Subgroups — C₄×#91290 #14411479

Order 1920.

1144 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	C₄×#6817	C₂	1	1	yes	maximal, characteristic
2	C₂×#91290	C₂	1	2	yes	maximal
3	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748324	C₂	1	2	yes	maximal, complemented
6	C₄×#6651	C₂	1	2	yes	maximal, complemented
7	#11746981	C₂	1	4	yes	maximal, complemented
8	C₄×#6648	C₂	1	2	yes	maximal, complemented
9	C₄×#4053	—	3	1	no	maximal, Hall
10	C₂×#6817	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₄×#1220	C₄	1	4	yes	—
13	#91815	C₂²	1	1	yes	characteristic
14	C₄×D₆₀	C₄	1	2	yes	—
15	#91761	C₂²	1	2	yes	—
16	C₄²×D₁₅	C₂²	1	2	yes	—
17	C₄×Dic₃₀	C₄	1	2	yes	—
18	C₂×C₅×#208	C₂²	1	2	yes	—
19	C₂²×C₃×#168	C₂²	1	2	yes	—
20	C₂×#6651	C₂²	1	4	yes	—
21	#91290	C₄	1	16	yes	direct product factor
22	C₂×#6648	C₂²	1	4	yes	—
23	C₅×#803	C₂²	1	2	yes	—
24	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
25	C₃×C₄×#168	C₂²	1	1	yes	characteristic
26	C₃×C₄×#168	C₂²	1	2	yes	—
27	C₃×C₄×#168	C₂²	1	1	yes	characteristic
28	#90954	C₂²	1	4	yes	—
29	#90951	C₂²	1	4	yes	—
30	C₄×#901	—	5	1	no	maximal, Hall
31	C₂×#4053	—	3	2	no	—
32	C₄×#709	—	3	1	no	—
33	C₄×C₅×M₁₆	—	3	1	no	—
34	#65411	—	3	2	no	—
35	C₂×C₄×#168	D₃	1	1	yes	characteristic, complemented
36	#64997	—	3	4	no	—
37	C₄×#676	—	3	2	no	—
38	C₄×C₈×D₅	—	3	2	no	—
39	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×#1220	C₂×C₄	1	2	yes	—
42	#6817	C₂×C₄	1	8	yes	—
43	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₅	C₂×C₄	1	4	yes	—
45	C₂×C₄×D₁₅	C₂³	1	2	yes	—
46	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	#6719	C₂×C₄	1	2	yes	—
51	#6714	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
53	#6712	C₂×C₄	1	4	yes	—
54	C₄×Dic₁₅	C₂×C₄	1	4	yes	—
55	C₄×Dic₁₅	C₂³	1	2	yes	—
56	C₅×#208	C₂×C₄	1	8	yes	complemented
57	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
58	C₂×C₃×#168	C₂×C₄	1	8	yes	complemented
59	C₂×C₃×#168	C₂³	1	2	yes	—
60	C₂×C₃×#168	C₂³	1	2	yes	—
61	#6651	C₂×C₄	1	16	yes	complemented
62	#6648	C₂×C₄	1	16	yes	complemented
63	C₂×#901	—	5	2	no	—
64	C₄×#235	—	5	1	no	—
65	C₂×C₃×C₄×C₈	—	5	1	no	—
66	#5101	—	5	2	no	—
67	C₄×#208	D₅	1	1	yes	characteristic, complemented
68	#4689	—	5	4	no	—
69	C₄×ΓL(1,𝔽₂₅)	—	5	2	no	—
70	C₄×D₃×C₈	—	5	2	no	—
71	C₂×#709	—	3	1	no	—
72	C₂×C₅×M₁₆	—	3	2	no	—
73	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
74	C₄×C₂²⋊D₅	—	3	4	no	—
75	#4072	—	3	1	no	—
76	C₂²×#168	C₂×D₃	1	2	yes	—
77	#4053	—	3	16	no	—
78	C₂×#676	—	3	4	no	—
79	C₂×C₈×D₅	—	3	4	no	—
80	C₄×D₂₀	—	3	2	no	—
81	#4018	—	3	2	no	—
82	C₄²×D₅	—	3	2	no	—
83	C₄×Dic₁₀	—	3	2	no	—
84	C₅×#97	—	3	2	no	—
85	C₄×C₅×C₈	—	3	2	no	—
86	#3947	—	3	4	no	—
87	C₈×Dic₅	—	3	4	no	—
88	C₄×#168	C₂×D₃	1	1	yes	characteristic
89	C₄×#168	C₂×D₃	1	1	yes	characteristic
90	C₄×#168	C₂×D₃	1	2	yes	—
91	C₄×#131	—	15	1	no	Sylow
