Subgroups — #25252202

Order 11008.

532 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145246	C₂	1	1	yes	maximal, characteristic, complemented
2	#19145242	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136871	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136844	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136837	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136697	C₂	1	2	yes	maximal, complemented
7	#19136680	C₂	1	2	yes	maximal, complemented
8	#19136468	C₂	1	2	yes	maximal, complemented
9	#19136439	C₂	1	2	yes	maximal, complemented
10	#19136193	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2244	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583505	—	2	1	no	—
13	#14583503	—	2	1	no	—
14	C₂×#13063266	C₂²	1	2	yes	—
15	C₂×#13063266	C₂²	1	2	yes	—
16	C₂×#13063264	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
18	#14583301	C₂²	1	1	yes	characteristic
19	#14583276	C₂²	1	1	yes	characteristic
20	#14583265	—	2	1	no	—
21	#14583263	—	2	1	no	—
22	#14583153	—	2	1	no	—
23	#14583149	—	2	1	no	—
24	#14583136	C₂²	1	2	yes	—
25	#14583132	C₂²	1	2	yes	—
26	#14583011	—	2	1	no	—
27	#14583010	—	2	1	no	—
28	D₈×Dic₄₃	C₂²	1	2	yes	—
29	#14582981	C₂²	1	2	yes	—
30	#14582853	C₂²	1	1	yes	characteristic
31	#14582726	C₂²	1	2	yes	—
32	#14582724	C₂²	1	2	yes	—
33	#14582701	C₂²	1	2	yes	—
34	#14582701	C₂²	1	2	yes	—
35	#14582674	C₂²	1	1	yes	characteristic
36	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#499	C₂²	1	1	yes	characteristic
39	C₄₃×#493	C₂²	1	1	yes	characteristic
40	C₄₃×#357	C₂²	1	2	yes	—
41	C₄₃×#357	C₂²	1	2	yes	—
42	C₄₃×#346	C₂²	1	1	yes	characteristic
43	C₂×D₃₄₄	—	2	1	no	—
44	C₂×D₃₄₄	C₂³	1	2	yes	—
45	C₂×#10621872	D₄	1	1	yes	characteristic
46	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
47	#13063322	—	2	1	no	—
48	#13063317	—	2	1	no	—
49	#13063310	—	4	2	no	—
50	#13063306	—	4	2	no	—
51	#13063300	—	2	2	no	—
52	#13063300	—	4	2	no	—
53	#13063297	—	4	2	no	—
54	#13063296	—	2	2	no	—
55	D₄×Dic₄₃	—	2	2	no	—
56	#13063287	—	2	2	no	—
57	#13063282	D₄	1	1	yes	characteristic
58	#13063266	—	2	4	no	—
59	#13063266	—	2	4	no	—
60	#13063264	—	2	2	no	—
61	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₁₆×D₄₃	—	2	2	no	—
64	#13063259	C₂³	1	2	yes	—
65	#13063257	—	2	1	no	—
66	#13063246	—	2	2	no	—
67	#13063246	—	2	2	no	—
68	#13063245	—	2	2	no	—
69	#13063245	—	2	2	no	—
70	#13063236	D₄	1	1	yes	characteristic
71	#13063235	D₄	1	1	yes	characteristic
72	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
73	C₂×D₈×C₄₃	—	2	1	no	—
74	C₂×D₈×C₄₃	C₂³	1	2	yes	—
75	#127×C₄₃	—	2	1	no	—
76	#125×C₄₃	—	2	1	no	—
77	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
79	#107×C₄₃	C₂³	1	2	yes	—
80	#106×C₄₃	—	2	1	no	—
81	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₄₃	—	2	2	no	—
83	C₂×D₄×C₄₃	—	4	2	no	—
84	C₂×C₂²⋊D₄₃	—	4	2	no	—
85	C₂²×Dic₄₃	—	4	2	no	—
86	C₂×D₁₇₂	—	2	2	no	—
87	C₂×D₁₇₂	—	4	2	no	—
88	C₂×D₁₇₂	—	4	2	no	—
89	C₂×C₄×D₄₃	—	4	2	no	—
90	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
91	D₈×C₄₃	—	2	2	no	—
92	D₈×C₄₃	—	2	2	no	—
93	D₈×C₄₃	—	4	2	no	—
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₄₃	—	2	2	no	—
97	C₄⋊C₄×C₄₃	—	4	2	no	—
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂²⋊Dic₄₃	—	4	2	no	—
