Subgroups — C₂×#11745885 #14312985

Order 1920.

448 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×#4047	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745885	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64735	—	3	1	no	maximal, Hall
6	C₂×C₃×#709	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×#681	C₄	1	1	yes	characteristic
9	C₃×#4047	—	2	1	no	—
10	C₃×#4047	—	2	1	no	—
11	C₃×#4047	C₄	1	2	yes	—
12	C₃×#4047	C₂²	1	2	yes	—
13	C₂×C₃×#676	—	2	1	no	—
14	C₂×C₃×C₈×D₅	—	2	1	no	—
15	C₂×C₃⋊#672	—	2	1	no	—
16	C₂×C₃⋊#672	—	2	1	no	—
17	C₂×C₃⋊#672	C₂²	1	2	yes	complemented
18	C₅×#812	—	2	1	no	—
19	C₅×#812	C₂²	1	2	yes	complemented
20	C₂×C₅×C₃⋊C₁₆	—	2	1	no	—
21	C₂×#4561	—	5	1	no	maximal, Hall
22	C₂×#4047	D₃	1	1	yes	characteristic
23	C₂×C₅×M₃₂	—	3	1	no	—
24	C₂²×#672	—	3	1	no	—
25	#64735	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
28	C₃×#709	—	2	1	no	—
29	C₃×#709	—	2	1	no	—
30	C₃×#709	C₈	1	2	yes	—
31	C₃×#709	C₂×C₄	1	2	yes	—
32	C₂×C₃×D₂₀	C₈	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	—	2	1	no	—
34	C₂×C₃×Dic₁₀	C₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₃×#681	—	2	1	no	—
42	C₃×#681	C₂×C₄	1	2	yes	—
43	C₂×C₃×#168	—	2	1	no	—
44	C₃×#676	—	2	2	no	—
45	C₃×#676	—	2	2	no	—
46	C₃×C₈×D₅	—	2	2	no	—
47	C₃×C₈×D₅	—	2	2	no	—
48	C₃⋊#672	—	2	2	no	—
49	C₅×C₃⋊C₁₆	—	2	2	no	—
50	C₂×C₃×#131	—	5	1	no	—
51	C₂×#812	D₅	1	1	yes	characteristic, complemented
52	C₂²×C₃⋊C₁₆	—	5	1	no	—
53	#4561	—	5	4	no	—
54	C₂×#709	Dic₃	1	1	yes	characteristic
55	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×#681	Dic₃	1	1	yes	characteristic
57	#4047	—	2	1	no	—
58	#4047	—	2	1	no	—
59	#4047	Dic₃	1	2	yes	—
60	#4047	C₂×D₃	1	2	yes	—
61	C₂×#676	—	2	1	no	—
62	C₂×C₈×D₅	—	2	1	no	—
63	C₅×M₃₂	—	3	2	no	—
64	C₅×M₃₂	—	6	1	no	—
65	C₂×C₅×C₁₆	—	6	1	no	—
66	C₂×#672	—	3	2	no	—
67	C₂×#672	—	6	1	no	—
68	C₂×#672	—	6	1	no	—
69	C₂×#350	—	15	1	no	Sylow
70	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
71	C₂²×C₃×D₅	—	2	1	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₃×C₂²⋊D₅	—	2	2	no	—
79	C₃×C₂²⋊D₅	—	2	2	no	—
80	C₂×C₃×Dic₅	—	2	1	no	—
81	C₃×D₂₀	—	2	1	no	—
82	C₃×D₂₀	C₂×C₈	1	2	yes	—
83	C₃×C₄×D₅	—	2	2	no	—
84	C₃×C₄×D₅	—	2	2	no	—
85	C₃×Dic₁₀	—	2	1	no	—
86	C₃×Dic₁₀	C₂×C₈	1	2	yes	—
87	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₃×#168	—	2	2	no	—
92	C₂×C₃×#41	—	5	1	no	—
93	C₃×#131	—	5	2	no	—
94	C₃×#131	—	5	2	no	—
95	C₃×#131	—	10	1	no	—
96	C₃×#131	—	10	1	no	—
97	C₂×C₃×M₁₆	—	5	1	no	—
98	C₂×C₃×M₁₆	—	10	1	no	—
99	C₂²×C₃×C₈	—	10	1	no	—
100	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
101	#812	—	2	1	no	—
102	#812	C₂×D₅	1	2	yes	complemented
