Subgroups — C₂×#19136487 #25250031

Order 11008.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583021	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583006	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136487	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582722	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582715	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582709	C₂	1	1	yes	maximal, characteristic
7	C₂×#14582676	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#510	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063332	—	2	1	no	—
10	C₂×Q₈×Dic₄₃	—	2	1	no	—
11	C₂×#13063265	C₂²	1	1	yes	characteristic
12	#14583021	—	2	2	no	—
13	#14583021	C₂²	1	4	yes	—
14	#14583006	—	2	2	no	—
15	#14583006	C₂²	1	4	yes	complemented
16	C₂×#13063261	C₂²	1	1	yes	characteristic
17	C₂×#13063257	C₂²	1	1	yes	characteristic
18	C₂×#13063252	—	2	1	no	—
19	C₂×#13063249	—	2	1	no	—
20	C₂×#13063235	C₂²	1	1	yes	characteristic
21	#14582722	C₂²	1	4	yes	complemented
22	#14582715	C₂²	1	4	yes	complemented
23	#14582709	C₂²	1	4	yes	—
24	#14582676	C₂²	1	4	yes	complemented
25	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#510	C₄	1	4	yes	complemented
28	C₄₃×#510	C₂²	1	4	yes	—
29	C₂×Q₃₂×C₄₃	C₄	1	2	yes	complemented
30	C₂×SD₃₂×C₄₃	C₄	1	2	yes	complemented
31	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
32	#13063332	—	2	4	no	—
33	#13063332	—	4	2	no	—
34	C₂×#10621877	—	2	1	no	—
35	C₂×#10621877	—	4	1	no	—
36	C₂²×C₄₃⋊C₈	—	4	1	no	—
37	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
38	Q₈×Dic₄₃	—	2	4	no	—
39	Q₈×Dic₄₃	—	4	2	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄⋊Dic₄₃	—	4	1	no	—
42	C₂×C₄×Dic₄₃	—	4	1	no	—
43	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
44	#13063265	—	2	1	no	—
45	#13063265	C₂³	1	2	yes	—
46	C₂×C₄₃⋊C₁₆	—	2	1	no	—
47	#13063261	—	2	1	no	—
48	#13063261	C₂³	1	2	yes	—
49	#13063257	—	2	1	no	—
50	#13063257	C₂³	1	2	yes	—
51	#13063252	—	2	4	no	—
52	#13063249	—	2	4	no	—
53	#13063235	D₄	1	2	yes	—
54	#13063235	C₂³	1	2	yes	—
55	C₂×#41×C₄₃	—	2	1	no	—
56	C₂²×Q₈×C₄₃	—	2	1	no	—
57	#135×C₄₃	—	2	1	no	—
58	#135×C₄₃	—	2	1	no	—
59	#135×C₄₃	C₂×C₄	1	2	yes	—
60	#135×C₄₃	C₂³	1	2	yes	—
61	C₂×Q₁₆×C₄₃	—	2	1	no	—
62	C₂×Q₁₆×C₄₃	—	2	1	no	—
63	C₂×Q₁₆×C₄₃	—	2	1	no	—
64	C₂×Q₁₆×C₄₃	C₂×C₄	1	2	yes	—
65	C₂×Q₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
66	C₂×Q₁₆×C₄₃	C₂×C₄	1	2	yes	—
67	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
68	C₂×SD₁₆×C₄₃	—	2	1	no	—
69	C₂×D₈×C₄₃	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	Q₃₂×C₄₃	C₂×C₄	1	8	yes	complemented
72	SD₃₂×C₄₃	C₂×C₄	1	8	yes	complemented
73	M₃₂×C₄₃	C₂×C₄	1	2	yes	—
74	M₃₂×C₄₃	C₂³	1	2	yes	—
75	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
76	#41×C₄₃	—	2	2	no	—
77	#41×C₄₃	—	2	2	no	—
78	#41×C₄₃	—	4	1	no	—