92	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×D₁₅	C₂²×C₄	1	2	yes	—
94	C₂×C₃×C₄×C₅	—	2	4	no	—
95	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
98	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
99	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	#1220	C₄²	1	8	yes	—
101	C₂×Dic₁₅	C₂²×C₄	1	4	yes	—
102	C₂×Dic₁₅	C₂²×C₄	1	2	yes	—
103	D₆₀	C₄²	1	4	yes	—
104	C₄×D₁₅	—	2	8	no	—
105	C₄×D₁₅	C₂²×C₄	1	8	yes	—
106	Dic₃₀	C₄²	1	4	yes	—
107	C₃×#168	C₂²×C₄	1	4	yes	—
108	C₃×#168	C₂²×C₄	1	4	yes	—
109	C₅×C₃⋊C₈	C₂²×C₄	1	8	yes	—
110	C₂×#235	—	5	1	no	—
111	C₂²×C₃×C₈	—	5	2	no	—
112	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
113	C₄×#67	—	5	4	no	—
114	#926	—	5	1	no	—
115	C₂×#208	C₂×D₅	1	2	yes	—
116	#901	—	5	16	no	—
117	C₂×ΓL(1,𝔽₂₅)	—	5	4	no	—
118	C₂×D₃×C₈	—	5	4	no	—
119	C₄×D₁₂	—	5	2	no	—
120	#872	—	5	2	no	—
121	C₄²×D₃	—	5	2	no	—
122	C₄×Dic₆	—	5	2	no	—
123	C₃×C₄×C₈	—	5	1	no	—
124	C₃×C₄×C₈	—	5	1	no	—
125	C₃×C₄×C₈	—	5	2	no	—
126	#815	—	5	4	no	—
127	C₈×Dic₃	—	5	4	no	—
128	#803	C₂×D₅	1	2	yes	—
129	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
130	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
131	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₂²⋊D₅	—	3	2	no	—
133	#709	—	3	8	no	—
134	C₂×D₂₀	—	3	1	no	—
135	C₂×C₄×D₅	—	3	2	no	—
136	C₂×C₄×D₅	—	3	4	no	—
137	C₂×Dic₁₀	—	3	1	no	—
138	C₅×M₁₆	—	3	8	no	—
139	C₂×C₅×C₈	—	3	4	no	—
140	C₄²×C₅	C₄×D₃	1	2	yes	—
141	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₂²⋊Dic₅	—	3	2	no	—
144	#685	—	3	4	no	—
145	C₄⋊Dic₅	—	3	2	no	—
146	#683	—	3	4	no	—
147	C₄×Dic₅	—	3	4	no	—
148	C₄×Dic₅	—	3	2	no	—
149	C₂×#168	C₄×D₃	1	8	yes	complemented
150	C₂×#168	C₂²×D₃	1	2	yes	—
151	C₂×#168	C₂²×D₃	1	2	yes	—
152	#676	—	3	16	no	—
153	C₈×D₅	—	3	16	no	—
154	C₂×#131	—	15	2	no	—
155	C₄×#41	—	15	1	no	—
156	#405	—	15	4	no	—
157	#405	—	15	2	no	—
158	C₄×M₁₆	—	3	1	no	—
159	C₄×M₁₆	—	15	2	no	—
160	C₂×C₄×C₈	—	5	1	no	—
161	C₂×C₄×C₈	—	15	2	no	—
162	C₂²×C₃×C₅	—	2	2	no	—
163	C₂²×C₃×C₅	C₂×C₄²	1	2	yes	—
164	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
165	C₂×D₁₅	—	2	4	no	—
166	C₂×D₁₅	C₂×C₄²	1	4	yes	—
167	C₃×C₄×C₅	—	2	2	no	—
168	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
169	C₃×C₄×C₅	#131	1	4	yes	complemented
170	C₃×C₄×C₅	C₂³×C₄	1	2	yes	—
171	Dic₁₅	—	2	4	no	—
172	Dic₁₅	C₂×C₄²	1	4	yes	—
173	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
174	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
175	C₂×#67	—	5	2	no	—
176	#235	—	5	8	no	—
177	C₂×D₁₂	—	5	1	no	—
178	C₂×C₄×D₃	—	5	2	no	—
179	C₂×C₄×D₃	—	5	4	no	—
180	C₂×Dic₆	—	5	1	no	—
181	C₂×C₃×C₈	—	5	8	no	—
182	C₂×C₃×C₈	—	5	2	no	—
183	C₂×C₃×C₈	—	5	2	no	—
184	C₃×C₄²	C₄×D₅	1	2	yes	—
185	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
186	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
187	#217	—	5	2	no	—
188	(C₂×D₃)⋊C₄	—	5	4	no	—
189	C₄⋊Dic₃	—	5	2	no	—
190	Dic₃⋊C₄	—	5	4	no	—
