Subgroups — #24992011

Order 11008.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143618	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582997	C₂	1	1	yes	maximal, characteristic
3	#19132990	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132985	C₂	1	1	yes	maximal, characteristic
5	#19132439	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132425	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621884	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
10	#14582997	—	2	2	no	—
11	#14582997	—	2	2	no	—
12	C₈×D₁₇₂	—	2	2	no	—
13	#14582964	—	2	2	no	—
14	#14582952	C₂²	1	1	yes	characteristic
15	#14582952	C₂²	1	1	yes	characteristic
16	C₂×#13063258	C₄	1	1	yes	characteristic
17	C₂×#13063249	C₄	1	1	yes	characteristic
18	C₂×#13063245	C₄	1	1	yes	characteristic
19	C₂²⋊C₈×D₄₃	—	2	2	no	—
20	C₂×#13063236	C₂²	1	1	yes	characteristic
21	Q₈⋊(C₄₃⋊C₈)	—	2	2	no	—
22	#14582626	—	2	2	no	—
23	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#431	C₂²	1	1	yes	characteristic
25	C₄₃×#431	C₂²	1	1	yes	characteristic
26	C₄₃×#324	—	2	2	no	—
27	C₂²×D₁₇₂	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₃	—	2	1	no	—
29	C₂×#10621884	—	2	2	no	—
30	C₂×#10621884	—	2	2	no	—
31	C₂×#10621884	C₈	1	4	yes	complemented
32	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₄₃	—	2	4	no	—
35	C₄×D₁₇₂	—	2	1	no	—
36	C₄×D₁₇₂	—	2	1	no	—
37	C₄×D₁₇₂	D₄	1	2	yes	—
38	C₄×D₁₇₂	D₄	1	2	yes	—
39	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
41	#13063258	—	2	2	no	—
42	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
44	#13063249	—	2	2	no	—
45	#13063245	—	2	2	no	—
46	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
47	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
48	(C₂×D₄₃)⋊C₈	—	2	4	no	—
49	#13063236	—	2	1	no	—
50	#13063236	—	2	1	no	—
51	C₄×C₄₃⋊C₈	—	2	1	no	—
52	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
53	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₄×Q₈×C₄₃	—	2	1	no	—
55	C₄×Q₈×C₄₃	—	2	1	no	—
56	C₄×Q₈×C₄₃	—	2	1	no	—
57	C₄×Q₈×C₄₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₃	—	2	1	no	—
61	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₄₃	—	2	1	no	—
63	C₄⋊C₈×C₄₃	—	2	1	no	—
64	C₂²⋊C₈×C₄₃	—	2	1	no	—
65	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
66	C₄×C₈×C₄₃	—	2	1	no	—
67	C₄×C₈×C₄₃	—	2	1	no	—
68	C₂³×D₄₃	—	2	1	no	—
69	C₂×Q₈×C₄₃	—	2	1	no	—
70	C₂×Q₈×C₄₃	—	2	1	no	—
71	C₂×Q₈×C₄₃	—	2	1	no	—
72	C₂×Q₈×C₄₃	—	2	1	no	—
73	C₂×Q₈×C₄₃	—	2	1	no	—
74	C₂×Q₈×C₄₃	C₂×C₈	1	2	yes	complemented
75	C₂²×C₄×C₄₃	—	2	1	no	—
76	C₂²×C₄×C₄₃	—	2	1	no	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
80	C₂²×Dic₄₃	—	2	1	no	—
81	C₂×D₁₇₂	—	2	1	no	—
82	C₂×D₁₇₂	—	2	1	no	—
83	C₂×D₁₇₂	—	2	1	no	—
84	C₂×D₁₇₂	C₂²⋊C₄	1	2	yes	—
85	C₂×D₁₇₂	C₂×C₈	1	2	yes	—
86	C₂×D₁₇₂	M₁₆	1	2	yes	—
87	C₂×C₄×D₄₃	—	2	2	no	—
88	C₂×C₄×D₄₃	—	2	2	no	—
89	C₂×C₄×D₄₃	—	8	2	no	—
90	C₂×C₈×C₄₃	—	2	2	no	—
91	C₂×C₈×C₄₃	—	4	1	no	—
92	C₄⋊C₄×C₄₃	—	2	1	no	—
93	C₄⋊C₄×C₄₃	—	2	1	no	—
94	C₄⋊C₄×C₄₃	—	4	1	no	—
95	C₄⋊C₄×C₄₃	—	4	1	no	—
96	C₄²×C₄₃	—	4	1	no	—
97	C₄²×C₄₃	—	4	1	no	—
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₃	#41	1	1	yes	characteristic
102	#10621884	—	2	8	no	—
103	(C₂×D₄₃)⋊C₄	—	2	2	no	—
104	(C₂×D₄₃)⋊C₄	—	2	2	no	—
105	C₄⋊Dic₄₃	—	2	1	no	—
106	C₄⋊Dic₄₃	C₂²⋊C₄	1	2	yes	—
107	C₂×C₄₃⋊C₈	—	2	1	no	—
108	C₂×C₄₃⋊C₈	—	2	1	no	—
109	C₂×C₄₃⋊C₈	—	2	1	no	—
110	C₂×C₄₃⋊C₈	—	2	2	no	—
111	C₂×C₄₃⋊C₈	—	2	1	no	—
112	C₂×C₄₃⋊C₈	—	2	1	no	—
113	C₂×C₄₃⋊C₈	—	4	1	no	—
114	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	complemented
