Subgroups — C₂×#11751217 #14427743

Order 1920.

1024 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	C₂³×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#91813	C₂	1	2	yes	maximal
3	C₂²×Dic₆₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751217	C₂	1	8	yes	maximal, direct product factor
5	C₂×#91102	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#91100	C₂	1	2	yes	maximal, complemented
7	C₂×#65405	—	3	1	no	maximal, Hall
8	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂²×Dic₃₀	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	—	2	4	no	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₅×C₈	C₂²	1	4	yes	—
13	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6719	—	2	2	no	—
15	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
16	#91813	—	2	4	no	—
17	#91813	C₂²	1	8	yes	—
18	C₂×#6712	—	2	2	no	—
19	C₂×Dic₆₀	—	2	4	no	—
20	C₂×Dic₆₀	C₂²	1	4	yes	complemented
21	#91102	C₂²	1	4	yes	complemented
22	#91100	C₂²	1	8	yes	complemented
23	C₂×#5116	—	5	1	no	maximal, Hall
24	C₂³×C₅×C₈	D₃	1	1	yes	characteristic
25	C₂×#4070	—	3	2	no	—
26	C₂²×Dic₂₀	—	3	1	no	—
27	#65405	—	3	8	no	—
28	C₂×#3950	—	3	1	no	—
29	C₂×#3948	—	3	2	no	—
30	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	—	2	2	no	—
32	C₂²×C₃×C₄×C₅	—	2	2	no	—
33	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₅	—	2	2	no	—
38	C₂²×Dic₁₅	—	2	2	no	—
39	C₂×Dic₃₀	—	2	4	no	—
40	C₂×Dic₃₀	—	2	4	no	—
41	C₂×Dic₃₀	—	2	2	no	—
42	C₂×Dic₃₀	C₂³	1	4	yes	—
43	C₂×C₃×C₅×C₈	—	2	2	no	—
44	C₂×C₃×C₅×C₈	—	2	8	no	—
45	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
46	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
47	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
48	#6719	—	2	8	no	—
49	C₄⋊Dic₁₅	—	2	2	no	—
50	C₄⋊Dic₁₅	C₂³	1	4	yes	—
51	#6712	—	2	8	no	—
52	Dic₆₀	—	2	8	no	—
53	C₂³×C₃×C₈	D₅	1	1	yes	characteristic
54	C₂×#924	—	5	2	no	—
55	C₂²×Dic₁₂	—	5	1	no	—
56	#5116	—	5	8	no	—
57	C₂×#818	—	5	1	no	—
58	C₂×#816	—	5	2	no	—
59	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
60	C₂²×Dic₁₀	—	3	2	no	—
61	C₂²×C₅×C₈	—	2	4	no	—
62	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₅×C₈	C₂×D₃	1	4	yes	—
64	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
65	C₂×C₂²⋊Dic₅	—	6	2	no	—
66	C₂×C₄⋊Dic₅	—	3	2	no	—
67	#4070	—	3	8	no	—
68	#4070	—	6	4	no	—
69	C₂×#683	—	6	2	no	—
70	C₂×Dic₂₀	—	3	4	no	—
71	C₂×Dic₂₀	—	6	4	no	—
72	#3950	—	3	4	no	—
73	#3948	—	3	8	no	—
74	C₂×#467	—	15	1	no	Sylow
75	C₂³×C₃×C₅	—	2	2	no	—
76	C₂³×C₃×C₅	—	2	2	no	—
77	C₂³×C₃×C₅	Q₁₆	1	4	yes	complemented
78	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
79	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	4	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	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₂×C₃×C₄×C₅	#41	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₂×Dic₁₅	—	2	4	no	—
90	C₂×Dic₁₅	—	2	4	no	—
91	C₂×Dic₁₅	—	4	4	no	—
92	C₂×Dic₁₅	—	4	4	no	—
93	Dic₃₀	—	2	4	no	—
94	Dic₃₀	—	2	4	no	—
95	Dic₃₀	—	4	4	no	—
