Subgroups — C₄×#91109 #14295582

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	C₄×#6817	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91109	C₂	1	2	yes	maximal
4	C₄²×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746376	C₂	1	2	yes	maximal, complemented
6	C₄×#3957	—	3	1	no	maximal, Hall
7	C₂×#6817	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₄×#1220	—	2	2	no	—
11	#91815	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₅	—	2	2	no	—
13	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
14	C₄×D₆₀	C₄	1	2	yes	complemented
15	#91761	—	2	1	no	—
16	C₄²×D₁₅	—	2	1	no	—
17	C₄×Dic₃₀	C₄	1	2	yes	complemented
18	C₃×C₅×#97	—	2	1	no	—
19	C₃×C₄×C₅×C₈	—	2	1	no	—
20	#91109	C₄	1	8	yes	direct product factor
21	C₄×#825	—	5	1	no	maximal, Hall
22	C₄×#709	—	3	1	no	—
23	C₄×C₅×M₁₆	D₃	1	1	yes	characteristic, complemented
24	C₂×#3957	—	3	2	no	—
25	C₄²×Dic₅	—	3	1	no	—
26	#64775	—	3	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×#1220	—	2	1	no	—
30	C₂²×Dic₁₅	—	2	2	no	—
31	#6817	—	2	2	no	—
32	#6817	C₂×C₄	1	4	yes	—
33	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₅	—	2	1	no	—
35	C₂×C₄×D₁₅	—	2	2	no	—
36	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×M₁₆	—	2	2	no	—
38	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	complemented
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	#6719	—	2	1	no	—
44	#6714	—	2	2	no	—
45	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
46	#6712	—	2	2	no	—
47	C₄×Dic₁₅	—	2	8	no	—
48	C₄×Dic₁₅	—	2	2	no	—
49	C₄×Dic₁₅	—	2	1	no	—
50	C₄×Dic₁₅	—	2	2	no	—
51	C₄×Dic₁₅	C₂×C₄	1	4	yes	complemented
52	C₄×#235	—	5	1	no	—
53	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
54	C₂×#825	—	5	2	no	—
55	C₄²×Dic₃	—	5	1	no	—
56	#4555	—	5	2	no	—
57	C₂×#709	—	3	1	no	—
58	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
59	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
60	C₄×C₂²⋊D₅	—	6	2	no	—
61	#4072	—	3	1	no	—
62	C₂×C₄×Dic₅	—	3	2	no	—
63	C₂×C₄×Dic₅	—	6	2	no	—
64	C₄×D₂₀	—	3	2	no	—
65	#4018	—	6	1	no	—
66	C₄²×D₅	—	6	1	no	—
67	C₄×Dic₁₀	—	3	2	no	—
68	C₅×#97	—	2	1	no	—
69	C₄×C₅×C₈	—	2	1	no	—
70	#3957	—	3	8	no	—
71	C₄×#104	—	15	1	no	Sylow
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×D₁₅	—	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	#1220	—	2	4	no	—
82	C₂×Dic₁₅	—	2	8	no	—
83	C₂×Dic₁₅	—	2	1	no	—
84	C₂×Dic₁₅	—	2	4	no	—
85	C₂×Dic₁₅	—	2	2	no	—
86	D₆₀	—	2	1	no	—
87	D₆₀	C₄²	1	2	yes	complemented
88	C₄×D₁₅	—	2	4	no	—
89	C₄×D₁₅	—	4	4	no	—
90	Dic₃₀	—	2	1	no	—
91	Dic₃₀	C₄²	1	2	yes	complemented
92	C₃×C₅×C₈	—	2	4	no	—