103	C₂×C₃⋊C₁₆	—	2	1	no	—
104	C₂×C₃⋊C₁₆	—	5	2	no	—
105	C₂×C₃⋊C₁₆	—	10	1	no	—
106	C₂×C₃⋊C₁₆	—	10	1	no	—
107	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
108	C₂×C₂²⋊D₅	—	2	1	no	—
109	#709	—	2	1	no	—
110	#709	—	2	1	no	—
111	#709	C₃⋊C₈	1	2	yes	—
112	#709	C₂×Dic₃	1	2	yes	—
113	C₂×D₂₀	C₃⋊C₈	1	1	yes	characteristic
114	C₂×C₄×D₅	—	2	1	no	—
115	C₂×Dic₁₀	C₃⋊C₈	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
117	C₂×C₅×C₈	#67	1	1	yes	characteristic
118	C₂×C₅×C₈	#67	1	1	yes	characteristic
119	C₂×C₅×C₈	#67	1	1	yes	characteristic
120	C₂×C₅×C₈	#67	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
122	#681	—	2	1	no	—
123	#681	C₂×Dic₃	1	2	yes	—
124	C₂×#168	—	2	1	no	—
125	#676	—	2	2	no	—
126	#676	—	2	2	no	—
127	C₈×D₅	—	2	2	no	—
128	C₈×D₅	—	2	2	no	—
129	C₅×C₁₆	—	6	2	no	—
130	#672	—	6	2	no	—
131	C₂×#131	—	5	1	no	—
132	C₂×M₃₂	—	3	1	no	—
133	C₂²×C₁₆	—	15	1	no	—
134	#350	—	15	4	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₃×D₅	—	2	2	no	—
140	C₂×C₃×D₅	—	4	2	no	—
141	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
145	C₃×Dic₅	—	2	2	no	—
146	C₃×#41	—	5	2	no	—
147	C₃×#41	—	5	2	no	—
148	C₃×#41	—	10	1	no	—
149	C₃×#41	—	10	1	no	—
150	C₂×C₃×Q₈	—	5	1	no	—
151	C₂×C₃×D₄	—	5	1	no	—
152	C₂×C₃×D₄	—	10	1	no	—
153	C₂²×C₃×C₄	—	10	1	no	—
154	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
155	C₃×M₁₆	—	5	2	no	—
156	C₃×M₁₆	—	10	1	no	—
157	C₃×M₁₆	—	10	2	no	—
158	C₃×M₁₆	—	10	2	no	—
159	C₂×C₃×C₈	—	10	1	no	—
160	C₂×C₃×C₈	—	10	2	no	—
161	C₂×C₃×C₈	—	10	2	no	—
162	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
163	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
164	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
165	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
166	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
167	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₃⋊C₁₆	—	2	2	no	—
169	C₃⋊C₁₆	—	10	2	no	—
170	C₂³×C₅	#208	1	1	yes	characteristic
171	C₂²×D₅	—	2	1	no	—
172	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
174	C₂×C₄×C₅	#208	1	1	yes	characteristic
175	C₂×C₄×C₅	#217	1	1	yes	characteristic
176	C₂×C₄×C₅	#217	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
178	C₂²⋊D₅	—	2	2	no	—
179	C₂²⋊D₅	—	2	2	no	—
180	C₂×Dic₅	—	2	1	no	—
181	D₂₀	—	2	1	no	—
182	D₂₀	C₂×C₃⋊C₈	1	2	yes	—
183	C₄×D₅	—	2	2	no	—
184	C₄×D₅	—	2	2	no	—
185	Dic₁₀	—	2	1	no	—
186	Dic₁₀	C₂×C₃⋊C₈	1	2	yes	—
187	C₅×C₈	#217	1	1	yes	characteristic
188	C₅×C₈	#217	1	1	yes	characteristic
189	C₅×C₈	C₂×#67	1	1	yes	characteristic
190	C₅×C₈	C₂×#67	1	1	yes	characteristic
191	#168	—	2	2	no	—
192	C₂×#41	—	5	1	no	—
193	#131	—	5	2	no	—
194	#131	—	5	2	no	—
