Subgroups — C₂×#11747943 #14427770

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	C₂²×Dic₆₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#91360	C₂	1	2	yes	maximal
4	C₂×C₅×#818	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90975	C₂	1	2	yes	maximal, complemented
6	#11747943	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65176	—	3	1	no	maximal, Hall
8	C₂²×Dic₃₀	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×Dic₆₀	—	2	4	no	—
12	C₂×Dic₆₀	C₂²	1	4	yes	complemented
13	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₈×D₅	—	2	4	no	—
16	C₂×C₃×C₈×D₅	C₂²	1	4	yes	—
17	C₂×Dic₃⋊Dic₅	—	2	2	no	—
18	C₂×#6665	—	2	2	no	—
19	#91360	—	2	4	no	—
20	#91360	C₂²	1	8	yes	—
21	C₅×#818	C₂²	1	4	yes	complemented
22	#90975	C₂²	1	8	yes	complemented
23	C₂×#5116	—	5	1	no	maximal, Hall
24	C₂²×Dic₂₀	—	3	1	no	—
25	C₂²×C₈×D₅	D₃	1	1	yes	characteristic
26	C₂×#4043	—	3	2	no	—
27	C₂×C₅×#107	—	3	1	no	—
28	C₂×#3942	—	3	2	no	—
29	#65176	—	3	8	no	—
30	C₂³×C₃×D₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₅	—	2	2	no	—
33	C₂×Dic₃₀	—	2	4	no	—
34	C₂×Dic₃₀	—	2	4	no	—
35	C₂×Dic₃₀	—	2	2	no	—
36	C₂×Dic₃₀	C₂³	1	4	yes	—
37	C₂²×C₅×Dic₃	—	2	2	no	—
38	C₂²×C₃×Dic₅	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₅	—	2	2	no	—
40	C₂×C₃×C₄×D₅	—	2	2	no	—
41	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
42	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
43	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	Dic₆₀	—	2	8	no	—
46	C₅×C₄⋊Dic₃	—	2	2	no	—
47	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
48	C₂×C₃×#168	—	2	2	no	—
49	C₂×C₃×#168	C₂³	1	2	yes	—
50	C₃×C₈×D₅	—	2	8	no	—
51	Dic₃⋊Dic₅	—	2	8	no	—
52	#6665	—	2	8	no	—
53	C₂³×C₃×C₈	—	5	1	no	—
54	C₂×#924	—	5	2	no	—
55	C₂²×Dic₁₂	—	5	1	no	—
56	#5116	—	5	8	no	—
57	C₂×#818	D₅	1	1	yes	characteristic, complemented
58	C₂×#816	—	5	2	no	—
59	C₂²×C₄×D₅	C₂×D₃	1	1	yes	characteristic
60	C₂²×Dic₁₀	—	3	2	no	—
61	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂×C₅×C₄⋊C₄	—	3	2	no	—
63	C₂×#685	—	6	2	no	—
64	C₂×C₄⋊Dic₅	—	6	2	no	—
65	C₂²×#168	C₂×D₃	1	1	yes	characteristic
66	C₂×Dic₂₀	—	3	4	no	—
67	C₂×Dic₂₀	—	6	4	no	—
68	C₂×C₈×D₅	—	2	4	no	—
69	C₂×C₈×D₅	C₂×D₃	1	4	yes	—
70	#4043	—	3	8	no	—
71	#4043	—	6	4	no	—
72	C₅×#107	—	3	4	no	—
73	#3942	—	3	8	no	—
74	C₂×#467	—	15	1	no	Sylow
75	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₃×D₅	—	2	2	no	—
77	C₂²×C₃×D₅	—	2	2	no	—
78	C₂²×C₃×D₅	Q₁₆	1	4	yes	complemented
79	C₂²×C₃×D₅	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	#41	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
84	C₂×Dic₁₅	—	2	4	no	—
85	C₂×Dic₁₅	—	4	4	no	—
86	Dic₃₀	—	2	4	no	—
87	Dic₃₀	—	2	4	no	—
88	Dic₃₀	—	4	4	no	—
89	C₂×C₅×Dic₃	—	2	4	no	—
90	C₂×C₅×Dic₃	—	4	4	no	—
91	C₂×C₃×Dic₅	—	2	2	no	—
