Subgroups — C₂×#11746331 #14314005

Order 1920.

448 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	C₂×#91793	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91103	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746331	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64730	—	3	1	no	maximal, Hall
6	C₂×#6817	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6710	—	2	1	no	—
9	C₂×Dic₆₀	C₄	1	1	yes	characteristic
10	#91793	—	2	1	no	—
11	#91793	—	2	1	no	—
12	#91793	C₄	1	2	yes	—
13	#91793	C₂²	1	2	yes	—
14	C₂×D₁₂₀	C₄	1	1	yes	characteristic
15	C₂×#6706	—	2	1	no	—
16	C₂×C₃×C₅×C₁₆	—	2	1	no	—
17	C₂×C₃×C₅×C₁₆	—	2	1	no	—
18	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	complemented
19	#91103	—	2	1	no	—
20	#91103	C₂²	1	2	yes	complemented
21	C₂×#4510	—	5	1	no	maximal, Hall
22	C₂×#4050	—	3	1	no	—
23	C₂²×C₅×C₁₆	D₃	1	1	yes	characteristic, complemented
24	C₂×#3951	—	3	1	no	—
25	#64730	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂×#1220	—	4	1	no	—
28	#6817	—	2	2	no	—
29	#6817	—	2	2	no	—
30	#6817	—	4	1	no	—
31	#6817	—	4	1	no	—
32	C₂×D₆₀	—	2	1	no	—
33	C₂×C₄×D₁₅	—	4	1	no	—
34	C₂×Dic₃₀	—	2	1	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	C₂×C₄	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	#6710	—	2	2	no	—
42	#6710	—	4	1	no	—
43	C₂×C₃⋊#168	—	4	1	no	—
44	Dic₆₀	—	2	1	no	—
45	Dic₆₀	C₂×C₄	1	2	yes	—
46	D₁₂₀	—	2	1	no	—
47	D₁₂₀	C₂×C₄	1	2	yes	—
48	#6706	—	2	2	no	—
49	#6706	—	2	2	no	—
50	C₃×C₅×C₁₆	—	2	2	no	—
51	C₂×#904	—	5	1	no	—
52	C₂²×C₃×C₁₆	D₅	1	1	yes	characteristic, complemented
53	C₂×#819	—	5	1	no	—
54	#4510	—	5	4	no	—
55	C₂×#709	—	6	1	no	—
56	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×#681	—	6	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	C₂×C₅×C₁₆	—	2	1	no	—
66	C₂×C₅×C₁₆	—	2	1	no	—
67	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	complemented
68	#3951	—	3	2	no	—
69	#3951	—	6	1	no	—
70	C₂×#359	—	15	1	no	Sylow
71	C₂³×C₃×C₅	SD₁₆	1	1	yes	characteristic
72	C₂²×D₁₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	#1220	—	4	2	no	—
80	#1220	—	4	2	no	—
81	C₂×Dic₁₅	—	4	1	no	—
82	D₆₀	—	2	2	no	—
83	D₆₀	—	4	1	no	—
84	C₄×D₁₅	—	4	2	no	—
85	C₄×D₁₅	—	4	2	no	—
86	Dic₃₀	—	2	2	no	—
87	Dic₃₀	—	4	1	no	—
88	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₃⋊#168	—	4	2	no	—
93	C₂×#235	—	10	1	no	—
94	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
95	C₂×#208	—	10	1	no	—
96	C₂×Dic₁₂	—	5	1	no	—
97	#904	—	5	2	no	—
98	#904	—	5	2	no	—
99	#904	—	10	1	no	—
100	#904	—	10	1	no	—
101	C₂×D₂₄	—	5	1	no	—
102	C₂×#204	—	10	1	no	—
103	C₂×C₃×C₁₆	—	2	1	no	—
104	C₂×C₃×C₁₆	—	2	1	no	—
105	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	complemented
106	#819	—	5	2	no	—
107	#819	—	10	1	no	—
108	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
109	C₂×C₂²⋊D₅	—	12	1	no	—
110	#709	—	6	2	no	—
111	#709	—	6	2	no	—
112	#709	—	12	1	no	—
113	#709	—	12	1	no	—
114	C₂×D₂₀	—	6	1	no	—
115	C₂×C₄×D₅	—	12	1	no	—
116	C₂×Dic₁₀	—	6	1	no	—
117	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
118	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
119	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
120	C₂×C₅×C₈	#67	1	1	yes	characteristic
121	C₂×C₅×C₈	#67	1	1	yes	characteristic
122	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
123	#681	—	6	2	no	—
124	#681	—	12	1	no	—
125	C₂×#168	—	12	1	no	—
126	Dic₂₀	—	3	2	no	—
127	Dic₂₀	—	6	1	no	—
128	D₄₀	—	3	2	no	—
129	D₄₀	—	6	1	no	—
130	#677	—	6	2	no	—
131	#677	—	6	2	no	—
132	C₅×C₁₆	—	2	2	no	—
133	C₂×#135	—	15	1	no	—
134	C₂²×C₁₆	D₁₅	1	1	yes	characteristic, complemented
135	C₂×#108	—	15	1	no	—
136	#359	—	15	4	no	—
137	C₂²×C₃×C₅	—	2	2	no	—
