Subgroups — C₂×#19134198 #25084253

Order 11008.

866 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583295	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583273	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583251	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14583247	C₂	1	1	yes	maximal, characteristic
5	C₂×#14583246	C₂	1	1	yes	maximal, characteristic
6	#19134198	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621884	—	2	2	no	—
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₂×#13063322	C₂²	1	1	yes	characteristic
12	#14583295	—	2	2	no	—
13	#14583295	C₂²	1	4	yes	complemented
14	#14583273	—	2	2	no	—
15	#14583273	C₂²	1	4	yes	—
16	#14583251	—	2	2	no	—
17	#14583251	C₂²	1	4	yes	complemented
18	#14583247	—	2	2	no	—
19	#14583247	C₂²	1	4	yes	—
20	#14583246	—	2	2	no	—
21	#14583246	C₂²	1	4	yes	—
22	C₂×#13063256	—	2	2	no	—
23	C₂×#13063248	—	2	2	no	—
24	C₂×#13063247	—	2	2	no	—
25	C₂×#13063245	—	2	2	no	—
26	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₈×C₄₃⋊C₈	C₂²	1	4	yes	complemented
29	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#487	—	2	2	no	—
32	C₄₃×#487	C₂²	1	4	yes	complemented
33	C₂×#103×C₄₃	—	2	2	no	—
34	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
35	C₂²×Dic₈₆	—	2	1	no	—
36	C₂²×D₁₇₂	—	2	1	no	—
37	C₂²×D₁₇₂	—	2	1	no	—
38	C₂²×D₁₇₂	—	2	1	no	—
39	C₂×#10621884	—	2	8	no	—
40	C₂×#10621884	—	4	8	no	—
41	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
42	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
43	#13063325	—	2	1	no	—
44	#13063325	—	2	1	no	—
45	#13063325	—	2	1	no	—
46	#13063325	C₂³	1	2	yes	—
47	#13063322	—	2	1	no	—
48	#13063322	—	2	1	no	—
49	#13063322	—	2	1	no	—
50	#13063322	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₄₃	—	2	1	no	—
52	C₂×C₄⋊Dic₄₃	—	2	1	no	—
53	#13063256	—	2	8	no	—
54	#13063248	—	2	8	no	—
55	#13063247	—	2	8	no	—
56	#13063245	—	2	8	no	—
57	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
58	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
59	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
60	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
61	C₂²×Q₈×C₄₃	—	2	1	no	—
62	C₂²×Q₈×C₄₃	—	2	1	no	—
63	C₂²×C₈×C₄₃	D₄	1	2	yes	—
64	C₄⋊Q₈×C₄₃	—	2	1	no	—
65	C₄⋊Q₈×C₄₃	—	2	1	no	—
66	C₄⋊Q₈×C₄₃	—	2	1	no	—
67	C₄⋊Q₈×C₄₃	—	2	1	no	—
68	C₄⋊Q₈×C₄₃	—	2	1	no	—
69	C₄⋊Q₈×C₄₃	—	2	1	no	—
70	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
71	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
76	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
77	#103×C₄₃	—	2	8	no	—
78	C₄×C₈×C₄₃	D₄	1	2	yes	—
79	C₄×C₈×C₄₃	C₂³	1	2	yes	—
80	C₂³×D₄₃	—	4	1	no	—
81	C₂×Q₈×C₄₃	—	2	2	no	—
82	C₂×Q₈×C₄₃	—	2	2	no	—