115	C₈×D₄₃	—	8	4	no	—
116	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₄₃	—	2	2	no	—
118	C₂²×D₄₃	—	2	2	no	—
119	C₂²×D₄₃	—	2	2	no	—
120	C₂²×D₄₃	—	8	1	no	—
121	C₂²×D₄₃	—	8	1	no	—
122	Q₈×C₄₃	—	2	2	no	—
123	Q₈×C₄₃	—	2	2	no	—
124	Q₈×C₄₃	—	4	1	no	—
125	Q₈×C₄₃	—	4	1	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	4	1	no	—
128	C₂×C₄×C₄₃	—	4	1	no	—
129	C₂×C₄×C₄₃	—	4	1	no	—
130	C₂×C₄×C₄₃	—	4	1	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	8	1	no	—
133	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
143	C₂×Dic₄₃	—	2	2	no	—
144	C₂×Dic₄₃	—	8	1	no	—
145	D₁₇₂	—	2	2	no	—
146	D₁₇₂	—	2	2	no	—
147	D₁₇₂	—	2	2	no	—
148	D₁₇₂	C₂²⋊C₈	1	4	yes	—
149	C₄×D₄₃	—	4	4	no	—
150	C₄×D₄₃	—	8	2	no	—
151	C₈×C₄₃	—	4	2	no	—
152	C₈×C₄₃	—	8	1	no	—
153	C₄₃⋊C₈	—	2	2	no	—
154	C₄₃⋊C₈	—	4	2	no	—
155	C₄₃⋊C₈	—	4	1	no	—
156	C₄₃⋊C₈	—	8	1	no	—
157	#9087	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	#386	1	1	yes	characteristic
161	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
162	C₂²×C₄₃	#435	1	1	yes	characteristic
163	C₂×D₄₃	—	2	4	no	—
164	C₂×D₄₃	—	4	2	no	—
165	C₂×D₄₃	—	4	2	no	—
166	C₂×D₄₃	—	8	2	no	—
167	C₂×D₄₃	—	8	2	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	—	4	2	no	—
173	C₄×C₄₃	—	8	1	no	—
174	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
175	C₄×C₄₃	#408	1	1	yes	characteristic
176	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
177	C₄×C₄₃	#435	1	1	yes	characteristic
178	Dic₄₃	—	4	2	no	—
179	#2972	—	43	1	no	—
180	C₂×#440	—	43	1	no	—
181	#1479	—	43	1	no	—
182	#1478	—	43	1	no	—
183	#1461	D₄₃	1	1	yes	characteristic, complemented
184	#1281	—	43	1	no	—
185	#1275	—	43	1	no	—
186	C₂×C₄₃	—	2	1	no	—
187	C₂×C₄₃	—	2	1	no	—
188	C₂×C₄₃	#1869	1	1	yes	characteristic
189	C₂×C₄₃	#1902	1	1	yes	characteristic
190	C₂×C₄₃	#1904	1	1	yes	characteristic
191	D₄₃	—	4	4	no	—
192	C₂²×SD₁₆	—	43	1	no	—
193	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
194	C₂×C₄×D₄	—	43	1	no	—
195	#440	—	86	2	no	—
196	#440	—	86	2	no	—
197	C₈×D₄	—	86	2	no	—
198	C₈×D₄	—	86	2	no	—
199	#431	—	43	1	no	—
200	#431	—	43	1	no	—
201	#431	C₂×D₄₃	1	1	yes	characteristic
202	#431	C₂×D₄₃	1	1	yes	characteristic
203	C₂×#107	—	43	1	no	—
204	C₂×#103	—	43	1	no	—
205	C₂×#102	—	43	1	no	—
206	C₂×C₂²⋊C₈	—	86	2	no	—
207	C₂×#97	—	43	1	no	—
208	Q₈⋊C₈	—	86	2	no	—
209	D₄⋊C₈	—	86	2	no	—
210	#324	—	2	2	no	—
211	C₄₃	#9087	1	1	yes	characteristic, complemented, Sylow
212	C₂²×Q₈	C₄×D₄₃	1	1	yes	characteristic
213	C₂²×D₄	—	43	1	no	—
214	C₂³×C₄	—	86	1	no	—
215	C₂×SD₁₆	—	43	4	no	—
216	C₂×SD₁₆	—	86	2	no	—
217	C₂×SD₁₆	—	86	2	no	—
218	C₂²×C₈	—	43	1	no	—
219	C₂²×C₈	—	43	1	no	—
220	C₂²×C₈	—	86	4	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	—	2	1	no	—
225	C₄×D₄	—	43	2	no	—
226	C₄×D₄	—	43	2	no	—
227	C₄×D₄	—	86	1	no	—
228	C₄×D₄	—	86	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	43	1	no	—
231	C₂×C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
232	C₂×C₂²⋊C₄	—	86	1	no	—
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
235	#107	—	86	2	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	86	1	no	—
239	C₄⋊C₈	—	86	1	no	—
240	#103	—	86	2	no	—
241	#102	—	86	2	no	—
242	C₂²⋊C₈	—	2	1	no	—
243	C₂²⋊C₈	—	86	4	no	—
244	C₂²⋊C₈	—	86	1	no	—
245	C₂²⋊C₈	—	86	1	no	—
246	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
247	#97	—	86	1	no	—
248	#97	—	86	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	—	43	2	no	—
252	C₄×C₈	—	86	1	no	—
253	C₂⁴	—	86	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	C₈×D₄₃	1	2	yes	complemented
260	C₂×D₄	—	43	2	no	—
