Subgroups — #25249944

Order 11008.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583340	C₂	1	1	yes	maximal, characteristic
2	#19143469	C₂	1	1	yes	maximal, characteristic
3	#19143463	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136463	C₂	1	2	yes	maximal, complemented
5	#19136462	C₂	1	2	yes	maximal, complemented
6	#19136451	C₂	1	2	yes	maximal, complemented
7	#19136444	C₂	1	2	yes	maximal, complemented
8	#19136381	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136303	C₂	1	1	yes	maximal, characteristic
10	#19136191	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	1	yes	characteristic
13	#14583380	—	2	1	no	—
14	#14583377	—	2	1	no	—
15	#14583340	C₄	1	8	yes	complemented
16	C₂×#13063271	C₂²	1	2	yes	—
17	C₂×#13063268	C₂²	1	2	yes	—
18	C₂×#13063265	C₂²	1	1	yes	characteristic
19	#14583006	—	2	2	no	—
20	#14582998	—	2	2	no	—
21	Q₁₆×Dic₄₃	C₂²	1	2	yes	—
22	Q₁₆×Dic₄₃	C₂²	1	2	yes	—
23	#14582993	C₂²	1	2	yes	—
24	#14582981	C₂²	1	2	yes	—
25	#14582937	C₂²	1	1	yes	characteristic
26	#14582908	—	2	1	no	—
27	#14582898	—	2	1	no	—
28	#14582854	C₂²	1	1	yes	characteristic
29	#14582726	C₂²	1	2	yes	—
30	#14582724	C₂²	1	2	yes	—
31	#14582704	C₂²	1	1	yes	characteristic
32	#14582704	C₂²	1	1	yes	characteristic
33	#14582701	C₂²	1	1	yes	characteristic
34	#14582701	C₂²	1	1	yes	characteristic
35	#14582674	C₂²	1	2	yes	—
36	C₁₆×Dic₄₃	C₂²	1	2	yes	—
37	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
38	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
39	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
40	C₄₃×#358	C₂²	1	2	yes	—
41	C₄₃×#358	C₂²	1	2	yes	—
42	C₂×#10621898	—	2	1	no	—
43	#13063347	—	2	2	no	—
44	#13063347	C₂×C₄	1	4	yes	—
45	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
46	C₂×#10621873	—	2	1	no	—
47	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
48	C₂×#10621877	D₄	1	1	yes	characteristic
49	Q₈×Dic₄₃	—	2	2	no	—
50	Q₈×Dic₄₃	—	2	2	no	—
51	Q₈×Dic₄₃	—	4	1	no	—
52	Q₈×Dic₄₃	—	4	1	no	—
53	#13063290	—	2	2	no	—
54	#13063287	—	2	2	no	—
55	C₄×C₂²⋊D₄₃	—	4	2	no	—
56	#13063284	—	2	1	no	—
57	#13063284	—	4	2	no	—
58	#13063284	D₄	1	1	yes	characteristic
59	#13063283	—	4	1	no	—
60	C₄⋊C₄×D₄₃	—	4	1	no	—
61	#13063271	C₂×C₄	1	8	yes	complemented
62	#13063268	C₂×C₄	1	8	yes	complemented
63	#13063265	C₂×C₄	1	4	yes	complemented
64	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
65	#13063259	C₂³	1	1	yes	characteristic
66	#13063259	C₂³	1	1	yes	characteristic
67	#13063257	—	2	1	no	—
68	#13063249	—	2	2	no	—
69	#13063249	—	2	2	no	—
70	#13063248	—	2	2	no	—
71	#13063245	—	2	2	no	—
72	#13063235	D₄	1	2	yes	—
73	C₈×Dic₄₃	C₂³	1	2	yes	—
74	C₂²×Q₈×C₄₃	—	2	1	no	—
75	C₂×Q₁₆×C₄₃	—	2	1	no	—
76	C₂×Q₁₆×C₄₃	—	2	2	no	—
77	C₂×Q₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
78	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
79	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
81	C₂×C₁₆×C₄₃	—	2	2	no	—
