Subgroups — C₂×#11745848 #14283874

Order 1920.

568 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×#91764	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745848	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64684	—	3	1	no	maximal, Hall
6	C₂²×D₆₀	C₄	1	1	yes	characteristic
7	C₂²×Dic₃₀	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×#6714	—	2	1	no	—
10	C₂×#6714	—	2	1	no	—
11	#91764	—	2	1	no	—
12	#91764	—	2	1	no	—
13	#91764	C₄	1	2	yes	—
14	#91764	C₂²	1	2	yes	—
15	C₂²×C₅×C₃⋊C₈	—	2	1	no	—
16	C₂²×C₃×#168	—	2	1	no	—
17	C₅×#803	—	2	1	no	—
18	C₅×#803	C₂²	1	2	yes	complemented
19	C₃×#3935	—	2	1	no	—
20	C₃×#3935	C₂²	1	2	yes	complemented
21	C₂×#4464	—	5	1	no	maximal, Hall
22	C₂×#4021	—	3	1	no	—
23	C₂×C₅×#97	—	3	1	no	—
24	C₂×#3935	D₃	1	1	yes	characteristic, complemented
25	#64684	—	3	4	no	—
26	C₂³×D₁₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×Dic₁₅	—	2	1	no	—
31	C₂×D₆₀	—	2	1	no	—
32	C₂×D₆₀	—	4	2	no	—
33	C₂×D₆₀	C₂×C₄	1	2	yes	—
34	C₂×Dic₃₀	—	2	1	no	—
35	C₂×Dic₃₀	—	4	2	no	—
36	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	#6714	—	2	2	no	—
42	#6714	—	2	2	no	—
43	#6714	—	2	2	no	—
44	#6714	—	2	2	no	—
45	C₂×C₅×C₃⋊C₈	—	2	4	no	—
46	C₂×C₅×C₃⋊C₈	—	2	2	no	—
47	C₂×C₃×#168	—	2	4	no	—
48	C₂×C₃×#168	—	2	2	no	—
49	C₂×#875	—	5	1	no	—
50	C₂×C₃×#97	—	5	1	no	—
51	C₂×#803	D₅	1	1	yes	characteristic, complemented
52	#4464	—	5	4	no	—
53	C₂²×D₂₀	—	3	1	no	—
54	C₂²×Dic₁₀	—	3	1	no	—
55	C₂²×C₅×C₈	—	6	1	no	—
56	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
57	C₂×#685	—	6	1	no	—
58	C₂×#685	—	6	1	no	—
59	C₂²×#168	—	2	1	no	—
60	#4021	—	3	2	no	—
61	#4021	—	3	2	no	—
62	#4021	—	6	1	no	—
63	#4021	—	6	1	no	—
64	C₅×#97	—	3	2	no	—
65	C₅×#97	—	6	1	no	—
66	#3935	—	2	1	no	—
67	#3935	C₂×D₃	1	2	yes	complemented
68	C₂×#329	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×D₁₅	—	2	2	no	—
71	C₂²×D₁₅	—	4	2	no	—
72	C₂²×D₁₅	—	4	2	no	—
73	C₂²×D₁₅	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₁₅	—	2	2	no	—
84	C₂×Dic₁₅	—	4	2	no	—
85	D₆₀	—	4	2	no	—
86	D₆₀	—	4	2	no	—
87	Dic₃₀	—	4	2	no	—
88	Dic₃₀	—	4	2	no	—
89	C₃×#168	—	2	4	no	—
90	C₅×C₃⋊C₈	—	2	4	no	—
91	C₂²×D₁₂	—	5	1	no	—
92	C₂²×Dic₆	—	5	1	no	—
93	C₂²×C₃×C₈	—	10	1	no	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₂×(C₂×D₃)⋊C₄	—	10	1	no	—
96	C₂×(C₂×D₃)⋊C₄	—	10	1	no	—
97	C₂²×C₃⋊C₈	—	2	1	no	—
98	#875	—	5	2	no	—
99	#875	—	5	2	no	—
100	#875	—	10	1	no	—
101	#875	—	10	1	no	—
102	C₃×#97	—	5	2	no	—
103	C₃×#97	—	10	1	no	—
104	#803	—	2	1	no	—
105	#803	C₂×D₅	1	2	yes	complemented
106	C₂³×D₅	—	6	1	no	—
107	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
108	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
109	C₂²×C₄×C₅	#67	1	1	yes	characteristic
110	C₂²×Dic₅	—	6	1	no	—
111	C₂×D₂₀	—	3	2	no	—
112	C₂×D₂₀	—	6	1	no	—
113	C₂×D₂₀	—	12	2	no	—
114	C₂×Dic₁₀	—	3	2	no	—
115	C₂×Dic₁₀	—	6	1	no	—
116	C₂×Dic₁₀	—	12	2	no	—
117	C₂×C₅×C₈	—	6	4	no	—
118	C₂×C₅×C₈	—	6	2	no	—
119	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
120	C₄²×C₅	D₁₂	1	1	yes	characteristic
121	C₄²×C₅	#67	1	1	yes	characteristic
122	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
123	#685	—	6	2	no	—
124	#685	—	6	2	no	—
125	#685	—	6	2	no	—
126	#685	—	6	2	no	—
127	C₂×#168	—	2	4	no	—
128	C₂×#168	—	2	2	no	—
129	C₂×#124	—	15	1	no	—
