Subgroups — C₂×#11747544 #14478363

Order 1920.

800 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×D₁₂₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4050	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90943	C₂	1	2	yes	maximal, complemented
5	#11747544	C₂	1	8	yes	maximal, direct product factor
6	C₂×#90938	C₂	1	2	yes	maximal, complemented
7	C₂×#65197	—	3	1	no	maximal, Hall
8	C₂²×D₆₀	—	2	1	no	—
9	C₂×C₃×#709	—	2	1	no	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×D₁₂₀	—	2	2	no	—
12	C₂×D₁₂₀	—	2	1	no	—
13	C₂×D₁₂₀	C₂²	1	2	yes	—
14	C₂×D₁₂₀	C₂²	1	4	yes	complemented
15	C₂×C₃×Dic₂₀	C₂²	1	1	yes	characteristic, complemented
16	C₃×#4050	—	2	2	no	—
17	C₃×#4050	C₂²	1	4	yes	complemented
18	C₂×C₃×D₄₀	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₃×#677	—	2	1	no	—
20	C₅×#812	C₂²	1	4	yes	complemented
21	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	complemented
22	#90943	C₂²	1	8	yes	complemented
23	#90938	C₂²	1	8	yes	complemented
24	C₂×#5193	—	5	1	no	maximal, Hall
25	C₂×#4050	D₃	1	1	yes	characteristic, complemented
26	C₂²×D₄₀	—	3	1	no	—
27	C₂×C₅×M₃₂	—	3	1	no	—
28	#65197	—	3	8	no	—
29	C₂×#3932	—	3	2	no	—
30	C₂×D₈₀	—	3	2	no	—
31	C₂³×D₁₅	—	4	1	no	—
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×D₆₀	—	2	2	no	—
34	C₂×D₆₀	—	2	4	no	—
35	C₂×D₆₀	—	4	1	no	—
36	C₂×D₆₀	—	4	2	no	—
37	C₂×C₃×C₂²⋊D₅	—	4	1	no	—
38	C₃×#709	—	2	4	no	—
39	C₃×#709	—	4	2	no	—
40	C₂×C₃×D₂₀	—	2	1	no	—
41	C₂×C₃×C₄×D₅	—	4	1	no	—
42	C₂×C₃×Dic₁₀	—	2	1	no	—
43	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
45	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
47	D₁₂₀	—	2	2	no	—
48	D₁₂₀	—	2	4	no	—
49	D₁₂₀	C₂³	1	4	yes	—
50	C₃×Dic₂₀	—	2	1	no	—
51	C₃×Dic₂₀	C₂³	1	2	yes	complemented
52	C₃×D₄₀	—	2	1	no	—
53	C₃×D₄₀	C₂³	1	2	yes	complemented
54	C₃×#677	—	2	4	no	—
55	C₅×C₃⋊C₁₆	C₂³	1	4	yes	complemented
56	C₂×C₃×#135	—	5	1	no	—
57	C₂²×D₂₄	—	5	1	no	—
58	#5193	—	5	8	no	—
59	C₂×#828	—	5	2	no	—
60	C₂×#826	—	5	2	no	—
61	C₂×#812	D₅	1	1	yes	characteristic, complemented
62	C₂×#709	—	2	1	no	—
63	C₂²×D₂₀	—	6	1	no	—
64	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
65	C₂×Dic₂₀	C₂×D₃	1	1	yes	characteristic, complemented
66	#4050	—	2	2	no	—
67	#4050	C₂×D₃	1	4	yes	complemented
68	C₂×D₄₀	—	3	2	no	—
69	C₂×D₄₀	—	3	4	no	—
70	C₂×D₄₀	—	6	1	no	—
