Subgroups — #25254310

Order 11008.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147463	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136836	C₂	1	2	yes	maximal, complemented
3	#19136818	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136752	C₂	1	2	yes	maximal, complemented
5	#19136742	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136627	C₂	1	2	yes	maximal, complemented
7	#19136627	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136504	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136469	C₂	1	2	yes	maximal, complemented
10	C₄×#13063266	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3391	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583609	—	2	1	no	—
13	C₂×#13063266	C₂²	1	1	yes	characteristic
14	C₂×#13063266	C₂²	1	2	yes	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583301	C₂²	1	2	yes	—
17	#14583200	—	2	2	no	—
18	#14583190	—	2	1	no	—
19	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
20	C₄×D₃₄₄	C₂²	1	2	yes	—
21	#14582726	C₂²	1	2	yes	—
22	#14582724	C₂²	1	1	yes	characteristic
23	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
25	#14582700	C₂²	1	2	yes	—
26	#14582700	C₂²	1	2	yes	—
27	#14582700	C₂²	1	1	yes	characteristic
28	#14582700	C₂²	1	2	yes	—
29	#14582700	C₂²	1	2	yes	—
30	#14582699	C₂²	1	1	yes	characteristic
31	#14582675	C₂²	1	2	yes	—
32	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄₃×#550	—	2	1	no	—
34	C₄₃×#493	C₂²	1	1	yes	characteristic
35	C₄₃×#493	C₂²	1	2	yes	—
36	C₄₃×#459	—	2	1	no	—
37	C₄₃×#459	—	2	2	no	—
38	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₄₃	C₂²	1	2	yes	—
41	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
42	#13063325	—	2	1	no	—
43	#13063317	—	2	2	no	—
44	#13063303	—	4	2	no	—
45	#13063301	—	4	1	no	—
46	C₄×D₁₇₂	—	2	1	no	—
47	C₄×D₁₇₂	—	2	2	no	—
48	C₄⋊C₄×D₄₃	—	4	1	no	—
49	#13063266	—	2	2	no	—
50	#13063266	—	2	4	no	—
51	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
53	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
54	#13063259	—	2	1	no	—
55	#13063259	C₂³	1	1	yes	characteristic
56	#13063259	C₂³	1	2	yes	—
57	#13063257	—	2	2	no	—
58	#13063247	—	2	2	no	—
59	#13063247	—	2	1	no	—
60	C₂×#41×C₄₃	—	4	1	no	—
61	C₂×D₈×C₄₃	—	2	2	no	—
62	C₂×D₈×C₄₃	—	2	1	no	—
63	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂×D₈×C₄₃	C₂³	1	2	yes	—
65	#127×C₄₃	—	2	2	no	—
66	#127×C₄₃	—	2	1	no	—
67	#121×C₄₃	—	4	1	no	—
68	#121×C₄₃	—	4	2	no	—
69	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
70	C₄×D₄×C₄₃	—	2	2	no	—
71	C₄×D₄×C₄₃	—	2	1	no	—
72	#107×C₄₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
75	#102×C₄₃	—	2	2	no	—
76	#102×C₄₃	—	2	1	no	—
77	C₄×C₈×C₄₃	C₂³	1	2	yes	—
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	#41×C₄₃	—	8	4	no	—
80	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
81	C₂×D₄×C₄₃	—	2	2	no	—
82	C₂×D₄×C₄₃	—	2	1	no	—
83	C₂×D₄×C₄₃	—	4	2	no	—
84	C₂×D₄×C₄₃	—	4	1	no	—
85	C₂×D₄×C₄₃	—	4	2	no	—
86	C₂×D₄×C₄₃	—	4	1	no	—
87	C₂²×C₄×C₄₃	—	4	2	no	—
88	C₂²×C₄×C₄₃	—	4	1	no	—
89	C₂×D₁₇₂	—	2	1	no	—
90	C₂×C₄×D₄₃	—	4	1	no	—
91	C₂×C₄×D₄₃	—	4	2	no	—
92	C₂×Dic₈₆	—	4	1	no	—
93	D₈×C₄₃	—	2	1	no	—
94	D₈×C₄₃	—	2	2	no	—
95	D₈×C₄₃	—	2	1	no	—
96	D₈×C₄₃	—	2	2	no	—
97	D₈×C₄₃	—	8	2	no	—
98	D₈×C₄₃	—	8	1	no	—
99	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₄₃	—	2	1	no	—
103	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
104	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
105	C₂²⋊C₄×C₄₃	—	4	2	no	—
106	C₂²⋊C₄×C₄₃	—	4	1	no	—
107	C₄²×C₄₃	C₂×D₄	1	2	yes	—
108	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
109	(C₂×D₄₃)⋊C₄	—	4	2	no	—
