Subgroups — C₄₃×#29778 #24949795

Order 11008.

838 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#2624	C₂	1	2	yes	maximal, complemented
4	C₄₃×#2087	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2067	C₂	1	4	yes	maximal, complemented
6	C₄₃×#2054	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1895	C₂	1	2	yes	maximal, complemented
8	C₄₃×#1808	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×#121×C₄₃	C₂²	1	2	yes	—
13	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#466	—	2	4	no	—
15	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₄₃	C₂²	1	2	yes	—
17	C₄₃×#424	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
20	C₂×#102×C₄₃	C₂²	1	2	yes	—
21	C₂×#102×C₄₃	C₂²	1	4	yes	—
22	C₄₃×#399	—	2	2	no	—
23	C₄₃×#397	—	2	2	no	—
24	C₄₃×#394	—	2	4	no	—
25	C₄₃×#390	C₂²	1	4	yes	—
26	C₄₃×#389	C₂²	1	4	yes	—
27	C₄₃×#380	C₂²	1	2	yes	—
28	C₄₃×#340	C₂²	1	2	yes	—
29	C₄₃×#340	C₂²	1	4	yes	—
30	C₂²×D₄×C₄₃	—	2	2	no	—
31	C₂²×D₄×C₄₃	C₂³	1	2	yes	—
32	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×D₈×C₄₃	—	2	2	no	—
34	C₂×D₈×C₄₃	—	4	2	no	—
35	C₂×M₁₆×C₄₃	—	2	2	no	—
36	C₂×M₁₆×C₄₃	—	4	2	no	—
37	C₂²×C₈×C₄₃	—	4	2	no	—
38	C₂²×C₈×C₄₃	D₄	1	4	yes	—
39	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
42	#121×C₄₃	—	2	4	no	—
43	#121×C₄₃	—	2	4	no	—
44	#117×C₄₃	—	2	2	no	—
45	#117×C₄₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
54	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
55	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
56	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
57	#107×C₄₃	—	2	2	no	—
58	#107×C₄₃	—	4	2	no	—
59	C₄⋊C₈×C₄₃	—	2	4	no	—
60	C₄⋊C₈×C₄₃	—	2	4	no	—
61	#102×C₄₃	—	2	4	no	—
62	#102×C₄₃	—	4	4	no	—
63	#95×C₄₃	—	2	2	no	—
64	#95×C₄₃	—	2	4	no	—
65	C₂⁴×C₄₃	—	2	2	no	—
66	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₄₃	—	2	2	no	—
68	C₂×D₄×C₄₃	—	2	2	no	—
69	C₂×D₄×C₄₃	—	2	4	no	—
70	C₂×D₄×C₄₃	—	2	4	no	—
71	C₂×D₄×C₄₃	—	4	4	no	—
72	C₂×D₄×C₄₃	—	8	4	no	—
73	C₂²×C₄×C₄₃	—	2	2	no	—
74	C₂²×C₄×C₄₃	—	2	2	no	—
75	C₂²×C₄×C₄₃	—	2	4	no	—
76	C₂²×C₄×C₄₃	—	2	2	no	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	#41	1	2	yes	—
82	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
83	D₈×C₄₃	—	4	4	no	—
84	M₁₆×C₄₃	—	4	4	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	—	2	2	no	—
88	C₂×C₈×C₄₃	—	2	4	no	—
89	C₂×C₈×C₄₃	—	4	4	no	—
90	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	2	2	no	—
94	C₄⋊C₄×C₄₃	—	2	2	no	—
95	C₄⋊C₄×C₄₃	—	2	2	no	—
96	C₄⋊C₄×C₄₃	—	2	4	no	—
97	C₄⋊C₄×C₄₃	—	2	4	no	—
98	C₄⋊C₄×C₄₃	—	8	4	no	—
99	C₂²⋊C₄×C₄₃	—	2	4	no	—
100	C₂²⋊C₄×C₄₃	—	4	4	no	—
101	C₂²⋊C₄×C₄₃	—	8	2	no	—
102	C₂²⋊C₄×C₄₃	—	8	4	no	—
103	C₂²⋊C₄×C₄₃	—	8	4	no	—
104	C₄²×C₄₃	—	2	2	no	—
105	C₄²×C₄₃	—	2	2	no	—
106	C₂³×C₄₃	—	2	2	no	—
107	C₂³×C₄₃	—	2	1	no	—
108	C₂³×C₄₃	—	2	2	no	—
109	C₂³×C₄₃	—	2	1	no	—
110	C₂³×C₄₃	—	4	2	no	—
111	C₂³×C₄₃	—	4	4	no	—
112	C₂³×C₄₃	—	4	2	no	—
113	C₂³×C₄₃	—	4	4	no	—
114	C₂³×C₄₃	—	4	2	no	—
115	C₂³×C₄₃	#121	1	2	yes	—
116	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	D₄×C₄₃	—	4	2	no	—
118	D₄×C₄₃	—	4	2	no	—
119	D₄×C₄₃	—	4	4	no	—
