Subgroups — #25066067

Order 11008.

678 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143676	C₂	1	1	yes	maximal, characteristic
2	#19143675	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#10621885	C₂	1	1	yes	maximal, characteristic
4	#19133411	C₂	1	4	yes	maximal, complemented
5	#19133399	C₂	1	4	yes	maximal, complemented
6	#19133341	C₂	1	1	yes	maximal, characteristic
7	#19132991	C₂	1	1	yes	maximal, characteristic
8	#19132879	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1479	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10621885	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
12	Q₈×C₄₃⋊C₈	—	2	2	no	—
13	Q₈×C₄₃⋊C₈	C₂²	1	4	yes	—
14	C₈×Dic₈₆	—	2	2	no	—
15	#14583023	C₂²	1	4	yes	—
16	C₄×#10621885	C₄	1	2	yes	—
17	C₄×#10621885	C₄	1	2	yes	—
18	C₄×#10621885	C₂²	1	2	yes	—
19	C₄×#10621885	C₂²	1	2	yes	—
20	#14582952	—	2	2	no	—
21	#14582952	C₂²	1	1	yes	characteristic
22	#14582952	C₂²	1	1	yes	characteristic
23	#14582949	—	2	2	no	—
24	C₂×#13063258	C₄	1	1	yes	characteristic
25	C₂×#13063249	C₄	1	1	yes	characteristic
26	C₂×#13063248	C₄	1	1	yes	characteristic
27	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
28	#14582651	C₂²	1	2	yes	—
29	#14582649	C₂²	1	2	yes	—
30	#14582631	C₂²	1	4	yes	—
31	#14582621	C₂²	1	2	yes	—
32	C₈×C₄₃⋊C₈	C₂²	1	2	yes	—
33	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#431	C₂²	1	1	yes	characteristic
35	C₄₃×#431	C₂²	1	1	yes	characteristic
36	C₄₃×Q₈⋊C₈	C₂²	1	4	yes	—
37	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
38	C₂×#10621885	C₈	1	8	yes	complemented
39	C₂×#10621885	C₂×C₄	1	2	yes	—
40	C₂×#10621885	C₂×C₄	1	2	yes	—
41	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₈₆	—	2	1	no	—
44	C₄×Dic₈₆	—	2	1	no	—
45	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
46	C₄×Dic₈₆	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂×Dic₄₃⋊C₄	—	2	1	no	—
49	C₂×C₄×Dic₄₃	—	2	1	no	—
50	#13063258	C₂×C₄	1	2	yes	—
51	#13063258	C₂×C₄	1	2	yes	—
52	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
53	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
54	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
55	Dic₄₃⋊C₈	—	2	4	no	—
56	#13063249	C₂×C₄	1	2	yes	—
57	#13063249	C₂×C₄	1	2	yes	—
58	#13063248	C₂×C₄	1	2	yes	—
59	#13063248	C₂×C₄	1	2	yes	—
60	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
61	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
62	C₄×C₄₃⋊C₈	—	2	1	no	—
63	C₄×C₄₃⋊C₈	—	2	4	no	—
64	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
67	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
69	C₈×Dic₄₃	—	2	4	no	—
70	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Q₈×C₄₃	—	2	1	no	—
72	C₄×Q₈×C₄₃	—	2	1	no	—
73	C₄×Q₈×C₄₃	C₂×C₄	1	2	yes	—
74	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₄₃	—	2	1	no	—
78	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₄₃	—	2	1	no	—
80	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
81	C₂²⋊C₈×C₄₃	—	2	1	no	—
82	C₂²⋊C₈×C₄₃	—	2	1	no	—
83	C₄×C₈×C₄₃	—	2	1	no	—
84	C₄×C₈×C₄₃	C₂³	1	2	yes	—
85	C₂×Q₈×C₄₃	—	2	1	no	—
86	C₂×Q₈×C₄₃	—	2	2	no	—
87	C₂×Q₈×C₄₃	C₂×C₈	1	4	yes	—
88	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₄₃	—	2	1	no	—
90	C₂²×C₄×C₄₃	—	2	1	no	—
91	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
94	C₂²×Dic₄₃	—	2	1	no	—
95	C₂²×Dic₄₃	—	2	1	no	—
96	C₂×Dic₈₆	—	2	1	no	—
97	C₂×Dic₈₆	—	2	2	no	—
98	C₂×Dic₈₆	C₂×C₈	1	4	yes	—
99	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	2	1	no	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₄₃	—	2	2	no	—
107	C₄²×C₄₃	—	2	2	no	—
