Subgroups — #25105388

Order 11008.

854 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583295	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	#19135556	C₂	1	1	yes	maximal, characteristic
5	#19135337	C₂	1	4	yes	maximal
6	#19134754	C₂	1	1	yes	maximal, characteristic
7	#19134491	C₂	1	4	yes	maximal
8	C₄×#13063235	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1990	C₂	1	1	yes	maximal, characteristic
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₂×Q₈×Dic₄₃	—	2	1	no	—
12	C₂×#13063290	—	2	1	no	—
13	#14583295	C₄	1	8	yes	complemented
14	C₂×#13063258	C₂²	1	4	yes	—
15	C₂×#13063256	C₂²	1	4	yes	—
16	C₂×#13063235	C₂²	1	2	yes	—
17	C₂×#13063235	C₂²	1	2	yes	—
18	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
19	#14582806	—	2	1	no	—
20	#14582804	—	2	1	no	—
21	#14582798	—	2	1	no	—
22	#14582798	C₂²	1	2	yes	—
23	#14582795	—	2	1	no	—
24	#14582795	C₂²	1	2	yes	—
25	#14582769	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×Dic₄₃	—	2	1	no	—
27	C₄⋊C₄×Dic₄₃	—	2	1	no	—
28	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
29	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
30	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
31	#14582662	C₂²	1	4	yes	—
32	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#384	C₂²	1	1	yes	characteristic
35	C₄₃×#340	C₂²	1	4	yes	—
36	C₂²×Dic₈₆	—	2	1	no	—
37	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
38	#13063325	—	2	2	no	—
39	#13063325	C₂×C₄	1	4	yes	—
40	Q₈×Dic₄₃	—	4	4	no	—
41	#13063290	—	4	4	no	—
42	C₂×C₄⋊Dic₄₃	—	2	1	no	—
43	C₂×C₄⋊Dic₄₃	—	2	1	no	—
44	C₂×C₄⋊Dic₄₃	—	2	1	no	—
45	C₂×C₄⋊Dic₄₃	—	2	2	no	—
46	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
48	C₂×Dic₄₃⋊C₄	—	2	2	no	—
49	C₂×C₄×Dic₄₃	—	2	2	no	—
50	C₂×C₄×Dic₄₃	—	2	2	no	—
51	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
52	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
53	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
54	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
55	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
57	#13063258	—	2	8	no	—
58	#13063256	C₂×C₄	1	16	yes	—
59	#13063235	—	2	4	no	—
60	#13063235	Q₈	1	8	yes	complemented
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	—	2	1	no	—
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	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₄₃	—	2	2	no	—
77	C₂×Q₈×C₄₃	—	2	4	no	—
78	C₂×Q₈×C₄₃	—	4	2	no	—
79	C₂×Q₈×C₄₃	—	4	2	no	—
80	C₂²×C₄×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	—	2	2	no	—
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₃	—	2	2	no	—
86	C₂²×Dic₄₃	—	2	2	no	—
87	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₈₆	—	2	4	no	—
92	C₂×Dic₈₆	—	4	2	no	—
