Subgroups — #25104758

Order 11008.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144918	C₂	1	1	yes	maximal, characteristic
2	#19143532	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143312	C₂	1	1	yes	maximal, characteristic
4	#19135575	C₂	1	1	yes	maximal, characteristic, complemented
5	#19135426	C₂	1	1	yes	maximal, characteristic
6	#19135304	C₂	1	2	yes	maximal, complemented
7	#19135297	C₂	1	2	yes	maximal, complemented
8	#19134438	C₂	1	2	yes	maximal, complemented
9	#19134431	C₂	1	2	yes	maximal, complemented
10	#19134281	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063333	C₂²	1	2	yes	—
13	#14583456	C₂²	1	1	yes	characteristic
14	#14583449	C₂²	1	1	yes	characteristic
15	#14583254	C₄	1	4	yes	complemented
16	#14583250	C₄	1	2	yes	complemented
17	#14583246	C₄	1	2	yes	complemented
18	#14583028	—	2	4	no	—
19	#14582969	—	2	4	no	—
20	#14582927	C₂²	1	2	yes	—
21	C₂×#13063251	C₂²	1	2	yes	—
22	C₂×#13063251	C₂²	1	2	yes	—
23	#14582896	C₂²	1	2	yes	—
24	#14582896	C₂²	1	2	yes	—
25	#14582867	C₂²	1	2	yes	—
26	#14582857	C₂²	1	2	yes	—
27	C₂×#13063236	C₂²	1	1	yes	characteristic
28	C₂×#13063235	C₂²	1	1	yes	characteristic
29	C₂×#13063235	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
32	#14582745	C₂²	1	1	yes	characteristic
33	#14582660	C₂²	1	2	yes	—
34	#14582660	C₂²	1	2	yes	—
35	C₄₃×#423	C₂²	1	1	yes	characteristic
36	C₄₃×#423	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#341	C₂²	1	2	yes	—
39	C₄₃×#341	C₂²	1	2	yes	—
40	C₂×#10621898	C₂³	1	2	yes	—
41	#13063333	—	2	4	no	—
42	#13063333	—	2	4	no	—
43	C₂×#10621872	—	2	2	no	—
44	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₈×D₄₃	—	2	2	no	—
47	#13063325	C₂×C₄	1	1	yes	characteristic
48	#13063325	C₂×C₄	1	2	yes	—
49	#13063322	C₂×C₄	1	1	yes	characteristic
50	#13063317	C₂×C₄	1	2	yes	—
51	#13063313	C₂×C₄	1	2	yes	—
52	#13063304	—	2	2	no	—
53	#13063299	—	2	2	no	—
54	C₄×Dic₈₆	—	2	2	no	—
55	C₄×D₁₇₂	—	2	2	no	—
56	#13063284	C₂³	1	2	yes	—
57	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
59	Dic₄₃⋊C₈	—	2	4	no	—
60	#13063251	—	2	4	no	—
61	#13063251	—	2	4	no	—
62	#13063250	C₂×C₄	1	4	yes	—
63	#13063250	C₂×C₄	1	4	yes	—
64	#13063247	C₂×C₄	1	4	yes	—
65	#13063247	C₂×C₄	1	4	yes	—
66	(C₂×D₄₃)⋊C₈	—	2	4	no	—
67	#13063236	D₄	1	4	yes	—
68	#13063235	—	2	2	no	—
69	#13063235	—	2	2	no	—
70	#13063235	—	2	2	no	—
71	C₈×Dic₄₃	—	2	2	no	—
72	C₈×Dic₄₃	—	2	2	no	—
73	C₈×Dic₄₃	—	2	2	no	—
74	C₄₃⋊#95	—	2	2	no	—
75	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
76	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₄₃	—	2	2	no	—
81	C₄⋊C₈×C₄₃	—	2	2	no	—
82	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
84	C₂²×C₄×C₄₃	#41	1	2	yes	—