108	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₃	#41	1	1	yes	characteristic
111	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
112	#10621885	C₂×C₈	1	8	yes	—
113	#10621885	C₂×C₈	1	8	yes	—
114	C₄⋊Dic₄₃	—	2	1	no	—
115	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
116	Dic₄₃⋊C₄	—	2	2	no	—
117	Dic₄₃⋊C₄	—	2	2	no	—
118	C₄×Dic₄₃	—	2	2	no	—
119	C₄×Dic₄₃	—	2	2	no	—
120	C₂×C₄₃⋊C₈	—	2	2	no	—
121	C₂×C₄₃⋊C₈	—	2	2	no	—
122	C₂×C₄₃⋊C₈	C₂×C₈	1	4	yes	—
123	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
128	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
129	Q₈×C₄₃	—	2	2	no	—
130	Q₈×C₄₃	—	2	4	no	—
131	Q₈×C₄₃	C₂²×C₈	1	4	yes	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	2	4	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	—	8	1	no	—
138	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
142	C₂×C₄×C₄₃	#131	1	2	yes	—
143	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	2	4	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	—	8	1	no	—
150	Dic₈₆	—	2	2	no	—
151	Dic₈₆	—	2	4	no	—
152	Dic₈₆	C₂²×C₈	1	4	yes	—
153	C₈×C₄₃	—	4	2	no	—
154	C₈×C₄₃	—	4	2	no	—
155	C₄₃⋊C₈	—	2	1	no	—
156	C₄₃⋊C₈	—	2	1	no	—
157	C₄₃⋊C₈	—	2	4	no	—
158	C₄₃⋊C₈	—	4	2	no	—
159	C₄₃⋊C₈	C₂²×C₈	1	2	yes	—
160	C₄₃⋊C₈	C₂²×C₈	1	2	yes	—
161	#17856	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	C₈×D₄	1	2	yes	—
165	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄×C₄₃	—	2	4	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	4	2	no	—
171	C₄×C₄₃	C₈×D₄	1	2	yes	—
172	C₄×C₄₃	C₂³×C₈	1	1	yes	characteristic
173	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
174	C₄×C₄₃	#578	1	2	yes	—
175	Dic₄₃	—	2	4	no	—
176	Dic₄₃	—	4	2	no	—
177	#2982	—	43	1	no	—
178	#2982	—	43	1	no	—
179	C₂×C₄×Q₁₆	—	43	1	no	—
180	#1579	—	43	4	no	—
181	C₈×Q₁₆	—	43	4	no	—
182	#1564	—	43	1	no	—
183	#1479	—	43	1	no	—
184	#1479	D₄₃	1	1	yes	characteristic
185	#1450	—	43	1	no	—
186	C₂×C₄₃	—	2	2	no	—
187	C₂×C₄₃	C₂×C₈×D₄	1	1	yes	characteristic
188	C₂×C₄₃	#2950	1	1	yes	characteristic
189	C₂×C₄₃	#2957	1	1	yes	characteristic
190	C₂²×Q₁₆	—	43	1	no	—
191	C₂×C₄×Q₈	—	43	1	no	—
192	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
193	C₈×Q₈	—	43	4	no	—
194	C₈×Q₈	—	43	4	no	—
195	C₈×Q₈	—	86	2	no	—
196	C₈×Q₈	—	86	2	no	—
197	C₄×Q₁₆	—	43	2	no	—
198	C₄×Q₁₆	—	43	2	no	—
199	C₄×Q₁₆	—	43	2	no	—
200	C₄×Q₁₆	—	43	2	no	—
201	#431	—	43	1	no	—
202	#431	—	43	1	no	—
203	#431	—	86	2	no	—
204	#431	—	86	2	no	—
205	#431	C₂×D₄₃	1	1	yes	characteristic
206	#431	C₂×D₄₃	1	1	yes	characteristic
207	C₂×#107	—	43	1	no	—
208	C₂×#103	—	43	1	no	—
209	C₂×#103	—	43	1	no	—
210	C₂×C₄×C₈	—	43	1	no	—
211	#335	—	43	2	no	—
212	#334	—	43	2	no	—
213	Q₈⋊C₈	—	43	4	no	—
214	Q₈⋊C₈	C₂×D₄₃	1	4	yes	—
215	#322	—	43	2	no	—
216	C₈²	—	43	2	no	—
217	C₄₃	#17856	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	—	43	1	no	—
219	C₂²×Q₈	C₄×D₄₃	1	1	yes	characteristic
220	C₂×Q₁₆	—	43	8	no	—
221	C₂×Q₁₆	—	43	2	no	—
222	C₂×Q₁₆	—	43	2	no	—
223	C₂²×C₈	—	43	1	no	—
224	C₂²×C₈	—	43	1	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	—	2	1	no	—
227	C₄×Q₈	—	43	2	no	—
228	C₄×Q₈	—	43	2	no	—
229	C₄×Q₈	—	86	1	no	—
230	C₄×Q₈	—	86	1	no	—
231	C₄×Q₈	C₄×D₄₃	1	2	yes	—
232	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	43	1	no	—
235	C₂×C₄⋊C₄	—	86	1	no	—
236	C₂×C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
237	C₂×C₄²	—	2	1	no	—
238	C₂×C₄²	—	86	1	no	—
239	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
240	#107	—	43	2	no	—
241	#107	—	43	2	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	43	2	no	—
244	C₄⋊C₈	—	43	2	no	—
245	C₄⋊C₈	—	86	4	no	—
246	C₄⋊C₈	—	86	4	no	—
