Subgroups — #25251677

Order 11008.

548 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063347	C₂	1	1	yes	maximal, characteristic
2	#19136800	C₂	1	2	yes	maximal, complemented
3	#19136744	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136739	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136642	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136637	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136533	C₂	1	2	yes	maximal
8	#19136464	C₂	1	2	yes	maximal, complemented
9	#19136457	C₂	1	2	yes	maximal, complemented
10	#19136398	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	1	yes	characteristic
13	C₄×#10621898	—	2	1	no	—
14	C₂×#13063271	C₂²	1	2	yes	—
15	C₂×#13063268	C₂²	1	2	yes	—
16	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
18	C₄×#10621873	—	2	1	no	—
19	C₄×#10621873	—	2	1	no	—
20	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
21	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
22	#14582937	C₂²	1	1	yes	characteristic
23	#14582896	—	2	1	no	—
24	#14582726	C₂²	1	2	yes	—
25	#14582724	C₂²	1	2	yes	—
26	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
28	#14582703	C₂²	1	2	yes	—
29	#14582703	C₂²	1	2	yes	—
30	#14582702	C₂²	1	2	yes	—
31	#14582699	C₂²	1	2	yes	—
32	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
33	#14582675	C₂²	1	2	yes	—
34	C₄₃×#533	—	2	1	no	—
35	C₄₃×#500	C₂²	1	1	yes	characteristic
36	C₄₃×#494	C₂²	1	1	yes	characteristic
37	C₄₃×#487	—	2	1	no	—
38	C₄₃×#467	—	2	1	no	—
39	C₄₃×#467	C₂²	1	2	yes	—
40	C₄×Q₁₆×C₄₃	—	2	1	no	—
41	C₄×Q₁₆×C₄₃	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
43	C₂×#10621898	—	2	1	no	—
44	#13063347	—	2	2	no	—
45	#13063347	—	2	2	no	—
46	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
47	C₂×#10621873	—	2	1	no	—
48	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
49	C₄×Dic₈₆	—	2	1	no	—
50	C₄×Dic₈₆	—	2	1	no	—
51	C₄×D₁₇₂	—	2	1	no	—
52	C₄×D₁₇₂	—	2	1	no	—
53	C₄×C₂²⋊D₄₃	—	4	1	no	—
54	C₄×C₂²⋊D₄₃	—	4	1	no	—
55	#13063284	—	2	1	no	—
56	#13063282	—	4	1	no	—
57	C₄²×D₄₃	—	4	1	no	—
58	#13063271	—	2	4	no	—
59	#13063268	—	2	4	no	—
60	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
61	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
62	#13063259	C₂³	1	1	yes	characteristic
63	#13063259	C₂³	1	1	yes	characteristic
64	#13063257	—	2	1	no	—
65	#13063250	—	2	1	no	—
66	#13063250	—	2	1	no	—
67	#13063247	—	2	1	no	—
68	#13063247	—	2	1	no	—
69	C₂×Q₁₆×C₄₃	—	2	1	no	—
70	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
71	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₄⋊Q₈×C₄₃	—	2	1	no	—
74	C₄⋊Q₈×C₄₃	—	2	1	no	—
75	#125×C₄₃	—	4	1	no	—
76	C₂²⋊Q₈×C₄₃	—	2	2	no	—
77	C₂²⋊Q₈×C₄₃	—	4	1	no	—
78	C₄×Q₈×C₄₃	—	2	2	no	—
79	C₄×Q₈×C₄₃	—	4	1	no	—
80	#117×C₄₃	—	4	1	no	—
81	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
82	#107×C₄₃	—	2	1	no	—
83	#107×C₄₃	C₂³	1	2	yes	—
84	#103×C₄₃	—	2	2	no	—
85	#103×C₄₃	—	2	2	no	—