71	C₂×D₄₀	—	6	2	no	—
72	C₂×D₄₀	C₂×D₃	1	1	yes	characteristic, complemented
73	C₂×#677	—	2	1	no	—
74	C₅×M₃₂	—	3	4	no	—
75	C₂×C₅×C₁₆	—	3	2	no	—
76	#3932	—	3	8	no	—
77	D₈₀	—	3	8	no	—
78	C₂×#509	—	15	1	no	Sylow
79	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
80	C₂²×D₁₅	—	4	4	no	—
81	C₂²×D₁₅	—	4	2	no	—
82	C₂²×D₁₅	—	8	4	no	—
83	C₂²×C₃×D₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
88	D₆₀	—	2	4	no	—
89	D₆₀	—	4	2	no	—
90	D₆₀	—	4	4	no	—
91	C₃×C₂²⋊D₅	—	4	4	no	—
92	C₂×C₃×Dic₅	—	4	1	no	—
93	C₃×D₂₀	—	2	2	no	—
94	C₃×D₂₀	—	4	1	no	—
95	C₃×C₄×D₅	—	4	4	no	—
96	C₃×Dic₁₀	—	2	2	no	—
97	C₃×Dic₁₀	—	4	1	no	—
98	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
101	C₂×C₃×#41	—	10	1	no	—
102	C₂²×D₁₂	—	10	1	no	—
103	C₃×#135	—	5	4	no	—
104	C₃×#135	—	10	2	no	—
105	C₂×C₃×Q₁₆	—	5	1	no	—
106	C₂×C₃×SD₁₆	—	10	1	no	—
107	C₂×C₃×D₈	—	5	1	no	—
108	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
109	C₂×D₂₄	—	5	2	no	—
110	C₂×D₂₄	—	5	4	no	—
111	C₂×D₂₄	—	10	1	no	—
112	C₂×D₂₄	—	10	2	no	—
113	#828	—	5	8	no	—
114	#826	—	5	8	no	—
115	#812	C₂×D₅	1	4	yes	complemented
116	C₂×C₃⋊C₁₆	C₂×D₅	1	2	yes	complemented
117	C₂³×D₅	—	12	1	no	—
118	C₂²×C₄×C₅	#67	1	1	yes	characteristic
119	C₂×C₂²⋊D₅	—	4	1	no	—
120	#709	—	2	4	no	—
121	#709	—	4	2	no	—
122	C₂×D₂₀	—	2	1	no	—
123	C₂×D₂₀	—	6	2	no	—
124	C₂×D₂₀	—	6	4	no	—
125	C₂×D₂₀	—	12	1	no	—
126	C₂×D₂₀	—	12	2	no	—
127	C₂×C₄×D₅	—	4	1	no	—
128	C₂×Dic₁₀	—	2	1	no	—
129	C₂×C₅×C₈	#67	1	1	yes	characteristic
130	C₂×C₅×C₈	#67	1	2	yes	—
131	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
133	Dic₂₀	—	2	1	no	—
134	Dic₂₀	C₂²×D₃	1	2	yes	complemented
135	D₄₀	—	2	1	no	—
136	D₄₀	—	3	4	no	—
137	D₄₀	—	6	4	no	—
138	D₄₀	—	6	2	no	—
139	D₄₀	C₂²×D₃	1	2	yes	complemented
140	#677	—	2	4	no	—
141	C₅×C₁₆	—	3	4	no	—
142	C₂×#135	—	5	1	no	—
143	C₂²×D₈	—	15	1	no	—
144	#509	—	15	8	no	—
145	C₂×SD₃₂	—	15	2	no	—
146	C₂×D₁₆	—	15	2	no	—
147	C₂×M₃₂	—	3	1	no	—
148	C₂²×C₃×C₅	—	2	2	no	—
149	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
150	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
151	C₂×D₁₅	—	4	4	no	—
152	C₂×D₁₅	—	8	8	no	—
153	C₂×D₁₅	—	8	4	no	—
154	C₂×C₃×D₅	—	4	2	no	—
155	C₂×C₃×D₅	—	8	2	no	—
156	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
157	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