195	#131	—	10	1	no	—
196	#131	—	10	1	no	—
197	C₂×M₁₆	—	5	1	no	—
198	C₂×M₁₆	—	10	1	no	—
199	C₂²×C₈	—	10	1	no	—
200	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
201	M₃₂	—	3	2	no	—
202	M₃₂	—	6	1	no	—
203	C₂×C₁₆	—	6	1	no	—
204	C₂×C₁₆	—	15	2	no	—
205	C₂×C₁₆	—	30	1	no	—
206	C₂×C₁₆	—	30	1	no	—
207	C₂×C₃×C₅	—	2	1	no	—
208	C₂×C₃×C₅	—	2	1	no	—
209	C₂×C₃×C₅	#350	1	2	yes	complemented
210	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
211	C₃×D₅	—	4	2	no	—
212	C₂³×C₃	—	10	1	no	—
213	C₂³×C₃	#676	1	1	yes	characteristic
214	C₃×Q₈	—	5	2	no	—
215	C₃×Q₈	—	10	1	no	—
216	C₃×D₄	—	5	2	no	—
217	C₃×D₄	—	10	2	no	—
218	C₃×D₄	—	10	2	no	—
219	C₃×D₄	—	10	1	no	—
220	C₂×C₃×C₄	—	10	1	no	—
221	C₂×C₃×C₄	—	10	2	no	—
222	C₂×C₃×C₄	—	10	2	no	—
223	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
224	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	#676	1	1	yes	characteristic
226	C₂×C₃×C₄	#685	1	1	yes	characteristic
227	C₂×C₃×C₄	#685	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
229	C₃×C₈	—	10	2	no	—
230	C₃×C₈	#685	1	1	yes	characteristic
231	C₃×C₈	#685	1	1	yes	characteristic
232	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
233	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	C₂²×C₅	—	2	2	no	—
235	C₂²×C₅	#830	1	1	yes	characteristic
236	C₂²×C₅	#830	1	1	yes	characteristic
237	C₂²×C₅	C₂×#208	1	1	yes	characteristic
238	C₂×D₅	—	2	2	no	—
239	C₂×D₅	—	4	2	no	—
240	C₄×C₅	#830	1	1	yes	characteristic
241	C₄×C₅	#830	1	1	yes	characteristic
242	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
243	C₄×C₅	C₂×#217	1	1	yes	characteristic
244	Dic₅	—	2	2	no	—
245	#41	—	5	2	no	—
246	#41	—	5	2	no	—
247	#41	—	10	1	no	—
248	#41	—	10	1	no	—
249	C₂×Q₈	—	5	1	no	—
250	C₂×D₄	—	5	1	no	—
251	C₂×D₄	—	10	1	no	—
252	C₂²×C₄	—	10	1	no	—
253	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
254	M₁₆	—	5	2	no	—
255	M₁₆	—	10	1	no	—
256	M₁₆	—	10	2	no	—
257	M₁₆	—	10	2	no	—
258	C₂×C₈	—	10	1	no	—
259	C₂×C₈	—	10	2	no	—
260	C₂×C₈	—	10	2	no	—
261	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
262	C₂×C₈	#1201	1	1	yes	characteristic
263	C₂×C₈	#1201	1	1	yes	characteristic
264	C₂×C₈	#1202	1	1	yes	characteristic
265	C₂×C₈	#1202	1	1	yes	characteristic
266	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
267	C₁₆	—	6	2	no	—
268	C₁₆	—	30	2	no	—
269	C₃×C₅	C₂×#350	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	10	2	no	—
272	C₂²×C₃	—	20	2	no	—
273	C₂²×C₃	#3952	1	1	yes	characteristic
274	C₂²×C₃	#3952	1	1	yes	characteristic
275	C₂²×C₃	C₂×#676	1	1	yes	characteristic
276	C₃×C₄	—	10	2	no	—
277	C₃×C₄	#3952	1	1	yes	characteristic
