Subgroups — C₂×#11746309 #14314085

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91793	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746309	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64708	—	3	1	no	maximal, Hall
6	C₂×#6817	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
9	C₂×Dic₆₀	C₄	1	1	yes	characteristic, complemented
10	#91793	—	2	1	no	—
11	#91793	—	2	1	no	—
12	#91793	C₄	1	2	yes	complemented
13	#91793	C₂²	1	2	yes	—
14	C₂×D₁₂₀	C₄	1	1	yes	characteristic, complemented
15	C₂×#6706	—	2	1	no	—
16	C₃×C₅×#106	—	2	1	no	—
17	C₃×C₅×#106	C₂²	1	2	yes	complemented
18	#91096	—	2	1	no	—
19	#91096	C₂²	1	2	yes	complemented
20	C₂×C₃⋊#672	—	2	1	no	—
21	C₂×#4488	—	5	1	no	maximal, Hall
22	C₂×#4050	—	3	1	no	—
23	C₂×C₅×#106	D₃	1	1	yes	characteristic, complemented
24	C₂×#3944	—	3	1	no	—
25	#64708	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	—	4	1	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂×#1220	—	4	1	no	—
29	#6817	—	2	2	no	—
30	#6817	—	2	2	no	—
31	#6817	—	4	1	no	—
32	#6817	—	4	1	no	—
33	C₂×D₆₀	—	2	1	no	—
34	C₂×C₄×D₁₅	—	4	1	no	—
35	C₂×Dic₃₀	—	2	1	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₄	—	2	2	no	—
43	C₃×C₅×C₄⋊C₄	—	4	1	no	—
44	Dic₆₀	—	2	1	no	—
45	Dic₆₀	C₂×C₄	1	2	yes	complemented
46	D₁₂₀	—	2	1	no	—
47	D₁₂₀	C₂×C₄	1	2	yes	complemented
48	#6706	—	2	2	no	—
49	#6706	—	2	2	no	—
50	C₃⋊#672	—	2	2	no	—
51	C₂×#904	—	5	1	no	—
52	C₂×C₃×#106	D₅	1	1	yes	characteristic, complemented
53	C₂×#812	—	5	1	no	—
54	#4488	—	5	4	no	—
55	C₂×#709	—	6	1	no	—
56	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₅×C₄⋊C₄	—	2	1	no	—
58	C₂×Dic₂₀	—	3	1	no	—
59	#4050	—	3	2	no	—
60	#4050	—	3	2	no	—
61	#4050	—	6	1	no	—
62	#4050	—	6	1	no	—
63	C₂×D₄₀	—	3	1	no	—
64	C₂×#677	—	6	1	no	—
65	C₅×#106	—	2	1	no	—
66	C₅×#106	C₂×D₃	1	2	yes	complemented
67	#3944	—	3	2	no	—
68	#3944	—	6	1	no	—
69	C₂×#672	—	6	1	no	—
70	C₂×#360	—	15	1	no	Sylow
71	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
72	C₂²×D₁₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	8	2	no	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	#1220	—	4	2	no	—
82	#1220	—	4	2	no	—
83	C₂×Dic₁₅	—	4	1	no	—
84	D₆₀	—	2	2	no	—
85	D₆₀	—	4	1	no	—
86	C₄×D₁₅	—	4	2	no	—
87	C₄×D₁₅	—	4	2	no	—
88	Dic₃₀	—	2	2	no	—
89	Dic₃₀	—	4	1	no	—
90	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×#235	—	10	1	no	—
95	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
96	C₂×C₃×C₄⋊C₄	—	2	1	no	—
97	C₂×Dic₁₂	—	5	1	no	—
98	#904	—	5	2	no	—
99	#904	—	5	2	no	—
100	#904	—	10	1	no	—
101	#904	—	10	1	no	—
102	C₂×D₂₄	—	5	1	no	—
103	C₂×#204	—	10	1	no	—
104	C₃×#106	—	2	1	no	—
105	C₃×#106	C₂×D₅	1	2	yes	complemented
106	#812	—	5	2	no	—
107	#812	—	10	1	no	—
108	C₂×C₃⋊C₁₆	—	10	1	no	—
109	C₂²×C₄×C₅	—	4	1	no	—
110	C₂²×C₄×C₅	#67	1	1	yes	characteristic
111	C₂×C₂²⋊D₅	—	12	1	no	—
