Subgroups — #25011713

Order 11008.

420 conjugacy classes of subgroups.

The subgroup lattice has 356 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	1	yes	maximal, characteristic
2	#19136205	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582676	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134372	C₂	1	1	yes	maximal, characteristic
5	#19132675	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
9	C₂×#13063265	C₂²	1	1	yes	characteristic
10	C₂×#13063265	C₂²	1	1	yes	characteristic
11	C₂×#13063264	C₄	1	2	yes	complemented
12	M₃₂×D₄₃	—	2	2	no	—
13	M₃₂×D₄₃	—	2	2	no	—
14	C₂×C₁₆×D₄₃	C₄	1	2	yes	complemented
15	C₂×(C₂×D₄₃)⋊C₈	C₄	1	1	yes	characteristic
16	C₂×#13063235	C₂²	1	1	yes	characteristic
17	C₄×(C₂×D₄₃)⋊C₄	C₄	1	1	yes	characteristic
18	(C₂×D₄₃)⋊C₁₆	C₄	1	2	yes	complemented
19	(C₂×D₄₃)⋊C₁₆	C₄	1	2	yes	complemented
20	#14582676	—	2	2	no	—
21	#14582654	C₄	1	1	yes	characteristic
22	#14582653	C₄	1	1	yes	characteristic
23	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#347	—	2	2	no	—
27	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₄₃	—	2	1	no	—
31	C₂×C₈×D₄₃	—	2	1	no	—
32	C₂×C₈×D₄₃	—	2	1	no	—
33	C₂×C₈×D₄₃	—	2	1	no	—
34	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
39	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
40	#13063265	—	2	1	no	—
41	#13063265	—	2	1	no	—
42	#13063265	—	2	1	no	—
43	#13063265	—	2	1	no	—
44	#13063264	—	2	4	no	—
45	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
47	C₁₆×D₄₃	—	2	4	no	—
48	(C₂×D₄₃)⋊C₈	—	2	1	no	—
49	(C₂×D₄₃)⋊C₈	—	2	1	no	—
50	#13063235	—	2	1	no	—
51	#13063235	—	2	1	no	—
52	C₄₃⋊#95	—	2	1	no	—
53	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₄₃	—	2	1	no	—
57	M₃₂×C₄₃	—	2	1	no	—
58	M₃₂×C₄₃	D₄	1	2	yes	—
59	M₃₂×C₄₃	D₄	1	2	yes	—
60	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
61	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
62	#97×C₄₃	—	2	1	no	—
63	#97×C₄₃	—	2	1	no	—
64	C₂³×D₄₃	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	—	2	1	no	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	1	no	—
68	C₂²×Dic₄₃	M₁₆	1	1	yes	characteristic
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	C₄²	1	1	yes	characteristic
74	C₂×C₄×D₄₃	C₄²	1	1	yes	characteristic
75	C₂×C₄×D₄₃	M₁₆	1	1	yes	characteristic
76	C₂×C₄×D₄₃	M₁₆	1	1	yes	characteristic
77	C₂×C₈×C₄₃	—	2	1	no	—
78	C₂×C₈×C₄₃	—	2	1	no	—
79	C₂×C₈×C₄₃	—	2	1	no	—
80	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
86	C₄²×C₄₃	—	2	1	no	—
87	C₄²×C₄₃	—	2	1	no	—
88	(C₂×D₄₃)⋊C₄	—	4	2	no	—
89	C₄×Dic₄₃	—	2	1	no	—
90	C₄×Dic₄₃	—	2	1	no	—
91	C₂×C₄₃⋊C₈	—	2	1	no	—
92	C₂×C₄₃⋊C₈	—	2	1	no	—
93	C₂×C₄₃⋊C₈	—	2	1	no	—
