Subgroups — #25252131

Order 11008.

602 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145171	C₂	1	2	yes	maximal, complemented
2	#19136869	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136843	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136817	C₂	1	2	yes	maximal, complemented
5	#19136682	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136680	C₂	1	2	yes	maximal, complemented
7	#19136678	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13063266	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583446	—	2	2	no	—
13	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
14	C₂×#13063266	C₂²	1	2	yes	—
15	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
17	#14583303	C₂²	1	2	yes	—
18	#14583274	C₂²	1	2	yes	—
19	#14583244	—	2	2	no	—
20	#14583136	C₂²	1	2	yes	—
21	#14583134	—	2	2	no	—
22	#14583132	C₂²	1	2	yes	—
23	C₄×D₃₄₄	C₂²	1	2	yes	—
24	C₄×#10621882	—	2	2	no	—
25	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
26	#14582725	C₂²	1	1	yes	characteristic
27	#14582724	C₂²	1	2	yes	—
28	#14582701	C₂²	1	2	yes	—
29	#14582700	C₂²	1	2	yes	—
30	#14582699	C₂²	1	2	yes	—
31	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
34	C₂×D₁₆×C₄₃	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₄₃	C₂²	1	2	yes	—
36	C₄₃×#366	C₂²	1	1	yes	characteristic
37	C₄₃×#357	C₂²	1	2	yes	—
38	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
39	C₂×#10621898	—	4	2	no	—
40	C₂×D₃₄₄	C₂³	1	2	yes	—
41	C₂×#10621882	—	2	2	no	—
42	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
43	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
44	#13063324	—	2	2	no	—
45	#13063321	—	2	2	no	—
46	#13063311	—	4	2	no	—
47	#13063300	—	2	2	no	—
48	#13063298	—	4	2	no	—
49	#13063296	—	2	2	no	—
50	C₄×D₁₇₂	—	2	2	no	—
51	C₄×C₂²⋊D₄₃	—	4	2	no	—
52	C₄²×D₄₃	D₄	1	1	yes	characteristic
53	D₆₈₈	—	2	2	no	—
54	#13063266	—	2	4	no	—
55	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
57	C₁₆×D₄₃	—	2	2	no	—
58	C₁₆×D₄₃	—	2	2	no	—
59	#13063259	C₂³	1	2	yes	—
60	#13063258	—	2	2	no	—
61	#13063247	—	2	2	no	—
62	#13063246	—	2	2	no	—
63	#13063245	—	2	2	no	—
64	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₄₃	D₄	1	1	yes	characteristic
66	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
67	C₂×D₈×C₄₃	C₂³	1	2	yes	—
68	C₄×D₄×C₄₃	—	2	2	no	—
69	D₁₆×C₄₃	—	2	2	no	—
70	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
72	#107×C₄₃	C₂³	1	2	yes	—
73	#102×C₄₃	—	2	2	no	—
74	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₄₃	—	2	2	no	—
76	C₂²×C₄×C₄₃	—	4	2	no	—
77	C₂×C₂²⋊D₄₃	—	4	2	no	—
78	C₂×C₂²⋊D₄₃	—	4	2	no	—
79	C₂×C₂²⋊D₄₃	—	4	2	no	—
80	#10621898	—	8	8	no	—
81	C₂×D₁₇₂	—	2	2	no	—
82	C₂×D₁₇₂	—	4	2	no	—
83	C₂×C₄×D₄₃	—	4	2	no	—
84	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
85	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
86	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₈₆	—	4	2	no	—
88	D₈×C₄₃	—	2	2	no	—
89	D₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₂²⋊C₄×C₄₃	—	4	2	no	—
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂²⋊Dic₄₃	—	4	2	no	—
96	#10621882	—	4	4	no	—
97	(C₂×D₄₃)⋊C₄	—	4	2	no	—
98	(C₂×D₄₃)⋊C₄	—	4	2	no	—
99	C₄⋊Dic₄₃	—	2	2	no	—
100	Dic₄₃⋊C₄	—	4	2	no	—
101	C₄×Dic₄₃	D₈	1	1	yes	characteristic
102	C₄×Dic₄₃	D₈	1	1	yes	characteristic
103	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
106	D₃₄₄	—	2	2	no	—
107	D₃₄₄	—	2	2	no	—
108	C₈×D₄₃	—	2	1	no	—
