Subgroups — #25066726

Order 11008.

626 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144627	C₂	1	1	yes	maximal, characteristic
2	#19143675	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143673	C₂	1	1	yes	maximal, characteristic
4	#19133472	C₂	1	2	yes	maximal, complemented
5	#19133437	C₂	1	2	yes	maximal, complemented
6	#19133406	C₂	1	2	yes	maximal, complemented
7	#19133394	C₂	1	2	yes	maximal, complemented
8	#19133041	C₂	1	1	yes	maximal, characteristic, complemented
9	#19133011	C₂	1	1	yes	maximal, characteristic
10	#19132890	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1572	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583449	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
14	#14583218	C₄	1	2	yes	—
15	#14583185	C₄	1	2	yes	—
16	#14583100	C₄	1	2	yes	—
17	#14583040	C₄	1	2	yes	—
18	C₈×Dic₈₆	—	2	2	no	—
19	#14583023	C₂²	1	2	yes	—
20	#14583023	C₂²	1	2	yes	—
21	C₄×#10621873	C₂²	1	2	yes	—
22	C₄×#10621873	C₂²	1	2	yes	—
23	#14582976	C₂²	1	2	yes	—
24	#14582972	—	2	2	no	—
25	#14582964	C₂²	1	2	yes	—
26	#14582952	C₂²	1	1	yes	characteristic
27	#14582949	—	2	2	no	—
28	#14582864	—	2	2	no	—
29	C₄×#10621877	C₂²	1	1	yes	characteristic
30	#14582629	C₂²	1	2	yes	—
31	#14582629	C₂²	1	2	yes	—
32	#14582626	C₂²	1	2	yes	—
33	#14582626	C₂²	1	2	yes	—
34	#14582621	C₂²	1	1	yes	characteristic
35	#14582621	C₂²	1	1	yes	characteristic
36	#14582620	C₂²	1	1	yes	characteristic
37	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₃×#432	C₂²	1	1	yes	characteristic
39	C₄₃×#431	C₂²	1	1	yes	characteristic
40	C₄₃×#431	C₂²	1	1	yes	characteristic
41	C₄₃×#335	C₂²	1	2	yes	—
42	C₄₃×#334	C₂²	1	2	yes	—
43	C₂×#10621898	—	2	1	no	—
44	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
45	C₂×#10621873	C₂×C₄	1	2	yes	—
46	C₂×#10621873	C₂×C₄	1	2	yes	—
47	C₂×#10621877	—	2	1	no	—
48	C₂×#10621872	—	2	2	no	—
49	C₂×C₈×D₄₃	—	2	2	no	—
50	#13063325	C₂×C₄	1	1	yes	characteristic
51	#13063322	C₂×C₄	1	1	yes	characteristic
52	#13063313	—	2	1	no	—
53	#13063299	—	2	1	no	—
54	#13063299	C₂×C₄	1	2	yes	—
55	C₄×Dic₈₆	—	2	1	no	—
56	C₄×Dic₈₆	—	2	1	no	—
57	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
58	C₄×Dic₈₆	C₂³	1	2	yes	—
59	C₄×D₁₇₂	—	2	1	no	—
60	C₄×D₁₇₂	C₂³	1	2	yes	—
61	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
62	C₂×Dic₄₃⋊C₄	—	2	1	no	—
63	C₂×C₄×Dic₄₃	—	2	1	no	—
64	#13063257	C₂×C₄	1	2	yes	—
65	#13063257	C₂×C₄	1	2	yes	—
66	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
67	Dic₄₃⋊C₈	—	2	2	no	—
68	Dic₄₃⋊C₈	—	2	2	no	—
69	#13063250	C₂×C₄	1	2	yes	—
70	#13063250	C₂×C₄	1	2	yes	—
71	#13063247	C₂×C₄	1	2	yes	—
72	#13063247	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
74	(C₂×D₄₃)⋊C₈	—	2	2	no	—
75	(C₂×D₄₃)⋊C₈	—	2	2	no	—
76	#13063236	—	2	1	no	—
77	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₄₃	—	2	2	no	—
82	C₈×Dic₄₃	—	2	2	no	—
83	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