79	#41×C₄₃	—	4	1	no	—
80	C₂×Q₈×C₄₃	—	2	1	no	—
81	C₂×Q₈×C₄₃	—	2	2	no	—
82	C₂×Q₈×C₄₃	—	2	2	no	—
83	C₂×Q₈×C₄₃	—	2	2	no	—
84	C₂×Q₈×C₄₃	—	4	1	no	—
85	C₂×Q₈×C₄₃	—	4	1	no	—
86	C₂×Q₈×C₄₃	—	4	1	no	—
87	C₂×D₄×C₄₃	—	2	1	no	—
88	C₂×D₄×C₄₃	—	4	1	no	—
89	C₂²×C₄×C₄₃	—	4	1	no	—
90	C₂²×C₄×C₄₃	—	4	1	no	—
91	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×Dic₄₃	—	2	1	no	—
93	C₂²×Dic₄₃	—	4	1	no	—
94	Q₁₆×C₄₃	—	2	2	no	—
95	Q₁₆×C₄₃	—	2	2	no	—
96	Q₁₆×C₄₃	—	2	1	no	—
97	Q₁₆×C₄₃	—	2	2	no	—
98	Q₁₆×C₄₃	C₂²×C₄	1	4	yes	—
99	Q₁₆×C₄₃	C₂²×C₄	1	2	yes	—
100	SD₁₆×C₄₃	—	2	2	no	—
101	SD₁₆×C₄₃	—	2	2	no	—
102	D₈×C₄₃	—	2	1	no	—
103	D₈×C₄₃	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
108	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
109	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	4	4	no	—
112	C₄⋊Dic₄₃	—	4	1	no	—
113	C₄×Dic₄₃	—	2	1	no	—
114	C₄×Dic₄₃	—	4	4	no	—
115	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
116	#10621877	—	2	2	no	—
117	#10621877	—	4	4	no	—
118	#10621877	—	4	1	no	—
119	C₂×C₄₃⋊C₈	—	2	1	no	—
120	C₂×C₄₃⋊C₈	—	2	1	no	—
121	C₂×C₄₃⋊C₈	—	4	4	no	—
122	C₂×C₄₃⋊C₈	—	4	1	no	—
123	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
124	C₁₆×C₄₃	C₂²×C₄	1	4	yes	—
125	C₄₃⋊C₁₆	—	2	2	no	—
126	C₂³×C₄₃	—	4	1	no	—
127	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	Q₈×C₄₃	—	2	2	no	—
129	Q₈×C₄₃	—	2	4	no	—
130	Q₈×C₄₃	—	4	2	no	—
131	Q₈×C₄₃	—	4	2	no	—
132	Q₈×C₄₃	—	4	2	no	—
133	Q₈×C₄₃	—	4	1	no	—
134	D₄×C₄₃	—	2	2	no	—
135	D₄×C₄₃	—	4	2	no	—
136	D₄×C₄₃	—	4	2	no	—
137	D₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	4	1	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	—	4	2	no	—
144	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₄₃	—	2	2	no	—
151	C₂×Dic₄₃	—	4	2	no	—
152	C₂×Dic₄₃	—	4	2	no	—
153	C₂×Dic₄₃	—	8	2	no	—
154	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
155	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
156	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
157	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
158	C₄₃⋊C₈	—	2	2	no	—
159	C₄₃⋊C₈	—	4	2	no	—
160	C₂×#2180	—	43	1	no	maximal, Sylow
161	C₂²×C₄₃	—	2	2	no	—
162	C₂²×C₄₃	—	4	2	no	—
163	C₂²×C₄₃	—	8	2	no	—
164	C₂²×C₄₃	#442	1	1	yes	characteristic
165	C₂²×C₄₃	#442	1	1	yes	characteristic
166	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₄×C₄₃	—	4	2	no	—
168	C₄×C₄₃	—	4	4	no	—
169	C₄×C₄₃	#442	1	1	yes	characteristic
170	C₄×C₄₃	#442	1	1	yes	characteristic
171	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	Dic₄₃	—	4	2	no	—
174	Dic₄₃	—	8	2	no	—
175	C₂×#510	D₄₃	1	1	yes	characteristic
176	C₂×#444	—	43	1	no	—
177	C₂×#441	—	43	1	no	—
178	#2180	—	43	8	no	—
179	C₂×#365	—	43	1	no	—
180	C₂×#362	—	43	1	no	—
181	C₂×#360	—	43	1	no	—
182	C₂×#347	—	43	1	no	—
183	C₂×C₄₃	—	2	1	no	—
184	C₂×C₄₃	—	2	1	no	—