110	(C₂×D₄₃)⋊C₄	—	4	1	no	—
111	C₄⋊Dic₄₃	—	2	1	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	C₄⋊Dic₄₃	—	4	1	no	—
114	Dic₄₃⋊C₄	—	4	2	no	—
115	Dic₄₃⋊C₄	—	4	2	no	—
116	D₃₄₄	—	2	1	no	—
117	D₃₄₄	#41	1	2	yes	complemented
118	C₄₃⋊C₁₆	—	2	1	no	—
119	C₄₃⋊C₁₆	—	2	2	no	—
120	C₄₃⋊C₁₆	—	2	1	no	—
121	C₂³×C₄₃	—	4	2	no	—
122	C₂³×C₄₃	—	4	1	no	—
123	C₂²×D₄₃	—	4	1	no	—
124	Q₈×C₄₃	—	2	2	no	—
125	D₄×C₄₃	—	4	1	no	—
126	D₄×C₄₃	—	4	2	no	—
127	D₄×C₄₃	—	4	1	no	—
128	D₄×C₄₃	—	4	2	no	—
129	D₄×C₄₃	—	8	4	no	—
130	D₄×C₄₃	—	8	2	no	—
131	D₄×C₄₃	—	16	1	no	—
132	D₄×C₄₃	—	16	2	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	8	4	no	—
135	C₂×C₄×C₄₃	—	8	2	no	—
136	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×Dic₄₃	—	4	1	no	—
142	C₂×Dic₄₃	—	4	2	no	—
143	D₁₇₂	—	2	2	no	—
144	D₁₇₂	—	4	1	no	—
145	C₄×D₄₃	—	8	2	no	—
146	C₄×D₄₃	—	16	2	no	—
147	Dic₈₆	—	16	1	no	—
148	C₈×C₄₃	—	2	2	no	—
149	C₈×C₄₃	—	2	1	no	—
150	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
151	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
152	#33933	—	43	1	no	maximal, Sylow
153	C₂²×C₄₃	—	8	2	no	—
154	C₂²×C₄₃	—	8	1	no	—
155	C₂²×C₄₃	—	8	2	no	—
156	C₂²×C₄₃	—	8	1	no	—
157	C₂²×C₄₃	—	8	2	no	—
158	C₂²×C₄₃	—	8	1	no	—
159	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
160	C₂×D₄₃	—	4	2	no	—
161	C₂×D₄₃	—	8	1	no	—
162	C₂×D₄₃	—	8	1	no	—
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	—	2	2	no	—
166	C₄×C₄₃	—	8	1	no	—
167	C₄×C₄₃	#509	1	2	yes	—
168	C₄×C₄₃	#548	1	1	yes	characteristic
169	C₄×C₄₃	#577	1	1	yes	characteristic
170	Dic₄₃	—	8	1	no	—
171	Dic₄₃	—	8	2	no	—
172	#3391	D₄₃	1	1	yes	characteristic, complemented
173	#3380	—	43	1	no	—
174	#2244	—	43	2	no	—
175	#2240	—	43	1	no	—
176	#2228	—	43	2	no	—
177	#2226	—	43	1	no	—
178	#2206	—	43	2	no	—
179	#2206	—	43	1	no	—
180	#2183	—	43	1	no	—
181	#2177	—	43	2	no	—
182	C₄×D₁₆	—	43	1	no	—
183	C₂×C₄₃	—	16	2	no	—
184	C₂×C₄₃	—	16	1	no	—
185	C₂×C₄₃	#3358	1	1	yes	characteristic
186	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
187	C₂×C₄₃	#3415	1	1	yes	characteristic
188	D₄₃	—	8	2	no	—
189	#554	—	86	1	no	—
190	#550	—	2	1	no	—
191	C₂×D₁₆	—	43	1	no	—
192	C₂×D₁₆	—	43	2	no	—
193	#500	—	43	1	no	—
194	#499	—	43	2	no	—
195	#493	C₂×D₄₃	1	1	yes	characteristic
196	#493	C₂×D₄₃	1	2	yes	—
197	#478	—	86	2	no	—
198	#476	—	86	1	no	—
199	#459	—	2	1	no	—
200	#459	—	2	2	no	—
201	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
202	C₄×D₈	—	43	1	no	—
203	C₄×D₈	—	43	2	no	—
204	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
205	C₄×D₈	C₂×D₄₃	1	2	yes	—
206	#367	—	43	2	no	—
207	#366	—	43	1	no	—
208	C₄⋊C₁₆	—	43	1	no	—
209	C₄⋊C₁₆	—	43	2	no	—
210	#357	—	43	1	no	—
211	#357	—	43	2	no	—
212	#357	—	43	2	no	—
213	#357	—	43	2	no	—
214	#357	—	43	2	no	—
215	#357	—	43	1	no	—
216	#346	—	43	2	no	—
217	C₄×C₁₆	—	43	1	no	—
218	C₄₃	#33933	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	4	1	no	—
220	C₂×D₈	—	2	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	—	43	1	no	—
223	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
224	C₂×D₈	C₂²×D₄₃	1	2	yes	—
225	C₄⋊Q₈	—	86	1	no	—
226	#127	—	2	2	no	—
227	#127	—	2	1	no	—
228	#125	—	86	2	no	—
229	C₂²⋊Q₈	—	172	1	no	—
230	C₂²⋊Q₈	—	172	2	no	—
231	#121	—	4	1	no	—
232	#121	—	4	2	no	—
233	C₄×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
234	C₄×D₄	—	2	2	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	—	86	1	no	—
237	C₄×D₄	—	86	2	no	—
238	C₂×C₄⋊C₄	—	172	1	no	—
239	D₁₆	—	86	2	no	—
240	D₁₆	—	86	4	no	—
241	C₂×C₁₆	—	43	1	no	—
242	C₂×C₁₆	—	43	1	no	—
243	C₂×C₁₆	—	43	2	no	—
244	#107	—	43	1	no	—
245	#107	—	43	2	no	—
