Subgroups — #25096966

Order 11008.

608 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143022	C₂	1	1	yes	maximal, characteristic
2	#19134745	C₂	1	1	yes	maximal, characteristic, complemented
3	#19134742	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134717	C₂	1	2	yes	maximal, complemented
5	#19134697	C₂	1	2	yes	maximal
6	#19134644	C₂	1	1	yes	maximal, characteristic
7	#19134640	C₂	1	1	yes	maximal, characteristic
8	#19134480	C₂	1	2	yes	maximal, complemented
9	#19134396	C₂	1	2	yes	maximal
10	#19134350	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063333	C₂²	1	1	yes	characteristic
13	C₂×#13063331	C₂²	1	1	yes	characteristic
14	#14583367	C₂²	1	1	yes	characteristic
15	C₂×#13063260	C₂²	1	2	yes	—
16	#14582935	C₂²	1	2	yes	—
17	#14582881	C₂²	1	2	yes	—
18	#14582872	C₂²	1	2	yes	—
19	#14582856	C₄	1	2	yes	complemented
20	#14582855	C₂²	1	1	yes	characteristic
21	#14582854	C₂²	1	2	yes	—
22	#14582853	C₄	1	2	yes	complemented
23	M₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
24	C₄×#10621872	C₄	1	2	yes	complemented
25	C₂×#13063235	C₂²	1	1	yes	characteristic
26	#97×D₄₃	C₄	1	2	yes	complemented
27	#14582671	C₂²	1	2	yes	—
28	#14582669	C₂²	1	2	yes	—
29	#14582660	C₂²	1	2	yes	—
30	#14582658	C₂²	1	2	yes	—
31	C₄₃⋊#338	C₂²	1	1	yes	characteristic
32	#14582656	C₂²	1	1	yes	characteristic
33	#14582655	C₂²	1	2	yes	—
34	C₄₃×#405	C₂²	1	2	yes	—
35	C₄₃×#405	C₂²	1	1	yes	characteristic
36	C₄₃×#338	C₂²	1	1	yes	characteristic
37	C₄₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
38	C₄₃×H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#10621898	C₂³	1	1	yes	characteristic
40	#13063333	—	2	2	no	—
41	#13063333	—	2	2	no	—
42	#13063331	—	2	2	no	—
43	#13063331	—	2	2	no	—
44	C₂×#10621877	C₂³	1	2	yes	—
45	C₂×#10621877	C₂³	1	1	yes	characteristic
46	C₂×#10621872	C₂×C₄	1	2	yes	—
47	C₂×#10621872	C₂×C₄	1	2	yes	—
48	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
51	#13063290	—	2	1	no	—
52	#13063287	—	2	1	no	—
53	(C₂²⋊D₄₃)⋊C₄	—	2	2	no	—
54	#13063284	C₂³	1	2	yes	—
55	#13063282	C₂×C₄	1	2	yes	—
56	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
57	C₄²×D₄₃	C₂×C₄	1	2	yes	—
58	#13063260	—	2	2	no	—
59	#13063260	—	2	2	no	—
60	#13063258	—	2	2	no	—
61	#13063256	—	2	2	no	—
62	#13063243	—	2	2	no	—
63	#13063241	—	2	2	no	—
64	#13063239	—	2	4	no	—
65	#13063236	C₂×C₄	1	2	yes	—
66	#13063235	—	2	2	no	—
67	#13063235	C₂×C₄	1	1	yes	characteristic
68	#13063235	C₂×C₄	1	2	yes	—
69	#13063235	C₂×C₄	1	1	yes	characteristic
70	#13063235	C₂×C₄	1	1	yes	characteristic
71	#13063235	C₂×C₄	1	1	yes	characteristic
72	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
73	C₈×Dic₄₃	—	2	2	no	—
74	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
75	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