185	C₂×C₄₃	—	8	2	no	—
186	C₂×C₄₃	#2180	1	2	yes	complemented
187	C₂×C₄₃	C₂×#442	1	1	yes	characteristic
188	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
189	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
190	C₂×#131	—	86	1	no	—
191	C₂×C₄×Q₈	—	86	1	no	—
192	#510	Dic₄₃	1	4	yes	complemented
193	#510	C₂×D₄₃	1	4	yes	—
194	C₂×Q₃₂	Dic₄₃	1	2	yes	complemented
195	C₂×SD₃₂	Dic₄₃	1	2	yes	complemented
196	C₂×M₃₂	—	43	1	no	—
197	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
198	#444	—	43	4	no	—
199	#444	—	86	2	no	—
200	#441	—	43	4	no	—
201	#441	—	86	2	no	—
202	C₂×#108	—	43	1	no	—
203	C₂×#106	—	43	1	no	—
204	C₂×#104	—	86	1	no	—
205	C₂×#103	—	86	1	no	—
206	C₂×#97	—	43	1	no	—
207	#365	—	43	4	no	—
208	#362	—	43	4	no	—
209	#360	—	43	4	no	—
210	#347	—	43	4	no	—
211	C₄₃	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
212	C₂×#41	—	2	1	no	—
213	C₂²×Q₈	—	2	1	no	—
214	#135	—	2	1	no	—
215	#135	—	2	1	no	—
216	#135	C₂×Dic₄₃	1	2	yes	—
217	#135	C₂²×D₄₃	1	2	yes	—
218	C₂×Q₁₆	—	2	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	C₂×Dic₄₃	1	2	yes	—
222	C₂×Q₁₆	C₂×Dic₄₃	1	2	yes	—
223	C₂×Q₁₆	C₂×Dic₄₃	1	1	yes	characteristic
224	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
225	C₂×SD₁₆	—	2	1	no	—
226	C₂×D₈	C₂×Dic₄₃	1	1	yes	characteristic
227	#131	—	86	4	no	—
228	#131	—	172	2	no	—
229	C₂×M₁₆	—	86	1	no	—
230	C₂×M₁₆	—	172	1	no	—
231	C₂²×C₈	—	43	1	no	—
232	C₂²×C₈	—	172	1	no	—
233	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₄×Q₈	—	86	4	no	—
235	C₄×Q₈	—	172	2	no	—
236	C₂×C₄⋊C₄	—	86	1	no	—
237	C₂×C₄⋊C₄	—	172	1	no	—
238	C₂×C₄²	—	43	1	no	—
239	C₂×C₄²	—	172	1	no	—
240	Q₃₂	C₂×Dic₄₃	1	8	yes	complemented
241	SD₃₂	C₂×Dic₄₃	1	8	yes	complemented
242	M₃₂	—	43	2	no	—
243	M₃₂	—	86	1	no	—
244	M₃₂	C₂×Dic₄₃	1	2	yes	—
245	M₃₂	C₂²×D₄₃	1	2	yes	—
246	C₂×C₁₆	—	86	1	no	—
247	C₂×C₁₆	C₂×Dic₄₃	1	2	yes	—
248	#108	—	43	2	no	—
249	#108	—	86	1	no	—
250	#106	—	43	2	no	—
251	#106	—	86	1	no	—
252	#104	—	86	4	no	—
253	#103	—	86	4	no	—
254	#97	—	43	2	no	—
255	#97	—	43	2	no	—
256	#41	—	2	2	no	—
257	#41	—	2	2	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	86	1	no	—
272	C₂²×C₄	—	172	1	no	—
273	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	2	2	no	—
278	Q₁₆	C₂²×Dic₄₃	1	4	yes	—
279	Q₁₆	C₂²×Dic₄₃	1	2	yes	—
280	SD₁₆	—	2	2	no	—
281	SD₁₆	—	2	2	no	—
282	D₈	—	2	1	no	—
283	D₈	C₂²×Dic₄₃	1	2	yes	—
284	M₁₆	—	86	2	no	—
285	M₁₆	—	172	4	no	—
286	M₁₆	—	172	1	no	—
287	C₂×C₈	—	43	2	no	—
288	C₂×C₈	—	86	1	no	—
289	C₂×C₈	—	86	1	no	—
290	C₂×C₈	—	172	4	no	—
291	C₂×C₈	—	172	1	no	—
292	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
293	C₂×C₈	#10621900	1	1	yes	characteristic
294	C₂×C₈	#10621900	1	1	yes	characteristic
295	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
296	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