94	C₂×C₄₃⋊C₈	—	2	1	no	—
95	C₂×C₄₃⋊C₈	—	2	1	no	—
96	C₂×C₄₃⋊C₈	C₄²	1	1	yes	characteristic
97	C₂×C₄₃⋊C₈	C₄²	1	1	yes	characteristic
98	C₈×D₄₃	—	2	1	no	—
99	C₈×D₄₃	—	2	1	no	—
100	C₈×D₄₃	—	2	1	no	—
101	C₈×D₄₃	—	2	1	no	—
102	C₈×D₄₃	—	2	1	no	—
103	C₈×D₄₃	—	2	1	no	—
104	C₈×D₄₃	—	2	1	no	—
105	C₈×D₄₃	—	2	1	no	—
106	C₁₆×C₄₃	—	2	2	no	—
107	C₁₆×C₄₃	C₂²⋊C₄	1	4	yes	complemented
108	C₄₃⋊C₁₆	—	2	2	no	—
109	C₄₃⋊C₁₆	—	2	2	no	—
110	C₂³×C₄₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂²×D₄₃	—	2	1	no	—
112	C₂²×D₄₃	—	2	1	no	—
113	C₂²×D₄₃	—	2	2	no	—
114	C₂²×D₄₃	—	4	1	no	—
115	C₂²×D₄₃	#97	1	1	yes	characteristic
116	C₂²×D₄₃	#97	1	1	yes	characteristic
117	C₂×C₄×C₄₃	—	4	1	no	—
118	C₂×C₄×C₄₃	—	4	1	no	—
119	C₂×C₄×C₄₃	—	4	1	no	—
120	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	4	1	no	—
129	C₂×Dic₄₃	—	4	1	no	—
130	C₂×Dic₄₃	—	4	1	no	—
131	C₂×Dic₄₃	#97	1	1	yes	characteristic
132	C₂×Dic₄₃	#97	1	1	yes	characteristic
133	C₄×D₄₃	—	2	1	no	—
134	C₄×D₄₃	—	2	1	no	—
135	C₄×D₄₃	—	2	1	no	—
136	C₄×D₄₃	—	2	1	no	—
137	C₄×D₄₃	—	2	1	no	—
138	C₄×D₄₃	—	2	1	no	—
139	C₄×D₄₃	—	2	1	no	—
140	C₄×D₄₃	—	2	1	no	—
141	C₈×C₄₃	—	4	1	no	—
142	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₈×C₄₃	#117	1	1	yes	characteristic
144	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
145	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
146	C₄₃⋊C₈	—	2	1	no	—
147	C₄₃⋊C₈	—	2	1	no	—
148	C₄₃⋊C₈	—	4	1	no	—
149	#12616	—	43	1	no	maximal, Sylow
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	—	2	1	no	—
152	C₂²×C₄₃	C₂×#97	1	1	yes	characteristic
153	C₂²×C₄₃	#435	1	1	yes	characteristic
154	C₂²×C₄₃	#435	1	1	yes	characteristic
155	C₂×D₄₃	—	2	1	no	—
156	C₂×D₄₃	—	2	1	no	—
157	C₂×D₄₃	—	2	1	no	—
158	C₂×D₄₃	—	2	1	no	—
159	C₂×D₄₃	—	2	1	no	—
160	C₂×D₄₃	—	4	2	no	—
161	C₂×D₄₃	—	4	2	no	—
162	C₂×D₄₃	#347	1	2	yes	complemented
163	C₄×C₄₃	—	8	1	no	—
164	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
165	C₄×C₄₃	#405	1	1	yes	characteristic
166	C₄×C₄₃	#435	1	1	yes	characteristic
167	C₄×C₄₃	#435	1	1	yes	characteristic
168	Dic₄₃	—	2	1	no	—
169	Dic₄₃	—	2	1	no	—
170	Dic₄₃	—	8	1	no	—
171	C₂²×M₃₂	—	43	1	no	—
172	#2112	—	43	1	no	—
173	C₂×#347	—	43	1	no	—
174	C₂×#347	D₄₃	1	1	yes	characteristic, complemented
175	#1752	—	43	1	no	—
176	#1375	—	43	1	no	—
177	#1375	—	43	1	no	—
178	C₂×C₄₃	—	2	1	no	—
179	C₂×C₄₃	—	2	1	no	—
180	C₂×C₄₃	#1752	1	1	yes	characteristic
181	C₂×C₄₃	C₂×#347	1	1	yes	characteristic
182	C₂×C₄₃	#2110	1	1	yes	characteristic
183	D₄₃	—	2	2	no	—
184	D₄₃	—	4	1	no	—
185	C₂³×C₈	—	43	1	no	—
186	C₂×M₃₂	—	43	1	no	—
187	C₂×M₃₂	—	43	1	no	—
188	C₂×M₃₂	—	43	2	no	—
189	C₂×M₃₂	—	86	2	no	—
