Subgroups — #25110385

Order 11008.

886 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583278	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#19137162	C₂	1	1	yes	maximal, characteristic
4	#19135397	C₂	1	1	yes	maximal, characteristic
5	#19134934	C₂	1	4	yes	maximal
6	#19134933	C₂	1	4	yes	maximal
7	#19134722	C₂	1	1	yes	maximal, characteristic
8	#19134340	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1956	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10621885	C₂²	1	4	yes	—
11	C₂×#13063325	C₂²	1	1	yes	characteristic
12	C₂×Q₈×Dic₄₃	C₂²	1	2	yes	—
13	C₂×#13063290	C₂²	1	2	yes	—
14	#14583278	C₄	1	8	yes	complemented
15	C₂×#13063250	C₂²	1	4	yes	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	#14582806	C₂²	1	2	yes	—
18	#14582804	C₂²	1	2	yes	—
19	#14582798	—	2	1	no	—
20	#14582798	—	2	1	no	—
21	#14582795	—	2	1	no	—
22	#14582795	—	2	1	no	—
23	#14582769	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×Dic₄₃	—	2	1	no	—
25	C₄⋊C₄×Dic₄₃	—	2	1	no	—
26	C₄⋊C₄×Dic₄₃	—	2	1	no	—
27	C₄⋊C₄×Dic₄₃	—	2	1	no	—
28	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
29	#14582661	C₂²	1	4	yes	—
30	#14582653	C₂²	1	2	yes	—
31	#14582653	C₂²	1	2	yes	—
32	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×#103×C₄₃	C₂²	1	4	yes	—
34	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
35	C₄₃×#384	C₂²	1	1	yes	characteristic
36	C₂²×Dic₈₆	C₂³	1	2	yes	—
37	C₂×#10621885	—	4	4	no	—
38	C₂×#10621885	—	4	4	no	—
39	C₂×#10621885	C₂×C₄	1	16	yes	—
40	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
41	#13063325	—	2	2	no	—
42	#13063325	C₂×C₄	1	4	yes	—
43	Q₈×Dic₄₃	—	4	4	no	—
44	#13063290	—	4	4	no	—
45	C₂×C₄⋊Dic₄₃	—	2	1	no	—
46	C₂×C₄⋊Dic₄₃	—	2	1	no	—
47	C₂×C₄⋊Dic₄₃	—	2	1	no	—
48	C₂×C₄⋊Dic₄₃	—	2	1	no	—
49	C₂×C₄⋊Dic₄₃	—	2	2	no	—
50	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
51	C₂×Dic₄₃⋊C₄	—	2	2	no	—
52	C₂×C₄×Dic₄₃	—	2	2	no	—
53	C₂×C₄×Dic₄₃	—	2	2	no	—
54	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
55	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
56	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
57	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
58	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
60	#13063250	—	4	4	no	—
61	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
62	C₄₃⋊#95	—	2	2	no	—
63	C₄₃⋊#95	—	2	2	no	—
64	C₄₃⋊#95	—	2	2	no	—
65	C₄₃⋊#95	—	2	2	no	—
66	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
67	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
68	C₄⋊Q₈×C₄₃	—	2	2	no	—
69	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
70	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	#103×C₄₃	—	4	4	no	—
77	C₄×C₈×C₄₃	—	2	2	no	—
78	C₂×Q₈×C₄₃	—	2	2	no	—
79	C₂×Q₈×C₄₃	—	2	4	no	—
80	C₂×Q₈×C₄₃	—	2	4	no	—
81	C₂×Q₈×C₄₃	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₄₃	—	2	2	no	—