83	C₂×Q₈×C₄₃	—	4	1	no	—
84	C₂×Q₈×C₄₃	—	4	1	no	—
85	C₂×Q₈×C₄₃	—	4	4	no	—
86	C₂×Q₈×C₄₃	—	4	4	no	—
87	C₂²×C₄×C₄₃	—	4	1	no	—
88	C₂²×C₄×C₄₃	—	4	1	no	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×Dic₄₃	—	4	1	no	—
93	C₂×D₁₇₂	—	2	2	no	—
94	C₂×D₁₇₂	—	2	2	no	—
95	C₂×D₁₇₂	—	2	2	no	—
96	C₂×D₁₇₂	—	4	1	no	—
97	C₂×D₁₇₂	—	4	1	no	—
98	C₂×D₁₇₂	—	4	1	no	—
99	C₂×D₁₇₂	—	4	4	no	—
100	C₂×D₁₇₂	—	4	4	no	—
101	C₂×D₁₇₂	—	4	4	no	—
102	C₂×Dic₈₆	—	2	2	no	—
103	C₂×Dic₈₆	—	4	4	no	—
104	C₂×Dic₈₆	—	4	1	no	—
105	C₂×C₈×C₄₃	—	2	4	no	—
106	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	—	2	2	no	—
109	C₄⋊C₄×C₄₃	—	2	2	no	—
110	C₄⋊C₄×C₄₃	—	2	2	no	—
111	C₄⋊C₄×C₄₃	—	4	1	no	—
112	C₄⋊C₄×C₄₃	—	4	1	no	—
113	C₄⋊C₄×C₄₃	—	4	1	no	—
114	C₄⋊C₄×C₄₃	—	4	1	no	—
115	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
119	#10621884	—	4	16	no	—
120	C₄⋊Dic₄₃	—	2	2	no	—
121	C₄⋊Dic₄₃	—	2	2	no	—
122	C₄⋊Dic₄₃	—	4	1	no	—
123	C₄⋊Dic₄₃	—	4	1	no	—
124	C₂×C₄₃⋊C₈	—	2	4	no	—
125	C₂×C₄₃⋊C₈	SD₁₆	1	8	yes	—
126	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
127	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
128	C₂³×C₄₃	#127	1	1	yes	characteristic
129	C₂²×D₄₃	—	4	2	no	—
130	C₂²×D₄₃	—	8	2	no	—
131	C₂²×D₄₃	—	8	2	no	—
132	C₂²×D₄₃	—	8	2	no	—
133	Q₈×C₄₃	—	4	4	no	—
134	Q₈×C₄₃	—	4	4	no	—
135	Q₈×C₄₃	—	8	2	no	—
136	Q₈×C₄₃	—	8	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	8	2	no	—
140	C₂×C₄×C₄₃	—	8	2	no	—
141	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
145	C₂×C₄×C₄₃	#135	1	4	yes	—
146	C₂×C₄×C₄₃	#135	1	4	yes	—
147	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₄₃	—	4	2	no	—
151	C₂×Dic₄₃	—	8	2	no	—
152	D₁₇₂	—	4	4	no	—
153	D₁₇₂	—	4	4	no	—
154	D₁₇₂	—	4	4	no	—
155	D₁₇₂	—	8	2	no	—
156	D₁₇₂	—	8	2	no	—
157	D₁₇₂	—	8	2	no	—
158	Dic₈₆	—	4	4	no	—
159	Dic₈₆	—	8	2	no	—
160	C₈×C₄₃	—	2	4	no	—
161	C₄₃⋊C₈	—	2	4	no	—
162	C₄₃⋊C₈	C₂×SD₁₆	1	8	yes	—
163	C₂×#1708	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	#492	1	2	yes	—
165	C₂²×C₄₃	#495	1	2	yes	—
166	C₂²×C₄₃	#495	1	2	yes	—
167	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
168	C₂×D₄₃	—	8	2	no	—
169	C₂×D₄₃	—	8	2	no	—
170	C₂×D₄₃	—	8	2	no	—
171	C₂×D₄₃	—	16	4	no	—
172	C₄×C₄₃	—	8	2	no	—
173	C₄×C₄₃	—	8	2	no	—
174	C₄×C₄₃	#492	1	2	yes	—
175	C₄×C₄₃	#495	1	2	yes	—
176	C₄×C₄₃	#495	1	2	yes	—
177	C₄×C₄₃	C₂²×SD₁₆	1	2	yes	—
178	C₄×C₄₃	C₂×#135	1	2	yes	—
179	C₄×C₄₃	C₂×#135	1	2	yes	—
180	Dic₄₃	—	8	2	no	—
181	C₂×#498	—	43	1	no	—
182	C₂×#492	—	43	1	no	—
183	C₂×#487	—	43	1	no	—
184	C₂×#487	D₄₃	1	1	yes	characteristic, complemented
185	C₂×#486	—	43	1	no	—
186	C₂×#486	—	43	1	no	—
187	#1708	—	43	8	no	—
188	C₂×C₈²	—	43	1	no	—
189	C₂×C₄₃	#1708	1	4	yes	complemented
190	C₂×C₄₃	C₂×#492	1	1	yes	characteristic
