Subgroups — C₂×#11748005 #14438789

Order 1920.

1520 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	C₂²×D₁₂₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91458	C₂	1	2	yes	maximal, complemented
4	C₂²×#6653	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#11748005	C₂	1	8	yes	maximal, direct product factor
7	C₂×#64984	—	3	1	no	maximal, Hall
8	C₂²×D₆₀	C₂²	1	2	yes	—
9	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
10	C₂²×#1202	—	2	4	no	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×D₁₂₀	—	2	4	no	—
13	C₂×D₁₂₀	C₂²	1	4	yes	complemented
14	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₄₀	—	2	4	no	—
17	C₂×C₃×D₄₀	C₂²	1	4	yes	complemented
18	#91458	—	2	4	no	—
19	#91458	C₂²	1	8	yes	complemented
20	C₂×#6653	—	2	8	no	—
21	C₂×#6653	C₂²	1	8	yes	complemented
22	C₅×C₈×Dic₃	C₂²	1	4	yes	complemented
23	C₂×#5151	—	5	1	no	maximal, Hall
24	C₂²×D₄₀	—	3	2	no	—
25	C₂²×D₄₀	—	3	1	no	—
26	C₂²×D₄₀	D₃	1	1	yes	characteristic, complemented
27	C₂×#4020	—	3	2	no	—
28	C₂×C₄×C₅×C₈	—	3	1	no	—
29	#64984	—	3	8	no	—
30	C₂³×D₁₅	—	2	2	no	—
31	C₂³×C₃×D₅	—	2	2	no	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×D₆₀	—	2	4	no	—
34	C₂×D₆₀	—	2	4	no	—
35	C₂×D₆₀	—	2	2	no	—
36	C₂×D₆₀	C₂³	1	4	yes	—
37	C₂²×C₅×Dic₃	D₄	1	2	yes	—
38	C₂×C₃×D₂₀	—	2	4	no	—
39	C₂×C₃×D₂₀	—	2	4	no	—
40	C₂×C₃×D₂₀	—	2	2	no	—
41	C₂×C₃×D₂₀	C₂³	1	4	yes	—
42	C₂×#1202	—	2	16	no	—
43	C₂×#1202	—	4	16	no	—
44	C₂×C₃×C₅×C₈	D₄	1	4	yes	complemented
45	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
46	D₁₂₀	—	2	8	no	—
47	C₄×C₅×Dic₃	D₄	1	2	yes	—
48	C₄×C₅×Dic₃	C₂³	1	2	yes	—
49	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	complemented
50	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
51	C₃×D₄₀	—	2	8	no	—
52	#6653	—	2	16	no	—
53	C₂²×C₃×D₈	—	5	1	no	—
54	C₂×#940	—	5	2	no	—
55	C₂²×#213	—	5	2	no	—
56	C₂²×D₂₄	—	5	1	no	—
57	#5151	—	5	8	no	—
58	C₂×C₈×Dic₃	D₅	1	1	yes	characteristic, complemented
59	C₂²×D₂₀	—	3	2	no	—
60	C₂²×D₂₀	—	6	4	no	—
61	C₂²×D₂₀	C₂×D₃	1	2	yes	—
62	C₂²×C₅×C₈	—	3	1	no	—
63	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
64	C₂×C₄²×C₅	—	3	1	no	—
65	C₂×D₄₀	—	2	4	no	—
66	C₂×D₄₀	—	3	8	no	—
67	C₂×D₄₀	—	3	4	no	—
68	C₂×D₄₀	—	6	8	no	—
69	C₂×D₄₀	—	6	4	no	—
70	C₂×D₄₀	C₂×D₃	1	4	yes	complemented
71	#4020	—	3	8	no	—
72	#4020	—	6	4	no	—
73	C₄×C₅×C₈	—	3	4	no	—
74	C₂×#493	—	15	1	no	Sylow
75	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×D₁₅	—	2	4	no	—
77	C₂²×D₁₅	—	4	4	no	—
78	C₂²×D₁₅	—	4	8	no	—
79	C₂²×C₃×D₅	—	2	4	no	—
80	C₂²×C₃×D₅	—	4	4	no	—
81	C₂²×C₃×D₅	—	4	8	no	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	D₆₀	—	2	4	no	—
87	D₆₀	—	2	4	no	—
88	D₆₀	—	4	4	no	—
89	C₂×C₅×Dic₃	D₈	1	8	yes	complemented
90	C₂×C₅×Dic₃	C₂×D₄	1	4	yes	—
91	C₃×D₂₀	—	2	4	no	—
92	C₃×D₂₀	—	2	4	no	—
93	C₃×D₂₀	—	4	4	no	—
94	#1202	—	4	32	no	—
95	C₃×C₅×C₈	C₂×D₄	1	4	yes	complemented
96	C₅×C₃⋊C₈	C₂×D₄	1	4	yes	complemented
