Subgroups — #25252201

Order 11008.

532 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145241	C₂	1	2	yes	maximal, complemented
2	#19136873	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136843	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136835	C₂	1	2	yes	maximal, complemented
5	#19136699	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136695	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136680	C₂	1	2	yes	maximal, complemented
8	#19136472	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13063266	C₂	1	2	yes	maximal, complemented
10	#19136193	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583500	—	2	2	no	—
13	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
14	C₂×#13063266	C₂²	1	2	yes	—
15	C₂×#13063264	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
17	#14583300	C₂²	1	2	yes	—
18	#14583274	C₂²	1	2	yes	—
19	#14583262	—	2	2	no	—
20	#14583151	—	2	2	no	—
21	#14583136	C₂²	1	2	yes	—
22	#14583132	C₂²	1	2	yes	—
23	#14583009	—	2	2	no	—
24	C₄×D₃₄₄	C₂²	1	2	yes	—
25	#14582853	C₂²	1	1	yes	characteristic
26	#14582727	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	#14582674	C₂²	1	1	yes	characteristic
32	C₄×C₄₃⋊C₁₆	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×#367	C₂²	1	1	yes	characteristic
37	C₄₃×#357	C₂²	1	2	yes	—
38	C₄₃×#346	C₂²	1	1	yes	characteristic
39	C₂×D₃₄₄	C₂³	1	2	yes	—
40	C₂×#10621882	—	2	2	no	—
41	C₂×#10621872	D₄	1	1	yes	characteristic
42	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
43	#13063321	—	2	2	no	—
44	#13063316	—	2	2	no	—
45	#13063308	—	4	2	no	—
46	#13063307	—	4	2	no	—
47	#13063300	—	2	2	no	—
48	#13063299	—	4	2	no	—
49	#13063298	—	4	2	no	—
50	#13063296	—	2	2	no	—
51	C₄×D₁₇₂	—	2	2	no	—
52	#13063282	D₄	1	1	yes	characteristic
53	D₆₈₈	—	2	2	no	—
54	#13063266	—	2	4	no	—
55	#13063264	—	2	2	no	—
56	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₁₆×D₄₃	—	2	2	no	—
59	#13063259	C₂³	1	2	yes	—
60	#13063256	—	2	2	no	—
61	#13063247	—	2	2	no	—
62	#13063246	—	2	2	no	—
63	#13063245	—	2	2	no	—
64	#13063236	D₄	1	1	yes	characteristic
65	#13063235	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	C₂×D₁₇₂	—	2	2	no	—
81	C₂×D₁₇₂	—	4	2	no	—
82	C₂×C₄×D₄₃	—	4	2	no	—
83	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
84	D₈×C₄₃	—	2	2	no	—
85	D₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₄₃	—	2	2	no	—
89	C₂²⋊C₄×C₄₃	—	4	2	no	—
90	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂²⋊Dic₄₃	—	4	2	no	—
92	#10621882	—	4	4	no	—
93	(C₂×D₄₃)⋊C₄	—	4	2	no	—
94	(C₂×D₄₃)⋊C₄	—	4	2	no	—
95	(C₂×D₄₃)⋊C₄	D₈	1	2	yes	—
96	C₄⋊Dic₄₃	—	2	2	no	—
97	C₄⋊Dic₄₃	—	4	2	no	—
98	Dic₄₃⋊C₄	—	4	2	no	—
99	Dic₄₃⋊C₄	D₈	1	2	yes	—
100	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
103	D₃₄₄	—	2	2	no	—
104	D₃₄₄	—	2	2	no	—
105	#10621872	—	2	2	no	—
106	C₈×D₄₃	—	2	1	no	—
107	C₈×D₄₃	—	2	1	no	—
108	C₁₆×C₄₃	—	2	1	no	—
109	C₁₆×C₄₃	#41	1	2	yes	—
110	C₄₃⋊C₁₆	—	2	1	no	—
111	C₄₃⋊C₁₆	#41	1	2	yes	—
112	C₂³×C₄₃	—	4	2	no	—
113	C₂²×D₄₃	—	4	2	no	—