100	C₂²⋊Dic₄₃	—	4	2	no	—
101	(C₂×D₄₃)⋊C₄	—	4	2	no	—
102	(C₂×D₄₃)⋊C₄	—	4	2	no	—
103	(C₂×D₄₃)⋊C₄	D₈	1	2	yes	—
104	C₄⋊Dic₄₃	—	2	2	no	—
105	Dic₄₃⋊C₄	—	4	2	no	—
106	Dic₄₃⋊C₄	D₈	1	2	yes	—
107	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
110	D₃₄₄	—	2	2	no	—
111	D₃₄₄	—	2	2	no	—
112	D₃₄₄	—	4	2	no	—
113	#10621872	—	2	2	no	—
114	C₈×D₄₃	—	2	1	no	—
115	C₈×D₄₃	—	2	1	no	—
116	C₁₆×C₄₃	—	2	1	no	—
117	C₁₆×C₄₃	—	2	1	no	—
118	C₄₃⋊C₁₆	#41	1	2	yes	—
119	C₄₃⋊C₁₆	#41	1	2	yes	—
120	C₂³×C₄₃	—	4	2	no	—
121	C₂²×D₄₃	—	4	2	no	—
122	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
123	D₄×C₄₃	—	4	2	no	—
124	D₄×C₄₃	—	4	2	no	—
125	D₄×C₄₃	—	16	2	no	—
126	C₂×C₄×C₄₃	—	4	2	no	—
127	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₄₃	—	16	2	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	C₂×Dic₄₃	—	16	2	no	—
132	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
133	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
134	D₁₇₂	—	4	2	no	—
135	D₁₇₂	—	4	2	no	—
136	D₁₇₂	—	16	2	no	—
137	D₁₇₂	—	16	2	no	—
138	C₄×D₄₃	—	2	1	no	—
139	C₄×D₄₃	—	2	1	no	—
140	C₄×D₄₃	—	16	2	no	—
141	C₈×C₄₃	#135	1	2	yes	—
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	—	2	1	no	—
146	#33740	—	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	2	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	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	8	2	no	—
158	C₄×C₄₃	#532	1	1	yes	characteristic
159	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
160	Dic₄₃	—	2	2	no	—
161	Dic₄₃	—	2	1	no	—
162	Dic₄₃	—	8	2	no	—
163	#3176	—	43	1	no	—
164	#3176	—	43	1	no	—
165	#2253	—	43	1	no	—
166	#2246	—	43	1	no	—
167	#2244	—	43	1	no	—
168	#2244	D₄₃	1	1	yes	characteristic, complemented
169	#2218	—	43	2	no	—
170	#2215	—	43	2	no	—
171	#2177	—	43	2	no	—
172	C₄×D₁₆	—	43	2	no	—
173	#2108	—	43	1	no	—
174	C₂×C₄₃	—	16	2	no	—
175	C₂×C₄₃	#3169	1	1	yes	characteristic
176	C₂×C₄₃	#3415	1	2	yes	—
177	D₄₃	—	4	1	no	—
178	D₄₃	—	16	2	no	—
179	#540	—	86	1	no	—
180	#540	—	86	1	no	—
181	C₂×D₁₆	—	43	2	no	—
182	C₂×D₁₆	—	43	2	no	—
183	C₂×M₃₂	—	43	1	no	—
184	C₂²×C₁₆	—	43	1	no	—
185	#499	—	43	1	no	—
186	#499	C₂×D₄₃	1	1	yes	characteristic
187	#493	—	43	1	no	—
188	#493	C₂×D₄₃	1	1	yes	characteristic
189	#490	—	86	1	no	—
190	#490	—	86	1	no	—
191	#469	—	86	1	no	—
192	#469	—	86	1	no	—
193	#466	—	43	2	no	—
194	#466	—	43	2	no	—
195	#442	—	86	1	no	—
196	#442	—	86	1	no	—
197	C₄×D₈	—	43	2	no	—
198	C₄×D₈	—	43	2	no	—
199	#405	—	43	1	no	—
200	#367	—	43	2	no	—
201	#366	—	43	2	no	—
202	#357	—	43	2	no	—
203	#357	—	43	2	no	—
204	#357	C₂×D₄₃	1	2	yes	—
205	#357	C₂×D₄₃	1	2	yes	—
206	#346	—	43	1	no	—
207	#346	C₂×D₄₃	1	1	yes	characteristic
208	C₄×C₁₆	—	43	1	no	—
209	C₄×C₁₆	—	43	1	no	—
210	C₄₃	#33740	1	1	yes	characteristic, complemented, Sylow
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	43	2	no	—
213	C₂×D₈	—	86	1	no	—
214	C₂×D₈	C₂²×D₄₃	1	2	yes	—
215	C₂×M₁₆	—	43	1	no	—
216	C₂²×C₈	—	43	1	no	—
217	#127	—	2	1	no	—
218	#127	—	86	1	no	—
219	#125	—	2	1	no	—
220	#125	—	86	1	no	—
221	#123	—	172	2	no	—
222	#123	—	172	2	no	—
223	#121	—	86	2	no	—
224	#121	—	86	2	no	—
225	#121	—	172	2	no	—
226	#121	—	172	2	no	—
227	C₄×D₄	—	86	2	no	—
228	C₄×D₄	—	86	2	no	—
229	#117	—	43	1	no	—
