Subgroups — #25196460

Order 11008.

722 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19142956	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#19135744	C₂	1	4	yes	maximal
4	#19135398	C₂	1	1	yes	maximal, characteristic
5	#19134723	C₂	1	1	yes	maximal, characteristic
6	#19134722	C₂	1	1	yes	maximal, characteristic
7	#19134608	C₂	1	4	yes	maximal
8	C₄×Dic₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1956	C₂	1	1	yes	maximal, characteristic
10	C₂×#13063316	C₂²	1	2	yes	—
11	C₂×Q₈×Dic₄₃	C₂²	1	2	yes	—
12	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
13	C₂×#13063249	C₂²	1	4	yes	—
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
15	#14582806	C₂²	1	2	yes	—
16	#14582798	—	2	1	no	—
17	#14582798	—	2	1	no	—
18	#14582798	—	2	1	no	—
19	#14582798	—	2	1	no	—
20	#14582795	—	2	1	no	—
21	#14582795	—	2	1	no	—
22	#14582769	C₂²	1	1	yes	characteristic
23	#14582769	C₂²	1	1	yes	characteristic
24	#14582764	C₂²	1	2	yes	—
25	C₄⋊C₄×Dic₄₃	—	2	1	no	—
26	C₄⋊C₄×Dic₄₃	—	2	1	no	—
27	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
28	#14582662	C₂²	1	4	yes	—
29	#14582661	C₂²	1	4	yes	—
30	#14582653	C₂²	1	2	yes	—
31	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×#103×C₄₃	C₂²	1	4	yes	—
33	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#384	C₂²	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
36	#13063316	—	4	4	no	—
37	Q₈×Dic₄₃	—	2	8	no	—
38	C₂×C₄⋊Dic₄₃	—	2	1	no	—
39	C₂×C₄⋊Dic₄₃	—	2	1	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄⋊Dic₄₃	—	2	1	no	—
42	C₂×C₄⋊Dic₄₃	—	2	2	no	—
43	C₂×C₄⋊Dic₄₃	—	2	2	no	—
44	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
46	C₂×Dic₄₃⋊C₄	—	2	2	no	—
47	C₂×C₄×Dic₄₃	—	2	2	no	—
48	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
49	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
51	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
52	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
54	Dic₄₃⋊C₈	—	2	4	no	—
55	#13063249	—	4	4	no	—
56	C₄×C₄₃⋊C₈	—	2	2	no	—
57	C₄₃⋊#95	—	2	2	no	—
58	C₄₃⋊#95	—	2	2	no	—
59	C₄₃⋊#95	—	2	2	no	—
60	C₄₃⋊#95	—	2	2	no	—
61	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₄₃	—	2	2	no	—
64	C₄⋊Q₈×C₄₃	Q₈	1	4	yes	—
65	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
69	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
71	#103×C₄₃	—	4	4	no	—
72	C₄×C₈×C₄₃	—	2	2	no	—
73	C₂×Q₈×C₄₃	—	2	2	no	—
74	C₂×Q₈×C₄₃	—	2	4	no	—
75	C₂×Q₈×C₄₃	—	2	4	no	—
76	C₂×Q₈×C₄₃	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₄₃	—	2	2	no	—
78	C₂²×C₄×C₄₃	—	2	2	no	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₃	—	2	2	no	—
83	C₂²×Dic₄₃	—	2	2	no	—
84	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	—	2	4	no	—
92	C₄⋊C₄×C₄₃	—	2	4	no	—
93	C₄⋊C₄×C₄₃	—	2	4	no	—
94	C₄⋊C₄×C₄₃	—	4	2	no	—
95	C₄⋊C₄×C₄₃	—	4	2	no	—
96	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
97	C₄²×C₄₃	#41	1	2	yes	—
98	C₄⋊Dic₄₃	—	2	4	no	—
99	C₄⋊Dic₄₃	—	2	4	no	—
100	C₄⋊Dic₄₃	—	4	4	no	—
101	C₄⋊Dic₄₃	—	4	2	no	—
102	C₄⋊Dic₄₃	—	4	2	no	—
103	C₄⋊Dic₄₃	—	4	2	no	—
104	C₄⋊Dic₄₃	—	4	2	no	—
105	C₄⋊Dic₄₃	—	4	4	no	—
106	Dic₄₃⋊C₄	—	4	4	no	—
107	C₄×Dic₄₃	—	2	2	no	—
108	C₄×Dic₄₃	—	2	2	no	—
109	C₄×Dic₄₃	—	2	2	no	—
110	C₄×Dic₄₃	—	2	2	no	—
111	C₄×Dic₄₃	—	2	2	no	—
112	C₄×Dic₄₃	—	4	4	no	—
113	C₄×Dic₄₃	#41	1	4	yes	—
114	C₂×C₄₃⋊C₈	—	2	2	no	—
115	C₂×C₄₃⋊C₈	—	2	2	no	—
116	C₂×C₄₃⋊C₈	—	2	2	no	—
117	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₄₃	—	2	8	no	—
