Subgroups — C₂×#11747539 #14478359

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	C₂×C₅×#904	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90940	C₂	1	2	yes	maximal, complemented
5	#11747539	C₂	1	8	yes	maximal, direct product factor
6	C₂×#90937	C₂	1	2	yes	maximal, complemented
7	C₂×#65482	—	3	1	no	maximal, Hall
8	C₂×C₅×#235	—	2	1	no	—
9	C₂²×C₃×D₂₀	—	2	1	no	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
12	C₅×#904	—	2	2	no	—
13	C₅×#904	C₂²	1	4	yes	complemented
14	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₅×#204	—	2	1	no	—
16	C₂×C₃×D₄₀	—	2	2	no	—
17	C₂×C₃×D₄₀	—	2	1	no	—
18	C₂×C₃×D₄₀	C₂²	1	2	yes	—
19	C₂×C₃×D₄₀	C₂²	1	4	yes	complemented
20	#91096	C₂²	1	4	yes	complemented
21	C₂×C₃⋊#672	C₂²	1	2	yes	complemented
22	#90940	C₂²	1	8	yes	complemented
23	#90937	C₂²	1	8	yes	complemented
24	C₂×#5147	—	5	1	no	maximal, Hall
25	C₂×C₅×#135	—	3	1	no	—
26	C₂²×D₄₀	D₃	1	1	yes	characteristic, complemented
27	#65482	—	3	8	no	—
28	C₂×#3960	—	3	2	no	—
29	C₂×#3958	—	3	2	no	—
30	C₂×#3944	—	3	1	no	—
31	C₂³×C₃×D₅	—	4	1	no	—
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×C₅×#67	—	4	1	no	—
34	C₅×#235	—	2	4	no	—
35	C₅×#235	—	4	2	no	—
36	C₂×C₅×D₁₂	—	2	1	no	—
37	C₂×C₄×C₅×D₃	—	4	1	no	—
38	C₂×C₅×Dic₆	—	2	1	no	—
39	C₂×C₃×D₂₀	—	2	2	no	—
40	C₂×C₃×D₂₀	—	2	4	no	—
41	C₂×C₃×D₂₀	—	4	1	no	—
42	C₂×C₃×D₂₀	—	4	2	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	C₅×Dic₁₂	—	2	1	no	—
48	C₅×Dic₁₂	C₂³	1	2	yes	complemented
49	C₅×D₂₄	—	2	1	no	—
50	C₅×D₂₄	C₂³	1	2	yes	complemented
51	C₅×#204	—	2	4	no	—
52	C₃×D₄₀	—	2	2	no	—
53	C₃×D₄₀	—	2	4	no	—
54	C₃×D₄₀	C₂³	1	4	yes	—
55	C₃⋊#672	C₂³	1	4	yes	complemented
56	C₂²×C₃×D₈	—	5	1	no	—
57	C₂×#904	D₅	1	1	yes	characteristic, complemented
58	C₂×#827	—	5	2	no	—
59	#5147	—	5	8	no	—
60	C₂×#826	—	5	2	no	—
61	C₂×#812	—	5	1	no	—
62	C₂×C₅×#41	—	6	1	no	—
63	C₂²×D₂₀	—	2	1	no	—
64	C₅×#135	—	3	4	no	—
65	C₅×#135	—	6	2	no	—
66	C₂×C₅×Q₁₆	—	3	1	no	—
67	C₂×C₅×SD₁₆	—	6	1	no	—
68	C₂×C₅×D₈	—	3	1	no	—
69	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
70	C₂×D₄₀	—	2	2	no	—
71	C₂×D₄₀	—	2	1	no	—
72	C₂×D₄₀	C₂×D₃	1	2	yes	—
73	C₂×D₄₀	C₂×D₃	1	4	yes	complemented
74	#3960	—	3	8	no	—
75	#3958	—	3	8	no	—
76	#3944	—	3	4	no	—
77	C₂×#672	—	3	2	no	—
78	C₂×#509	—	15	1	no	Sylow
79	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
80	C₂²×C₅×D₃	—	4	1	no	—
81	C₂²×C₃×D₅	—	4	4	no	—
82	C₂²×C₃×D₅	—	4	2	no	—
83	C₂²×C₃×D₅	—	8	4	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	C₅×#67	—	4	4	no	—
89	C₂×C₅×Dic₃	—	4	1	no	—
90	C₅×D₁₂	—	2	2	no	—
91	C₅×D₁₂	—	4	1	no	—
92	C₄×C₅×D₃	—	4	4	no	—
93	C₅×Dic₆	—	2	2	no	—
94	C₅×Dic₆	—	4	1	no	—
95	C₃×D₂₀	—	2	4	no	—
96	C₃×D₂₀	—	4	2	no	—