158	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₃×Dic₅	—	4	2	no	—
160	C₂³×D₃	—	20	1	no	—
161	C₃×#41	—	10	4	no	—
162	C₃×#41	—	20	2	no	—
163	C₂×C₃×Q₈	—	10	1	no	—
164	C₂×C₃×D₄	—	10	1	no	—
165	C₂×C₃×D₄	—	20	1	no	—
166	C₂²×C₃×C₄	—	20	1	no	—
167	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
168	C₂×D₁₂	—	10	2	no	—
169	C₂×D₁₂	—	10	4	no	—
170	C₂×D₁₂	—	20	1	no	—
171	C₂×D₁₂	—	20	2	no	—
172	C₃×Q₁₆	—	5	2	no	—
173	C₃×Q₁₆	—	10	1	no	—
174	C₃×SD₁₆	—	10	4	no	—
175	C₃×D₈	—	5	2	no	—
176	C₃×D₈	—	10	1	no	—
177	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
178	C₂×C₃×C₈	D₂₀	1	2	yes	—
179	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
180	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
181	D₂₄	—	5	4	no	—
182	D₂₄	—	10	2	no	—
183	D₂₄	—	10	4	no	—
184	C₃⋊C₁₆	C₂²×D₅	1	4	yes	complemented
185	C₂³×C₅	#213	1	1	yes	characteristic
186	C₂²×D₅	—	4	1	no	—
187	C₂²×D₅	—	12	4	no	—
188	C₂²×D₅	—	12	2	no	—
189	C₂²×D₅	—	24	4	no	—
190	C₂×C₄×C₅	#213	1	1	yes	characteristic
191	C₂×C₄×C₅	#213	1	2	yes	—
192	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂×#67	1	2	yes	—
194	C₂²⋊D₅	—	4	4	no	—
195	C₂×Dic₅	—	4	1	no	—
196	D₂₀	—	2	2	no	—
197	D₂₀	—	4	1	no	—
198	D₂₀	—	6	4	no	—
199	D₂₀	—	12	2	no	—
200	D₂₀	—	12	4	no	—
201	C₄×D₅	—	4	4	no	—
202	Dic₁₀	—	2	2	no	—
203	Dic₁₀	—	4	1	no	—
204	C₅×C₈	C₂×#67	1	2	yes	—
205	C₅×C₈	C₂×#67	1	1	yes	characteristic
206	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
207	C₂×#41	—	10	1	no	—
208	C₂²×D₄	—	30	1	no	—
209	#135	—	5	4	no	—
210	#135	—	10	2	no	—
211	C₂×Q₁₆	—	5	1	no	—
212	C₂×SD₁₆	—	10	1	no	—
213	C₂×D₈	—	5	1	no	—
214	C₂×D₈	—	15	2	no	—
215	C₂×D₈	—	15	4	no	—
216	C₂×D₈	—	30	1	no	—
217	C₂×D₈	—	30	2	no	—
218	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
219	SD₃₂	—	15	8	no	—
220	D₁₆	—	15	8	no	—
221	M₃₂	—	3	4	no	—
222	C₂×C₁₆	—	3	2	no	—
223	C₂×C₃×C₅	—	2	2	no	—
224	C₂×C₃×C₅	#509	1	2	yes	complemented
225	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
226	D₁₅	—	8	4	no	—
227	C₃×D₅	—	8	2	no	—
228	C₂³×C₃	—	20	1	no	—
229	C₂³×C₃	D₄₀	1	1	yes	characteristic
230	C₂²×D₃	—	20	4	no	—
231	C₂²×D₃	—	20	2	no	—
232	C₂²×D₃	—	40	4	no	—
233	C₃×Q₈	—	10	2	no	—
234	C₃×Q₈	—	20	1	no	—
235	C₃×D₄	—	10	2	no	—
236	C₃×D₄	—	20	4	no	—
237	C₃×D₄	—	20	1	no	—
238	C₂×C₃×C₄	—	20	1	no	—
239	C₂×C₃×C₄	—	20	4	no	—
240	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
