Subgroups — C₄₃×#2415 #19131232

Order 5504.

604 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#117×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#397	C₂	1	2	yes	maximal, complemented
6	C₄₃×#394	C₂	1	2	yes	maximal
7	C₄₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#389	C₂	1	2	yes	maximal, complemented
9	C₄₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₃×#388	C₂	1	2	yes	maximal, complemented
11	C₄₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈×C₄₃	—	2	4	no	—
17	C₂²⋊Q₈×C₄₃	—	2	4	no	—
18	C₄×D₄×C₄₃	—	2	4	no	—
19	#117×C₄₃	—	2	4	no	—
20	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
33	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
34	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
35	#95×C₄₃	C₂²	1	2	yes	—
36	#95×C₄₃	C₂²	1	2	yes	—
37	#95×C₄₃	C₂²	1	2	yes	—
38	#95×C₄₃	C₂²	1	2	yes	—
39	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
40	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₄₃	—	2	2	no	—
42	C₂×Q₈×C₄₃	—	2	2	no	—
43	C₂×Q₈×C₄₃	—	2	2	no	—
44	C₂×D₄×C₄₃	—	2	2	no	—
45	C₂×D₄×C₄₃	—	2	2	no	—
46	C₂×D₄×C₄₃	—	2	2	no	—
47	C₂²×C₄×C₄₃	—	2	2	no	—
48	C₂²×C₄×C₄₃	—	2	2	no	—
49	C₂²×C₄×C₄₃	—	2	2	no	—
50	C₂²×C₄×C₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	—	2	2	no	—
52	C₂²×C₄×C₄₃	D₄	1	4	yes	—
53	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₄₃	—	2	2	no	—
63	C₄⋊C₄×C₄₃	—	2	2	no	—
64	C₄⋊C₄×C₄₃	—	2	2	no	—
65	C₄⋊C₄×C₄₃	—	2	4	no	—
66	C₄⋊C₄×C₄₃	—	2	4	no	—
67	C₄⋊C₄×C₄₃	—	2	4	no	—
68	C₄⋊C₄×C₄₃	D₄	1	4	yes	complemented
69	C₂²⋊C₄×C₄₃	—	2	4	no	—
70	C₂²⋊C₄×C₄₃	—	2	2	no	—
71	C₂²⋊C₄×C₄₃	—	2	4	no	—
72	C₂²⋊C₄×C₄₃	—	2	2	no	—
73	C₂²⋊C₄×C₄₃	—	2	4	no	—
74	C₂²⋊C₄×C₄₃	—	2	4	no	—
75	C₄²×C₄₃	—	2	4	no	—
76	C₄²×C₄₃	—	2	2	no	—
77	C₂³×C₄₃	—	2	1	no	—
78	C₂³×C₄₃	—	2	1	no	—
79	C₂³×C₄₃	—	2	1	no	—
80	C₂³×C₄₃	—	2	1	no	—
81	C₂³×C₄₃	—	2	1	no	—
82	C₂³×C₄₃	—	2	1	no	—
83	C₂³×C₄₃	—	2	1	no	—
84	C₂³×C₄₃	—	2	1	no	—
85	C₂³×C₄₃	—	2	1	no	—
86	C₂³×C₄₃	—	2	1	no	—
87	C₂³×C₄₃	—	2	1	no	—
88	C₂³×C₄₃	—	2	1	no	—
89	C₂³×C₄₃	C₂×D₄	1	2	yes	—
90	C₂³×C₄₃	#41	1	2	yes	—
91	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
92	Q₈×C₄₃	—	4	4	no	—
93	D₄×C₄₃	—	2	8	no	—
94	C₂×C₄×C₄₃	—	2	1	no	—
95	C₂×C₄×C₄₃	—	2	1	no	—
96	C₂×C₄×C₄₃	—	2	2	no	—
97	C₂×C₄×C₄₃	—	2	1	no	—
98	C₂×C₄×C₄₃	—	2	1	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	8	no	—
101	C₂×C₄×C₄₃	—	2	1	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	4	4	no	—
117	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
118	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₄₃	#41	1	2	yes	—
122	C₂×C₄×C₄₃	#41	1	2	yes	—
123	C₂²×C₄₃	—	2	2	no	—
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	—	2	2	no	—
