Subgroups — #25080532

Order 11008.

638 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583302	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144411	C₂	1	1	yes	maximal, characteristic
3	#19144404	C₂	1	1	yes	maximal, characteristic
4	#19143551	C₂	1	1	yes	maximal, characteristic
5	#19133998	C₂	1	4	yes	maximal, complemented
6	#19133993	C₂	1	4	yes	maximal, complemented
7	#19133353	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133275	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₄×#10621898	C₂²	1	1	yes	characteristic
12	#14583302	—	2	1	no	—
13	#14583302	—	2	1	no	—
14	#14583302	C₂²	1	4	yes	—
15	#14583195	C₂²	1	2	yes	—
16	#14583191	C₂²	1	2	yes	—
17	#14583190	C₂²	1	2	yes	—
18	#14583186	C₂²	1	2	yes	—
19	#14583185	—	2	1	no	—
20	#14583181	—	2	1	no	—
21	#14583180	—	2	1	no	—
22	#14583176	—	2	1	no	—
23	#14583023	C₂²	1	2	yes	—
24	#14582964	C₂²	1	2	yes	—
25	(C₂²⋊D₄₃)⋊C₈	—	2	2	no	—
26	#14582950	—	2	1	no	—
27	#14582941	—	2	1	no	—
28	#14582939	—	2	1	no	—
29	C₂×#13063258	—	2	2	no	—
30	#14582923	C₂²	1	1	yes	characteristic
31	#14582901	—	2	1	no	—
32	#14582898	—	2	1	no	—
33	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₃	—	2	1	no	—
35	#14582651	C₂²	1	4	yes	—
36	#14582647	C₂²	1	4	yes	—
37	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#431	C₂²	1	1	yes	characteristic
39	C₄₃×#423	C₂²	1	1	yes	characteristic
40	C₄₃×#333	C₂²	1	4	yes	—
41	C₂×#10621898	D₄	1	1	yes	characteristic
42	C₂²×Dic₈₆	—	2	1	no	—
43	C₂×#10621877	—	2	1	no	—
44	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
45	C₂×C₈×D₄₃	—	2	2	no	—
46	#13063325	—	2	1	no	—
47	#13063325	—	2	1	no	—
48	#13063325	—	2	1	no	—
49	#13063325	C₂³	1	2	yes	—
50	C₄×Dic₈₆	D₄	1	1	yes	characteristic
51	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
52	C₄×D₁₇₂	D₄	1	1	yes	characteristic
53	C₄×D₁₇₂	C₂³	1	1	yes	characteristic
54	C₄×C₂²⋊D₄₃	—	2	1	no	—
55	C₄×C₂²⋊D₄₃	—	2	1	no	—
56	#13063284	D₄	1	1	yes	characteristic
57	#13063282	—	2	1	no	—
58	C₂×C₄⋊Dic₄₃	—	2	1	no	—
59	C₂×C₄⋊Dic₄₃	—	2	1	no	—
60	C₄²×D₄₃	—	2	1	no	—
61	#13063259	—	2	2	no	—
62	#13063258	—	2	2	no	—
63	#13063258	—	2	4	no	—
64	#13063258	—	2	4	no	—
65	#13063257	—	2	2	no	—
66	#13063256	—	2	2	no	—
67	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
68	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
69	Dic₄₃⋊C₈	—	2	2	no	—
70	#13063250	—	2	2	no	—
71	#13063248	—	2	2	no	—
72	#13063247	—	2	2	no	—
73	#13063245	—	2	2	no	—
74	(C₂×D₄₃)⋊C₈	—	2	2	no	—
75	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
76	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
77	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
78	C₈×Dic₄₃	—	2	2	no	—
79	C₂²×Q₈×C₄₃	—	2	1	no	—
80	C₂×M₁₆×C₄₃	—	2	1	no	—
81	C₄⋊Q₈×C₄₃	—	2	1	no	—
82	C₄⋊Q₈×C₄₃	—	2	1	no	—
83	C₄⋊Q₈×C₄₃	—	2	1	no	—
84	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
87	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₄₃	—	2	1	no	—
89	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
90	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
