Subgroups — C₂×#19136382 #25249630

Order 11008.

718 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063264	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582936	C₂	1	1	yes	maximal, characteristic
3	C₂×#107×D₄₃	C₂	1	1	yes	maximal, characteristic
4	#19136382	C₂	1	8	yes	maximal, direct product factor
5	C₂×#14582726	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582725	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582724	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
10	C₂×#13063283	—	2	1	no	—
11	C₂×C₄⋊C₄×D₄₃	—	2	1	no	—
12	C₂×#13063264	C₄	1	8	yes	complemented
13	C₂×#13063264	C₂²	1	4	yes	—
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	#14582936	—	2	2	no	—
16	#14582936	C₂²	1	4	yes	—
17	C₂×#13063259	C₂²	1	1	yes	characteristic
18	C₂×#13063259	C₂²	1	1	yes	characteristic
19	C₂×#13063258	—	2	1	no	—
20	#107×D₄₃	—	2	2	no	—
21	#107×D₄₃	C₂²	1	4	yes	—
22	C₂×#13063256	—	2	1	no	—
23	#14582726	C₂²	1	4	yes	complemented
24	#14582725	C₂²	1	4	yes	complemented
25	#14582724	C₂²	1	4	yes	complemented
26	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#367	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₄₃	—	4	1	no	—
31	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
32	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₄₃	D₄	1	2	yes	—
35	C₂×C₈×D₄₃	Q₈	1	4	yes	—
36	C₂×C₈×D₄₃	C₂³	1	2	yes	—
37	#13063283	—	2	4	no	—
38	#13063283	—	4	2	no	—
39	C₂×C₄⋊Dic₄₃	—	2	1	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×Dic₄₃⋊C₄	—	4	1	no	—
42	C₄⋊C₄×D₄₃	—	2	4	no	—
43	C₄⋊C₄×D₄₃	—	4	2	no	—
44	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
45	C₂×C₄×Dic₄₃	—	4	1	no	—
46	#13063264	C₂×C₄	1	16	yes	complemented
47	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
48	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
49	#13063259	—	2	1	no	—
50	#13063259	—	2	1	no	—
51	#13063259	C₂³	1	2	yes	—
52	#13063259	C₂³	1	2	yes	—
53	#13063258	—	2	4	no	—
54	#13063256	—	2	4	no	—
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
58	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
60	#107×C₄₃	—	2	1	no	—
61	#107×C₄₃	—	2	1	no	—
62	#107×C₄₃	C₂³	1	2	yes	—
63	#107×C₄₃	C₂³	1	2	yes	—
64	C₂³×D₄₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	—	4	1	no	—
66	C₂²×C₄×C₄₃	—	4	1	no	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₄₃	—	4	1	no	—
69	C₂²×Dic₄₃	—	4	1	no	—
70	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
71	C₂×C₄×D₄₃	—	4	4	no	—
72	C₂×C₄×D₄₃	—	8	4	no	—
73	C₂×C₄×D₄₃	C₄⋊C₄	1	4	yes	—
74	C₂×C₄×D₄₃	D₈	1	2	yes	—
75	C₂×C₄×D₄₃	Q₁₆	1	4	yes	—
76	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	—	2	2	no	—
82	C₄⋊C₄×C₄₃	—	2	2	no	—
83	C₄⋊C₄×C₄₃	—	4	1	no	—
84	C₄⋊C₄×C₄₃	—	4	1	no	—
85	(C₂×D₄₃)⋊C₄	—	4	4	no	—
86	C₄⋊Dic₄₃	—	2	2	no	—
87	C₄⋊Dic₄₃	—	2	2	no	—
88	C₄⋊Dic₄₃	—	4	1	no	—
89	C₄⋊Dic₄₃	—	4	1	no	—
90	Dic₄₃⋊C₄	—	4	4	no	—
91	C₄×Dic₄₃	—	4	4	no	—
92	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
93	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
94	C₈×D₄₃	C₄⋊C₄	1	8	yes	—
95	C₈×D₄₃	C₂²×C₄	1	4	yes	—
96	C₈×D₄₃	C₂×Q₈	1	4	yes	—
97	C₁₆×C₄₃	C₂²×C₄	1	4	yes	complemented
98	C₄₃⋊C₁₆	C₂²×C₄	1	4	yes	complemented
99	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₄₃	—	2	2	no	—
101	C₂²×D₄₃	—	2	2	no	—
102	C₂²×D₄₃	#107	1	4	yes	—