241	C₂×C₃×C₄	D₄₀	1	2	yes	—
242	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
244	D₁₂	—	10	4	no	—
245	D₁₂	—	20	4	no	—
246	D₁₂	—	20	2	no	—
247	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
248	C₃×C₈	C₂×D₂₀	1	2	yes	—
249	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
250	C₂²×C₅	—	2	2	no	—
251	C₂²×C₅	C₂×#213	1	2	yes	—
252	C₂²×C₅	C₂×#213	1	1	yes	characteristic
253	C₂×D₅	—	4	2	no	—
254	C₂×D₅	—	8	2	no	—
255	C₂×D₅	—	12	4	no	—
256	C₂×D₅	—	24	8	no	—
257	C₂×D₅	—	24	4	no	—
258	C₄×C₅	C₂×#213	1	1	yes	characteristic
259	C₄×C₅	C₂×#213	1	2	yes	—
260	C₄×C₅	C₂²×#67	1	1	yes	characteristic
261	Dic₅	—	4	2	no	—
262	C₂⁴	—	60	1	no	—
263	#41	—	10	4	no	—
264	#41	—	20	2	no	—
265	C₂×Q₈	—	10	1	no	—
266	C₂×D₄	—	10	1	no	—
267	C₂×D₄	—	20	1	no	—
268	C₂×D₄	—	30	4	no	—
269	C₂×D₄	—	30	2	no	—
270	C₂×D₄	—	60	2	no	—
271	C₂×D₄	—	60	1	no	—
272	C₂²×C₄	—	20	1	no	—
273	C₂²×C₄	#1202	1	1	yes	characteristic
274	Q₁₆	—	5	2	no	—
275	Q₁₆	—	10	1	no	—
276	SD₁₆	—	10	4	no	—
277	D₈	—	5	2	no	—
278	D₈	—	10	1	no	—
279	D₈	—	15	4	no	—
280	D₈	—	30	2	no	—
281	D₈	—	30	4	no	—
282	C₂×C₈	#1202	1	1	yes	characteristic
283	C₂×C₈	#1202	1	2	yes	—
284	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
285	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
286	C₁₆	—	3	4	no	—
287	C₃×C₅	C₂×#509	1	1	yes	characteristic, complemented, Hall
288	C₂²×C₃	—	2	2	no	—
289	C₂²×C₃	—	20	2	no	—
290	C₂²×C₃	—	40	2	no	—
291	C₂²×C₃	C₂×D₄₀	1	2	yes	—
292	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
293	C₂×D₃	—	20	4	no	—
294	C₂×D₃	—	40	8	no	—
295	C₂×D₃	—	40	4	no	—
296	C₃×C₄	—	20	2	no	—
297	C₃×C₄	C₂×D₄₀	1	2	yes	—
298	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
299	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
300	C₂×C₅	—	2	2	no	—
301	C₂×C₅	#5193	1	2	yes	complemented
302	C₂×C₅	C₂²×#213	1	1	yes	characteristic
303	D₅	—	8	2	no	—
304	D₅	—	24	4	no	—
305	C₂³	—	20	1	no	—
306	C₂³	—	60	4	no	—
307	C₂³	—	60	2	no	—
308	C₂³	—	120	4	no	—
309	C₂³	#6653	1	1	yes	characteristic
310	Q₈	—	10	2	no	—
311	Q₈	—	20	1	no	—
312	D₄	—	10	2	no	—
313	D₄	—	20	4	no	—
314	D₄	—	20	1	no	—
315	D₄	—	30	4	no	—
316	D₄	—	60	4	no	—
317	D₄	—	60	2	no	—
318	C₂×C₄	—	20	1	no	—
319	C₂×C₄	—	20	4	no	—
320	C₂×C₄	#6653	1	2	yes	—
321	C₂×C₄	#6653	1	1	yes	characteristic
322	C₂×C₄	C₂×#1202	1	2	yes	—
