Subgroups — C₂×#19136786 #25251367

Order 11008.

638 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583135	C₂	1	1	yes	maximal, characteristic
2	#19136786	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14582726	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582724	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582699	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×#13063299	—	2	1	no	—
11	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14583135	—	2	2	no	—
14	#14583135	C₂²	1	4	yes	—
15	C₂×#13063259	C₂²	1	1	yes	characteristic
16	C₂×#13063247	—	2	1	no	—
17	#14582726	C₂²	1	4	yes	complemented
18	#14582724	C₂²	1	4	yes	complemented
19	#14582699	C₂²	1	4	yes	complemented
20	#14582699	C₂²	1	4	yes	complemented
21	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	complemented
22	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#117×C₄₃	—	2	1	no	—
24	C₄₃×#427	—	2	2	no	—
25	C₄₃×#427	C₂²	1	4	yes	complemented
26	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#106×C₄₃	—	2	1	no	—
29	C₂²×D₁₇₂	—	2	1	no	—
30	C₂²×C₂²⋊D₄₃	—	4	1	no	—
31	C₂×D₃₄₄	—	2	2	no	—
32	C₂×D₃₄₄	—	2	2	no	—
33	C₂×D₃₄₄	—	2	1	no	—
34	C₂×D₃₄₄	C₂³	1	2	yes	—
35	#13063299	—	2	4	no	—
36	#13063299	—	4	2	no	—
37	C₂×C₄⋊Dic₄₃	—	2	1	no	—
38	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
39	C₂×C₄₃⋊C₁₆	—	2	2	no	—
40	C₂×C₄₃⋊C₁₆	—	2	2	no	—
41	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
42	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
43	#13063259	—	2	1	no	—
44	#13063259	C₂³	1	2	yes	—
45	#13063247	—	2	4	no	—
46	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	—	2	2	no	—
48	C₂²×C₈×C₄₃	—	2	2	no	—
49	C₂²×C₈×C₄₃	D₄	1	2	yes	—
50	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	#117×C₄₃	—	2	4	no	—
54	#117×C₄₃	—	4	2	no	—
55	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₄₃	—	4	1	no	—
58	C₂×C₄²×C₄₃	—	4	1	no	—
59	#107×C₄₃	—	2	1	no	—
60	#107×C₄₃	—	2	1	no	—
61	#107×C₄₃	C₂³	1	2	yes	—
62	#107×C₄₃	C₂³	1	2	yes	—
63	#106×C₄₃	—	2	4	no	—
64	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
65	C₂³×D₄₃	—	4	1	no	—
66	C₂²×C₄×C₄₃	—	2	2	no	—
67	C₂²×C₄×C₄₃	—	2	2	no	—
68	C₂²×C₄×C₄₃	—	4	1	no	—
69	C₂²×C₄×C₄₃	—	4	1	no	—
70	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	D₈	1	2	yes	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₃	—	4	4	no	—
75	C₂×C₂²⋊D₄₃	—	8	4	no	—
76	C₂²×Dic₄₃	—	4	1	no	—
77	C₂×D₁₇₂	—	2	2	no	—
78	C₂×D₁₇₂	—	4	1	no	—
79	C₂×D₁₇₂	—	4	2	no	—
80	C₂×D₁₇₂	—	4	2	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	2	4	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	#41	1	2	yes	—
88	C₂×C₈×C₄₃	#41	1	2	yes	—
89	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	—	2	2	no	—
92	C₄⋊C₄×C₄₃	—	4	1	no	—
93	C₄⋊C₄×C₄₃	—	4	1	no	—
94	C₂²⋊C₄×C₄₃	—	4	4	no	—
95	C₄²×C₄₃	—	4	4	no	—
96	(C₂×D₄₃)⋊C₄	—	4	4	no	—
97	C₄⋊Dic₄₃	—	2	2	no	—
98	C₄⋊Dic₄₃	—	4	1	no	—
99	D₃₄₄	—	2	2	no	—
100	D₃₄₄	—	2	2	no	—
101	D₃₄₄	—	4	2	no	—
102	C₄₃⋊C₁₆	—	2	2	no	—
103	C₄₃⋊C₁₆	—	2	2	no	—
104	C₂³×C₄₃	—	2	2	no	—
105	C₂³×C₄₃	—	2	2	no	—
106	C₂³×C₄₃	—	2	2	no	—
107	C₂³×C₄₃	C₂×D₈	1	2	yes	—
108	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
109	C₂²×D₄₃	—	4	2	no	—