76	C₂×M₁₆×C₄₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	#117×C₄₃	C₂³	1	2	yes	—
79	#117×C₄₃	C₂³	1	1	yes	characteristic
80	#97×C₄₃	—	2	2	no	—
81	#97×C₄₃	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₄₃	—	2	2	no	—
83	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₂×C₂²⋊D₄₃	#41	1	2	yes	—
86	C₂²×Dic₄₃	—	2	1	no	—
87	#10621898	—	2	2	no	—
88	#10621898	—	2	2	no	—
89	C₂×D₁₇₂	#41	1	1	yes	characteristic
90	C₂×C₄×D₄₃	—	2	2	no	—
91	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
92	C₂×Dic₈₆	#41	1	1	yes	characteristic
93	M₁₆×C₄₃	—	2	2	no	—
94	M₁₆×C₄₃	—	2	2	no	—
95	M₁₆×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	#41	1	2	yes	—
101	C₂×C₈×C₄₃	#41	1	2	yes	—
102	C₄⋊C₄×C₄₃	—	2	1	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₂²⋊C₄×C₄₃	—	2	1	no	—
105	C₂²⋊C₄×C₄₃	—	2	2	no	—
106	C₄²×C₄₃	—	2	2	no	—
107	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
108	C₂²⋊Dic₄₃	—	2	2	no	—
109	(C₂×D₄₃)⋊C₄	—	2	2	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	Dic₄₃⋊C₄	—	2	2	no	—
112	C₄×Dic₄₃	—	2	1	no	—
113	C₄×Dic₄₃	—	2	2	no	—
114	C₄×Dic₄₃	—	2	2	no	—
115	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
117	#10621877	—	2	2	no	—
118	#10621877	—	2	2	no	—
119	#10621877	—	2	2	no	—
120	C₂×C₄₃⋊C₈	—	2	2	no	—
121	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
122	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄₃⋊C₈	#41	1	2	yes	—
125	C₂×C₄₃⋊C₈	#41	1	2	yes	—
126	#10621872	—	2	2	no	—
127	#10621872	—	2	2	no	—
128	#10621872	—	2	4	no	—
129	C₈×D₄₃	—	2	4	no	—
130	C₈×D₄₃	—	2	2	no	—
131	C₈×D₄₃	—	2	2	no	—
132	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
133	C₂²×D₄₃	#117	1	2	yes	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
138	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
140	C₂²⋊D₄₃	—	2	4	no	—
141	C₂×Dic₄₃	—	2	1	no	—
142	C₂×Dic₄₃	—	2	1	no	—
143	C₂×Dic₄₃	—	4	2	no	—
144	C₂×Dic₄₃	—	8	1	no	—
145	C₂×Dic₄₃	#117	1	2	yes	—
146	D₁₇₂	—	2	2	no	—
147	C₄×D₄₃	—	2	2	no	—
148	C₄×D₄₃	—	2	2	no	—
149	C₄×D₄₃	—	8	2	no	—
150	Dic₈₆	—	2	2	no	—
151	C₈×C₄₃	—	2	2	no	—
152	C₈×C₄₃	—	4	2	no	—
153	C₈×C₄₃	#117	1	2	yes	—
154	C₈×C₄₃	#117	1	1	yes	characteristic
155	C₈×C₄₃	#117	1	1	yes	characteristic
156	C₄₃⋊C₈	—	2	2	no	—
157	C₄₃⋊C₈	—	4	2	no	—
158	C₄₃⋊C₈	#117	1	1	yes	characteristic
159	C₄₃⋊C₈	#117	1	1	yes	characteristic
160	C₄₃⋊C₈	#117	1	2	yes	—
161	#20788	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	—	4	1	no	—
163	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
164	C₂²×C₄₃	#529	1	2	yes	—
165	C₂×D₄₃	—	2	2	no	—
166	C₂×D₄₃	—	4	2	no	—
167	C₄×C₄₃	—	4	2	no	—
168	C₄×C₄₃	—	8	2	no	—
169	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
170	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
171	C₄×C₄₃	#529	1	2	yes	—
172	Dic₄₃	—	2	2	no	—
173	Dic₄₃	—	4	1	no	—
174	Dic₄₃	—	4	1	no	—
