Subgroups — #24993433

Order 11008.

480 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144404	C₂	1	1	yes	maximal, characteristic
2	#19143703	C₂	1	1	yes	maximal, characteristic
3	#19135544	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134612	C₂	1	1	yes	maximal, characteristic
5	#19132497	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10621898	C₂²	1	1	yes	characteristic
9	#14583191	—	2	1	no	—
10	#14583186	—	2	1	no	—
11	#14583181	C₄	1	2	yes	complemented
12	#14583176	C₄	1	2	yes	complemented
13	C₈×Dic₈₆	—	2	1	no	—
14	C₈×D₁₇₂	—	2	1	no	—
15	(C₂²⋊D₄₃)⋊C₈	—	2	2	no	—
16	#14582951	—	2	1	no	—
17	#14582939	—	2	1	no	—
18	#14582923	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×D₄₃	—	2	1	no	—
21	C₂×#13063251	—	2	1	no	—
22	#14582898	—	2	1	no	—
23	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
24	#14582661	—	2	2	no	—
25	#14582660	—	2	1	no	—
26	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#431	C₂²	1	1	yes	characteristic
28	C₄₃×#423	C₂²	1	1	yes	characteristic
29	C₄₃×#333	—	2	1	no	—
30	C₄₃×#333	—	2	1	no	—
31	C₂×#10621898	D₄	1	1	yes	characteristic
32	C₂×#10621877	—	2	1	no	—
33	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
34	C₂×C₈×D₄₃	—	2	2	no	—
35	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₈₆	D₄	1	1	yes	characteristic
37	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
38	C₄×D₁₇₂	D₄	1	1	yes	characteristic
39	C₄×C₂²⋊D₄₃	—	2	1	no	—
40	C₄×C₂²⋊D₄₃	—	2	1	no	—
41	#13063284	D₄	1	1	yes	characteristic
42	#13063282	—	2	1	no	—
43	C₂×C₄⋊Dic₄₃	—	2	1	no	—
44	C₂×C₄×Dic₄₃	—	2	1	no	—
45	C₄²×D₄₃	—	2	1	no	—
46	#13063258	—	2	2	no	—
47	#13063256	—	2	2	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
50	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
51	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
52	Dic₄₃⋊C₈	—	2	2	no	—
53	#13063251	—	4	2	no	—
54	#13063248	—	2	2	no	—
55	#13063245	—	2	2	no	—
56	(C₂×D₄₃)⋊C₈	—	2	2	no	—
57	C₈×Dic₄₃	—	2	2	no	—
58	C₄₃⋊#95	—	2	1	no	—
59	C₂²×Q₈×C₄₃	—	2	1	no	—
60	C₂×M₁₆×C₄₃	—	2	1	no	—
61	C₄⋊Q₈×C₄₃	—	2	1	no	—
62	C₄⋊Q₈×C₄₃	—	2	1	no	—
63	C₄⋊Q₈×C₄₃	—	2	1	no	—
64	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
67	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₄₃	—	2	1	no	—
69	C₄⋊C₈×C₄₃	—	2	1	no	—
70	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
71	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
72	C₂²⋊C₈×C₄₃	—	2	1	no	—
73	C₄×C₈×C₄₃	—	2	1	no	—
74	C₂×Q₈×C₄₃	—	2	2	no	—
75	C₂×Q₈×C₄₃	—	4	1	no	—
76	C₂×Q₈×C₄₃	—	8	2	no	—
77	C₂²×C₄×C₄₃	—	4	1	no	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
81	C₂×C₂²⋊D₄₃	—	2	1	no	—
82	C₂²×Dic₄₃	—	4	1	no	—
83	#10621898	—	2	1	no	—
84	#10621898	—	2	1	no	—
85	#10621898	SD₁₆	1	2	yes	—
86	#10621898	Q₁₆	1	2	yes	—
87	C₂×D₁₇₂	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₄₃	—	2	1	no	—
89	C₂×C₄×D₄₃	—	2	1	no	—