278	C₃×C₄	#3952	1	1	yes	characteristic
279	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
280	C₃×C₄	C₂×#685	1	1	yes	characteristic
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	2	1	no	—
283	C₂×C₅	#4561	1	2	yes	complemented
284	C₂×C₅	C₂×#830	1	1	yes	characteristic
285	D₅	—	4	2	no	—
286	C₂³	—	10	1	no	—
287	C₂³	#6649	1	1	yes	characteristic
288	Q₈	—	5	2	no	—
289	Q₈	—	10	1	no	—
290	D₄	—	5	2	no	—
291	D₄	—	10	2	no	—
292	D₄	—	10	2	no	—
293	D₄	—	10	1	no	—
294	C₂×C₄	—	10	1	no	—
295	C₂×C₄	—	10	2	no	—
296	C₂×C₄	—	10	2	no	—
297	C₂×C₄	D₅×C₃⋊C₈	1	1	yes	characteristic
298	C₂×C₄	D₅×C₃⋊C₈	1	1	yes	characteristic
299	C₂×C₄	#6649	1	1	yes	characteristic
300	C₂×C₄	#6665	1	1	yes	characteristic
301	C₂×C₄	#6665	1	1	yes	characteristic
302	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
303	C₈	—	10	2	no	—
304	C₈	#6665	1	1	yes	characteristic
305	C₈	#6665	1	1	yes	characteristic
306	C₈	C₂×#1201	1	1	yes	characteristic
307	C₈	C₂×#1202	1	1	yes	characteristic
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	20	2	no	—
311	C₂×C₃	#64735	1	2	yes	complemented
312	C₂×C₃	C₂×#3952	1	1	yes	characteristic
313	C₅	C₂×#4561	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	2	2	no	—
315	C₂²	—	10	2	no	—
316	C₂²	—	20	2	no	—
317	C₂²	#90955	1	1	yes	characteristic
318	C₂²	#90955	1	1	yes	characteristic
319	C₂²	C₂×#6649	1	1	yes	characteristic
320	C₄	—	10	2	no	—
321	C₄	#90955	1	1	yes	characteristic
322	C₄	#90955	1	1	yes	characteristic
323	C₄	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
324	C₄	C₂×#6665	1	1	yes	characteristic
325	C₃	C₂×#64735	1	1	yes	characteristic, complemented, Sylow
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	20	2	no	—
329	C₂	#11745885	1	2	yes	direct product factor
330	C₂	C₂×#90955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#1201
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²⋊C₈
prime radical	C₂×C₃×#709	C₄
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#350
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊#672	C₂	semidirect
3. C₂×C₅×#812	C₂	semidirect
4. #11745885	C₂	direct (N × H)
17. C₂×C₃⋊#672	C₂²	semidirect
19. C₅×#812	C₂²	semidirect
51. C₂×#812	D₅	semidirect
102. #812	C₂×D₅	semidirect
209. C₂×C₃×C₅	#350	semidirect
269. C₃×C₅	C₂×#350	semidirect
283. C₂×C₅	#4561	semidirect
311. C₂×C₃	#64735	semidirect
313. C₅	C₂×#4561	semidirect
325. C₃	C₂×#64735	semidirect
329. C₂	#11745885	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×#4047	2
C₂²×C₃⋊#672	2
C₂×C₅×#812	2
#11745885	2
C₂×#64735	3
C₂×#4561	5

Subgroups — C2×#11745885 #14312985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11745885 #14312985