Subgroups — #25253559

Order 11008.

568 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147019	C₂	1	1	yes	maximal, characteristic, complemented
2	#19146354	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136819	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136788	C₂	1	2	yes	maximal, complemented
5	#19136747	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136680	C₂	1	2	yes	maximal, complemented
7	#19136630	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136517	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136301	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063348	C₂²	1	2	yes	—
13	#14583614	—	2	1	no	—
14	#14583571	—	2	1	no	—
15	C₂×#13063266	C₂²	1	2	yes	—
16	C₂×C₁₆×D₄₃	C₂²	1	2	yes	—
17	#14583304	C₂²	1	1	yes	characteristic
18	#14583276	C₂²	1	1	yes	characteristic
19	#14583221	—	2	2	no	—
20	#14583189	—	2	1	no	—
21	#14583136	C₂²	1	2	yes	—
22	#14583132	C₂²	1	2	yes	—
23	#14583099	—	2	1	no	—
24	#14583047	—	2	2	no	—
25	C₈×D₁₇₂	C₂²	1	1	yes	characteristic
26	#14582936	C₂²	1	2	yes	—
27	#14582724	C₂²	1	2	yes	—
28	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
29	#14582703	C₂²	1	2	yes	—
30	#14582701	C₂²	1	2	yes	—
31	#14582701	C₂²	1	2	yes	—
32	#14582699	C₂²	1	2	yes	—
33	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₄₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₄₃×#367	C₂²	1	1	yes	characteristic
38	C₄₃×#358	C₂²	1	1	yes	characteristic
39	C₄₃×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×#10621902	—	2	2	no	—
42	#13063348	—	2	4	no	—
43	#13063348	—	4	2	no	—
44	C₂×#10621884	—	2	2	no	—
45	C₂×D₃₄₄	—	2	1	no	—
46	C₂×D₃₄₄	C₂³	1	2	yes	—
47	C₂×C₈×D₄₃	C₂³	1	2	yes	—
48	#13063325	—	2	1	no	—
49	#13063322	—	2	1	no	—
50	#13063322	—	4	1	no	—
51	#13063311	—	4	2	no	—
52	#13063300	—	2	2	no	—
53	#13063300	—	4	2	no	—
54	#13063299	—	4	1	no	—
55	#13063296	—	2	2	no	—
56	C₄×D₁₇₂	—	4	1	no	—
57	C₄×D₁₇₂	D₄	1	1	yes	characteristic
58	#13063284	—	4	1	no	—
59	#13063283	—	2	2	no	—
60	#13063266	—	4	2	no	—
61	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
62	C₁₆×D₄₃	—	4	2	no	—
63	#13063259	—	2	1	no	—
64	#13063259	C₂³	1	2	yes	—
65	#13063256	—	2	2	no	—
66	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
67	#13063246	—	2	2	no	—
68	#13063245	—	2	2	no	—
69	(C₂×D₄₃)⋊C₈	D₄	1	2	yes	—
70	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₄₃	—	2	1	no	—
73	C₄×D₄×C₄₃	—	2	1	no	—
74	SD₃₂×C₄₃	—	2	2	no	—
75	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
77	#107×C₄₃	C₂³	1	1	yes	characteristic
78	#107×C₄₃	C₂³	1	1	yes	characteristic
79	#103×C₄₃	—	2	1	no	—
80	#102×C₄₃	—	2	1	no	—
81	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₃	—	2	1	no	—
83	C₂×D₄×C₄₃	—	2	1	no	—
84	C₂²×C₄×C₄₃	—	4	1	no	—
85	C₂×C₂²⋊D₄₃	—	4	2	no	—