91	C₂²⋊C₈×C₄₃	—	2	1	no	—
92	C₄×C₈×C₄₃	—	2	1	no	—
93	C₂×Q₈×C₄₃	—	2	2	no	—
94	C₂×Q₈×C₄₃	—	4	1	no	—
95	C₂×Q₈×C₄₃	—	8	2	no	—
96	C₂²×C₄×C₄₃	—	4	1	no	—
97	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
100	C₂×C₂²⋊D₄₃	—	2	1	no	—
101	C₂²×Dic₄₃	—	4	1	no	—
102	#10621898	—	2	2	no	—
103	#10621898	Q₁₆	1	4	yes	—
104	C₂×D₁₇₂	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₄₃	—	2	1	no	—
106	C₂×C₄×D₄₃	—	2	1	no	—
107	C₂×C₄×D₄₃	—	2	1	no	—
108	C₂×Dic₈₆	—	2	2	no	—
109	C₂×Dic₈₆	—	4	1	no	—
110	C₂×Dic₈₆	—	8	2	no	—
111	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
112	M₁₆×C₄₃	—	4	2	no	—
113	C₂×C₈×C₄₃	—	2	2	no	—
114	C₂×C₈×C₄₃	—	2	2	no	—
115	C₄⋊C₄×C₄₃	—	2	2	no	—
116	C₄⋊C₄×C₄₃	—	2	2	no	—
117	C₄⋊C₄×C₄₃	—	4	1	no	—
118	C₄⋊C₄×C₄₃	—	4	1	no	—
119	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₄₃	#41	1	1	yes	characteristic
122	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊Dic₄₃	—	2	1	no	—
124	(C₂×D₄₃)⋊C₄	—	2	1	no	—
125	(C₂×D₄₃)⋊C₄	—	2	1	no	—
126	C₄⋊Dic₄₃	—	2	2	no	—
127	C₄⋊Dic₄₃	—	2	2	no	—
128	C₄⋊Dic₄₃	—	4	1	no	—
129	C₄⋊Dic₄₃	—	4	1	no	—
130	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
131	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
132	Dic₄₃⋊C₄	—	2	1	no	—
133	Dic₄₃⋊C₄	—	2	1	no	—
134	C₄×Dic₄₃	—	2	1	no	—
135	C₄×Dic₄₃	—	2	1	no	—
136	C₄×Dic₄₃	—	2	1	no	—
137	#10621877	—	4	2	no	—
138	C₂×C₄₃⋊C₈	—	2	4	no	—
139	C₂×C₄₃⋊C₈	—	2	2	no	—
140	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
141	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
142	C₈×D₄₃	—	4	4	no	—
143	C₂³×C₄₃	#121	1	1	yes	characteristic
144	C₂²×D₄₃	—	2	1	no	—
145	Q₈×C₄₃	—	8	2	no	—
146	Q₈×C₄₃	—	8	2	no	—
147	C₂×C₄×C₄₃	—	2	2	no	—
148	C₂×C₄×C₄₃	—	2	2	no	—
149	C₂×C₄×C₄₃	—	4	2	no	—
150	C₂×C₄×C₄₃	—	8	2	no	—
151	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
155	C₂×C₄×C₄₃	#135	1	2	yes	—
156	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
159	C₂²⋊D₄₃	—	2	4	no	—
160	C₂×Dic₄₃	—	2	1	no	—
161	C₂×Dic₄₃	—	2	1	no	—
162	C₂×Dic₄₃	—	2	1	no	—
163	C₂×Dic₄₃	—	4	2	no	—
164	C₂×Dic₄₃	—	8	2	no	—
165	D₁₇₂	—	2	1	no	—
166	D₁₇₂	C₂×Q₁₆	1	2	yes	—
167	C₄×D₄₃	—	2	4	no	—
168	C₄×D₄₃	—	4	2	no	—
169	C₄×D₄₃	—	4	2	no	—
170	Dic₈₆	—	2	1	no	—
171	Dic₈₆	—	8	2	no	—
172	Dic₈₆	—	8	2	no	—
173	Dic₈₆	C₂×Q₁₆	1	2	yes	—
174	C₈×C₄₃	—	4	2	no	—
175	C₈×C₄₃	—	4	2	no	—
176	C₄₃⋊C₈	—	2	2	no	—
177	C₄₃⋊C₈	—	4	2	no	—
178	C₄₃⋊C₈	#137	1	4	yes	—
179	#19272	—	43	1	no	maximal, Sylow
180	C₂²×C₄₃	—	2	1	no	—
181	C₂²×C₄₃	—	2	1	no	—
182	C₂²×C₄₃	#467	1	2	yes	—
183	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
184	C₂×D₄₃	—	2	2	no	—
185	C₂×D₄₃	—	4	1	no	—
186	C₂×D₄₃	—	4	1	no	—
187	C₄×C₄₃	—	2	1	no	—
188	C₄×C₄₃	—	2	1	no	—
189	C₄×C₄₃	—	8	2	no	—
190	C₄×C₄₃	#467	1	2	yes	—
191	C₄×C₄₃	#472	1	2	yes	—
192	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
193	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
194	C₄×C₄₃	C₂×#137	1	2	yes	—
195	Dic₄₃	—	2	2	no	—
196	Dic₄₃	—	4	1	no	—
197	Dic₄₃	—	4	1	no	—
198	Dic₄₃	—	8	2	no	—
199	C₂×#500	—	43	1	no	—
200	#3079	—	43	1	no	—
201	#3079	—	43	1	no	—
