Subgroups — C₂×#19136808 #25251480

Order 11008.

604 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583149	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582946	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136808	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582723	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582722	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582712	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621872	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×D₄₃	—	2	1	no	—
11	C₂×#13063296	—	2	1	no	—
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	#14583149	—	2	2	no	—
14	#14583149	C₂²	1	4	yes	—
15	#14582946	—	2	2	no	—
16	#14582946	C₂²	1	4	yes	complemented
17	C₂×#13063261	C₂²	1	1	yes	characteristic
18	C₂×#13063260	—	2	1	no	—
19	C₂×#13063257	C₂²	1	1	yes	characteristic
20	C₂×#13063246	—	2	1	no	—
21	#14582723	C₂²	1	4	yes	complemented
22	#14582722	C₂²	1	4	yes	complemented
23	#14582712	C₂²	1	4	yes	complemented
24	#14582680	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#361	C₂²	1	4	yes	complemented
29	C₂²×C₂²⋊D₄₃	—	4	1	no	—
30	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
31	C₂×#10621877	—	2	1	no	—
32	C₂×#10621872	—	2	1	no	—
33	C₂×#10621872	—	2	1	no	—
34	C₂×#10621872	—	2	1	no	—
35	C₂×#10621872	—	2	1	no	—
36	C₂×#10621872	—	4	1	no	—
37	C₂×#10621872	D₄	1	2	yes	—
38	C₂×#10621872	C₂³	1	2	yes	—
39	M₁₆×D₄₃	—	2	4	no	—
40	M₁₆×D₄₃	—	4	2	no	—
41	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
42	C₂×C₈×D₄₃	—	4	1	no	—
43	#13063296	—	2	4	no	—
44	#13063296	—	4	2	no	—
45	C₂×C₄⋊Dic₄₃	—	2	1	no	—
46	C₂×C₂²⋊Dic₄₃	—	4	1	no	—
47	#13063265	—	2	1	no	—
48	#13063265	C₂³	1	2	yes	—
49	C₂×C₄₃⋊C₁₆	—	2	1	no	—
50	#13063261	—	2	1	no	—
51	#13063261	C₂³	1	2	yes	—
52	#13063260	—	2	4	no	—
53	#13063257	—	2	1	no	—
54	#13063257	C₂³	1	2	yes	—
55	#13063246	—	2	4	no	—
56	C₂²×D₄×C₄₃	—	2	1	no	—
57	C₂×D₈×C₄₃	—	2	1	no	—
58	C₂×D₈×C₄₃	—	2	1	no	—
59	C₂×D₈×C₄₃	—	2	1	no	—
60	C₂×D₈×C₄₃	—	2	1	no	—
61	C₂×D₈×C₄₃	—	2	1	no	—
62	C₂×D₈×C₄₃	C₂³	1	2	yes	—
63	C₂×M₁₆×C₄₃	—	2	1	no	—
64	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	M₃₂×C₄₃	—	2	1	no	—
66	M₃₂×C₄₃	C₂³	1	2	yes	—
67	C₂×C₁₆×C₄₃	—	2	1	no	—
68	#108×C₄₃	—	2	1	no	—
69	#108×C₄₃	C₂³	1	2	yes	—
70	C₂⁴×C₄₃	—	4	1	no	—
71	C₂³×D₄₃	SD₁₆	1	1	yes	characteristic
72	C₂×D₄×C₄₃	—	2	2	no	—
73	C₂×D₄×C₄₃	—	4	1	no	—
74	C₂×D₄×C₄₃	—	4	1	no	—
75	C₂×D₄×C₄₃	—	4	1	no	—
76	C₂×D₄×C₄₃	—	4	1	no	—
77	C₂×D₄×C₄₃	—	4	1	no	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₄₃	—	4	4	no	—
80	C₂×C₂²⋊D₄₃	—	8	4	no	—
81	C₂²×Dic₄₃	—	4	1	no	—
82	C₂²×Dic₄₃	SD₁₆	1	1	yes	characteristic
83	C₂×C₄×D₄₃	—	2	1	no	—
84	C₂×C₄×D₄₃	—	2	1	no	—
85	C₂×C₄×D₄₃	—	2	1	no	—
86	C₂×C₄×D₄₃	—	2	1	no	—
87	C₂×C₄×D₄₃	SD₁₆	1	2	yes	—
