Subgroups — C₂×#19136813 #25251486

Order 11008.

552 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583154	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582946	C₂	1	1	yes	maximal, characteristic
3	#19136813	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×#14582715	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582680	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621872	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×D₄₃	—	2	1	no	—
11	C₂×#13063301	—	2	1	no	—
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	#14583154	—	2	2	no	—
14	#14583154	C₂²	1	4	yes	—
15	#14582946	—	2	2	no	—
16	#14582946	C₂²	1	4	yes	—
17	C₂×#13063261	C₂²	1	1	yes	characteristic
18	C₂×#13063260	—	2	1	no	—
19	C₂×#13063257	C₂²	1	1	yes	characteristic
20	C₂×#13063249	—	2	1	no	—
21	#14582723	C₂²	1	4	yes	complemented
22	#14582722	C₂²	1	4	yes	complemented
23	#14582715	C₂²	1	4	yes	complemented
24	#14582680	C₂²	1	4	yes	—
25	C₂²×Q₁₆×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×#362	C₂²	1	4	yes	complemented
29	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
30	C₂×#10621877	—	2	1	no	—
31	C₂×#10621872	—	2	1	no	—
32	C₂×#10621872	—	2	1	no	—
33	C₂×#10621872	—	2	1	no	—
34	C₂×#10621872	—	2	1	no	—
35	C₂×#10621872	—	4	1	no	—
36	C₂×#10621872	D₄	1	2	yes	—
37	C₂×#10621872	C₂³	1	2	yes	—
38	M₁₆×D₄₃	—	2	4	no	—
39	M₁₆×D₄₃	—	4	2	no	—
40	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₈×D₄₃	—	4	1	no	—
42	#13063301	—	2	4	no	—
43	#13063301	—	4	2	no	—
44	C₂×C₄⋊Dic₄₃	—	2	1	no	—
45	C₂×Dic₄₃⋊C₄	—	4	1	no	—
46	C₂×(C₂×D₄₃)⋊C₄	—	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	#13063249	—	2	4	no	—
56	C₂²×Q₈×C₄₃	—	2	1	no	—
57	C₂×Q₁₆×C₄₃	—	2	1	no	—
58	C₂×Q₁₆×C₄₃	—	2	1	no	—
59	C₂×Q₁₆×C₄₃	—	2	1	no	—
60	C₂×Q₁₆×C₄₃	—	2	1	no	—
61	C₂×Q₁₆×C₄₃	—	2	1	no	—
62	C₂×Q₁₆×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	C₂³×D₄₃	SD₁₆	1	1	yes	characteristic
71	C₂×Q₈×C₄₃	—	2	2	no	—
72	C₂×Q₈×C₄₃	—	4	1	no	—
73	C₂×Q₈×C₄₃	—	4	1	no	—
74	C₂×Q₈×C₄₃	—	4	1	no	—
75	C₂×Q₈×C₄₃	—	4	1	no	—
76	C₂×Q₈×C₄₃	—	4	1	no	—
77	C₂²×C₄×C₄₃	—	4	1	no	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₄₃	—	4	1	no	—
80	C₂²×Dic₄₃	SD₁₆	1	1	yes	characteristic
81	C₂×C₄×D₄₃	—	2	1	no	—
82	C₂×C₄×D₄₃	—	2	1	no	—
83	C₂×C₄×D₄₃	—	2	1	no	—
84	C₂×C₄×D₄₃	—	2	1	no	—
85	C₂×C₄×D₄₃	SD₁₆	1	2	yes	—
86	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
87	Q₁₆×C₄₃	—	2	2	no	—
88	Q₁₆×C₄₃	—	2	2	no	—
89	Q₁₆×C₄₃	—	4	1	no	—
90	Q₁₆×C₄₃	—	4	1	no	—
91	M₁₆×C₄₃	—	2	2	no	—
92	M₁₆×C₄₃	—	4	1	no	—
93	C₂×C₈×C₄₃	—	4	1	no	—
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₄₃	#41	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	C₂⁴	1	1	yes	characteristic
100	(C₂×D₄₃)⋊C₄	—	4	4	no	—
101	C₄⋊Dic₄₃	—	2	2	no	—
102	C₄⋊Dic₄₃	—	4	1	no	—
103	Dic₄₃⋊C₄	—	4	4	no	—
104	#10621877	—	2	2	no	—
105	#10621877	—	4	1	no	—
106	C₂×C₄₃⋊C₈	—	2	1	no	—
107	C₂×C₄₃⋊C₈	—	2	1	no	—
108	C₂×C₄₃⋊C₈	—	4	1	no	—
109	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
110	#10621872	—	2	2	no	—
111	#10621872	—	2	2	no	—
112	#10621872	—	2	2	no	—
113	#10621872	—	2	2	no	—
114	#10621872	—	4	4	no	—
115	C₈×D₄₃	—	4	4	no	—
116	C₁₆×C₄₃	—	2	2	no	—
117	C₄₃⋊C₁₆	—	2	2	no	—
118	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂²×D₄₃	—	2	1	no	—
120	C₂²×D₄₃	—	2	1	no	—
121	C₂²×D₄₃	—	4	2	no	—
122	C₂²×D₄₃	C₂×SD₁₆	1	2	yes	—
123	Q₈×C₄₃	—	4	2	no	—
124	Q₈×C₄₃	—	4	2	no	—
