Subgroups — C₂×#19136816 #25251489

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×#14583158	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582946	C₂	1	1	yes	maximal, characteristic
3	#19136816	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582721	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582716	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582714	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582680	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂²×#10621872	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₄₃	—	2	1	no	—
12	C₂×#13063305	—	2	1	no	—
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	#14583158	—	2	2	no	—
15	#14583158	C₂²	1	4	yes	—
16	#14582946	—	2	2	no	—
17	#14582946	C₂²	1	4	yes	—
18	C₂×#13063261	C₂²	1	1	yes	characteristic
19	C₂×#13063260	—	2	1	no	—
20	C₂×#13063248	—	2	1	no	—
21	#14582721	C₂²	1	4	yes	complemented
22	#14582716	C₂²	1	4	yes	complemented
23	#14582714	C₂²	1	4	yes	complemented
24	#14582680	C₂²	1	4	yes	—
25	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#365	C₂²	1	4	yes	complemented
29	C₂²×Dic₈₆	—	2	1	no	—
30	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
31	C₂×Dic₁₇₂	—	2	1	no	—
32	C₂×Dic₁₇₂	—	2	1	no	—
33	C₂×Dic₁₇₂	—	2	1	no	—
34	C₂×Dic₁₇₂	—	2	1	no	—
35	C₂×Dic₁₇₂	—	2	1	no	—
36	C₂×Dic₁₇₂	C₂³	1	2	yes	—
37	C₂×#10621877	—	2	1	no	—
38	C₂×#10621872	—	2	1	no	—
39	C₂×#10621872	—	2	1	no	—
40	C₂×#10621872	—	2	1	no	—
41	C₂×#10621872	—	2	1	no	—
42	C₂×#10621872	—	4	1	no	—
43	C₂×#10621872	D₄	1	2	yes	—
44	C₂×#10621872	C₂³	1	2	yes	—
45	M₁₆×D₄₃	—	2	4	no	—
46	M₁₆×D₄₃	—	4	2	no	—
47	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
48	C₂×C₈×D₄₃	—	4	1	no	—
49	#13063305	—	2	4	no	—
50	#13063305	—	4	2	no	—
51	C₂×C₄⋊Dic₄₃	—	4	1	no	—
52	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
53	#13063265	—	2	1	no	—
54	#13063265	C₂³	1	2	yes	—
55	C₂×C₄₃⋊C₁₆	—	2	1	no	—
56	#13063261	—	2	1	no	—
57	#13063261	C₂³	1	2	yes	—
58	#13063260	—	2	4	no	—
59	#13063248	—	2	4	no	—
60	C₂×M₁₆×C₄₃	—	2	1	no	—
61	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
63	M₃₂×C₄₃	—	2	1	no	—
64	M₃₂×C₄₃	C₂³	1	2	yes	—
65	C₂×C₁₆×C₄₃	—	2	1	no	—
66	#108×C₄₃	—	2	1	no	—
67	#108×C₄₃	C₂³	1	2	yes	—
68	#106×C₄₃	—	2	1	no	—
69	#106×C₄₃	C₂³	1	2	yes	—
70	C₂³×D₄₃	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	—	4	1	no	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₄₃	—	4	1	no	—
74	C₂²×Dic₄₃	SD₁₆	1	1	yes	characteristic
75	C₂×C₄×D₄₃	—	2	1	no	—
76	C₂×C₄×D₄₃	—	2	1	no	—
77	C₂×C₄×D₄₃	—	2	1	no	—
78	C₂×C₄×D₄₃	—	2	1	no	—
79	C₂×C₄×D₄₃	SD₁₆	1	2	yes	—
80	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
81	C₂×Dic₈₆	—	2	2	no	—
82	C₂×Dic₈₆	—	4	1	no	—
83	C₂×Dic₈₆	—	4	1	no	—
84	C₂×Dic₈₆	—	4	1	no	—
85	C₂×Dic₈₆	—	4	1	no	—
86	C₂×Dic₈₆	—	4	1	no	—
87	M₁₆×C₄₃	—	2	2	no	—
88	M₁₆×C₄₃	—	4	1	no	—
89	C₂×C₈×C₄₃	—	4	1	no	—
90	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₄₃	—	2	2	no	—
97	C₄⋊C₄×C₄₃	—	4	1	no	—
98	(C₂×D₄₃)⋊C₄	—	4	4	no	—
99	C₄⋊Dic₄₃	—	4	4	no	—
100	#10621877	—	2	2	no	—
101	#10621877	—	4	1	no	—
102	C₂×C₄₃⋊C₈	—	2	1	no	—
103	C₂×C₄₃⋊C₈	—	2	1	no	—
104	C₂×C₄₃⋊C₈	—	4	1	no	—
105	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
106	Dic₁₇₂	—	2	2	no	—
107	Dic₁₇₂	—	2	2	no	—
108	Dic₁₇₂	—	4	1	no	—
109	Dic₁₇₂	—	4	1	no	—
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	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	8	2	no	—
125	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₄₃	—	2	1	no	—
132	C₂×Dic₄₃	—	2	1	no	—
133	C₂×Dic₄₃	—	4	2	no	—
134	C₂×Dic₄₃	—	8	1	no	—
135	C₂×Dic₄₃	—	8	1	no	—
136	C₂×Dic₄₃	C₂×SD₁₆	1	2	yes	—
137	C₄×D₄₃	—	2	2	no	—
138	C₄×D₄₃	—	2	2	no	—