96	C₃×C₅×C₈	—	2	2	no	—
97	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
98	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
99	C₂²×Dic₆	—	5	2	no	—
100	C₂²×C₃×C₈	—	2	4	no	—
101	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
102	C₂²×C₃×C₈	C₂×D₅	1	4	yes	—
103	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
104	C₂×#217	—	10	2	no	—
105	C₂×C₄⋊Dic₃	—	5	2	no	—
106	#924	—	5	8	no	—
107	#924	—	10	4	no	—
108	C₂×Dic₃⋊C₄	—	10	2	no	—
109	C₂×Dic₁₂	—	5	4	no	—
110	C₂×Dic₁₂	—	10	4	no	—
111	#818	—	5	4	no	—
112	#816	—	5	8	no	—
113	C₂⁴×C₅	D₁₂	1	1	yes	characteristic
114	C₂²×C₄×C₅	—	2	2	no	—
115	C₂²×C₄×C₅	—	2	2	no	—
116	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
117	C₂²×C₄×C₅	D₁₂	1	2	yes	—
118	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
119	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
120	C₂²×Dic₅	—	6	2	no	—
121	C₂²×Dic₅	—	6	2	no	—
122	C₂×Dic₁₀	—	3	4	no	—
123	C₂×Dic₁₀	—	6	4	no	—
124	C₂×Dic₁₀	—	6	4	no	—
125	C₂×Dic₁₀	—	6	2	no	—
126	C₂×C₅×C₈	—	2	2	no	—
127	C₂×C₅×C₈	—	2	8	no	—
128	C₂×C₅×C₈	#67	1	4	yes	—
129	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
130	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
131	C₂²⋊Dic₅	—	6	8	no	—
132	C₄⋊Dic₅	—	3	4	no	—
133	C₄⋊Dic₅	—	6	2	no	—
134	#683	—	6	8	no	—
135	Dic₂₀	—	6	8	no	—
136	C₂²×Q₁₆	—	15	1	no	—
137	C₂³×C₈	D₁₅	1	1	yes	characteristic
138	C₂×C₂²⋊Q₈	—	15	2	no	—
139	#467	—	15	8	no	—
140	C₂×#107	—	15	1	no	—
141	C₂×#103	—	15	2	no	—
142	C₂²×C₃×C₅	—	2	2	no	—
143	C₂²×C₃×C₅	—	2	2	no	—
144	C₂²×C₃×C₅	—	2	8	no	—
145	C₂²×C₃×C₅	#121	1	2	yes	—
146	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
147	C₂²×C₃×C₅	C₂×Q₁₆	1	4	yes	complemented
148	C₂²×C₃×C₅	#135	1	2	yes	—
149	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₃×C₄×C₅	—	2	2	no	—
151	C₃×C₄×C₅	#121	1	2	yes	—
152	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
154	Dic₁₅	—	4	4	no	—
155	Dic₁₅	—	4	4	no	—
156	C₂⁴×C₃	D₂₀	1	1	yes	characteristic
157	C₂²×C₃×C₄	—	2	2	no	—
158	C₂²×C₃×C₄	—	2	2	no	—
159	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
160	C₂²×C₃×C₄	D₂₀	1	2	yes	—
161	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
162	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
163	C₂²×Dic₃	—	10	2	no	—
164	C₂²×Dic₃	—	10	2	no	—
165	C₂×Dic₆	—	5	4	no	—
166	C₂×Dic₆	—	10	4	no	—
167	C₂×Dic₆	—	10	4	no	—
168	C₂×Dic₆	—	10	2	no	—
169	C₂×C₃×C₈	—	2	2	no	—
170	C₂×C₃×C₈	—	2	8	no	—
171	C₂×C₃×C₈	C₂²⋊D₅	1	4	yes	—
172	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
173	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
174	#217	—	10	8	no	—
175	C₄⋊Dic₃	—	5	4	no	—
176	C₄⋊Dic₃	—	10	2	no	—
177	Dic₃⋊C₄	—	10	8	no	—
178	Dic₁₂	—	10	8	no	—
179	C₂³×C₅	—	2	2	no	—
180	C₂³×C₅	—	2	2	no	—
181	C₂³×C₅	Dic₁₂	1	4	yes	complemented
182	C₂³×C₅	C₂×D₁₂	1	2	yes	—
183	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
184	C₂×C₄×C₅	—	2	2	no	—
185	C₂×C₄×C₅	—	2	4	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
189	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
