Subgroups — C₂×#11746904 #14437891

Order 1920.

872 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#875	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91087	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90972	C₂	1	2	yes	maximal, complemented
5	C₂×#90966	C₂	1	2	yes	maximal
6	#11746904	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65344	—	3	1	no	maximal, Hall
8	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃×Dic₁₀	—	2	1	no	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊#168	C₂²	1	2	yes	—
13	C₂×C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
14	C₂×C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
15	C₅×#875	—	2	2	no	—
16	C₅×#875	C₂²	1	4	yes	—
17	C₂×C₃×C₄⋊Dic₅	—	2	1	no	—
18	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
19	C₃×#4015	—	2	2	no	—
20	C₃×#4015	C₂²	1	4	yes	complemented
21	#91087	C₂²	1	4	yes	complemented
22	#90972	C₂²	1	8	yes	complemented
23	#90966	C₂²	1	8	yes	—
24	C₂×#5089	—	5	1	no	maximal, Hall
25	C₂×C₅×#124	—	3	1	no	—
26	C₂×#4015	D₃	1	1	yes	characteristic, complemented
27	C₂×#3967	—	3	2	no	—
28	C₂×#3964	—	3	2	no	—
29	#65344	—	3	8	no	—
30	C₂×#3935	—	3	1	no	—
31	C₂³×C₅×D₃	—	2	1	no	—
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₅×Dic₃	—	2	1	no	—
36	C₂×C₅×D₁₂	—	2	4	no	—
37	C₂×C₅×D₁₂	—	2	1	no	—
38	C₂×C₅×D₁₂	C₂³	1	2	yes	—
39	C₂×C₅×Dic₆	—	2	4	no	—
40	C₂×C₅×Dic₆	—	2	1	no	—
41	C₂×C₅×Dic₆	C₂³	1	2	yes	—
42	C₂²×C₃×Dic₅	—	2	2	no	—
43	C₂×C₃×Dic₁₀	—	2	4	no	—
44	C₂×C₃×Dic₁₀	—	4	4	no	—
45	C₃×C₄²×C₅	D₄	1	2	yes	—
46	C₃×C₄²×C₅	C₂³	1	2	yes	—
47	C₂×C₃⋊#168	—	2	4	no	—
48	C₂×C₃⋊#168	C₂³	1	4	yes	—
49	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
50	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
51	C₃×C₄⋊Dic₅	—	2	2	no	—
52	C₃×C₄⋊Dic₅	—	2	4	no	—
53	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
54	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
55	C₂×#875	D₅	1	1	yes	characteristic
56	#5089	—	5	8	no	—
57	C₂×#810	—	5	2	no	—
58	C₂×#809	—	5	2	no	—
59	C₂×#803	—	5	1	no	—
60	C₂²×C₅×Q₈	—	3	1	no	—
61	C₂²×C₅×D₄	—	3	1	no	—
62	C₂²×Dic₁₀	—	2	1	no	—
63	C₅×#124	—	3	4	no	—
64	C₅×#124	—	6	2	no	—
65	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
66	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
67	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
68	C₂×C₄⋊Dic₅	—	2	1	no	—
69	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
70	C₂²×#168	—	3	2	no	—
71	#4015	—	2	2	no	—
72	#4015	C₂×D₃	1	4	yes	complemented
73	#3967	—	3	8	no	—
74	#3964	—	3	8	no	—
75	#3935	—	3	4	no	—
76	C₂×#489	—	15	1	no	Sylow
77	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₅×D₃	—	2	2	no	—
79	C₂²×C₅×D₃	—	4	2	no	—
80	C₂²×C₅×D₃	—	4	2	no	—
81	C₂²×C₅×D₃	—	4	2	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	4	yes	—
