Subgroups — #25081940

Order 11008.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145725	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583194	C₂	1	1	yes	maximal, characteristic, complemented
3	#19144067	C₂	1	1	yes	maximal, characteristic
4	#19134011	C₂	1	4	yes	maximal, complemented
5	#19133977	C₂	1	4	yes	maximal, complemented
6	#19133366	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133275	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133242	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063356	—	2	1	no	—
11	#14583449	C₂²	1	1	yes	characteristic
12	C₂×#13063325	C₂²	1	1	yes	characteristic
13	#14583297	C₂²	1	2	yes	—
14	#14583246	C₂²	1	2	yes	—
15	#14583218	—	2	2	no	—
16	#14583194	—	2	2	no	—
17	#14583194	C₂²	1	4	yes	—
18	#14583190	—	2	2	no	—
19	#14583121	—	2	2	no	—
20	#14583109	C₂²	1	2	yes	—
21	#14583099	C₂²	1	2	yes	—
22	C₂×#13063258	—	2	1	no	—
23	#14582923	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
25	#14582898	—	2	1	no	—
26	C₂×#13063249	—	2	1	no	—
27	#14582650	C₂²	1	4	yes	—
28	#14582647	C₂²	1	4	yes	—
29	#14582644	C₂²	1	2	yes	—
30	#14582642	C₂²	1	2	yes	—
31	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₈×Q₈×C₄₃	—	2	2	no	—
33	C₈×Q₈×C₄₃	C₂²	1	4	yes	—
34	C₄₃×#431	—	2	2	no	—
35	C₄₃×#431	C₂²	1	1	yes	characteristic
36	C₄₃×#431	C₂²	1	1	yes	characteristic
37	C₂×#10621898	—	2	1	no	—
38	C₂²×Dic₈₆	—	2	1	no	—
39	#13063356	—	4	4	no	—
40	C₂×#10621885	—	2	2	no	—
41	C₂×#10621884	—	2	2	no	—
42	C₂×#10621877	—	2	1	no	—
43	C₂²×C₄₃⋊C₈	—	2	1	no	—
44	#13063325	—	2	1	no	—
45	#13063325	—	2	1	no	—
46	#13063325	—	2	1	no	—
47	#13063325	C₂³	1	1	yes	characteristic
48	#13063325	C₂³	1	2	yes	—
49	#13063322	C₂³	1	1	yes	characteristic
50	#13063313	—	2	1	no	—
51	#13063299	—	2	1	no	—
52	#13063299	—	2	1	no	—
53	C₄×D₁₇₂	—	2	1	no	—
54	C₄×D₁₇₂	—	2	1	no	—
55	C₂×C₄⋊Dic₄₃	—	2	1	no	—
56	C₂×C₄⋊Dic₄₃	—	2	1	no	—
57	#13063258	—	2	4	no	—
58	#13063257	—	2	4	no	—
59	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
60	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
61	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
62	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
63	#13063249	—	2	4	no	—
64	#13063248	—	2	2	no	—
65	#13063247	—	2	4	no	—
66	#13063245	—	2	2	no	—
67	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
68	C₄×Q₈×C₄₃	—	2	1	no	—
69	C₄×Q₈×C₄₃	—	2	1	no	—
70	C₄×Q₈×C₄₃	D₄	1	2	yes	—
71	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₄₃	—	2	1	no	—
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₃	—	2	4	no	—
77	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
78	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
79	C₂²⋊C₈×C₄₃	—	2	1	no	—
80	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
81	C₄×C₈×C₄₃	—	2	4	no	—
82	C₄×C₈×C₄₃	—	2	1	no	—
83	C₄×C₈×C₄₃	C₂³	1	2	yes	—
84	C₂×Q₈×C₄₃	—	2	1	no	—
85	C₂×Q₈×C₄₃	—	2	2	no	—
