Subgroups — C₂×#19136359 #25249595

Order 11008.

698 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	C₂×M₃₂×D₄₃	C₂	1	2	yes	maximal
2	C₂×#14582853	C₂	1	1	yes	maximal, characteristic
3	#19136359	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582720	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582719	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10621872	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
9	C₂×#13063282	C₄	1	1	yes	characteristic
10	C₂×#13063265	C₂²	1	2	yes	—
11	C₂×#13063264	—	2	2	no	—
12	M₃₂×D₄₃	—	2	4	no	—
13	M₃₂×D₄₃	C₂²	1	8	yes	—
14	C₂×C₁₆×D₄₃	—	2	2	no	—
15	#14582853	C₄	1	4	yes	—
16	#14582853	C₂²	1	4	yes	—
17	C₂×#13063236	C₄	1	1	yes	characteristic
18	C₂×#13063235	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
20	#14582720	C₂²	1	4	yes	complemented
21	#14582719	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#364	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#10621872	C₈	1	4	yes	—
27	C₂×#10621872	C₈	1	4	yes	—
28	C₂×#10621872	C₂×C₄	1	4	yes	—
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₄₃	D₄	1	2	yes	—
33	C₂×C₈×D₄₃	D₄	1	2	yes	—
34	C₂×C₈×D₄₃	Q₈	1	2	yes	—
35	C₂×C₈×D₄₃	Q₈	1	2	yes	—
36	C₂×C₈×D₄₃	C₂³	1	2	yes	—
37	#13063282	—	2	2	no	—
38	#13063282	C₂×C₄	1	4	yes	—
39	C₂×Dic₄₃⋊C₄	—	2	1	no	—
40	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
41	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
42	#13063265	—	2	2	no	—
43	#13063265	C₂³	1	4	yes	—
44	#13063264	—	2	8	no	—
45	C₂×C₄₃⋊C₁₆	—	2	2	no	—
46	C₁₆×D₄₃	—	2	8	no	—
47	#13063236	C₂×C₄	1	4	yes	—
48	#13063235	C₂×C₄	1	4	yes	—
49	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
50	C₈×Dic₄₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₄₃	—	2	2	no	—
55	M₃₂×C₄₃	C₂³	1	4	yes	—
56	C₂×C₁₆×C₄₃	—	2	2	no	—
57	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₄₃	C₂³	1	2	yes	—
59	C₂³×D₄₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₄₃	—	2	1	no	—
63	C₂²×Dic₄₃	M₁₆	1	1	yes	characteristic
64	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₄₃	M₁₆	1	2	yes	—
69	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
79	C₄²×C₄₃	—	2	1	no	—
80	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
81	(C₂×D₄₃)⋊C₄	—	2	4	no	—
82	Dic₄₃⋊C₄	—	2	4	no	—
83	C₄×Dic₄₃	—	2	1	no	—
84	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
88	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
89	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
91	#10621872	C₂×C₈	1	16	yes	—
92	C₈×D₄₃	C₄⋊C₄	1	4	yes	—
93	C₈×D₄₃	C₄⋊C₄	1	4	yes	—
94	C₈×D₄₃	C₂×D₄	1	4	yes	—
95	C₈×D₄₃	C₂×Q₈	1	4	yes	—
96	C₁₆×C₄₃	—	2	4	no	—
97	C₄₃⋊C₁₆	—	2	4	no	—
98	C₂³×C₄₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂²×D₄₃	—	4	2	no	—
100	C₂²×D₄₃	C₄⋊C₈	1	2	yes	—
101	C₂²×D₄₃	C₄⋊C₈	1	2	yes	—
102	C₂²×D₄₃	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₄₃	—	2	2	no	—
112	C₂×Dic₄₃	—	4	2	no	—
113	C₂×Dic₄₃	C₄⋊C₈	1	2	yes	—
114	C₂×Dic₄₃	C₄⋊C₈	1	2	yes	—
115	C₂×Dic₄₃	C₂×M₁₆	1	2	yes	—
116	C₄×D₄₃	C₄⋊C₈	1	4	yes	—
117	C₄×D₄₃	C₄⋊C₈	1	4	yes	—