120	D₄×C₄₃	—	8	8	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	4	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	4	no	—
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	4	4	no	—
132	C₂×C₄×C₄₃	—	4	4	no	—
133	C₂×C₄×C₄₃	—	4	4	no	—
134	C₂×C₄×C₄₃	—	4	4	no	—
135	C₂×C₄×C₄₃	#121	1	2	yes	—
136	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
139	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
140	C₈×C₄₃	—	2	2	no	—
141	C₈×C₄₃	—	2	2	no	—
142	C₈×C₄₃	—	4	2	no	—
143	#29778	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	2	2	no	—
147	C₂²×C₄₃	—	4	2	no	—
148	C₂²×C₄₃	—	4	2	no	—
149	C₂²×C₄₃	—	8	2	no	—
150	C₂²×C₄₃	—	8	4	no	—
151	C₂²×C₄₃	—	8	4	no	—
152	C₂²×C₄₃	—	8	2	no	—
153	C₂²×C₄₃	—	8	2	no	—
154	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
155	C₂²×C₄₃	#537	1	2	yes	—
156	C₂²×C₄₃	#576	1	1	yes	characteristic
157	C₂²×C₄₃	#576	1	2	yes	—
158	C₂²×C₄₃	#578	1	1	yes	characteristic
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	4	2	no	—
161	C₄×C₄₃	—	4	2	no	—
162	C₄×C₄₃	—	8	4	no	—
163	C₄×C₄₃	—	8	2	no	—
164	C₄×C₄₃	#534	1	1	yes	characteristic
165	C₄×C₄₃	#536	1	1	yes	characteristic
166	C₄×C₄₃	#539	1	2	yes	—
167	C₂×#466	C₂×C₄₃	1	1	yes	characteristic
168	C₂×#424	C₂×C₄₃	1	1	yes	characteristic, complemented
169	#2624	C₂×C₄₃	1	2	yes	complemented
170	#2087	C₂×C₄₃	1	2	yes	complemented
171	#2067	C₂×C₄₃	1	4	yes	complemented
172	#2054	C₂×C₄₃	1	2	yes	complemented
173	#1895	C₂×C₄₃	1	2	yes	complemented
174	#1808	C₂×C₄₃	1	1	yes	characteristic, complemented
175	C₂×C₄₃	—	4	2	no	—
176	C₂×C₄₃	—	16	2	no	—
177	C₂×C₄₃	#2618	1	1	yes	characteristic
178	C₂×C₄₃	#3058	1	1	yes	characteristic
179	C₂×C₄₃	#3061	1	1	yes	characteristic
180	C₂×C₄₃	#3259	1	2	yes	—
181	C₂×C₄₃	#3262	1	2	yes	—
182	C₂²×D₈	C₂²×C₄₃	1	1	yes	characteristic
183	C₂²×M₁₆	C₂²×C₄₃	1	1	yes	characteristic
184	C₂³×C₈	C₂²×C₄₃	1	1	yes	characteristic
185	C₂×#121	C₂²×C₄₃	1	2	yes	—
186	C₂×#117	C₂²×C₄₃	1	1	yes	characteristic
187	#466	—	2	4	no	—
188	C₂×#107	C₂²×C₄₃	1	1	yes	characteristic
189	C₂×#107	C₂²×C₄₃	1	2	yes	—
190	#424	—	2	4	no	—
191	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
192	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
193	C₂×#102	C₂²×C₄₃	1	2	yes	—
194	C₂×#102	C₂²×C₄₃	1	4	yes	—
195	#399	—	2	2	no	—
196	#397	—	2	2	no	—
197	#394	—	2	4	no	—
198	#390	C₂²×C₄₃	1	4	yes	—
199	#389	C₂²×C₄₃	1	4	yes	—
200	#380	C₂²×C₄₃	1	2	yes	—
201	#340	C₂²×C₄₃	1	2	yes	—
202	#340	C₂²×C₄₃	1	4	yes	—
203	C₄₃	#29778	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²×D₄	—	2	2	no	—
205	C₂²×D₄	C₂³×C₄₃	1	2	yes	—
206	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
207	C₂×D₈	—	2	2	no	—
208	C₂×D₈	—	4	2	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	4	2	no	—
211	C₂²×C₈	—	4	2	no	—
212	C₂²×C₈	D₄×C₄₃	1	4	yes	—
213	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
214	C₂²×C₈	C₂³×C₄₃	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₄₃	1	1	yes	characteristic
216	#121	—	2	4	no	—
217	#121	—	2	4	no	—
218	#117	—	2	2	no	—
219	#117	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	4	no	—
221	C₂×C₄⋊C₄	D₄×C₄₃	1	1	yes	characteristic
222	C₂×C₄⋊C₄	D₄×C₄₃	1	1	yes	characteristic