119	Q₈×C₄₃	—	4	4	no	—
120	C₂×C₄×C₄₃	—	2	4	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₄₃	#135	1	2	yes	—
129	C₂×C₄×C₄₃	#135	1	2	yes	—
130	C₂×C₄×C₄₃	#137	1	2	yes	—
131	C₂×C₄×C₄₃	#137	1	2	yes	—
132	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
134	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	2	1	no	—
138	C₂×Dic₄₃	—	2	1	no	—
139	C₂×Dic₄₃	—	2	1	no	—
140	C₂×Dic₄₃	—	2	1	no	—
141	C₂×Dic₄₃	—	2	1	no	—
142	C₂×Dic₄₃	—	2	1	no	—
143	C₂×Dic₄₃	—	2	1	no	—
144	C₂×Dic₄₃	—	2	1	no	—
145	C₂×Dic₄₃	—	2	1	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	4	2	no	—
151	C₂×Dic₄₃	—	4	2	no	—
152	C₂×Dic₄₃	#124	1	2	yes	—
153	C₂×Dic₄₃	#124	1	2	yes	—
154	C₈×C₄₃	—	4	2	no	—
155	C₄₃⋊C₈	—	4	2	no	—
156	#29144	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
159	C₂²×C₄₃	#535	1	1	yes	characteristic
160	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
161	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
162	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
163	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
164	C₄×C₄₃	—	4	4	no	—
165	C₄×C₄₃	—	8	2	no	—
166	C₄×C₄₃	#479	1	4	yes	—
167	C₄×C₄₃	#479	1	4	yes	—
168	C₄×C₄₃	#554	1	2	yes	—
169	C₄×C₄₃	#555	1	2	yes	—
170	Dic₄₃	—	2	2	no	—
171	Dic₄₃	—	2	2	no	—
172	Dic₄₃	—	4	1	no	—
173	Dic₄₃	—	4	1	no	—
174	Dic₄₃	—	8	2	no	—
175	Dic₄₃	—	8	2	no	—
176	#2859	—	43	1	no	—
177	C₄×C₄⋊Q₈	—	43	1	no	—
178	#2026	—	43	4	no	—
179	#1956	—	43	1	no	—
180	#1956	D₄₃	1	1	yes	characteristic
181	#1834	—	43	1	no	—
182	#1834	—	43	1	no	—
183	#1805	—	43	4	no	—
184	C₄×C₄⋊C₈	—	43	1	no	—
185	C₂×C₄₃	#2563	1	1	yes	characteristic
186	C₂×C₄₃	C₂×#479	1	1	yes	characteristic
187	C₂×C₄₃	C₂×#479	1	1	yes	characteristic
188	C₂×C₄₃	#3131	1	1	yes	characteristic
189	C₂×C₄₃	#3139	1	1	yes	characteristic
190	C₂×C₄₃	#3210	1	1	yes	characteristic
191	C₂×C₄₃	#3210	1	1	yes	characteristic
192	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
193	C₂×#125	—	43	2	no	—
194	C₂×C₄×Q₈	—	43	2	no	—
195	C₂×C₄⋊C₈	—	43	2	no	—
196	C₂×#103	—	43	4	no	—
197	C₂×#103	C₂×D₄₃	1	4	yes	—
198	C₂×C₄×C₈	—	43	1	no	—
199	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
200	#391	—	43	2	no	—
201	#389	—	86	1	no	—
202	#389	—	86	1	no	—
203	#389	—	86	1	no	—
204	#389	—	86	1	no	—
205	#389	—	86	1	no	—
206	#389	—	86	1	no	—
207	#384	—	43	1	no	—
208	#384	—	43	1	no	—
209	#384	C₂×D₄₃	1	1	yes	characteristic
210	#382	—	43	2	no	—
211	C₄×C₄⋊C₄	—	86	1	no	—
212	C₄×C₄⋊C₄	—	86	1	no	—
213	C₄³	—	43	1	no	—
214	#340	—	43	4	no	—
215	#340	—	43	4	no	—
216	#336	—	43	2	no	—
217	C₄₃	#29144	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
219	C₂²×C₈	—	43	2	no	—
220	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	Dic₈₆	1	4	yes	—
223	#125	—	172	4	no	—
224	C₄×Q₈	—	86	8	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	43	2	no	—
230	C₂×C₄⋊C₄	—	43	2	no	—
231	C₂×C₄⋊C₄	—	86	2	no	—
232	C₂×C₄⋊C₄	—	86	2	no	—
233	C₂×C₄⋊C₄	—	86	2	no	—
234	C₂×C₄⋊C₄	—	86	1	no	—
235	C₂×C₄⋊C₄	—	86	1	no	—
236	C₂×C₄⋊C₄	—	86	1	no	—
237	C₂×C₄⋊C₄	—	86	1	no	—
238	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
239	C₂×C₄²	—	43	1	no	—
240	C₂×C₄²	—	43	1	no	—
241	C₂×C₄²	—	43	1	no	—
242	C₂×C₄²	—	43	1	no	—
243	C₂×C₄²	—	43	1	no	—
244	C₂×C₄²	—	43	1	no	—
245	C₂×C₄²	—	86	2	no	—
246	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
247	C₄⋊C₈	—	86	4	no	—
248	#103	—	4	4	no	—
249	#103	—	172	4	no	—
250	C₄×C₈	—	2	2	no	—
