Subgroups — #25081879

Order 11008.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145520	C₂	1	1	yes	maximal, characteristic
2	#19144067	C₂	1	1	yes	maximal, characteristic
3	#19143676	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134023	C₂	1	4	yes	maximal, complemented
5	#19133945	C₂	1	4	yes	maximal, complemented
6	#19133365	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133280	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133245	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063356	—	2	1	no	—
11	#14583449	C₂²	1	1	yes	characteristic
12	#14583273	C₂²	1	2	yes	—
13	#14583249	C₂²	1	2	yes	—
14	#14583121	—	2	2	no	—
15	#14583109	C₂²	1	2	yes	—
16	#14583101	—	2	2	no	—
17	#14583099	C₂²	1	2	yes	—
18	#14583038	—	2	2	no	—
19	Q₈×C₄₃⋊C₈	—	2	2	no	—
20	Q₈×C₄₃⋊C₈	C₂²	1	4	yes	—
21	#14582952	—	2	2	no	—
22	#14582952	C₂²	1	1	yes	characteristic
23	#14582952	C₂²	1	1	yes	characteristic
24	#14582923	C₂²	1	1	yes	characteristic
25	#14582898	—	2	1	no	—
26	#14582649	C₂²	1	4	yes	—
27	#14582646	C₂²	1	2	yes	—
28	#14582645	C₂²	1	4	yes	—
29	#14582642	C₂²	1	2	yes	—
30	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₄₃×#475	—	2	2	no	—
33	C₄₃×#475	C₂²	1	4	yes	—
34	C₂×#106×C₄₃	—	2	1	no	—
35	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#103×C₄₃	—	2	1	no	—
37	C₂×#10621898	—	2	1	no	—
38	#13063356	—	4	4	no	—
39	C₂×#10621885	—	2	2	no	—
40	C₂×#10621884	—	2	2	no	—
41	C₂×#10621884	—	2	4	no	—
42	C₂×#10621877	—	2	1	no	—
43	#13063325	C₂³	1	1	yes	characteristic
44	#13063322	C₂³	1	1	yes	characteristic
45	#13063313	—	2	1	no	—
46	#13063299	—	2	1	no	—
47	#13063299	—	2	1	no	—
48	C₄×D₁₇₂	—	2	1	no	—
49	C₄×D₁₇₂	—	2	1	no	—
50	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
51	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
52	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
53	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
54	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
55	#13063249	—	2	4	no	—
56	#13063248	—	2	2	no	—
57	#13063245	—	2	2	no	—
58	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
59	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
60	C₄×C₄₃⋊C₈	—	2	1	no	—
61	C₄×C₄₃⋊C₈	—	2	4	no	—
62	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
63	C₂²×Q₈×C₄₃	—	2	1	no	—
64	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₄₃	—	2	1	no	—
66	C₄⋊Q₈×C₄₃	—	2	1	no	—
67	C₄⋊Q₈×C₄₃	—	2	1	no	—
68	C₄⋊Q₈×C₄₃	—	2	1	no	—
69	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
70	C₄×Q₈×C₄₃	—	2	1	no	—
71	C₄×Q₈×C₄₃	—	2	1	no	—
72	C₄×Q₈×C₄₃	D₄	1	2	yes	—
73	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₄₃	—	2	1	no	—
79	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
80	#106×C₄₃	—	2	4	no	—
81	C₄⋊C₈×C₄₃	—	2	1	no	—
82	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
83	#103×C₄₃	—	2	4	no	—
84	C₂×Q₈×C₄₃	—	2	1	no	—
85	C₂×Q₈×C₄₃	—	2	2	no	—
