Subgroups — #25249640

Order 11008.

658 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583317	C₂	1	1	yes	maximal, characteristic
2	#19143192	C₂	1	1	yes	maximal, characteristic
3	#19143191	C₂	1	1	yes	maximal, characteristic
4	#19136383	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136382	C₂	1	2	yes	maximal, complemented
6	#19136380	C₂	1	2	yes	maximal, complemented
7	#19136378	C₂	1	2	yes	maximal, complemented
8	#19136377	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14582724	C₂	1	1	yes	maximal, characteristic, complemented
10	#366×D₄₃	C₂	1	2	yes	maximal, complemented
11	C₄₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063331	C₂²	1	1	yes	characteristic
13	#14583380	—	2	1	no	—
14	#14583379	—	2	1	no	—
15	#14583317	C₄	1	8	yes	complemented
16	C₂×#13063264	C₂²	1	2	yes	—
17	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₄₃	C₂²	1	2	yes	—
19	#14582942	—	2	1	no	—
20	#14582940	—	2	2	no	—
21	#14582939	—	2	1	no	—
22	#14582938	—	2	2	no	—
23	#14582937	C₂²	1	1	yes	characteristic
24	#14582936	C₂²	1	2	yes	—
25	#14582936	C₂²	1	2	yes	—
26	C₂×#13063259	C₂²	1	1	yes	characteristic
27	#107×D₄₃	C₂²	1	2	yes	—
28	#107×D₄₃	C₂²	1	2	yes	—
29	#14582727	C₂²	1	2	yes	—
30	#14582726	C₂²	1	1	yes	characteristic
31	#14582726	C₂²	1	1	yes	characteristic
32	#14582725	C₂²	1	2	yes	—
33	#14582724	C₂²	1	1	yes	characteristic
34	#14582724	C₂²	1	2	yes	—
35	#14582724	C₂²	1	2	yes	—
36	#14582724	C₂²	1	1	yes	characteristic
37	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#427	C₂²	1	1	yes	characteristic
39	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
40	C₄₃×#367	C₂²	1	2	yes	—
41	C₄₃×#366	C₂²	1	2	yes	—
42	C₂×#10621898	D₄	1	1	yes	characteristic
43	#13063331	C₂×C₄	1	4	yes	—
44	#13063331	Q₈	1	4	yes	—
45	C₂×#10621877	D₄	1	1	yes	characteristic
46	C₂×#10621872	D₄	1	2	yes	—
47	C₂×C₈×D₄₃	C₂³	1	2	yes	—
48	C₄×Dic₈₆	—	4	1	no	—
49	#13063290	—	4	1	no	—
50	C₄×D₁₇₂	—	4	1	no	—
51	#13063287	—	4	1	no	—
52	C₄×C₂²⋊D₄₃	—	4	2	no	—
53	(C₂²⋊D₄₃)⋊C₄	—	4	2	no	—
54	#13063284	—	2	1	no	—
55	#13063283	—	2	2	no	—
56	#13063283	—	2	2	no	—
57	C₂×C₄⋊Dic₄₃	—	2	1	no	—
58	C₄⋊C₄×D₄₃	—	2	2	no	—
59	C₄⋊C₄×D₄₃	—	2	2	no	—
60	#13063264	C₂×C₄	1	8	yes	complemented
61	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
62	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₁₆×D₄₃	C₂×C₄	1	8	yes	complemented
65	#13063259	—	2	1	no	—
66	#13063259	C₂³	1	1	yes	characteristic
67	#13063259	C₂³	1	2	yes	—
68	#13063259	C₂³	1	1	yes	characteristic
69	#13063258	—	2	2	no	—
70	#13063258	—	2	2	no	—
71	#13063257	—	2	1	no	—
72	#13063256	—	2	2	no	—
73	#13063256	—	2	2	no	—
74	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	#117×C₄₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
77	M₃₂×C₄₃	C₂×C₄	1	4	yes	complemented