88	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₂×C₅×Dic₃	—	2	2	no	—
90	C₂×C₅×Dic₃	—	4	2	no	—
91	C₅×D₁₂	—	2	4	no	—
92	C₅×D₁₂	—	4	2	no	—
93	C₅×Dic₆	—	2	4	no	—
94	C₅×Dic₆	—	4	2	no	—
95	C₂×C₃×Dic₅	—	2	4	no	—
96	C₂×C₃×Dic₅	—	4	4	no	—
97	C₃×Dic₁₀	—	4	8	no	—
98	C₃⋊#168	—	2	4	no	—
99	C₂²×C₃×Q₈	—	10	1	no	—
100	C₂²×D₁₂	C₂×D₅	1	1	yes	characteristic
101	C₂²×Dic₆	C₂×D₅	1	1	yes	characteristic
102	C₃×C₄⋊Q₈	—	5	4	no	—
103	C₃×C₄⋊Q₈	—	10	2	no	—
104	C₂×C₃×C₄⋊C₄	—	5	2	no	—
105	C₂×C₃×C₄⋊C₄	—	10	1	no	—
106	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
107	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
108	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
109	C₂²×C₃⋊C₈	—	5	2	no	—
110	#875	—	2	2	no	—
111	#875	C₂×D₅	1	4	yes	—
112	#810	—	5	8	no	—
113	#809	—	5	8	no	—
114	#803	—	5	4	no	—
115	C₂⁴×C₅	—	6	1	no	—
116	C₂×C₅×Q₈	—	3	2	no	—
117	C₂×C₅×Q₈	—	6	1	no	—
118	C₂×C₅×Q₈	—	6	4	no	—
119	C₂×C₅×D₄	—	3	2	no	—
120	C₂×C₅×D₄	—	6	4	no	—
121	C₂×C₅×D₄	—	6	1	no	—
122	C₂²×C₄×C₅	—	6	1	no	—
123	C₂²×C₄×C₅	#67	1	1	yes	characteristic
124	C₂²×C₄×C₅	#67	1	1	yes	characteristic
125	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
126	C₂²×Dic₅	—	2	2	no	—
127	C₂×Dic₁₀	—	2	4	no	—
128	C₂×Dic₁₀	—	4	4	no	—
129	C₅×C₂²⋊C₄	—	6	4	no	—
130	C₅×C₂²⋊C₄	—	6	4	no	—
131	C₄²×C₅	#67	1	2	yes	—
132	C₄²×C₅	C₂²×D₃	1	2	yes	—
133	C₄⋊Dic₅	—	2	2	no	—
134	C₄⋊Dic₅	—	2	4	no	—
135	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
136	C₂×#168	—	3	4	no	—
137	C₂×#168	—	6	4	no	—
138	C₂×C₄⋊Q₈	—	5	1	no	—
139	C₂×#124	—	3	1	no	—
140	#489	—	15	8	no	—
141	C₂×#103	—	15	2	no	—
142	C₂×#102	—	15	2	no	—
143	C₂×#97	—	15	1	no	—
144	C₂²×C₃×C₅	#124	1	2	yes	—
145	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
146	C₂²×C₃×C₅	#137	1	2	yes	—
147	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₅×D₃	—	4	2	no	—
149	C₂×C₅×D₃	—	4	2	no	—
150	C₂×C₅×D₃	—	4	2	no	—
151	C₂×C₅×D₃	—	8	4	no	—
152	C₃×C₄×C₅	—	2	2	no	—
153	C₃×C₄×C₅	—	2	2	no	—
154	C₃×C₄×C₅	#124	1	2	yes	—
155	C₃×C₄×C₅	C₂×#41	1	2	yes	—
156	C₃×Dic₅	—	4	4	no	—
157	C₅×Dic₃	—	4	2	no	—
158	C₂³×D₃	—	2	1	no	—
159	C₂×C₃×Q₈	—	10	4	no	—
160	C₂×C₃×Q₈	—	20	4	no	—
161	C₂²×C₃×C₄	—	10	2	no	—
162	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
163	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
164	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
165	C₂²×Dic₃	—	2	1	no	—
166	C₂×D₁₂	—	2	4	no	—
167	C₂×D₁₂	—	2	1	no	—
168	C₂×D₁₂	C₂²×D₅	1	2	yes	—
169	C₂×Dic₆	—	2	4	no	—
170	C₂×Dic₆	—	2	1	no	—
171	C₂×Dic₆	C₂²×D₅	1	2	yes	—
172	C₃×C₄⋊C₄	—	5	4	no	—
173	C₃×C₄⋊C₄	—	10	2	no	—
174	C₃×C₄⋊C₄	—	10	4	no	—
175	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
176	C₃×C₄²	C₂²×D₅	1	2	yes	—
177	(C₂×D₃)⋊C₄	—	2	4	no	—
178	(C₂×D₃)⋊C₄	—	2	4	no	—
179	C₂×C₃⋊C₈	—	5	4	no	—
180	C₂×C₃⋊C₈	—	10	4	no	—
181	C₂³×C₅	—	6	2	no	—
182	C₂³×C₅	—	12	2	no	—