190	C₂×C₄×C₅	#235	1	2	yes	—
191	C₂×C₄×C₅	C₂×#67	1	2	yes	—
192	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
193	C₂×Dic₅	—	6	4	no	—
194	C₂×Dic₅	—	6	4	no	—
195	C₂×Dic₅	—	12	4	no	—
196	C₂×Dic₅	—	12	4	no	—
197	Dic₁₀	—	6	4	no	—
198	Dic₁₀	—	6	4	no	—
199	Dic₁₀	—	12	4	no	—
200	C₅×C₈	—	2	2	no	—
201	C₅×C₈	C₂×#67	1	4	yes	—
202	C₂²×Q₈	—	15	2	no	—
203	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
204	C₂×Q₁₆	—	15	4	no	—
205	C₂×Q₁₆	—	30	4	no	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
208	C₂²×C₈	C₂×D₁₅	1	4	yes	—
209	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
210	C₂²⋊Q₈	—	15	8	no	—
211	C₂²⋊Q₈	—	30	4	no	—
212	C₂×C₄⋊C₄	—	15	2	no	—
213	C₂×C₄⋊C₄	—	30	2	no	—
214	C₂×C₂²⋊C₄	—	30	2	no	—
215	#107	—	15	4	no	—
216	#103	—	15	8	no	—
217	C₂×C₃×C₅	—	2	4	no	—
218	C₂×C₃×C₅	#467	1	4	yes	complemented
219	C₂×C₃×C₅	C₂×#121	1	1	yes	characteristic
220	C₂×C₃×C₅	C₂²×Q₁₆	1	1	yes	characteristic
221	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
222	C₂³×C₃	—	2	2	no	—
223	C₂³×C₃	—	2	2	no	—
224	C₂³×C₃	Dic₂₀	1	4	yes	complemented
225	C₂³×C₃	C₂×D₂₀	1	2	yes	—
226	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	—	2	2	no	—
230	C₂×C₃×C₄	—	2	4	no	—
231	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
232	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
233	C₂×C₃×C₄	#709	1	2	yes	—
234	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
235	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
236	C₂×Dic₃	—	10	4	no	—
237	C₂×Dic₃	—	10	4	no	—
238	C₂×Dic₃	—	20	4	no	—
239	C₂×Dic₃	—	20	4	no	—
240	Dic₆	—	10	4	no	—
241	Dic₆	—	10	4	no	—
242	Dic₆	—	20	4	no	—
243	C₃×C₈	—	2	2	no	—
244	C₃×C₈	C₂×C₂²⋊D₅	1	4	yes	—
245	C₂²×C₅	—	2	2	no	—
246	C₂²×C₅	—	2	2	no	—
247	C₂²×C₅	—	2	8	no	—
248	C₂²×C₅	#904	1	2	yes	—
249	C₂²×C₅	C₂×Dic₁₂	1	2	yes	—
250	C₂²×C₅	C₂×Dic₁₂	1	4	yes	complemented
251	C₂²×C₅	#930	1	2	yes	—
252	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
253	C₄×C₅	—	2	2	no	—
254	C₄×C₅	#930	1	2	yes	—
255	C₄×C₅	C₂×#235	1	1	yes	characteristic
256	C₄×C₅	C₂²×#67	1	1	yes	characteristic
257	Dic₅	—	12	4	no	—
258	Dic₅	—	12	4	no	—
259	C₂⁴	D₆₀	1	1	yes	characteristic
260	C₂×Q₈	—	15	4	no	—
261	C₂×Q₈	—	30	2	no	—
262	C₂×Q₈	—	30	4	no	—
263	C₂×Q₈	—	30	4	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	30	2	no	—
267	C₂²×C₄	—	30	2	no	—
268	C₂²×C₄	D₆₀	1	2	yes	—
269	C₂²×C₄	D₆₀	1	1	yes	characteristic
270	C₂²×C₄	C₂²×D₁₅	1	2	yes	—
271	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
272	Q₁₆	—	30	8	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	8	no	—
275	C₂×C₈	#1220	1	4	yes	—
276	C₂×C₈	C₂²×D₁₅	1	2	yes	—
277	C₂×C₈	C₂²×D₁₅	1	2	yes	—
278	C₄⋊C₄	—	15	4	no	—
279	C₄⋊C₄	—	30	2	no	—
280	C₄⋊C₄	—	30	8	no	—
281	C₂²⋊C₄	—	30	8	no	—
282	C₃×C₅	C₂×#467	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	2	2	no	—
284	C₂²×C₃	—	2	2	no	—
285	C₂²×C₃	—	2	8	no	—
286	C₂²×C₃	#4050	1	2	yes	—
287	C₂²×C₃	C₂×Dic₂₀	1	2	yes	—
288	C₂²×C₃	C₂×Dic₂₀	1	4	yes	complemented
289	C₂²×C₃	#4076	1	2	yes	—