130	C₂×#97	—	3	1	no	—
131	C₂×#97	—	5	1	no	—
132	#329	—	15	4	no	—
133	C₂²×C₃×C₅	#100	1	2	yes	—
134	C₂²×C₃×C₅	#104	1	2	yes	—
135	C₂²×C₃×C₅	#104	1	2	yes	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×D₁₅	—	4	2	no	—
138	C₂×D₁₅	—	4	2	no	—
139	C₂×D₁₅	—	4	2	no	—
140	C₂×D₁₅	—	8	4	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	—	2	1	no	—
147	Dic₁₅	—	4	2	no	—
148	C₂³×D₃	—	10	1	no	—
149	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
150	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
151	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
152	C₂²×Dic₃	—	10	1	no	—
153	C₂×D₁₂	—	5	2	no	—
154	C₂×D₁₂	—	10	1	no	—
155	C₂×D₁₂	—	20	2	no	—
156	C₂×Dic₆	—	5	2	no	—
157	C₂×Dic₆	—	10	1	no	—
158	C₂×Dic₆	—	20	2	no	—
159	C₂×C₃×C₈	—	10	4	no	—
160	C₂×C₃×C₈	—	10	2	no	—
161	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
162	C₃×C₄²	D₂₀	1	1	yes	characteristic
163	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
164	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
165	(C₂×D₃)⋊C₄	—	10	2	no	—
166	(C₂×D₃)⋊C₄	—	10	2	no	—
167	(C₂×D₃)⋊C₄	—	10	2	no	—
168	(C₂×D₃)⋊C₄	—	10	2	no	—
169	C₂×C₃⋊C₈	—	2	4	no	—
170	C₂×C₃⋊C₈	—	2	2	no	—
171	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂²×D₅	—	6	2	no	—
173	C₂²×D₅	—	12	2	no	—
174	C₂²×D₅	—	12	2	no	—
175	C₂²×D₅	—	12	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	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	(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
185	C₂×Dic₅	—	6	2	no	—
186	C₂×Dic₅	—	12	2	no	—
187	D₂₀	—	12	2	no	—
188	D₂₀	—	12	2	no	—
189	Dic₁₀	—	12	2	no	—
190	Dic₁₀	—	12	2	no	—
191	C₅×C₈	—	6	4	no	—
192	#168	—	2	4	no	—
193	C₂²×Q₈	—	15	1	no	—
194	C₂²×D₄	—	15	1	no	—
195	C₂²×C₈	—	6	1	no	—
196	C₂²×C₈	—	10	1	no	—
197	#124	—	15	2	no	—
198	#124	—	15	2	no	—
199	#124	—	30	1	no	—
200	#124	—	30	1	no	—
201	C₂×C₂²⋊C₄	—	30	1	no	—
202	C₂×C₂²⋊C₄	—	30	1	no	—
203	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
204	#97	—	3	2	no	—
205	#97	—	5	2	no	—
206	#97	—	6	1	no	—
207	#97	—	10	1	no	—
208	C₂×C₃×C₅	#329	1	4	yes	complemented
209	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
210	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
211	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
212	D₁₅	—	8	2	no	—
213	C₂³×C₃	#685	1	1	yes	characteristic
214	C₂²×D₃	—	10	2	no	—
215	C₂²×D₃	—	20	2	no	—
216	C₂²×D₃	—	20	2	no	—
217	C₂²×D₃	—	20	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	#685	1	1	yes	characteristic
222	C₂×C₃×C₄	#685	1	1	yes	characteristic
223	C₂×C₃×C₄	#685	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₂×Dic₃	—	10	2	no	—
228	C₂×Dic₃	—	20	2	no	—
229	D₁₂	—	20	2	no	—
230	D₁₂	—	20	2	no	—
231	Dic₆	—	20	2	no	—
232	Dic₆	—	20	2	no	—
233	C₃×C₈	—	10	4	no	—
234	C₃⋊C₈	—	2	4	no	—
235	C₂²×C₅	#805	1	2	yes	—
236	C₂²×C₅	#823	1	2	yes	—
237	C₂²×C₅	#825	1	2	yes	—
238	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
239	C₂×D₅	—	12	2	no	—
240	C₂×D₅	—	12	2	no	—
241	C₂×D₅	—	12	2	no	—
242	C₂×D₅	—	24	4	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	2	1	no	—
249	Dic₅	—	12	2	no	—
250	C₂⁴	—	30	1	no	—
251	C₂×Q₈	—	15	2	no	—
252	C₂×Q₈	—	30	1	no	—
253	C₂×Q₈	—	60	2	no	—
254	C₂×D₄	—	15	2	no	—
255	C₂×D₄	—	30	1	no	—
256	C₂×D₄	—	60	2	no	—
257	C₂²×C₄	—	30	1	no	—
258	C₂²×C₄	#1200	1	1	yes	characteristic