97	C₃×D₂₀	—	4	4	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	C₂²×C₃×D₄	—	10	1	no	—
102	C₂×#235	—	2	1	no	—
103	C₂×C₃×D₈	—	5	2	no	—
104	C₂×C₃×D₈	—	5	4	no	—
105	C₂×C₃×D₈	—	10	1	no	—
106	C₂×C₃×D₈	—	10	2	no	—
107	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
108	C₂×Dic₁₂	C₂×D₅	1	1	yes	characteristic, complemented
109	#904	—	2	2	no	—
110	#904	C₂×D₅	1	4	yes	complemented
111	C₂×D₂₄	C₂×D₅	1	1	yes	characteristic, complemented
112	C₂×#204	—	2	1	no	—
113	#827	—	5	8	no	—
114	#826	—	5	8	no	—
115	#812	—	5	4	no	—
116	C₂×C₃⋊C₁₆	—	5	2	no	—
117	C₂³×D₅	—	4	1	no	—
118	C₅×#41	—	6	4	no	—
119	C₅×#41	—	12	2	no	—
120	C₂×C₅×Q₈	—	6	1	no	—
121	C₂×C₅×D₄	—	6	1	no	—
122	C₂×C₅×D₄	—	12	1	no	—
123	C₂²×C₄×C₅	—	12	1	no	—
124	C₂²×C₄×C₅	#67	1	1	yes	characteristic
125	C₂×D₂₀	—	2	2	no	—
126	C₂×D₂₀	—	2	4	no	—
127	C₂×D₂₀	—	4	1	no	—
128	C₂×D₂₀	—	4	2	no	—
129	C₅×Q₁₆	—	3	2	no	—
130	C₅×Q₁₆	—	6	1	no	—
131	C₅×SD₁₆	—	6	4	no	—
132	C₅×D₈	—	3	2	no	—
133	C₅×D₈	—	6	1	no	—
134	C₂×C₅×C₈	#67	1	1	yes	characteristic
135	C₂×C₅×C₈	#67	1	2	yes	—
136	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
138	D₄₀	—	2	2	no	—
139	D₄₀	—	2	4	no	—
140	D₄₀	C₂²×D₃	1	4	yes	—
141	#672	—	3	4	no	—
142	C₂×#135	—	3	1	no	—
143	C₂²×D₈	—	5	1	no	—
144	#509	—	15	8	no	—
145	C₂×SD₃₂	—	15	2	no	—
146	C₂×D₁₆	—	15	2	no	—
147	C₂×M₃₂	—	15	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×C₅×D₃	—	4	2	no	—
152	C₂×C₅×D₃	—	8	2	no	—
153	C₂×C₃×D₅	—	4	4	no	—
154	C₂×C₃×D₅	—	8	8	no	—
155	C₂×C₃×D₅	—	8	4	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	C₅×Dic₃	—	4	2	no	—
160	C₂⁴×C₃	—	20	1	no	—
161	C₂×C₃×D₄	—	10	2	no	—
162	C₂×C₃×D₄	—	10	4	no	—
163	C₂×C₃×D₄	—	20	1	no	—
164	C₂×C₃×D₄	—	20	2	no	—
165	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
166	C₂×#67	—	4	1	no	—
167	#235	—	2	4	no	—
168	#235	—	4	2	no	—
169	C₂×D₁₂	—	2	1	no	—
170	C₂×C₄×D₃	—	4	1	no	—
171	C₂×Dic₆	—	2	1	no	—
172	C₃×D₈	—	5	4	no	—
173	C₃×D₈	—	10	4	no	—
174	C₃×D₈	—	10	2	no	—
175	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
176	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
177	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
179	Dic₁₂	—	2	1	no	—
180	Dic₁₂	C₂²×D₅	1	2	yes	complemented
181	D₂₄	—	2	1	no	—
182	D₂₄	C₂²×D₅	1	2	yes	complemented
183	#204	—	2	4	no	—
184	C₃⋊C₁₆	—	5	4	no	—
185	C₂³×C₅	—	12	1	no	—
186	C₂³×C₅	#213	1	1	yes	characteristic
187	C₂²×D₅	—	4	4	no	—
188	C₂²×D₅	—	4	2	no	—
189	C₂²×D₅	—	8	4	no	—
190	C₅×Q₈	—	6	2	no	—
191	C₅×Q₈	—	12	1	no	—
192	C₅×D₄	—	6	2	no	—
193	C₅×D₄	—	12	4	no	—
194	C₅×D₄	—	12	1	no	—
195	C₂×C₄×C₅	—	12	1	no	—
196	C₂×C₄×C₅	—	12	4	no	—
197	C₂×C₄×C₅	#213	1	1	yes	characteristic
198	C₂×C₄×C₅	#213	1	2	yes	—
199	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂×#67	1	2	yes	—
