₂²×C₃⋊C₈	—	5	2	no	—
98	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
99	#830	—	5	8	no	—
100	C₄×C₃⋊C₈	—	2	2	no	—
101	C₄×C₃⋊C₈	—	10	2	no	—
102	C₂³×D₅	C₂×Dic₃	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
104	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
105	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
106	C₂²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
107	C₂×C₄×D₅	—	2	4	no	—
108	C₂×C₄×D₅	—	2	4	no	—
109	C₂×C₄×D₅	—	2	2	no	—
110	C₂×C₄×D₅	—	2	1	no	—
111	C₂×C₄×D₅	—	2	1	no	—
112	C₂×C₄×D₅	C₃⋊C₈	1	8	yes	complemented
113	C₂×C₄×D₅	C₄×D₃	1	4	yes	—
114	C₂×C₅×C₈	—	3	4	no	—
115	C₂×C₅×C₈	—	6	2	no	—
116	C₂×C₅×C₈	—	6	2	no	—
117	C₄²×C₅	#67	1	2	yes	—
118	C₄²×C₅	#67	1	2	yes	—
119	C₄×Dic₅	—	2	4	no	—
120	C₄×Dic₅	—	2	1	no	—
121	C₄×Dic₅	—	2	1	no	—
122	C₂×#168	—	3	4	no	—
123	C₂×#168	—	6	2	no	—
124	C₂×#168	—	6	2	no	—
125	C₂²×C₄²	—	5	1	no	—
126	C₂×C₂²⋊C₈	—	15	2	no	—
127	C₂×C₄×C₈	—	3	1	no	—
128	C₂×C₄×C₈	—	15	1	no	—
129	#336	—	15	2	no	—
130	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
131	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₃×C₅	#117	1	1	yes	characteristic
133	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
134	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
135	C₂×C₃×D₅	—	2	2	no	—
136	C₂×C₃×D₅	—	2	2	no	—
137	C₂×C₃×D₅	—	2	4	no	—
138	C₂×C₃×D₅	—	2	4	no	—
139	C₂×C₃×D₅	C₄×C₈	1	4	yes	complemented
140	C₃×C₄×C₅	C₂²⋊C₈	1	8	yes	complemented
141	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
142	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
143	C₃×Dic₅	—	2	2	no	—
144	C₃×Dic₅	—	2	4	no	—
145	C₂⁴×C₃	—	5	1	no	—
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	5	2	no	—
148	C₂²×C₃×C₄	—	5	8	no	—
149	C₂²×C₃×C₄	—	5	4	no	—
150	C₂²×C₃×C₄	—	10	4	no	—
151	C₂²×C₃×C₄	—	10	4	no	—
152	C₂²×C₃×C₄	—	10	1	no	—
153	C₂²×C₃×C₄	—	10	2	no	—
154	C₂²×C₃×C₄	—	10	1	no	—
155	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
156	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
157	C₃×C₄²	—	10	4	no	—
158	C₃×C₄²	—	10	1	no	—
159	C₃×C₄²	—	10	1	no	—
160	C₃×C₄²	D₂₀	1	2	yes	—
161	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
162	C₂×C₃⋊C₈	—	2	2	no	—
163	C₂×C₃⋊C₈	—	2	2	no	—
164	C₂×C₃⋊C₈	—	5	4	no	—
165	C₂×C₃⋊C₈	—	10	2	no	—
166	C₂×C₃⋊C₈	—	10	2	no	—
167	C₂×C₃⋊C₈	C₄×D₅	1	4	yes	complemented
168	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
169	C₂²×D₅	—	2	2	no	—
170	C₂²×D₅	—	2	1	no	—
171	C₂²×D₅	—	2	1	no	—
172	C₂²×D₅	C₂×C₃⋊C₈	1	4	yes	—
173	C₂²×D₅	C₄×Dic₃	1	2	yes	—
174	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
175	C₂×C₄×C₅	#208	1	4	yes	—
176	C₂×C₄×C₅	#217	1	4	yes	—
177	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
178	C₂×C₄×C₅	#235	1	2	yes	—
179	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
181	C₂×Dic₅	—	2	2	no	—
182	C₂×Dic₅	—	2	1	no	—
183	C₂×Dic₅	—	2	2	no	—
184	C₂×Dic₅	—	2	1	no	—
185	C₂×Dic₅	C₂×C₃⋊C₈	1	4	yes	—