191	C₄×Dic₃	—	5	4	no	—
192	C₄×Dic₃	—	5	2	no	—
193	#208	C₄×D₅	1	8	yes	complemented
194	C₂×C₃⋊C₈	C₂²×D₅	1	4	yes	—
195	ΓL(1,𝔽₂₅)	—	5	16	no	—
196	D₃×C₈	—	5	16	no	—
197	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂²×D₅	—	3	2	no	—
199	C₂×C₄×C₅	—	2	4	no	—
200	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
201	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
202	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
203	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
204	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₂²⋊D₅	—	3	8	no	—
206	C₂×Dic₅	—	3	2	no	—
207	C₂×Dic₅	—	3	4	no	—
208	D₂₀	—	3	4	no	—
209	C₄×D₅	—	3	8	no	—
210	C₄×D₅	—	6	8	no	—
211	Dic₁₀	—	3	4	no	—
212	C₅×C₈	—	3	8	no	—
213	#168	C₂×C₄×D₃	1	4	yes	—
214	#168	C₂×C₄×D₃	1	4	yes	—
215	C₂×#41	—	15	1	no	—
216	#131	—	15	16	no	—
217	C₂×M₁₆	—	3	2	no	—
218	C₂×M₁₆	—	15	4	no	—
219	C₂²×C₈	—	5	2	no	—
220	C₂²×C₈	—	15	4	no	—
221	C₄×Q₈	—	15	2	no	—
222	C₄×D₄	—	15	4	no	—
223	C₄×D₄	—	15	2	no	—
224	#117	—	15	1	no	—
225	#117	—	15	2	no	—
226	C₂×C₄²	—	15	2	no	—
227	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
228	#97	—	3	2	no	—
229	#97	—	15	4	no	—
230	C₄×C₈	—	3	2	no	—
231	C₄×C₈	—	5	2	no	—
232	C₄×C₈	—	5	1	no	—
233	C₄×C₈	—	5	1	no	—
234	C₄×C₈	—	15	4	no	—
235	C₂×C₃×C₅	—	2	2	no	—
236	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
237	C₂×C₃×C₅	C₂×#131	1	2	yes	—
238	D₁₅	—	2	4	no	—
239	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
240	C₂²×D₃	—	5	2	no	—
241	C₂×C₃×C₄	—	2	4	no	—
242	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
243	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
244	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
246	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
247	#67	—	5	8	no	—
248	C₂×Dic₃	—	5	2	no	—
249	C₂×Dic₃	—	5	4	no	—
250	D₁₂	—	5	4	no	—
251	C₄×D₃	—	5	8	no	—
252	C₄×D₃	—	10	8	no	—
253	Dic₆	—	5	4	no	—
254	C₃×C₈	—	5	4	no	—
255	C₃×C₈	—	5	4	no	—
256	C₃⋊C₈	C₂×C₄×D₅	1	8	yes	—
257	C₂²×C₅	—	2	2	no	—
258	C₂²×C₅	C₄²×D₃	1	2	yes	—
259	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
260	C₂×D₅	—	3	4	no	—
261	C₂×D₅	—	6	4	no	—
262	C₄×C₅	—	2	2	no	—
263	C₄×C₅	C₄²×D₃	1	2	yes	—
264	C₄×C₅	#901	1	4	yes	complemented
265	C₄×C₅	C₂²×C₄×D₃	1	2	yes	—
266	Dic₅	—	3	4	no	—
267	Dic₅	—	6	4	no	—
268	#41	—	15	8	no	—
269	C₂×Q₈	—	15	1	no	—
270	C₂×D₄	—	15	2	no	—
271	C₂×D₄	—	15	1	no	—
272	C₂²×C₄	—	15	2	no	—
273	C₂²×C₄	—	15	4	no	—
274	C₂²×C₄	#1200	1	2	yes	—
275	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
276	M₁₆	—	3	8	no	—
277	M₁₆	—	15	16	no	—
278	C₂×C₈	—	3	4	no	—
279	C₂×C₈	—	5	8	no	—
280	C₂×C₈	—	5	2	no	—
281	C₂×C₈	—	5	2	no	—
282	C₂×C₈	—	15	16	no	—
283	C₄⋊C₄	—	15	4	no	—
284	C₄⋊C₄	—	15	2	no	—
285	C₂²⋊C₄	—	15	2	no	—
286	C₂²⋊C₄	—	15	4	no	—
287	C₄²	—	15	4	no	—
288	C₄²	—	15	2	no	—
289	C₄²	#1200	1	2	yes	—