202	C₈×#41	—	43	1	no	—
203	#1676	—	43	4	no	—
204	#1675	—	43	4	no	—
205	#1567	—	43	1	no	—
206	#1548	—	43	1	no	—
207	#1548	D₄₃	1	1	yes	characteristic, complemented
208	C₂×C₄₃	—	2	2	no	—
209	C₂×C₄₃	#3039	1	1	yes	characteristic
210	C₂×C₄₃	C₂×#467	1	1	yes	characteristic
211	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
212	D₄₃	—	4	2	no	—
213	C₂×C₄⋊Q₈	—	43	1	no	—
214	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
215	C₄×#41	—	43	1	no	—
216	#500	—	43	4	no	—
217	#500	—	86	1	no	—
218	#500	—	86	1	no	—
219	#477	—	43	2	no	—
220	#477	—	43	2	no	—
221	#476	—	43	2	no	—
222	#476	—	43	2	no	—
223	#475	—	86	1	no	—
224	#475	—	86	1	no	—
225	#474	—	86	1	no	—
226	#474	—	86	1	no	—
227	C₈×Q₈	—	43	2	no	—
228	C₈×D₄	—	43	2	no	—
229	C₈×D₄	—	86	2	no	—
230	#431	—	86	1	no	—
231	#431	C₂×D₄₃	1	1	yes	characteristic
232	#428	—	86	1	no	—
233	#428	—	86	1	no	—
234	C₂×#107	—	86	2	no	—
235	#423	—	43	1	no	—
236	#423	C₂×D₄₃	1	1	yes	characteristic
237	#417	—	86	1	no	—
238	#417	—	86	1	no	—
239	C₂×C₄×C₈	—	43	1	no	—
240	C₂×C₄×C₈	—	86	1	no	—
241	#335	—	43	4	no	—
242	#333	—	43	4	no	—
243	#333	C₂×D₄₃	1	4	yes	—
244	C₄₃	#19272	1	1	yes	characteristic, complemented, Sylow
245	C₂×#41	—	43	1	no	—
246	C₂²×Q₈	—	2	1	no	—
247	C₂²×Q₈	—	86	1	no	—
248	C₂×M₁₆	—	2	1	no	—
249	C₂×M₁₆	—	86	1	no	—
250	C₂²×C₈	—	43	2	no	—
251	C₂²×C₈	—	86	2	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	—	43	2	no	—
256	C₄⋊Q₈	—	86	1	no	—
257	C₄⋊Q₈	—	86	1	no	—
258	C₄⋊Q₈	—	86	1	no	—
259	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
260	C₄×Q₈	—	43	1	no	—
261	C₄×Q₈	—	43	1	no	—
262	C₄×D₄	—	43	1	no	—
263	C₄×D₄	—	43	1	no	—
264	C₄×D₄	—	86	1	no	—
265	C₄×D₄	—	86	1	no	—
266	#117	—	43	1	no	—
267	#117	—	86	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	86	1	no	—
271	C₂×C₄⋊C₄	—	86	1	no	—
272	C₂×C₄²	—	86	1	no	—
273	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
274	#107	—	86	2	no	—
275	#107	—	86	4	no	—
276	#107	—	86	4	no	—
277	#107	—	86	2	no	—
278	#106	—	86	2	no	—
279	#106	—	86	2	no	—
280	C₄⋊C₈	—	2	1	no	—
281	C₄⋊C₈	—	43	2	no	—
282	C₄⋊C₈	—	86	2	no	—
283	C₄⋊C₈	—	86	1	no	—
284	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
285	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
286	#103	—	86	2	no	—
287	#103	—	86	2	no	—
288	#102	—	86	2	no	—
289	#102	—	86	2	no	—
290	C₂²⋊C₈	—	2	1	no	—
291	C₂²⋊C₈	—	86	2	no	—
292	C₄×C₈	—	2	1	no	—
293	C₄×C₈	—	43	1	no	—
294	C₄×C₈	—	43	1	no	—
295	C₄×C₈	—	43	2	no	—
296	C₄×C₈	—	86	2	no	—
297	#41	—	43	4	no	—
298	#41	—	86	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	4	1	no	—
301	C₂×Q₈	—	8	2	no	—
302	C₂×Q₈	—	43	1	no	—
303	C₂×Q₈	—	86	2	no	—
304	C₂×Q₈	—	344	2	no	—
305	C₂×Q₈	—	172	1	no	—
306	C₂×D₄	—	43	1	no	—
307	C₂×D₄	—	86	1	no	—
308	C₂²×C₄	—	4	1	no	—
309	C₂²×C₄	—	86	1	no	—
310	C₂²×C₄	—	86	1	no	—
311	C₂²×C₄	—	86	1	no	—
312	C₂²×C₄	—	172	1	no	—
313	C₂²×C₄	#10621898	1	1	yes	characteristic
314	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
315	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
316	M₁₆	—	4	2	no	—
