Subgroups — C₄×#90977 #14295600

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#709	C₂	1	1	yes	maximal, characteristic
2	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90977	C₂	1	2	yes	maximal
5	#11745974	C₂	1	2	yes	maximal, complemented
6	C₄×#3937	—	3	1	no	maximal, Hall
7	C₂×C₃×#709	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×#6710	C₂²	1	2	yes	—
10	C₂×C₄×C₅×Dic₃	—	2	2	no	—
11	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
12	C₃×C₄×C₂²⋊D₅	—	2	2	no	—
13	C₃×#4072	C₂²	1	1	yes	characteristic
14	C₃×C₄×D₂₀	C₄	1	2	yes	complemented
15	C₃×#4018	—	2	1	no	—
16	C₃×C₄²×D₅	—	2	1	no	—
17	C₃×C₄×Dic₁₀	C₄	1	2	yes	complemented
18	#91087	—	2	1	no	—
19	C₄×C₃⋊#168	—	2	1	no	—
20	#90977	C₄	1	8	yes	direct product factor
21	C₄×#837	—	5	1	no	maximal, Hall
22	C₄×#709	D₃	1	1	yes	characteristic
23	C₄³×C₅	—	3	1	no	—
24	C₂×#3937	—	3	2	no	—
25	C₄×#681	—	3	1	no	—
26	#64743	—	3	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₅×Dic₃	—	2	2	no	—
30	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
31	C₃×#709	—	2	2	no	—
32	C₃×#709	C₂×C₄	1	4	yes	—
33	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	—	2	1	no	—
35	C₂×C₃×C₄×D₅	—	2	2	no	—
36	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	#6710	—	2	2	no	—
41	#6710	C₂×C₄	1	4	yes	complemented
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₄×C₅×Dic₃	—	2	8	no	—
44	C₄×C₅×Dic₃	—	2	2	no	—
45	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	complemented
46	C₃×C₂²⋊Dic₅	—	2	1	no	—
47	C₃×#685	—	2	2	no	—
48	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
49	C₃×#683	—	2	2	no	—
50	C₃×C₄×Dic₅	—	2	2	no	—
51	C₃×C₄×Dic₅	—	2	1	no	—
52	C₃×C₄×#41	—	5	1	no	—
53	C₂×#837	—	5	2	no	—
54	C₄²×Dic₃	D₅	1	1	yes	characteristic, complemented
55	C₄×#208	—	5	1	no	—
56	#4532	—	5	2	no	—
57	C₂×#709	C₂×D₃	1	1	yes	characteristic
58	C₂×C₄²×C₅	—	3	2	no	—
59	C₂×C₄²×C₅	—	6	2	no	—
60	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
61	C₄×C₂²⋊D₅	—	2	2	no	—
62	#4072	C₂×D₃	1	1	yes	characteristic
63	C₂×#681	—	3	2	no	—
64	C₄×D₂₀	Dic₃	1	2	yes	complemented
65	#4018	—	2	1	no	—
66	C₄²×D₅	—	2	1	no	—
67	C₄×Dic₁₀	Dic₃	1	2	yes	complemented
68	#3937	—	3	8	no	—
69	#3935	—	6	1	no	—
70	C₄×#168	—	6	1	no	—
71	C₄×#104	—	15	1	no	Sylow
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₃×D₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	2	yes	—
81	C₂×C₅×Dic₃	—	2	8	no	—
82	C₂×C₅×Dic₃	—	2	4	no	—
83	C₃×C₂²⋊D₅	—	2	4	no	—
84	C₂×C₃×Dic₅	—	2	1	no	—
85	C₂×C₃×Dic₅	—	2	2	no	—
86	C₃×D₂₀	—	2	1	no	—
87	C₃×D₂₀	C₄²	1	2	yes	complemented
88	C₃×C₄×D₅	—	2	4	no	—
89	C₃×C₄×D₅	—	4	4	no	—
90	C₃×Dic₁₀	—	2	1	no	—
91	C₃×Dic₁₀	C₄²	1	2	yes	complemented
92	C₃⋊#168	—	2	4	no	—