114	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
115	D₄×C₄₃	—	4	2	no	—
116	D₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	16	2	no	—
119	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
121	C₂²⋊D₄₃	—	16	2	no	—
122	C₂²⋊D₄₃	—	16	2	no	—
123	C₂²⋊D₄₃	—	16	2	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
126	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
127	D₁₇₂	—	4	2	no	—
128	D₁₇₂	—	4	2	no	—
129	D₁₇₂	—	16	2	no	—
130	C₄×D₄₃	—	2	1	no	—
131	C₄×D₄₃	—	2	1	no	—
132	C₄×D₄₃	—	16	2	no	—
133	C₈×C₄₃	—	2	1	no	—
134	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
135	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
136	C₄₃⋊C₈	—	2	1	no	—
137	C₄₃⋊C₈	#135	1	2	yes	—
138	#33740	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	—	8	2	no	—
140	C₂²×C₄₃	—	8	2	no	—
141	C₂²×C₄₃	—	8	2	no	—
142	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
143	C₂×D₄₃	—	2	2	no	—
144	C₂×D₄₃	—	2	1	no	—
145	C₂×D₄₃	—	8	2	no	—
146	C₂×D₄₃	—	8	2	no	—
147	C₂×D₄₃	—	8	2	no	—
148	C₄×C₄₃	—	2	2	no	—
149	C₄×C₄₃	—	8	2	no	—
150	C₄×C₄₃	#532	1	1	yes	characteristic
151	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
152	Dic₄₃	—	2	2	no	—
153	Dic₄₃	—	2	1	no	—
154	Dic₄₃	—	8	2	no	—
155	#3176	—	43	2	no	—
156	#2253	—	43	1	no	—
157	#2246	—	43	1	no	—
158	#2244	—	43	2	no	—
159	#2218	—	43	1	no	—
160	#2218	—	43	1	no	—
161	#2215	—	43	2	no	—
162	#2177	—	43	1	no	—
163	#2177	D₄₃	1	1	yes	characteristic, complemented
164	C₄×D₁₆	—	43	2	no	—
165	#2108	—	43	1	no	—
166	C₂×C₄₃	—	16	2	no	—
167	C₂×C₄₃	#3169	1	1	yes	characteristic
168	C₂×C₄₃	#3415	1	2	yes	—
169	D₄₃	—	4	1	no	—
170	D₄₃	—	16	2	no	—
171	#540	—	86	2	no	—
172	C₂×D₁₆	—	43	2	no	—
173	C₂×D₁₆	—	43	1	no	—
174	C₂×D₁₆	C₂×D₄₃	1	1	yes	characteristic
175	C₂×M₃₂	—	43	1	no	—
176	C₂²×C₁₆	—	43	1	no	—
177	#499	—	43	2	no	—
178	#493	—	43	2	no	—
179	#490	—	86	2	no	—
180	#469	—	86	2	no	—
181	#466	—	43	2	no	—
182	#466	—	43	2	no	—
183	#442	—	86	2	no	—
184	C₄×D₈	—	43	2	no	—
185	C₄×D₈	C₂×D₄₃	1	2	yes	—
186	#405	—	43	1	no	—
187	#367	—	43	1	no	—
188	#367	C₂×D₄₃	1	1	yes	characteristic
189	#366	—	43	2	no	—
190	#357	—	43	2	no	—
191	#357	—	43	2	no	—
192	#357	—	43	2	no	—
193	#357	C₂×D₄₃	1	2	yes	—
194	#346	—	43	1	no	—
195	#346	C₂×D₄₃	1	1	yes	characteristic
196	C₄×C₁₆	—	43	1	no	—
197	C₄×C₁₆	—	43	1	no	—
198	C₄₃	#33740	1	1	yes	characteristic, complemented, Sylow
199	C₂×D₈	—	43	2	no	—
200	C₂×D₈	—	86	2	no	—
201	C₂×D₈	C₂²×D₄₃	1	2	yes	—
202	C₂×M₁₆	—	43	1	no	—
203	C₂²×C₈	—	43	1	no	—
204	#127	—	86	2	no	—
205	#125	—	86	2	no	—
206	#123	—	172	2	no	—
207	#123	—	172	2	no	—
208	#121	—	86	2	no	—
209	#121	—	86	2	no	—
210	#121	—	172	2	no	—
211	#121	—	172	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	86	2	no	—
214	#117	—	43	1	no	—
215	D₁₆	—	2	2	no	—
216	D₁₆	—	86	4	no	—
217	D₁₆	—	86	2	no	—
218	M₃₂	—	86	2	no	—
219	C₂×C₁₆	—	43	1	no	—
220	C₂×C₁₆	—	43	1	no	—
221	C₂×C₁₆	—	86	2	no	—
222	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
223	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