97	C₂²×C₃×D₄	—	5	2	no	—
98	C₂²×#67	—	10	4	no	—
99	C₂²×D₁₂	—	5	2	no	—
100	C₂×C₃×D₈	—	5	4	no	—
101	C₂×C₃×D₈	—	10	4	no	—
102	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
103	#940	—	5	8	no	—
104	#940	—	10	4	no	—
105	C₂×#213	—	5	8	no	—
106	C₂×#213	—	10	8	no	—
107	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
108	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
109	C₂×D₂₄	—	5	4	no	—
110	C₂×D₂₄	—	10	4	no	—
111	C₈×Dic₃	C₂×D₅	1	4	yes	complemented
112	C₂³×D₅	—	2	2	no	—
113	C₂³×D₅	—	6	2	no	—
114	C₂²×C₄×C₅	—	3	2	no	—
115	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₂×D₂₀	—	2	4	no	—
117	C₂×D₂₀	—	2	4	no	—
118	C₂×D₂₀	—	2	2	no	—
119	C₂×D₂₀	—	3	4	no	—
120	C₂×D₂₀	—	6	4	no	—
121	C₂×D₂₀	—	6	4	no	—
122	C₂×D₂₀	—	6	16	no	—
123	C₂×D₂₀	—	6	2	no	—
124	C₂×D₂₀	—	12	16	no	—
125	C₂×D₂₀	C₂²×D₃	1	4	yes	—
126	C₂×C₅×C₈	—	3	4	no	—
127	C₂×C₅×C₈	—	3	2	no	—
128	C₂×C₅×C₈	#67	1	4	yes	complemented
129	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
130	C₄²×C₅	—	3	2	no	—
131	C₄²×C₅	—	3	2	no	—
132	D₄₀	—	2	8	no	—
133	D₄₀	—	6	16	no	—
134	D₄₀	—	6	8	no	—
135	C₂²×D₈	—	5	1	no	—
136	C₂²×D₈	—	15	2	no	—
137	C₂²×D₈	—	15	1	no	—
138	C₂×#127	—	15	2	no	—
139	#493	—	15	8	no	—
140	C₂×C₄×C₈	—	3	1	no	—
141	C₂²×C₃×C₅	#127	1	2	yes	—
142	C₂²×C₃×C₅	C₂×D₈	1	4	yes	—
143	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₅	—	4	4	no	—
145	C₂×D₁₅	—	4	8	no	—
146	C₂×D₁₅	—	8	8	no	—
147	C₂×C₃×D₅	—	4	4	no	—
148	C₂×C₃×D₅	—	4	8	no	—
149	C₂×C₃×D₅	—	8	8	no	—
150	C₃×C₄×C₅	#127	1	2	yes	—
151	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₅×Dic₃	C₂×D₈	1	8	yes	complemented
154	C₂⁴×C₃	—	10	2	no	—
155	C₂³×D₃	—	10	2	no	—
156	C₂×C₃×D₄	—	5	4	no	—
157	C₂×C₃×D₄	—	10	4	no	—
158	C₂×C₃×D₄	—	10	4	no	—
159	C₂×C₃×D₄	—	10	2	no	—
160	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
161	C₂×#67	—	10	16	no	—
162	C₂×#67	—	20	16	no	—
163	C₂²×Dic₃	D₂₀	1	2	yes	—
164	C₂×D₁₂	—	5	4	no	—
165	C₂×D₁₂	—	10	4	no	—
166	C₂×D₁₂	—	10	4	no	—
167	C₂×D₁₂	—	10	2	no	—
168	C₃×D₈	—	10	8	no	—
169	C₂×C₃×C₈	D₂₀	1	4	yes	complemented
170	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
171	#213	—	10	16	no	—
172	C₄×Dic₃	D₂₀	1	2	yes	—
173	C₄×Dic₃	C₂²×D₅	1	2	yes	—
174	C₂×C₃⋊C₈	D₂₀	1	4	yes	complemented
175	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
176	D₂₄	—	10	8	no	—
177	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
178	C₂²×D₅	—	2	4	no	—
179	C₂²×D₅	—	4	8	no	—
180	C₂²×D₅	—	4	4	no	—
181	C₂²×D₅	—	6	4	no	—
182	C₂²×D₅	—	12	8	no	—
183	C₂²×D₅	—	12	4	no	—
184	C₂×C₄×C₅	—	3	8	no	—
185	C₂×C₄×C₅	—	3	4	no	—
186	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
188	C₂×C₄×C₅	C₂×#67	1	2	yes	—
189	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
190	D₂₀	—	2	4	no	—
191	D₂₀	—	2	4	no	—
192	D₂₀	—	4	4	no	—
193	D₂₀	—	6	4	no	—
194	D₂₀	—	6	4	no	—
195	D₂₀	—	12	32	no	—
196	D₂₀	—	12	4	no	—
197	C₅×C₈	—	3	4	no	—