83	C₂²×C₄×C₄₃	—	2	2	no	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₃	—	2	2	no	—
88	C₂²×Dic₄₃	—	2	2	no	—
89	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₈₆	—	2	4	no	—
94	C₂×Dic₈₆	—	2	4	no	—
95	C₂×Dic₈₆	—	2	2	no	—
96	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	2	4	no	—
101	C₄⋊C₄×C₄₃	—	2	4	no	—
102	C₄⋊C₄×C₄₃	—	2	4	no	—
103	C₄⋊C₄×C₄₃	—	4	2	no	—
104	C₄⋊C₄×C₄₃	—	4	2	no	—
105	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₄₃	#41	1	2	yes	—
107	#10621885	—	4	16	no	—
108	C₄⋊Dic₄₃	—	2	4	no	—
109	C₄⋊Dic₄₃	—	2	4	no	—
110	C₄⋊Dic₄₃	—	2	4	no	—
111	C₄⋊Dic₄₃	—	4	4	no	—
112	C₄⋊Dic₄₃	—	4	2	no	—
113	C₄⋊Dic₄₃	—	4	2	no	—
114	Dic₄₃⋊C₄	—	4	4	no	—
115	C₄×Dic₄₃	—	2	2	no	—
116	C₄×Dic₄₃	—	2	2	no	—
117	C₄×Dic₄₃	—	2	2	no	—
118	C₄×Dic₄₃	—	2	2	no	—
119	C₄×Dic₄₃	—	2	2	no	—
120	C₄×Dic₄₃	—	4	4	no	—
121	C₄×Dic₄₃	—	4	4	no	—
122	C₄×Dic₄₃	Q₁₆	1	4	yes	complemented
123	C₂×C₄₃⋊C₈	—	2	2	no	—
124	C₂×C₄₃⋊C₈	—	2	2	no	—
125	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
126	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	Q₈×C₄₃	—	2	8	no	—
128	Q₈×C₄₃	—	4	4	no	—
129	C₂×C₄×C₄₃	—	2	4	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	4	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	C₂×Q₁₆	1	2	yes	—
138	C₂×C₄×C₄₃	#135	1	2	yes	—
139	C₂×C₄×C₄₃	#137	1	2	yes	—
140	C₂×C₄×C₄₃	#137	1	2	yes	—
141	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
144	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
145	C₂×Dic₄₃	—	2	1	no	—
146	C₂×Dic₄₃	—	2	1	no	—
147	C₂×Dic₄₃	—	2	1	no	—
148	C₂×Dic₄₃	—	2	1	no	—
149	C₂×Dic₄₃	—	2	4	no	—
150	C₂×Dic₄₃	—	2	1	no	—
151	C₂×Dic₄₃	—	2	1	no	—
152	C₂×Dic₄₃	—	2	1	no	—
153	C₂×Dic₄₃	—	2	1	no	—
154	C₂×Dic₄₃	—	2	1	no	—
155	C₂×Dic₄₃	—	2	1	no	—
156	C₂×Dic₄₃	—	4	2	no	—
157	C₂×Dic₄₃	—	4	2	no	—
158	C₂×Dic₄₃	—	4	2	no	—
159	C₂×Dic₄₃	—	4	2	no	—
160	C₂×Dic₄₃	—	4	2	no	—
161	C₂×Dic₄₃	C₂×Q₁₆	1	2	yes	—
162	C₂×Dic₄₃	C₂×Q₁₆	1	2	yes	—
163	Dic₈₆	—	2	8	no	—
164	Dic₈₆	—	4	4	no	—
165	C₈×C₄₃	—	4	2	no	—
166	C₄₃⋊C₈	—	2	4	no	—
167	#22270	—	43	1	no	maximal, Sylow
168	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₄₃	#535	1	1	yes	characteristic
170	C₂²×C₄₃	#535	1	1	yes	characteristic
171	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
172	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
173	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
174	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
175	C₄×C₄₃	—	4	4	no	—
176	C₄×C₄₃	—	8	2	no	—
177	C₄×C₄₃	C₄×Q₁₆	1	4	yes	—
178	C₄×C₄₃	#441	1	4	yes	—
179	C₄×C₄₃	#518	1	2	yes	—
180	C₄×C₄₃	#540	1	2	yes	—
181	Dic₄₃	—	2	2	no	—
182	Dic₄₃	—	2	2	no	—
183	Dic₄₃	—	4	4	no	—
184	Dic₄₃	—	4	1	no	—
185	Dic₄₃	—	4	1	no	—
186	Dic₄₃	—	8	2	no	—
187	C₂×#494	—	43	1	no	—
188	C₄×C₄⋊Q₈	—	43	1	no	—
189	C₄×C₄⋊Q₈	—	43	1	no	—
190	#1956	—	43	1	no	—
191	#1956	D₄₃	1	1	yes	characteristic
192	#1878	—	43	4	no	—
193	#1878	—	43	4	no	—
194	#1834	—	43	1	no	—