92	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
93	C₃×C₄×D₅	—	2	4	no	—
94	C₃×C₄×D₅	—	2	2	no	—
95	C₃×C₄×D₅	—	2	2	no	—
96	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
97	C₃×#168	—	2	2	no	—
98	C₂³×C₃×C₄	—	5	1	no	—
99	C₂²×Dic₆	—	5	2	no	—
100	C₂²×C₃×C₈	—	5	1	no	—
101	C₂²×C₃×C₈	—	5	4	no	—
102	C₂²×C₃×C₈	—	10	4	no	—
103	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
104	C₂×#217	—	10	2	no	—
105	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
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	C₂×D₅	1	4	yes	complemented
112	#816	—	5	8	no	—
113	C₂³×D₅	D₁₂	1	1	yes	characteristic
114	C₂²×C₄×C₅	—	6	2	no	—
115	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₂²×Dic₅	—	6	2	no	—
117	C₂²×Dic₅	D₁₂	1	1	yes	characteristic
118	C₂×C₄×D₅	—	2	2	no	—
119	C₂×C₄×D₅	—	2	2	no	—
120	C₂×C₄×D₅	D₁₂	1	2	yes	—
121	C₂×C₄×D₅	C₂²×D₃	1	2	yes	—
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	#67	1	4	yes	—
127	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
128	C₅×C₄⋊C₄	—	3	4	no	—
129	C₅×C₄⋊C₄	—	6	2	no	—
130	#685	—	6	8	no	—
131	C₄⋊Dic₅	—	6	8	no	—
132	C₂×#168	—	2	2	no	—
133	C₂×#168	C₂²×D₃	1	2	yes	—
134	Dic₂₀	—	6	8	no	—
135	C₈×D₅	—	2	8	no	—
136	C₂²×Q₁₆	—	15	1	no	—
137	C₂³×C₈	—	5	1	no	—
138	C₂×C₂²⋊Q₈	—	15	2	no	—
139	#467	—	15	8	no	—
140	C₂×#107	—	3	1	no	—
141	C₂×#103	—	15	2	no	—
142	C₂²×C₃×C₅	#121	1	2	yes	—
143	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
144	C₂²×C₃×C₅	#135	1	2	yes	—
145	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₃×D₅	—	2	2	no	—
147	C₂×C₃×D₅	—	2	2	no	—
148	C₂×C₃×D₅	—	2	8	no	—
149	C₂×C₃×D₅	C₂×Q₁₆	1	4	yes	complemented
150	C₃×C₄×C₅	#121	1	2	yes	—
151	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
153	Dic₁₅	—	4	4	no	—
154	C₃×Dic₅	—	2	2	no	—
155	C₅×Dic₃	—	4	4	no	—
156	C₂⁴×C₃	—	5	1	no	—
157	C₂²×C₃×C₄	—	5	1	no	—
158	C₂²×C₃×C₄	—	5	2	no	—
159	C₂²×C₃×C₄	—	5	2	no	—
160	C₂²×C₃×C₄	—	10	2	no	—
161	C₂²×C₃×C₄	—	10	2	no	—
162	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
163	C₂²×Dic₃	—	2	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	—	5	2	no	—
170	C₂×C₃×C₈	—	10	2	no	—
171	C₂×C₃×C₈	—	10	8	no	—
172	C₂×C₃×C₈	D₂₀	1	4	yes	—
173	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
174	#217	—	10	8	no	—
175	C₄⋊Dic₃	—	2	2	no	—
176	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
177	Dic₃⋊C₄	—	10	8	no	—
178	Dic₁₂	—	10	8	no	—
179	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
180	C₂²×D₅	—	2	2	no	—
181	C₂²×D₅	—	2	2	no	—
182	C₂²×D₅	Dic₁₂	1	4	yes	complemented
183	C₂²×D₅	C₂×D₁₂	1	2	yes	—
184	C₂×C₄×C₅	—	6	4	no	—
185	C₂×C₄×C₅	—	12	4	no	—
186	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
187	C₂×C₄×C₅	#235	1	2	yes	—
188	C₂×C₄×C₅	C₂×#67	1	2	yes	—
189	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
190	C₂×Dic₅	—	2	2	no	—