201	D₂₀	—	2	4	no	—
202	D₂₀	—	4	4	no	—
203	D₂₀	—	4	2	no	—
204	C₅×C₈	C₂×#67	1	1	yes	characteristic
205	C₅×C₈	C₂×#67	1	2	yes	—
206	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
207	C₂×#41	—	6	1	no	—
208	C₂²×D₄	—	10	1	no	—
209	#135	—	3	4	no	—
210	#135	—	6	2	no	—
211	C₂×Q₁₆	—	3	1	no	—
212	C₂×SD₁₆	—	6	1	no	—
213	C₂×D₈	—	3	1	no	—
214	C₂×D₈	—	5	2	no	—
215	C₂×D₈	—	5	4	no	—
216	C₂×D₈	—	10	1	no	—
217	C₂×D₈	—	10	2	no	—
218	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
219	SD₃₂	—	15	8	no	—
220	D₁₆	—	15	8	no	—
221	M₃₂	—	15	4	no	—
222	C₂×C₁₆	—	15	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	C₃×D₅	—	8	4	no	—
227	C₅×D₃	—	8	2	no	—
228	C₂³×C₃	—	20	4	no	—
229	C₂³×C₃	—	20	2	no	—
230	C₂³×C₃	—	40	4	no	—
231	C₂³×C₃	#686	1	1	yes	characteristic
232	C₂²×D₃	—	4	1	no	—
233	C₃×D₄	—	10	4	no	—
234	C₃×D₄	—	20	2	no	—
235	C₃×D₄	—	20	4	no	—
236	C₂×C₃×C₄	#686	1	1	yes	characteristic
237	C₂×C₃×C₄	#686	1	2	yes	—
238	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
240	#67	—	4	4	no	—
241	C₂×Dic₃	—	4	1	no	—
242	D₁₂	—	2	2	no	—
243	D₁₂	—	4	1	no	—
244	C₄×D₃	—	4	4	no	—
245	Dic₆	—	2	2	no	—
246	Dic₆	—	4	1	no	—
247	C₃×C₈	C₂×C₂²⋊D₅	1	2	yes	—
248	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
249	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
250	C₂²×C₅	—	2	2	no	—
251	C₂²×C₅	—	12	2	no	—
252	C₂²×C₅	—	24	2	no	—
253	C₂²×C₅	C₂×#213	1	2	yes	—
254	C₂²×C₅	C₂×#213	1	1	yes	characteristic
255	C₂×D₅	—	4	4	no	—
256	C₂×D₅	—	8	8	no	—
257	C₂×D₅	—	8	4	no	—
258	C₄×C₅	—	12	2	no	—
259	C₄×C₅	C₂×#213	1	2	yes	—
260	C₄×C₅	C₂×#213	1	1	yes	characteristic
261	C₄×C₅	C₂²×#67	1	1	yes	characteristic
262	C₂⁴	—	20	1	no	—
263	#41	—	6	4	no	—
264	#41	—	12	2	no	—
265	C₂×Q₈	—	6	1	no	—
266	C₂×D₄	—	6	1	no	—
267	C₂×D₄	—	10	4	no	—
268	C₂×D₄	—	10	2	no	—
269	C₂×D₄	—	12	1	no	—
270	C₂×D₄	—	20	2	no	—
271	C₂×D₄	—	20	1	no	—
272	C₂²×C₄	—	12	1	no	—
273	C₂²×C₄	#1201	1	1	yes	characteristic
274	Q₁₆	—	3	2	no	—
275	Q₁₆	—	6	1	no	—
276	SD₁₆	—	6	4	no	—
277	D₈	—	3	2	no	—
278	D₈	—	5	4	no	—
279	D₈	—	6	1	no	—
280	D₈	—	10	2	no	—
281	D₈	—	10	4	no	—
282	C₂×C₈	#1201	1	1	yes	characteristic
283	C₂×C₈	#1201	1	2	yes	—
284	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
285	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
286	C₁₆	—	15	4	no	—
287	C₃×C₅	C₂×#509	1	1	yes	characteristic, complemented, Hall
288	C₂²×C₃	—	2	2	no	—
289	C₂²×C₃	—	20	4	no	—
290	C₂²×C₃	—	40	8	no	—
291	C₂²×C₃	—	40	4	no	—
292	C₂²×C₃	C₂×#686	1	2	yes	—
293	C₂²×C₃	C₂×#686	1	1	yes	characteristic
294	C₂×D₃	—	4	2	no	—
295	C₂×D₃	—	8	2	no	—
296	C₃×C₄	C₂×#686	1	1	yes	characteristic
297	C₃×C₄	C₂×#686	1	2	yes	—
298	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
299	Dic₃	—	4	2	no	—
