Subgroups — #25096965

Order 11008.

608 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143023	C₂	1	1	yes	maximal, characteristic
2	#19134741	C₂	1	2	yes	maximal, complemented
3	#19134717	C₂	1	2	yes	maximal, complemented
4	#19134699	C₂	1	1	yes	maximal, characteristic
5	#19134643	C₂	1	1	yes	maximal, characteristic
6	#19134639	C₂	1	1	yes	maximal, characteristic
7	#19134478	C₂	1	2	yes	maximal, complemented
8	#19134398	C₂	1	1	yes	maximal, characteristic
9	#19134397	C₂	1	1	yes	maximal, characteristic
10	#19134349	C₂	1	2	yes	maximal, complemented
11	C₄₃×#1827	C₂	1	1	yes	maximal, characteristic
12	C₂×#13063332	C₂²	1	1	yes	characteristic
13	C₂×#13063332	C₂²	1	1	yes	characteristic
14	C₂×#13063260	C₂²	1	2	yes	—
15	#14582935	C₂²	1	2	yes	—
16	#14582883	C₂²	1	1	yes	characteristic
17	#14582873	C₂²	1	1	yes	characteristic
18	#14582857	C₂²	1	2	yes	—
19	#14582855	C₄	1	2	yes	complemented
20	#14582855	C₄	1	2	yes	complemented
21	#14582855	C₂²	1	2	yes	—
22	#14582855	C₂²	1	1	yes	characteristic
23	#14582855	C₂²	1	1	yes	characteristic
24	C₄×#10621877	C₄	1	2	yes	complemented
25	C₄×#10621877	C₂²	1	1	yes	characteristic
26	C₂×#13063236	C₄	1	2	yes	complemented
27	C₂×#13063236	C₂²	1	1	yes	characteristic
28	#14582671	C₂²	1	2	yes	—
29	#14582669	C₂²	1	2	yes	—
30	#14582659	C₂²	1	2	yes	—
31	C₄₃⋊#338	C₂²	1	2	yes	—
32	#14582655	C₂²	1	2	yes	—
33	#14582655	C₂²	1	2	yes	—
34	C₄×#41×C₄₃	C₂²	1	1	yes	characteristic
35	C₄₃×#413	C₂²	1	1	yes	characteristic
36	C₄₃×#410	C₂²	1	1	yes	characteristic
37	C₄₃×#339	C₂²	1	2	yes	—
38	C₄₃×H(3,ℤ/4)	C₂²	1	2	yes	—
39	#13063332	—	2	2	no	—
40	#13063332	—	2	2	no	—
41	#13063332	—	2	2	no	—
42	#13063332	—	2	2	no	—
43	C₂×#10621877	C₂×C₄	1	2	yes	—
44	C₂×#10621877	C₂×C₄	1	2	yes	—
45	C₂×#10621877	C₂³	1	2	yes	—
46	C₂×#10621877	C₂³	1	1	yes	characteristic
47	C₂×#10621877	C₂³	1	1	yes	characteristic
48	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
50	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
52	#13063284	C₂³	1	2	yes	—
53	#13063260	—	2	2	no	—
54	#13063260	—	2	2	no	—
55	#13063258	—	2	2	no	—
56	#13063256	—	2	2	no	—
57	#13063244	—	2	1	no	—
58	#13063242	—	2	1	no	—
59	#13063240	—	2	2	no	—
60	#13063236	—	2	2	no	—
61	#13063236	C₂×C₄	1	1	yes	characteristic
62	#13063236	C₂×C₄	1	2	yes	—
63	#13063236	C₂×C₄	1	2	yes	—
64	#13063236	C₂×C₄	1	2	yes	—
65	#13063236	C₂×C₄	1	1	yes	characteristic
66	#13063236	C₂×C₄	1	1	yes	characteristic
67	#13063236	C₂×C₄	1	1	yes	characteristic
68	#13063235	—	2	2	no	—
69	C₄×C₄₃⋊C₈	—	2	2	no	—
70	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
71	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
72	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
73	C₈×Dic₄₃	—	2	2	no	—