93	C₂×Dic₈₆	—	4	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
97	C₄⋊C₄×C₄₃	—	2	4	no	—
98	C₄⋊C₄×C₄₃	—	2	2	no	—
99	C₄⋊C₄×C₄₃	—	2	4	no	—
100	C₄⋊C₄×C₄₃	—	4	2	no	—
101	C₄⋊C₄×C₄₃	—	4	2	no	—
102	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
103	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₄₃	#41	1	2	yes	—
105	C₄⋊Dic₄₃	—	2	4	no	—
106	C₄⋊Dic₄₃	—	2	2	no	—
107	C₄⋊Dic₄₃	—	2	4	no	—
108	C₄⋊Dic₄₃	—	4	4	no	—
109	C₄⋊Dic₄₃	—	4	2	no	—
110	C₄⋊Dic₄₃	—	4	2	no	—
111	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
112	Dic₄₃⋊C₄	—	4	4	no	—
113	C₄×Dic₄₃	—	2	2	no	—
114	C₄×Dic₄₃	—	2	2	no	—
115	C₄×Dic₄₃	—	2	2	no	—
116	C₄×Dic₄₃	—	2	2	no	—
117	C₄×Dic₄₃	—	2	2	no	—
118	C₄×Dic₄₃	—	4	4	no	—
119	C₄×Dic₄₃	—	4	4	no	—
120	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
121	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
122	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
123	C₂×C₄₃⋊C₈	#41	1	4	yes	—
124	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	Q₈×C₄₃	—	4	8	no	—
126	C₂×C₄×C₄₃	—	2	4	no	—
127	C₂×C₄×C₄₃	—	4	2	no	—
128	C₂×C₄×C₄₃	—	4	2	no	—
129	C₂×C₄×C₄₃	—	4	2	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₄×C₄₃	#137	1	2	yes	—
137	C₂×C₄×C₄₃	#137	1	2	yes	—
138	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
140	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
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	—	2	4	no	—
147	C₂×Dic₄₃	—	2	1	no	—
148	C₂×Dic₄₃	—	2	1	no	—
149	C₂×Dic₄₃	—	2	1	no	—
150	C₂×Dic₄₃	—	2	1	no	—
151	C₂×Dic₄₃	—	2	1	no	—
152	C₂×Dic₄₃	—	2	1	no	—
153	C₂×Dic₄₃	—	4	2	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	C₂×Dic₄₃	—	4	2	no	—
157	C₂×Dic₄₃	—	4	2	no	—
158	C₂×Dic₄₃	C₄⋊Q₈	1	2	yes	—
159	C₂×Dic₄₃	C₄⋊Q₈	1	2	yes	—
160	Dic₈₆	—	4	8	no	—
161	C₈×C₄₃	—	2	4	no	—
162	C₄₃⋊C₈	C₄×Q₈	1	8	yes	complemented
163	#21727	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
166	C₂²×C₄₃	#532	1	1	yes	characteristic
167	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
170	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
171	C₄×C₄₃	—	4	4	no	—
172	C₄×C₄₃	—	8	2	no	—
173	C₄×C₄₃	#440	1	4	yes	—
174	C₄×C₄₃	#440	1	4	yes	—
175	C₄×C₄₃	C₂×C₄×Q₈	1	2	yes	—
176	C₄×C₄₃	#533	1	2	yes	—
177	Dic₄₃	—	2	2	no	—
178	Dic₄₃	—	2	2	no	—
179	Dic₄₃	—	4	1	no	—
180	Dic₄₃	—	4	1	no	—
181	Dic₄₃	—	4	4	no	—
182	Dic₄₃	—	8	2	no	—
183	C₂×#498	—	43	1	no	—
184	C₄×C₄⋊Q₈	—	43	1	no	—
185	C₄×C₄⋊Q₈	—	43	1	no	—
186	#1990	—	43	1	no	—
187	#1990	D₄₃	1	1	yes	characteristic
188	#1944	—	43	4	no	—
189	#1845	—	43	1	no	—
190	#1776	—	43	4	no	—
191	C₄×#97	—	43	1	no	—
192	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
193	C₂×C₄₃	C₂×#440	1	1	yes	characteristic
194	C₂×C₄₃	C₂×#440	1	1	yes	characteristic
195	C₂×C₄₃	#3165	1	1	yes	characteristic