323	C₂×C₄	C₂×#1202	1	1	yes	characteristic
324	C₈	C₂×#1202	1	1	yes	characteristic
325	C₈	C₂×#1202	1	2	yes	—
326	C₈	C₂²×D₃×D₅	1	1	yes	characteristic
327	C₂×C₃	—	2	2	no	—
328	C₂×C₃	—	40	2	no	—
329	C₂×C₃	#65197	1	2	yes	complemented
330	C₂×C₃	C₂²×D₄₀	1	1	yes	characteristic
331	D₃	—	40	4	no	—
332	C₅	C₂×#5193	1	1	yes	characteristic, complemented, Sylow
333	C₂²	—	2	2	no	—
334	C₂²	—	20	2	no	—
335	C₂²	—	40	2	no	—
336	C₂²	—	60	4	no	—
337	C₂²	—	120	8	no	—
338	C₂²	—	120	4	no	—
339	C₂²	C₂×#6653	1	2	yes	—
340	C₂²	C₂×#6653	1	1	yes	characteristic
341	C₄	—	20	2	no	—
342	C₄	C₂×#6653	1	2	yes	—
343	C₄	C₂×#6653	1	1	yes	characteristic
344	C₄	C₂²×#1202	1	1	yes	characteristic
345	C₃	C₂×#65197	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	2	2	no	—
347	C₂	—	40	2	no	—
348	C₂	—	120	4	no	—
349	C₂	#11747544	1	2	yes	direct product factor
350	C₂	C₂²×#6653	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6653
commutator, cosocle	C₃×C₅×C₈	C₂⁴
Frattini	C₈	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#509
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ 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. C₂×C₃×#4050	C₂	semidirect
3. C₂×C₅×#812	C₂	semidirect
4. C₂×#90943	C₂	semidirect
5. #11747544	C₂	direct (N × H)
6. C₂×#90938	C₂	semidirect
14. C₂×D₁₂₀	C₂²	semidirect
15. C₂×C₃×Dic₂₀	C₂²	semidirect
17. C₃×#4050	C₂²	semidirect
18. C₂×C₃×D₄₀	C₂²	semidirect
20. C₅×#812	C₂²	semidirect
21. C₂×C₅×C₃⋊C₁₆	C₂²	semidirect
22. #90943	C₂²	semidirect
23. #90938	C₂²	semidirect
25. C₂×#4050	D₃	semidirect
51. C₃×Dic₂₀	C₂³	semidirect
53. C₃×D₄₀	C₂³	semidirect
55. C₅×C₃⋊C₁₆	C₂³	semidirect
61. C₂×#812	D₅	semidirect
65. C₂×Dic₂₀	C₂×D₃	semidirect
67. #4050	C₂×D₃	semidirect
72. C₂×D₄₀	C₂×D₃	semidirect
115. #812	C₂×D₅	semidirect
116. C₂×C₃⋊C₁₆	C₂×D₅	semidirect
134. Dic₂₀	C₂²×D₃	semidirect
139. D₄₀	C₂²×D₃	semidirect
184. C₃⋊C₁₆	C₂²×D₅	semidirect
224. C₂×C₃×C₅	#509	semidirect
287. C₃×C₅	C₂×#509	semidirect
301. C₂×C₅	#5193	semidirect
329. C₂×C₃	#65197	semidirect
332. C₅	C₂×#5193	semidirect
345. C₃	C₂×#65197	semidirect
349. C₂	#11747544	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
C₂×C₃×#4050	2
C₂×C₅×#812	2
C₂×#90943	2
#11747544	2
C₂×#90938	2
C₂×#65197	3
C₂×#5193	5

Subgroups — C2×#11747544 #14478363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11747544 #14478363