74	C₂×#41×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
76	C₄×Q₈×C₄₃	—	2	1	no	—
77	C₄×D₄×C₄₃	—	2	2	no	—
78	C₄×D₄×C₄₃	—	2	1	no	—
79	#117×C₄₃	C₂×C₄	1	2	yes	—
80	#117×C₄₃	C₂³	1	2	yes	—
81	#117×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
83	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
84	#101×C₄₃	—	2	1	no	—
85	#100×C₄₃	—	2	1	no	—
86	#99×C₄₃	—	2	2	no	—
87	#41×C₄₃	—	2	2	no	—
88	#41×C₄₃	—	2	2	no	—
89	C₂×Q₈×C₄₃	#41	1	1	yes	characteristic
90	C₂×D₄×C₄₃	#41	1	1	yes	characteristic
91	C₂×D₄×C₄₃	#41	1	2	yes	—
92	C₂²×C₄×C₄₃	—	2	2	no	—
93	C₂²×C₄×C₄₃	—	2	1	no	—
94	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
96	M₁₆×C₄₃	—	2	2	no	—
97	M₁₆×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	2	1	no	—
101	C₄⋊C₄×C₄₃	—	2	2	no	—
102	C₂²⋊C₄×C₄₃	—	2	2	no	—
103	C₂²⋊C₄×C₄₃	—	2	1	no	—
104	C₂²⋊C₄×C₄₃	—	2	2	no	—
105	C₄²×C₄₃	—	2	2	no	—
106	C₄²×C₄₃	—	2	1	no	—
107	C₄²×C₄₃	—	2	2	no	—
108	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
110	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	C₄×Dic₄₃	—	2	2	no	—
114	#10621877	—	2	2	no	—
115	#10621877	—	2	2	no	—
116	#10621877	—	2	4	no	—
117	#10621877	—	2	4	no	—
118	#10621877	—	2	2	no	—
119	#10621877	—	2	2	no	—
120	C₂×C₄₃⋊C₈	—	2	2	no	—
121	C₂×C₄₃⋊C₈	—	2	2	no	—
122	C₂×C₄₃⋊C₈	—	2	2	no	—
123	C₂×C₄₃⋊C₈	—	2	2	no	—
124	C₂×C₄₃⋊C₈	—	2	2	no	—
125	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
126	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
127	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄₃⋊C₈	#41	1	2	yes	—
132	C₂×C₄₃⋊C₈	#41	1	2	yes	—
133	C₂×C₄₃⋊C₈	#41	1	2	yes	—
134	C₂×C₄₃⋊C₈	#41	1	2	yes	—
135	C₂³×C₄₃	#117	1	2	yes	—
136	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
137	Q₈×C₄₃	—	2	2	no	—
138	D₄×C₄₃	—	2	4	no	—
139	D₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	2	2	no	—
141	C₂×C₄×C₄₃	—	2	2	no	—
142	C₂×C₄×C₄₃	—	2	1	no	—
143	C₂×C₄×C₄₃	—	2	2	no	—
144	C₂×C₄×C₄₃	—	2	1	no	—
145	C₂×C₄×C₄₃	—	4	2	no	—
146	C₂×C₄×C₄₃	—	8	1	no	—
147	C₂×C₄×C₄₃	—	8	2	no	—
148	C₂×C₄×C₄₃	#117	1	2	yes	—
149	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
151	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
152	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
153	C₂×Dic₄₃	—	4	2	no	—
154	C₈×C₄₃	—	4	2	no	—
155	C₄₃⋊C₈	—	2	2	no	—
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	1	yes	characteristic
161	C₄₃⋊C₈	#117	1	1	yes	characteristic
162	C₄₃⋊C₈	#117	1	2	yes	—
163	C₄₃⋊C₈	#117	1	2	yes	—
164	#20788	—	43	1	no	maximal, Sylow
165	C₂²×C₄₃	—	2	2	no	—
166	C₂²×C₄₃	—	4	1	no	—
167	C₂²×C₄₃	—	4	2	no	—
168	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
169	C₂²×C₄₃	#529	1	2	yes	—
170	C₄×C₄₃	—	2	2	no	—
171	C₄×C₄₃	—	4	2	no	—
172	C₄×C₄₃	—	4	1	no	—
173	C₄×C₄₃	—	4	1	no	—