198	C₅×C₈	C₂×#67	1	4	yes	complemented
199	C₂²×D₄	—	5	2	no	—
200	C₂²×D₄	—	15	2	no	—
201	C₂²×D₄	—	30	4	no	—
202	C₂×D₈	—	5	4	no	—
203	C₂×D₈	—	10	4	no	—
204	C₂×D₈	—	15	8	no	—
205	C₂×D₈	—	15	4	no	—
206	C₂×D₈	—	30	8	no	—
207	C₂×D₈	—	30	4	no	—
208	C₂²×C₈	—	3	1	no	—
209	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
210	#127	—	15	8	no	—
211	#127	—	30	4	no	—
212	C₂×C₄²	—	3	1	no	—
213	C₄×C₈	—	3	4	no	—
214	C₂×C₃×C₅	#493	1	4	yes	complemented
215	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
216	C₂×C₃×C₅	C₂²×D₈	1	2	yes	—
217	D₁₅	—	8	4	no	—
218	C₃×D₅	—	8	4	no	—
219	C₂³×C₃	—	10	4	no	—
220	C₂³×C₃	—	20	4	no	—
221	C₂³×C₃	—	20	8	no	—
222	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
223	C₂²×D₃	—	10	4	no	—
224	C₂²×D₃	—	20	4	no	—
225	C₂²×D₃	—	20	8	no	—
226	C₃×D₄	—	10	4	no	—
227	C₃×D₄	—	10	4	no	—
228	C₃×D₄	—	20	4	no	—
229	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
230	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
232	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
233	#67	—	20	32	no	—
234	C₂×Dic₃	D₄₀	1	8	yes	complemented
235	C₂×Dic₃	C₂×D₂₀	1	4	yes	—
236	D₁₂	—	10	4	no	—
237	D₁₂	—	10	4	no	—
238	D₁₂	—	20	4	no	—
239	C₃×C₈	C₂×D₂₀	1	4	yes	complemented
240	C₃⋊C₈	C₂×D₂₀	1	4	yes	complemented
241	C₂²×C₅	D₃×D₈	1	4	yes	—
242	C₂²×C₅	#940	1	2	yes	—
243	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
244	C₂×D₅	—	4	8	no	—
245	C₂×D₅	—	4	4	no	—
246	C₂×D₅	—	8	8	no	—
247	C₂×D₅	—	12	8	no	—
248	C₂×D₅	—	12	4	no	—
249	C₂×D₅	—	24	8	no	—
250	C₄×C₅	—	3	8	no	—
251	C₄×C₅	#940	1	2	yes	—
252	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
253	C₄×C₅	C₂²×#67	1	1	yes	characteristic
254	C₂⁴	—	10	2	no	—
255	C₂⁴	—	30	2	no	—
256	C₂×D₄	—	5	4	no	—
257	C₂×D₄	—	10	4	no	—
258	C₂×D₄	—	10	2	no	—
259	C₂×D₄	—	10	4	no	—
260	C₂×D₄	—	15	4	no	—
261	C₂×D₄	—	30	16	no	—
262	C₂×D₄	—	30	4	no	—
263	C₂×D₄	—	30	2	no	—
264	C₂×D₄	—	30	4	no	—
265	C₂×D₄	—	60	16	no	—
266	C₂²×C₄	—	3	2	no	—
267	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
268	D₈	—	10	8	no	—
269	D₈	—	30	16	no	—
270	D₈	—	30	8	no	—
271	C₂×C₈	—	3	4	no	—
272	C₂×C₈	—	3	2	no	—
273	C₂×C₈	#1202	1	4	yes	complemented
274	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
275	C₄²	—	3	2	no	—
276	C₄²	—	3	2	no	—
277	C₃×C₅	C₂×#493	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	20	4	no	—
279	C₂²×C₃	—	20	8	no	—
280	C₂²×C₃	—	40	8	no	—
281	C₂²×C₃	#4020	1	2	yes	—
282	C₂²×C₃	C₂×D₄₀	1	4	yes	—
283	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
284	C₂×D₃	—	20	4	no	—
285	C₂×D₃	—	20	8	no	—
286	C₂×D₃	—	40	8	no	—
287	C₃×C₄	#4020	1	2	yes	—
288	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
289	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
290	Dic₃	C₂×D₄₀	1	8	yes	complemented
291	C₂×C₅	#5151	1	4	yes	complemented
292	C₂×C₅	C₂×D₃×D₈	1	2	yes	—
293	C₂×C₅	C₂×#940	1	1	yes	characteristic
294	D₅	—	8	4	no	—