85	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂×C₂²⋊D₄₃	—	2	2	no	—
87	C₂²×Dic₄₃	—	2	1	no	—
88	C₂²×Dic₄₃	—	2	1	no	—
89	#10621898	—	2	2	no	—
90	#10621898	—	2	2	no	—
91	#10621898	—	2	2	no	—
92	#10621898	—	2	2	no	—
93	C₂×D₁₇₂	—	2	2	no	—
94	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₄₃	—	2	2	no	—
96	C₂×Dic₈₆	—	2	2	no	—
97	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
98	M₁₆×C₄₃	—	2	2	no	—
99	M₁₆×C₄₃	—	2	2	no	—
100	M₁₆×C₄₃	—	2	2	no	—
101	M₁₆×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	1	no	—
103	C₂×C₈×C₄₃	—	2	1	no	—
104	C₂×C₈×C₄₃	—	2	1	no	—
105	C₂×C₈×C₄₃	—	2	1	no	—
106	C₂×C₈×C₄₃	—	2	2	no	—
107	C₂×C₈×C₄₃	—	2	2	no	—
108	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
110	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₄₃	C₂×D₄	1	2	yes	—
113	C₂²⋊Dic₄₃	—	2	2	no	—
114	(C₂×D₄₃)⋊C₄	—	2	4	no	—
115	C₄⋊Dic₄₃	—	2	2	no	—
116	C₄⋊Dic₄₃	—	2	2	no	—
117	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
118	Dic₄₃⋊C₄	—	2	4	no	—
119	C₄×Dic₄₃	—	2	2	no	—
120	C₄×Dic₄₃	—	2	1	no	—
121	C₄×Dic₄₃	—	2	1	no	—
122	C₄×Dic₄₃	#41	1	2	yes	—
123	C₄×Dic₄₃	#41	1	2	yes	—
124	C₂×C₄₃⋊C₈	—	2	1	no	—
125	C₂×C₄₃⋊C₈	—	2	1	no	—
126	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
127	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
128	C₂×C₄₃⋊C₈	#41	1	2	yes	—
129	C₂×C₄₃⋊C₈	#41	1	2	yes	—
130	#10621872	—	4	4	no	—
131	C₈×D₄₃	—	4	4	no	—
132	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
133	C₂²×D₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
143	C₂²⋊D₄₃	—	4	2	no	—
144	C₂²⋊D₄₃	—	4	2	no	—
145	C₂×Dic₄₃	—	2	4	no	—
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	4	2	no	—
151	D₁₇₂	—	2	2	no	—
152	D₁₇₂	—	2	2	no	—
153	D₁₇₂	—	8	2	no	—
154	C₄×D₄₃	—	4	2	no	—
155	C₄×D₄₃	—	4	2	no	—
156	Dic₈₆	—	2	2	no	—
157	Dic₈₆	—	2	2	no	—
158	Dic₈₆	—	8	2	no	—
159	C₈×C₄₃	—	2	2	no	—
160	C₈×C₄₃	—	2	2	no	—
161	C₈×C₄₃	—	4	2	no	—
162	C₈×C₄₃	—	4	2	no	—
163	C₄₃⋊C₈	—	2	2	no	—
164	C₄₃⋊C₈	#124	1	4	yes	—
165	#21659	—	43	1	no	maximal, Sylow
166	C₂²×C₄₃	—	2	2	no	—
167	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₄₃	#529	1	1	yes	characteristic
169	C₂²×C₄₃	#529	1	1	yes	characteristic
170	C₂×D₄₃	—	4	4	no	—
171	C₂×D₄₃	—	4	2	no	—
172	C₄×C₄₃	—	2	2	no	—
173	C₄×C₄₃	—	2	2	no	—
174	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
175	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
176	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
177	C₄×C₄₃	#532	1	1	yes	characteristic
178	Dic₄₃	—	4	2	no	—
179	Dic₄₃	—	4	2	no	—
180	Dic₄₃	—	4	4	no	—
181	Dic₄₃	—	4	2	no	—
182	#3131	—	43	1	no	—
183	#2961	—	43	1	no	—
184	#2932	—	43	1	no	—
185	#1995	—	43	1	no	—
186	#1963	—	43	1	no	—
187	#1938	—	43	2	no	—
188	#1937	—	43	2	no	—