191	C₂×C₄₃	C₂×#495	1	1	yes	characteristic
192	C₂×C₄₃	C₂×#495	1	1	yes	characteristic
193	D₄₃	—	16	2	no	—
194	C₂²×SD₁₆	—	86	2	no	—
195	C₂×C₄⋊Q₈	—	43	1	no	—
196	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
197	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
198	C₂×#127	—	43	1	no	—
199	#498	—	43	4	no	—
200	#498	—	86	2	no	—
201	#492	—	43	4	no	—
202	#492	—	86	2	no	—
203	#487	—	2	2	no	—
204	#487	—	43	4	no	—
205	#487	—	86	2	no	—
206	#487	C₂×D₄₃	1	4	yes	complemented
207	#486	—	43	4	no	—
208	#486	—	43	4	no	—
209	#486	—	86	2	no	—
210	#486	—	86	2	no	—
211	C₂×#106	—	86	2	no	—
212	C₂×#103	—	2	2	no	—
213	C₂×#103	—	86	2	no	—
214	C₂×#102	—	86	2	no	—
215	C₂×#102	—	86	2	no	—
216	C₂×C₄×C₈	—	43	1	no	—
217	C₂×C₄×C₈	—	43	1	no	—
218	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
219	C₈²	—	43	4	no	—
220	C₄₃	C₂×#1708	1	1	yes	characteristic, complemented, Sylow
221	C₂²×Q₈	—	2	1	no	—
222	C₂²×Q₈	—	2	1	no	—
223	C₂²×Q₈	—	86	1	no	—
224	C₂²×D₄	—	86	1	no	—
225	C₂²×D₄	—	86	1	no	—
226	C₂²×D₄	—	86	1	no	—
227	C₂×SD₁₆	—	86	8	no	—
228	C₂×SD₁₆	—	172	8	no	—
229	C₂²×C₈	—	43	2	no	—
230	C₂²×C₈	—	43	2	no	—
231	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	43	2	no	—
239	C₄⋊Q₈	—	86	1	no	—
240	C₄⋊Q₈	—	86	1	no	—
241	C₄⋊Q₈	—	86	1	no	—
242	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
243	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
244	#127	—	43	2	no	—
245	#127	—	86	1	no	—
246	#127	—	86	1	no	—
247	#127	—	86	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	86	1	no	—
253	C₂×C₄⋊C₄	—	86	1	no	—
254	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
255	#106	—	86	8	no	—
256	#103	—	2	8	no	—
257	#103	—	86	8	no	—
258	#102	—	86	8	no	—
259	#102	—	86	8	no	—
260	C₄×C₈	—	43	2	no	—
261	C₄×C₈	—	43	2	no	—
262	C₄×C₈	—	43	2	no	—
263	C₄×C₈	—	43	2	no	—
264	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
265	C₄×C₈	C₂²×D₄₃	1	2	yes	—
266	C₂⁴	—	172	1	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂×Q₈	—	4	1	no	—
271	C₂×Q₈	—	4	4	no	—
272	C₂×Q₈	—	4	4	no	—
273	C₂×Q₈	—	86	2	no	—
274	C₂×Q₈	—	172	4	no	—
275	C₂×Q₈	—	172	1	no	—
276	C₂×D₄	—	86	2	no	—
277	C₂×D₄	—	86	2	no	—
278	C₂×D₄	—	86	2	no	—
279	C₂×D₄	—	172	1	no	—
280	C₂×D₄	—	172	1	no	—
281	C₂×D₄	—	172	1	no	—
282	C₂×D₄	—	172	4	no	—
283	C₂×D₄	—	172	4	no	—
284	C₂×D₄	—	172	4	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	172	1	no	—
288	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
291	SD₁₆	—	172	16	no	—
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	43	8	no	—
294	C₂×C₈	—	43	4	no	—
295	C₂×C₈	—	43	4	no	—
296	C₂×C₈	—	86	4	no	—
297	C₂×C₈	C₂×C₂²⋊D₄₃	1	4	yes	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	86	2	no	—