86	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
87	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
88	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
90	C₂×Q₈×C₄₃	—	2	2	no	—
91	C₂×Q₈×C₄₃	—	4	1	no	—
92	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₂²⋊D₄₃	—	4	1	no	—
96	#10621898	—	2	4	no	—
97	#10621898	—	4	2	no	—
98	C₂×D₁₇₂	—	2	1	no	—
99	C₂×C₄×D₄₃	—	4	1	no	—
100	C₂×C₄×D₄₃	—	4	1	no	—
101	C₂×C₄×D₄₃	—	4	1	no	—
102	C₂×Dic₈₆	—	2	1	no	—
103	Q₁₆×C₄₃	—	2	2	no	—
104	Q₁₆×C₄₃	—	2	2	no	—
105	Q₁₆×C₄₃	—	4	2	no	—
106	C₂×C₈×C₄₃	—	2	1	no	—
107	C₂×C₈×C₄₃	—	2	1	no	—
108	C₂×C₈×C₄₃	—	4	1	no	—
109	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₄₃	—	2	2	no	—
114	C₄⋊C₄×C₄₃	—	4	1	no	—
115	C₄⋊C₄×C₄₃	—	4	1	no	—
116	C₄⋊C₄×C₄₃	—	4	2	no	—
117	C₄⋊C₄×C₄₃	—	4	2	no	—
118	C₄⋊C₄×C₄₃	—	4	1	no	—
119	C₂²⋊C₄×C₄₃	—	4	2	no	—
120	C₄²×C₄₃	—	4	2	no	—
121	C₄²×C₄₃	D₈	1	1	yes	characteristic
122	C₄²×C₄₃	D₈	1	1	yes	characteristic
123	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂²⋊Dic₄₃	—	4	1	no	—
126	(C₂×D₄₃)⋊C₄	—	4	1	no	—
127	(C₂×D₄₃)⋊C₄	—	4	1	no	—
128	C₄⋊Dic₄₃	—	2	1	no	—
129	C₄⋊Dic₄₃	—	2	1	no	—
130	Dic₄₃⋊C₄	—	4	1	no	—
131	Dic₄₃⋊C₄	—	4	1	no	—
132	C₄×Dic₄₃	—	4	1	no	—
133	C₄×Dic₄₃	—	4	1	no	—
134	C₄×Dic₄₃	—	4	1	no	—
135	Dic₁₇₂	—	2	1	no	—
136	Dic₁₇₂	#41	1	2	yes	—
137	D₃₄₄	—	2	1	no	—
138	D₃₄₄	#41	1	2	yes	—
139	#10621873	—	2	4	no	—
140	C₄₃⋊C₁₆	—	2	2	no	—
141	C₄₃⋊C₁₆	—	2	2	no	—
142	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
143	C₂²×D₄₃	—	4	1	no	—
144	Q₈×C₄₃	—	4	2	no	—
145	Q₈×C₄₃	—	4	2	no	—
146	Q₈×C₄₃	—	8	2	no	—
147	C₂×C₄×C₄₃	—	2	1	no	—
148	C₂×C₄×C₄₃	—	2	1	no	—
149	C₂×C₄×C₄₃	—	2	2	no	—
150	C₂×C₄×C₄₃	—	4	1	no	—
151	C₂×C₄×C₄₃	—	4	2	no	—
152	C₂×C₄×C₄₃	—	4	2	no	—
153	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
159	C₂²⋊D₄₃	—	4	4	no	—
160	C₂×Dic₄₃	—	4	1	no	—
161	C₂×Dic₄₃	—	4	1	no	—
162	C₂×Dic₄₃	—	4	1	no	—
163	D₁₇₂	—	2	2	no	—
164	D₁₇₂	—	4	1	no	—
165	C₄×D₄₃	—	4	4	no	—
166	C₄×D₄₃	—	8	2	no	—
167	C₄×D₄₃	—	8	2	no	—
168	Dic₈₆	—	2	2	no	—
169	Dic₈₆	—	4	1	no	—
170	C₈×C₄₃	—	2	1	no	—
171	C₈×C₄₃	—	2	1	no	—
172	C₈×C₄₃	—	2	1	no	—
173	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
174	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
175	#33676	—	43	1	no	maximal, Sylow
176	C₂²×C₄₃	—	2	1	no	—
177	C₂²×C₄₃	—	2	1	no	—
178	C₂²×C₄₃	—	2	1	no	—
179	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
180	C₂×D₄₃	—	4	2	no	—
181	C₂×D₄₃	—	8	1	no	—
182	C₂×D₄₃	—	8	1	no	—
183	C₄×C₄₃	—	2	1	no	—
184	C₄×C₄₃	—	2	1	no	—
185	C₄×C₄₃	—	2	1	no	—
186	C₄×C₄₃	—	2	1	no	—
187	C₄×C₄₃	—	8	2	no	—
188	C₄×C₄₃	—	8	2	no	—
189	C₄×C₄₃	#510	1	2	yes	—
190	C₄×C₄₃	#525	1	1	yes	characteristic
191	C₄×C₄₃	#575	1	1	yes	characteristic
192	Dic₄₃	—	4	2	no	—
193	Dic₄₃	—	8	1	no	—
194	Dic₄₃	—	8	1	no	—
195	#3170	D₄₃	1	1	yes	characteristic, complemented
196	C₄×#135	—	43	1	no	—
197	#2237	—	43	2	no	—
198	#2227	—	43	1	no	—
199	#2226	—	43	1	no	—
200	#2209	—	43	1	no	—