224	#107	—	43	2	no	—
225	#107	C₂²×D₄₃	1	2	yes	—
226	#106	—	86	2	no	—
227	#102	—	2	2	no	—
228	#102	—	86	2	no	—
229	#102	—	86	2	no	—
230	#102	—	86	2	no	—
231	#97	—	43	1	no	—
232	#97	—	43	1	no	—
233	C₄×C₈	—	43	1	no	—
234	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	86	2	no	—
237	C₂×D₄	—	172	2	no	—
238	C₂×D₄	—	172	2	no	—
239	C₂×D₄	—	172	2	no	—
240	C₂×D₄	—	172	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	43	1	no	—
243	C₂²×C₄	—	172	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	D₈	—	86	2	no	—
247	D₈	—	86	2	no	—
248	D₈	—	172	4	no	—
249	M₁₆	—	86	2	no	—
250	C₂×C₈	—	43	1	no	—
251	C₂×C₈	—	43	1	no	—
252	C₂×C₈	—	86	1	no	—
253	C₂×C₈	—	86	1	no	—
254	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	43	2	no	—
258	C₄⋊C₄	—	86	2	no	—
259	C₄⋊C₄	—	172	2	no	—
260	C₄⋊C₄	—	172	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	43	2	no	—
263	C₂²⋊C₄	—	172	2	no	—
264	C₂²⋊C₄	—	172	2	no	—
265	C₂²⋊C₄	—	172	2	no	—
266	C₄²	—	43	1	no	—
267	C₄²	D₄×D₄₃	1	1	yes	characteristic
268	C₁₆	—	2	1	no	—
269	C₁₆	—	43	2	no	—
270	C₁₆	—	86	1	no	—
271	C₁₆	#10621898	1	2	yes	—
272	C₂³	—	4	2	no	—
273	C₂³	—	43	1	no	—
274	C₂³	—	172	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	172	2	no	—
278	D₄	—	172	2	no	—
279	D₄	—	688	2	no	—
280	D₄	—	688	2	no	—
281	D₄	—	688	2	no	—
282	D₄	—	688	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	16	2	no	—
285	C₂×C₄	—	43	2	no	—
286	C₂×C₄	—	43	1	no	—
287	C₂×C₄	—	86	1	no	—
288	C₂×C₄	—	86	1	no	—
289	C₂×C₄	—	172	2	no	—
290	C₂×C₄	—	688	2	no	—
291	C₂×C₄	D₈×D₄₃	1	2	yes	—
292	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	43	2	no	—
295	C₈	—	86	1	no	—
296	C₈	#41×D₄₃	1	1	yes	characteristic
297	C₈	C₂×#10621898	1	1	yes	characteristic
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	86	1	no	—
302	C₂²	—	86	2	no	—
303	C₂²	—	344	2	no	—
304	C₂²	—	344	2	no	—
305	C₂²	—	344	2	no	—
306	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	86	1	no	—
310	C₄	—	86	2	no	—
311	C₄	—	344	2	no	—
312	C₄	#14583446	1	1	yes	characteristic
313	C₄	#135×D₄₃	1	1	yes	characteristic
314	C₂	—	16	2	no	—
315	C₂	—	172	1	no	—
316	C₂	—	688	2	no	—
317	C₂	#19145171	1	1	yes	characteristic
318	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×#2177	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#2177	D₄₃
hypocenter	C₄₃	#33740
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
1. #19145241	C₂	semidirect
2. #19136873	C₂	semidirect
3. #19136843	C₂	semidirect
4. #19136835	C₂	semidirect
5. #19136699	C₂	semidirect
6. #19136695	C₂	semidirect
7. #19136680	C₂	semidirect
8. #19136472	C₂	semidirect
9. C₄×#13063266	C₂	semidirect
10. #19136193	C₂	semidirect
11. C₄₃×#2177	C₂	semidirect
163. #2177	D₄₃	semidirect
198. C₄₃	#33740	semidirect

Maximal subgroups

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

subgroup	index
#19145241	2
#19136873	2
#19136843	2
#19136835	2
#19136699	2
#19136695	2
#19136680	2
#19136472	2
C₄×#13063266	2
#19136193	2
C₄₃×#2177	2
#33740	43