Subgroups — #25081946

Order 11008.

676 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145716	C₂	1	1	yes	maximal, characteristic
2	#19144067	C₂	1	1	yes	maximal, characteristic
3	#19143676	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134041	C₂	1	4	yes	maximal, complemented
5	#19133987	C₂	1	4	yes	maximal, complemented
6	#19133369	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133267	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133245	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063356	—	2	1	no	—
11	#14583449	C₂²	1	1	yes	characteristic
12	#14583302	C₂²	1	2	yes	—
13	#14583247	C₂²	1	2	yes	—
14	#14583212	—	2	2	no	—
15	#14583176	—	2	2	no	—
16	#14583121	—	2	2	no	—
17	#14583109	C₂²	1	2	yes	—
18	#14583099	C₂²	1	2	yes	—
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	#14582651	C₂²	1	4	yes	—
27	#14582646	C₂²	1	2	yes	—
28	#14582645	C₂²	1	4	yes	—
29	#14582642	C₂²	1	2	yes	—
30	C₂²×Q₁₆×C₄₃	—	2	1	no	—
31	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₄₃×#462	—	2	2	no	—
34	C₄₃×#462	C₂²	1	4	yes	—
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×#10621877	—	2	1	no	—
42	#13063325	C₂³	1	1	yes	characteristic
43	#13063322	C₂³	1	1	yes	characteristic
44	#13063313	—	2	1	no	—
45	#13063299	—	2	1	no	—
46	#13063299	—	2	1	no	—
47	C₄×D₁₇₂	—	2	1	no	—
48	C₄×D₁₇₂	—	2	1	no	—
49	#13063258	—	2	4	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	#13063248	—	2	2	no	—
56	#13063245	—	2	2	no	—
57	#13063245	—	2	4	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	C₂×Q₁₆×C₄₃	—	2	4	no	—
66	C₂×Q₁₆×C₄₃	—	4	4	no	—
67	C₂²×C₈×C₄₃	—	2	1	no	—
68	C₄⋊Q₈×C₄₃	—	2	1	no	—
69	C₄⋊Q₈×C₄₃	—	2	1	no	—
70	C₄⋊Q₈×C₄₃	—	2	1	no	—
71	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
72	C₄×Q₈×C₄₃	—	2	1	no	—
73	C₄×Q₈×C₄₃	—	2	1	no	—
74	C₄×Q₈×C₄₃	D₄	1	2	yes	—
75	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₄₃	—	2	1	no	—
81	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₄₃	—	2	1	no	—
83	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
84	#103×C₄₃	—	2	4	no	—
85	C₂×Q₈×C₄₃	—	2	1	no	—
86	C₂×Q₈×C₄₃	—	2	2	no	—
87	C₂×Q₈×C₄₃	—	2	2	no	—
88	C₂×Q₈×C₄₃	—	4	1	no	—
89	C₂×Q₈×C₄₃	—	4	2	no	—
90	C₂×Q₈×C₄₃	—	4	2	no	—
91	C₂×Q₈×C₄₃	D₈	1	4	yes	—
92	C₂×Q₈×C₄₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₄₃	—	2	1	no	—
94	C₂²×C₄×C₄₃	—	2	1	no	—
95	C₂²×C₄×C₄₃	—	4	1	no	—
96	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
99	C₂×C₂²⋊D₄₃	—	4	1	no	—
100	#10621898	—	4	2	no	—
101	#10621898	—	4	2	no	—
102	C₂×D₁₇₂	—	2	1	no	—
103	C₂×D₁₇₂	—	2	1	no	—
104	C₂×D₁₇₂	—	2	1	no	—
105	C₂×C₄×D₄₃	—	4	1	no	—
106	C₂×Dic₈₆	—	2	1	no	—
107	Q₁₆×C₄₃	—	4	8	no	—
108	C₂×C₈×C₄₃	—	2	2	no	—
109	C₂×C₈×C₄₃	—	4	2	no	—
110	C₄⋊C₄×C₄₃	—	2	2	no	—
111	C₄⋊C₄×C₄₃	—	2	1	no	—
112	C₄⋊C₄×C₄₃	—	2	2	no	—
113	C₄⋊C₄×C₄₃	—	2	2	no	—
114	C₄⋊C₄×C₄₃	—	2	2	no	—
115	C₄⋊C₄×C₄₃	—	4	1	no	—
116	C₄⋊C₄×C₄₃	—	4	1	no	—
117	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
118	C₄²×C₄₃	—	2	2	no	—
119	C₄²×C₄₃	—	2	2	no	—
120	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₄₃	#41	1	1	yes	characteristic
123	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
124	#10621885	—	4	4	no	—
125	#10621884	—	4	4	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	#10621877	—	4	2	no	—
131	C₂×C₄₃⋊C₈	—	2	2	no	—
132	C₂×C₄₃⋊C₈	—	2	2	no	—
133	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
134	C₂³×C₄₃	#121	1	1	yes	characteristic
135	C₂²×D₄₃	—	4	1	no	—
136	Q₈×C₄₃	—	2	2	no	—
137	Q₈×C₄₃	—	2	4	no	—
138	Q₈×C₄₃	—	4	2	no	—
139	Q₈×C₄₃	—	4	2	no	—
140	Q₈×C₄₃	—	8	2	no	—
141	Q₈×C₄₃	C₂×D₈	1	4	yes	—