110	C₂²×D₄₃	—	8	2	no	—
111	C₂²×D₄₃	—	8	2	no	—
112	C₂²×D₄₃	—	8	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	4	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	—	8	2	no	—
120	C₂×C₄×C₄₃	—	8	2	no	—
121	C₂×C₄×C₄₃	#123	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#137	1	2	yes	—
125	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₄₃	—	8	8	no	—
127	C₂×Dic₄₃	—	4	2	no	—
128	C₂×Dic₄₃	—	8	2	no	—
129	D₁₇₂	—	4	2	no	—
130	D₁₇₂	—	4	2	no	—
131	D₁₇₂	—	8	2	no	—
132	C₈×C₄₃	—	2	2	no	—
133	C₈×C₄₃	#123	1	2	yes	—
134	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
135	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
136	C₂×#2234	—	43	1	no	maximal, Sylow
137	C₂²×C₄₃	—	2	2	no	—
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	4	4	no	—
141	C₂²×C₄₃	#480	1	2	yes	—
142	C₂²×C₄₃	#508	1	2	yes	—
143	C₂²×C₄₃	#509	1	2	yes	—
144	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₄₃	—	8	2	no	—
146	C₂×D₄₃	—	8	2	no	—
147	C₂×D₄₃	—	8	2	no	—
148	C₂×D₄₃	—	16	4	no	—
149	C₄×C₄₃	—	2	2	no	—
150	C₄×C₄₃	—	8	2	no	—
151	C₄×C₄₃	—	8	2	no	—
152	C₄×C₄₃	#480	1	2	yes	—
153	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
154	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
155	Dic₄₃	—	8	2	no	—
156	C₂×#466	—	43	1	no	—
157	C₂×#427	D₄₃	1	1	yes	characteristic, complemented
158	#2234	—	43	8	no	—
159	C₂×#367	—	43	1	no	—
160	C₂×#366	—	43	1	no	—
161	C₂×#357	—	43	1	no	—
162	C₂×#357	—	43	1	no	—
163	C₂×C₂²⋊C₁₆	—	43	1	no	—
164	C₂×C₄₃	—	4	2	no	—
165	C₂×C₄₃	#2234	1	4	yes	complemented
166	C₂×C₄₃	C₂×#480	1	1	yes	characteristic
167	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
169	D₄₃	—	16	2	no	—
170	C₂²×D₈	—	43	1	no	—
171	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
172	C₂×#121	—	86	1	no	—
173	C₂×#117	—	2	1	no	—
174	C₂²×C₁₆	—	43	1	no	—
175	C₂²×C₁₆	—	43	1	no	—
176	#466	—	43	4	no	—
177	#466	—	86	2	no	—
178	#427	—	2	2	no	—
179	#427	C₂×D₄₃	1	4	yes	complemented
180	C₂×#107	—	43	1	no	—
181	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
182	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
183	C₂×#106	—	2	1	no	—
184	C₂×#102	—	86	1	no	—
185	#367	—	43	4	no	—
186	#366	—	43	4	no	—
187	#357	—	43	4	no	—
188	#357	—	43	4	no	—
189	C₂²⋊C₁₆	—	43	4	no	—
190	C₄₃	C₂×#2234	1	1	yes	characteristic, complemented, Sylow
191	C₂²×D₄	—	86	1	no	—
192	C₂²×D₄	—	172	1	no	—
193	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
194	C₂×D₈	—	43	2	no	—
195	C₂×D₈	—	86	2	no	—
196	C₂×D₈	—	86	2	no	—
197	C₂×D₈	—	86	1	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
202	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
203	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
204	#121	—	86	4	no	—
205	#121	—	172	2	no	—
206	#117	—	2	4	no	—
207	#117	—	4	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	86	1	no	—
211	C₂×C₂²⋊C₄	—	4	1	no	—
212	C₂×C₂²⋊C₄	—	172	1	no	—
213	C₂×C₄²	—	4	1	no	—
214	C₂×C₁₆	—	43	2	no	—
215	C₂×C₁₆	—	43	2	no	—
216	C₂×C₁₆	—	86	2	no	—
217	C₂×C₁₆	—	86	2	no	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	43	2	no	—
221	#107	—	86	1	no	—
222	#107	C₂²×D₄₃	1	2	yes	—
223	#107	C₂²×D₄₃	1	2	yes	—
224	#106	—	2	4	no	—