93	C₂×C₃×#41	—	5	1	no	—
94	C₃×C₄×Q₈	—	5	2	no	—
95	C₃×C₄×D₄	—	5	2	no	—
96	C₃×C₄×D₄	—	10	2	no	—
97	C₃×#117	—	5	1	no	—
98	C₃×#117	—	10	1	no	—
99	C₂×C₃×C₄²	—	10	1	no	—
100	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
101	C₂×C₄×Dic₃	—	2	2	no	—
102	C₂×C₄×Dic₃	C₂×D₅	1	2	yes	—
103	C₂×#208	—	5	2	no	—
104	#837	—	5	8	no	—
105	#803	—	10	1	no	—
106	C₄×C₃⋊C₈	—	10	1	no	—
107	C₂²×C₄×C₅	—	6	2	no	—
108	C₂²×C₄×C₅	#67	1	2	yes	—
109	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
110	C₂×C₂²⋊D₅	—	2	1	no	—
111	#709	—	2	2	no	—
112	#709	C₄×D₃	1	4	yes	—
113	C₂×D₂₀	C₂×Dic₃	1	1	yes	characteristic
114	C₂×C₄×D₅	—	2	1	no	—
115	C₂×C₄×D₅	—	2	2	no	—
116	C₂×Dic₁₀	C₂×Dic₃	1	1	yes	characteristic
117	C₄²×C₅	—	3	4	no	—
118	C₄²×C₅	—	6	8	no	—
119	C₄²×C₅	—	6	2	no	—
120	C₄²×C₅	C₂×Dic₃	1	2	yes	—
121	C₄²×C₅	#67	1	1	yes	characteristic
122	C₄²×C₅	#67	1	1	yes	characteristic
123	C₂²⋊Dic₅	—	2	1	no	—
124	#685	—	2	2	no	—
125	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
126	#683	—	2	2	no	—
127	C₄×Dic₅	—	2	2	no	—
128	C₄×Dic₅	—	2	1	no	—
129	#681	—	3	4	no	—
130	#681	—	6	2	no	—
131	C₂×#168	—	6	2	no	—
132	C₄×#41	—	5	1	no	—
133	C₂×#104	—	15	2	no	—
134	C₄×M₁₆	—	15	1	no	—
135	C₄³	—	3	1	no	—
136	#339	—	15	2	no	—
137	C₂²×C₃×C₅	—	2	2	no	—
138	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
140	C₂×C₃×D₅	—	2	2	no	—
141	C₂×C₃×D₅	—	4	2	no	—
142	C₃×C₄×C₅	—	2	2	no	—
143	C₃×C₄×C₅	#104	1	4	yes	complemented
144	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
145	C₃×C₄×C₅	#117	1	1	yes	characteristic
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×Dic₅	—	2	2	no	—
148	C₃×Dic₅	—	4	2	no	—
149	C₅×Dic₃	—	2	8	no	—
150	C₃×#41	—	5	4	no	—
151	C₃×#41	—	10	2	no	—
152	C₂×C₃×Q₈	—	5	1	no	—
153	C₂×C₃×D₄	—	5	1	no	—
154	C₂×C₃×D₄	—	10	1	no	—
155	C₂²×C₃×C₄	—	10	1	no	—
156	C₂²×C₃×C₄	—	10	2	no	—
157	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
158	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
159	C₂²×Dic₃	—	2	2	no	—
160	C₃×C₄⋊C₄	—	5	2	no	—
161	C₃×C₄⋊C₄	—	10	2	no	—
162	C₃×C₂²⋊C₄	—	10	1	no	—
163	C₃×C₂²⋊C₄	—	10	2	no	—
164	C₃×C₄²	—	10	2	no	—
165	C₃×C₄²	—	10	1	no	—
166	C₃×C₄²	C₄×D₅	1	2	yes	—
167	C₃×C₄²	D₂₀	1	1	yes	characteristic
168	C₃×C₄²	D₂₀	1	1	yes	characteristic
169	C₄×Dic₃	—	2	8	no	—
170	C₄×Dic₃	—	2	2	no	—
171	C₄×Dic₃	C₄×D₅	1	4	yes	complemented
172	#208	—	5	4	no	—
173	#208	—	10	2	no	—
174	C₂×C₃⋊C₈	—	10	2	no	—
175	C₂³×C₅	C₂×#67	1	1	yes	characteristic
176	C₂²×D₅	—	2	1	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	6	8	no	—
179	C₂×C₄×C₅	—	6	4	no	—
180	C₂×C₄×C₅	#217	1	2	yes	—
181	C₂×C₄×C₅	#217	1	2	yes	—
182	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
183	C₂×C₄×C₅	#235	1	2	yes	—
184	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