295	D₅	—	24	4	no	—
296	C₂³	—	10	4	no	—
297	C₂³	—	20	4	no	—
298	C₂³	—	20	8	no	—
299	C₂³	—	30	4	no	—
300	C₂³	—	60	4	no	—
301	C₂³	—	60	8	no	—
302	C₂³	D₃×D₂₀	1	1	yes	characteristic
303	D₄	—	10	4	no	—
304	D₄	—	10	4	no	—
305	D₄	—	20	4	no	—
306	D₄	—	30	4	no	—
307	D₄	—	30	4	no	—
308	D₄	—	60	32	no	—
309	D₄	—	60	4	no	—
310	C₂×C₄	—	3	4	no	—
311	C₂×C₄	—	3	8	no	—
312	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
313	C₂×C₄	D₃×D₂₀	1	2	yes	—
314	C₂×C₄	C₂×#1202	1	2	yes	—
315	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
316	C₈	—	3	4	no	—
317	C₈	C₂×#1202	1	4	yes	complemented
318	C₂×C₃	—	40	4	no	—
319	C₂×C₃	#64984	1	4	yes	complemented
320	C₂×C₃	C₂×#4020	1	1	yes	characteristic
321	C₂×C₃	C₂²×D₄₀	1	2	yes	—
322	D₃	—	40	4	no	—
323	C₅	C₂×#5151	1	1	yes	characteristic, complemented, Sylow
324	C₂²	—	20	4	no	—
325	C₂²	—	20	8	no	—
326	C₂²	—	40	8	no	—
327	C₂²	—	60	4	no	—
328	C₂²	—	60	8	no	—
329	C₂²	—	120	8	no	—
330	C₂²	D₃×D₄₀	1	4	yes	—
331	C₂²	#91458	1	2	yes	—
332	C₂²	C₂×D₃×D₂₀	1	1	yes	characteristic
333	C₄	—	3	8	no	—
334	C₄	#91458	1	2	yes	—
335	C₄	C₂×D₃×D₂₀	1	1	yes	characteristic
336	C₄	C₂²×#1202	1	1	yes	characteristic
337	C₃	C₂×#64984	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	40	4	no	—
339	C₂	—	120	4	no	—
340	C₂	#11748005	1	4	yes	direct product factor
341	C₂	C₂×D₃×D₄₀	1	2	yes	—
342	C₂	C₂×#91458	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃×D₂₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#493
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₂₀	C₂	semidirect
2. C₂²×C₃×D₄₀	C₂	semidirect
3. C₂×#91458	C₂	semidirect
4. C₂²×#6653	C₂	semidirect
5. C₂×C₅×C₈×Dic₃	C₂	semidirect
6. #11748005	C₂	direct (N × H)
13. C₂×D₁₂₀	C₂²	semidirect
17. C₂×C₃×D₄₀	C₂²	semidirect
19. #91458	C₂²	semidirect
21. C₂×#6653	C₂²	semidirect
22. C₅×C₈×Dic₃	C₂²	semidirect
26. C₂²×D₄₀	D₃	semidirect
44. C₂×C₃×C₅×C₈	D₄	semidirect
49. C₂×C₅×C₃⋊C₈	D₄	semidirect
58. C₂×C₈×Dic₃	D₅	semidirect
70. C₂×D₄₀	C₂×D₃	semidirect
89. C₂×C₅×Dic₃	D₈	semidirect
95. C₃×C₅×C₈	C₂×D₄	semidirect
96. C₅×C₃⋊C₈	C₂×D₄	semidirect
111. C₈×Dic₃	C₂×D₅	semidirect
128. C₂×C₅×C₈	#67	semidirect
153. C₅×Dic₃	C₂×D₈	semidirect
169. C₂×C₃×C₈	D₂₀	semidirect
174. C₂×C₃⋊C₈	D₂₀	semidirect
198. C₅×C₈	C₂×#67	semidirect
214. C₂×C₃×C₅	#493	semidirect
234. C₂×Dic₃	D₄₀	semidirect
239. C₃×C₈	C₂×D₂₀	semidirect
240. C₃⋊C₈	C₂×D₂₀	semidirect
273. C₂×C₈	#1202	semidirect
277. C₃×C₅	C₂×#493	semidirect
290. Dic₃	C₂×D₄₀	semidirect
291. C₂×C₅	#5151	semidirect
317. C₈	C₂×#1202	semidirect
319. C₂×C₃	#64984	semidirect
323. C₅	C₂×#5151	semidirect
337. C₃	C₂×#64984	semidirect
340. C₂	#11748005	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₁₂₀	2
C₂²×C₃×D₄₀	2
C₂×#91458	2
C₂²×#6653	2
C₂×C₅×C₈×Dic₃	2
#11748005	2
C₂×#64984	3
C₂×#5151	5

Subgroups — C2×#11748005 #14438789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11748005 #14438789