190	C₂×M₃₂	—	86	2	no	—
191	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
192	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
193	C₂²×C₁₆	—	43	2	no	—
194	C₂×C₂²⋊C₈	—	43	1	no	—
195	C₂×#97	—	43	1	no	—
196	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
197	C₄×C₂²⋊C₄	—	43	1	no	—
198	C₂²⋊C₁₆	—	43	2	no	—
199	C₂²⋊C₁₆	—	43	2	no	—
200	#347	—	2	2	no	—
201	#347	—	86	2	no	—
202	#336	—	43	1	no	—
203	#336	—	43	1	no	—
204	C₄₃	#12616	1	1	yes	characteristic, complemented, Sylow
205	C₂³×C₄	—	43	1	no	—
206	C₂²×C₈	—	43	1	no	—
207	C₂²×C₈	—	43	1	no	—
208	C₂²×C₈	—	43	1	no	—
209	C₂²×C₈	—	43	1	no	—
210	C₂²×C₈	—	43	1	no	—
211	C₂²×C₈	—	43	1	no	—
212	C₂²×C₈	—	86	1	no	—
213	C₂²×C₈	—	86	1	no	—
214	C₂²×C₈	—	86	1	no	—
215	C₂²×C₈	—	86	1	no	—
216	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	—	86	1	no	—
219	C₂×C₄²	—	43	1	no	—
220	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	86	1	no	—
224	M₃₂	—	86	1	no	—
225	M₃₂	—	86	1	no	—
226	M₃₂	—	86	1	no	—
227	M₃₂	—	86	4	no	—
228	M₃₂	D₁₇₂	1	2	yes	—
229	M₃₂	C₂²⋊D₄₃	1	2	yes	—
230	C₂×C₁₆	—	43	2	no	—
231	C₂×C₁₆	—	43	2	no	—
232	C₂×C₁₆	—	86	4	no	—
233	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
234	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
235	C₂²⋊C₈	—	86	1	no	—
236	C₂²⋊C₈	—	86	1	no	—
237	#97	—	2	1	no	—
238	#97	—	2	1	no	—
239	#97	—	86	1	no	—
240	#97	—	86	1	no	—
241	#95	—	86	1	no	—
242	C₂⁴	—	43	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	43	1	no	—
245	C₂²×C₄	—	43	1	no	—
246	C₂²×C₄	—	43	1	no	—
247	C₂²×C₄	—	43	1	no	—
248	C₂²×C₄	—	43	1	no	—
249	C₂²×C₄	—	86	1	no	—
250	C₂²×C₄	—	86	1	no	—
251	C₂²×C₄	—	86	1	no	—
252	C₂²×C₄	—	86	1	no	—
253	C₂²×C₄	—	86	1	no	—
254	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	43	1	no	—
259	C₂×C₈	—	43	1	no	—
260	C₂×C₈	—	86	1	no	—
261	C₂×C₈	—	86	1	no	—
262	C₂×C₈	—	86	1	no	—
263	C₂×C₈	—	86	1	no	—
264	C₂×C₈	—	86	1	no	—
265	C₂×C₈	—	86	1	no	—
266	C₂×C₈	—	86	1	no	—
267	C₂×C₈	—	86	1	no	—
268	C₂×C₈	—	86	1	no	—
269	C₂×C₈	—	86	1	no	—
270	C₂×C₈	—	86	1	no	—
271	C₂×C₈	—	86	1	no	—
272	C₂×C₈	—	86	1	no	—
273	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
275	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
276	C₂×C₈	#10621898	1	1	yes	characteristic
277	C₂×C₈	#10621898	1	1	yes	characteristic
278	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
279	C₂²⋊C₄	—	172	2	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	86	1	no	—
283	C₄²	—	86	1	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	86	2	no	—
286	C₁₆	—	86	2	no	—
287	C₁₆	(C₂×D₄₃)⋊C₄	1	4	yes	complemented
288	C₂³	—	43	1	no	—
289	C₂³	—	43	1	no	—
290	C₂³	—	86	2	no	—
291	C₂³	—	86	1	no	—