86	C₂×Q₈×C₄₃	SD₁₆	1	4	yes	—
87	C₂×Q₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₃	—	2	1	no	—
89	C₂²×C₄×C₄₃	—	2	1	no	—
90	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
93	C₂×C₂²⋊D₄₃	—	4	1	no	—
94	C₂²×Dic₄₃	—	4	1	no	—
95	#10621898	—	4	2	no	—
96	#10621898	—	4	2	no	—
97	C₂×D₁₇₂	—	2	1	no	—
98	C₂×D₁₇₂	—	2	1	no	—
99	C₂×D₁₇₂	—	2	1	no	—
100	C₂×C₄×D₄₃	—	4	1	no	—
101	C₂×Dic₈₆	—	2	1	no	—
102	C₂×Dic₈₆	—	2	2	no	—
103	C₂×Dic₈₆	—	4	1	no	—
104	C₂×Dic₈₆	—	8	2	no	—
105	C₂×C₈×C₄₃	—	2	2	no	—
106	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
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	C₂×D₄	1	2	yes	—
111	C₄²×C₄₃	—	2	2	no	—
112	C₄²×C₄₃	—	2	2	no	—
113	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₃	#41	1	1	yes	characteristic
116	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
117	#10621885	—	4	4	no	—
118	#10621884	—	4	4	no	—
119	(C₂×D₄₃)⋊C₄	—	4	1	no	—
120	(C₂×D₄₃)⋊C₄	—	4	1	no	—
121	C₄⋊Dic₄₃	—	2	2	no	—
122	C₄⋊Dic₄₃	—	2	2	no	—
123	C₄⋊Dic₄₃	—	2	1	no	—
124	C₄⋊Dic₄₃	—	2	1	no	—
125	C₄⋊Dic₄₃	—	4	1	no	—
126	C₄⋊Dic₄₃	—	4	1	no	—
127	#10621877	—	4	2	no	—
128	C₂×C₄₃⋊C₈	—	2	2	no	—
129	C₂×C₄₃⋊C₈	—	2	2	no	—
130	C₂×C₄₃⋊C₈	—	4	2	no	—
131	C₂³×C₄₃	#121	1	1	yes	characteristic
132	C₂²×D₄₃	—	4	1	no	—
133	Q₈×C₄₃	—	2	2	no	—
134	Q₈×C₄₃	—	2	4	no	—
135	Q₈×C₄₃	C₂×SD₁₆	1	4	yes	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	4	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	4	1	no	—
141	C₂×C₄×C₄₃	—	8	1	no	—
142	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₄₃	#135	1	2	yes	—
147	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
150	C₂²⋊D₄₃	—	8	2	no	—
151	C₂×Dic₄₃	—	4	2	no	—
152	C₂×Dic₄₃	—	4	1	no	—
153	C₂×Dic₄₃	—	8	2	no	—
154	D₁₇₂	—	4	1	no	—
155	D₁₇₂	—	4	1	no	—
156	D₁₇₂	—	4	2	no	—
157	D₁₇₂	—	8	1	no	—
158	C₄×D₄₃	—	8	2	no	—
159	Dic₈₆	—	4	1	no	—
160	Dic₈₆	—	4	1	no	—
161	Dic₈₆	—	8	2	no	—
162	Dic₈₆	—	8	2	no	—
163	C₈×C₄₃	—	2	4	no	—
164	C₈×C₄₃	—	4	2	no	—
165	C₄₃⋊C₈	—	4	2	no	—
166	C₄₃⋊C₈	—	4	2	no	—
167	#19403	—	43	1	no	maximal, Sylow
168	C₂²×C₄₃	—	2	1	no	—
169	C₂²×C₄₃	—	2	1	no	—
170	C₂²×C₄₃	#465	1	2	yes	—
171	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
172	C₂×D₄₃	—	8	1	no	—
173	C₂×D₄₃	—	8	1	no	—
174	C₂×D₄₃	—	8	1	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	2	4	no	—
177	C₄×C₄₃	—	2	1	no	—
178	C₄×C₄₃	—	2	1	no	—
179	C₄×C₄₃	—	4	2	no	—
180	C₄×C₄₃	#465	1	2	yes	—
181	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
182	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
183	C₄×C₄₃	#578	1	2	yes	—
184	Dic₄₃	—	8	1	no	—
185	Dic₄₃	—	8	2	no	—
186	#3227	—	43	1	no	—
187	C₂×#477	—	43	1	no	—
188	#3037	—	43	1	no	—
189	#2982	D₄₃	1	1	yes	characteristic, complemented
190	#1678	—	43	4	no	—
191	#1672	—	43	4	no	—
192	#1570	—	43	1	no	—
193	#1548	—	43	1	no	—
194	#1540	—	43	1	no	—
195	C₂×C₄₃	—	2	2	no	—
196	C₂×C₄₃	#3044	1	1	yes	characteristic