93	C₂×#235	—	5	1	no	—
94	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
95	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
96	C₄×#67	—	10	2	no	—
97	#926	—	5	1	no	—
98	C₂×C₄×Dic₃	—	5	2	no	—
99	C₂×C₄×Dic₃	—	10	2	no	—
100	C₄×D₁₂	—	5	2	no	—
101	#872	—	10	1	no	—
102	C₄²×D₃	—	10	1	no	—
103	C₄×Dic₆	—	5	2	no	—
104	C₃×#97	—	2	1	no	—
105	C₃×C₄×C₈	—	2	1	no	—
106	#825	—	5	8	no	—
107	C₂²×C₄×C₅	D₁₂	1	2	yes	—
108	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₂²⋊D₅	—	6	1	no	—
110	C₂²×Dic₅	—	6	2	no	—
111	#709	—	3	4	no	—
112	#709	—	6	2	no	—
113	C₂×D₂₀	—	3	1	no	—
114	C₂×C₄×D₅	—	6	1	no	—
115	C₂×C₄×D₅	—	6	2	no	—
116	C₂×Dic₁₀	—	3	1	no	—
117	C₅×M₁₆	—	2	2	no	—
118	C₅×M₁₆	C₄×D₃	1	4	yes	complemented
119	C₂×C₅×C₈	—	2	2	no	—
120	C₄²×C₅	C₄×D₃	1	2	yes	—
121	C₄²×C₅	#67	1	1	yes	characteristic
122	C₄²×C₅	#67	1	1	yes	characteristic
123	C₂²⋊Dic₅	—	6	1	no	—
124	#685	—	6	2	no	—
125	C₄⋊Dic₅	—	3	2	no	—
126	#683	—	6	2	no	—
127	C₄×Dic₅	—	3	4	no	—
128	C₄×Dic₅	—	6	8	no	—
129	C₄×Dic₅	—	6	2	no	—
130	C₄×Dic₅	—	6	2	no	—
131	C₄×Dic₅	—	6	1	no	—
132	C₄×#41	—	15	1	no	—
133	C₂×#104	—	15	2	no	—
134	C₄×M₁₆	D₁₅	1	1	yes	characteristic, complemented
135	C₄³	—	15	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×D₁₅	—	2	2	no	—
141	C₂×D₁₅	—	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	Dic₁₅	—	2	8	no	—
148	Dic₁₅	—	2	2	no	—
149	Dic₁₅	—	4	2	no	—
150	C₂²×C₃×C₄	D₂₀	1	2	yes	—
151	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
152	C₂×#67	—	10	1	no	—
153	C₂²×Dic₃	—	10	2	no	—
154	#235	—	5	4	no	—
155	#235	—	10	2	no	—
156	C₂×D₁₂	—	5	1	no	—
157	C₂×C₄×D₃	—	10	1	no	—
158	C₂×C₄×D₃	—	10	2	no	—
159	C₂×Dic₆	—	5	1	no	—
160	C₃×M₁₆	—	2	2	no	—
161	C₃×M₁₆	C₄×D₅	1	4	yes	complemented
162	C₂×C₃×C₈	—	2	2	no	—
163	C₃×C₄²	C₄×D₅	1	2	yes	—
164	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
165	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
166	#217	—	10	1	no	—
167	(C₂×D₃)⋊C₄	—	10	2	no	—
168	C₄⋊Dic₃	—	5	2	no	—
169	Dic₃⋊C₄	—	10	2	no	—
170	C₄×Dic₃	—	5	4	no	—
171	C₄×Dic₃	—	10	8	no	—
172	C₄×Dic₃	—	10	2	no	—
173	C₄×Dic₃	—	10	2	no	—
174	C₄×Dic₃	—	10	1	no	—
175	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
176	C₂²×D₅	—	6	1	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
179	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
180	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
182	C₂×C₄×C₅	#235	1	2	yes	—
183	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
184	C₂²⋊D₅	—	6	4	no	—
185	C₂×Dic₅	—	6	8	no	—
186	C₂×Dic₅	—	6	4	no	—
187	C₂×Dic₅	—	6	1	no	—
188	C₂×Dic₅	—	6	2	no	—
189	D₂₀	—	3	2	no	—