82	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
83	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
84	#107×C₄₃	—	2	1	no	—
85	#107×C₄₃	C₂³	1	2	yes	—
86	C₂×Q₈×C₄₃	—	2	4	no	—
87	C₂×Q₈×C₄₃	—	2	2	no	—
88	C₂×Q₈×C₄₃	—	4	1	no	—
89	C₂×Q₈×C₄₃	—	4	2	no	—
90	C₂²×C₄×C₄₃	—	4	1	no	—
91	C₂²×C₄×C₄₃	—	4	1	no	—
92	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₄₃	—	4	1	no	—
94	#10621898	—	2	4	no	—
95	#10621898	—	4	2	no	—
96	C₂×D₁₇₂	—	2	1	no	—
97	C₂×C₄×D₄₃	—	4	1	no	—
98	C₂×C₄×D₄₃	—	4	2	no	—
99	C₂×Dic₈₆	—	2	1	no	—
100	Q₁₆×C₄₃	—	2	4	no	—
101	Q₁₆×C₄₃	—	2	2	no	—
102	Q₁₆×C₄₃	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	#41	1	2	yes	—
106	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	—	4	1	no	—
109	C₂²⋊Dic₄₃	—	4	1	no	—
110	C₂²⋊Dic₄₃	D₈	1	2	yes	—
111	(C₂×D₄₃)⋊C₄	—	4	2	no	—
112	C₄⋊Dic₄₃	—	2	1	no	—
113	C₄⋊Dic₄₃	—	2	1	no	—
114	C₄⋊Dic₄₃	—	4	2	no	—
115	C₄⋊Dic₄₃	—	4	2	no	—
116	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
117	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
118	Dic₄₃⋊C₄	—	4	2	no	—
119	C₄×Dic₄₃	—	4	1	no	—
120	C₄×Dic₄₃	—	4	2	no	—
121	C₄×Dic₄₃	—	4	2	no	—
122	C₄×Dic₄₃	—	4	2	no	—
123	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
124	#10621877	—	2	2	no	—
125	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
126	Dic₁₇₂	—	2	1	no	—
127	Dic₁₇₂	C₂²×C₄	1	2	yes	—
128	D₃₄₄	—	2	1	no	—
129	D₃₄₄	C₂²×C₄	1	2	yes	—
130	#10621873	—	2	4	no	—
131	C₁₆×C₄₃	—	2	1	no	—
132	C₁₆×C₄₃	—	2	1	no	—
133	C₄₃⋊C₁₆	C₂²×C₄	1	4	yes	—
134	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₄₃	—	4	1	no	—
136	Q₈×C₄₃	—	2	4	no	—
137	Q₈×C₄₃	—	4	2	no	—
138	Q₈×C₄₃	—	4	4	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	4	4	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	—	16	1	no	—
143	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#135	1	2	yes	—
146	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₄₃	—	4	4	no	—
148	C₂×Dic₄₃	—	4	1	no	—
149	C₂×Dic₄₃	—	4	1	no	—
150	C₂×Dic₄₃	—	4	1	no	—
151	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
152	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
153	D₁₇₂	—	2	2	no	—
154	D₁₇₂	—	4	1	no	—
155	C₄×D₄₃	—	4	4	no	—
156	C₄×D₄₃	—	16	2	no	—
157	Dic₈₆	—	2	2	no	—
158	Dic₈₆	—	4	1	no	—
159	C₈×C₄₃	C₄×D₄	1	2	yes	—
160	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
161	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
162	C₄₃⋊C₈	—	2	2	no	—
163	#33455	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	C₄×D₈	1	2	yes	—
167	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₄₃	—	4	2	no	—
169	C₂×D₄₃	—	8	1	no	—
170	C₂×D₄₃	—	8	1	no	—
171	C₄×C₄₃	—	4	4	no	—
172	C₄×C₄₃	—	8	2	no	—
173	C₄×C₄₃	C₄×D₈	1	2	yes	—
174	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
176	Dic₄₃	—	2	2	no	—
177	Dic₄₃	—	2	2	no	—
178	Dic₄₃	—	4	2	no	—
179	Dic₄₃	—	8	1	no	—
180	Dic₄₃	—	8	1	no	—
181	C₂×#510	—	43	1	no	—
182	#2951	—	43	1	no	—
183	#2950	—	43	1	no	—
184	#2176	—	43	2	no	—
185	#2175	—	43	2	no	—