90	C₂×C₄×D₄₃	—	2	1	no	—
91	C₂×Dic₈₆	C₂²⋊C₄	1	1	yes	characteristic
92	M₁₆×C₄₃	—	4	1	no	—
93	M₁₆×C₄₃	—	4	1	no	—
94	C₂×C₈×C₄₃	—	2	1	no	—
95	C₂×C₈×C₄₃	—	2	1	no	—
96	C₂×C₈×C₄₃	—	4	1	no	—
97	C₄⋊C₄×C₄₃	—	2	2	no	—
98	C₄⋊C₄×C₄₃	—	2	2	no	—
99	C₄⋊C₄×C₄₃	—	4	1	no	—
100	C₄⋊C₄×C₄₃	—	4	1	no	—
101	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₃	#41	1	1	yes	characteristic
105	C₂²⋊Dic₄₃	—	2	1	no	—
106	(C₂×D₄₃)⋊C₄	—	2	1	no	—
107	(C₂×D₄₃)⋊C₄	—	2	1	no	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	4	1	no	—
110	C₄⋊Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
111	C₄⋊Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
112	Dic₄₃⋊C₄	—	2	1	no	—
113	Dic₄₃⋊C₄	—	2	1	no	—
114	C₄×Dic₄₃	—	2	1	no	—
115	C₄×Dic₄₃	—	2	1	no	—
116	C₄×Dic₄₃	—	2	1	no	—
117	C₄×Dic₄₃	—	4	1	no	—
118	C₄×Dic₄₃	—	4	1	no	—
119	#10621877	—	4	1	no	—
120	#10621877	—	4	1	no	—
121	C₂×C₄₃⋊C₈	—	2	2	no	—
122	C₂×C₄₃⋊C₈	—	2	2	no	—
123	C₂×C₄₃⋊C₈	—	4	2	no	—
124	C₈×D₄₃	—	8	2	no	—
125	C₂³×C₄₃	#121	1	1	yes	characteristic
126	C₂²×D₄₃	—	2	1	no	—
127	Q₈×C₄₃	—	8	2	no	—
128	Q₈×C₄₃	—	8	2	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	8	2	no	—
134	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
144	C₂²⋊D₄₃	—	2	2	no	—
145	C₂²⋊D₄₃	—	2	2	no	—
146	C₂×Dic₄₃	—	2	1	no	—
147	C₂×Dic₄₃	—	2	1	no	—
148	C₂×Dic₄₃	—	2	1	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	8	1	no	—
151	C₂×Dic₄₃	—	8	1	no	—
152	D₁₇₂	—	2	1	no	—
153	D₁₇₂	#103	1	2	yes	—
154	C₄×D₄₃	—	2	2	no	—
155	C₄×D₄₃	—	2	2	no	—
156	C₄×D₄₃	—	4	1	no	—
157	C₄×D₄₃	—	4	1	no	—
158	C₄×D₄₃	—	8	1	no	—
159	Dic₈₆	—	2	1	no	—
160	Dic₈₆	#103	1	2	yes	—
161	C₈×C₄₃	—	4	1	no	—
162	C₈×C₄₃	—	4	1	no	—
163	C₈×C₄₃	—	8	1	no	—
164	C₄₃⋊C₈	—	4	2	no	—
165	C₄₃⋊C₈	—	4	2	no	—
166	#9315	—	43	1	no	maximal, Sylow
167	C₂²×C₄₃	—	2	1	no	—
168	C₂²×C₄₃	—	2	1	no	—
169	C₂²×C₄₃	#386	1	1	yes	characteristic
170	C₂²×C₄₃	#465	1	1	yes	characteristic
171	C₂²×C₄₃	#467	1	1	yes	characteristic
172	C₂×D₄₃	—	2	2	no	—
173	C₂×D₄₃	—	4	1	no	—
174	C₂×D₄₃	—	4	1	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	2	1	no	—
177	C₄×C₄₃	—	2	1	no	—
178	C₄×C₄₃	—	8	2	no	—
179	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
180	C₄×C₄₃	#416	1	1	yes	characteristic
181	C₄×C₄₃	#455	1	1	yes	characteristic
182	C₄×C₄₃	#456	1	1	yes	characteristic
183	C₄×C₄₃	#465	1	1	yes	characteristic
184	C₄×C₄₃	#467	1	1	yes	characteristic
185	Dic₄₃	—	2	2	no	—
186	Dic₄₃	—	4	1	no	—
187	Dic₄₃	—	4	1	no	—
188	Dic₄₃	—	8	2	no	—
189	#3079	—	43	1	no	—
190	#2988	—	43	1	no	—
191	#1987	—	43	1	no	—
192	#1806	—	43	1	no	—
193	#1548	D₄₃	1	1	yes	characteristic, complemented
194	#1303	—	43	1	no	—
195	#1292	—	43	1	no	—
196	C₂×C₄₃	—	2	1	no	—
197	C₂×C₄₃	—	2	1	no	—
198	C₂×C₄₃	#1882	1	1	yes	characteristic
199	C₂×C₄₃	#1892	1	1	yes	characteristic
200	C₂×C₄₃	#1903	1	1	yes	characteristic
201	D₄₃	—	4	2	no	—
202	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