190	D₂₀	—	6	1	no	—
191	C₄×D₅	—	6	4	no	—
192	C₄×D₅	—	12	4	no	—
193	Dic₁₀	—	3	2	no	—
194	Dic₁₀	—	6	1	no	—
195	C₅×C₈	—	2	4	no	—
196	C₂×#41	—	15	1	no	—
197	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
198	C₄×Q₈	—	15	2	no	—
199	C₄×D₄	—	15	2	no	—
200	C₄×D₄	—	30	2	no	—
201	#117	—	15	1	no	—
202	#117	—	30	1	no	—
203	C₂×C₄²	—	15	2	no	—
204	C₂×C₄²	—	30	2	no	—
205	C₂×C₄²	—	30	1	no	—
206	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
207	#104	—	15	8	no	—
208	#97	—	2	1	no	—
209	C₄×C₈	—	2	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	D₁₅	—	4	2	no	—
214	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
215	C₂²×D₃	—	10	1	no	—
216	C₂×C₃×C₄	—	2	4	no	—
217	C₂×C₃×C₄	#685	1	2	yes	—
218	C₂×C₃×C₄	#685	1	2	yes	—
219	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
221	C₂×C₃×C₄	#709	1	2	yes	—
222	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
223	#67	—	10	4	no	—
224	C₂×Dic₃	—	10	8	no	—
225	C₂×Dic₃	—	10	4	no	—
226	C₂×Dic₃	—	10	1	no	—
227	C₂×Dic₃	—	10	2	no	—
228	D₁₂	—	5	2	no	—
229	D₁₂	—	10	1	no	—
230	C₄×D₃	—	10	4	no	—
231	C₄×D₃	—	20	4	no	—
232	Dic₆	—	5	2	no	—
233	Dic₆	—	10	1	no	—
234	C₃×C₈	—	2	4	no	—
235	C₂²×C₅	—	2	2	no	—
236	C₂²×C₅	C₄×D₁₂	1	2	yes	—
237	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
238	C₂×D₅	—	6	2	no	—
239	C₂×D₅	—	12	2	no	—
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	#825	1	4	yes	complemented
242	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
243	C₄×C₅	#872	1	1	yes	characteristic
244	C₄×C₅	C₄×#67	1	2	yes	—
245	Dic₅	—	6	8	no	—
246	Dic₅	—	6	2	no	—
247	Dic₅	—	12	2	no	—
248	#41	—	15	4	no	—
249	#41	—	30	2	no	—
250	C₂×Q₈	—	15	1	no	—
251	C₂×D₄	—	15	1	no	—
252	C₂×D₄	—	30	1	no	—
253	C₂²×C₄	—	30	2	no	—
254	C₂²×C₄	—	30	1	no	—
255	C₂²×C₄	—	30	2	no	—
256	C₂²×C₄	D₆₀	1	2	yes	—
257	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
258	M₁₆	—	2	2	no	—
259	M₁₆	C₄×D₁₅	1	4	yes	complemented
260	C₂×C₈	—	2	2	no	—
261	C₄⋊C₄	—	15	2	no	—
262	C₄⋊C₄	—	30	2	no	—
263	C₂²⋊C₄	—	30	1	no	—
264	C₂²⋊C₄	—	30	2	no	—
265	C₄²	—	15	4	no	—
266	C₄²	—	30	8	no	—
267	C₄²	—	30	2	no	—
268	C₄²	—	30	2	no	—
269	C₄²	—	30	1	no	—
270	C₄²	C₄×D₁₅	1	2	yes	—
271	C₄²	#1220	1	1	yes	characteristic
272	C₄²	#1220	1	1	yes	characteristic
273	C₃×C₅	C₄×#104	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	2	2	no	—
275	C₂²×C₃	C₄×D₂₀	1	2	yes	—
276	C₂²×C₃	C₂×#685	1	1	yes	characteristic
277	C₂×D₃	—	10	2	no	—
278	C₂×D₃	—	20	2	no	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	#3957	1	4	yes	complemented
281	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