138	C₂²×C₃×C₅	#102	1	1	yes	characteristic
139	C₂²×C₃×C₅	#102	1	1	yes	characteristic
140	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×D₁₅	—	4	2	no	—
142	C₂×D₁₅	—	8	2	no	—
143	C₃×C₄×C₅	#102	1	1	yes	characteristic
144	C₃×C₄×C₅	#102	1	1	yes	characteristic
145	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
147	Dic₁₅	—	4	2	no	—
148	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
149	C₂×#67	—	20	1	no	—
150	#235	—	10	2	no	—
151	#235	—	10	2	no	—
152	#235	—	20	1	no	—
153	#235	—	20	1	no	—
154	C₂×D₁₂	—	10	1	no	—
155	C₂×C₄×D₃	—	20	1	no	—
156	C₂×Dic₆	—	10	1	no	—
157	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
158	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
159	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
160	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
161	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
162	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
163	#208	—	10	2	no	—
164	#208	—	20	1	no	—
165	C₂×C₃⋊C₈	—	20	1	no	—
166	Dic₁₂	—	5	2	no	—
167	Dic₁₂	—	10	1	no	—
168	D₂₄	—	5	2	no	—
169	D₂₄	—	10	1	no	—
170	#204	—	10	2	no	—
171	#204	—	10	2	no	—
172	C₃×C₁₆	—	2	2	no	—
173	C₂³×C₅	#204	1	1	yes	characteristic
174	C₂²×D₅	—	12	1	no	—
175	C₂×C₄×C₅	#204	1	1	yes	characteristic
176	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
177	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
178	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
181	C₂²⋊D₅	—	12	2	no	—
182	C₂²⋊D₅	—	12	2	no	—
183	C₂×Dic₅	—	12	1	no	—
184	D₂₀	—	6	2	no	—
185	D₂₀	—	12	1	no	—
186	C₄×D₅	—	12	2	no	—
187	C₄×D₅	—	12	2	no	—
188	Dic₁₀	—	6	2	no	—
189	Dic₁₀	—	12	1	no	—
190	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
191	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
192	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
193	C₅×C₈	C₂×#67	1	1	yes	characteristic
194	#168	—	12	2	no	—
195	C₂×#41	—	30	1	no	—
196	#135	—	15	2	no	—
197	#135	—	15	2	no	—
198	#135	—	30	1	no	—
199	#135	—	30	1	no	—
200	C₂×Q₁₆	—	15	1	no	—
201	C₂×SD₁₆	—	30	1	no	—
202	C₂×D₈	—	15	1	no	—
203	C₂×M₁₆	—	30	1	no	—
204	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	C₂×D₁₅	1	2	yes	complemented
208	#108	—	15	2	no	—
209	#108	—	30	1	no	—
210	C₂×C₃×C₅	—	2	1	no	—
211	C₂×C₃×C₅	—	2	1	no	—
212	C₂×C₃×C₅	#359	1	2	yes	complemented
213	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
214	D₁₅	—	8	2	no	—
215	C₂³×C₃	#677	1	1	yes	characteristic
216	C₂²×D₃	—	20	1	no	—
217	C₂×C₃×C₄	#677	1	1	yes	characteristic
218	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
219	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
220	C₂×C₃×C₄	#685	1	1	yes	characteristic
221	C₂×C₃×C₄	#685	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
223	#67	—	20	2	no	—
224	#67	—	20	2	no	—
225	C₂×Dic₃	—	20	1	no	—
226	D₁₂	—	10	2	no	—
227	D₁₂	—	20	1	no	—
228	C₄×D₃	—	20	2	no	—
229	C₄×D₃	—	20	2	no	—
230	Dic₆	—	10	2	no	—
231	Dic₆	—	20	1	no	—
232	C₃×C₈	#685	1	1	yes	characteristic
233	C₃×C₈	#685	1	1	yes	characteristic
234	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
235	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
236	C₃⋊C₈	—	20	2	no	—
237	C₂²×C₅	—	2	2	no	—
238	C₂²×C₅	#821	1	1	yes	characteristic
239	C₂²×C₅	#821	1	1	yes	characteristic
240	C₂²×C₅	C₂×#204	1	1	yes	characteristic
241	C₂×D₅	—	12	2	no	—
242	C₂×D₅	—	24	2	no	—
243	C₄×C₅	#821	1	1	yes	characteristic
244	C₄×C₅	#821	1	1	yes	characteristic
245	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
246	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
247	Dic₅	—	12	2	no	—
248	#41	—	30	2	no	—
249	#41	—	30	2	no	—
250	#41	—	60	1	no	—
251	#41	—	60	1	no	—