118	C₄×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₄×D₄₃	C₂²×C₈	1	4	yes	—
120	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₈×C₄₃	C₂²×C₈	1	4	yes	—
123	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
125	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
126	C₄₃⋊C₈	C₂²×C₈	1	4	yes	—
127	C₂×#2153	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	2	no	—
129	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂²×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
132	C₂×D₄₃	—	2	4	no	—
133	C₂×D₄₃	—	4	2	no	—
134	C₂×D₄₃	—	4	2	no	—
135	C₂×D₄₃	C₂×C₄⋊C₈	1	4	yes	—
136	C₄×C₄₃	—	4	2	no	—
137	C₄×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₄×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
139	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₄₃	C₂³×C₈	1	1	yes	characteristic
141	Dic₄₃	—	4	2	no	—
142	Dic₄₃	C₂×C₄⋊C₈	1	4	yes	—
143	C₂²×M₃₂	—	43	2	no	—
144	C₂×#405	—	43	1	no	—
145	#2153	—	43	8	no	—
146	C₂×#364	—	43	2	no	—
147	C₂×#364	—	43	1	no	—
148	C₂×#364	D₄₃	1	1	yes	characteristic, complemented
149	C₂×C₄₃	—	2	1	no	—
150	C₂×C₄₃	—	2	1	no	—
151	C₂×C₄₃	#2153	1	2	yes	complemented
152	C₂×C₄₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
153	D₄₃	—	2	4	no	—
154	C₂²×M₁₆	—	43	1	no	—
155	C₂³×C₈	—	43	1	no	—
156	C₂×#117	—	43	1	no	—
157	C₂×M₃₂	—	43	2	no	—
158	C₂×M₃₂	—	43	8	no	—
159	C₂×M₃₂	—	86	2	no	—
160	C₂×M₃₂	—	86	4	no	—
161	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
162	C₂²×C₁₆	—	86	2	no	—
163	#405	—	43	4	no	—
164	#405	—	43	4	no	—
165	C₂×#97	—	43	1	no	—
166	C₂×#97	—	43	1	no	—
167	C₂×C₄×C₈	—	43	1	no	—
168	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
169	#364	—	43	8	no	—
170	#364	—	43	4	no	—
171	#364	C₂×D₄₃	1	4	yes	complemented
172	C₄₃	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	43	1	no	—
174	C₂×M₁₆	—	43	4	no	—
175	C₂×M₁₆	—	43	4	no	—
176	C₂×M₁₆	—	43	4	no	—
177	C₂²×C₈	—	43	1	no	—
178	C₂²×C₈	—	43	1	no	—
179	C₂²×C₈	—	43	2	no	—
180	C₂²×C₈	—	43	2	no	—
181	C₂²×C₈	—	43	2	no	—
182	C₂²×C₈	—	43	2	no	—
183	C₂²×C₈	—	43	2	no	—
184	C₂²×C₈	—	43	2	no	—
185	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
187	#117	—	43	4	no	—
188	#117	—	86	2	no	—
189	C₂×C₄⋊C₄	—	86	1	no	—
190	C₂×C₂²⋊C₄	—	86	1	no	—
191	C₂×C₄²	—	43	1	no	—
192	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
193	M₃₂	—	2	2	no	—
194	M₃₂	—	43	4	no	—
195	M₃₂	—	86	8	no	—
196	M₃₂	—	86	2	no	—
197	M₃₂	C₂²×D₄₃	1	4	yes	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	86	8	no	—
200	C₂×C₁₆	—	86	2	no	—
201	#97	—	43	4	no	—
202	#97	—	43	4	no	—
203	C₄×C₈	—	43	2	no	—
204	C₄×C₈	—	43	2	no	—
205	C₄×C₈	C₄×D₄₃	1	2	yes	—
206	C₄×C₈	C₂²×D₄₃	1	2	yes	—
207	C₂⁴	—	43	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	43	1	no	—
210	C₂²×C₄	—	43	2	no	—
211	C₂²×C₄	—	43	2	no	—
212	C₂²×C₄	—	43	2	no	—
213	C₂²×C₄	—	43	2	no	—
214	C₂²×C₄	—	43	2	no	—
215	C₂²×C₄	—	43	2	no	—
216	C₂²×C₄	—	86	1	no	—
217	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
218	M₁₆	—	43	16	no	—
219	C₂×C₈	—	43	2	no	—
220	C₂×C₈	—	43	2	no	—