197	C₂×C₄₃	C₂×#465	1	1	yes	characteristic
198	C₂×C₄₃	#3068	1	1	yes	characteristic
199	D₄₃	—	16	1	no	—
200	C₂×#137	—	86	1	no	—
201	#532	—	43	1	no	—
202	C₂×C₄⋊Q₈	—	43	1	no	—
203	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
204	#498	—	43	2	no	—
205	#486	—	43	2	no	—
206	#481	—	86	2	no	—
207	#477	—	43	4	no	—
208	#477	—	86	2	no	—
209	#476	—	86	2	no	—
210	#464	—	86	2	no	—
211	#462	—	43	2	no	—
212	#460	—	43	2	no	—
213	C₈×Q₈	—	2	2	no	—
214	C₈×Q₈	C₂×D₄₃	1	4	yes	—
215	#431	—	2	2	no	—
216	#431	C₂×D₄₃	1	1	yes	characteristic
217	#431	C₂×D₄₃	1	1	yes	characteristic
218	C₂×#107	—	86	1	no	—
219	#423	—	43	1	no	—
220	C₂×C₄⋊C₈	—	43	1	no	—
221	#417	—	86	1	no	—
222	C₂×#103	—	86	1	no	—
223	#334	—	43	4	no	—
224	#333	—	43	4	no	—
225	#332	—	43	2	no	—
226	#331	—	43	2	no	—
227	C₄₃	#19403	1	1	yes	characteristic, complemented, Sylow
228	C₂×#41	—	86	1	no	—
229	C₂²×Q₈	—	86	1	no	—
230	C₂²×Q₈	D₁₇₂	1	1	yes	characteristic
231	#137	—	172	4	no	—
232	C₂×Q₁₆	—	86	2	no	—
233	C₂×SD₁₆	—	86	2	no	—
234	C₂×M₁₆	—	86	1	no	—
235	C₂²×C₈	—	86	1	no	—
236	C₄⋊Q₈	—	43	2	no	—
237	C₄⋊Q₈	—	43	1	no	—
238	C₄⋊Q₈	—	86	1	no	—
239	C₄⋊Q₈	—	86	1	no	—
240	C₄⋊Q₈	—	86	1	no	—
241	#127	—	43	1	no	—
242	#124	—	86	1	no	—
243	#121	—	86	1	no	—
244	#121	—	86	1	no	—
245	C₄×Q₈	—	2	1	no	—
246	C₄×Q₈	—	2	1	no	—
247	C₄×Q₈	D₁₇₂	1	2	yes	—
248	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
249	C₄×D₄	—	86	1	no	—
250	C₄×D₄	—	86	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	86	1	no	—
253	C₂×C₄⋊C₄	—	86	1	no	—
254	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
255	C₂×C₄²	—	2	1	no	—
256	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
257	#107	—	86	4	no	—
258	#106	—	86	4	no	—
259	C₄⋊C₈	—	2	4	no	—
260	C₄⋊C₈	—	43	2	no	—
261	C₄⋊C₈	—	43	2	no	—
262	C₄⋊C₈	—	86	1	no	—
263	C₄⋊C₈	—	86	1	no	—
264	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
265	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
266	#103	—	86	4	no	—
267	#103	—	86	2	no	—
268	#102	—	86	4	no	—
269	#102	—	86	2	no	—
270	C₂²⋊C₈	—	2	1	no	—
271	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
272	C₄×C₈	—	2	4	no	—
273	C₄×C₈	—	2	1	no	—
274	C₄×C₈	C₂²×D₄₃	1	2	yes	—
275	#41	—	172	2	no	—
276	#41	—	172	2	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	86	1	no	—
280	C₂×Q₈	—	86	2	no	—
281	C₂×Q₈	—	344	2	no	—
282	C₂×Q₈	—	172	1	no	—
283	C₂×Q₈	#10621873	1	4	yes	—
284	C₂×Q₈	C₂×D₁₇₂	1	2	yes	—
285	C₂×D₄	—	86	1	no	—
286	C₂×D₄	—	86	1	no	—
287	C₂×D₄	—	86	1	no	—
288	C₂×D₄	—	172	1	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	172	1	no	—
292	C₂²×C₄	—	172	1	no	—
293	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
294	C₂²×C₄	#10621898	1	1	yes	characteristic
295	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
296	Q₁₆	—	172	4	no	—
297	SD₁₆	—	172	4	no	—
298	M₁₆	—	172	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	86	2	no	—