142	C₂×C₄×C₄₃	—	2	2	no	—
143	C₂×C₄×C₄₃	—	2	4	no	—
144	C₂×C₄×C₄₃	—	2	2	no	—
145	C₂×C₄×C₄₃	—	2	2	no	—
146	C₂×C₄×C₄₃	—	4	1	no	—
147	C₂×C₄×C₄₃	—	4	2	no	—
148	C₂×C₄×C₄₃	—	8	1	no	—
149	C₂×C₄×C₄₃	—	8	2	no	—
150	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₄₃	#135	1	2	yes	—
155	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
158	C₂²⋊D₄₃	—	8	2	no	—
159	C₂×Dic₄₃	—	4	1	no	—
160	D₁₇₂	—	4	1	no	—
161	D₁₇₂	—	4	1	no	—
162	D₁₇₂	—	4	2	no	—
163	D₁₇₂	—	8	1	no	—
164	C₄×D₄₃	—	8	2	no	—
165	Dic₈₆	—	4	1	no	—
166	Dic₈₆	—	4	1	no	—
167	C₈×C₄₃	—	4	2	no	—
168	C₄₃⋊C₈	—	2	4	no	—
169	C₄₃⋊C₈	—	4	2	no	—
170	C₄₃⋊C₈	—	4	2	no	—
171	#19404	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	2	1	no	—
173	C₂²×C₄₃	—	2	1	no	—
174	C₂²×C₄₃	#466	1	2	yes	—
175	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
176	C₂×D₄₃	—	8	1	no	—
177	C₂×D₄₃	—	8	1	no	—
178	C₂×D₄₃	—	8	1	no	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	2	4	no	—
181	C₄×C₄₃	—	2	1	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	4	2	no	—
184	C₄×C₄₃	—	8	2	no	—
185	C₄×C₄₃	#466	1	2	yes	—
186	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
187	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
188	C₄×C₄₃	#578	1	2	yes	—
189	Dic₄₃	—	8	1	no	—
190	#3226	—	43	1	no	—
191	#3037	—	43	1	no	—
192	C₂×#462	D₄₃	1	1	yes	characteristic, complemented
193	#2982	—	43	1	no	—
194	#1682	—	43	4	no	—
195	#1674	—	43	4	no	—
196	#1571	—	43	1	no	—
197	#1546	—	43	1	no	—
198	#1540	—	43	1	no	—
199	C₂×C₄₃	—	2	2	no	—
200	C₂×C₄₃	#3044	1	1	yes	characteristic
201	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
202	C₂×C₄₃	#3069	1	1	yes	characteristic
203	D₄₃	—	16	1	no	—
204	C₂×#137	—	86	1	no	—
205	C₂²×Q₁₆	—	2	1	no	—
206	#532	—	43	1	no	—
207	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
208	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
209	#500	—	43	2	no	—
210	#486	—	43	2	no	—
211	#480	—	86	2	no	—
212	#474	—	86	2	no	—
213	#464	—	86	2	no	—
214	#462	—	2	2	no	—
215	#462	—	43	2	no	—
216	#462	C₂×D₄₃	1	4	yes	—
217	#460	—	43	2	no	—
218	C₈×Q₈	—	43	4	no	—
219	C₈×Q₈	—	86	2	no	—
220	#431	—	43	1	no	—
221	#431	—	43	1	no	—
222	#431	—	86	2	no	—
223	#423	—	43	1	no	—
224	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
225	#417	—	86	1	no	—
226	C₂×#103	—	2	1	no	—
227	#335	—	43	4	no	—
228	#332	—	43	4	no	—
229	#332	—	43	2	no	—
230	#331	—	43	2	no	—
231	C₄₃	#19404	1	1	yes	characteristic, complemented, Sylow
232	C₂×#41	—	86	1	no	—
233	C₂²×Q₈	—	2	1	no	—
234	C₂²×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
235	#137	—	172	4	no	—
236	C₂×Q₁₆	—	2	4	no	—
237	C₂×Q₁₆	—	4	4	no	—
238	C₂×Q₁₆	—	86	2	no	—
239	C₂×SD₁₆	—	86	2	no	—
240	C₂×M₁₆	—	86	1	no	—
241	C₂²×C₈	—	2	1	no	—
242	C₄⋊Q₈	—	2	1	no	—
243	C₄⋊Q₈	—	2	1	no	—
244	C₄⋊Q₈	—	2	1	no	—
245	C₄⋊Q₈	—	43	1	no	—
246	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
247	#127	—	43	1	no	—
248	#124	—	86	1	no	—
249	#121	—	86	1	no	—
250	#121	—	86	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₄×Q₈	C₂²⋊D₄₃	1	2	yes	—
254	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
255	C₄×D₄	—	86	1	no	—
256	C₄×D₄	—	86	1	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	—	2	1	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
261	C₂×C₄²	—	2	1	no	—
262	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
263	#107	—	86	4	no	—
264	C₄⋊C₈	—	2	1	no	—
265	C₄⋊C₈	—	43	2	no	—
266	C₄⋊C₈	—	43	2	no	—
267	C₄⋊C₈	—	43	2	no	—
268	C₄⋊C₈	—	86	4	no	—
269	C₄⋊C₈	—	86	1	no	—
270	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
271	#103	—	2	4	no	—
272	#103	—	86	2	no	—