183	C₂³×C₅	—	12	2	no	—
184	C₂³×C₅	—	12	2	no	—
185	C₂³×C₅	C₂×#67	1	1	yes	characteristic
186	C₅×Q₈	—	6	4	no	—
187	C₅×Q₈	—	12	2	no	—
188	C₅×D₄	—	6	4	no	—
189	C₅×D₄	—	12	2	no	—
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	—	2	2	no	—
192	C₂×C₄×C₅	—	6	2	no	—
193	C₂×C₄×C₅	—	12	2	no	—
194	C₂×C₄×C₅	#239	1	4	yes	—
195	C₂×C₄×C₅	C₂×#67	1	2	yes	—
196	C₂×C₄×C₅	C₂×#67	1	2	yes	—
197	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₂×Dic₅	—	2	4	no	—
200	C₂×Dic₅	—	4	4	no	—
201	Dic₁₀	—	4	8	no	—
202	#168	—	6	4	no	—
203	C₂²×Q₈	—	3	1	no	—
204	C₂²×Q₈	—	10	1	no	—
205	C₂²×D₄	—	3	1	no	—
206	C₂²×C₈	—	15	2	no	—
207	C₄⋊Q₈	—	5	4	no	—
208	C₄⋊Q₈	—	10	2	no	—
209	#124	—	3	4	no	—
210	#124	—	6	2	no	—
211	C₂×C₄⋊C₄	—	5	2	no	—
212	C₂×C₄⋊C₄	—	10	1	no	—
213	C₂×C₂²⋊C₄	—	6	1	no	—
214	C₂×C₂²⋊C₄	—	6	1	no	—
215	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
216	#103	—	15	8	no	—
217	#102	—	15	8	no	—
218	#97	—	15	4	no	—
219	C₂×C₃×C₅	#489	1	4	yes	complemented
220	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
221	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂×C₃×C₅	C₂×#137	1	1	yes	characteristic
223	C₅×D₃	—	8	2	no	—
224	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
225	C₂²×D₃	—	2	2	no	—
226	C₂²×D₃	—	4	2	no	—
227	C₂²×D₃	—	4	2	no	—
228	C₂²×D₃	—	4	2	no	—
229	C₃×Q₈	—	20	8	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	—	10	4	no	—
233	C₂×C₃×C₄	—	20	4	no	—
234	C₂×C₃×C₄	#711	1	4	yes	—
235	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
236	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
237	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
239	C₂×Dic₃	—	2	2	no	—
240	C₂×Dic₃	—	4	2	no	—
241	D₁₂	—	2	4	no	—
242	D₁₂	—	4	2	no	—
243	Dic₆	—	2	4	no	—
244	Dic₆	—	4	2	no	—
245	C₃⋊C₈	—	10	4	no	—
246	C₂²×C₅	—	12	2	no	—
247	C₂²×C₅	—	12	2	no	—
248	C₂²×C₅	—	12	2	no	—
249	C₂²×C₅	—	24	4	no	—
250	C₂²×C₅	#932	1	2	yes	—
251	C₂²×C₅	#942	1	2	yes	—
252	C₂²×C₅	#947	1	2	yes	—
253	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
254	C₄×C₅	—	2	2	no	—
255	C₄×C₅	—	2	2	no	—
256	C₄×C₅	—	12	2	no	—
257	C₄×C₅	#947	1	2	yes	—
258	C₄×C₅	C₂×#239	1	2	yes	—
259	Dic₅	—	4	4	no	—
260	C₂⁴	—	6	1	no	—
261	C₂×Q₈	—	3	2	no	—
262	C₂×Q₈	—	6	1	no	—
263	C₂×Q₈	—	6	4	no	—
264	C₂×Q₈	—	10	4	no	—
265	C₂×Q₈	—	20	4	no	—
266	C₂×D₄	—	3	2	no	—
267	C₂×D₄	—	6	4	no	—
268	C₂×D₄	—	6	1	no	—
269	C₂²×C₄	—	6	1	no	—
270	C₂²×C₄	—	10	2	no	—
271	C₂²×C₄	#1201	1	1	yes	characteristic
272	C₂²×C₄	#1201	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
274	C₂×C₈	—	15	4	no	—
275	C₂×C₈	—	30	4	no	—
276	C₄⋊C₄	—	5	4	no	—
277	C₄⋊C₄	—	10	2	no	—
278	C₄⋊C₄	—	10	4	no	—
279	C₂²⋊C₄	—	6	4	no	—
280	C₂²⋊C₄	—	6	4	no	—
281	C₄²	#1201	1	2	yes	—
282	C₄²	C₂×D₃×D₅	1	2	yes	—
283	C₃×C₅	C₂×#489	1	1	yes	characteristic, complemented, Hall
284	C₂²×C₃	#4082	1	2	yes	—