298	C₄⋊C₄	—	86	2	no	—
299	C₄⋊C₄	—	172	4	no	—
300	C₄⋊C₄	—	172	1	no	—
301	C₄²	—	43	2	no	—
302	C₄²	—	86	1	no	—
303	C₄²	—	172	4	no	—
304	C₁₆	—	86	2	no	—
305	C₁₆	C₂²×Dic₄₃	1	4	yes	—
306	C₂³	—	4	1	no	—
307	C₂³	#13063349	1	1	yes	characteristic
308	Q₈	—	2	2	no	—
309	Q₈	—	2	4	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	86	2	no	—
325	C₂×C₄	—	344	2	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	—	172	2	no	—
328	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
329	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
330	C₂×C₄	#13063349	1	1	yes	characteristic
331	C₂×C₄	#13063349	1	1	yes	characteristic
332	C₂×C₄	#13063349	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
334	C₈	—	86	2	no	—
335	C₈	—	172	2	no	—
336	C₈	D₄×Dic₄₃	1	1	yes	characteristic
337	C₈	D₄×Dic₄₃	1	1	yes	characteristic
338	C₈	C₂³×Dic₄₃	1	1	yes	characteristic
339	C₈	C₂×#10621900	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	#14583011	1	1	yes	characteristic
344	C₂²	#14583011	1	1	yes	characteristic
345	C₂²	C₂×#13063349	1	1	yes	characteristic
346	C₄	—	4	2	no	—
347	C₄	—	4	4	no	—
348	C₄	—	344	2	no	—
349	C₄	—	172	2	no	—
350	C₄	#14583011	1	1	yes	characteristic
351	C₄	#14583011	1	1	yes	characteristic
352	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
353	C₄	C₂×#13063349	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	8	2	no	—
357	C₂	#19136487	1	2	yes	direct product factor
358	C₂	C₂×#14583011	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14583011
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#510	C₂
socle	C₂²×C₄₃	#442
prime radical	C₂×#135×C₄₃	C₂²
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#510	D₄₃
hypocenter	C₄₃	C₂×#2180
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#510

Complements

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

N	H ≅ G/N	type
2. C₂×#14583006	C₂	semidirect
3. #19136487	C₂	direct (N × H)
4. C₂×#14582722	C₂	semidirect
5. C₂×#14582715	C₂	semidirect
7. C₂×#14582676	C₂	semidirect
15. #14583006	C₂²	semidirect
21. #14582722	C₂²	semidirect
22. #14582715	C₂²	semidirect
24. #14582676	C₂²	semidirect
27. C₄₃×#510	C₄	semidirect
29. C₂×Q₃₂×C₄₃	C₄	semidirect
30. C₂×SD₃₂×C₄₃	C₄	semidirect
71. Q₃₂×C₄₃	C₂×C₄	semidirect
72. SD₃₂×C₄₃	C₂×C₄	semidirect
186. C₂×C₄₃	#2180	semidirect
192. #510	Dic₄₃	semidirect
194. C₂×Q₃₂	Dic₄₃	semidirect
195. C₂×SD₃₂	Dic₄₃	semidirect
211. C₄₃	C₂×#2180	semidirect
240. Q₃₂	C₂×Dic₄₃	semidirect
241. SD₃₂	C₂×Dic₄₃	semidirect
357. C₂	#19136487	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×#14583021	2
C₂×#14583006	2
#19136487	2
C₂×#14582722	2
C₂×#14582715	2
C₂×#14582709	2
C₂×#14582676	2
C₂×C₄₃×#510	2
C₂×#2180	43

Subgroups — C2×#19136487 #25250031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136487 #25250031