112	#709	—	6	2	no	—
113	#709	—	6	2	no	—
114	#709	—	12	1	no	—
115	#709	—	12	1	no	—
116	C₂×D₂₀	—	6	1	no	—
117	C₂×C₄×D₅	—	12	1	no	—
118	C₂×Dic₁₀	—	6	1	no	—
119	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
120	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
121	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
122	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
123	C₂×C₅×C₈	#67	1	1	yes	characteristic
124	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₅×C₄⋊C₄	—	2	2	no	—
126	C₅×C₄⋊C₄	—	4	1	no	—
127	Dic₂₀	—	3	2	no	—
128	Dic₂₀	—	6	1	no	—
129	D₄₀	—	3	2	no	—
130	D₄₀	—	6	1	no	—
131	#677	—	6	2	no	—
132	#677	—	6	2	no	—
133	#672	—	6	2	no	—
134	C₂×#135	—	15	1	no	—
135	C₂×M₃₂	—	15	1	no	—
136	C₂×#106	D₁₅	1	1	yes	characteristic, complemented
137	#360	—	15	4	no	—
138	C₂²×C₃×C₅	—	2	2	no	—
139	C₂²×C₃×C₅	#102	1	1	yes	characteristic
140	C₂²×C₃×C₅	#102	1	1	yes	characteristic
141	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
142	C₂×D₁₅	—	4	2	no	—
143	C₂×D₁₅	—	8	2	no	—
144	C₃×C₄×C₅	—	8	2	no	—
145	C₃×C₄×C₅	#102	1	1	yes	characteristic
146	C₃×C₄×C₅	#102	1	1	yes	characteristic
147	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
149	Dic₁₅	—	4	2	no	—
150	C₂²×C₃×C₄	—	4	1	no	—
151	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
152	C₂×#67	—	20	1	no	—
153	#235	—	10	2	no	—
154	#235	—	10	2	no	—
155	#235	—	20	1	no	—
156	#235	—	20	1	no	—
157	C₂×D₁₂	—	10	1	no	—
158	C₂×C₄×D₃	—	20	1	no	—
159	C₂×Dic₆	—	10	1	no	—
160	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
161	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
162	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
163	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
164	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
165	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
166	C₃×C₄⋊C₄	—	2	2	no	—
167	C₃×C₄⋊C₄	—	4	1	no	—
168	Dic₁₂	—	5	2	no	—
169	Dic₁₂	—	10	1	no	—
170	D₂₄	—	5	2	no	—
171	D₂₄	—	10	1	no	—
172	#204	—	10	2	no	—
173	#204	—	10	2	no	—
174	C₃⋊C₁₆	—	10	2	no	—
175	C₂³×C₅	#213	1	1	yes	characteristic
176	C₂²×D₅	—	12	1	no	—
177	C₂×C₄×C₅	—	4	2	no	—
178	C₂×C₄×C₅	—	8	2	no	—
179	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	#213	1	1	yes	characteristic
182	C₂×C₄×C₅	#215	1	1	yes	characteristic
183	C₂×C₄×C₅	#215	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
185	C₂²⋊D₅	—	12	2	no	—
186	C₂²⋊D₅	—	12	2	no	—
187	C₂×Dic₅	—	12	1	no	—
188	D₂₀	—	6	2	no	—
189	D₂₀	—	12	1	no	—
190	C₄×D₅	—	12	2	no	—
191	C₄×D₅	—	12	2	no	—
192	Dic₁₀	—	6	2	no	—
193	Dic₁₀	—	12	1	no	—
194	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
195	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
196	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
197	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
198	C₂×#41	—	30	1	no	—