86	C₂×Q₈×C₄₃	—	2	2	no	—
87	C₂×Q₈×C₄₃	—	4	1	no	—
88	C₂×Q₈×C₄₃	—	8	2	no	—
89	C₂×Q₈×C₄₃	SD₁₆	1	4	yes	complemented
90	C₂×Q₈×C₄₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₄₃	—	2	1	no	—
92	C₂²×C₄×C₄₃	—	2	1	no	—
93	C₂²×C₄×C₄₃	—	4	1	no	—
94	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
97	C₂×C₂²⋊D₄₃	—	4	1	no	—
98	#10621898	—	4	2	no	—
99	#10621898	—	4	2	no	—
100	C₂×D₁₇₂	—	2	1	no	—
101	C₂×D₁₇₂	—	2	1	no	—
102	C₂×D₁₇₂	—	2	1	no	—
103	C₂×C₄×D₄₃	—	4	1	no	—
104	C₂×Dic₈₆	—	2	1	no	—
105	C₂×C₈×C₄₃	—	2	2	no	—
106	C₂×C₈×C₄₃	—	4	2	no	—
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	—	2	1	no	—
109	C₄⋊C₄×C₄₃	—	2	2	no	—
110	C₄⋊C₄×C₄₃	—	2	2	no	—
111	C₄⋊C₄×C₄₃	—	2	2	no	—
112	C₄⋊C₄×C₄₃	—	4	1	no	—
113	C₄⋊C₄×C₄₃	—	4	1	no	—
114	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
115	C₄²×C₄₃	—	2	2	no	—
116	C₄²×C₄₃	—	2	2	no	—
117	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₃	#41	1	1	yes	characteristic
120	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
121	#10621885	—	4	4	no	—
122	#10621884	—	4	4	no	—
123	#10621884	—	8	4	no	—
124	(C₂×D₄₃)⋊C₄	—	4	1	no	—
125	(C₂×D₄₃)⋊C₄	—	4	1	no	—
126	C₄⋊Dic₄₃	—	2	1	no	—
127	C₄⋊Dic₄₃	—	2	1	no	—
128	#10621877	—	4	2	no	—
129	C₂×C₄₃⋊C₈	—	2	2	no	—
130	C₂×C₄₃⋊C₈	—	2	2	no	—
131	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
132	C₂³×C₄₃	#121	1	1	yes	characteristic
133	C₂²×D₄₃	—	4	1	no	—
134	Q₈×C₄₃	—	2	2	no	—
135	Q₈×C₄₃	—	2	4	no	—
136	Q₈×C₄₃	—	4	4	no	—
137	Q₈×C₄₃	—	8	2	no	—
138	Q₈×C₄₃	C₂×SD₁₆	1	4	yes	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	2	4	no	—
141	C₂×C₄×C₄₃	—	2	2	no	—
142	C₂×C₄×C₄₃	—	2	2	no	—
143	C₂×C₄×C₄₃	—	4	1	no	—
144	C₂×C₄×C₄₃	—	4	2	no	—
145	C₂×C₄×C₄₃	—	8	1	no	—
146	C₂×C₄×C₄₃	—	8	2	no	—
147	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
151	C₂×C₄×C₄₃	#135	1	2	yes	—
152	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
155	C₂²⋊D₄₃	—	8	2	no	—
156	C₂×Dic₄₃	—	4	1	no	—
157	D₁₇₂	—	4	1	no	—
158	D₁₇₂	—	4	1	no	—
159	D₁₇₂	—	4	2	no	—
160	D₁₇₂	—	8	1	no	—
161	C₄×D₄₃	—	8	2	no	—
162	Dic₈₆	—	4	1	no	—
163	Dic₈₆	—	4	1	no	—
164	C₈×C₄₃	—	4	2	no	—
165	C₄₃⋊C₈	—	2	4	no	—
166	C₄₃⋊C₈	—	4	2	no	—
167	C₄₃⋊C₈	—	4	2	no	—
168	#19397	—	43	1	no	maximal, Sylow
169	C₂²×C₄₃	—	2	1	no	—
170	C₂²×C₄₃	—	2	1	no	—
171	C₂²×C₄₃	#465	1	2	yes	—
172	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
173	C₂×D₄₃	—	8	1	no	—
174	C₂×D₄₃	—	8	1	no	—
175	C₂×D₄₃	—	8	1	no	—
176	C₄×C₄₃	—	2	1	no	—
177	C₄×C₄₃	—	2	4	no	—
178	C₄×C₄₃	—	2	1	no	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	4	2	no	—
181	C₄×C₄₃	—	8	2	no	—
182	C₄×C₄₃	#465	1	2	yes	—
183	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
184	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
185	C₄×C₄₃	#577	1	2	yes	—
186	Dic₄₃	—	8	1	no	—
187	#3206	—	43	1	no	—
188	C₂×#475	D₄₃	1	1	yes	characteristic, complemented
189	#3037	—	43	1	no	—
190	#2982	—	43	1	no	—