86	#10621902	—	4	4	no	—
87	#10621902	—	8	2	no	—
88	C₂×D₁₇₂	—	2	2	no	—
89	C₂×D₁₇₂	—	4	1	no	—
90	C₂×D₁₇₂	—	4	1	no	—
91	C₂×D₁₇₂	—	4	2	no	—
92	C₂×D₁₇₂	—	4	2	no	—
93	C₂×D₁₇₂	D₈	1	1	yes	characteristic
94	C₂×C₄×D₄₃	—	4	2	no	—
95	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
96	C₂×Dic₈₆	—	4	1	no	—
97	Q₁₆×C₄₃	—	2	1	no	—
98	Q₁₆×C₄₃	#41	1	2	yes	—
99	D₈×C₄₃	—	2	1	no	—
100	D₈×C₄₃	—	2	1	no	—
101	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₄₃	—	2	1	no	—
104	C₄⋊C₄×C₄₃	—	2	1	no	—
105	C₄⋊C₄×C₄₃	—	4	1	no	—
106	C₂²⋊C₄×C₄₃	—	4	1	no	—
107	C₄²×C₄₃	—	4	1	no	—
108	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂²⋊Dic₄₃	—	4	2	no	—
110	#10621884	—	4	4	no	—
111	(C₂×D₄₃)⋊C₄	—	4	2	no	—
112	(C₂×D₄₃)⋊C₄	—	4	2	no	—
113	(C₂×D₄₃)⋊C₄	D₈	1	2	yes	—
114	C₄⋊Dic₄₃	—	2	2	no	—
115	C₄⋊Dic₄₃	—	4	1	no	—
116	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
117	C₄×Dic₄₃	—	4	2	no	—
118	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
119	D₃₄₄	—	2	2	no	—
120	D₃₄₄	—	2	2	no	—
121	D₃₄₄	—	8	1	no	—
122	C₈×D₄₃	—	2	2	no	—
123	C₈×D₄₃	—	2	2	no	—
124	C₁₆×C₄₃	—	2	1	no	—
125	C₁₆×C₄₃	—	2	1	no	—
126	C₄₃⋊C₁₆	—	2	2	no	—
127	C₂³×C₄₃	—	4	1	no	—
128	C₂²×D₄₃	—	4	2	no	—
129	C₂²×D₄₃	C₂×D₈	1	2	yes	—
130	Q₈×C₄₃	—	2	2	no	—
131	Q₈×C₄₃	—	4	1	no	—
132	D₄×C₄₃	—	4	1	no	—
133	D₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	—	8	2	no	—
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	C₂²⋊D₄₃	—	16	2	no	—
142	C₂×Dic₄₃	—	4	2	no	—
143	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
144	D₁₇₂	—	2	2	no	—
145	D₁₇₂	—	4	2	no	—
146	D₁₇₂	—	4	2	no	—
147	D₁₇₂	—	8	4	no	—
148	D₁₇₂	—	8	2	no	—
149	D₁₇₂	—	16	1	no	—
150	D₁₇₂	—	16	2	no	—
151	C₄×D₄₃	—	2	2	no	—
152	C₄×D₄₃	—	2	2	no	—
153	C₄×D₄₃	—	8	4	no	—
154	Dic₈₆	—	16	1	no	—
155	C₈×C₄₃	—	2	1	no	—
156	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
157	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
158	C₄₃⋊C₈	—	2	2	no	—
159	#33872	—	43	1	no	maximal, Sylow
160	C₂²×C₄₃	—	8	1	no	—
161	C₂²×C₄₃	—	8	1	no	—
162	C₂²×C₄₃	—	8	1	no	—
163	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
164	C₂×D₄₃	—	2	2	no	—
165	C₂×D₄₃	—	2	2	no	—
166	C₂×D₄₃	—	2	2	no	—
167	C₂×D₄₃	—	8	2	no	—
168	C₂×D₄₃	—	8	2	no	—
169	C₂×D₄₃	—	8	2	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	—	4	2	no	—
172	C₄×C₄₃	—	8	1	no	—
173	C₄×C₄₃	—	8	1	no	—
174	C₄×C₄₃	#508	1	2	yes	—
175	C₄×C₄₃	#546	1	1	yes	characteristic
176	C₄×C₄₃	#577	1	1	yes	characteristic
177	Dic₄₃	—	2	2	no	—
178	Dic₄₃	—	8	2	no	—
179	#3341	—	43	1	no	—
180	#3287	—	43	1	no	—
181	#2240	—	43	1	no	—
182	#2234	—	43	2	no	—
183	#2227	—	43	1	no	—
184	#2215	—	43	2	no	—
185	#2206	—	43	1	no	—
186	#2186	—	43	2	no	—
187	C₄×SD₃₂	D₄₃	1	1	yes	characteristic, complemented
188	D₄×C₁₆	—	43	1	no	—
189	#2138	—	43	2	no	—