186	C₂×Dic₅	C₄×Dic₃	1	2	yes	—
187	C₄×D₅	—	2	16	no	—
188	C₄×D₅	—	2	4	no	—
189	C₄×D₅	—	2	4	no	—
190	C₅×C₈	—	6	4	no	—
191	#168	—	6	4	no	—
192	C₂³×C₄	—	5	2	no	—
193	C₂³×C₄	—	5	1	no	—
194	C₂²×C₈	—	3	2	no	—
195	C₂²×C₈	—	15	2	no	—
196	C₂×C₄²	—	5	2	no	—
197	C₂×C₄²	—	5	1	no	—
198	C₂×C₄²	—	10	2	no	—
199	C₂×C₄²	—	10	2	no	—
200	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
201	C₂²⋊C₈	—	15	8	no	—
202	C₄×C₈	—	6	2	no	—
203	C₄×C₈	—	30	2	no	—
204	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
206	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
207	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	2	yes	—
208	C₂×C₃×C₅	#431	1	2	yes	—
209	C₃×D₅	—	2	4	no	—
210	C₂³×C₃	—	5	2	no	—
211	C₂³×C₃	—	5	4	no	—
212	C₂³×C₃	—	10	1	no	—
213	C₂³×C₃	—	10	2	no	—
214	C₂³×C₃	—	10	1	no	—
215	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂×C₃×C₄	—	5	2	no	—
217	C₂×C₃×C₄	—	5	4	no	—
218	C₂×C₃×C₄	—	10	1	no	—
219	C₂×C₃×C₄	—	10	2	no	—
220	C₂×C₃×C₄	—	10	2	no	—
221	C₂×C₃×C₄	—	10	1	no	—
222	C₂×C₃×C₄	—	10	16	no	—
223	C₂×C₃×C₄	—	10	4	no	—
224	C₂×C₃×C₄	—	10	4	no	—
225	C₂×C₃×C₄	C₈×D₅	1	4	yes	—
226	C₂×C₃×C₄	#676	1	4	yes	—
227	C₂×C₃×C₄	#685	1	4	yes	—
228	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
229	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
230	C₂×C₃×C₄	#709	1	2	yes	—
231	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	C₃⋊C₈	—	2	4	no	—
233	C₃⋊C₈	—	10	4	no	—
234	C₂²×C₅	C₂²×C₃⋊C₈	1	2	yes	—
235	C₂²×C₅	C₂×#208	1	2	yes	—
236	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
237	C₂²×C₅	#926	1	1	yes	characteristic
238	C₂²×C₅	C₂×#217	1	1	yes	characteristic
239	C₂×D₅	—	2	2	no	—
240	C₂×D₅	—	2	4	no	—
241	C₂×D₅	—	2	4	no	—
242	C₂×D₅	—	2	2	no	—
243	C₂×D₅	C₄×C₃⋊C₈	1	4	yes	complemented
244	C₄×C₅	#830	1	8	yes	complemented
245	C₄×C₅	C₄×#67	1	2	yes	—
246	C₄×C₅	C₄×#67	1	2	yes	—
247	Dic₅	—	2	4	no	—
248	Dic₅	—	2	2	no	—
249	C₂⁴	—	5	1	no	—
250	C₂²×C₄	—	5	2	no	—
251	C₂²×C₄	—	5	1	no	—
252	C₂²×C₄	—	5	4	no	—
253	C₂²×C₄	—	5	8	no	—
254	C₂²×C₄	—	10	4	no	—
255	C₂²×C₄	—	10	2	no	—
256	C₂²×C₄	—	10	1	no	—
257	C₂²×C₄	—	10	1	no	—
258	C₂²×C₄	—	10	4	no	—
259	C₂²×C₄	D₅×Dic₃	1	2	yes	—
260	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
261	C₂×C₈	—	3	4	no	—
262	C₂×C₈	—	6	2	no	—
263	C₂×C₈	—	6	2	no	—
264	C₂×C₈	—	15	4	no	—
265	C₂×C₈	—	30	2	no	—
266	C₂×C₈	—	30	2	no	—
267	C₄²	—	10	4	no	—
268	C₄²	—	10	1	no	—
269	C₄²	—	10	1	no	—
270	C₄²	#1201	1	2	yes	—
271	C₄²	#1202	1	2	yes	—
272	C₃×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
273	C₂²×C₃	—	5	4	no	—
274	C₂²×C₃	—	10	4	no	—
275	C₂²×C₃	—	10	2	no	—
276	C₂²×C₃	—	10	4	no	—
277	C₂²×C₃	—	10	2	no	—
278	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
279	C₂²×C₃	#4018	1	1	yes	characteristic
280	C₂²×C₃	C₂×C₈×D₅	1	2	yes	—
281	C₂²×C₃	C₂×#676	1	2	yes	—
282	C₂²×C₃	C₂×#685	1	1	yes	characteristic
283	C₃×C₄	—	10	4	no	—
284	C₃×C₄	—	10	2	no	—
285	C₃×C₄	#3952	1	8	yes	complemented