223	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₄₃	1	4	yes	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	—	2	4	no	—
228	C₂×C₂²⋊C₄	—	2	4	no	—
229	C₂×C₂²⋊C₄	D₄×C₄₃	1	2	yes	—
230	C₂×C₄²	C₂³×C₄₃	1	2	yes	—
231	#107	—	2	2	no	—
232	#107	—	4	2	no	—
233	C₄⋊C₈	—	2	4	no	—
234	C₄⋊C₈	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	4	4	no	—
237	#95	—	2	2	no	—
238	#95	—	2	4	no	—
239	C₂⁴	—	2	2	no	—
240	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	8	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
255	C₂²×C₄	#41×C₄₃	1	2	yes	—
256	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
257	D₈	—	4	4	no	—
258	M₁₆	—	4	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	4	no	—
264	C₂×C₈	C₂×D₄×C₄₃	1	2	yes	—
265	C₂×C₈	C₂×D₄×C₄₃	1	2	yes	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	8	4	no	—
273	C₂²⋊C₄	—	2	4	no	—
274	C₂²⋊C₄	—	4	4	no	—
275	C₂²⋊C₄	—	8	4	no	—
276	C₂²⋊C₄	—	8	4	no	—
277	C₂²⋊C₄	—	8	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	4	4	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	4	4	no	—
288	C₂³	—	4	2	no	—
289	C₂³	#121×C₄₃	1	2	yes	—
290	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	4	no	—
294	D₄	—	8	8	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	#121×C₄₃	1	2	yes	—
310	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
311	C₂×C₄	C₂×#41×C₄₃	1	1	yes	characteristic
312	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
313	C₂×C₄	2₋¹⁺⁴×C₄₃	1	2	yes	—
314	C₈	—	2	2	no	—
315	C₈	—	2	2	no	—
316	C₈	—	4	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	4	no	—
324	C₂²	—	8	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	C₂×#121×C₄₃	1	1	yes	characteristic
328	C₂²	C₄₃×#537	1	2	yes	—
329	C₂²	C₄₃×#576	1	1	yes	characteristic
330	C₂²	C₄₃×#576	1	2	yes	—
331	C₂²	C₄₃×#578	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	4	no	—
336	C₄	—	8	2	no	—
337	C₄	C₄₃×#534	1	1	yes	characteristic
338	C₄	C₄₃×#536	1	1	yes	characteristic
339	C₄	C₄₃×#539	1	2	yes	—
340	C₂	—	4	2	no	—
341	C₂	—	16	2	no	—
342	C₂	C₄₃×#2618	1	1	yes	characteristic
343	C₂	C₄₃×#3058	1	1	yes	characteristic
344	C₂	C₄₃×#3061	1	1	yes	characteristic
345	C₂	C₄₃×#3259	1	2	yes	—
346	C₂	C₄₃×#3262	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
prime radical	C₂×C₄₃×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₃ ◁ C₂×#117×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₃×#424	C₂	semidirect
3. C₄₃×#2624	C₂	semidirect
4. C₄₃×#2087	C₂	semidirect
5. C₄₃×#2067	C₂	semidirect
6. C₄₃×#2054	C₂	semidirect
7. C₄₃×#1895	C₂	semidirect
8. C₄₃×#1808	C₂	semidirect
143. #29778	C₄₃	direct (N × H)
168. C₂×#424	C₂×C₄₃	semidirect
169. #2624	C₂×C₄₃	semidirect
170. #2087	C₂×C₄₃	semidirect
171. #2067	C₂×C₄₃	semidirect
172. #2054	C₂×C₄₃	semidirect
173. #1895	C₂×C₄₃	semidirect
174. #1808	C₂×C₄₃	semidirect
203. C₄₃	#29778	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×C₄₃×#466	2
C₂×C₄₃×#424	2
C₄₃×#2624	2
C₄₃×#2087	2
C₄₃×#2067	2
C₄₃×#2054	2
C₄₃×#1895	2
C₄₃×#1808	2
#29778	43

Subgroups — C43×#29778 #24949795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#29778 #24949795