290	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
291	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
292	C₃×C₅	C₄×#131	1	1	yes	characteristic, complemented, Hall
293	C₂²×C₃	—	2	2	no	—
294	C₂²×C₃	C₄²×D₅	1	2	yes	—
295	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
296	C₂×D₃	—	5	4	no	—
297	C₂×D₃	—	10	4	no	—
298	C₃×C₄	—	2	2	no	—
299	C₃×C₄	C₄²×D₅	1	2	yes	—
300	C₃×C₄	#4053	1	4	yes	complemented
301	C₃×C₄	C₂²×C₄×D₅	1	2	yes	—
302	Dic₃	—	5	4	no	—
303	Dic₃	—	10	4	no	—
304	C₂×C₅	—	2	2	no	—
305	C₂×C₅	C₂×C₄²×D₃	1	1	yes	characteristic
306	C₂×C₅	C₂×#901	1	2	yes	—
307	D₅	—	6	4	no	—
308	C₂³	—	15	2	no	—
309	C₂³	C₂×#1200	1	1	yes	characteristic
310	Q₈	—	15	4	no	—
311	D₄	—	15	8	no	—
312	D₄	—	15	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	15	2	no	—
315	C₂×C₄	—	15	8	no	—
316	C₂×C₄	—	15	4	no	—
317	C₂×C₄	—	30	8	no	—
318	C₂×C₄	C₄×D₃×D₅	1	4	yes	—
319	C₂×C₄	C₂×#1200	1	2	yes	—
320	C₂×C₄	C₂×#1200	1	1	yes	characteristic
321	C₂×C₄	C₂×#1200	1	2	yes	—
322	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
323	C₈	—	3	8	no	—
324	C₈	—	5	4	no	—
325	C₈	—	5	4	no	—
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	C₂×C₄²×D₅	1	1	yes	characteristic
328	C₂×C₃	C₂×#4053	1	2	yes	—
329	D₃	—	10	4	no	—
330	C₅	C₄×#901	1	1	yes	characteristic, complemented, Sylow
331	C₂²	—	2	2	no	—
332	C₂²	—	15	4	no	—
333	C₂²	—	30	4	no	—
334	C₂²	C₄×#1200	1	2	yes	—
335	C₂²	C₂²×#1200	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	15	4	no	—
338	C₄	—	30	4	no	—
339	C₄	#91290	1	4	yes	direct product factor
340	C₄	C₄×#1200	1	2	yes	—
341	C₄	C₂×C₄×D₃×D₅	1	2	yes	—
342	C₃	C₄×#4053	1	1	yes	characteristic, complemented, Sylow
343	C₂	—	2	2	no	—
344	C₂	—	30	4	no	—
345	C₂	C₂×#91290	1	2	yes	—
346	C₂	C₂×C₄×#1200	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂²×C₄²
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂×#6817	C₂²
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₄×#131
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×#208	C₂	semidirect
4. C₂×C₃×C₄×#168	C₂	semidirect
5. #11748324	C₂	semidirect
6. C₄×#6651	C₂	semidirect
7. #11746981	C₂	semidirect
8. C₄×#6648	C₂	semidirect
21. #91290	C₄	direct (N × H)
35. C₂×C₄×#168	D₃	semidirect
56. C₅×#208	C₂×C₄	semidirect
58. C₂×C₃×#168	C₂×C₄	semidirect
61. #6651	C₂×C₄	semidirect
62. #6648	C₂×C₄	semidirect
67. C₄×#208	D₅	semidirect
149. C₂×#168	C₄×D₃	semidirect
169. C₃×C₄×C₅	#131	semidirect
193. #208	C₄×D₅	semidirect
264. C₄×C₅	#901	semidirect
292. C₃×C₅	C₄×#131	semidirect
300. C₃×C₄	#4053	semidirect
330. C₅	C₄×#901	semidirect
339. C₄	#91290	direct (N × H)
342. C₃	C₄×#4053	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6817	2
C₂×#91290	2
C₄×C₅×#208	2
C₂×C₃×C₄×#168	2
#11748324	2
C₄×#6651	2
#11746981	2
C₄×#6648	2
C₄×#4053	3
C₄×#901	5

Subgroups — C4×#91290 #14411479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#91290 #14411479