88	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
89	D₈×C₄₃	—	2	2	no	—
90	D₈×C₄₃	—	2	2	no	—
91	D₈×C₄₃	—	4	1	no	—
92	D₈×C₄₃	—	4	1	no	—
93	M₁₆×C₄₃	—	2	2	no	—
94	M₁₆×C₄₃	—	4	1	no	—
95	C₂×C₈×C₄₃	—	4	1	no	—
96	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
100	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
101	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
102	C₂²⋊Dic₄₃	—	4	4	no	—
103	C₄⋊Dic₄₃	—	2	2	no	—
104	C₄⋊Dic₄₃	—	4	1	no	—
105	#10621877	—	2	2	no	—
106	#10621877	—	4	1	no	—
107	C₂×C₄₃⋊C₈	—	2	1	no	—
108	C₂×C₄₃⋊C₈	—	2	1	no	—
109	C₂×C₄₃⋊C₈	—	4	1	no	—
110	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
111	#10621872	—	2	2	no	—
112	#10621872	—	2	2	no	—
113	#10621872	—	2	2	no	—
114	#10621872	—	2	2	no	—
115	#10621872	—	4	4	no	—
116	C₈×D₄₃	—	4	4	no	—
117	C₁₆×C₄₃	—	2	2	no	—
118	C₄₃⋊C₁₆	—	2	2	no	—
119	C₂³×C₄₃	—	4	2	no	—
120	C₂³×C₄₃	—	8	2	no	—
121	C₂³×C₄₃	—	8	2	no	—
122	C₂³×C₄₃	—	8	1	no	—
123	C₂³×C₄₃	—	8	1	no	—
124	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂²×D₄₃	—	2	1	no	—
126	C₂²×D₄₃	—	2	1	no	—
127	C₂²×D₄₃	—	4	2	no	—
128	C₂²×D₄₃	C₂×SD₁₆	1	2	yes	—
129	D₄×C₄₃	—	4	2	no	—
130	D₄×C₄₃	—	4	2	no	—
131	D₄×C₄₃	—	8	1	no	—
132	D₄×C₄₃	—	8	1	no	—
133	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₄₃	—	8	8	no	—
140	C₂×Dic₄₃	—	2	1	no	—
141	C₂×Dic₄₃	—	2	1	no	—
142	C₂×Dic₄₃	—	4	2	no	—
143	C₂×Dic₄₃	—	8	2	no	—
144	C₂×Dic₄₃	C₂×SD₁₆	1	2	yes	—
145	C₄×D₄₃	—	2	2	no	—
146	C₄×D₄₃	—	2	2	no	—
147	C₄×D₄₃	—	2	2	no	—
148	C₄×D₄₃	—	2	2	no	—
149	C₈×C₄₃	—	4	2	no	—
150	C₈×C₄₃	#123	1	1	yes	characteristic
151	C₈×C₄₃	#123	1	1	yes	characteristic
152	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
153	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
154	C₄₃⋊C₈	—	2	2	no	—
155	C₄₃⋊C₈	—	4	2	no	—
156	C₂×#2238	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	—	2	2	no	—
158	C₂²×C₄₃	—	8	2	no	—
159	C₂²×C₄₃	—	8	2	no	—
160	C₂²×C₄₃	—	8	2	no	—
161	C₂²×C₄₃	—	16	2	no	—
162	C₂²×C₄₃	—	16	2	no	—
163	C₂²×C₄₃	#481	1	1	yes	characteristic
164	C₂²×C₄₃	#481	1	1	yes	characteristic
165	C₂²×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
166	C₂×D₄₃	—	2	2	no	—
167	C₂×D₄₃	—	4	2	no	—
168	C₂×D₄₃	—	4	2	no	—
169	C₂×D₄₃	—	8	1	no	—
170	C₄×C₄₃	#481	1	1	yes	characteristic
171	C₄×C₄₃	#481	1	1	yes	characteristic
172	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
173	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	Dic₄₃	—	2	2	no	—
175	Dic₄₃	—	8	2	no	—
176	C₂×#469	—	43	1	no	—
177	C₂×#430	—	43	1	no	—
178	#2238	—	43	8	no	—
179	C₂×#365	—	43	1	no	—
180	C₂×#365	—	43	1	no	—
181	C₂×#361	—	43	1	no	—
182	C₂×#361	D₄₃	1	1	yes	characteristic, complemented
183	C₂×#349	—	43	1	no	—
184	C₂×C₄₃	—	2	1	no	—
185	C₂×C₄₃	—	2	1	no	—
186	C₂×C₄₃	—	16	2	no	—
187	C₂×C₄₃	#2238	1	2	yes	complemented