230	D₁₆	—	86	4	no	—
231	D₁₆	—	86	4	no	—
232	M₃₂	—	86	2	no	—
233	C₂×C₁₆	—	43	1	no	—
234	C₂×C₁₆	—	43	1	no	—
235	C₂×C₁₆	—	86	2	no	—
236	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
238	#107	—	43	2	no	—
239	#107	C₂²×D₄₃	1	2	yes	—
240	#106	—	2	1	no	—
241	#106	—	86	1	no	—
242	#102	—	86	2	no	—
243	#102	—	86	2	no	—
244	#102	—	86	2	no	—
245	#102	—	86	2	no	—
246	#97	—	43	1	no	—
247	#97	—	43	1	no	—
248	C₄×C₈	—	43	1	no	—
249	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	86	2	no	—
253	C₂×D₄	—	172	2	no	—
254	C₂×D₄	—	172	2	no	—
255	C₂×D₄	—	172	2	no	—
256	C₂²×C₄	—	43	1	no	—
257	C₂²×C₄	—	172	2	no	—
258	C₂²×C₄	—	172	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	4	2	no	—
262	D₈	—	86	2	no	—
263	D₈	—	86	2	no	—
264	D₈	—	172	2	no	—
265	M₁₆	—	86	2	no	—
266	C₂×C₈	—	43	1	no	—
267	C₂×C₈	—	43	1	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	—	4	2	no	—
274	C₄⋊C₄	—	43	2	no	—
275	C₄⋊C₄	—	86	2	no	—
276	C₄⋊C₄	—	172	2	no	—
277	C₂²⋊C₄	—	43	2	no	—
278	C₂²⋊C₄	—	172	2	no	—
279	C₂²⋊C₄	—	172	2	no	—
280	C₂²⋊C₄	—	172	2	no	—
281	C₂²⋊C₄	—	172	2	no	—
282	C₄²	—	43	1	no	—
283	C₄²	D₄×D₄₃	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	43	2	no	—
287	C₁₆	—	43	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	43	1	no	—
290	C₂³	—	172	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	16	2	no	—
294	D₄	—	172	2	no	—
295	D₄	—	172	2	no	—
296	D₄	—	688	2	no	—
297	D₄	—	688	2	no	—
298	D₄	—	688	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	43	2	no	—
301	C₂×C₄	—	43	1	no	—
302	C₂×C₄	—	86	1	no	—
303	C₂×C₄	—	86	1	no	—
304	C₂×C₄	—	172	2	no	—
305	C₂×C₄	—	688	2	no	—
306	C₂×C₄	—	688	2	no	—
307	C₂×C₄	D₈×D₄₃	1	2	yes	—
308	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
309	C₈	—	86	1	no	—
310	C₈	—	86	1	no	—
311	C₈	#13063346	1	2	yes	—
312	C₈	#41×D₄₃	1	1	yes	characteristic
313	C₈	#41×D₄₃	1	1	yes	characteristic
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	86	1	no	—
318	C₂²	—	86	2	no	—
319	C₂²	—	344	2	no	—
320	C₂²	—	344	2	no	—
321	C₂²	—	344	2	no	—
322	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	86	1	no	—
326	C₄	—	86	2	no	—
327	C₄	—	344	2	no	—
328	C₄	#14583448	1	1	yes	characteristic
329	C₄	C₂×#13063346	1	1	yes	characteristic
330	C₂	—	16	2	no	—
331	C₂	—	172	1	no	—
332	C₂	—	688	2	no	—
333	C₂	#19145173	1	1	yes	characteristic
334	C₂	#19147735	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2244	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#2244	D₄₃
hypocenter	C₄₃	#33740
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2244

Complements

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

N	H ≅ G/N	type
1. #19145246	C₂	semidirect
2. #19145242	C₂	semidirect
3. #19136871	C₂	semidirect
4. #19136844	C₂	semidirect
5. #19136837	C₂	semidirect
6. #19136697	C₂	semidirect
7. #19136680	C₂	semidirect
8. #19136468	C₂	semidirect
9. #19136439	C₂	semidirect
10. #19136193	C₂	semidirect
11. C₄₃×#2244	C₂	semidirect
168. #2244	D₄₃	semidirect
210. C₄₃	#33740	semidirect

Maximal subgroups

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

subgroup	index
#19145246	2
#19145242	2
#19136871	2
#19136844	2
#19136837	2
#19136697	2
#19136680	2
#19136468	2
#19136439	2
#19136193	2
C₄₃×#2244	2
#33740	43