195	#1744	—	43	1	no	—
196	C₂×C₄₃	#2328	1	1	yes	characteristic
197	C₂×C₄₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
198	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
199	C₂×C₄₃	#3209	1	1	yes	characteristic
200	C₂×C₄₃	#3209	1	1	yes	characteristic
201	C₂×C₄₃	#3210	1	1	yes	characteristic
202	C₂×C₄₃	#3210	1	1	yes	characteristic
203	C₂²×Q₁₆	—	43	4	no	—
204	C₂×C₄⋊Q₈	—	43	1	no	—
205	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
206	C₂×C₄×Q₈	—	43	2	no	—
207	C₂×C₄×Q₈	—	43	2	no	—
208	#494	—	43	8	no	—
209	C₂×#103	—	43	4	no	—
210	C₂×#103	C₂×D₄₃	1	4	yes	—
211	C₂×C₄×C₈	—	43	1	no	—
212	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
213	#391	—	43	2	no	—
214	#391	—	43	2	no	—
215	#389	—	86	1	no	—
216	#389	—	86	1	no	—
217	#389	—	86	1	no	—
218	#389	—	86	1	no	—
219	#384	—	43	1	no	—
220	#384	C₂×D₄₃	1	1	yes	characteristic
221	C₄×C₄⋊C₄	—	86	1	no	—
222	C₄×C₄⋊C₄	—	86	1	no	—
223	C₄×C₄⋊C₄	—	86	1	no	—
224	C₄×C₄⋊C₄	—	86	1	no	—
225	C₄³	—	43	1	no	—
226	#340	—	43	4	no	—
227	#336	—	43	2	no	—
228	#336	—	43	2	no	—
229	C₄₃	#22270	1	1	yes	characteristic, complemented, Sylow
230	C₂²×Q₈	—	43	2	no	—
231	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
232	C₂×Q₁₆	—	43	16	no	—
233	C₂×Q₁₆	—	172	4	no	—
234	C₂×Q₁₆	—	172	4	no	—
235	C₂²×C₈	—	43	2	no	—
236	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
237	C₄⋊Q₈	—	2	2	no	—
238	C₄⋊Q₈	—	43	4	no	—
239	C₄⋊Q₈	—	86	2	no	—
240	C₄⋊Q₈	C₄×D₄₃	1	4	yes	—
241	C₄×Q₈	—	172	4	no	—
242	C₄×Q₈	—	172	4	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	43	2	no	—
248	C₂×C₄⋊C₄	—	86	1	no	—
249	C₂×C₄⋊C₄	—	86	1	no	—
250	C₂×C₄⋊C₄	—	86	2	no	—
251	C₂×C₄⋊C₄	—	86	2	no	—
252	C₂×C₄⋊C₄	—	86	1	no	—
253	C₂×C₄⋊C₄	—	86	1	no	—
254	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
255	C₂×C₄²	—	43	1	no	—
256	C₂×C₄²	—	43	1	no	—
257	C₂×C₄²	—	43	1	no	—
258	C₂×C₄²	—	43	1	no	—
259	C₂×C₄²	—	43	1	no	—
260	C₂×C₄²	—	43	1	no	—
261	C₂×C₄²	—	86	2	no	—
262	C₂×C₄²	—	86	2	no	—
263	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
264	#103	—	4	4	no	—
265	#103	—	172	4	no	—
266	C₄×C₈	—	2	2	no	—
267	C₄×C₈	—	43	4	no	—
268	#95	—	86	2	no	—
269	#95	—	86	2	no	—
270	#95	—	86	2	no	—
271	#95	—	86	2	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	4	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	43	4	no	—
276	C₂×Q₈	—	86	2	no	—
277	C₂×Q₈	—	86	4	no	—
278	C₂×Q₈	—	86	4	no	—
279	C₂×Q₈	C₂×C₄×D₄₃	1	4	yes	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	43	1	no	—
283	C₂²×C₄	—	43	1	no	—
284	C₂²×C₄	—	43	1	no	—
285	C₂²×C₄	—	43	1	no	—
286	C₂²×C₄	—	86	2	no	—
287	C₂²×C₄	—	86	2	no	—
288	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
290	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
291	Q₁₆	—	172	16	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	43	4	no	—
296	C₂×C₈	—	86	2	no	—
297	C₂×C₈	—	86	2	no	—