203	C₄×#41	—	43	1	no	—
204	#477	—	86	1	no	—
205	#476	—	86	1	no	—
206	#475	—	43	2	no	—
207	#474	—	43	2	no	—
208	C₈×Q₈	—	86	1	no	—
209	C₈×D₄	—	86	1	no	—
210	C₈×D₄	—	86	2	no	—
211	#431	—	86	1	no	—
212	#431	C₂×D₄₃	1	1	yes	characteristic
213	#428	—	86	1	no	—
214	#423	—	43	1	no	—
215	#423	C₂×D₄₃	1	1	yes	characteristic
216	C₂×C₄⋊C₈	—	43	1	no	—
217	C₂×C₄⋊C₈	—	86	1	no	—
218	C₂×#104	—	86	1	no	—
219	#417	—	86	1	no	—
220	C₄×C₄⋊C₄	—	43	1	no	—
221	#340	—	86	2	no	—
222	#339	—	86	1	no	—
223	#333	—	2	1	no	—
224	#333	—	2	1	no	—
225	C₄₃	#9315	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	43	1	no	—
227	C₂²×Q₈	—	2	1	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	—	86	1	no	—
230	C₂²×C₈	—	43	2	no	—
231	C₂²×C₈	—	86	2	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	C₄×D₄₃	1	2	yes	—
236	C₄×Q₈	—	43	1	no	—
237	C₄×Q₈	—	43	1	no	—
238	C₄×D₄	—	43	1	no	—
239	C₄×D₄	—	43	1	no	—
240	C₄×D₄	—	86	1	no	—
241	C₄×D₄	—	86	1	no	—
242	#117	—	43	1	no	—
243	#117	—	86	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	86	1	no	—
247	C₂×C₄²	—	86	1	no	—
248	C₂×C₄²	—	86	1	no	—
249	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
250	#107	—	86	2	no	—
251	#106	—	86	2	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	—	43	2	no	—
255	C₄⋊C₈	—	86	2	no	—
256	C₄⋊C₈	—	86	1	no	—
257	C₄⋊C₈	—	86	1	no	—
258	C₄⋊C₈	—	86	1	no	—
259	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
260	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
261	#104	—	172	2	no	—
262	#103	—	86	2	no	—
263	#102	—	86	2	no	—
264	C₂²⋊C₈	—	2	1	no	—
265	C₂²⋊C₈	—	86	2	no	—
266	C₄×C₈	—	2	1	no	—
267	C₄×C₈	—	86	2	no	—
268	#95	—	86	1	no	—
269	#41	—	43	2	no	—
270	#41	—	43	2	no	—
271	#41	—	86	1	no	—
272	#41	—	86	1	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	—	4	1	no	—
275	C₂×Q₈	—	8	2	no	—
276	C₂×Q₈	—	43	1	no	—
277	C₂×D₄	—	43	1	no	—
278	C₂×D₄	—	86	1	no	—
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	—	86	1	no	—
281	C₂²×C₄	—	86	1	no	—
282	C₂²×C₄	—	86	1	no	—
283	C₂²×C₄	—	172	1	no	—
284	C₂²×C₄	#10621898	1	1	yes	characteristic
285	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
286	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
287	M₁₆	—	4	1	no	—
288	M₁₆	—	4	1	no	—
289	M₁₆	—	172	1	no	—
290	M₁₆	—	172	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	4	1	no	—
294	C₂×C₈	—	86	2	no	—
295	C₂×C₈	—	86	2	no	—
296	C₂×C₈	—	344	2	no	—
297	C₂×C₈	—	172	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	43	1	no	—
303	C₄⋊C₄	—	43	1	no	—
304	C₄⋊C₄	—	86	2	no	—
305	C₄⋊C₄	—	86	1	no	—
306	C₄⋊C₄	—	86	1	no	—
307	C₄⋊C₄	—	172	1	no	—
308	C₂²⋊C₄	—	86	1	no	—
309	C₂²⋊C₄	—	86	1	no	—
310	C₂²⋊C₄	—	86	1	no	—
311	C₄²	—	86	1	no	—
312	C₄²	—	86	1	no	—
313	C₄²	—	86	1	no	—
314	C₄²	—	172	1	no	—
315	C₄²	—	172	1	no	—
316	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