196	C₂×C₄₃	#3165	1	1	yes	characteristic
197	C₂×C₄₃	#3184	1	1	yes	characteristic
198	C₂×C₄₃	#3185	1	1	yes	characteristic
199	C₂×C₄⋊Q₈	—	43	1	no	—
200	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
201	C₂×C₄×Q₈	—	86	1	no	—
202	C₂×C₄×Q₈	—	86	1	no	—
203	#498	—	43	8	no	—
204	C₂×#107	—	43	4	no	—
205	C₂×#106	—	43	4	no	—
206	C₂×#97	—	43	2	no	—
207	C₂×#97	—	43	2	no	—
208	C₂×C₄×C₈	—	43	1	no	—
209	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
210	#391	—	86	1	no	—
211	#391	—	86	1	no	—
212	#389	—	43	2	no	—
213	#389	—	43	2	no	—
214	#389	—	86	1	no	—
215	#389	—	86	1	no	—
216	#384	—	43	1	no	—
217	#384	C₂×D₄₃	1	1	yes	characteristic
218	C₄×C₄⋊C₄	—	43	2	no	—
219	C₄×C₄⋊C₄	—	43	2	no	—
220	C₄×C₄⋊C₄	—	86	1	no	—
221	C₄×C₄⋊C₄	—	86	1	no	—
222	C₄³	—	43	1	no	—
223	#340	—	43	4	no	—
224	#340	C₂×D₄₃	1	4	yes	—
225	C₄₃	#21727	1	1	yes	characteristic, complemented, Sylow
226	C₂²×Q₈	—	2	1	no	—
227	C₂²×Q₈	—	86	1	no	—
228	C₂²×C₈	—	43	2	no	—
229	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
230	C₄⋊Q₈	—	2	2	no	—
231	C₄⋊Q₈	—	43	4	no	—
232	C₄⋊Q₈	—	86	2	no	—
233	C₄⋊Q₈	C₄×D₄₃	1	4	yes	—
234	C₄×Q₈	—	172	4	no	—
235	C₄×Q₈	—	172	4	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	43	2	no	—
240	C₂×C₄⋊C₄	—	43	2	no	—
241	C₂×C₄⋊C₄	—	86	1	no	—
242	C₂×C₄⋊C₄	—	86	2	no	—
243	C₂×C₄⋊C₄	—	86	2	no	—
244	C₂×C₄⋊C₄	—	86	1	no	—
245	C₂×C₄⋊C₄	—	86	1	no	—
246	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
247	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
248	C₂×C₄²	—	43	1	no	—
249	C₂×C₄²	—	43	1	no	—
250	C₂×C₄²	—	43	1	no	—
251	C₂×C₄²	—	43	1	no	—
252	C₂×C₄²	—	43	1	no	—
253	C₂×C₄²	—	43	1	no	—
254	C₂×C₄²	—	86	2	no	—
255	C₂×C₄²	—	86	2	no	—
256	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
257	#107	—	86	8	no	—
258	#106	—	43	16	no	—
259	#97	—	43	8	no	—
260	#97	—	86	4	no	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	43	4	no	—
263	#95	—	86	2	no	—
264	#95	—	86	2	no	—
265	#95	—	86	2	no	—
266	#95	—	86	2	no	—
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	—	4	2	no	—
270	C₂×Q₈	—	86	4	no	—
271	C₂×Q₈	—	172	2	no	—
272	C₂×Q₈	—	172	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	43	1	no	—
276	C₂²×C₄	—	43	1	no	—
277	C₂²×C₄	—	43	1	no	—
278	C₂²×C₄	—	43	1	no	—
279	C₂²×C₄	—	86	2	no	—
280	C₂²×C₄	—	86	2	no	—
281	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
282	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
283	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	43	4	no	—
287	C₂×C₈	—	43	4	no	—
288	C₂×C₈	—	43	4	no	—
289	C₂×C₈	Q₈×D₄₃	1	4	yes	—
290	C₄⋊C₄	—	2	4	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	43	4	no	—
296	C₄⋊C₄	—	86	4	no	—
297	C₄⋊C₄	—	86	2	no	—
298	C₄⋊C₄	—	86	4	no	—
299	C₄⋊C₄	—	172	4	no	—
300	C₄⋊C₄	—	172	4	no	—