127	C₂²×C₄₃	—	2	2	no	—
128	C₂²×C₄₃	—	2	2	no	—
129	C₂²×C₄₃	—	4	1	no	—
130	C₂²×C₄₃	—	4	1	no	—
131	C₂²×C₄₃	—	4	1	no	—
132	C₂²×C₄₃	—	4	1	no	—
133	C₂²×C₄₃	—	4	1	no	—
134	C₂²×C₄₃	—	4	1	no	—
135	C₂²×C₄₃	—	4	1	no	—
136	C₂²×C₄₃	—	4	1	no	—
137	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
140	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
141	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
142	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₄₃	—	2	2	no	—
145	C₄×C₄₃	—	2	2	no	—
146	C₄×C₄₃	—	2	4	no	—
147	C₄×C₄₃	—	4	1	no	—
148	C₄×C₄₃	—	4	2	no	—
149	C₄×C₄₃	—	4	2	no	—
150	C₄×C₄₃	—	4	1	no	—
151	C₄×C₄₃	—	4	2	no	—
152	C₄×C₄₃	#121	1	4	yes	—
153	#2415	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₄₃	—	4	2	no	—
155	C₂×C₄₃	—	4	2	no	—
156	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
157	C₂×C₄₃	#525	1	1	yes	characteristic
158	C₂×C₄₃	#536	1	1	yes	characteristic
159	C₂×C₄₃	#547	1	1	yes	characteristic
160	C₂×C₄₃	#548	1	1	yes	characteristic
161	C₂×C₄₃	#552	1	1	yes	characteristic
162	C₂×C₄₃	#556	1	1	yes	characteristic
163	C₂×C₂²⋊Q₈	C₂×C₄₃	1	1	yes	characteristic, complemented
164	C₂×C₂²⋊Q₈	C₂×C₄₃	1	1	yes	characteristic, complemented
165	C₂×C₄×D₄	C₂×C₄₃	1	1	yes	characteristic
166	C₂×#117	C₂×C₄₃	1	1	yes	characteristic, complemented
167	#397	C₂×C₄₃	1	2	yes	complemented
168	#394	C₂×C₄₃	1	2	yes	—
169	#391	C₂×C₄₃	1	1	yes	characteristic, complemented
170	#389	C₂×C₄₃	1	2	yes	complemented
171	#389	C₂×C₄₃	1	1	yes	characteristic, complemented
172	#388	C₂×C₄₃	1	2	yes	complemented
173	#380	C₂×C₄₃	1	1	yes	characteristic, complemented
174	C₄₃	#2415	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²×Q₈	C₂²×C₄₃	1	1	yes	characteristic
176	C₂²×D₄	C₂²×C₄₃	1	1	yes	characteristic
177	C₂³×C₄	C₂²×C₄₃	1	1	yes	characteristic
178	C₂³×C₄	C₂²×C₄₃	1	1	yes	characteristic
179	C₂²⋊Q₈	—	2	4	no	—
180	C₂²⋊Q₈	—	2	4	no	—
181	C₄×D₄	—	2	4	no	—
182	#117	—	2	4	no	—
183	C₂×C₄⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
187	C₂×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
188	C₂×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
189	C₂×C₄⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	C₂²×C₄₃	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	C₂²×C₄₃	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂²×C₄₃	1	2	yes	—
195	C₂×C₂²⋊C₄	C₂²×C₄₃	1	2	yes	—
196	C₂×C₄²	C₂²×C₄₃	1	2	yes	—
197	C₂×C₄²	C₂²×C₄₃	1	1	yes	characteristic
198	#95	C₂²×C₄₃	1	2	yes	—
199	#95	C₂²×C₄₃	1	2	yes	—
200	#95	C₂²×C₄₃	1	2	yes	—
201	#95	C₂²×C₄₃	1	2	yes	—
202	C₂⁴	C₂³×C₄₃	1	1	yes	characteristic
203	C₂⁴	C₂³×C₄₃	1	1	yes	characteristic
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	D₄×C₄₃	1	4	yes	—