191	C₂×Dic₅	—	6	4	no	—
192	C₂×Dic₅	—	12	4	no	—
193	C₂×Dic₅	C₂×D₁₂	1	2	yes	—
194	C₄×D₅	—	2	4	no	—
195	C₄×D₅	—	2	2	no	—
196	C₄×D₅	—	2	2	no	—
197	Dic₁₀	—	6	4	no	—
198	Dic₁₀	—	6	4	no	—
199	Dic₁₀	—	12	4	no	—
200	C₅×C₈	C₂×#67	1	4	yes	—
201	#168	—	2	2	no	—
202	C₂²×Q₈	—	15	2	no	—
203	C₂³×C₄	—	5	1	no	—
204	C₂×Q₁₆	—	15	4	no	—
205	C₂×Q₁₆	—	30	4	no	—
206	C₂²×C₈	—	5	1	no	—
207	C₂²×C₈	—	5	4	no	—
208	C₂²×C₈	—	10	4	no	—
209	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
210	C₂²⋊Q₈	—	15	8	no	—
211	C₂²⋊Q₈	—	30	4	no	—
212	C₂×C₄⋊C₄	—	3	2	no	—
213	C₂×C₄⋊C₄	—	30	2	no	—
214	C₂×C₂²⋊C₄	—	30	2	no	—
215	#107	—	3	4	no	—
216	#103	—	15	8	no	—
217	C₂×C₃×C₅	#467	1	4	yes	complemented
218	C₂×C₃×C₅	C₂×#121	1	1	yes	characteristic
219	C₂×C₃×C₅	C₂²×Q₁₆	1	1	yes	characteristic
220	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
221	C₃×D₅	—	2	4	no	—
222	C₂³×C₃	—	5	2	no	—
223	C₂³×C₃	—	5	4	no	—
224	C₂³×C₃	—	10	2	no	—
225	C₂³×C₃	—	10	2	no	—
226	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
227	C₂×C₃×C₄	—	5	2	no	—
228	C₂×C₃×C₄	—	10	2	no	—
229	C₂×C₃×C₄	—	10	2	no	—
230	C₂×C₃×C₄	—	10	2	no	—
231	C₂×C₃×C₄	—	10	4	no	—
232	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
233	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
234	C₂×C₃×C₄	#713	1	2	yes	—
235	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
236	C₂×Dic₃	—	2	4	no	—
237	C₂×Dic₃	—	4	4	no	—
238	C₂×Dic₃	—	10	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	—	10	2	no	—
244	C₃×C₈	C₂×D₂₀	1	4	yes	—
245	C₂²×C₅	#904	1	2	yes	—
246	C₂²×C₅	C₂×Dic₁₂	1	2	yes	—
247	C₂²×C₅	#930	1	2	yes	—
248	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
249	C₂×D₅	—	2	2	no	—
250	C₂×D₅	—	2	2	no	—
251	C₂×D₅	—	2	8	no	—
252	C₂×D₅	C₂×Dic₁₂	1	4	yes	complemented
253	C₄×C₅	—	12	4	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	—	2	2	no	—
258	Dic₅	—	12	4	no	—
259	C₂⁴	—	5	1	no	—
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	—	5	1	no	—
265	C₂²×C₄	—	5	2	no	—
266	C₂²×C₄	—	5	2	no	—
267	C₂²×C₄	—	6	2	no	—
268	C₂²×C₄	—	10	2	no	—
269	C₂²×C₄	—	10	2	no	—
270	C₂²×C₄	—	30	2	no	—
271	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
272	Q₁₆	—	30	8	no	—
273	C₂×C₈	—	5	2	no	—
274	C₂×C₈	—	10	2	no	—
275	C₂×C₈	—	10	8	no	—
276	C₂×C₈	#1202	1	4	yes	—
277	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
278	C₄⋊C₄	—	3	4	no	—
279	C₄⋊C₄	—	6	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	—	5	4	no	—
284	C₂²×C₃	—	10	2	no	—
285	C₂²×C₃	—	10	2	no	—
286	C₂²×C₃	—	10	8	no	—
287	C₂²×C₃	#4041	1	2	yes	—
288	C₂²×C₃	#4058	1	2	yes	—
289	C₂²×C₃	D₅×Q₁₆	1	2	yes	—
290	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
291	C₃×C₄	—	10	2	no	—
292	C₃×C₄	#4041	1	2	yes	—
293	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
294	C₃×C₄	C₂×#713	1	1	yes	characteristic