301	C₂×C₈	—	86	2	no	—
302	C₂×C₈	—	172	2	no	—
303	C₂×C₈	C₂×C₂²⋊D₄₃	1	4	yes	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	86	2	no	—
308	C₄⋊C₄	—	86	1	no	—
309	C₄⋊C₄	—	86	2	no	—
310	C₄⋊C₄	—	86	1	no	—
311	C₄⋊C₄	—	172	1	no	—
312	C₄⋊C₄	—	172	1	no	—
313	C₄⋊C₄	C₂×D₁₇₂	1	2	yes	—
314	C₂²⋊C₄	—	172	1	no	—
315	C₂²⋊C₄	—	172	1	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
319	C₄²	#10621898	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
321	C₄²	C₂³×D₄₃	1	1	yes	characteristic
322	C₂³	—	172	1	no	—
323	C₂³	#13063299	1	1	yes	characteristic
324	Q₈	—	2	4	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	344	2	no	—
327	Q₈	—	344	2	no	—
328	Q₈	—	172	1	no	—
329	Q₈	—	172	1	no	—
330	Q₈	C₂×#10621873	1	4	yes	—
331	D₄	—	344	1	no	—
332	D₄	—	344	2	no	—
333	D₄	—	172	1	no	—
334	D₄	—	172	1	no	—
335	D₄	—	172	2	no	—
336	C₂×C₄	—	2	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	8	1	no	—
342	C₂×C₄	—	344	2	no	—
343	C₂×C₄	—	344	2	no	—
344	C₂×C₄	—	172	1	no	—
345	C₂×C₄	—	172	2	no	—
346	C₂×C₄	#13063299	1	1	yes	characteristic
347	C₂×C₄	#13063299	1	1	yes	characteristic
348	C₂×C₄	#13063299	1	1	yes	characteristic
349	C₂×C₄	C₂×#10621873	1	2	yes	—
350	C₂×C₄	#13063347	1	2	yes	—
351	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
352	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
353	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
354	C₈	—	2	4	no	—
355	C₈	—	4	2	no	—
356	C₈	—	172	2	no	—
357	C₈	—	172	2	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	344	1	no	—
361	C₂²	—	344	1	no	—
362	C₂²	—	344	1	no	—
363	C₂²	#14583128	1	2	yes	—
364	C₂²	C₂×#13063299	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	4	no	—
369	C₄	—	4	2	no	—
370	C₄	—	344	2	no	—
371	C₄	—	344	1	no	—
372	C₄	#14583128	1	2	yes	—
373	C₄	C₂²×#10621873	1	1	yes	characteristic
374	C₄	C₂×#13063347	1	1	yes	characteristic
375	C₄	#14583714	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	688	1	no	—
378	C₂	#19144125	1	1	yes	characteristic
379	C₂	C₂×#14583128	1	1	yes	characteristic
380	C₂	#19144323	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063299
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#2982	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	#14583449	C₂²
hypercenter	#2982	D₄₃
hypocenter	C₄₃	#19403
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2982

Complements

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

N	H ≅ G/N	type
2. C₂×#14583194	C₂	semidirect
4. #19134011	C₂	semidirect
5. #19133977	C₂	semidirect
6. #19133366	C₂	semidirect
7. #19133275	C₂	semidirect
9. C₄₃×#2982	C₂	semidirect
189. #2982	D₄₃	semidirect
227. C₄₃	#19403	semidirect

Maximal subgroups

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

subgroup	index
#19145725	2
C₂×#14583194	2
#19144067	2
#19134011	2
#19133977	2
#19133366	2
#19133275	2
#19133242	2
C₄₃×#2982	2
#19403	43