109	C₈×D₄₃	—	2	1	no	—
110	C₈×D₄₃	—	2	2	no	—
111	C₁₆×C₄₃	—	2	1	no	—
112	C₁₆×C₄₃	#41	1	2	yes	complemented
113	C₄₃⋊C₁₆	—	2	1	no	—
114	C₄₃⋊C₁₆	#41	1	2	yes	complemented
115	C₂³×C₄₃	—	4	2	no	—
116	C₂²×D₄₃	—	4	2	no	—
117	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
118	D₄×C₄₃	—	4	2	no	—
119	D₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	8	4	no	—
122	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₄₃	—	8	4	no	—
126	C₂²⋊D₄₃	—	8	4	no	—
127	C₂²⋊D₄₃	—	8	4	no	—
128	C₂×Dic₄₃	—	4	2	no	—
129	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
132	D₁₇₂	—	4	2	no	—
133	D₁₇₂	—	4	2	no	—
134	D₁₇₂	—	8	4	no	—
135	C₄×D₄₃	—	2	1	no	—
136	C₄×D₄₃	—	2	1	no	—
137	C₄×D₄₃	—	2	2	no	—
138	C₄×D₄₃	—	8	4	no	—
139	C₄×D₄₃	D₁₆	1	4	yes	complemented
140	Dic₈₆	—	8	4	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
143	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
144	C₄₃⋊C₈	—	2	1	no	—
145	C₄₃⋊C₈	#135	1	2	yes	—
146	#33733	—	43	1	no	maximal, Sylow
147	C₂²×C₄₃	—	8	2	no	—
148	C₂²×C₄₃	—	8	2	no	—
149	C₂²×C₄₃	—	8	2	no	—
150	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₄₃	—	2	1	no	—
152	C₂×D₄₃	—	2	1	no	—
153	C₂×D₄₃	—	8	2	no	—
154	C₂×D₄₃	—	8	2	no	—
155	C₂×D₄₃	—	8	2	no	—
156	C₂×D₄₃	C₂×D₁₆	1	2	yes	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	8	2	no	—
159	C₄×C₄₃	C₂×D₁₆	1	2	yes	—
160	C₄×C₄₃	#532	1	1	yes	characteristic
161	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
162	Dic₄₃	—	2	1	no	—
163	Dic₄₃	—	8	2	no	—
164	Dic₄₃	C₂×D₁₆	1	2	yes	—
165	Dic₄₃	#508	1	2	yes	—
166	#3169	—	43	2	no	—
167	#2252	—	43	1	no	—
168	#2246	—	43	1	no	—
169	#2240	—	43	2	no	—
170	#2215	—	43	1	no	—
171	#2215	—	43	1	no	—
172	#2215	—	43	2	no	—
173	C₄×D₁₆	—	43	2	no	—
174	C₄×D₁₆	—	43	1	no	—
175	C₄×D₁₆	D₄₃	1	1	yes	characteristic, complemented
176	C₂×C₄×C₁₆	—	43	1	no	—
177	C₂×C₄₃	—	16	2	no	—
178	C₂×C₄₃	#3169	1	1	yes	characteristic
179	C₂×C₄₃	C₂²×D₁₆	1	1	yes	characteristic
180	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
181	D₄₃	—	2	2	no	—
182	D₄₃	—	16	2	no	—
183	#532	—	86	2	no	—
184	C₂×D₁₆	—	43	2	no	—
185	C₂×D₁₆	—	43	1	no	—
186	C₂×D₁₆	C₂×D₄₃	1	1	yes	characteristic
187	C₂²×C₁₆	—	43	1	no	—
188	C₂²×C₁₆	—	43	1	no	—
189	#500	—	43	2	no	—
190	#493	—	43	2	no	—
191	#486	—	86	2	no	—
192	#466	—	43	2	no	—
193	#466	—	43	2	no	—
194	#466	—	86	2	no	—
195	C₄×D₈	—	43	2	no	—
196	C₄×D₈	—	86	2	no	—
197	C₄×D₈	C₂×D₄₃	1	2	yes	—
198	C₂×C₄×C₈	—	43	1	no	—
199	#366	—	43	1	no	—
200	#366	—	43	2	no	—
201	#366	C₂×D₄₃	1	1	yes	characteristic
202	#357	—	43	2	no	—
203	#357	—	43	2	no	—
204	#357	—	43	2	no	—
205	#357	C₂×D₄₃	1	2	yes	—
206	C₄×C₁₆	—	43	1	no	—
207	C₄×C₁₆	—	43	1	no	—
208	C₄×C₁₆	—	43	1	no	—
209	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
210	C₄₃	#33733	1	1	yes	characteristic, complemented, Sylow
211	C₂×#41	—	172	2	no	—
212	C₂×D₈	—	43	2	no	—
213	C₂×D₈	—	86	2	no	—
214	C₂×D₈	C₂²×D₄₃	1	2	yes	—
215	C₂²×C₈	—	43	1	no	—
216	C₂²×C₈	—	43	1	no	—
217	C₄⋊Q₈	—	86	2	no	—
218	#127	—	86	2	no	—
219	#124	—	172	2	no	—
220	#121	—	86	2	no	—
221	#121	—	86	2	no	—
222	#121	—	172	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	86	2	no	—
225	C₄×D₄	—	172	2	no	—
226	C₂×C₄²	—	43	1	no	—
227	D₁₆	—	2	2	no	—
228	D₁₆	—	86	4	no	—
229	D₁₆	—	86	2	no	—
230	C₂×C₁₆	—	43	1	no	—
231	C₂×C₁₆	—	43	1	no	—
232	C₂×C₁₆	—	86	2	no	—
233	C₂×C₁₆	—	86	2	no	—
234	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
236	#107	—	43	2	no	—
237	#107	—	86	2	no	—