125	Q₈×C₄₃	—	8	1	no	—
126	Q₈×C₄₃	—	8	1	no	—
127	C₂×C₄×C₄₃	—	4	2	no	—
128	C₂×C₄×C₄₃	—	8	1	no	—
129	C₂×C₄×C₄₃	—	8	1	no	—
130	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	2	1	no	—
138	C₂×Dic₄₃	—	4	2	no	—
139	C₂×Dic₄₃	—	8	2	no	—
140	C₂×Dic₄₃	C₂×SD₁₆	1	2	yes	—
141	C₄×D₄₃	—	2	2	no	—
142	C₄×D₄₃	—	2	2	no	—
143	C₄×D₄₃	—	2	2	no	—
144	C₄×D₄₃	—	2	2	no	—
145	C₈×C₄₃	—	4	2	no	—
146	C₈×C₄₃	#123	1	1	yes	characteristic
147	C₈×C₄₃	#123	1	1	yes	characteristic
148	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
149	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
150	C₄₃⋊C₈	—	2	2	no	—
151	C₄₃⋊C₈	—	4	2	no	—
152	C₂×#2239	—	43	1	no	maximal, Sylow
153	C₂²×C₄₃	—	2	2	no	—
154	C₂²×C₄₃	#481	1	1	yes	characteristic
155	C₂²×C₄₃	#481	1	1	yes	characteristic
156	C₂²×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
157	C₂×D₄₃	—	2	2	no	—
158	C₂×D₄₃	—	4	2	no	—
159	C₂×D₄₃	—	4	2	no	—
160	C₂×D₄₃	—	8	1	no	—
161	C₄×C₄₃	—	8	2	no	—
162	C₄×C₄₃	#481	1	1	yes	characteristic
163	C₄×C₄₃	#481	1	1	yes	characteristic
164	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
165	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	Dic₄₃	—	2	2	no	—
167	Dic₄₃	—	8	2	no	—
168	C₂×#470	—	43	1	no	—
169	C₂×#430	—	43	1	no	—
170	#2239	—	43	8	no	—
171	C₂×#365	—	43	1	no	—
172	C₂×#365	—	43	1	no	—
173	C₂×#362	—	43	1	no	—
174	C₂×#362	D₄₃	1	1	yes	characteristic, complemented
175	C₂×#349	—	43	1	no	—
176	C₂×C₄₃	—	2	1	no	—
177	C₂×C₄₃	—	2	1	no	—
178	C₂×C₄₃	#2239	1	2	yes	complemented
179	C₂×C₄₃	C₂×#481	1	1	yes	characteristic
180	D₄₃	—	4	2	no	—
181	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
182	C₂²×M₁₆	—	43	1	no	—
183	C₂²×M₁₆	—	86	1	no	—
184	C₂×C₂²⋊Q₈	—	86	1	no	—
185	C₂×M₃₂	—	43	1	no	—
186	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
187	#470	—	43	4	no	—
188	#470	—	86	2	no	—
189	#430	—	43	4	no	—
190	#430	—	86	2	no	—
191	C₂×#108	—	43	1	no	—
192	C₂×#108	—	86	1	no	—
193	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
194	C₂×#106	—	43	1	no	—
195	C₂×#103	—	86	1	no	—
196	#365	—	43	4	no	—
197	#365	—	43	4	no	—
198	#362	—	43	4	no	—
199	#362	C₂×D₄₃	1	4	yes	complemented
200	#349	—	43	4	no	—
201	C₄₃	C₂×#2239	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	2	1	no	—
203	C₂³×C₄	—	43	1	no	—
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	43	2	no	—
212	C₂×M₁₆	—	43	2	no	—
213	C₂×M₁₆	—	86	1	no	—
214	C₂×M₁₆	—	86	1	no	—
215	C₂×M₁₆	—	86	1	no	—
216	C₂×M₁₆	—	86	1	no	—
217	C₂×M₁₆	—	86	4	no	—
218	C₂×M₁₆	—	86	1	no	—
219	C₂×M₁₆	—	172	1	no	—
220	C₂×M₁₆	—	172	2	no	—
221	C₂²×C₈	—	43	1	no	—
222	C₂²×C₈	—	172	1	no	—
223	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
224	C₂²⋊Q₈	—	86	4	no	—
225	C₂²⋊Q₈	—	172	2	no	—
226	C₂×C₄⋊C₄	—	86	1	no	—
227	C₂×C₄⋊C₄	—	172	1	no	—
228	C₂×C₂²⋊C₄	—	172	1	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	43	2	no	—
231	M₃₂	—	86	1	no	—
232	M₃₂	C₂²×D₄₃	1	2	yes	—
233	C₂×C₁₆	—	2	1	no	—
234	C₂×C₁₆	—	86	1	no	—
235	#108	—	2	1	no	—
236	#108	—	43	2	no	—
237	#108	—	86	4	no	—
238	#108	—	86	1	no	—
239	#108	C₂²×D₄₃	1	2	yes	—
240	#106	—	43	2	no	—
241	#106	—	86	1	no	—
242	#103	—	86	4	no	—
243	C₂⁴	—	43	1	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	43	1	no	—
252	C₂²×C₄	—	43	2	no	—
253	C₂²×C₄	—	43	2	no	—
254	C₂²×C₄	—	86	1	no	—
255	C₂²×C₄	—	86	1	no	—
256	C₂²×C₄	—	86	1	no	—
257	C₂²×C₄	—	86	1	no	—
258	C₂²×C₄	—	172	1	no	—