317	M₁₆	—	172	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	43	2	no	—
321	C₂×C₈	—	43	2	no	—
322	C₂×C₈	—	86	4	no	—
323	C₂×C₈	—	86	2	no	—
324	C₂×C₈	—	172	4	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	43	1	no	—
330	C₄⋊C₄	—	43	1	no	—
331	C₄⋊C₄	—	86	1	no	—
332	C₄⋊C₄	—	86	1	no	—
333	C₄⋊C₄	—	86	2	no	—
334	C₄⋊C₄	—	86	2	no	—
335	C₄⋊C₄	—	172	1	no	—
336	C₄⋊C₄	—	172	1	no	—
337	C₂²⋊C₄	—	86	1	no	—
338	C₂²⋊C₄	—	86	1	no	—
339	C₂²⋊C₄	—	86	1	no	—
340	C₄²	—	86	1	no	—
341	C₄²	—	86	1	no	—
342	C₄²	—	86	1	no	—
343	C₄²	#10621898	1	1	yes	characteristic
344	C₄²	D₄×D₄₃	1	1	yes	characteristic
345	C₄²	D₄×D₄₃	1	1	yes	characteristic
346	C₄²	C₂³×D₄₃	1	1	yes	characteristic
347	C₂³	—	86	1	no	—
348	C₂³	#13063298	1	1	yes	characteristic
349	Q₈	—	8	2	no	—
350	Q₈	—	8	2	no	—
351	Q₈	—	43	2	no	—
352	Q₈	—	86	1	no	—
353	Q₈	—	344	2	no	—
354	Q₈	—	344	2	no	—
355	D₄	—	43	2	no	—
356	D₄	—	86	4	no	—
357	D₄	—	86	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	8	2	no	—
362	C₂×C₄	—	86	1	no	—
363	C₂×C₄	—	86	4	no	—
364	C₂×C₄	—	86	1	no	—
365	C₂×C₄	—	86	1	no	—
366	C₂×C₄	—	344	2	no	—
367	C₂×C₄	—	172	2	no	—
368	C₂×C₄	—	172	2	no	—
369	C₂×C₄	—	172	2	no	—
370	C₂×C₄	#13063298	1	1	yes	characteristic
371	C₂×C₄	#13063298	1	1	yes	characteristic
372	C₂×C₄	#13063298	1	1	yes	characteristic
373	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
374	C₂×C₄	#13063346	1	2	yes	—
375	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
376	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
377	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
378	C₈	—	4	2	no	—
379	C₈	—	4	2	no	—
380	C₈	—	43	4	no	—
381	C₈	—	86	2	no	—
382	C₈	—	172	2	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	86	2	no	—
386	C₂²	—	172	1	no	—
387	C₂²	—	172	1	no	—
388	C₂²	#14583139	1	2	yes	—
389	C₂²	C₂×#13063298	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	8	2	no	—
393	C₄	—	86	2	no	—
394	C₄	—	344	2	no	—
395	C₄	—	172	1	no	—
396	C₄	—	172	1	no	—
397	C₄	#14583139	1	2	yes	—
398	C₄	#14583168	1	2	yes	—
399	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
400	C₄	C₂×#13063346	1	1	yes	characteristic
401	C₄	#137×D₄₃	1	2	yes	—
402	C₂	—	2	2	no	—
403	C₂	—	172	2	no	—
404	C₂	#19144081	1	1	yes	characteristic
405	C₂	C₂×#14583139	1	1	yes	characteristic
406	C₂	C₂×#14583168	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063298
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1548	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂×#10621898	D₄
hypercenter	#1548	D₄₃
hypocenter	C₄₃	#19272
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1548

Complements

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

N	H ≅ G/N	type
1. C₂×#14583302	C₂	semidirect
5. #19133998	C₂	semidirect
6. #19133993	C₂	semidirect
7. #19133353	C₂	semidirect
8. #19133275	C₂	semidirect
9. C₄₃×#1548	C₂	semidirect
207. #1548	D₄₃	semidirect
244. C₄₃	#19272	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583302	2
#19144411	2
#19144404	2
#19143551	2
#19133998	2
#19133993	2
#19133353	2
#19133275	2
C₄₃×#1548	2
#19272	43