251	C₄×C₈	—	86	2	no	—
252	#95	—	86	2	no	—
253	#95	—	86	2	no	—
254	#95	—	86	2	no	—
255	#95	—	86	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	4	no	—
258	C₂×Q₈	—	2	4	no	—
259	C₂×Q₈	C₂×Dic₈₆	1	4	yes	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	43	1	no	—
263	C₂²×C₄	—	43	1	no	—
264	C₂²×C₄	—	43	1	no	—
265	C₂²×C₄	—	43	1	no	—
266	C₂²×C₄	—	86	2	no	—
267	C₂²×C₄	—	86	2	no	—
268	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
269	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
270	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	86	2	no	—
275	C₂×C₈	—	86	2	no	—
276	C₂×C₈	—	86	2	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	86	4	no	—
283	C₄⋊C₄	—	86	4	no	—
284	C₄⋊C₄	—	172	4	no	—
285	C₄⋊C₄	—	172	4	no	—
286	C₄⋊C₄	—	172	4	no	—
287	C₄⋊C₄	—	172	2	no	—
288	C₄⋊C₄	—	172	2	no	—
289	C₄⋊C₄	—	172	2	no	—
290	C₄⋊C₄	—	172	2	no	—
291	C₄²	—	43	4	no	—
292	C₄²	—	86	2	no	—
293	C₄²	—	86	2	no	—
294	C₄²	—	86	2	no	—
295	C₄²	—	86	2	no	—
296	C₄²	—	86	2	no	—
297	C₄²	—	172	4	no	—
298	C₄²	C₂×Dic₈₆	1	2	yes	—
299	C₄²	#10621900	1	2	yes	—
300	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
301	Q₈	—	2	8	no	—
302	Q₈	—	4	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	43	2	no	—
310	C₂×C₄	—	43	2	no	—
311	C₂×C₄	—	86	1	no	—
312	C₂×C₄	—	86	1	no	—
313	C₂×C₄	—	86	1	no	—
314	C₂×C₄	—	86	1	no	—
315	C₂×C₄	—	86	1	no	—
316	C₂×C₄	—	86	1	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	86	1	no	—
319	C₂×C₄	—	86	1	no	—
320	C₂×C₄	—	86	1	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	—	172	2	no	—
323	C₂×C₄	—	172	2	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	#13063302	1	2	yes	—
328	C₂×C₄	#13063302	1	2	yes	—
329	C₂×C₄	#13063346	1	2	yes	—
330	C₂×C₄	#13063348	1	2	yes	—
331	C₂×C₄	#13063354	1	2	yes	—
332	C₂×C₄	#13063357	1	2	yes	—
333	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
334	C₂×C₄	#41×D₄₃	1	2	yes	—
335	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
336	C₂×C₄	#13063370	1	2	yes	—
337	C₈	—	4	2	no	—
338	C₈	—	172	2	no	—
339	C₂²	C₂×#13063302	1	1	yes	characteristic
340	C₂²	#124×D₄₃	1	1	yes	characteristic
341	C₂²	#14583462	1	1	yes	characteristic
342	C₂²	C₂×#13063346	1	1	yes	characteristic
343	C₂²	C₂×#13063348	1	1	yes	characteristic
344	C₂²	C₂×#13063354	1	1	yes	characteristic
345	C₂²	C₂×#13063357	1	1	yes	characteristic
346	C₄	—	4	4	no	—
347	C₄	—	8	2	no	—
348	C₄	—	86	2	no	—
349	C₄	—	86	2	no	—
350	C₄	—	344	2	no	—
351	C₄	—	344	2	no	—
352	C₄	—	172	1	no	—
353	C₄	—	172	1	no	—
354	C₄	#14583204	1	4	yes	—
355	C₄	#14583208	1	4	yes	—
356	C₄	#14583608	1	2	yes	—
357	C₄	#14583613	1	2	yes	—
358	C₂	#19139766	1	1	yes	characteristic
359	C₂	C₂×#14583204	1	1	yes	characteristic
360	C₂	C₂×#14583208	1	1	yes	characteristic
361	C₂	#19144919	1	1	yes	characteristic
362	C₂	#19144972	1	1	yes	characteristic
363	C₂	#19145546	1	1	yes	characteristic
364	C₂	#19145548	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1956	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	#1956	D₄₃
hypocenter	C₄₃	#29144
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

Complements

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

N	H ≅ G/N	type
217. C₄₃	#29144	semidirect

Maximal subgroups

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

subgroup	index
#19142956	2
C₄⋊Q₈×Dic₄₃	2
#19135744	2
#19135398	2
#19134723	2
#19134722	2
#19134608	2
C₄×Dic₄₃⋊C₈	2
C₄₃×#1956	2
#29144	43