317	C₄²	D₄×D₄₃	1	1	yes	characteristic
318	C₄²	#10621900	1	1	yes	characteristic
319	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
320	C₂³	—	86	1	no	—
321	C₂³	#13063298	1	1	yes	characteristic
322	Q₈	—	8	2	no	—
323	Q₈	—	8	2	no	—
324	Q₈	—	43	2	no	—
325	Q₈	—	86	1	no	—
326	D₄	—	43	2	no	—
327	D₄	—	86	2	no	—
328	D₄	—	86	2	no	—
329	D₄	—	86	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	86	1	no	—
336	C₂×C₄	—	86	2	no	—
337	C₂×C₄	—	86	2	no	—
338	C₂×C₄	—	86	1	no	—
339	C₂×C₄	—	86	1	no	—
340	C₂×C₄	—	344	1	no	—
341	C₂×C₄	—	344	1	no	—
342	C₂×C₄	—	344	1	no	—
343	C₂×C₄	—	172	2	no	—
344	C₂×C₄	—	172	1	no	—
345	C₂×C₄	—	172	1	no	—
346	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
347	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
348	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
349	C₂×C₄	#13063293	1	1	yes	characteristic
350	C₂×C₄	#13063296	1	1	yes	characteristic
351	C₂×C₄	#13063307	1	1	yes	characteristic
352	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
353	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
354	C₂×C₄	#13063346	1	1	yes	characteristic
355	C₂×C₄	#13063348	1	1	yes	characteristic
356	C₈	—	4	1	no	—
357	C₈	—	4	1	no	—
358	C₈	—	8	1	no	—
359	C₈	—	172	2	no	—
360	C₈	—	172	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	86	2	no	—
364	C₂²	—	172	1	no	—
365	C₂²	—	172	1	no	—
366	C₂²	#14582775	1	1	yes	characteristic
367	C₂²	#14583131	1	1	yes	characteristic
368	C₂²	#14583139	1	1	yes	characteristic
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	8	2	no	—
373	C₄	—	86	2	no	—
374	C₄	—	344	2	no	—
375	C₄	—	172	1	no	—
376	C₄	—	172	1	no	—
377	C₄	#103×D₄₃	1	1	yes	characteristic
378	C₄	#14582894	1	1	yes	characteristic
379	C₄	#14583075	1	1	yes	characteristic
380	C₄	#14583080	1	1	yes	characteristic
381	C₄	#14583125	1	1	yes	characteristic
382	C₄	#14583137	1	1	yes	characteristic
383	C₂	—	2	1	no	—
384	C₂	—	2	1	no	—
385	C₂	—	172	2	no	—
386	C₂	#19134961	1	1	yes	characteristic
387	C₂	#19135015	1	1	yes	characteristic
388	C₂	#19135078	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582775
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1548	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂×#10621898	D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1548	D₄₃
hypocenter	C₄₃	#9315
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1548

Complements

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

N	H ≅ G/N	type
3. #19135544	C₂	semidirect
5. #19132497	C₂	semidirect
6. #19132469	C₂	semidirect
7. C₄₃×#1548	C₂	semidirect
11. #14583181	C₄	semidirect
12. #14583176	C₄	semidirect
193. #1548	D₄₃	semidirect
225. C₄₃	#9315	semidirect

Maximal subgroups

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

subgroup	index
#19144404	2
#19143703	2
#19135544	2
#19134612	2
#19132497	2
#19132469	2
C₄₃×#1548	2
#9315	43