191	#1680	—	43	4	no	—
192	#1666	—	43	4	no	—
193	#1570	—	43	1	no	—
194	#1549	—	43	1	no	—
195	#1540	—	43	1	no	—
196	C₂×C₄₃	—	2	2	no	—
197	C₂×C₄₃	#3043	1	1	yes	characteristic
198	C₂×C₄₃	C₂×#465	1	1	yes	characteristic
199	C₂×C₄₃	#3069	1	1	yes	characteristic
200	D₄₃	—	16	1	no	—
201	C₂×#137	—	86	1	no	—
202	#532	—	43	1	no	—
203	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
204	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
205	#492	—	43	2	no	—
206	#487	—	43	2	no	—
207	#475	—	2	2	no	—
208	#475	C₂×D₄₃	1	4	yes	—
209	#464	—	86	2	no	—
210	#462	—	43	2	no	—
211	#461	—	86	2	no	—
212	#460	—	43	2	no	—
213	#448	—	86	2	no	—
214	C₈×Q₈	—	43	4	no	—
215	C₈×Q₈	—	86	2	no	—
216	#431	—	43	1	no	—
217	#431	—	43	1	no	—
218	#431	—	86	2	no	—
219	C₂×#106	—	2	1	no	—
220	#423	—	43	1	no	—
221	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
222	#417	—	86	1	no	—
223	C₂×#103	—	2	1	no	—
224	#334	—	43	4	no	—
225	#332	—	43	4	no	—
226	#332	—	43	2	no	—
227	#331	—	43	2	no	—
228	C₄₃	#19397	1	1	yes	characteristic, complemented, Sylow
229	C₂×#41	—	86	1	no	—
230	C₂²×Q₈	—	2	1	no	—
231	C₂²×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
232	#137	—	172	4	no	—
233	C₂×Q₁₆	—	86	2	no	—
234	C₂×SD₁₆	—	86	4	no	—
235	C₂×SD₁₆	—	86	2	no	—
236	C₂×M₁₆	—	86	1	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	1	no	—
240	C₄⋊Q₈	—	2	1	no	—
241	C₄⋊Q₈	—	43	1	no	—
242	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
243	#127	—	43	1	no	—
244	#124	—	86	1	no	—
245	#121	—	86	1	no	—
246	#121	—	86	1	no	—
247	C₄×Q₈	—	2	1	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×Q₈	C₂²⋊D₄₃	1	2	yes	—
250	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
251	C₄×D₄	—	86	1	no	—
252	C₄×D₄	—	86	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
257	C₂×C₄²	—	2	1	no	—
258	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
259	#106	—	2	4	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	43	2	no	—
262	C₄⋊C₈	—	43	2	no	—
263	C₄⋊C₈	—	43	2	no	—
264	C₄⋊C₈	—	86	4	no	—
265	C₄⋊C₈	—	86	1	no	—
266	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
267	#103	—	2	4	no	—
268	#103	—	86	4	no	—
269	#103	—	86	2	no	—
270	#102	—	86	2	no	—
271	C₂²⋊C₈	—	43	2	no	—
272	C₂²⋊C₈	—	86	1	no	—
273	C₄×C₈	—	43	2	no	—
274	C₄×C₈	—	86	4	no	—
275	C₄×C₈	—	86	1	no	—
276	#41	—	172	2	no	—
277	#41	—	172	2	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	4	1	no	—
282	C₂×Q₈	—	8	2	no	—
283	C₂×Q₈	—	86	1	no	—
284	C₂×Q₈	#10621884	1	4	yes	complemented
285	C₂×Q₈	C₂×C₂²⋊D₄₃	1	2	yes	—
286	C₂×D₄	—	86	1	no	—
287	C₂×D₄	—	86	1	no	—
288	C₂×D₄	—	86	1	no	—
289	C₂×D₄	—	172	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	172	1	no	—
294	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
295	C₂²×C₄	#10621900	1	1	yes	characteristic
296	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
297	Q₁₆	—	172	4	no	—
298	SD₁₆	—	344	4	no	—