78	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
79	#107×C₄₃	—	2	1	no	—
80	#107×C₄₃	C₂³	1	1	yes	characteristic
81	#107×C₄₃	C₂³	1	2	yes	—
82	#107×C₄₃	C₂³	1	1	yes	characteristic
83	#106×C₄₃	—	2	1	no	—
84	C₂²×C₄×C₄₃	—	4	1	no	—
85	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₂²⋊D₄₃	D₈	1	2	yes	—
87	C₂²×Dic₄₃	—	4	1	no	—
88	#10621898	C₄⋊C₄	1	4	yes	—
89	#10621898	Q₁₆	1	4	yes	—
90	C₂×D₁₇₂	D₈	1	1	yes	characteristic
91	C₂×C₄×D₄₃	—	4	2	no	—
92	C₂×C₄×D₄₃	—	4	2	no	—
93	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
94	C₂×Dic₈₆	D₈	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₄₃	—	2	1	no	—
100	C₄⋊C₄×C₄₃	—	2	1	no	—
101	C₄⋊C₄×C₄₃	—	2	2	no	—
102	C₄⋊C₄×C₄₃	—	4	1	no	—
103	C₂²⋊C₄×C₄₃	—	4	1	no	—
104	C₄²×C₄₃	—	4	1	no	—
105	C₂²⋊Dic₄₃	—	4	1	no	—
106	(C₂×D₄₃)⋊C₄	—	4	2	no	—
107	(C₂×D₄₃)⋊C₄	—	4	2	no	—
108	C₄⋊Dic₄₃	—	2	1	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	2	1	no	—
111	C₄⋊Dic₄₃	—	4	1	no	—
112	Dic₄₃⋊C₄	—	4	2	no	—
113	Dic₄₃⋊C₄	—	4	2	no	—
114	C₄×Dic₄₃	—	4	1	no	—
115	C₄×Dic₄₃	—	4	2	no	—
116	C₄×Dic₄₃	—	4	2	no	—
117	#10621877	C₄⋊C₄	1	4	yes	—
118	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
119	#10621872	C₄⋊C₄	1	8	yes	—
120	C₈×D₄₃	C₂²×C₄	1	4	yes	—
121	C₈×D₄₃	C₂×Q₈	1	4	yes	—
122	C₁₆×C₄₃	C₂²×C₄	1	4	yes	—
123	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
124	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
125	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
126	C₂²×D₄₃	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₄₃	—	4	1	no	—
128	C₂×C₄×C₄₃	—	4	2	no	—
129	C₂×C₄×C₄₃	—	4	1	no	—
130	C₂×C₄×C₄₃	—	16	1	no	—
131	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₄₃	#107	1	8	yes	—
136	C₂×Dic₄₃	—	4	1	no	—
137	C₂×Dic₄₃	—	4	1	no	—
138	C₂×Dic₄₃	—	4	2	no	—
139	C₂×Dic₄₃	—	16	1	no	—
140	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
141	D₁₇₂	#107	1	4	yes	—
142	C₄×D₄₃	—	16	2	no	—
143	C₄×D₄₃	—	16	2	no	—
144	C₄×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
145	C₄×D₄₃	C₂×Q₁₆	1	4	yes	—
146	Dic₈₆	#107	1	4	yes	—
147	C₈×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
148	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
149	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
150	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
151	#33410	—	43	1	no	maximal, Sylow
152	C₂²×C₄₃	—	2	1	no	—
153	C₂²×C₄₃	—	2	1	no	—
154	C₂²×C₄₃	C₂×#107	1	2	yes	—
155	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₄₃	—	2	2	no	—
157	C₂×D₄₃	—	2	2	no	—
158	C₂×D₄₃	C₂×#107	1	4	yes	—
159	C₄×C₄₃	—	8	1	no	—
160	C₄×C₄₃	—	8	2	no	—
161	C₄×C₄₃	—	8	1	no	—
162	C₄×C₄₃	C₂×#107	1	2	yes	—
163	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
165	Dic₄₃	—	8	1	no	—
166	Dic₄₃	—	8	2	no	—
167	Dic₄₃	—	8	1	no	—
168	Dic₄₃	C₂×#107	1	4	yes	—
169	C₂×#504	—	43	1	no	—