103	C₂²×D₄₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	—	4	2	no	—
106	C₂×C₄×C₄₃	—	8	2	no	—
107	C₂×C₄×C₄₃	—	8	2	no	—
108	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₄₃	—	4	2	no	—
113	C₂×Dic₄₃	—	4	2	no	—
114	C₂×Dic₄₃	—	8	2	no	—
115	C₂×Dic₄₃	—	8	2	no	—
116	C₂×Dic₄₃	#107	1	4	yes	—
117	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
118	C₄×D₄₃	—	8	8	no	—
119	C₄×D₄₃	#107	1	8	yes	—
120	C₄×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
121	C₄×D₄₃	C₂×Q₁₆	1	4	yes	—
122	C₈×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
125	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×#2159	—	43	1	no	maximal, Sylow
127	C₂²×C₄₃	C₂×#107	1	2	yes	—
128	C₂²×C₄₃	#509	1	2	yes	—
129	C₂²×C₄₃	#510	1	2	yes	—
130	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
131	C₂×D₄₃	—	2	2	no	—
132	C₂×D₄₃	—	2	2	no	—
133	C₂×D₄₃	—	2	8	no	—
134	C₂×D₄₃	C₂×#107	1	4	yes	—
135	C₄×C₄₃	—	8	2	no	—
136	C₄×C₄₃	—	8	2	no	—
137	C₄×C₄₃	C₂×#107	1	2	yes	—
138	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
140	Dic₄₃	—	8	2	no	—
141	Dic₄₃	—	8	2	no	—
142	Dic₄₃	C₂×#107	1	4	yes	—
143	C₂²×M₃₂	—	43	1	no	—
144	C₂×#427	—	43	1	no	—
145	C₂²×#107	—	43	1	no	—
146	#2159	—	43	8	no	—
147	C₂×#367	—	43	1	no	—
148	C₂×#367	D₄₃	1	1	yes	characteristic, complemented
149	C₂×#366	—	43	1	no	—
150	C₂×#366	—	43	1	no	—
151	C₂×C₄₃	#2159	1	4	yes	complemented
152	C₂×C₄₃	C₂²×#107	1	1	yes	characteristic
153	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
154	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
155	D₄₃	—	2	4	no	—
156	C₂³×C₈	—	43	1	no	—
157	C₂×#117	—	86	1	no	—
158	C₂²×C₄⋊C₄	—	86	1	no	—
159	C₂×M₃₂	—	43	4	no	—
160	C₂×M₃₂	—	43	8	no	—
161	C₂²×C₁₆	—	43	1	no	—
162	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
163	#427	—	43	4	no	—
164	#427	—	86	2	no	—
165	C₂×#107	—	43	1	no	—
166	C₂×#107	—	43	1	no	—
167	C₂×#107	—	43	4	no	—
168	C₂×#107	—	86	2	no	—
169	C₂×#107	—	86	1	no	—
170	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
171	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
172	C₂×#106	—	86	1	no	—
173	#367	—	43	4	no	—
174	#367	C₂×D₄₃	1	4	yes	complemented
175	#366	—	43	4	no	—
176	#366	—	43	4	no	—
177	C₄₃	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	43	1	no	—
179	C₂³×C₄	—	172	1	no	—
180	C₂²×C₈	—	43	1	no	—
181	C₂²×C₈	—	43	4	no	—
182	C₂²×C₈	—	43	2	no	—
183	C₂²×C₈	—	43	4	no	—
184	C₂²×C₈	—	43	2	no	—
185	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
186	#117	—	86	4	no	—
187	#117	—	172	2	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	86	4	no	—
191	C₂×C₄⋊C₄	—	86	1	no	—
192	C₂×C₄⋊C₄	—	86	1	no	—
193	C₂×C₄⋊C₄	—	172	2	no	—
194	C₂×C₄⋊C₄	—	172	1	no	—
195	C₂×C₂²⋊C₄	—	172	1	no	—
196	C₂×C₄²	—	172	1	no	—
197	M₃₂	—	43	16	no	—
198	C₂×C₁₆	—	43	4	no	—
199	C₂×C₁₆	—	43	2	no	—
200	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
201	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
202	#107	—	2	1	no	—
203	#107	—	2	1	no	—
204	#107	—	43	2	no	—
205	#107	—	43	2	no	—
206	#107	—	86	4	no	—
207	#107	—	86	1	no	—
208	#107	—	86	1	no	—
209	#107	C₂²×D₄₃	1	2	yes	—
210	#107	C₂²×D₄₃	1	2	yes	—
211	#106	—	86	4	no	—
212	C₂⁴	—	43	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	43	1	no	—
216	C₂²×C₄	—	43	4	no	—
217	C₂²×C₄	—	43	2	no	—
218	C₂²×C₄	—	43	4	no	—
219	C₂²×C₄	—	43	2	no	—