259	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	4	1	no	—
263	Q₁₆	—	4	1	no	—
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	86	2	no	—
267	M₁₆	—	86	2	no	—
268	M₁₆	—	86	2	no	—
269	M₁₆	—	86	2	no	—
270	M₁₆	—	86	2	no	—
271	M₁₆	—	172	4	no	—
272	M₁₆	—	172	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	43	2	no	—
275	C₂×C₈	—	86	1	no	—
276	C₂×C₈	—	86	1	no	—
277	C₂×C₈	—	172	4	no	—
278	C₂×C₈	—	172	1	no	—
279	C₂×C₈	#10621898	1	1	yes	characteristic
280	C₂×C₈	#10621898	1	1	yes	characteristic
281	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
282	C₂×C₈	#10621900	1	1	yes	characteristic
283	C₂×C₈	#10621900	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
285	C₄⋊C₄	—	86	2	no	—
286	C₄⋊C₄	—	172	4	no	—
287	C₄⋊C₄	—	172	1	no	—
288	C₂²⋊C₄	—	172	4	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	86	2	no	—
291	C₂³	—	43	2	no	—
292	C₂³	—	86	1	no	—
293	C₂³	—	86	1	no	—
294	C₂³	—	172	2	no	—
295	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
296	Q₈	—	4	2	no	—
297	Q₈	—	4	2	no	—
298	Q₈	—	8	1	no	—
299	Q₈	—	8	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	43	2	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	1	no	—
306	C₂×C₄	—	86	2	no	—
307	C₂×C₄	—	86	2	no	—
308	C₂×C₄	—	86	2	no	—
309	C₂×C₄	—	86	2	no	—
310	C₂×C₄	—	344	2	no	—
311	C₂×C₄	—	172	2	no	—
312	C₂×C₄	#13063307	1	1	yes	characteristic
313	C₂×C₄	#13063307	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
315	C₂×C₄	#13063349	1	1	yes	characteristic
316	C₂×C₄	#13063349	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
318	C₈	—	4	2	no	—
319	C₈	—	86	2	no	—
320	C₈	—	172	2	no	—
321	C₈	#13063307	1	1	yes	characteristic
322	C₈	#13063307	1	1	yes	characteristic
323	C₈	C₂×#10621900	1	1	yes	characteristic
324	C₈	C₂×#10621898	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	86	2	no	—
327	C₂²	—	344	1	no	—
328	C₂²	—	172	2	no	—
329	C₂²	—	172	2	no	—
330	C₂²	#14583216	1	1	yes	characteristic
331	C₂²	#14583216	1	1	yes	characteristic
332	C₂²	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
333	C₄	—	8	2	no	—
334	C₄	—	86	2	no	—
335	C₄	—	344	2	no	—
336	C₄	#14583216	1	1	yes	characteristic
337	C₄	#14583216	1	1	yes	characteristic
338	C₄	C₂×#13063307	1	1	yes	characteristic
339	C₄	C₂×#13063349	1	1	yes	characteristic
340	C₂	—	2	1	no	—
341	C₂	—	2	1	no	—
342	C₂	—	172	2	no	—
343	C₂	#19136813	1	2	yes	direct product factor
344	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×#362	C₂
socle	C₂²×C₄₃	#481
prime radical	C₂³×D₄₃	SD₁₆
hypercenter	C₂×#362	D₄₃
hypocenter	C₄₃	C₂×#2239
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

Complements

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

N	H ≅ G/N	type
3. #19136813	C₂	direct (N × H)
4. C₂×#14582723	C₂	semidirect
5. C₂×#14582722	C₂	semidirect
6. C₂×#14582715	C₂	semidirect
8. C₂×C₄₃×#362	C₂	semidirect
21. #14582723	C₂²	semidirect
22. #14582722	C₂²	semidirect
23. #14582715	C₂²	semidirect
28. C₄₃×#362	C₂²	semidirect
174. C₂×#362	D₄₃	semidirect
178. C₂×C₄₃	#2239	semidirect
199. #362	C₂×D₄₃	semidirect
201. C₄₃	C₂×#2239	semidirect
343. C₂	#19136813	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×#14583154	2
C₂×#14582946	2
#19136813	2
C₂×#14582723	2
C₂×#14582722	2
C₂×#14582715	2
C₂×#14582680	2
C₂×C₄₃×#362	2
C₂×#2239	43

Subgroups — C2×#19136813 #25251486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136813 #25251486