299	SD₁₆	—	172	4	no	—
300	M₁₆	—	172	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	2	no	—
303	C₂×C₈	—	43	4	no	—
304	C₂×C₈	—	86	2	no	—
305	C₂×C₈	—	86	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	86	1	no	—
314	C₄⋊C₄	—	86	1	no	—
315	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
316	C₂²⋊C₄	—	172	1	no	—
317	C₂²⋊C₄	—	172	1	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	2	no	—
320	C₄²	D₄×D₄₃	1	1	yes	characteristic
321	C₄²	#10621900	1	1	yes	characteristic
322	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
323	C₄²	C₂³×D₄₃	1	1	yes	characteristic
324	C₂³	—	172	1	no	—
325	C₂³	#13063297	1	1	yes	characteristic
326	Q₈	—	2	4	no	—
327	Q₈	—	2	2	no	—
328	Q₈	—	4	4	no	—
329	Q₈	—	8	2	no	—
330	Q₈	—	172	1	no	—
331	Q₈	—	172	1	no	—
332	Q₈	C₂×#10621884	1	4	yes	—
333	D₄	—	344	1	no	—
334	D₄	—	344	2	no	—
335	D₄	—	172	1	no	—
336	D₄	—	172	1	no	—
337	D₄	—	172	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	8	1	no	—
345	C₂×C₄	—	8	2	no	—
346	C₂×C₄	—	344	2	no	—
347	C₂×C₄	—	172	1	no	—
348	C₂×C₄	#13063297	1	1	yes	characteristic
349	C₂×C₄	#13063297	1	1	yes	characteristic
350	C₂×C₄	#13063297	1	1	yes	characteristic
351	C₂×C₄	C₂×#10621884	1	2	yes	—
352	C₂×C₄	#13063345	1	2	yes	—
353	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
355	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
356	C₈	—	4	2	no	—
357	C₈	—	86	4	no	—
358	C₈	—	172	2	no	—
359	C₈	—	172	2	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	344	1	no	—
363	C₂²	—	344	1	no	—
364	C₂²	—	344	1	no	—
365	C₂²	#14583130	1	2	yes	—
366	C₂²	C₂×#13063297	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	4	no	—
371	C₄	—	4	2	no	—
372	C₄	—	8	2	no	—
373	C₄	—	344	1	no	—
374	C₄	#14583130	1	2	yes	—
375	C₄	C₂²×#10621884	1	1	yes	characteristic
376	C₄	C₂×#13063345	1	1	yes	characteristic
377	C₄	#14583705	1	2	yes	—
378	C₂	—	2	2	no	—
379	C₂	—	688	1	no	—
380	C₂	#19144113	1	1	yes	characteristic
381	C₂	C₂×#14583130	1	1	yes	characteristic
382	C₂	#19144329	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063297
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#475	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	#14583449	C₂²
hypercenter	C₂×#475	D₄₃
hypocenter	C₄₃	#19397
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#475

Complements

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

N	H ≅ G/N	type
3. #19143676	C₂	semidirect
4. #19134023	C₂	semidirect
5. #19133945	C₂	semidirect
6. #19133365	C₂	semidirect
7. #19133280	C₂	semidirect
9. C₂×C₄₃×#475	C₂	semidirect
89. C₂×Q₈×C₄₃	SD₁₆	semidirect
188. C₂×#475	D₄₃	semidirect
228. C₄₃	#19397	semidirect
284. C₂×Q₈	#10621884	semidirect

Maximal subgroups

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

subgroup	index
#19145520	2
#19144067	2
#19143676	2
#19134023	2
#19133945	2
#19133365	2
#19133280	2
#19133245	2
C₂×C₄₃×#475	2
#19397	43