189	#1837	D₄₃	1	1	yes	characteristic, complemented
190	#1765	—	43	2	no	—
191	#1764	—	43	2	no	—
192	#1726	—	43	1	no	—
193	C₂×C₄₃	—	2	1	no	—
194	C₂×C₄₃	—	2	1	no	—
195	C₂×C₄₃	C₄×#124	1	1	yes	characteristic
196	C₂×C₄₃	#2957	1	2	yes	—
197	D₄₃	—	8	2	no	—
198	C₂×#131	—	43	2	no	—
199	#533	—	43	1	no	—
200	#532	—	43	1	no	—
201	#488	—	43	4	no	—
202	#487	—	43	2	no	—
203	#486	—	43	2	no	—
204	#446	—	86	4	no	—
205	#435	—	86	4	no	—
206	#424	—	43	2	no	—
207	#423	C₂×D₄₃	1	1	yes	characteristic
208	#423	C₂×D₄₃	1	1	yes	characteristic
209	C₂×#104	—	43	2	no	—
210	C₂×#104	—	43	2	no	—
211	#416	—	43	2	no	—
212	#416	—	43	2	no	—
213	#408	—	43	2	no	—
214	#405	—	43	2	no	—
215	C₂×#97	—	43	1	no	—
216	C₂×#97	—	43	1	no	—
217	C₂×#97	—	43	1	no	—
218	C₂×C₄×C₈	—	43	1	no	—
219	C₂×C₄×C₈	—	43	1	no	—
220	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
221	#376	—	43	1	no	—
222	#341	C₂×D₄₃	1	2	yes	—
223	#341	C₂×D₄₃	1	2	yes	—
224	#339	—	43	2	no	—
225	#339	—	43	2	no	—
226	C₄₃	#21659	1	1	yes	characteristic, complemented, Sylow
227	C₂×#41	—	43	2	no	—
228	#131	—	86	4	no	—
229	#131	—	86	4	no	—
230	C₂×M₁₆	—	86	2	no	—
231	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
232	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
233	C₂²×C₈	—	43	1	no	—
234	C₂²×C₈	—	43	1	no	—
235	C₂²×C₈	—	86	2	no	—
236	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
238	C₄⋊Q₈	—	43	2	no	—
239	C₄⋊Q₈	—	43	1	no	—
240	#127	—	43	1	no	—
241	#125	—	43	2	no	—
242	#124	—	43	2	no	—
243	C₂²⋊Q₈	—	86	2	no	—
244	#121	—	86	2	no	—
245	C₄×Q₈	—	86	2	no	—
246	C₄×D₄	—	86	2	no	—
247	#117	—	43	2	no	—
248	C₂×C₄²	—	43	1	no	—
249	C₂×C₄²	—	43	1	no	—
250	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	86	4	no	—
254	#104	—	86	4	no	—
255	#104	—	86	4	no	—
256	#103	—	43	4	no	—
257	#103	—	43	4	no	—
258	#102	—	43	4	no	—
259	#102	—	43	4	no	—
260	C₂²⋊C₈	—	86	4	no	—
261	#97	—	43	4	no	—
262	#97	—	86	2	no	—
263	#97	—	86	2	no	—
264	#97	—	86	2	no	—
265	C₄×C₈	—	86	2	no	—
266	C₄×C₈	—	86	2	no	—
267	C₄×C₈	—	86	2	no	—
268	C₄×C₈	C₄×D₄₃	1	2	yes	—
269	C₄×C₈	C₄×D₄₃	1	2	yes	—
270	#95	—	86	2	no	—
271	#41	—	86	2	no	—
272	#41	—	86	2	no	—
273	#41	—	86	2	no	—
274	#41	—	86	2	no	—
275	C₂×Q₈	—	43	2	no	—
276	C₂×Q₈	—	86	2	no	—
277	C₂×D₄	—	43	2	no	—
278	C₂×D₄	—	86	2	no	—
279	C₂×D₄	—	86	2	no	—
280	C₂²×C₄	—	86	1	no	—
281	C₂²×C₄	—	86	1	no	—
282	C₂²×C₄	—	86	2	no	—
283	C₂²×C₄	#10621902	1	2	yes	—
284	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	172	4	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	43	2	no	—
297	C₂×C₈	—	43	2	no	—
298	C₂×C₈	—	43	2	no	—
299	C₂×C₈	—	43	2	no	—
300	C₂×C₈	—	86	1	no	—
301	C₂×C₈	—	86	1	no	—
302	C₂×C₈	—	172	4	no	—
303	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