201	#2208	—	43	1	no	—
202	#2189	—	43	2	no	—
203	#2176	—	43	2	no	—
204	#2175	—	43	2	no	—
205	#2163	—	43	1	no	—
206	C₂×C₄₃	—	4	1	no	—
207	C₂×C₄₃	#3100	1	1	yes	characteristic
208	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
209	C₂×C₄₃	#3414	1	1	yes	characteristic
210	D₄₃	—	8	2	no	—
211	C₂×#135	—	43	1	no	—
212	#533	—	2	1	no	—
213	C₄×#41	—	86	1	no	—
214	C₂×Q₃₂	—	43	2	no	—
215	C₂×SD₃₂	—	43	2	no	—
216	#500	C₂×D₄₃	1	1	yes	characteristic
217	#494	C₂×D₄₃	1	1	yes	characteristic
218	#487	—	2	1	no	—
219	#467	—	2	1	no	—
220	#467	C₂×D₄₃	1	2	yes	—
221	C₄×Q₁₆	—	2	1	no	—
222	C₄×Q₁₆	—	43	1	no	—
223	C₄×Q₁₆	—	43	1	no	—
224	C₄×Q₁₆	C₂×D₄₃	1	2	yes	—
225	C₄×SD₁₆	—	86	1	no	—
226	C₄×SD₁₆	—	86	1	no	—
227	C₄×D₈	—	43	1	no	—
228	C₄×D₈	—	43	1	no	—
229	#427	—	43	1	no	—
230	#416	—	86	1	no	—
231	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
232	#367	—	43	2	no	—
233	#366	—	43	2	no	—
234	C₄⋊C₁₆	—	43	1	no	—
235	C₄⋊C₁₆	—	43	1	no	—
236	#358	—	43	2	no	—
237	#358	—	43	2	no	—
238	#358	—	43	2	no	—
239	#357	—	43	2	no	—
240	C₂²⋊C₁₆	—	43	2	no	—
241	#346	—	43	2	no	—
242	C₄₃	#33676	1	1	yes	characteristic, complemented, Sylow
243	C₂×#41	—	86	1	no	—
244	#135	—	86	2	no	—
245	#135	—	86	2	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	—	43	1	no	—
248	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
249	C₂×SD₁₆	—	86	1	no	—
250	C₂×D₈	—	43	1	no	—
251	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
252	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	#125	—	4	1	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₂²⋊Q₈	—	4	1	no	—
258	C₄×Q₈	—	2	2	no	—
259	C₄×Q₈	—	4	1	no	—
260	C₄×Q₈	—	86	1	no	—
261	C₄×Q₈	—	86	1	no	—
262	C₄×D₄	—	86	1	no	—
263	C₄×D₄	—	86	1	no	—
264	C₄×D₄	—	172	1	no	—
265	C₄×D₄	—	172	1	no	—
266	#117	—	4	1	no	—
267	#117	—	86	1	no	—
268	#117	—	172	1	no	—
269	C₂×C₄²	—	172	1	no	—
270	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
271	Q₃₂	—	86	4	no	—
272	SD₃₂	—	86	4	no	—
273	C₂×C₁₆	—	43	2	no	—
274	C₂×C₁₆	—	43	2	no	—
275	#107	—	2	1	no	—
276	#107	—	43	1	no	—
277	#107	—	43	1	no	—
278	#107	C₂²×D₄₃	1	2	yes	—
279	#106	—	86	1	no	—
280	#103	—	2	2	no	—
281	#103	—	2	2	no	—
282	#103	—	86	1	no	—
283	#103	—	86	1	no	—
284	#102	—	86	1	no	—
285	#102	—	86	1	no	—
286	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
287	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
290	#41	—	86	4	no	—
291	#41	—	172	2	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	4	1	no	—
294	C₂×Q₈	—	86	1	no	—
295	C₂×D₄	—	86	1	no	—
296	C₂×D₄	—	172	1	no	—
297	C₂²×C₄	—	172	1	no	—
298	C₂²×C₄	—	172	1	no	—
299	C₂²×C₄	—	172	1	no	—
300	C₂²×C₄	#10621882	1	1	yes	characteristic
301	C₂²×C₄	#10621882	1	1	yes	characteristic
302	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
303	Q₁₆	—	2	2	no	—
304	Q₁₆	—	2	2	no	—
305	Q₁₆	—	4	2	no	—
306	Q₁₆	—	43	2	no	—
307	Q₁₆	—	86	1	no	—
308	SD₁₆	—	86	4	no	—
309	D₈	—	43	2	no	—