188	C₂×C₄₃	C₂×#481	1	1	yes	characteristic
189	D₄₃	—	4	2	no	—
190	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
191	C₂²×M₁₆	—	43	1	no	—
192	C₂²×M₁₆	—	86	1	no	—
193	C₂×#121	—	86	1	no	—
194	C₂×M₃₂	—	43	1	no	—
195	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
196	#469	—	43	4	no	—
197	#469	—	86	2	no	—
198	#430	—	43	4	no	—
199	#430	—	86	2	no	—
200	C₂×#108	—	43	1	no	—
201	C₂×#108	—	86	1	no	—
202	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
203	C₂×#106	—	43	1	no	—
204	C₂×#102	—	86	1	no	—
205	#365	—	43	4	no	—
206	#365	—	43	4	no	—
207	#361	—	43	4	no	—
208	#361	C₂×D₄₃	1	4	yes	complemented
209	#349	—	43	4	no	—
210	C₄₃	C₂×#2238	1	1	yes	characteristic, complemented, Sylow
211	C₂²×D₄	—	2	1	no	—
212	C₂²×D₄	—	172	1	no	—
213	C₂³×C₄	—	43	1	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	—	2	1	no	—
219	C₂×D₈	C₂²×D₄₃	1	2	yes	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	43	2	no	—
222	C₂×M₁₆	—	43	2	no	—
223	C₂×M₁₆	—	86	1	no	—
224	C₂×M₁₆	—	86	1	no	—
225	C₂×M₁₆	—	86	1	no	—
226	C₂×M₁₆	—	86	1	no	—
227	C₂×M₁₆	—	86	4	no	—
228	C₂×M₁₆	—	86	1	no	—
229	C₂×M₁₆	—	172	1	no	—
230	C₂×M₁₆	—	172	2	no	—
231	C₂²×C₈	—	43	1	no	—
232	C₂²×C₈	—	172	1	no	—
233	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
234	#121	—	86	4	no	—
235	#121	—	172	2	no	—
236	C₂×C₄⋊C₄	—	86	1	no	—
237	C₂×C₂²⋊C₄	—	172	1	no	—
238	M₃₂	—	2	1	no	—
239	M₃₂	—	43	2	no	—
240	M₃₂	—	86	1	no	—
241	M₃₂	C₂²×D₄₃	1	2	yes	—
242	C₂×C₁₆	—	2	1	no	—
243	C₂×C₁₆	—	86	1	no	—
244	#108	—	2	1	no	—
245	#108	—	43	2	no	—
246	#108	—	86	4	no	—
247	#108	—	86	1	no	—
248	#108	C₂²×D₄₃	1	2	yes	—
249	#106	—	43	2	no	—
250	#106	—	86	1	no	—
251	#102	—	86	4	no	—
252	C₂⁴	—	4	1	no	—
253	C₂⁴	—	43	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	344	4	no	—
261	C₂×D₄	—	172	4	no	—
262	C₂²×C₄	—	43	1	no	—
263	C₂²×C₄	—	43	2	no	—
264	C₂²×C₄	—	43	2	no	—
265	C₂²×C₄	—	86	1	no	—
266	C₂²×C₄	—	86	1	no	—
267	C₂²×C₄	—	86	1	no	—
268	C₂²×C₄	—	86	1	no	—
269	C₂²×C₄	—	172	1	no	—
270	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
271	D₈	—	2	2	no	—
272	D₈	—	2	2	no	—
273	D₈	—	4	1	no	—
274	D₈	—	4	1	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	86	2	no	—
278	M₁₆	—	86	2	no	—
279	M₁₆	—	86	2	no	—
280	M₁₆	—	86	2	no	—
281	M₁₆	—	86	2	no	—
282	M₁₆	—	172	4	no	—
283	M₁₆	—	172	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	43	2	no	—
286	C₂×C₈	—	86	1	no	—
287	C₂×C₈	—	86	1	no	—
288	C₂×C₈	—	172	4	no	—
289	C₂×C₈	—	172	1	no	—
290	C₂×C₈	#10621898	1	1	yes	characteristic
291	C₂×C₈	#10621898	1	1	yes	characteristic
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	C₂³×D₄₃	1	1	yes	characteristic
296	C₄⋊C₄	—	86	2	no	—
297	C₄⋊C₄	—	172	1	no	—
298	C₂²⋊C₄	—	172	4	no	—
299	C₁₆	—	2	2	no	—
300	C₁₆	—	86	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	8	2	no	—