246	#107	—	86	1	no	—
247	#107	C₂²×D₄₃	1	1	yes	characteristic
248	#106	—	86	2	no	—
249	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
250	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
251	#102	—	2	2	no	—
252	#102	—	2	1	no	—
253	#102	—	86	2	no	—
254	#102	—	86	1	no	—
255	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
256	C₄×C₈	C₂²×D₄₃	1	2	yes	—
257	#41	—	8	4	no	—
258	C₂×Q₈	—	172	1	no	—
259	C₂×Q₈	#10621882	1	1	yes	characteristic
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	86	1	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	172	1	no	—
270	C₂²×C₄	—	172	2	no	—
271	D₈	—	2	1	no	—
272	D₈	—	2	2	no	—
273	D₈	—	2	1	no	—
274	D₈	—	2	2	no	—
275	D₈	—	8	2	no	—
276	D₈	—	8	1	no	—
277	D₈	—	43	2	no	—
278	D₈	—	86	1	no	—
279	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
280	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
281	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	86	1	no	—
284	C₄⋊C₄	—	86	2	no	—
285	C₄⋊C₄	—	172	2	no	—
286	C₄⋊C₄	—	172	1	no	—
287	C₄⋊C₄	—	172	2	no	—
288	C₄⋊C₄	#10621882	1	1	yes	characteristic
289	C₄⋊C₄	#10621882	1	2	yes	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	172	1	no	—
293	C₂²⋊C₄	—	172	2	no	—
294	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
295	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
296	C₁₆	—	86	1	no	—
297	C₁₆	—	86	2	no	—
298	C₁₆	—	86	1	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	1	no	—
301	C₂³	—	172	1	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	688	1	no	—
304	D₄	—	4	1	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	1	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	4	no	—
309	D₄	—	8	2	no	—
310	D₄	—	16	2	no	—
311	D₄	—	16	1	no	—
312	D₄	—	86	2	no	—
313	D₄	—	172	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	344	2	no	—
318	C₂×C₄	—	172	1	no	—
319	C₂×C₄	—	172	2	no	—
320	C₂×C₄	—	688	2	no	—
321	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
322	C₂×C₄	C₂×#10621882	1	2	yes	—
323	C₂×C₄	C₂×#10621882	1	2	yes	—
324	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	1	no	—
328	C₈	#41×D₄₃	1	1	yes	characteristic
329	C₈	#13063372	1	1	yes	characteristic
330	C₂²	—	8	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	344	1	no	—
337	C₂²	—	344	1	no	—
338	C₂²	—	172	2	no	—
339	C₂²	C₂²×#10621882	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	344	1	no	—
345	C₄	—	344	2	no	—
346	C₄	#14583335	1	2	yes	—
347	C₄	#14583556	1	1	yes	characteristic
348	C₄	#14583710	1	1	yes	characteristic
349	C₂	—	16	2	no	—
350	C₂	—	16	1	no	—
351	C₂	—	344	2	no	—
352	C₂	#19147226	1	1	yes	characteristic
353	C₂	C₂×#14583335	1	1	yes	characteristic
354	C₂	#19147734	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3391	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3391	D₄₃
hypocenter	C₄₃	#33933
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

Complements

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

N	H ≅ G/N	type
1. #19147463	C₂	semidirect
2. #19136836	C₂	semidirect
3. #19136818	C₂	semidirect
4. #19136752	C₂	semidirect
5. #19136742	C₂	semidirect
6. #19136627	C₂	semidirect
7. #19136627	C₂	semidirect
8. #19136504	C₂	semidirect
9. #19136469	C₂	semidirect
10. C₄×#13063266	C₂	semidirect
11. C₄₃×#3391	C₂	semidirect
117. D₃₄₄	#41	semidirect
172. #3391	D₄₃	semidirect
218. C₄₃	#33933	semidirect

Maximal subgroups

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

subgroup	index
#19147463	2
#19136836	2
#19136818	2
#19136752	2
#19136742	2
#19136627	2
#19136504	2
#19136469	2
C₄×#13063266	2
C₄₃×#3391	2
#33933	43