301	C₄⋊C₄	—	172	2	no	—
302	C₄⋊C₄	—	172	2	no	—
303	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
304	C₄²	—	43	4	no	—
305	C₄²	—	86	2	no	—
306	C₄²	—	86	2	no	—
307	C₄²	—	86	2	no	—
308	C₄²	—	86	2	no	—
309	C₄²	—	86	2	no	—
310	C₄²	—	172	4	no	—
311	C₄²	—	172	4	no	—
312	C₄²	C₂×C₄×D₄₃	1	2	yes	—
313	C₄²	#10621900	1	2	yes	—
314	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
315	Q₈	—	4	8	no	—
316	Q₈	—	172	8	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	43	2	no	—
324	C₂×C₄	—	43	2	no	—
325	C₂×C₄	—	86	1	no	—
326	C₂×C₄	—	86	1	no	—
327	C₂×C₄	—	86	1	no	—
328	C₂×C₄	—	86	1	no	—
329	C₂×C₄	—	86	4	no	—
330	C₂×C₄	—	86	1	no	—
331	C₂×C₄	—	86	1	no	—
332	C₂×C₄	—	86	1	no	—
333	C₂×C₄	—	86	1	no	—
334	C₂×C₄	—	86	1	no	—
335	C₂×C₄	—	86	1	no	—
336	C₂×C₄	—	172	2	no	—
337	C₂×C₄	—	172	2	no	—
338	C₂×C₄	—	172	2	no	—
339	C₂×C₄	—	172	2	no	—
340	C₂×C₄	—	172	2	no	—
341	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
342	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
343	C₂×C₄	#13063349	1	2	yes	—
344	C₂×C₄	#13063352	1	2	yes	—
345	C₂×C₄	#13063354	1	2	yes	—
346	C₂×C₄	#13063357	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
348	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
349	C₂×C₄	#41×D₄₃	1	2	yes	—
350	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
351	C₈	—	2	4	no	—
352	C₈	—	43	8	no	—
353	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
354	C₂²	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
355	C₂²	#14583448	1	1	yes	characteristic
356	C₂²	C₂×#13063349	1	1	yes	characteristic
357	C₂²	C₂×#13063352	1	1	yes	characteristic
358	C₂²	C₂×#13063354	1	1	yes	characteristic
359	C₂²	C₂×#13063357	1	1	yes	characteristic
360	C₄	—	4	4	no	—
361	C₄	—	8	2	no	—
362	C₄	—	86	2	no	—
363	C₄	—	86	2	no	—
364	C₄	—	344	2	no	—
365	C₄	—	172	1	no	—
366	C₄	—	172	1	no	—
367	C₄	—	172	4	no	—
368	C₄	#14582996	1	4	yes	—
369	C₄	#14583002	1	4	yes	—
370	C₄	C₄×Q₈×D₄₃	1	2	yes	—
371	C₄	#14583451	1	2	yes	—
372	C₂	#19137164	1	1	yes	characteristic
373	C₂	C₂×#14582996	1	1	yes	characteristic
374	C₂	C₂×#14583002	1	1	yes	characteristic
375	C₂	#19145136	1	1	yes	characteristic
376	C₂	#19145138	1	1	yes	characteristic
377	C₂	#19145332	1	1	yes	characteristic
378	C₂	#19145339	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×#1990	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1990	D₄₃
hypocenter	C₄₃	#21727
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
13. #14583295	C₄	semidirect
60. #13063235	Q₈	semidirect
162. C₄₃⋊C₈	C₄×Q₈	semidirect
225. C₄₃	#21727	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583295	2
C₄⋊Q₈×Dic₄₃	2
#19137162	2
#19135556	2
#19135337	2
#19134754	2
#19134491	2
C₄×#13063235	2
C₄₃×#1990	2
#21727	43