175	Dic₄₃	—	8	2	no	—
176	#2875	—	43	1	no	—
177	#1841	—	43	1	no	—
178	#1841	—	43	1	no	—
179	#1833	—	43	2	no	—
180	#1827	—	43	2	no	—
181	#1813	—	43	1	no	—
182	#1812	—	43	1	no	—
183	#1773	—	43	2	no	—
184	#1757	—	43	2	no	—
185	#1747	—	43	1	no	—
186	#1747	D₄₃	1	1	yes	characteristic, complemented
187	C₂×C₄₃	—	2	1	no	—
188	C₂×C₄₃	—	2	2	no	—
189	C₂×C₄₃	#2283	1	1	yes	characteristic
190	D₄₃	—	4	2	no	—
191	C₂×#131	—	43	1	no	—
192	C₂×#131	—	43	1	no	—
193	C₄×#41	—	43	1	no	—
194	C₂×#108	—	43	2	no	—
195	#427	—	43	2	no	—
196	#413	—	43	2	no	—
197	#410	—	43	2	no	—
198	#405	—	43	1	no	—
199	#405	—	43	2	no	—
200	#405	—	43	2	no	—
201	#405	—	43	2	no	—
202	#405	C₂×D₄₃	1	2	yes	—
203	#405	C₂×D₄₃	1	1	yes	characteristic
204	C₄×M₁₆	—	43	1	no	—
205	C₄×M₁₆	—	43	2	no	—
206	C₂×#97	—	43	1	no	—
207	C₂×#97	—	43	2	no	—
208	#344	—	43	2	no	—
209	#344	—	43	2	no	—
210	#339	—	43	2	no	—
211	#339	—	43	2	no	—
212	#338	—	43	1	no	—
213	#338	C₂×D₄₃	1	1	yes	characteristic
214	H(3,ℤ/4)	—	43	1	no	—
215	H(3,ℤ/4)	—	43	2	no	—
216	H(3,ℤ/4)	C₂×D₄₃	1	1	yes	characteristic
217	H(3,ℤ/4)	C₂×D₄₃	1	2	yes	—
218	C₄₃	#20788	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	43	1	no	—
220	#131	—	86	2	no	—
221	#131	—	86	2	no	—
222	#131	—	86	2	no	—
223	#131	—	86	2	no	—
224	C₂×M₁₆	—	43	1	no	—
225	C₂×M₁₆	—	43	2	no	—
226	C₂×M₁₆	—	43	2	no	—
227	C₂×M₁₆	—	43	2	no	—
228	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
229	C₂×M₁₆	C₂²×D₄₃	1	1	yes	characteristic
230	C₂²×C₈	—	43	1	no	—
231	C₂²×C₈	—	43	2	no	—
232	C₂²×C₈	—	43	2	no	—
233	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₄×Q₈	—	86	1	no	—
235	C₄×D₄	—	86	2	no	—
236	C₄×D₄	—	86	1	no	—
237	#117	—	43	2	no	—
238	#117	—	43	2	no	—
239	#117	C₂²×D₄₃	1	2	yes	—
240	#117	C₂²×D₄₃	1	1	yes	characteristic
241	C₂×C₄²	—	43	1	no	—
242	C₂×C₄²	—	43	2	no	—
243	#108	—	86	2	no	—
244	#108	—	86	2	no	—
245	#107	—	86	2	no	—
246	#106	—	86	2	no	—
247	#101	—	86	2	no	—
248	#100	—	86	2	no	—
249	#99	—	86	4	no	—
250	#97	—	2	2	no	—
251	#97	—	43	1	no	—
252	#97	—	43	1	no	—
253	#97	—	43	2	no	—
254	#97	—	43	1	no	—
255	#97	—	43	1	no	—
256	#97	—	43	2	no	—
257	#97	—	86	2	no	—
258	#97	C₄×D₄₃	1	2	yes	—
259	C₄×C₈	—	2	2	no	—
260	C₄×C₈	—	43	2	no	—
261	C₄×C₈	—	43	2	no	—
262	C₄×C₈	—	86	2	no	—
263	C₄×C₈	C₄×D₄₃	1	2	yes	—
264	#41	—	86	2	no	—
265	#41	—	86	2	no	—
266	C₂×Q₈	—	43	1	no	—
267	C₂×D₄	—	43	2	no	—
268	C₂×D₄	—	43	1	no	—
269	C₂²×C₄	—	43	2	no	—
270	C₂²×C₄	—	86	2	no	—
271	C₂²×C₄	—	86	1	no	—
272	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	86	4	no	—
277	M₁₆	—	86	2	no	—
278	M₁₆	—	86	2	no	—
279	M₁₆	—	86	2	no	—
280	M₁₆	—	86	2	no	—
281	M₁₆	—	86	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	43	1	no	—
284	C₂×C₈	—	43	1	no	—
285	C₂×C₈	—	43	2	no	—
286	C₂×C₈	—	43	2	no	—
287	C₂×C₈	—	43	2	no	—
288	C₂×C₈	—	86	2	no	—
289	C₂×C₈	—	86	2	no	—