190	C₂×C₄₃	—	16	1	no	—
191	C₂×C₄₃	#3294	1	1	yes	characteristic
192	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
193	C₂×C₄₃	#3415	1	1	yes	characteristic
194	D₄₃	—	4	2	no	—
195	D₄₃	—	16	2	no	—
196	C₂×#135	—	43	2	no	—
197	#555	—	86	1	no	—
198	#550	—	86	1	no	—
199	C₂×SD₃₂	C₂×D₄₃	1	1	yes	characteristic
200	C₂×D₁₆	—	43	2	no	—
201	C₂²×C₁₆	—	43	2	no	—
202	#500	—	43	1	no	—
203	#493	—	43	1	no	—
204	#482	—	86	2	no	—
205	#476	—	86	1	no	—
206	#466	—	43	2	no	—
207	#466	—	43	2	no	—
208	#460	—	86	1	no	—
209	#449	—	86	2	no	—
210	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
211	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
212	C₈×D₄	—	43	1	no	—
213	#427	—	43	2	no	—
214	#367	C₂×D₄₃	1	1	yes	characteristic
215	#366	—	43	2	no	—
216	C₄⋊C₁₆	—	43	1	no	—
217	#358	—	43	2	no	—
218	#358	C₂×D₄₃	1	1	yes	characteristic
219	#357	—	43	2	no	—
220	#357	—	43	2	no	—
221	#357	—	43	2	no	—
222	#357	C₂×D₄₃	1	1	yes	characteristic
223	C₂²⋊C₁₆	—	43	2	no	—
224	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
225	C₄₃	#33872	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	86	2	no	—
227	#135	—	86	4	no	—
228	#135	—	172	2	no	—
229	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
230	C₂×SD₁₆	—	86	2	no	—
231	C₂×D₈	—	43	2	no	—
232	C₂×D₈	—	86	1	no	—
233	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₂²×C₈	—	43	2	no	—
235	C₄⋊Q₈	—	86	1	no	—
236	#127	—	86	1	no	—
237	#127	—	172	1	no	—
238	#124	—	172	2	no	—
239	#121	—	86	2	no	—
240	#121	—	86	2	no	—
241	#121	—	172	2	no	—
242	#121	—	172	1	no	—
243	C₄×Q₈	—	2	1	no	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	—	43	1	no	—
246	C₄×D₄	—	172	1	no	—
247	#117	—	86	2	no	—
248	#117	—	172	1	no	—
249	SD₃₂	—	2	2	no	—
250	D₁₆	—	172	2	no	—
251	C₂×C₁₆	—	43	2	no	—
252	C₂×C₁₆	—	172	2	no	—
253	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
254	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
255	#107	—	43	2	no	—
256	#107	—	86	1	no	—
257	#107	C₂²×D₄₃	1	1	yes	characteristic
258	#107	C₂²×D₄₃	1	1	yes	characteristic
259	#106	—	86	2	no	—
260	C₄⋊C₈	—	43	1	no	—
261	#103	—	2	1	no	—
262	#102	—	2	1	no	—
263	#102	—	86	2	no	—
264	#102	—	86	2	no	—
265	C₂²⋊C₈	—	43	2	no	—
266	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
267	#41	—	344	2	no	—
268	#41	—	172	4	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	172	1	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	43	1	no	—
273	C₂×D₄	—	86	2	no	—
274	C₂×D₄	—	172	2	no	—
275	C₂×D₄	—	172	2	no	—
276	C₂×D₄	—	172	2	no	—
277	C₂×D₄	—	172	1	no	—
278	C₂×D₄	—	172	1	no	—
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	—	43	2	no	—
281	C₂²×C₄	—	172	2	no	—
282	Q₁₆	—	2	1	no	—
283	Q₁₆	#10621898	1	2	yes	—
284	SD₁₆	—	172	4	no	—
285	D₈	—	2	1	no	—
286	D₈	—	2	1	no	—
287	D₈	—	86	2	no	—