261	C₂×D₄	—	43	2	no	—
262	C₂×D₄	—	43	2	no	—
263	C₂×D₄	—	86	1	no	—
264	C₂×D₄	—	86	1	no	—
265	C₂×D₄	—	86	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	86	1	no	—
269	C₂²×C₄	—	86	2	no	—
270	C₂²×C₄	—	86	2	no	—
271	C₂²×C₄	—	344	2	no	—
272	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
273	C₂²×C₄	#10621898	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
275	SD₁₆	—	86	8	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	43	2	no	—
279	C₂×C₈	—	86	1	no	—
280	C₂×C₈	—	86	1	no	—
281	C₂×C₈	—	86	1	no	—
282	C₂×C₈	—	86	2	no	—
283	C₂×C₈	—	86	1	no	—
284	C₂×C₈	—	86	1	no	—
285	C₂×C₈	—	344	4	no	—
286	C₂×C₈	—	172	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	43	2	no	—
292	C₄⋊C₄	—	86	1	no	—
293	C₂²⋊C₄	—	86	2	no	—
294	C₂²⋊C₄	—	86	2	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	4	1	no	—
297	C₄²	D₄×D₄₃	1	1	yes	characteristic
298	C₄²	D₄×D₄₃	1	1	yes	characteristic
299	C₄²	D₄×D₄₃	1	1	yes	characteristic
300	C₄²	#10621900	1	1	yes	characteristic
301	C₂³	—	86	2	no	—
302	C₂³	—	86	2	no	—
303	C₂³	—	86	2	no	—
304	C₂³	—	344	1	no	—
305	C₂³	—	344	1	no	—
306	C₂³	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	D₄	—	43	4	no	—
312	D₄	—	86	2	no	—
313	D₄	—	86	2	no	—
314	D₄	—	86	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	8	1	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	344	2	no	—
324	C₂×C₄	—	344	1	no	—
325	C₂×C₄	—	172	4	no	—
326	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
327	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
328	C₂×C₄	#13063293	1	1	yes	characteristic
329	C₂×C₄	#13063296	1	1	yes	characteristic
330	C₂×C₄	#13063298	1	1	yes	characteristic
331	C₂×C₄	#13063307	1	1	yes	characteristic
332	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
333	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
334	C₂×C₄	#13063331	1	1	yes	characteristic
335	C₂×C₄	#13063333	1	1	yes	characteristic
336	C₈	—	4	2	no	—
337	C₈	—	8	1	no	—
338	C₈	—	86	2	no	—
339	C₈	—	344	1	no	—
340	C₈	—	172	2	no	—
341	C₈	—	172	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	86	4	no	—
345	C₂²	—	344	2	no	—
346	C₂²	—	344	2	no	—
347	C₂²	—	172	2	no	—
348	C₂²	—	172	2	no	—
349	C₂²	#14582775	1	1	yes	characteristic
350	C₂²	C₈×C₂²⋊D₄₃	1	1	yes	characteristic
351	C₂²	#14582968	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	8	1	no	—
358	C₄	—	172	2	no	—
359	C₄	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
360	C₄	#14582864	1	1	yes	characteristic
361	C₄	(C₂²⋊D₄₃)⋊C₈	1	1	yes	characteristic
362	C₄	#14582973	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	172	4	no	—
366	C₂	#19134884	1	1	yes	characteristic
367	C₂	#19135071	1	1	yes	characteristic
368	C₂	#19135085	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582775
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1461	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂²×D₁₇₂	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1461	D₄₃
hypocenter	C₄₃	#9087
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1461

Complements

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

N	H ≅ G/N	type
3. #19132990	C₂	semidirect
5. #19132439	C₂	semidirect
6. #19132425	C₂	semidirect
7. C₄₃×#1461	C₂	semidirect
31. C₂×#10621884	C₈	semidirect
74. C₂×Q₈×C₄₃	C₂×C₈	semidirect
114. C₂×C₄₃⋊C₈	C₂×C₈	semidirect
183. #1461	D₄₃	semidirect
211. C₄₃	#9087	semidirect
259. C₂×Q₈	C₈×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19143618	2
C₂×#14582997	2
#19132990	2
#19132985	2
#19132439	2
#19132425	2
C₄₃×#1461	2
#9087	43