139	C₄×D₄₃	—	2	2	no	—
140	C₄×D₄₃	—	2	2	no	—
141	Dic₈₆	—	4	2	no	—
142	Dic₈₆	—	4	2	no	—
143	Dic₈₆	—	8	1	no	—
144	Dic₈₆	—	8	1	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	D₄₃	1	1	yes	characteristic, complemented
173	C₂×#362	—	43	1	no	—
174	C₂×#362	—	43	1	no	—
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	—	43	1	no	—
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	C₂×D₄₃	1	1	yes	characteristic
195	C₂×#103	—	86	1	no	—
196	#365	—	43	4	no	—
197	#365	C₂×D₄₃	1	4	yes	complemented
198	#362	—	43	4	no	—
199	#362	—	43	4	no	—
200	#349	—	43	4	no	—
201	C₄₃	C₂×#2239	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	86	1	no	—
203	C₂³×C₄	—	43	1	no	—
204	C₂×Q₁₆	—	43	2	no	—
205	C₂×Q₁₆	—	86	1	no	—
206	C₂×Q₁₆	—	86	1	no	—
207	C₂×Q₁₆	—	86	1	no	—
208	C₂×Q₁₆	—	86	1	no	—
209	C₂×Q₁₆	—	86	1	no	—
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	—	2	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	—	2	1	no	—
241	#106	C₂²×D₄₃	1	2	yes	—
242	#103	—	86	4	no	—
243	C₂⁴	—	43	1	no	—
244	C₂×Q₈	—	86	2	no	—
245	C₂×Q₈	—	172	1	no	—
246	C₂×Q₈	—	172	1	no	—
247	C₂×Q₈	—	172	1	no	—
248	C₂×Q₈	—	172	1	no	—
249	C₂×Q₈	—	172	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	—	86	2	no	—
261	Q₁₆	—	86	2	no	—
262	Q₁₆	—	172	1	no	—
263	Q₁₆	—	172	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	#10621902	1	1	yes	characteristic
283	C₂×C₈	#10621902	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	172	4	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	—	344	1	no	—
297	Q₈	—	344	1	no	—
298	Q₈	—	172	2	no	—
299	Q₈	—	172	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	43	2	no	—
303	C₂×C₄	—	86	1	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	2	no	—
306	C₂×C₄	—	86	2	no	—
307	C₂×C₄	—	86	2	no	—
308	C₂×C₄	—	86	2	no	—
309	C₂×C₄	—	344	1	no	—
310	C₂×C₄	—	344	1	no	—
311	C₂×C₄	—	172	2	no	—
312	C₂×C₄	#13063310	1	1	yes	characteristic
313	C₂×C₄	#13063310	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
315	C₂×C₄	#13063352	1	1	yes	characteristic
316	C₂×C₄	#13063352	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	#13063310	1	1	yes	characteristic
322	C₈	#13063310	1	1	yes	characteristic
323	C₈	C₂×#10621898	1	1	yes	characteristic
324	C₈	C₂×#10621902	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	#14583219	1	1	yes	characteristic
331	C₂²	#14583219	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	#14583219	1	1	yes	characteristic
337	C₄	#14583219	1	1	yes	characteristic
338	C₄	C₂×#13063310	1	1	yes	characteristic
339	C₄	C₂×#13063352	1	1	yes	characteristic
340	C₂	—	2	1	no	—
341	C₂	—	2	1	no	—
342	C₂	—	172	2	no	—
343	C₂	#19136816	1	2	yes	direct product factor
344	C₂	C₂×#14583219	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14583219
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#365	C₂
socle	C₂²×C₄₃	#481
prime radical	C₂³×D₄₃	SD₁₆
hypercenter	C₂×#365	D₄₃
hypocenter	C₄₃	C₂×#2239
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
3. #19136816	C₂	direct (N × H)
4. C₂×#14582721	C₂	semidirect
5. C₂×#14582716	C₂	semidirect
6. C₂×#14582714	C₂	semidirect
8. C₂×C₄₃×#365	C₂	semidirect
21. #14582721	C₂²	semidirect
22. #14582716	C₂²	semidirect
23. #14582714	C₂²	semidirect
28. C₄₃×#365	C₂²	semidirect
172. C₂×#365	D₄₃	semidirect
178. C₂×C₄₃	#2239	semidirect
197. #365	C₂×D₄₃	semidirect
201. C₄₃	C₂×#2239	semidirect
343. C₂	#19136816	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×#14583158	2
C₂×#14582946	2
#19136816	2
C₂×#14582721	2
C₂×#14582716	2
C₂×#14582714	2
C₂×#14582680	2
C₂×C₄₃×#365	2
C₂×#2239	43

Subgroups — C2×#19136816 #25251489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136816 #25251489