225	#102	—	86	4	no	—
226	C₂⁴	—	172	1	no	—
227	C₂⁴	#10621882	1	1	yes	characteristic
228	C₂×D₄	—	86	2	no	—
229	C₂×D₄	—	344	4	no	—
230	C₂×D₄	—	172	4	no	—
231	C₂×D₄	—	172	2	no	—
232	C₂×D₄	—	172	1	no	—
233	C₂×D₄	—	172	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	172	1	no	—
239	C₂²×C₄	#10621882	1	2	yes	—
240	C₂²×C₄	#10621882	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
243	D₈	—	86	2	no	—
244	D₈	—	86	2	no	—
245	D₈	—	172	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	#10621898	1	2	yes	—
251	C₂×C₈	#10621898	1	2	yes	—
252	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
253	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	86	2	no	—
260	C₄⋊C₄	—	172	1	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₂²⋊C₄	—	172	4	no	—
263	C₄²	—	4	4	no	—
264	C₁₆	—	86	2	no	—
265	C₁₆	—	86	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	344	2	no	—
270	C₂³	—	344	2	no	—
271	C₂³	—	344	2	no	—
272	C₂³	—	172	2	no	—
273	C₂³	C₂×#10621882	1	2	yes	—
274	C₂³	C₂×#10621882	1	1	yes	characteristic
275	D₄	—	344	8	no	—
276	D₄	—	344	2	no	—
277	D₄	—	172	2	no	—
278	D₄	—	172	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	344	2	no	—
288	C₂×C₄	—	172	2	no	—
289	C₂×C₄	#13063308	1	2	yes	—
290	C₂×C₄	C₂×#10621882	1	2	yes	—
291	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
292	C₂×C₄	#13063356	1	2	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	#13063308	1	2	yes	—
296	C₈	C₂×#10621898	1	1	yes	characteristic
297	C₈	C₂×#10621898	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	344	2	no	—
303	C₂²	—	344	2	no	—
304	C₂²	—	344	2	no	—
305	C₂²	—	688	4	no	—
306	C₂²	#14583212	1	2	yes	—
307	C₂²	#14583329	1	2	yes	—
308	C₂²	#14583334	1	2	yes	—
309	C₂²	C₂²×#10621882	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	344	2	no	—
314	C₄	#14583212	1	2	yes	—
315	C₄	C₂×#13063308	1	1	yes	characteristic
316	C₄	C₂×#13063356	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	—	688	2	no	—
319	C₂	#19136786	1	4	yes	direct product factor
320	C₂	C₂×#14583212	1	1	yes	characteristic
321	C₂	C₂×#14583329	1	1	yes	characteristic
322	C₂	C₂×#14583334	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#427	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂×#14583135	C₂
hypercenter	C₂×#427	D₄₃
hypocenter	C₄₃	C₂×#2234
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #19136786	C₂	direct (N × H)
3. C₂×#14582726	C₂	semidirect
4. C₂×#14582724	C₂	semidirect
5. C₂×#14582699	C₂	semidirect
6. C₂×#14582699	C₂	semidirect
7. C₂×C₂²⋊(C₄₃⋊C₁₆)	C₂	semidirect
8. C₂×C₄₃×#427	C₂	semidirect
17. #14582726	C₂²	semidirect
18. #14582724	C₂²	semidirect
19. #14582699	C₂²	semidirect
20. #14582699	C₂²	semidirect
21. C₂²⋊(C₄₃⋊C₁₆)	C₂²	semidirect
25. C₄₃×#427	C₂²	semidirect
157. C₂×#427	D₄₃	semidirect
165. C₂×C₄₃	#2234	semidirect
179. #427	C₂×D₄₃	semidirect
190. C₄₃	C₂×#2234	semidirect
319. C₂	#19136786	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×#14583135	2
#19136786	2
C₂×#14582726	2
C₂×#14582724	2
C₂×#14582699	2
C₂×C₂²⋊(C₄₃⋊C₁₆)	2
C₂×C₄₃×#427	2
C₂×#2234	43

Subgroups — C2×#19136786 #25251367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136786 #25251367