310	D₈	—	86	1	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	4	1	no	—
314	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
316	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
317	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	4	2	no	—
322	C₄⋊C₄	—	4	2	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	86	1	no	—
325	C₄⋊C₄	—	86	1	no	—
326	C₄⋊C₄	—	172	1	no	—
327	C₄⋊C₄	—	172	1	no	—
328	C₂²⋊C₄	—	4	2	no	—
329	C₂²⋊C₄	—	172	1	no	—
330	C₂²⋊C₄	—	172	1	no	—
331	C₂²⋊C₄	—	172	1	no	—
332	C₄²	—	4	2	no	—
333	C₄²	—	172	1	no	—
334	C₄²	—	172	1	no	—
335	C₄²	—	172	1	no	—
336	C₄²	#10621882	1	1	yes	characteristic
337	C₄²	#10621882	1	1	yes	characteristic
338	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
339	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
340	C₁₆	—	86	2	no	—
341	C₁₆	—	86	2	no	—
342	C₂³	—	172	1	no	—
343	C₂³	C₂×#10621882	1	1	yes	characteristic
344	Q₈	—	4	2	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	8	2	no	—
347	Q₈	—	86	2	no	—
348	Q₈	—	172	1	no	—
349	D₄	—	86	2	no	—
350	D₄	—	172	4	no	—
351	D₄	—	172	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	344	2	no	—
359	C₂×C₄	—	344	2	no	—
360	C₂×C₄	—	172	4	no	—
361	C₂×C₄	—	172	1	no	—
362	C₂×C₄	—	172	1	no	—
363	C₂×C₄	—	172	1	no	—
364	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
365	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
366	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
367	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
368	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
369	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	#41×D₄₃	1	1	yes	characteristic
374	C₈	#41×D₄₃	1	1	yes	characteristic
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	344	1	no	—
379	C₂²	—	344	1	no	—
380	C₂²	—	172	2	no	—
381	C₂²	C₂²×#10621882	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	8	2	no	—
387	C₄	—	8	2	no	—
388	C₄	—	344	1	no	—
389	C₄	—	344	1	no	—
390	C₄	—	172	2	no	—
391	C₄	#14583340	1	2	yes	—
392	C₄	#14583406	1	1	yes	characteristic
393	C₄	#14583697	1	1	yes	characteristic
394	C₂	—	4	1	no	—
395	C₂	—	344	2	no	—
396	C₂	#19144591	1	1	yes	characteristic
397	C₂	C₂×#14583340	1	1	yes	characteristic
398	C₂	#19147729	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3170	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#13063347	C₂²
hypercenter	#3170	D₄₃
hypocenter	C₄₃	#33676
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #19136800	C₂	semidirect
3. #19136744	C₂	semidirect
4. #19136739	C₂	semidirect
5. #19136642	C₂	semidirect
6. #19136637	C₂	semidirect
8. #19136464	C₂	semidirect
9. #19136457	C₂	semidirect
10. #19136398	C₂	semidirect
11. C₄₃×#3170	C₂	semidirect
195. #3170	D₄₃	semidirect
242. C₄₃	#33676	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13063347	2
#19136800	2
#19136744	2
#19136739	2
#19136642	2
#19136637	2
#19136533	2
#19136464	2
#19136457	2
#19136398	2
C₄₃×#3170	2
#33676	43