290	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
291	C₃×C₄	—	2	2	no	—
292	C₃×C₄	#4076	1	2	yes	—
293	C₃×C₄	C₂×#709	1	1	yes	characteristic
294	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
295	Dic₃	—	20	4	no	—
296	Dic₃	—	20	4	no	—
297	C₂×C₅	—	2	4	no	—
298	C₂×C₅	#5116	1	4	yes	complemented
299	C₂×C₅	C₂×#904	1	1	yes	characteristic
300	C₂×C₅	C₂²×Dic₁₂	1	1	yes	characteristic
301	C₂×C₅	C₂×#930	1	1	yes	characteristic
302	C₂³	—	2	2	no	—
303	C₂³	—	2	2	no	—
304	C₂³	Dic₆₀	1	4	yes	complemented
305	C₂³	C₂×D₆₀	1	2	yes	—
306	C₂³	C₂×D₆₀	1	1	yes	characteristic
307	Q₈	—	30	4	no	—
308	Q₈	—	30	4	no	—
309	Q₈	—	60	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	30	4	no	—
315	C₂×C₄	—	30	4	no	—
316	C₂×C₄	—	60	4	no	—
317	C₂×C₄	—	60	4	no	—
318	C₂×C₄	C₂×D₆₀	1	2	yes	—
319	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
320	C₂×C₄	#6817	1	2	yes	—
321	C₂×C₄	C₂×#1220	1	2	yes	—
322	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
323	C₈	—	2	2	no	—
324	C₈	C₂×#1220	1	4	yes	—
325	C₂×C₃	—	2	4	no	—
326	C₂×C₃	#65405	1	4	yes	complemented
327	C₂×C₃	C₂×#4050	1	1	yes	characteristic
328	C₂×C₃	C₂²×Dic₂₀	1	1	yes	characteristic
329	C₂×C₃	C₂×#4076	1	1	yes	characteristic
330	C₅	C₂×#5116	1	1	yes	characteristic, complemented, Sylow
331	C₂²	—	2	2	no	—
332	C₂²	—	2	8	no	—
333	C₂²	—	2	2	no	—
334	C₂²	#91793	1	2	yes	—
335	C₂²	C₂×Dic₆₀	1	2	yes	—
336	C₂²	C₂×Dic₆₀	1	4	yes	complemented
337	C₂²	#91819	1	2	yes	—
338	C₂²	C₂²×D₆₀	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	60	4	no	—
341	C₄	—	60	4	no	—
342	C₄	#91819	1	2	yes	—
343	C₄	C₂×#6817	1	1	yes	characteristic
344	C₄	C₂²×#1220	1	1	yes	characteristic
345	C₃	C₂×#65405	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	2	4	no	—
347	C₂	#11751217	1	4	yes	direct product factor
348	C₂	C₂×#91793	1	1	yes	characteristic
349	C₂	C₂²×Dic₆₀	1	1	yes	characteristic
350	C₂	C₂×#91819	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂³×C₃×C₅×C₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
hypercenter	C₂³×C₈	D₁₅
hypocenter	C₃×C₅	C₂×#467
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×Dic₆₀	C₂	semidirect
4. #11751217	C₂	direct (N × H)
5. C₂×#91102	C₂	semidirect
6. C₂×#91100	C₂	semidirect
20. C₂×Dic₆₀	C₂²	semidirect
21. #91102	C₂²	semidirect
22. #91100	C₂²	semidirect
77. C₂³×C₃×C₅	Q₁₆	semidirect
147. C₂²×C₃×C₅	C₂×Q₁₆	semidirect
181. C₂³×C₅	Dic₁₂	semidirect
218. C₂×C₃×C₅	#467	semidirect
224. C₂³×C₃	Dic₂₀	semidirect
250. C₂²×C₅	C₂×Dic₁₂	semidirect
282. C₃×C₅	C₂×#467	semidirect
288. C₂²×C₃	C₂×Dic₂₀	semidirect
298. C₂×C₅	#5116	semidirect
304. C₂³	Dic₆₀	semidirect
326. C₂×C₃	#65405	semidirect
330. C₅	C₂×#5116	semidirect
336. C₂²	C₂×Dic₆₀	semidirect
345. C₃	C₂×#65405	semidirect
347. C₂	#11751217	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×C₈	2
C₂×#91813	2
C₂²×Dic₆₀	2
#11751217	2
C₂×#91102	2
C₂×#91100	2
C₂×#65405	3
C₂×#5116	5

Subgroups — C2×#11751217 #14427743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751217 #14427743