307	C₄⋊C₄	—	86	2	no	—
308	C₄⋊C₄	—	172	1	no	—
309	C₄⋊C₄	—	172	1	no	—
310	C₄²	D₄×D₄₃	1	1	yes	characteristic
311	C₄²	D₄×D₄₃	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
313	C₄²	C₂³×D₄₃	1	1	yes	characteristic
314	C₂³	—	344	2	no	—
315	C₂³	—	344	2	no	—
316	C₂³	—	344	2	no	—
317	C₂³	—	172	2	no	—
318	C₂³	#13063321	1	1	yes	characteristic
319	Q₈	—	4	4	no	—
320	Q₈	—	4	4	no	—
321	Q₈	—	8	2	no	—
322	Q₈	—	8	2	no	—
323	Q₈	—	344	2	no	—
324	Q₈	—	172	4	no	—
325	D₄	—	344	2	no	—
326	D₄	—	344	2	no	—
327	D₄	—	344	2	no	—
328	D₄	—	172	4	no	—
329	D₄	—	172	4	no	—
330	D₄	—	172	4	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	344	2	no	—
336	C₂×C₄	—	172	2	no	—
337	C₂×C₄	#13063321	1	1	yes	characteristic
338	C₂×C₄	#13063321	1	1	yes	characteristic
339	C₂×C₄	#13063321	1	1	yes	characteristic
340	C₂×C₄	SD₁₆×D₄₃	1	4	yes	—
341	C₂×C₄	#13063345	1	4	yes	—
342	C₂×C₄	#13063348	1	4	yes	—
343	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
344	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
346	C₈	—	2	4	no	—
347	C₈	—	43	8	no	—
348	C₈	—	86	4	no	—
349	C₂²	—	344	2	no	—
350	C₂²	—	344	2	no	—
351	C₂²	—	344	2	no	—
352	C₂²	—	688	4	no	—
353	C₂²	#14583270	1	2	yes	—
354	C₂²	#14583281	1	2	yes	—
355	C₂²	#14583284	1	2	yes	—
356	C₂²	C₂×#13063321	1	1	yes	characteristic
357	C₄	—	8	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	344	2	no	—
360	C₄	#14583270	1	2	yes	—
361	C₄	#14583281	1	2	yes	—
362	C₄	#14583284	1	2	yes	—
363	C₄	C₂×SD₁₆×D₄₃	1	2	yes	—
364	C₄	C₂×#13063345	1	2	yes	—
365	C₄	C₂×#13063348	1	2	yes	—
366	C₂	—	688	2	no	—
367	C₂	#19134198	1	4	yes	direct product factor
368	C₂	C₂×#14583270	1	1	yes	characteristic
369	C₂	C₂×#14583281	1	1	yes	characteristic
370	C₂	C₂×#14583284	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063321
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#487	C₂
socle	C₂³×C₄₃	#127
prime radical	C₂×#13063322	C₂²
hypercenter	C₂×#487	D₄₃
hypocenter	C₄₃	C₂×#1708
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
1. C₂×#14583295	C₂	semidirect
3. C₂×#14583251	C₂	semidirect
6. #19134198	C₂	direct (N × H)
7. C₂×C₈×C₄₃⋊C₈	C₂	semidirect
8. C₂×C₄₃×#487	C₂	semidirect
13. #14583295	C₂²	semidirect
17. #14583251	C₂²	semidirect
28. C₈×C₄₃⋊C₈	C₂²	semidirect
32. C₄₃×#487	C₂²	semidirect
184. C₂×#487	D₄₃	semidirect
189. C₂×C₄₃	#1708	semidirect
206. #487	C₂×D₄₃	semidirect
220. C₄₃	C₂×#1708	semidirect
367. C₂	#19134198	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×#14583295	2
C₂×#14583273	2
C₂×#14583251	2
C₂×#14583247	2
C₂×#14583246	2
#19134198	2
C₂×C₈×C₄₃⋊C₈	2
C₂×C₄₃×#487	2
C₂×#1708	43

Subgroups — C2×#19134198 #25084253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19134198 #25084253