186	C₂²⋊D₅	—	2	4	no	—
187	C₂×Dic₅	—	2	1	no	—
188	C₂×Dic₅	—	2	2	no	—
189	D₂₀	—	2	1	no	—
190	D₂₀	C₄×Dic₃	1	2	yes	complemented
191	C₄×D₅	—	2	4	no	—
192	C₄×D₅	—	4	4	no	—
193	Dic₁₀	—	2	1	no	—
194	Dic₁₀	C₄×Dic₃	1	2	yes	complemented
195	#168	—	6	4	no	—
196	C₂×#41	—	5	1	no	—
197	C₂×M₁₆	—	15	2	no	—
198	C₄×Q₈	—	5	2	no	—
199	C₄×D₄	—	5	2	no	—
200	C₄×D₄	—	10	2	no	—
201	#117	—	5	1	no	—
202	#117	—	10	1	no	—
203	C₂×C₄²	—	3	2	no	—
204	C₂×C₄²	—	6	2	no	—
205	C₂×C₄²	—	10	1	no	—
206	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
207	#104	—	15	8	no	—
208	#97	—	30	1	no	—
209	C₄×C₈	—	30	1	no	—
210	C₂×C₃×C₅	—	2	2	no	—
211	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₃×C₅	C₂×#104	1	2	yes	—
213	C₃×D₅	—	4	2	no	—
214	C₂³×C₃	—	10	1	no	—
215	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
216	C₃×Q₈	—	5	2	no	—
217	C₃×Q₈	—	10	1	no	—
218	C₃×D₄	—	5	2	no	—
219	C₃×D₄	—	10	4	no	—
220	C₃×D₄	—	10	1	no	—
221	C₂×C₃×C₄	—	2	4	no	—
222	C₂×C₃×C₄	—	10	1	no	—
223	C₂×C₃×C₄	—	10	4	no	—
224	C₂×C₃×C₄	—	10	2	no	—
225	C₂×C₃×C₄	—	20	4	no	—
226	C₂×C₃×C₄	#685	1	2	yes	—
227	C₂×C₃×C₄	#685	1	2	yes	—
228	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
229	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
231	C₂×C₃×C₄	#709	1	2	yes	—
232	C₂×Dic₃	—	2	8	no	—
233	C₂×Dic₃	—	2	4	no	—
234	C₃⋊C₈	—	10	4	no	—
235	C₂²×C₅	—	2	2	no	—
236	C₂²×C₅	C₄×#67	1	2	yes	—
237	C₂²×C₅	C₂×#217	1	1	yes	characteristic
238	C₂×D₅	—	2	2	no	—
239	C₂×D₅	—	4	2	no	—
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	—	6	8	no	—
242	C₄×C₅	#837	1	4	yes	complemented
243	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
244	C₄×C₅	#926	1	1	yes	characteristic
245	C₄×C₅	C₄×#67	1	2	yes	—
246	Dic₅	—	2	2	no	—
247	Dic₅	—	4	2	no	—
248	#41	—	5	4	no	—
249	#41	—	10	2	no	—
250	C₂×Q₈	—	5	1	no	—
251	C₂×D₄	—	5	1	no	—
252	C₂×D₄	—	10	1	no	—
253	C₂²×C₄	—	6	2	no	—
254	C₂²×C₄	—	10	1	no	—
255	C₂²×C₄	—	10	2	no	—
256	C₂²×C₄	#1201	1	2	yes	—
257	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
258	M₁₆	—	15	4	no	—
259	M₁₆	—	30	2	no	—
260	C₂×C₈	—	30	2	no	—
261	C₄⋊C₄	—	5	2	no	—
262	C₄⋊C₄	—	10	2	no	—
263	C₂²⋊C₄	—	10	1	no	—
264	C₂²⋊C₄	—	10	2	no	—
265	C₄²	—	3	4	no	—
266	C₄²	—	6	8	no	—
267	C₄²	—	6	2	no	—
268	C₄²	—	10	2	no	—
269	C₄²	—	10	1	no	—
270	C₄²	D₅×Dic₃	1	2	yes	—
271	C₄²	#1202	1	1	yes	characteristic
272	C₄²	#1202	1	1	yes	characteristic
273	C₃×C₅	C₄×#104	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	2	2	no	—
275	C₂²×C₃	—	10	2	no	—
276	C₂²×C₃	—	20	2	no	—
277	C₂²×C₃	C₂×#685	1	1	yes	characteristic
278	C₂²×C₃	C₄×C₂²⋊D₅	1	2	yes	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	10	2	no	—
281	C₃×C₄	—	20	2	no	—