85	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
86	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
87	C₂²⋊C₈×C₄₃	—	2	1	no	—
88	C₂²⋊C₈×C₄₃	—	2	1	no	—
89	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
90	#97×C₄₃	—	2	1	no	—
91	C₄×C₈×C₄₃	—	2	1	no	—
92	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
97	C₂×C₂²⋊D₄₃	—	2	2	no	—
98	C₂²×Dic₄₃	—	2	1	no	—
99	C₂²×Dic₄₃	—	2	1	no	—
100	#10621898	—	8	2	no	—
101	C₂×D₁₇₂	—	2	1	no	—
102	C₂×D₁₇₂	—	2	1	no	—
103	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₄₃	—	2	2	no	—
105	C₂×Dic₈₆	—	2	1	no	—
106	C₂×Dic₈₆	—	2	4	no	—
107	C₂×Dic₈₆	—	2	1	no	—
108	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₄₃	—	2	2	no	—
110	C₂×C₈×C₄₃	—	2	2	no	—
111	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
113	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₃	#41	1	1	yes	characteristic
116	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
117	(C₂×D₄₃)⋊C₄	—	2	2	no	—
118	(C₂×D₄₃)⋊C₄	—	2	2	no	—
119	C₄⋊Dic₄₃	—	2	1	no	—
120	C₄⋊Dic₄₃	—	2	1	no	—
121	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
122	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
123	Dic₄₃⋊C₄	—	2	2	no	—
124	Dic₄₃⋊C₄	—	2	2	no	—
125	C₄×Dic₄₃	—	2	2	no	—
126	C₄×Dic₄₃	—	2	2	no	—
127	#10621877	—	4	2	no	—
128	C₂×C₄₃⋊C₈	—	2	1	no	—
129	C₂×C₄₃⋊C₈	—	2	1	no	—
130	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
131	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
132	#10621873	—	2	4	no	—
133	#10621873	—	2	4	no	—
134	#10621872	—	8	2	no	—
135	C₈×D₄₃	—	8	2	no	—
136	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	4	1	no	—
141	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
147	C₂²⋊D₄₃	—	8	2	no	—
148	C₂×Dic₄₃	—	2	4	no	—
149	C₂×Dic₄₃	—	2	2	no	—
150	C₂×Dic₄₃	—	2	2	no	—
151	C₂×Dic₄₃	—	2	2	no	—
152	C₂×Dic₄₃	—	8	1	no	—
153	D₁₇₂	—	2	2	no	—
154	D₁₇₂	—	2	2	no	—
155	D₁₇₂	—	8	1	no	—
156	D₁₇₂	—	8	1	no	—
157	C₄×D₄₃	—	4	2	no	—
158	C₄×D₄₃	—	4	2	no	—
159	Dic₈₆	—	2	4	no	—
160	Dic₈₆	—	2	2	no	—
161	Dic₈₆	—	8	1	no	—
162	Dic₈₆	#131	1	4	yes	—
163	C₈×C₄₃	—	2	2	no	—
164	C₈×C₄₃	—	2	2	no	—
165	C₈×C₄₃	—	4	2	no	—
166	C₈×C₄₃	—	4	2	no	—
167	C₄₃⋊C₈	—	2	2	no	—
168	C₄₃⋊C₈	—	2	2	no	—
169	C₄₃⋊C₈	—	2	2	no	—
170	C₄₃⋊C₈	—	4	1	no	—
171	#17923	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	2	1	no	—
173	C₂²×C₄₃	—	2	1	no	—
174	C₂²×C₄₃	—	2	1	no	—
175	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₄₃	—	4	2	no	—
177	C₂×D₄₃	—	4	2	no	—
178	C₂×D₄₃	—	4	2	no	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	2	1	no	—
181	C₄×C₄₃	—	2	1	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	C₂×#131	1	2	yes	—
184	C₄×C₄₃	#577	1	2	yes	—
185	Dic₄₃	—	2	4	no	—
186	Dic₄₃	—	4	2	no	—
187	Dic₄₃	—	4	2	no	—
188	#3103	—	43	1	no	—
189	#2982	—	43	1	no	—
190	#2981	—	43	1	no	—
191	#1588	—	43	2	no	—
192	#1583	—	43	2	no	—
193	SD₁₆⋊C₈	—	43	2	no	—