252	C₂×Q₈	—	30	1	no	—
253	C₂×D₄	—	30	1	no	—
254	C₂×D₄	—	60	1	no	—
255	C₂²×C₄	—	60	1	no	—
256	C₂²×C₄	D₆₀	1	1	yes	characteristic
257	Q₁₆	—	15	2	no	—
258	Q₁₆	—	30	1	no	—
259	SD₁₆	—	30	2	no	—
260	SD₁₆	—	30	2	no	—
261	D₈	—	15	2	no	—
262	D₈	—	30	1	no	—
263	M₁₆	—	30	2	no	—
264	M₁₆	—	60	1	no	—
265	C₂×C₈	—	60	1	no	—
266	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
267	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
268	C₂×C₈	D₆₀	1	1	yes	characteristic
269	C₂×C₈	#1220	1	1	yes	characteristic
270	C₂×C₈	#1220	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₃×C₅	C₂×#359	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	2	2	no	—
275	C₂²×C₃	#3953	1	1	yes	characteristic
276	C₂²×C₃	#3953	1	1	yes	characteristic
277	C₂²×C₃	C₂×#677	1	1	yes	characteristic
278	C₂×D₃	—	20	2	no	—
279	C₂×D₃	—	40	2	no	—
280	C₃×C₄	#3953	1	1	yes	characteristic
281	C₃×C₄	#3953	1	1	yes	characteristic
282	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
283	C₃×C₄	C₂×#685	1	1	yes	characteristic
284	Dic₃	—	20	2	no	—
285	C₂×C₅	—	2	1	no	—
286	C₂×C₅	—	2	1	no	—
287	C₂×C₅	#4510	1	2	yes	complemented
288	C₂×C₅	C₂×#821	1	1	yes	characteristic
289	D₅	—	24	2	no	—
290	C₂³	—	60	1	no	—
291	C₂³	#6706	1	1	yes	characteristic
292	Q₈	—	30	2	no	—
293	Q₈	—	60	1	no	—
294	D₄	—	30	2	no	—
295	D₄	—	60	2	no	—
296	D₄	—	60	2	no	—
297	D₄	—	60	1	no	—
298	C₂×C₄	—	60	1	no	—
299	C₂×C₄	—	60	2	no	—
300	C₂×C₄	—	60	2	no	—
301	C₂×C₄	#6706	1	1	yes	characteristic
302	C₂×C₄	D₁₂₀	1	1	yes	characteristic
303	C₂×C₄	D₁₂₀	1	1	yes	characteristic
304	C₂×C₄	#6714	1	1	yes	characteristic
305	C₂×C₄	#6714	1	1	yes	characteristic
306	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
307	C₈	—	60	2	no	—
308	C₈	#6714	1	1	yes	characteristic
309	C₈	#6714	1	1	yes	characteristic
310	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
311	C₈	C₂×#1220	1	1	yes	characteristic
312	C₂×C₃	—	2	1	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	#64730	1	2	yes	complemented
315	C₂×C₃	C₂×#3953	1	1	yes	characteristic
316	D₃	—	40	2	no	—
317	C₅	C₂×#4510	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	2	2	no	—
319	C₂²	—	60	2	no	—
320	C₂²	—	120	2	no	—
321	C₂²	#91105	1	1	yes	characteristic
322	C₂²	#91105	1	1	yes	characteristic
323	C₂²	C₂×#6706	1	1	yes	characteristic
324	C₄	—	60	2	no	—
325	C₄	#91105	1	1	yes	characteristic
326	C₄	#91105	1	1	yes	characteristic
327	C₄	C₂×D₁₂₀	1	1	yes	characteristic
328	C₄	C₂×#6714	1	1	yes	characteristic
329	C₃	C₂×#64730	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	120	2	no	—
333	C₂	#11746331	1	2	yes	direct product factor
334	C₂	C₂×#91105	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₂₀
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂²×C₃×C₅×C₁₆	C₂
socle	C₂²×C₃×C₅	#102
prime radical	C₂×#91793	C₂
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₁₆	D₁₅
hypocenter	C₃×C₅	C₂×#359
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₅×C₁₆	C₂	semidirect
3. C₂×#91103	C₂	semidirect
4. #11746331	C₂	direct (N × H)
18. C₂×C₃×C₅×C₁₆	C₂²	semidirect
20. #91103	C₂²	semidirect
23. C₂²×C₅×C₁₆	D₃	semidirect
52. C₂²×C₃×C₁₆	D₅	semidirect
67. C₂×C₅×C₁₆	C₂×D₃	semidirect
105. C₂×C₃×C₁₆	C₂×D₅	semidirect
134. C₂²×C₁₆	D₁₅	semidirect
207. C₂×C₁₆	C₂×D₁₅	semidirect
212. C₂×C₃×C₅	#359	semidirect
273. C₃×C₅	C₂×#359	semidirect
287. C₂×C₅	#4510	semidirect
314. C₂×C₃	#64730	semidirect
317. C₅	C₂×#4510	semidirect
329. C₃	C₂×#64730	semidirect
333. C₂	#11746331	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
C₂²×C₃×C₅×C₁₆	2
C₂×#91103	2
#11746331	2
C₂×#64730	3
C₂×#4510	5

Subgroups — C2×#11746331 #14314005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746331 #14314005