199	#135	—	15	2	no	—
200	#135	—	15	2	no	—
201	#135	—	30	1	no	—
202	#135	—	30	1	no	—
203	C₂×Q₁₆	—	15	1	no	—
204	C₂×SD₁₆	—	30	1	no	—
205	C₂×D₈	—	15	1	no	—
206	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	2	1	no	—
208	M₃₂	—	15	2	no	—
209	M₃₂	—	30	1	no	—
210	C₂×C₁₆	—	30	1	no	—
211	#106	—	2	1	no	—
212	#106	C₂×D₁₅	1	2	yes	complemented
213	C₂×C₃×C₅	—	2	1	no	—
214	C₂×C₃×C₅	—	2	1	no	—
215	C₂×C₃×C₅	#360	1	2	yes	complemented
216	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
217	D₁₅	—	8	2	no	—
218	C₂³×C₃	#686	1	1	yes	characteristic
219	C₂²×D₃	—	20	1	no	—
220	C₂×C₃×C₄	—	4	2	no	—
221	C₂×C₃×C₄	—	8	2	no	—
222	C₂×C₃×C₄	#685	1	1	yes	characteristic
223	C₂×C₃×C₄	#685	1	1	yes	characteristic
224	C₂×C₃×C₄	#686	1	1	yes	characteristic
225	C₂×C₃×C₄	#688	1	1	yes	characteristic
226	C₂×C₃×C₄	#688	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
228	#67	—	20	2	no	—
229	#67	—	20	2	no	—
230	C₂×Dic₃	—	20	1	no	—
231	D₁₂	—	10	2	no	—
232	D₁₂	—	20	1	no	—
233	C₄×D₃	—	20	2	no	—
234	C₄×D₃	—	20	2	no	—
235	Dic₆	—	10	2	no	—
236	Dic₆	—	20	1	no	—
237	C₃×C₈	#685	1	1	yes	characteristic
238	C₃×C₈	#685	1	1	yes	characteristic
239	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
240	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
241	C₂²×C₅	—	2	2	no	—
242	C₂²×C₅	#809	1	1	yes	characteristic
243	C₂²×C₅	#809	1	1	yes	characteristic
244	C₂²×C₅	C₂×#213	1	1	yes	characteristic
245	C₂×D₅	—	12	2	no	—
246	C₂×D₅	—	24	2	no	—
247	C₄×C₅	—	8	2	no	—
248	C₄×C₅	#809	1	1	yes	characteristic
249	C₄×C₅	#809	1	1	yes	characteristic
250	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
251	C₄×C₅	C₂×#215	1	1	yes	characteristic
252	Dic₅	—	12	2	no	—
253	#41	—	30	2	no	—
254	#41	—	30	2	no	—
255	#41	—	60	1	no	—
256	#41	—	60	1	no	—
257	C₂×Q₈	—	30	1	no	—
258	C₂×D₄	—	30	1	no	—
259	C₂×D₄	—	60	1	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	60	1	no	—
262	C₂²×C₄	#1220	1	1	yes	characteristic
263	Q₁₆	—	15	2	no	—
264	Q₁₆	—	30	1	no	—
265	SD₁₆	—	30	2	no	—
266	SD₁₆	—	30	2	no	—
267	D₈	—	15	2	no	—
268	D₈	—	30	1	no	—
269	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
270	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
271	C₂×C₈	D₆₀	1	1	yes	characteristic
272	C₂×C₈	D₆₀	1	1	yes	characteristic
273	C₂×C₈	#1220	1	1	yes	characteristic
274	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₁₆	—	30	2	no	—
278	C₃×C₅	C₂×#360	1	1	yes	characteristic, complemented, Hall
279	C₂²×C₃	—	2	2	no	—
280	C₂²×C₃	#3941	1	1	yes	characteristic
281	C₂²×C₃	#3941	1	1	yes	characteristic
282	C₂²×C₃	C₂×#686	1	1	yes	characteristic
283	C₂×D₃	—	20	2	no	—
284	C₂×D₃	—	40	2	no	—
285	C₃×C₄	—	8	2	no	—
286	C₃×C₄	#3941	1	1	yes	characteristic
287	C₃×C₄	#3941	1	1	yes	characteristic
288	C₃×C₄	C₂×#685	1	1	yes	characteristic
289	C₃×C₄	C₂×#688	1	1	yes	characteristic
290	Dic₃	—	20	2	no	—