285	C₂²×C₃	#4095	1	2	yes	—
286	C₂²×C₃	#4097	1	2	yes	—
287	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
288	C₂×D₃	—	4	2	no	—
289	C₂×D₃	—	4	2	no	—
290	C₂×D₃	—	4	2	no	—
291	C₂×D₃	—	8	4	no	—
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	2	no	—
294	C₃×C₄	—	20	4	no	—
295	C₃×C₄	#4082	1	2	yes	—
296	C₃×C₄	C₂×#711	1	2	yes	—
297	Dic₃	—	4	2	no	—
298	C₂×C₅	—	24	2	no	—
299	C₂×C₅	#5089	1	4	yes	complemented
300	C₂×C₅	C₂×#932	1	1	yes	characteristic
301	C₂×C₅	C₂×#942	1	1	yes	characteristic
302	C₂×C₅	C₂×#947	1	1	yes	characteristic
303	C₂³	—	6	2	no	—
304	C₂³	—	12	2	no	—
305	C₂³	—	12	2	no	—
306	C₂³	—	12	2	no	—
307	C₂³	C₂×#1201	1	1	yes	characteristic
308	Q₈	—	6	4	no	—
309	Q₈	—	12	2	no	—
310	Q₈	—	20	8	no	—
311	D₄	—	6	4	no	—
312	D₄	—	12	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	6	2	no	—
316	C₂×C₄	—	10	4	no	—
317	C₂×C₄	—	12	2	no	—
318	C₂×C₄	—	20	4	no	—
319	C₂×C₄	#6768	1	4	yes	—
320	C₂×C₄	C₂×#1201	1	2	yes	—
321	C₂×C₄	C₂×#1201	1	2	yes	—
322	C₂×C₄	C₂×#1201	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
324	C₈	—	30	4	no	—
325	C₂×C₃	#65344	1	4	yes	complemented
326	C₂×C₃	C₂×#4082	1	1	yes	characteristic
327	C₂×C₃	C₂×#4095	1	1	yes	characteristic
328	C₂×C₃	C₂×#4097	1	1	yes	characteristic
329	D₃	—	8	2	no	—
330	C₅	C₂×#5089	1	1	yes	characteristic, complemented, Sylow
331	C₂²	—	12	2	no	—
332	C₂²	—	12	2	no	—
333	C₂²	—	12	2	no	—
334	C₂²	—	24	4	no	—
335	C₂²	#91298	1	2	yes	—
336	C₂²	#91309	1	2	yes	—
337	C₂²	#91370	1	2	yes	—
338	C₂²	C₂²×#1201	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	12	2	no	—
342	C₄	—	20	4	no	—
343	C₄	#91370	1	2	yes	—
344	C₄	C₂×#6768	1	2	yes	—
345	C₃	C₂×#65344	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	24	2	no	—
347	C₂	#11746904	1	4	yes	direct product factor
348	C₂	C₂×#91298	1	1	yes	characteristic
349	C₂	C₂×#91309	1	1	yes	characteristic
350	C₂	C₂×#91370	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1201
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₅×D₁₂	C₂²
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₂×#489
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#4015	C₂	semidirect
3. C₂×#91087	C₂	semidirect
4. C₂×#90972	C₂	semidirect
6. #11746904	C₂	direct (N × H)
20. C₃×#4015	C₂²	semidirect
21. #91087	C₂²	semidirect
22. #90972	C₂²	semidirect
26. C₂×#4015	D₃	semidirect
72. #4015	C₂×D₃	semidirect
219. C₂×C₃×C₅	#489	semidirect
283. C₃×C₅	C₂×#489	semidirect
299. C₂×C₅	#5089	semidirect
325. C₂×C₃	#65344	semidirect
330. C₅	C₂×#5089	semidirect
345. C₃	C₂×#65344	semidirect
347. C₂	#11746904	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×C₅×#875	2
C₂×C₃×#4015	2
C₂×#91087	2
C₂×#90972	2
C₂×#90966	2
#11746904	2
C₂×#65344	3
C₂×#5089	5

Subgroups — C2×#11746904 #14437891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746904 #14437891