194	C₈×SD₁₆	—	43	2	no	—
195	#1572	D₄₃	1	1	yes	characteristic, complemented
196	#1491	—	43	1	no	—
197	#1484	—	43	1	no	—
198	#1453	—	43	1	no	—
199	C₂×C₄₃	—	4	1	no	—
200	C₂×C₄₃	#2929	1	1	yes	characteristic
201	C₂×C₄₃	#2949	1	1	yes	characteristic
202	C₂×C₄₃	#2956	1	1	yes	characteristic
203	D₄₃	—	8	2	no	—
204	#532	—	43	1	no	—
205	C₂×C₄×Q₈	—	43	1	no	—
206	#481	—	43	2	no	—
207	#475	—	43	2	no	—
208	#460	—	43	2	no	—
209	#448	—	43	2	no	—
210	C₈×Q₈	—	43	2	no	—
211	C₈×Q₈	—	43	2	no	—
212	C₈×Q₈	—	86	2	no	—
213	C₄×SD₁₆	—	43	2	no	—
214	C₄×SD₁₆	—	43	2	no	—
215	#436	—	43	2	no	—
216	#435	—	86	2	no	—
217	C₈×D₄	—	43	2	no	—
218	#432	C₂×D₄₃	1	1	yes	characteristic
219	#431	—	43	1	no	—
220	#431	—	86	2	no	—
221	#431	C₂×D₄₃	1	1	yes	characteristic
222	#431	C₂×D₄₃	1	1	yes	characteristic
223	#408	—	86	2	no	—
224	C₄×M₁₆	—	43	1	no	—
225	#335	C₂×D₄₃	1	2	yes	—
226	#334	C₂×D₄₃	1	2	yes	—
227	Q₈⋊C₈	—	43	2	no	—
228	Q₈⋊C₈	—	43	2	no	—
229	D₄⋊C₈	—	43	2	no	—
230	D₄⋊C₈	—	43	2	no	—
231	#322	—	43	1	no	—
232	#322	—	43	1	no	—
233	#322	—	43	1	no	—
234	C₈²	—	43	1	no	—
235	C₄₃	#17923	1	1	yes	characteristic, complemented, Sylow
236	C₂×#41	—	86	1	no	—
237	C₂²×Q₈	—	43	1	no	—
238	C₂×SD₁₆	—	43	2	no	—
239	C₂×SD₁₆	—	43	2	no	—
240	C₂×M₁₆	—	86	2	no	—
241	C₂×M₁₆	—	86	1	no	—
242	C₂²×C₈	—	86	2	no	—
243	C₄⋊Q₈	—	43	1	no	—
244	#127	—	43	1	no	—
245	#124	—	86	1	no	—
246	#121	—	43	2	no	—
247	#121	—	86	1	no	—
248	C₄×Q₈	—	43	2	no	—
249	C₄×Q₈	—	43	2	no	—
250	C₄×Q₈	—	86	1	no	—
251	C₄×Q₈	—	86	1	no	—
252	C₄×D₄	—	43	2	no	—
253	C₄×D₄	—	86	1	no	—
254	C₂×C₄⋊C₄	—	43	1	no	—
255	C₂×C₄⋊C₄	—	86	1	no	—
256	C₂×C₄²	—	86	1	no	—
257	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
258	#106	—	43	2	no	—
259	#106	—	43	2	no	—
260	C₄⋊C₈	—	43	2	no	—
261	C₄⋊C₈	—	86	2	no	—
262	C₄⋊C₈	—	86	2	no	—
263	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
264	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
265	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
266	#103	—	43	2	no	—
267	#103	—	43	2	no	—
268	#102	—	43	2	no	—
269	#102	—	43	2	no	—
270	C₂²⋊C₈	—	2	1	no	—
271	C₂²⋊C₈	—	2	1	no	—
272	C₂²⋊C₈	—	43	2	no	—
273	C₂²⋊C₈	—	86	2	no	—
274	C₂²⋊C₈	—	86	2	no	—
275	C₂²⋊C₈	C₄×D₄₃	1	2	yes	—
276	#97	—	2	1	no	—
277	#97	—	86	1	no	—
278	C₄×C₈	—	2	1	no	—
279	C₄×C₈	—	43	1	no	—
280	C₄×C₈	—	43	1	no	—
281	C₄×C₈	—	43	1	no	—
282	C₄×C₈	—	43	1	no	—
283	C₄×C₈	—	86	2	no	—
284	C₄×C₈	—	86	2	no	—
285	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
287	#41	—	344	2	no	—
288	C₂×Q₈	—	43	2	no	—
289	C₂×Q₈	—	86	4	no	—
290	C₂×Q₈	—	86	1	no	—
291	C₂×Q₈	—	86	1	no	—
292	C₂×D₄	—	43	2	no	—
293	C₂×D₄	—	86	2	no	—
294	C₂×D₄	—	86	1	no	—
295	C₂×D₄	—	86	1	no	—
296	C₂²×C₄	—	86	1	no	—
297	C₂²×C₄	—	86	1	no	—
298	C₂²×C₄	—	86	2	no	—
299	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
300	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
301	C₂²×C₄	#10621902	1	1	yes	characteristic