259	C₂²×C₄	#1202	1	1	yes	characteristic
260	C₂²×C₄	#1203	1	1	yes	characteristic
261	C₂×C₈	—	6	4	no	—
262	C₂×C₈	—	6	2	no	—
263	C₂×C₈	—	10	4	no	—
264	C₂×C₈	—	10	2	no	—
265	C₂²⋊C₄	—	30	2	no	—
266	C₂²⋊C₄	—	30	2	no	—
267	C₂²⋊C₄	—	30	2	no	—
268	C₂²⋊C₄	—	30	2	no	—
269	C₄²	#1200	1	1	yes	characteristic
270	C₄²	#1202	1	1	yes	characteristic
271	C₄²	#1203	1	1	yes	characteristic
272	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
273	C₃×C₅	C₂×#329	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	#3937	1	2	yes	—
275	C₂²×C₃	#3955	1	2	yes	—
276	C₂²×C₃	#3957	1	2	yes	—
277	C₂²×C₃	C₂×#685	1	1	yes	characteristic
278	C₂×D₃	—	20	2	no	—
279	C₂×D₃	—	20	2	no	—
280	C₂×D₃	—	20	2	no	—
281	C₂×D₃	—	40	4	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	1	no	—
288	Dic₃	—	20	2	no	—
289	C₂×C₅	#4464	1	4	yes	complemented
290	C₂×C₅	C₂×#805	1	1	yes	characteristic
291	C₂×C₅	C₂×#823	1	1	yes	characteristic
292	C₂×C₅	C₂×#825	1	1	yes	characteristic
293	D₅	—	24	2	no	—
294	C₂³	—	30	2	no	—
295	C₂³	—	60	2	no	—
296	C₂³	—	60	2	no	—
297	C₂³	—	60	2	no	—
298	C₂³	#6667	1	1	yes	characteristic
299	Q₈	—	60	2	no	—
300	Q₈	—	60	2	no	—
301	D₄	—	60	2	no	—
302	D₄	—	60	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	30	2	no	—
307	C₂×C₄	—	60	2	no	—
308	C₂×C₄	#6667	1	1	yes	characteristic
309	C₂×C₄	#6667	1	1	yes	characteristic
310	C₂×C₄	#6667	1	1	yes	characteristic
311	C₂×C₄	C₂×#1200	1	1	yes	characteristic
312	C₂×C₄	C₂×#1202	1	1	yes	characteristic
313	C₂×C₄	C₂×#1203	1	1	yes	characteristic
314	C₈	—	6	4	no	—
315	C₈	—	10	4	no	—
316	C₂×C₃	#64684	1	4	yes	complemented
317	C₂×C₃	C₂×#3937	1	1	yes	characteristic
318	C₂×C₃	C₂×#3955	1	1	yes	characteristic
319	C₂×C₃	C₂×#3957	1	1	yes	characteristic
320	D₃	—	40	2	no	—
321	C₅	C₂×#4464	1	1	yes	characteristic, complemented, Sylow
322	C₂²	—	60	2	no	—
323	C₂²	—	60	2	no	—
324	C₂²	—	60	2	no	—
325	C₂²	—	120	4	no	—
326	C₂²	#90960	1	2	yes	—
327	C₂²	#90980	1	2	yes	—
328	C₂²	#90981	1	2	yes	—
329	C₂²	C₂×#6667	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	60	2	no	—
337	C₃	C₂×#64684	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	120	2	no	—
339	C₂	#11745848	1	4	yes	direct product factor
340	C₂	C₂×#90960	1	1	yes	characteristic
341	C₂	C₂×#90980	1	1	yes	characteristic
342	C₂	C₂×#90981	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6667
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂³×C₃×C₅	C₂²⋊C₄
prime radical	C₂²×D₆₀	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₂×#329
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×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×C₅×#803	C₂	semidirect
3. C₂×C₃×#3935	C₂	semidirect
4. #11745848	C₂	direct (N × H)
18. C₅×#803	C₂²	semidirect
20. C₃×#3935	C₂²	semidirect
24. C₂×#3935	D₃	semidirect
51. C₂×#803	D₅	semidirect
67. #3935	C₂×D₃	semidirect
105. #803	C₂×D₅	semidirect
208. C₂×C₃×C₅	#329	semidirect
273. C₃×C₅	C₂×#329	semidirect
289. C₂×C₅	#4464	semidirect
316. C₂×C₃	#64684	semidirect
321. C₅	C₂×#4464	semidirect
337. C₃	C₂×#64684	semidirect
339. C₂	#11745848	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×#91764	2
C₂×C₅×#803	2
C₂×C₃×#3935	2
#11745848	2
C₂×#64684	3
C₂×#4464	5

Subgroups — C2×#11745848 #14283874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11745848 #14283874