292	C₂³	—	86	1	no	—
293	C₂³	—	172	1	no	—
294	C₂³	M₁₆×D₄₃	1	1	yes	characteristic
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	43	1	no	—
299	C₂×C₄	—	43	1	no	—
300	C₂×C₄	—	86	1	no	—
301	C₂×C₄	—	86	1	no	—
302	C₂×C₄	—	86	1	no	—
303	C₂×C₄	—	86	1	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	1	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	86	1	no	—
308	C₂×C₄	—	86	1	no	—
309	C₂×C₄	—	86	1	no	—
310	C₂×C₄	—	172	1	no	—
311	C₂×C₄	—	172	1	no	—
312	C₂×C₄	—	172	1	no	—
313	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
314	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
315	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
316	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
317	C₂×C₄	#13063333	1	1	yes	characteristic
318	C₂×C₄	#13063333	1	1	yes	characteristic
319	C₈	—	4	1	no	—
320	C₈	—	86	1	no	—
321	C₈	—	86	1	no	—
322	C₈	—	172	1	no	—
323	C₈	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
324	C₈	#13063282	1	1	yes	characteristic
325	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
326	C₈	C₄×D₁₇₂	1	1	yes	characteristic
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	43	2	no	—
330	C₂²	—	86	1	no	—
331	C₂²	—	86	1	no	—
332	C₂²	—	86	1	no	—
333	C₂²	—	86	1	no	—
334	C₂²	—	86	1	no	—
335	C₂²	—	172	2	no	—
336	C₂²	—	172	2	no	—
337	C₂²	#97×D₄₃	1	1	yes	characteristic
338	C₂²	#14582968	1	1	yes	characteristic
339	C₂²	#14582972	1	1	yes	characteristic
340	C₄	—	8	1	no	—
341	C₄	—	86	1	no	—
342	C₄	—	86	1	no	—
343	C₄	—	344	1	no	—
344	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
345	C₄	#14582856	1	1	yes	characteristic
346	C₄	#14582968	1	1	yes	characteristic
347	C₄	#14582972	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	86	2	no	—
351	C₂	—	172	1	no	—
352	C₂	#19134372	1	1	yes	characteristic
353	C₂	#347×D₄₃	1	1	yes	characteristic
354	C₂	#19136198	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#347	C₂
socle	C₂²×C₄₃	C₂×#97
prime radical	C₂³×D₄₃	M₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#347	D₄₃
hypocenter	C₄₃	#12616
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. C₂×#14582676	C₂	semidirect
5. #19132675	C₂	semidirect
6. #19132674	C₂	semidirect
7. C₂×C₄₃×#347	C₂	semidirect
11. C₂×#13063264	C₄	semidirect
14. C₂×C₁₆×D₄₃	C₄	semidirect
18. (C₂×D₄₃)⋊C₁₆	C₄	semidirect
19. (C₂×D₄₃)⋊C₁₆	C₄	semidirect
107. C₁₆×C₄₃	C₂²⋊C₄	semidirect
162. C₂×D₄₃	#347	semidirect
174. C₂×#347	D₄₃	semidirect
204. C₄₃	#12616	semidirect
287. C₁₆	(C₂×D₄₃)⋊C₄	semidirect

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
#19136205	2
C₂×#14582676	2
#19134372	2
#19132675	2
#19132674	2
C₂×C₄₃×#347	2
#12616	43