174	C₄×C₄₃	—	8	2	no	—
175	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
176	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
177	C₄×C₄₃	#529	1	2	yes	—
178	Dic₄₃	—	8	2	no	—
179	#2875	—	43	1	no	—
180	#1841	—	43	2	no	—
181	#1833	—	43	2	no	—
182	#1827	—	43	1	no	—
183	#1827	D₄₃	1	1	yes	characteristic
184	#1813	—	43	1	no	—
185	#1812	—	43	1	no	—
186	#1773	—	43	2	no	—
187	#1757	—	43	1	no	—
188	#1757	—	43	1	no	—
189	#1747	—	43	2	no	—
190	C₂×C₄₃	—	2	2	no	—
191	C₂×C₄₃	—	2	1	no	—
192	C₂×C₄₃	—	4	2	no	—
193	C₂×C₄₃	#2283	1	1	yes	characteristic
194	C₂×#131	—	43	1	no	—
195	C₂×#131	—	43	1	no	—
196	C₄×#41	C₂×D₄₃	1	1	yes	characteristic
197	C₂×#108	—	43	2	no	—
198	#427	—	43	2	no	—
199	#413	—	43	1	no	—
200	#413	C₂×D₄₃	1	1	yes	characteristic
201	#410	—	43	1	no	—
202	#410	C₂×D₄₃	1	1	yes	characteristic
203	#405	—	43	1	no	—
204	#405	—	43	1	no	—
205	#405	—	43	2	no	—
206	#405	—	43	2	no	—
207	#405	—	43	2	no	—
208	#405	—	43	2	no	—
209	C₄×M₁₆	—	43	1	no	—
210	C₄×M₁₆	—	43	2	no	—
211	C₂×#97	—	43	1	no	—
212	C₂×#97	—	43	2	no	—
213	#344	—	43	2	no	—
214	#344	—	43	2	no	—
215	#339	—	43	2	no	—
216	#339	C₂×D₄₃	1	2	yes	—
217	#338	—	43	2	no	—
218	H(3,ℤ/4)	—	43	2	no	—
219	H(3,ℤ/4)	—	43	2	no	—
220	H(3,ℤ/4)	C₂×D₄₃	1	2	yes	—
221	C₄₃	#20788	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	C₂²×D₄₃	1	1	yes	characteristic
223	#131	—	86	2	no	—
224	#131	—	86	2	no	—
225	#131	—	86	2	no	—
226	#131	—	86	2	no	—
227	C₂×M₁₆	—	43	2	no	—
228	C₂×M₁₆	—	43	2	no	—
229	C₂×M₁₆	—	43	1	no	—
230	C₂×M₁₆	—	43	1	no	—
231	C₂×M₁₆	—	43	2	no	—
232	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
233	C₂²×C₈	—	43	1	no	—
234	C₂²×C₈	—	43	1	no	—
235	C₂²×C₈	—	43	2	no	—
236	C₂²×C₈	—	43	2	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	—	2	1	no	—
240	#117	—	43	2	no	—
241	#117	C₄×D₄₃	1	2	yes	—
242	#117	C₂²×D₄₃	1	2	yes	—
243	#117	C₂²×D₄₃	1	1	yes	characteristic
244	C₂×C₄²	C₄×D₄₃	1	2	yes	—
245	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
246	#108	—	86	2	no	—
247	#108	—	86	2	no	—
248	#107	—	86	2	no	—
249	#106	—	86	2	no	—
250	#101	—	2	1	no	—
251	#101	—	86	1	no	—
252	#100	—	2	1	no	—
253	#100	—	86	1	no	—
254	#99	—	2	2	no	—
255	#99	—	86	2	no	—
256	#97	—	43	1	no	—
257	#97	—	43	1	no	—
258	#97	—	43	2	no	—
259	#97	—	43	1	no	—
260	#97	—	43	1	no	—
261	#97	—	43	2	no	—
262	#97	—	43	2	no	—
263	#97	—	86	2	no	—
264	#97	—	86	2	no	—
265	C₄×C₈	—	43	2	no	—
266	C₄×C₈	—	43	2	no	—
267	C₄×C₈	—	43	2	no	—
268	C₄×C₈	—	86	2	no	—
269	C₄×C₈	—	86	2	no	—
270	#41	—	2	2	no	—
271	#41	—	2	2	no	—
272	C₂×Q₈	#10621900	1	1	yes	characteristic
273	C₂×D₄	#10621900	1	2	yes	—
274	C₂×D₄	#10621900	1	1	yes	characteristic
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
278	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	86	4	no	—