291	C₂×C₅	—	2	1	no	—
292	C₂×C₅	—	2	1	no	—
293	C₂×C₅	#4488	1	2	yes	complemented
294	C₂×C₅	C₂×#809	1	1	yes	characteristic
295	D₅	—	24	2	no	—
296	C₂³	—	60	1	no	—
297	C₂³	#6715	1	1	yes	characteristic
298	Q₈	—	30	2	no	—
299	Q₈	—	60	1	no	—
300	D₄	—	30	2	no	—
301	D₄	—	60	2	no	—
302	D₄	—	60	2	no	—
303	D₄	—	60	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	60	1	no	—
307	C₂×C₄	—	60	2	no	—
308	C₂×C₄	—	60	2	no	—
309	C₂×C₄	#6714	1	1	yes	characteristic
310	C₂×C₄	#6714	1	1	yes	characteristic
311	C₂×C₄	#6715	1	1	yes	characteristic
312	C₂×C₄	#6717	1	1	yes	characteristic
313	C₂×C₄	#6717	1	1	yes	characteristic
314	C₂×C₄	C₂×#1220	1	1	yes	characteristic
315	C₈	#6714	1	1	yes	characteristic
316	C₈	#6714	1	1	yes	characteristic
317	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
318	C₈	C₂×D₆₀	1	1	yes	characteristic
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	#64708	1	2	yes	complemented
322	C₂×C₃	C₂×#3941	1	1	yes	characteristic
323	D₃	—	40	2	no	—
324	C₅	C₂×#4488	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	2	2	no	—
326	C₂²	—	60	2	no	—
327	C₂²	—	120	2	no	—
328	C₂²	#91093	1	1	yes	characteristic
329	C₂²	#91093	1	1	yes	characteristic
330	C₂²	C₂×#6715	1	1	yes	characteristic
331	C₄	—	8	2	no	—
332	C₄	—	60	2	no	—
333	C₄	#91093	1	1	yes	characteristic
334	C₄	#91093	1	1	yes	characteristic
335	C₄	C₂×#6714	1	1	yes	characteristic
336	C₄	C₂×#6717	1	1	yes	characteristic
337	C₃	C₂×#64708	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	2	1	no	—
339	C₂	—	2	1	no	—
340	C₂	—	120	2	no	—
341	C₂	#11746309	1	2	yes	direct product factor
342	C₂	C₂×#91093	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#91093
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂×C₃×C₅×#106	C₂
socle	C₂²×C₃×C₅	#102
prime radical	C₂×#91793	C₂
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×#106	D₁₅
hypocenter	C₃×C₅	C₂×#360
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#106	C₂	semidirect
3. C₂×#91096	C₂	semidirect
4. #11746309	C₂	direct (N × H)
9. C₂×Dic₆₀	C₄	semidirect
12. #91793	C₄	semidirect
14. C₂×D₁₂₀	C₄	semidirect
17. C₃×C₅×#106	C₂²	semidirect
19. #91096	C₂²	semidirect
23. C₂×C₅×#106	D₃	semidirect
45. Dic₆₀	C₂×C₄	semidirect
47. D₁₂₀	C₂×C₄	semidirect
52. C₂×C₃×#106	D₅	semidirect
66. C₅×#106	C₂×D₃	semidirect
105. C₃×#106	C₂×D₅	semidirect
136. C₂×#106	D₁₅	semidirect
212. #106	C₂×D₁₅	semidirect
215. C₂×C₃×C₅	#360	semidirect
278. C₃×C₅	C₂×#360	semidirect
293. C₂×C₅	#4488	semidirect
321. C₂×C₃	#64708	semidirect
324. C₅	C₂×#4488	semidirect
337. C₃	C₂×#64708	semidirect
341. C₂	#11746309	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×#91793	2
C₂×C₃×C₅×#106	2
C₂×#91096	2
#11746309	2
C₂×#64708	3
C₂×#4488	5

Subgroups — C2×#11746309 #14314085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746309 #14314085