282	C₃×C₄	#3937	1	4	yes	complemented
283	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
284	C₃×C₄	#4018	1	1	yes	characteristic
285	C₃×C₄	C₄×D₂₀	1	2	yes	—
286	Dic₃	—	2	8	no	—
287	C₂×C₅	—	2	2	no	—
288	C₂×C₅	C₄×#217	1	1	yes	characteristic
289	C₂×C₅	C₂×#837	1	2	yes	—
290	D₅	—	4	2	no	—
291	C₂³	—	10	1	no	—
292	C₂³	C₂×#1201	1	1	yes	characteristic
293	Q₈	—	5	2	no	—
294	Q₈	—	10	1	no	—
295	D₄	—	5	2	no	—
296	D₄	—	10	4	no	—
297	D₄	—	10	1	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	6	8	no	—
300	C₂×C₄	—	6	4	no	—
301	C₂×C₄	—	10	1	no	—
302	C₂×C₄	—	10	4	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	20	4	no	—
305	C₂×C₄	#6665	1	2	yes	—
306	C₂×C₄	#6665	1	2	yes	—
307	C₂×C₄	#6771	1	2	yes	—
308	C₂×C₄	C₄×D₃×D₅	1	2	yes	—
309	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
310	C₂×C₄	C₂×#1202	1	1	yes	characteristic
311	C₈	—	30	4	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	—	20	2	no	—
314	C₂×C₃	C₂×#3937	1	2	yes	—
315	C₂×C₃	C₄×#685	1	1	yes	characteristic
316	C₅	C₄×#837	1	1	yes	characteristic, complemented, Sylow
317	C₂²	—	2	2	no	—
318	C₂²	—	10	2	no	—
319	C₂²	—	20	2	no	—
320	C₂²	C₄×#1201	1	2	yes	—
321	C₂²	C₂×#6665	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	6	8	no	—
324	C₄	—	10	2	no	—
325	C₄	—	20	2	no	—
326	C₄	#90977	1	4	yes	direct product factor
327	C₄	#91357	1	1	yes	characteristic
328	C₄	C₄×D₅×Dic₃	1	1	yes	characteristic
329	C₄	C₄×#1202	1	2	yes	—
330	C₃	C₄×#3937	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	20	2	no	—
333	C₂	C₂×#90977	1	2	yes	—
334	C₂	C₄×#6665	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#1202
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₃×#709	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₄×#104
LCS	G ▷ C₃×C₄×C₅ ▷ 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
2. C₄×#6710	C₂	semidirect
3. C₄²×C₅×Dic₃	C₂	semidirect
5. #11745974	C₂	semidirect
14. C₃×C₄×D₂₀	C₄	semidirect
17. C₃×C₄×Dic₁₀	C₄	semidirect
20. #90977	C₄	direct (N × H)
41. #6710	C₂×C₄	semidirect
45. C₄×C₅×Dic₃	C₂×C₄	semidirect
54. C₄²×Dic₃	D₅	semidirect
64. C₄×D₂₀	Dic₃	semidirect
67. C₄×Dic₁₀	Dic₃	semidirect
87. C₃×D₂₀	C₄²	semidirect
91. C₃×Dic₁₀	C₄²	semidirect
143. C₃×C₄×C₅	#104	semidirect
171. C₄×Dic₃	C₄×D₅	semidirect
190. D₂₀	C₄×Dic₃	semidirect
194. Dic₁₀	C₄×Dic₃	semidirect
242. C₄×C₅	#837	semidirect
273. C₃×C₅	C₄×#104	semidirect
282. C₃×C₄	#3937	semidirect
316. C₅	C₄×#837	semidirect
326. C₄	#90977	direct (N × H)
330. C₃	C₄×#3937	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#709	2
C₄×#6710	2
C₄²×C₅×Dic₃	2
C₂×#90977	2
#11745974	2
C₄×#3937	3
C₄×#837	5

Subgroups — C4×#90977 #14295600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#90977 #14295600