282	C₃×C₄	#4018	1	1	yes	characteristic
283	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
284	Dic₃	—	10	8	no	—
285	Dic₃	—	10	2	no	—
286	Dic₃	—	20	2	no	—
287	C₂×C₅	—	2	2	no	—
288	C₂×C₅	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
289	C₂×C₅	C₂×#825	1	2	yes	—
290	D₅	—	12	2	no	—
291	C₂³	—	30	1	no	—
292	C₂³	C₂×D₆₀	1	1	yes	characteristic
293	Q₈	—	15	2	no	—
294	Q₈	—	30	1	no	—
295	D₄	—	15	2	no	—
296	D₄	—	30	4	no	—
297	D₄	—	30	1	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	30	8	no	—
300	C₂×C₄	—	30	4	no	—
301	C₂×C₄	—	30	1	no	—
302	C₂×C₄	—	30	4	no	—
303	C₂×C₄	—	30	2	no	—
304	C₂×C₄	—	60	4	no	—
305	C₂×C₄	#6714	1	2	yes	—
306	C₂×C₄	#6714	1	2	yes	—
307	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
309	C₂×C₄	#6817	1	2	yes	—
310	C₂×C₄	C₂×#1220	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	C₄×#685	1	1	yes	characteristic
314	C₂×C₃	C₂×#3957	1	2	yes	—
315	D₃	—	20	2	no	—
316	C₅	C₄×#825	1	1	yes	characteristic, complemented, Sylow
317	C₂²	—	2	2	no	—
318	C₂²	—	30	2	no	—
319	C₂²	—	60	2	no	—
320	C₂²	C₄×D₆₀	1	2	yes	—
321	C₂²	C₂×#6714	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	30	8	no	—
324	C₄	—	30	2	no	—
325	C₄	—	60	2	no	—
326	C₄	#91109	1	4	yes	direct product factor
327	C₄	C₄²×D₁₅	1	1	yes	characteristic
328	C₄	#91761	1	1	yes	characteristic
329	C₄	C₄×#1220	1	2	yes	—
330	C₃	C₄×#3957	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	60	2	no	—
333	C₂	C₄×#6714	1	1	yes	characteristic
334	C₂	C₂×#91109	1	2	yes	—

	subgroup	G/H
center	C₄²	#1220
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×#6817	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₄×M₁₆	D₁₅
hypocenter	C₃×C₅	C₄×#104
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×M₁₆	C₂	semidirect
4. C₄²×Dic₁₅	C₂	semidirect
5. #11746376	C₂	semidirect
14. C₄×D₆₀	C₄	semidirect
17. C₄×Dic₃₀	C₄	semidirect
20. #91109	C₄	direct (N × H)
23. C₄×C₅×M₁₆	D₃	semidirect
38. C₃×C₅×M₁₆	C₂×C₄	semidirect
51. C₄×Dic₁₅	C₂×C₄	semidirect
53. C₃×C₄×M₁₆	D₅	semidirect
87. D₆₀	C₄²	semidirect
91. Dic₃₀	C₄²	semidirect
118. C₅×M₁₆	C₄×D₃	semidirect
134. C₄×M₁₆	D₁₅	semidirect
143. C₃×C₄×C₅	#104	semidirect
161. C₃×M₁₆	C₄×D₅	semidirect
241. C₄×C₅	#825	semidirect
259. M₁₆	C₄×D₁₅	semidirect
273. C₃×C₅	C₄×#104	semidirect
280. C₃×C₄	#3957	semidirect
316. C₅	C₄×#825	semidirect
326. C₄	#91109	direct (N × H)
330. C₃	C₄×#3957	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
C₃×C₄×C₅×M₁₆	2
C₂×#91109	2
C₄²×Dic₁₅	2
#11746376	2
C₄×#3957	3
C₄×#825	5

Subgroups — C4×#91109 #14295582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#91109 #14295582