303	C₂³	—	8	2	no	—
304	C₂³	—	8	1	no	—
305	C₂³	—	8	1	no	—
306	C₂³	—	43	2	no	—
307	C₂³	—	86	1	no	—
308	C₂³	—	86	1	no	—
309	C₂³	—	172	2	no	—
310	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	8	1	no	—
314	D₄	—	8	1	no	—
315	D₄	—	344	8	no	—
316	C₂×C₄	—	43	2	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	86	1	no	—
319	C₂×C₄	—	86	2	no	—
320	C₂×C₄	—	86	2	no	—
321	C₂×C₄	—	86	2	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	344	2	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	#13063307	1	1	yes	characteristic
326	C₂×C₄	#13063307	1	1	yes	characteristic
327	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
328	C₂×C₄	#13063349	1	1	yes	characteristic
329	C₂×C₄	#13063349	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	86	2	no	—
333	C₈	—	172	2	no	—
334	C₈	#13063307	1	1	yes	characteristic
335	C₈	#13063307	1	1	yes	characteristic
336	C₈	C₂×#10621900	1	1	yes	characteristic
337	C₈	C₂×#10621898	1	1	yes	characteristic
338	C₂²	—	2	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	16	2	no	—
343	C₂²	—	16	2	no	—
344	C₂²	—	86	2	no	—
345	C₂²	—	344	1	no	—
346	C₂²	—	172	2	no	—
347	C₂²	—	172	2	no	—
348	C₂²	#14583216	1	1	yes	characteristic
349	C₂²	#14583216	1	1	yes	characteristic
350	C₂²	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
351	C₄	—	86	2	no	—
352	C₄	—	344	2	no	—
353	C₄	#14583216	1	1	yes	characteristic
354	C₄	#14583216	1	1	yes	characteristic
355	C₄	C₂×#13063307	1	1	yes	characteristic
356	C₄	C₂×#13063349	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	16	2	no	—
360	C₂	—	172	2	no	—
361	C₂	#19136808	1	2	yes	direct product factor
362	C₂	C₂×#14583216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14583216
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#361	C₂
socle	C₂²×C₄₃	#481
prime radical	C₂×#14583149	C₂
hypercenter	C₂×#361	D₄₃
hypocenter	C₄₃	C₂×#2238
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
2. C₂×#14582946	C₂	semidirect
3. #19136808	C₂	direct (N × H)
4. C₂×#14582723	C₂	semidirect
5. C₂×#14582722	C₂	semidirect
6. C₂×#14582712	C₂	semidirect
7. C₂×#14582680	C₂	semidirect
8. C₂×C₄₃×#361	C₂	semidirect
16. #14582946	C₂²	semidirect
21. #14582723	C₂²	semidirect
22. #14582722	C₂²	semidirect
23. #14582712	C₂²	semidirect
24. #14582680	C₂²	semidirect
28. C₄₃×#361	C₂²	semidirect
182. C₂×#361	D₄₃	semidirect
187. C₂×C₄₃	#2238	semidirect
208. #361	C₂×D₄₃	semidirect
210. C₄₃	C₂×#2238	semidirect
361. C₂	#19136808	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×#14583149	2
C₂×#14582946	2
#19136808	2
C₂×#14582723	2
C₂×#14582722	2
C₂×#14582712	2
C₂×#14582680	2
C₂×C₄₃×#361	2
C₂×#2238	43

Subgroups — C2×#19136808 #25251480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136808 #25251480