273	#102	—	86	4	no	—
274	#102	—	86	2	no	—
275	C₂²⋊C₈	—	43	2	no	—
276	C₂²⋊C₈	—	86	1	no	—
277	C₄×C₈	—	43	2	no	—
278	C₄×C₈	—	86	4	no	—
279	C₄×C₈	—	86	1	no	—
280	#41	—	172	2	no	—
281	#41	—	172	2	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	—	2	2	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	4	1	no	—
286	C₂×Q₈	—	4	2	no	—
287	C₂×Q₈	—	4	2	no	—
288	C₂×Q₈	—	86	1	no	—
289	C₂×Q₈	#10621882	1	4	yes	—
290	C₂×Q₈	C₂×C₂²⋊D₄₃	1	2	yes	—
291	C₂×D₄	—	86	1	no	—
292	C₂×D₄	—	86	1	no	—
293	C₂×D₄	—	86	1	no	—
294	C₂×D₄	—	172	1	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	4	1	no	—
298	C₂²×C₄	—	172	1	no	—
299	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
300	C₂²×C₄	#10621900	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
302	Q₁₆	—	4	8	no	—
303	Q₁₆	—	172	4	no	—
304	SD₁₆	—	172	4	no	—
305	M₁₆	—	172	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	4	2	no	—
308	C₂×C₈	—	43	4	no	—
309	C₂×C₈	—	86	2	no	—
310	C₂×C₈	—	86	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	—	86	1	no	—
319	C₄⋊C₄	—	86	1	no	—
320	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
321	C₂²⋊C₄	—	172	1	no	—
322	C₂²⋊C₄	—	172	1	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	D₄×D₄₃	1	1	yes	characteristic
326	C₄²	#10621900	1	1	yes	characteristic
327	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
328	C₄²	C₂³×D₄₃	1	1	yes	characteristic
329	C₂³	—	172	1	no	—
330	C₂³	#13063297	1	1	yes	characteristic
331	Q₈	—	2	4	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	8	2	no	—
336	Q₈	—	172	1	no	—
337	Q₈	—	172	1	no	—
338	Q₈	C₂×#10621882	1	4	yes	—
339	D₄	—	344	1	no	—
340	D₄	—	344	2	no	—
341	D₄	—	172	1	no	—
342	D₄	—	172	1	no	—
343	D₄	—	172	2	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	—	8	2	no	—
352	C₂×C₄	—	344	2	no	—
353	C₂×C₄	—	172	1	no	—
354	C₂×C₄	#13063297	1	1	yes	characteristic
355	C₂×C₄	#13063297	1	1	yes	characteristic
356	C₂×C₄	#13063297	1	1	yes	characteristic
357	C₂×C₄	C₂×#10621882	1	2	yes	—
358	C₂×C₄	#13063345	1	2	yes	—
359	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
360	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
361	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
362	C₈	—	4	2	no	—
363	C₈	—	86	4	no	—
364	C₈	—	172	2	no	—
365	C₈	—	172	2	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	344	1	no	—
369	C₂²	—	344	1	no	—
370	C₂²	—	344	1	no	—
371	C₂²	#14583133	1	2	yes	—
372	C₂²	C₂×#13063297	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	4	no	—
377	C₄	—	4	2	no	—
378	C₄	—	8	2	no	—
379	C₄	—	344	1	no	—
380	C₄	#14583133	1	2	yes	—
381	C₄	C₂²×#10621882	1	1	yes	characteristic
382	C₄	C₂×#13063345	1	1	yes	characteristic
383	C₄	#14583712	1	2	yes	—
384	C₂	—	2	2	no	—
385	C₂	—	688	1	no	—
386	C₂	#19144122	1	1	yes	characteristic
387	C₂	C₂×#14583133	1	1	yes	characteristic
388	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×#462	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	#14583449	C₂²
hypercenter	C₂×#462	D₄₃
hypocenter	C₄₃	#19404
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#462

Complements

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

N	H ≅ G/N	type
3. #19143676	C₂	semidirect
4. #19134041	C₂	semidirect
5. #19133987	C₂	semidirect
6. #19133369	C₂	semidirect
7. #19133267	C₂	semidirect
9. C₂×C₄₃×#462	C₂	semidirect
192. C₂×#462	D₄₃	semidirect
231. C₄₃	#19404	semidirect

Maximal subgroups

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

subgroup	index
#19145716	2
#19144067	2
#19143676	2
#19134041	2
#19133987	2
#19133369	2
#19133267	2
#19133245	2
C₂×C₄₃×#462	2
#19404	43