220	C₂²×C₄	—	344	4	no	—
221	C₂²×C₄	—	172	4	no	—
222	C₂²×C₄	—	172	1	no	—
223	C₂²×C₄	—	172	1	no	—
224	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
225	C₂×C₈	—	43	4	no	—
226	C₂×C₈	—	43	2	no	—
227	C₂×C₈	—	43	8	no	—
228	C₂×C₈	—	43	4	no	—
229	C₂×C₈	—	43	4	no	—
230	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
231	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
232	C₂×C₈	Q₈×D₄₃	1	2	yes	—
233	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	86	2	no	—
239	C₄⋊C₄	—	86	2	no	—
240	C₄⋊C₄	—	172	4	no	—
241	C₄⋊C₄	—	172	1	no	—
242	C₄⋊C₄	—	172	1	no	—
243	C₂²⋊C₄	—	172	4	no	—
244	C₄²	—	172	4	no	—
245	C₁₆	—	43	4	no	—
246	C₁₆	C₂×C₄×D₄₃	1	4	yes	complemented
247	C₂³	—	43	2	no	—
248	C₂³	—	43	4	no	—
249	C₂³	—	86	2	no	—
250	C₂³	—	86	2	no	—
251	C₂³	D₈×D₄₃	1	1	yes	characteristic
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	43	2	no	—
257	C₂×C₄	—	43	4	no	—
258	C₂×C₄	—	43	4	no	—
259	C₂×C₄	—	43	4	no	—
260	C₂×C₄	—	43	8	no	—
261	C₂×C₄	—	344	8	no	—
262	C₂×C₄	—	344	2	no	—
263	C₂×C₄	—	344	2	no	—
264	C₂×C₄	—	172	2	no	—
265	C₂×C₄	—	172	2	no	—
266	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
267	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
268	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
269	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
270	C₈	—	43	4	no	—
271	C₈	C₄⋊C₄×D₄₃	1	2	yes	—
272	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
273	C₈	C₂×Q₈×D₄₃	1	1	yes	characteristic
274	C₂²	—	43	4	no	—
275	C₂²	—	86	8	no	—
276	C₂²	—	86	2	no	—
277	C₂²	—	86	2	no	—
278	C₂²	#107×D₄₃	1	2	yes	—
279	C₂²	#14583336	1	2	yes	—
280	C₂²	#14583338	1	2	yes	—
281	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	43	4	no	—
285	C₄	—	344	2	no	—
286	C₄	—	344	2	no	—
287	C₄	#107×D₄₃	1	2	yes	—
288	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
290	C₂	—	86	4	no	—
291	C₂	#19136382	1	4	yes	direct product factor
292	C₂	C₂×#107×D₄₃	1	1	yes	characteristic
293	C₂	C₂×#14583336	1	1	yes	characteristic
294	C₂	C₂×#14583338	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#367	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂³×D₄₃	D₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#367	D₄₃
hypocenter	C₄₃	C₂×#2159
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
4. #19136382	C₂	direct (N × H)
5. C₂×#14582726	C₂	semidirect
6. C₂×#14582725	C₂	semidirect
7. C₂×#14582724	C₂	semidirect
8. C₂×C₄₃×#367	C₂	semidirect
12. C₂×#13063264	C₄	semidirect
23. #14582726	C₂²	semidirect
24. #14582725	C₂²	semidirect
25. #14582724	C₂²	semidirect
29. C₄₃×#367	C₂²	semidirect
46. #13063264	C₂×C₄	semidirect
47. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
58. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
97. C₁₆×C₄₃	C₂²×C₄	semidirect
98. C₄₃⋊C₁₆	C₂²×C₄	semidirect
148. C₂×#367	D₄₃	semidirect
151. C₂×C₄₃	#2159	semidirect
174. #367	C₂×D₄₃	semidirect
177. C₄₃	C₂×#2159	semidirect
200. C₂×C₁₆	C₄×D₄₃	semidirect
246. C₁₆	C₂×C₄×D₄₃	semidirect
291. C₂	#19136382	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13063264	2
C₂×#14582936	2
C₂×#107×D₄₃	2
#19136382	2
C₂×#14582726	2
C₂×#14582725	2
C₂×#14582724	2
C₂×C₄₃×#367	2
C₂×#2159	43

Subgroups — C2×#19136382 #25249630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136382 #25249630