238	#107	C₂²×D₄₃	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	86	2	no	—
241	#102	—	86	2	no	—
242	#102	—	86	2	no	—
243	C₄×C₈	—	43	1	no	—
244	C₄×C₈	—	43	1	no	—
245	C₄×C₈	—	43	1	no	—
246	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
247	#41	—	344	8	no	—
248	C₂×Q₈	—	172	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	86	2	no	—
251	C₂×D₄	—	172	2	no	—
252	C₂×D₄	—	172	2	no	—
253	C₂×D₄	—	172	2	no	—
254	C₂×D₄	—	172	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	43	1	no	—
257	C₂²×C₄	—	43	1	no	—
258	C₂²×C₄	—	43	1	no	—
259	C₂²×C₄	—	172	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	86	2	no	—
263	D₈	—	86	2	no	—
264	D₈	—	172	4	no	—
265	C₂×C₈	—	43	1	no	—
266	C₂×C₈	—	43	1	no	—
267	C₂×C₈	—	86	2	no	—
268	C₂×C₈	—	86	1	no	—
269	C₂×C₈	—	86	1	no	—
270	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	86	2	no	—
274	C₄⋊C₄	—	172	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	172	2	no	—
277	C₂²⋊C₄	—	172	2	no	—
278	C₂²⋊C₄	—	172	2	no	—
279	C₄²	—	43	1	no	—
280	C₄²	—	43	1	no	—
281	C₄²	—	43	1	no	—
282	C₄²	D₄×D₄₃	1	1	yes	characteristic
283	C₁₆	—	2	1	no	—
284	C₁₆	—	43	2	no	—
285	C₁₆	—	86	1	no	—
286	C₁₆	#10621898	1	2	yes	complemented
287	C₂³	—	4	2	no	—
288	C₂³	—	43	1	no	—
289	C₂³	—	172	2	no	—
290	Q₈	—	344	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	344	4	no	—
294	D₄	—	344	4	no	—
295	D₄	—	344	4	no	—
296	D₄	—	344	4	no	—
297	D₄	—	172	2	no	—
298	D₄	—	172	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	—	43	1	no	—
302	C₂×C₄	—	43	1	no	—
303	C₂×C₄	—	43	4	no	—
304	C₂×C₄	—	43	1	no	—
305	C₂×C₄	—	86	2	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	86	1	no	—
308	C₂×C₄	—	344	4	no	—
309	C₂×C₄	—	172	2	no	—
310	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
311	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	43	2	no	—
315	C₈	—	86	1	no	—
316	C₈	#41×D₄₃	1	1	yes	characteristic
317	C₈	C₂×#10621898	1	1	yes	characteristic
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	43	2	no	—
322	C₂²	—	86	1	no	—
323	C₂²	—	86	1	no	—
324	C₂²	—	344	2	no	—
325	C₂²	—	344	2	no	—
326	C₂²	—	344	2	no	—
327	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	43	2	no	—
331	C₄	—	43	2	no	—
332	C₄	—	86	1	no	—
333	C₄	—	344	2	no	—
334	C₄	D₁₆×D₄₃	1	2	yes	—
335	C₄	#14583446	1	1	yes	characteristic
336	C₄	#135×D₄₃	1	1	yes	characteristic
337	C₂	—	16	2	no	—
338	C₂	—	86	2	no	—
339	C₂	—	688	2	no	—
340	C₂	#19145171	1	1	yes	characteristic
341	C₂	C₂×D₁₆×D₄₃	1	1	yes	characteristic
342	C₂	#508×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×D₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₄×D₁₆	D₄₃
hypocenter	C₄₃	#33733
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #19145171	C₂	semidirect
2. #19136869	C₂	semidirect
3. #19136843	C₂	semidirect
4. #19136817	C₂	semidirect
5. #19136682	C₂	semidirect
6. #19136680	C₂	semidirect
7. #19136678	C₂	semidirect
8. C₄×D₆₈₈	C₂	semidirect
9. C₄×#13063266	C₂	semidirect
10. C₄×C₁₆×D₄₃	C₂	semidirect
11. C₄×D₁₆×C₄₃	C₂	semidirect
112. C₁₆×C₄₃	#41	semidirect
114. C₄₃⋊C₁₆	#41	semidirect
139. C₄×D₄₃	D₁₆	semidirect
175. C₄×D₁₆	D₄₃	semidirect
210. C₄₃	#33733	semidirect
286. C₁₆	#10621898	semidirect

Maximal subgroups

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

subgroup	index
#19145171	2
#19136869	2
#19136843	2
#19136817	2
#19136682	2
#19136680	2
#19136678	2
C₄×D₆₈₈	2
C₄×#13063266	2
C₄×C₁₆×D₄₃	2
C₄×D₁₆×C₄₃	2
#33733	43