186	C₄×Q₃₂	—	43	2	no	—
187	C₄×SD₃₂	—	43	2	no	—
188	#2159	—	43	1	no	—
189	#2138	—	43	1	no	—
190	C₂×#358	D₄₃	1	1	yes	characteristic, complemented
191	#2108	—	43	1	no	—
192	C₂×C₄₃	—	2	2	no	—
193	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
194	C₂×C₄₃	#3416	1	1	yes	characteristic
195	C₂×C₄₃	#3417	1	1	yes	characteristic
196	D₄₃	—	8	2	no	—
197	C₂×#135	—	43	1	no	—
198	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
199	#520	—	86	1	no	—
200	#519	—	86	1	no	—
201	#510	—	43	8	no	—
202	C₂×Q₃₂	—	43	2	no	—
203	C₂×SD₃₂	—	43	2	no	—
204	C₂×M₃₂	—	43	1	no	—
205	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
206	#441	—	86	2	no	—
207	#440	—	86	2	no	—
208	C₄×Q₁₆	—	43	2	no	—
209	C₄×Q₁₆	—	43	2	no	—
210	C₄×Q₁₆	—	43	2	no	—
211	C₄×D₈	—	43	2	no	—
212	#427	—	43	1	no	—
213	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
214	#419	—	86	1	no	—
215	#417	—	86	1	no	—
216	#405	—	43	1	no	—
217	#367	—	43	2	no	—
218	#366	—	43	2	no	—
219	#358	—	43	1	no	—
220	#358	—	43	1	no	—
221	#358	C₂×D₄₃	1	2	yes	—
222	#358	C₂×D₄₃	1	2	yes	—
223	#357	—	43	1	no	—
224	#357	—	43	1	no	—
225	#346	—	43	2	no	—
226	C₄×C₁₆	—	43	2	no	—
227	C₄₃	#33455	1	1	yes	characteristic, complemented, Sylow
228	C₂×#41	—	86	1	no	—
229	C₂²×Q₈	—	2	1	no	—
230	#135	—	43	4	no	—
231	#135	—	86	2	no	—
232	C₂×Q₁₆	—	2	1	no	—
233	C₂×Q₁₆	—	2	2	no	—
234	C₂×Q₁₆	—	43	1	no	—
235	C₂×Q₁₆	C₄×D₄₃	1	4	yes	complemented
236	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
237	C₂×SD₁₆	—	86	1	no	—
238	C₂×D₈	—	43	1	no	—
239	C₂×M₁₆	—	43	1	no	—
240	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	C₄×Q₈	—	86	2	no	—
242	C₄×Q₈	—	86	2	no	—
243	C₄×Q₈	—	86	2	no	—
244	C₄×Q₈	—	172	1	no	—
245	C₄×Q₈	—	172	1	no	—
246	C₄×D₄	—	86	2	no	—
247	C₄×D₄	—	172	2	no	—
248	#117	—	43	1	no	—
249	#117	—	86	1	no	—
250	#117	—	172	2	no	—
251	#117	—	172	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	172	1	no	—
254	Q₃₂	—	43	8	no	—
255	SD₃₂	—	43	8	no	—
256	M₃₂	—	43	4	no	—
257	C₂×C₁₆	—	2	2	no	—
258	C₂×C₁₆	—	43	2	no	—
259	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
261	#107	—	2	1	no	—
262	#107	—	43	1	no	—
263	#107	—	43	1	no	—
264	#107	C₂²×D₄₃	1	2	yes	—
265	#106	—	86	1	no	—
266	#103	—	86	2	no	—
267	#103	—	86	2	no	—
268	#103	—	86	2	no	—
269	#102	—	86	2	no	—
270	#97	—	43	2	no	—
271	C₄×C₈	—	43	2	no	—
272	#41	—	86	4	no	—
273	#41	—	172	2	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	86	1	no	—
279	C₂×D₄	—	86	1	no	—
280	C₂×D₄	—	172	1	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	172	2	no	—
284	C₂²×C₄	—	172	1	no	—
285	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
286	Q₁₆	—	2	4	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	43	2	no	—
289	Q₁₆	—	86	1	no	—
290	Q₁₆	C₂×C₄×D₄₃	1	4	yes	—
291	SD₁₆	—	86	4	no	—
292	D₈	—	43	2	no	—
293	D₈	—	86	1	no	—
294	M₁₆	—	86	2	no	—
295	C₂×C₈	—	43	2	no	—
296	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
297	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