170	#2913	—	43	1	no	—
171	#2913	—	43	1	no	—
172	#2159	—	43	1	no	—
173	#2159	—	43	2	no	—
174	#2159	—	43	2	no	—
175	#2159	D₄₃	1	1	yes	characteristic, complemented
176	#2158	—	43	1	no	—
177	#2158	—	43	2	no	—
178	C₂×#366	—	43	1	no	—
179	C₂×#366	—	43	2	no	—
180	C₂×C₄₃	—	2	2	no	—
181	C₂×C₄₃	C₂²×#107	1	1	yes	characteristic
182	C₂×C₄₃	#3415	1	1	yes	characteristic
183	C₂×C₄₃	#3417	1	1	yes	characteristic
184	D₄₃	—	2	4	no	—
185	C₂×#131	—	43	1	no	—
186	#520	—	86	1	no	—
187	#520	—	86	1	no	—
188	#504	—	43	8	no	—
189	C₂×M₃₂	—	43	2	no	—
190	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
191	C₂²×C₁₆	—	43	1	no	—
192	C₂²×C₁₆	—	43	2	no	—
193	#428	—	86	2	no	—
194	#428	—	86	2	no	—
195	#428	—	86	1	no	—
196	#428	—	86	1	no	—
197	#427	—	43	1	no	—
198	#427	—	43	2	no	—
199	#427	—	43	2	no	—
200	#427	C₂×D₄₃	1	1	yes	characteristic
201	C₂×#107	—	43	1	no	—
202	C₂×#107	—	43	2	no	—
203	C₂×#107	—	43	2	no	—
204	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
205	#367	—	43	2	no	—
206	#367	—	43	1	no	—
207	#367	—	43	1	no	—
208	#367	C₂×D₄₃	1	2	yes	—
209	#366	—	43	2	no	—
210	#366	—	43	2	no	—
211	#366	—	43	1	no	—
212	#366	—	43	2	no	—
213	#366	—	43	1	no	—
214	#366	C₂×D₄₃	1	2	yes	—
215	C₄₃	#33410	1	1	yes	characteristic, complemented, Sylow
216	C₂×#41	—	43	1	no	—
217	#131	—	43	4	no	—
218	#131	—	43	4	no	—
219	C₂×M₁₆	—	43	1	no	—
220	C₂×M₁₆	—	43	2	no	—
221	C₂²×C₈	—	43	2	no	—
222	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
223	C₄×Q₈	—	172	1	no	—
224	C₄×Q₈	—	172	1	no	—
225	C₄×D₄	—	172	1	no	—
226	C₄×D₄	—	172	2	no	—
227	C₄×D₄	—	172	1	no	—
228	C₄×D₄	—	172	2	no	—
229	#117	—	2	1	no	—
230	#117	—	86	2	no	—
231	#117	—	86	2	no	—
232	#117	—	86	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	86	2	no	—
235	C₂×C₄⋊C₄	—	86	2	no	—
236	C₂×C₄⋊C₄	—	86	1	no	—
237	M₃₂	—	43	8	no	—
238	M₃₂	C₄×D₄₃	1	4	yes	complemented
239	C₂×C₁₆	—	43	4	no	—
240	C₂×C₁₆	—	43	1	no	—
241	C₂×C₁₆	—	43	1	no	—
242	C₂×C₁₆	—	43	8	no	—
243	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
244	#107	—	2	1	no	—
245	#107	—	43	1	no	—
246	#107	—	43	2	no	—
247	#107	—	43	1	no	—
248	#107	—	86	2	no	—
249	#107	—	86	2	no	—
250	#107	—	86	1	no	—
251	#107	C₂²×D₄₃	1	2	yes	—
252	#107	C₂²×D₄₃	1	1	yes	characteristic
253	#107	C₂²×D₄₃	1	1	yes	characteristic
254	#106	—	2	1	no	—
255	#106	—	86	2	no	—
256	#106	—	86	2	no	—
257	#106	—	86	1	no	—
258	#41	—	43	4	no	—
259	#41	—	43	4	no	—
260	C₂×Q₈	—	43	1	no	—
261	C₂×D₄	—	43	2	no	—
262	C₂×D₄	—	43	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	43	2	no	—
265	C₂²×C₄	—	172	2	no	—
266	C₂²×C₄	—	172	2	no	—
267	C₂²×C₄	—	172	1	no	—
268	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
269	M₁₆	—	43	4	no	—
270	M₁₆	—	43	8	no	—
271	C₂×C₈	—	43	2	no	—
272	C₂×C₈	—	43	4	no	—
273	C₂×C₈	—	43	4	no	—