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	—	2	4	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	86	4	no	—
304	C₄⋊C₄	—	86	4	no	—
305	C₄⋊C₄	—	86	4	no	—
306	C₄⋊C₄	—	172	4	no	—
307	C₄⋊C₄	—	172	4	no	—
308	C₄⋊C₄	—	172	2	no	—
309	C₄⋊C₄	—	172	2	no	—
310	C₄²	—	43	4	no	—
311	C₄²	—	86	2	no	—
312	C₄²	—	86	2	no	—
313	C₄²	—	86	2	no	—
314	C₄²	—	86	2	no	—
315	C₄²	—	86	2	no	—
316	C₄²	—	172	4	no	—
317	C₄²	—	172	4	no	—
318	C₄²	C₂×C₄×D₄₃	1	2	yes	—
319	C₄²	#10621900	1	2	yes	—
320	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
321	Q₈	—	2	8	no	—
322	Q₈	—	4	4	no	—
323	Q₈	—	86	8	no	—
324	Q₈	—	172	4	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	43	2	no	—
332	C₂×C₄	—	43	2	no	—
333	C₂×C₄	—	86	1	no	—
334	C₂×C₄	—	86	1	no	—
335	C₂×C₄	—	86	1	no	—
336	C₂×C₄	—	86	1	no	—
337	C₂×C₄	—	86	4	no	—
338	C₂×C₄	—	86	1	no	—
339	C₂×C₄	—	86	1	no	—
340	C₂×C₄	—	86	1	no	—
341	C₂×C₄	—	86	1	no	—
342	C₂×C₄	—	86	1	no	—
343	C₂×C₄	—	86	1	no	—
344	C₂×C₄	—	172	2	no	—
345	C₂×C₄	—	172	2	no	—
346	C₂×C₄	—	172	2	no	—
347	C₂×C₄	—	172	2	no	—
348	C₂×C₄	—	172	2	no	—
349	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
350	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
351	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
352	C₂×C₄	#13063346	1	2	yes	—
353	C₂×C₄	#13063354	1	2	yes	—
354	C₂×C₄	#13063357	1	2	yes	—
355	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
356	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
357	C₂×C₄	#13063370	1	2	yes	—
358	C₂×C₄	#13063371	1	2	yes	—
359	C₈	—	4	2	no	—
360	C₈	—	86	4	no	—
361	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
362	C₂²	#14583462	1	1	yes	characteristic
363	C₂²	#14583467	1	1	yes	characteristic
364	C₂²	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
365	C₂²	C₂×#13063346	1	1	yes	characteristic
366	C₂²	C₂×#13063354	1	1	yes	characteristic
367	C₂²	C₂×#13063357	1	1	yes	characteristic
368	C₄	—	4	4	no	—
369	C₄	—	8	2	no	—
370	C₄	—	86	2	no	—
371	C₄	—	86	2	no	—
372	C₄	—	344	2	no	—
373	C₄	—	172	4	no	—
374	C₄	—	172	1	no	—
375	C₄	—	172	1	no	—
376	C₄	#14582991	1	4	yes	—
377	C₄	#14583003	1	4	yes	—
378	C₄	#14583371	1	2	yes	—
379	C₄	#14583502	1	2	yes	—
380	C₂	#19137319	1	1	yes	characteristic
381	C₂	C₂×#14582991	1	1	yes	characteristic
382	C₂	C₂×#14583003	1	1	yes	characteristic
383	C₂	#19145539	1	1	yes	characteristic
384	C₂	#19145544	1	1	yes	characteristic
385	C₂	#19145546	1	1	yes	characteristic
386	C₂	#19145554	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1956	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1956	D₄₃
hypocenter	C₄₃	#22270
LCS	G ▷ C₂×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
14. #14583278	C₄	semidirect
122. C₄×Dic₄₃	Q₁₆	semidirect
229. C₄₃	#22270	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583278	2
C₄⋊Q₈×Dic₄₃	2
#19137162	2
#19135397	2
#19134934	2
#19134933	2
#19134722	2
#19134340	2
C₄₃×#1956	2
#22270	43