298	C₂×C₈	#10621900	1	2	yes	—
299	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	43	1	no	—
303	C₄⋊C₄	—	43	1	no	—
304	C₄⋊C₄	—	86	1	no	—
305	C₄⋊C₄	—	86	1	no	—
306	C₄⋊C₄	—	172	2	no	—
307	C₄⋊C₄	—	172	2	no	—
308	C₄⋊C₄	—	172	2	no	—
309	C₂²⋊C₄	—	43	2	no	—
310	C₂²⋊C₄	—	172	2	no	—
311	C₂²⋊C₄	—	172	1	no	—
312	C₄²	—	43	2	no	—
313	C₄²	—	172	2	no	—
314	C₄²	—	172	2	no	—
315	C₄²	—	172	2	no	—
316	C₄²	—	172	1	no	—
317	C₁₆	—	2	1	no	—
318	C₁₆	—	2	1	no	—
319	C₁₆	—	43	4	no	—
320	C₂³	—	172	1	no	—
321	C₂³	D₈×D₄₃	1	1	yes	characteristic
322	Q₈	—	2	4	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	4	4	no	—
325	Q₈	—	86	2	no	—
326	Q₈	—	172	1	no	—
327	D₄	—	86	2	no	—
328	D₄	—	172	4	no	—
329	D₄	—	172	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	16	1	no	—
334	C₂×C₄	—	43	2	no	—
335	C₂×C₄	—	43	2	no	—
336	C₂×C₄	—	172	1	no	—
337	C₂×C₄	—	172	1	no	—
338	C₂×C₄	—	172	1	no	—
339	C₂×C₄	—	172	4	no	—
340	C₂×C₄	—	688	2	no	—
341	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
342	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
343	C₂×C₄	#13063346	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
345	C₈	—	86	2	no	—
346	C₈	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
347	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
348	C₈	C₂×#10621900	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	344	1	no	—
352	C₂²	—	344	1	no	—
353	C₂²	—	172	2	no	—
354	C₂²	#14582979	1	2	yes	—
355	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
356	C₄	—	4	4	no	—
357	C₄	—	8	2	no	—
358	C₄	—	86	2	no	—
359	C₄	—	86	2	no	—
360	C₄	—	344	1	no	—
361	C₄	—	344	1	no	—
362	C₄	—	172	2	no	—
363	C₄	#14582979	1	2	yes	—
364	C₄	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
365	C₄	C₂×#13063346	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	344	2	no	—
368	C₂	C₂×#14582979	1	1	yes	characteristic
369	C₂	#19147737	1	1	yes	characteristic
370	C₂	#19147744	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#358	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#13063347	C₂²
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#358	D₄₃
hypocenter	C₄₃	#33455
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

Complements

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

N	H ≅ G/N	type
3. #19143463	C₂	semidirect
4. #19136463	C₂	semidirect
5. #19136462	C₂	semidirect
6. #19136451	C₂	semidirect
7. #19136444	C₂	semidirect
8. #19136381	C₂	semidirect
10. #19136191	C₂	semidirect
11. C₂×C₄₃×#358	C₂	semidirect
15. #14583340	C₄	semidirect
61. #13063271	C₂×C₄	semidirect
62. #13063268	C₂×C₄	semidirect
63. #13063265	C₂×C₄	semidirect
77. C₂×Q₁₆×C₄₃	C₂×C₄	semidirect
190. C₂×#358	D₄₃	semidirect
227. C₄₃	#33455	semidirect
235. C₂×Q₁₆	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583340	2
#19143469	2
#19143463	2
#19136463	2
#19136462	2
#19136451	2
#19136444	2
#19136381	2
#19136303	2
#19136191	2
C₂×C₄₃×#358	2
#33455	43