304	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
305	C₄⋊C₄	—	43	4	no	—
306	C₄⋊C₄	—	86	2	no	—
307	C₄⋊C₄	—	86	4	no	—
308	C₄⋊C₄	—	86	2	no	—
309	C₂²⋊C₄	—	86	2	no	—
310	C₂²⋊C₄	—	86	4	no	—
311	C₄²	—	43	2	no	—
312	C₄²	—	43	2	no	—
313	C₄²	—	86	2	no	—
314	C₄²	—	86	1	no	—
315	C₄²	—	86	1	no	—
316	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
318	C₄²	D₄×D₄₃	1	2	yes	—
319	C₂³	—	86	2	no	—
320	C₂³	C₂×#10621902	1	1	yes	characteristic
321	Q₈	—	86	2	no	—
322	Q₈	—	86	2	no	—
323	Q₈	—	344	2	no	—
324	D₄	—	86	2	no	—
325	D₄	—	86	2	no	—
326	D₄	—	344	2	no	—
327	D₄	—	172	2	no	—
328	D₄	—	172	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	86	4	no	—
331	C₂×C₄	—	86	2	no	—
332	C₂×C₄	—	86	2	no	—
333	C₂×C₄	—	86	2	no	—
334	C₂×C₄	—	172	2	no	—
335	C₂×C₄	—	172	2	no	—
336	C₂×C₄	—	172	2	no	—
337	C₂×C₄	—	172	2	no	—
338	C₂×C₄	#13063287	1	2	yes	—
339	C₂×C₄	#13063287	1	2	yes	—
340	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
342	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
343	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
344	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
345	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	43	4	no	—
351	C₈	—	86	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	172	4	no	—
354	C₂²	—	172	2	no	—
355	C₂²	C₂×#13063287	1	1	yes	characteristic
356	C₂²	#14583434	1	1	yes	characteristic
357	C₂²	#14583434	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	172	2	no	—
361	C₄	—	172	2	no	—
362	C₄	—	172	4	no	—
363	C₄	—	172	2	no	—
364	C₄	#14583367	1	1	yes	characteristic
365	C₄	#14583367	1	1	yes	characteristic
366	C₄	#124×D₄₃	1	1	yes	characteristic
367	C₄	#14583448	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	344	2	no	—
371	C₂	#19137134	1	1	yes	characteristic
372	C₂	#19143511	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1837	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#14583449	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1837	D₄₃
hypocenter	C₄₃	#21659
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
2. #19143532	C₂	semidirect
4. #19135575	C₂	semidirect
6. #19135304	C₂	semidirect
7. #19135297	C₂	semidirect
8. #19134438	C₂	semidirect
9. #19134431	C₂	semidirect
10. #19134281	C₂	semidirect
11. C₄₃×#1837	C₂	semidirect
15. #14583254	C₄	semidirect
16. #14583250	C₄	semidirect
17. #14583246	C₄	semidirect
189. #1837	D₄₃	semidirect
226. C₄₃	#21659	semidirect

Maximal subgroups

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

subgroup	index
#19144918	2
#19143532	2
#19143312	2
#19135575	2
#19135426	2
#19135304	2
#19135297	2
#19134438	2
#19134431	2
#19134281	2
C₄₃×#1837	2
#21659	43