221	C₂×C₈	—	43	2	no	—
222	C₂×C₈	—	43	2	no	—
223	C₂×C₈	—	43	2	no	—
224	C₂×C₈	—	43	2	no	—
225	C₂×C₈	—	43	4	no	—
226	C₂×C₈	—	43	4	no	—
227	C₂×C₈	—	43	4	no	—
228	C₂×C₈	—	43	4	no	—
229	C₂×C₈	C₈×D₄₃	1	2	yes	—
230	C₂×C₈	C₈×D₄₃	1	2	yes	—
231	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
233	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
234	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
235	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
236	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
238	C₄⋊C₄	—	86	4	no	—
239	C₂²⋊C₄	—	86	4	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	43	2	no	—
242	C₄²	—	86	1	no	—
243	C₄²	C₂×C₄×D₄₃	1	2	yes	—
244	C₁₆	—	2	4	no	—
245	C₁₆	—	86	4	no	—
246	C₂³	—	43	2	no	—
247	C₂³	—	43	2	no	—
248	C₂³	—	43	2	no	—
249	C₂³	—	172	2	no	—
250	C₂³	M₁₆×D₄₃	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	43	2	no	—
254	C₂×C₄	—	43	2	no	—
255	C₂×C₄	—	43	2	no	—
256	C₂×C₄	—	43	4	no	—
257	C₂×C₄	—	43	4	no	—
258	C₂×C₄	—	43	4	no	—
259	C₂×C₄	—	43	4	no	—
260	C₂×C₄	—	86	2	no	—
261	C₂×C₄	—	172	2	no	—
262	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
263	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
266	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
268	C₈	—	43	4	no	—
269	C₈	—	43	4	no	—
270	C₈	C₄⋊C₄×D₄₃	1	1	yes	characteristic
271	C₈	C₄⋊C₄×D₄₃	1	1	yes	characteristic
272	C₈	C₂×C₈×D₄₃	1	4	yes	—
273	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
274	C₈	C₂×Q₈×D₄₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	43	4	no	—
277	C₂²	—	86	4	no	—
278	C₂²	—	172	2	no	—
279	C₂²	—	172	2	no	—
280	C₂²	C₄⋊C₈×D₄₃	1	1	yes	characteristic
281	C₂²	C₄⋊C₈×D₄₃	1	1	yes	characteristic
282	C₂²	C₂×M₁₆×D₄₃	1	1	yes	characteristic
283	C₄	—	4	2	no	—
284	C₄	—	43	4	no	—
285	C₄	—	172	2	no	—
286	C₄	C₄⋊C₈×D₄₃	1	1	yes	characteristic
287	C₄	C₄⋊C₈×D₄₃	1	1	yes	characteristic
288	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
289	C₄	C₂²×C₈×D₄₃	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	86	4	no	—
293	C₂	#19136359	1	2	yes	direct product factor
294	C₂	C₂×C₄⋊C₈×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₃
commutator	C₄×C₄₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#364	C₂
socle	C₂²×C₄₃	C₂×C₄⋊C₈
prime radical	C₂³×D₄₃	M₁₆
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#364	D₄₃
hypocenter	C₄₃	C₂×#2153
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #19136359	C₂	direct (N × H)
4. C₂×#14582720	C₂	semidirect
5. C₂×#14582719	C₂	semidirect
6. C₂×C₄₃×#364	C₂	semidirect
20. #14582720	C₂²	semidirect
21. #14582719	C₂²	semidirect
24. C₄₃×#364	C₂²	semidirect
148. C₂×#364	D₄₃	semidirect
151. C₂×C₄₃	#2153	semidirect
171. #364	C₂×D₄₃	semidirect
172. C₄₃	C₂×#2153	semidirect
293. C₂	#19136359	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×M₃₂×D₄₃	2
C₂×#14582853	2
#19136359	2
C₂×#14582720	2
C₂×#14582719	2
C₂×C₄₃×#364	2
C₂×#2153	43

Subgroups — C2×#19136359 #25249595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136359 #25249595