300	C₂×C₅	—	2	2	no	—
301	C₂×C₅	—	24	2	no	—
302	C₂×C₅	#5147	1	2	yes	complemented
303	C₂×C₅	C₂²×#213	1	1	yes	characteristic
304	D₅	—	8	4	no	—
305	C₂³	—	12	1	no	—
306	C₂³	—	20	4	no	—
307	C₂³	—	20	2	no	—
308	C₂³	—	40	4	no	—
309	C₂³	#6652	1	1	yes	characteristic
310	Q₈	—	6	2	no	—
311	Q₈	—	12	1	no	—
312	D₄	—	6	2	no	—
313	D₄	—	10	4	no	—
314	D₄	—	12	4	no	—
315	D₄	—	12	1	no	—
316	D₄	—	20	4	no	—
317	D₄	—	20	2	no	—
318	C₂×C₄	—	12	1	no	—
319	C₂×C₄	—	12	4	no	—
320	C₂×C₄	#6652	1	2	yes	—
321	C₂×C₄	#6652	1	1	yes	characteristic
322	C₂×C₄	C₂×#1201	1	2	yes	—
323	C₂×C₄	C₂×#1201	1	1	yes	characteristic
324	C₈	C₂×#1201	1	1	yes	characteristic
325	C₈	C₂×#1201	1	2	yes	—
326	C₈	C₂²×D₃×D₅	1	1	yes	characteristic
327	C₂×C₃	—	2	2	no	—
328	C₂×C₃	—	40	4	no	—
329	C₂×C₃	#65482	1	2	yes	complemented
330	C₂×C₃	C₂²×#686	1	1	yes	characteristic
331	D₃	—	8	2	no	—
332	C₅	C₂×#5147	1	1	yes	characteristic, complemented, Sylow
333	C₂²	—	2	2	no	—
334	C₂²	—	12	2	no	—
335	C₂²	—	20	4	no	—
336	C₂²	—	24	2	no	—
337	C₂²	—	40	8	no	—
338	C₂²	—	40	4	no	—
339	C₂²	C₂×#6652	1	2	yes	—
340	C₂²	C₂×#6652	1	1	yes	characteristic
341	C₄	—	12	2	no	—
342	C₄	C₂×#6652	1	2	yes	—
343	C₄	C₂×#6652	1	1	yes	characteristic
344	C₄	C₂²×#1201	1	1	yes	characteristic
345	C₃	C₂×#65482	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	2	2	no	—
347	C₂	—	24	2	no	—
348	C₂	—	40	4	no	—
349	C₂	#11747539	1	2	yes	direct product factor
350	C₂	C₂²×#6652	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6652
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. C₂×C₅×#904	C₂	semidirect
2. C₂²×C₃×D₄₀	C₂	semidirect
3. C₂×#91096	C₂	semidirect
4. C₂×#90940	C₂	semidirect
5. #11747539	C₂	direct (N × H)
6. C₂×#90937	C₂	semidirect
11. C₂×C₅×Dic₁₂	C₂²	semidirect
13. C₅×#904	C₂²	semidirect
14. C₂×C₅×D₂₄	C₂²	semidirect
19. C₂×C₃×D₄₀	C₂²	semidirect
20. #91096	C₂²	semidirect
21. C₂×C₃⋊#672	C₂²	semidirect
22. #90940	C₂²	semidirect
23. #90937	C₂²	semidirect
26. C₂²×D₄₀	D₃	semidirect
48. C₅×Dic₁₂	C₂³	semidirect
50. C₅×D₂₄	C₂³	semidirect
55. C₃⋊#672	C₂³	semidirect
57. C₂×#904	D₅	semidirect
73. C₂×D₄₀	C₂×D₃	semidirect
108. C₂×Dic₁₂	C₂×D₅	semidirect
110. #904	C₂×D₅	semidirect
111. C₂×D₂₄	C₂×D₅	semidirect
180. Dic₁₂	C₂²×D₅	semidirect
182. D₂₄	C₂²×D₅	semidirect
224. C₂×C₃×C₅	#509	semidirect
287. C₃×C₅	C₂×#509	semidirect
302. C₂×C₅	#5147	semidirect
329. C₂×C₃	#65482	semidirect
332. C₅	C₂×#5147	semidirect
345. C₃	C₂×#65482	semidirect
349. C₂	#11747539	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×C₅×#904	2
C₂²×C₃×D₄₀	2
C₂×#91096	2
C₂×#90940	2
#11747539	2
C₂×#90937	2
C₂×#65482	3
C₂×#5147	5

Subgroups — C2×#11747539 #14478359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11747539 #14478359