216	C₂²×C₄	C₂³×C₄₃	1	1	yes	characteristic
217	C₂²×C₄	C₂³×C₄₃	1	2	yes	—
218	C₂²×C₄	C₂³×C₄₃	1	2	yes	—
219	C₂²×C₄	C₂³×C₄₃	1	2	yes	—
220	C₂²×C₄	C₂³×C₄₃	1	1	yes	characteristic
221	C₂²×C₄	C₂³×C₄₃	1	2	yes	—
222	C₂²×C₄	C₂³×C₄₃	1	1	yes	characteristic
223	C₂²×C₄	C₂³×C₄₃	1	1	yes	characteristic
224	C₂²×C₄	C₂³×C₄₃	1	1	yes	characteristic
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	D₄×C₄₃	1	4	yes	complemented
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	2	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	C₂×D₄×C₄₃	1	2	yes	—
253	C₂³	#41×C₄₃	1	2	yes	—
254	C₂³	C₂⁴×C₄₃	1	1	yes	characteristic
255	Q₈	—	4	4	no	—
256	D₄	—	2	8	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×D₄×C₄₃	1	4	yes	—
281	C₂×C₄	C₂×D₄×C₄₃	1	2	yes	—
282	C₂×C₄	C₂×D₄×C₄₃	1	2	yes	—
283	C₂×C₄	C₂×D₄×C₄₃	1	2	yes	—
284	C₂×C₄	#41×C₄₃	1	2	yes	—
285	C₂×C₄	#41×C₄₃	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	C₂²×D₄×C₄₃	1	1	yes	characteristic
301	C₂²	C₂²×D₄×C₄₃	1	1	yes	characteristic
302	C₂²	C₂×#41×C₄₃	1	1	yes	characteristic
303	C₂²	C₂×#41×C₄₃	1	1	yes	characteristic
304	C₂²	C₂×#41×C₄₃	1	1	yes	characteristic
305	C₂²	2₋¹⁺⁴×C₄₃	1	1	yes	characteristic
306	C₂²	2₋¹⁺⁴×C₄₃	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	4	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	#121×C₄₃	1	4	yes	—
316	C₂	—	4	2	no	—
317	C₂	—	4	2	no	—
318	C₂	C₂×#121×C₄₃	1	1	yes	characteristic
319	C₂	C₄₃×#525	1	1	yes	characteristic
320	C₂	C₄₃×#536	1	1	yes	characteristic
321	C₂	C₄₃×#547	1	1	yes	characteristic
322	C₂	C₄₃×#548	1	1	yes	characteristic
323	C₂	C₄₃×#552	1	1	yes	characteristic
324	C₂	C₄₃×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₄₃
prime radical	C₂²×D₄×C₄₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊Q₈×C₄₃	C₂	semidirect
2. C₂×C₂²⋊Q₈×C₄₃	C₂	semidirect
4. C₂×#117×C₄₃	C₂	semidirect
5. C₄₃×#397	C₂	semidirect
7. C₄₃×#391	C₂	semidirect
8. C₄₃×#389	C₂	semidirect
9. C₄₃×#389	C₂	semidirect
10. C₄₃×#388	C₂	semidirect
11. C₄₃×#380	C₂	semidirect
68. C₄⋊C₄×C₄₃	D₄	semidirect
153. #2415	C₄₃	direct (N × H)
163. C₂×C₂²⋊Q₈	C₂×C₄₃	semidirect
164. C₂×C₂²⋊Q₈	C₂×C₄₃	semidirect
166. C₂×#117	C₂×C₄₃	semidirect
167. #397	C₂×C₄₃	semidirect
169. #391	C₂×C₄₃	semidirect
170. #389	C₂×C₄₃	semidirect
171. #389	C₂×C₄₃	semidirect
172. #388	C₂×C₄₃	semidirect
173. #380	C₂×C₄₃	semidirect
174. C₄₃	#2415	direct (N × H)
231. C₄⋊C₄	D₄×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₄₃	2
C₂×C₄×D₄×C₄₃	2
C₂×#117×C₄₃	2
C₄₃×#397	2
C₄₃×#394	2
C₄₃×#391	2
C₄₃×#389	2
C₄₃×#388	2
C₄₃×#380	2
#2415	43

Subgroups — C43×#2415 #19131232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#2415 #19131232