302	SD₁₆	—	86	4	no	—
303	SD₁₆	—	86	4	no	—
304	M₁₆	—	344	2	no	—
305	M₁₆	—	172	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	43	2	no	—
309	C₂×C₈	—	43	2	no	—
310	C₂×C₈	—	86	1	no	—
311	C₂×C₈	—	86	1	no	—
312	C₂×C₈	—	344	2	no	—
313	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
314	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
315	C₄⋊C₄	—	43	2	no	—
316	C₄⋊C₄	—	43	2	no	—
317	C₄⋊C₄	—	86	1	no	—
318	C₄⋊C₄	—	86	2	no	—
319	C₄⋊C₄	—	86	1	no	—
320	C₄⋊C₄	—	86	2	no	—
321	C₂²⋊C₄	—	86	2	no	—
322	C₂²⋊C₄	—	86	2	no	—
323	C₄²	—	86	2	no	—
324	C₄²	—	86	2	no	—
325	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
326	C₄²	D₄×D₄₃	1	1	yes	characteristic
327	C₄²	#10621902	1	1	yes	characteristic
328	C₄²	C₂³×D₄₃	1	1	yes	characteristic
329	C₂³	—	86	2	no	—
330	C₂³	#13063287	1	1	yes	characteristic
331	Q₈	—	43	4	no	—
332	Q₈	—	86	4	no	—
333	Q₈	—	86	2	no	—
334	Q₈	—	344	1	no	—
335	D₄	—	86	2	no	—
336	D₄	—	86	2	no	—
337	D₄	—	344	1	no	—
338	D₄	—	344	1	no	—
339	D₄	—	344	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	86	4	no	—
344	C₂×C₄	—	86	2	no	—
345	C₂×C₄	—	86	2	no	—
346	C₂×C₄	—	86	2	no	—
347	C₂×C₄	—	344	1	no	—
348	C₂×C₄	—	172	2	no	—
349	C₂×C₄	—	172	2	no	—
350	C₂×C₄	#13063287	1	1	yes	characteristic
351	C₂×C₄	#13063287	1	1	yes	characteristic
352	C₂×C₄	#13063287	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
355	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	2	2	no	—
358	C₈	—	4	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	86	2	no	—
361	C₈	—	86	2	no	—
362	C₈	—	86	2	no	—
363	C₈	—	172	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	172	2	no	—
368	C₂²	—	172	2	no	—
369	C₂²	—	172	2	no	—
370	C₂²	C₂×#13063287	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	86	4	no	—
376	C₄	—	172	2	no	—
377	C₄	—	172	2	no	—
378	C₄	#131×D₄₃	1	2	yes	—
379	C₄	#14583706	1	2	yes	—
380	C₂	—	4	1	no	—
381	C₂	—	344	2	no	—
382	C₂	#19143288	1	1	yes	characteristic
383	C₂	#19143453	1	1	yes	characteristic
384	C₂	#19143504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1572	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#14583449	C₂²
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1572	D₄₃
hypocenter	C₄₃	#17923
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1572

Complements

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

N	H ≅ G/N	type
2. #19143675	C₂	semidirect
4. #19133472	C₂	semidirect
5. #19133437	C₂	semidirect
6. #19133406	C₂	semidirect
7. #19133394	C₂	semidirect
8. #19133041	C₂	semidirect
10. #19132890	C₂	semidirect
11. C₄₃×#1572	C₂	semidirect
195. #1572	D₄₃	semidirect
235. C₄₃	#17923	semidirect

Maximal subgroups

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

subgroup	index
#19144627	2
#19143675	2
#19143673	2
#19133472	2
#19133437	2
#19133406	2
#19133394	2
#19133041	2
#19133011	2
#19132890	2
C₄₃×#1572	2
#17923	43