247	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
248	#103	—	43	2	no	—
249	#103	—	43	2	no	—
250	#103	—	43	2	no	—
251	#103	—	43	2	no	—
252	C₂²⋊C₈	—	2	1	no	—
253	C₂²⋊C₈	—	2	1	no	—
254	C₂²⋊C₈	—	43	2	no	—
255	C₂²⋊C₈	—	86	1	no	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	43	1	no	—
258	C₄×C₈	—	43	1	no	—
259	C₄×C₈	—	43	1	no	—
260	C₄×C₈	—	43	2	no	—
261	C₄×C₈	—	43	1	no	—
262	C₄×C₈	—	86	4	no	—
263	C₄×C₈	—	86	4	no	—
264	C₄×C₈	—	86	1	no	—
265	C₄×C₈	C₂²×D₄₃	1	2	yes	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	43	4	no	—
269	C₂×Q₈	—	43	2	no	—
270	C₂×Q₈	—	86	1	no	—
271	C₂×Q₈	—	86	2	no	—
272	C₂×Q₈	C₈×D₄₃	1	4	yes	—
273	C₂×Q₈	C₂×C₄×D₄₃	1	2	yes	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	86	1	no	—
277	C₂²×C₄	—	86	1	no	—
278	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
280	C₂²×C₄	#10621900	1	1	yes	characteristic
281	Q₁₆	—	43	8	no	—
282	Q₁₆	—	43	8	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	43	4	no	—
286	C₂×C₈	—	43	1	no	—
287	C₂×C₈	—	43	1	no	—
288	C₂×C₈	—	43	1	no	—
289	C₂×C₈	—	43	1	no	—
290	C₂×C₈	—	43	4	no	—
291	C₂×C₈	—	86	2	no	—
292	C₂×C₈	—	86	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	43	2	no	—
297	C₄⋊C₄	—	86	1	no	—
298	C₄⋊C₄	—	86	2	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₄⋊C₄	C₂×C₄×D₄₃	1	2	yes	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	86	2	no	—
304	C₄²	—	86	2	no	—
305	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
306	C₄²	D₄×D₄₃	1	1	yes	characteristic
307	C₄²	#10621900	1	1	yes	characteristic
308	C₄²	C₂³×D₄₃	1	1	yes	characteristic
309	C₂³	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
310	Q₈	—	2	2	no	—
311	Q₈	—	2	4	no	—
312	Q₈	—	43	4	no	—
313	Q₈	—	86	2	no	—
314	Q₈	—	86	4	no	—
315	Q₈	C₂×C₈×D₄₃	1	4	yes	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	8	1	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	86	4	no	—
324	C₂×C₄	—	86	2	no	—
325	C₂×C₄	—	344	1	no	—
326	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
327	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
328	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
329	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
330	C₂×C₄	#13063333	1	2	yes	—
331	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
333	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₈	—	43	2	no	—
337	C₈	—	43	2	no	—
338	C₈	—	86	1	no	—
339	C₈	—	86	1	no	—
340	C₈	—	86	4	no	—
341	C₈	—	172	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	(C₂²⋊D₄₃)⋊C₈	1	2	yes	—
345	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	4	no	—
350	C₄	—	4	2	no	—
351	C₄	—	86	4	no	—
352	C₄	—	172	2	no	—
353	C₄	(C₂²⋊D₄₃)⋊C₈	1	2	yes	—
354	C₄	C₂²×C₈×D₄₃	1	1	yes	characteristic
355	C₄	C₂×#13063333	1	1	yes	characteristic
356	C₄	#14583715	1	2	yes	—
357	C₂	—	2	2	no	—
358	C₂	C₂×(C₂²⋊D₄₃)⋊C₈	1	1	yes	characteristic
359	C₂	#19143459	1	1	yes	characteristic
360	C₂	#19143508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₄×C₄₃	C₂³×C₈
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1479	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1479	D₄₃
hypocenter	C₄₃	#17856
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1479

Complements

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

N	H ≅ G/N	type
4. #19133411	C₂	semidirect
5. #19133399	C₂	semidirect
38. C₂×#10621885	C₈	semidirect
217. C₄₃	#17856	semidirect

Maximal subgroups

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

subgroup	index
#19143676	2
#19143675	2
C₂×C₄×#10621885	2
#19133411	2
#19133399	2
#19133341	2
#19132991	2
#19132879	2
C₄₃×#1479	2
#17856	43