274	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
275	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
276	C₂×C₈	Q₈×D₄₃	1	2	yes	—
277	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	86	1	no	—
283	C₄⋊C₄	—	86	2	no	—
284	C₄⋊C₄	—	86	1	no	—
285	C₄⋊C₄	—	172	2	no	—
286	C₄⋊C₄	—	172	2	no	—
287	C₄⋊C₄	—	172	1	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	172	2	no	—
290	C₂²⋊C₄	—	172	2	no	—
291	C₂²⋊C₄	—	172	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	172	2	no	—
294	C₄²	—	172	2	no	—
295	C₄²	—	172	1	no	—
296	C₁₆	—	43	2	no	—
297	C₁₆	—	43	2	no	—
298	C₁₆	C₂×C₄×D₄₃	1	4	yes	—
299	C₂³	—	43	2	no	—
300	C₂³	D₈×D₄₃	1	1	yes	characteristic
301	Q₈	—	43	4	no	—
302	D₄	—	43	8	no	—
303	D₄	—	43	4	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	43	2	no	—
309	C₂×C₄	—	43	4	no	—
310	C₂×C₄	—	43	4	no	—
311	C₂×C₄	—	172	1	no	—
312	C₂×C₄	—	172	2	no	—
313	C₂×C₄	—	172	1	no	—
314	C₂×C₄	—	688	2	no	—
315	C₂×C₄	—	688	2	no	—
316	C₂×C₄	—	688	1	no	—
317	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
318	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
319	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
320	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
321	C₈	—	43	4	no	—
322	C₈	C₄⋊C₄×D₄₃	1	2	yes	—
323	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
324	C₈	C₂×Q₈×D₄₃	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	43	4	no	—
328	C₂²	—	86	2	no	—
329	C₂²	—	86	2	no	—
330	C₂²	#107×D₄₃	1	2	yes	—
331	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	43	4	no	—
336	C₄	—	344	1	no	—
337	C₄	—	344	2	no	—
338	C₄	—	344	1	no	—
339	C₄	#107×D₄₃	1	2	yes	—
340	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
341	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	86	4	no	—
344	C₂	C₂×#107×D₄₃	1	1	yes	characteristic
345	C₂	#19147735	1	1	yes	characteristic
346	C₂	#19147747	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2159	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#10621898	D₄
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2159	D₄₃
hypocenter	C₄₃	#33410
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
4. #19136383	C₂	semidirect
5. #19136382	C₂	semidirect
6. #19136380	C₂	semidirect
7. #19136378	C₂	semidirect
8. #19136377	C₂	semidirect
9. C₂×#14582724	C₂	semidirect
10. #366×D₄₃	C₂	semidirect
11. C₄₃×#2159	C₂	semidirect
15. #14583317	C₄	semidirect
60. #13063264	C₂×C₄	semidirect
61. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
64. C₁₆×D₄₃	C₂×C₄	semidirect
77. M₃₂×C₄₃	C₂×C₄	semidirect
175. #2159	D₄₃	semidirect
215. C₄₃	#33410	semidirect
238. M₃₂	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583317	2
#19143192	2
#19143191	2
#19136383	2
#19136382	2
#19136380	2
#19136378	2
#19136377	2
C₂×#14582724	2
#366×D₄₃	2
C₄₃×#2159	2
#33410	43