295	Dic₃	—	4	4	no	—
296	Dic₃	—	20	4	no	—
297	C₂×C₅	#5116	1	4	yes	complemented
298	C₂×C₅	C₂×#904	1	1	yes	characteristic
299	C₂×C₅	C₂²×Dic₁₂	1	1	yes	characteristic
300	C₂×C₅	C₂×#930	1	1	yes	characteristic
301	D₅	—	2	4	no	—
302	C₂³	—	5	2	no	—
303	C₂³	—	5	4	no	—
304	C₂³	—	10	2	no	—
305	C₂³	—	10	2	no	—
306	C₂³	D₅×D₁₂	1	1	yes	characteristic
307	Q₈	—	30	4	no	—
308	Q₈	—	30	4	no	—
309	Q₈	—	60	4	no	—
310	C₂×C₄	—	5	2	no	—
311	C₂×C₄	—	6	4	no	—
312	C₂×C₄	—	10	2	no	—
313	C₂×C₄	—	10	4	no	—
314	C₂×C₄	—	10	2	no	—
315	C₂×C₄	—	10	2	no	—
316	C₂×C₄	—	12	4	no	—
317	C₂×C₄	—	30	4	no	—
318	C₂×C₄	—	60	4	no	—
319	C₂×C₄	#6773	1	2	yes	—
320	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
321	C₂×C₄	C₂×#1202	1	2	yes	—
322	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
323	C₈	—	10	2	no	—
324	C₈	C₂×#1202	1	4	yes	—
325	C₂×C₃	—	10	4	no	—
326	C₂×C₃	#65176	1	4	yes	complemented
327	C₂×C₃	C₂×#4041	1	1	yes	characteristic
328	C₂×C₃	C₂×#4058	1	1	yes	characteristic
329	C₂×C₃	C₂×D₅×Q₁₆	1	1	yes	characteristic
330	C₅	C₂×#5116	1	1	yes	characteristic, complemented, Sylow
331	C₂²	—	5	4	no	—
332	C₂²	—	10	2	no	—
333	C₂²	—	10	2	no	—
334	C₂²	—	10	8	no	—
335	C₂²	D₅×Dic₁₂	1	2	yes	—
336	C₂²	#91270	1	2	yes	—
337	C₂²	#91450	1	2	yes	—
338	C₂²	C₂×D₅×D₁₂	1	1	yes	characteristic
339	C₄	—	10	2	no	—
340	C₄	—	12	4	no	—
341	C₄	—	60	4	no	—
342	C₄	#91450	1	2	yes	—
343	C₄	C₂×#6773	1	1	yes	characteristic
344	C₄	C₂²×#1202	1	1	yes	characteristic
345	C₃	C₂×#65176	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	10	4	no	—
347	C₂	#11747943	1	4	yes	direct product factor
348	C₂	C₂×D₅×Dic₁₂	1	1	yes	characteristic
349	C₂	C₂×#91270	1	1	yes	characteristic
350	C₂	C₂×#91450	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₅×D₁₂
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
prime radical	C₂³×C₃×D₅	D₄
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#467
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×Dic₆₀	C₂	semidirect
4. C₂×C₅×#818	C₂	semidirect
5. C₂×#90975	C₂	semidirect
6. #11747943	C₂	direct (N × H)
12. C₂×Dic₆₀	C₂²	semidirect
21. C₅×#818	C₂²	semidirect
22. #90975	C₂²	semidirect
57. C₂×#818	D₅	semidirect
78. C₂²×C₃×D₅	Q₁₆	semidirect
111. #818	C₂×D₅	semidirect
149. C₂×C₃×D₅	C₂×Q₁₆	semidirect
182. C₂²×D₅	Dic₁₂	semidirect
217. C₂×C₃×C₅	#467	semidirect
252. C₂×D₅	C₂×Dic₁₂	semidirect
282. C₃×C₅	C₂×#467	semidirect
297. C₂×C₅	#5116	semidirect
326. C₂×C₃	#65176	semidirect
330. C₅	C₂×#5116	semidirect
345. C₃	C₂×#65176	semidirect
347. C₂	#11747943	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×Dic₆₀	2
C₂²×C₃×C₈×D₅	2
C₂×#91360	2
C₂×C₅×#818	2
C₂×#90975	2
#11747943	2
C₂×#65176	3
C₂×#5116	5

Subgroups — C2×#11747943 #14427770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11747943 #14427770