288	D₈	—	86	2	no	—
289	D₈	—	344	1	no	—
290	C₂×C₈	—	43	2	no	—
291	C₂×C₈	—	86	2	no	—
292	C₂×C₈	—	86	2	no	—
293	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
294	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	43	1	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₄⋊C₄	—	172	1	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₂²⋊C₄	—	43	2	no	—
303	C₂²⋊C₄	—	172	2	no	—
304	C₂²⋊C₄	—	172	2	no	—
305	C₂²⋊C₄	—	172	2	no	—
306	C₄²	—	4	1	no	—
307	C₄²	—	172	2	no	—
308	C₄²	D₄×D₄₃	1	1	yes	characteristic
309	C₁₆	—	2	1	no	—
310	C₁₆	—	2	1	no	—
311	C₁₆	—	86	2	no	—
312	C₂³	—	4	1	no	—
313	C₂³	—	43	2	no	—
314	C₂³	—	172	2	no	—
315	Q₈	—	2	2	no	—
316	Q₈	—	4	1	no	—
317	Q₈	—	688	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	4	1	no	—
320	D₄	—	86	2	no	—
321	D₄	—	344	4	no	—
322	D₄	—	344	2	no	—
323	D₄	—	172	2	no	—
324	D₄	—	172	2	no	—
325	D₄	—	688	1	no	—
326	D₄	—	688	2	no	—
327	D₄	—	688	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	43	2	no	—
333	C₂×C₄	—	86	2	no	—
334	C₂×C₄	—	86	2	no	—
335	C₂×C₄	—	344	4	no	—
336	C₂×C₄	—	172	2	no	—
337	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
338	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
339	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	86	2	no	—
342	C₈	C₂×#10621898	1	1	yes	characteristic
343	C₈	#13063371	1	1	yes	characteristic
344	C₂²	—	8	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	8	1	no	—
347	C₂²	—	86	2	no	—
348	C₂²	—	86	2	no	—
349	C₂²	—	86	2	no	—
350	C₂²	—	344	2	no	—
351	C₂²	—	344	2	no	—
352	C₂²	—	344	2	no	—
353	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	8	1	no	—
357	C₄	—	8	1	no	—
358	C₄	—	86	2	no	—
359	C₄	—	344	2	no	—
360	C₄	#14583330	1	2	yes	—
361	C₄	#14583545	1	1	yes	characteristic
362	C₄	#14583706	1	1	yes	characteristic
363	C₂	—	16	1	no	—
364	C₂	—	172	2	no	—
365	C₂	—	688	2	no	—
366	C₂	#19146410	1	1	yes	characteristic
367	C₂	C₂×#14583330	1	1	yes	characteristic
368	C₂	#19147733	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×SD₃₂×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₄×SD₃₂	D₄₃
hypocenter	C₄₃	#33872
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #19147019	C₂	semidirect
2. #19146354	C₂	semidirect
3. #19136819	C₂	semidirect
4. #19136788	C₂	semidirect
5. #19136747	C₂	semidirect
6. #19136680	C₂	semidirect
7. #19136630	C₂	semidirect
8. #19136517	C₂	semidirect
9. C₁₆×D₁₇₂	C₂	semidirect
10. #19136301	C₂	semidirect
11. C₄×SD₃₂×C₄₃	C₂	semidirect
187. C₄×SD₃₂	D₄₃	semidirect
225. C₄₃	#33872	semidirect

Maximal subgroups

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

subgroup	index
#19147019	2
#19146354	2
#19136819	2
#19136788	2
#19136747	2
#19136680	2
#19136630	2
#19136517	2
C₁₆×D₁₇₂	2
#19136301	2
C₄×SD₃₂×C₄₃	2
#33872	43