286	C₃×C₄	C₄×D₂₀	1	2	yes	—
287	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
288	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
289	C₂×C₅	C₄×#208	1	1	yes	characteristic
290	C₂×C₅	#4925	1	2	yes	—
291	C₂×C₅	C₂×#830	1	2	yes	—
292	C₂×C₅	C₄×#217	1	1	yes	characteristic
293	D₅	—	2	4	no	—
294	C₂³	—	5	4	no	—
295	C₂³	—	5	2	no	—
296	C₂³	—	10	1	no	—
297	C₂³	—	10	2	no	—
298	C₂³	—	10	1	no	—
299	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
300	C₂×C₄	—	5	4	no	—
301	C₂×C₄	—	5	2	no	—
302	C₂×C₄	—	10	1	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	10	2	no	—
305	C₂×C₄	—	10	1	no	—
306	C₂×C₄	—	10	4	no	—
307	C₂×C₄	—	10	4	no	—
308	C₂×C₄	—	10	16	no	—
309	C₂×C₄	D₅×C₃⋊C₈	1	4	yes	—
310	C₂×C₄	#6649	1	4	yes	—
311	C₂×C₄	#6665	1	4	yes	—
312	C₂×C₄	#6771	1	2	yes	—
313	C₂×C₄	C₄×D₃×D₅	1	2	yes	—
314	C₂×C₄	C₂×#1201	1	1	yes	characteristic
315	C₂×C₄	C₂×#1202	1	1	yes	characteristic
316	C₈	—	6	4	no	—
317	C₈	—	30	4	no	—
318	C₂×C₃	—	10	4	no	—
319	C₂×C₃	C₄×C₈×D₅	1	1	yes	characteristic
320	C₂×C₃	#64974	1	2	yes	—
321	C₂×C₃	C₄×#676	1	1	yes	characteristic
322	C₂×C₃	C₄×#685	1	1	yes	characteristic
323	C₂×C₃	C₂×#3952	1	2	yes	—
324	C₅	C₄×#830	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	5	4	no	—
326	C₂²	—	10	4	no	—
327	C₂²	—	10	2	no	—
328	C₂²	—	10	2	no	—
329	C₂²	—	10	4	no	—
330	C₂²	C₂×D₅×C₃⋊C₈	1	2	yes	—
331	C₂²	C₂×#6649	1	2	yes	—
332	C₂²	#91357	1	1	yes	characteristic
333	C₂²	C₄×D₅×Dic₃	1	1	yes	characteristic
334	C₂²	C₂×#6665	1	1	yes	characteristic
335	C₄	—	10	2	no	—
336	C₄	—	10	4	no	—
337	C₄	#90955	1	8	yes	direct product factor
338	C₄	C₄×#1201	1	2	yes	—
339	C₄	C₄×#1202	1	2	yes	—
340	C₃	C₄×#3952	1	1	yes	characteristic, complemented, Sylow
341	C₂	—	10	4	no	—
342	C₂	#11746868	1	2	yes	—
343	C₂	C₄×D₅×C₃⋊C₈	1	1	yes	characteristic
344	C₂	C₄×#6649	1	1	yes	characteristic
345	C₂	C₂×#90955	1	2	yes	—
346	C₂	C₄×#6665	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₃×D₅	C₂×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypocenter	C₃×C₅	C₄×C₂²⋊C₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₃⋊#168	C₂	semidirect
3. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
5. #11745966	C₂	semidirect
18. #90955	C₄	direct (N × H)
35. C₂×C₃×C₄×D₅	C₈	semidirect
41. C₂×C₃⋊#168	C₂×C₄	semidirect
44. C₂×C₅×C₃⋊C₈	C₂×C₄	semidirect
51. C₂×C₄×C₃⋊C₈	D₅	semidirect
112. C₂×C₄×D₅	C₃⋊C₈	semidirect
139. C₂×C₃×D₅	C₄×C₈	semidirect
140. C₃×C₄×C₅	C₂²⋊C₈	semidirect
167. C₂×C₃⋊C₈	C₄×D₅	semidirect
243. C₂×D₅	C₄×C₃⋊C₈	semidirect
244. C₄×C₅	#830	semidirect
272. C₃×C₅	C₄×C₂²⋊C₈	semidirect
285. C₃×C₄	#3952	semidirect
324. C₅	C₄×#830	semidirect
337. C₄	#90955	direct (N × H)
340. C₃	C₄×#3952	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₄²×D₅	2
C₂×C₄×C₃⋊#168	2
C₂×C₄×C₅×C₃⋊C₈	2
C₂×#90955	2
#11745966	2
C₄×#3952	3
C₄×#830	5

Subgroups — C4×#90955 #14295022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#90955 #14295022