282	M₁₆	—	86	4	no	—
283	M₁₆	—	86	2	no	—
284	M₁₆	—	86	2	no	—
285	M₁₆	—	86	2	no	—
286	M₁₆	—	86	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	43	1	no	—
289	C₂×C₈	—	43	1	no	—
290	C₂×C₈	—	43	1	no	—
291	C₂×C₈	—	43	1	no	—
292	C₂×C₈	—	43	2	no	—
293	C₂×C₈	—	43	2	no	—
294	C₂×C₈	—	43	2	no	—
295	C₂×C₈	—	43	2	no	—
296	C₂×C₈	—	43	2	no	—
297	C₂×C₈	—	43	2	no	—
298	C₂×C₈	—	86	2	no	—
299	C₂×C₈	—	86	2	no	—
300	C₂×C₈	—	86	2	no	—
301	C₂×C₈	—	86	2	no	—
302	C₂×C₈	—	86	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	1	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	86	2	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	2	1	no	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	86	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	86	2	no	—
315	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₄₃	1	2	yes	—
317	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
318	C₂³	#13063281	1	2	yes	—
319	C₂³	C₂×#10621900	1	1	yes	characteristic
320	Q₈	—	2	2	no	—
321	D₄	—	2	4	no	—
322	D₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	#13063281	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
334	C₂×C₄	#41×D₄₃	1	2	yes	—
335	C₂×C₄	#41×D₄₃	1	2	yes	—
336	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
337	C₈	—	4	2	no	—
338	C₈	—	43	1	no	—
339	C₈	—	43	1	no	—
340	C₈	—	43	2	no	—
341	C₈	—	43	1	no	—
342	C₈	—	43	1	no	—
343	C₈	—	43	2	no	—
344	C₈	—	86	2	no	—
345	C₈	—	86	2	no	—
346	C₈	—	172	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	2	no	—
350	C₂²	C₂×#13063281	1	1	yes	characteristic
351	C₂²	#14583426	1	2	yes	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	344	2	no	—
358	C₄	#117×D₄₃	1	1	yes	characteristic
359	C₄	#117×D₄₃	1	1	yes	characteristic
360	C₄	#14583426	1	2	yes	—
361	C₂	—	2	1	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	2	no	—
364	C₂	#19136986	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1827	C₂
socle	C₂×C₄₃	#2283
prime radical	C₂×#41×C₄₃	C₂³
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1827	D₄₃
hypocenter	C₄₃	#20788
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ #117 ◁ #1827

Complements

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

N	H ≅ G/N	type
2. #19134741	C₂	semidirect
3. #19134717	C₂	semidirect
7. #19134478	C₂	semidirect
10. #19134349	C₂	semidirect
19. #14582855	C₄	semidirect
20. #14582855	C₄	semidirect
24. C₄×#10621877	C₄	semidirect
26. C₂×#13063236	C₄	semidirect
221. C₄₃	#20788	semidirect

Maximal subgroups

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

subgroup	index
#19143023	2
#19134741	2
#19134717	2
#19134699	2
#19134643	2
#19134639	2
#19134478	2
#19134398	2
#19134397	2
#19134349	2
C₄₃×#1827	2
#20788	43