290	C₂×C₈	—	86	4	no	—
291	C₂×C₈	—	86	2	no	—
292	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
293	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
294	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
295	C₂×C₈	#10621898	1	2	yes	—
296	C₂×C₈	#10621898	1	2	yes	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₄⋊C₄	—	86	2	no	—
301	C₂²⋊C₄	—	2	1	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	86	2	no	—
304	C₂²⋊C₄	—	86	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	43	1	no	—
307	C₄²	—	43	1	no	—
308	C₄²	—	86	2	no	—
309	C₄²	—	86	1	no	—
310	C₄²	—	86	2	no	—
311	C₄²	C₂×C₄×D₄₃	1	2	yes	—
312	C₂³	—	43	2	no	—
313	C₂³	#41×D₄₃	1	1	yes	characteristic
314	Q₈	—	86	2	no	—
315	D₄	—	86	4	no	—
316	D₄	—	86	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	43	2	no	—
320	C₂×C₄	—	86	2	no	—
321	C₂×C₄	—	86	2	no	—
322	C₂×C₄	—	86	1	no	—
323	C₂×C₄	—	86	1	no	—
324	C₂×C₄	—	344	2	no	—
325	C₂×C₄	—	344	1	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
328	C₂×C₄	#41×D₄₃	1	2	yes	—
329	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
330	C₂×C₄	C₂×#10621898	1	2	yes	—
331	C₈	—	2	2	no	—
332	C₈	—	4	2	no	—
333	C₈	—	43	1	no	—
334	C₈	—	43	1	no	—
335	C₈	—	43	2	no	—
336	C₈	—	86	2	no	—
337	C₈	—	172	2	no	—
338	C₈	#13063282	1	2	yes	—
339	C₈	#13063282	1	1	yes	characteristic
340	C₈	#13063282	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	86	2	no	—
343	C₂²	—	172	2	no	—
344	C₂²	#117×D₄₃	1	1	yes	characteristic
345	C₂²	#14583427	1	2	yes	—
346	C₄	—	4	2	no	—
347	C₄	—	8	2	no	—
348	C₄	—	86	2	no	—
349	C₄	—	344	2	no	—
350	C₄	—	172	1	no	—
351	C₄	—	172	1	no	—
352	C₄	#117×D₄₃	1	1	yes	characteristic
353	C₄	C₂×#13063282	1	1	yes	characteristic
354	C₄	#14583427	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	172	2	no	—
358	C₂	#19136987	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13063282
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1747	C₂
socle	C₂×C₄₃	#2283
prime radical	C₂×#10621898	C₂³
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1747	D₄₃
hypocenter	C₄₃	#20788
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ #405 ◁ #1747

Complements

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

N	H ≅ G/N	type
2. #19134745	C₂	semidirect
3. #19134742	C₂	semidirect
4. #19134717	C₂	semidirect
8. #19134480	C₂	semidirect
10. #19134350	C₂	semidirect
11. C₄₃×#1747	C₂	semidirect
19. #14582856	C₄	semidirect
22. #14582853	C₄	semidirect
24. C₄×#10621872	C₄	semidirect
26. #97×D₄₃	C₄	semidirect
186. #1747	D₄₃	semidirect
218. C₄₃	#20788	semidirect

Maximal subgroups

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

subgroup	index
#19143022	2
#19134745	2
#19134742	2
#19134717	2
#19134697	2
#19134644	2
#19134640	2
#19134480	2
#19134396	2
#19134350	2
C₄₃×#1747	2
#20788	43