Subgroups — #24996358

Order 11008.

726 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583275	C₂	1	1	yes	maximal, characteristic
2	#19135402	C₂	1	1	yes	maximal, characteristic, complemented
3	#19135401	C₂	1	1	yes	maximal, characteristic, complemented
4	#19135400	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134754	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1957	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621882	—	2	2	no	—
9	C₂×#13063322	C₂²	1	1	yes	characteristic
10	#14583275	—	2	2	no	—
11	#14583275	C₄	1	4	yes	complemented
12	C₂×#13063258	—	2	2	no	—
13	C₂×#13063247	—	2	2	no	—
14	C₂×#13063246	—	2	2	no	—
15	C₂×#13063245	—	2	2	no	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	#14582769	C₂²	1	1	yes	characteristic
19	C₈×C₄₃⋊C₈	—	2	2	no	—
20	C₂×#127×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×#102×C₄₃	—	2	2	no	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#384	C₂²	1	1	yes	characteristic
24	C₂²×D₁₇₂	—	2	1	no	—
25	C₂²×D₁₇₂	—	2	1	no	—
26	C₂²×D₁₇₂	—	2	1	no	—
27	C₂×#10621882	—	2	8	no	—
28	C₂×#10621882	—	4	4	no	—
29	C₂×#10621882	—	4	4	no	—
30	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
31	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
32	#13063322	—	2	1	no	—
33	#13063322	—	2	1	no	—
34	#13063322	—	2	1	no	—
35	#13063322	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₄₃	—	2	1	no	—
37	C₂×C₄⋊Dic₄₃	—	2	1	no	—
38	C₂×C₄⋊Dic₄₃	—	2	1	no	—
39	#13063258	—	4	4	no	—
40	#13063247	—	4	4	no	—
41	#13063246	—	4	4	no	—
42	#13063245	—	4	4	no	—
43	C₄×C₄₃⋊C₈	—	2	1	no	—
44	C₄×C₄₃⋊C₈	—	2	1	no	—
45	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
46	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
47	C₂²×D₄×C₄₃	—	2	1	no	—
48	C₂²×D₄×C₄₃	—	2	1	no	—
49	C₂²×D₄×C₄₃	—	2	1	no	—
50	C₂²×C₈×C₄₃	D₄	1	2	yes	—
51	#127×C₄₃	—	2	1	no	—
52	#127×C₄₃	—	2	1	no	—
53	#127×C₄₃	—	2	1	no	—
54	#127×C₄₃	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	#102×C₄₃	—	4	4	no	—
60	C₄×C₈×C₄₃	—	2	1	no	—
61	C₄×C₈×C₄₃	—	2	1	no	—
62	C₂⁴×C₄₃	—	4	1	no	—
63	C₂³×D₄₃	—	4	1	no	—
64	C₂×D₄×C₄₃	—	2	2	no	—
65	C₂×D₄×C₄₃	—	2	2	no	—
66	C₂×D₄×C₄₃	—	2	2	no	—
67	C₂×D₄×C₄₃	—	4	1	no	—
68	C₂×D₄×C₄₃	—	4	1	no	—
69	C₂×D₄×C₄₃	—	4	1	no	—
70	C₂×D₄×C₄₃	—	4	2	no	—
71	C₂×D₄×C₄₃	—	4	2	no	—
72	C₂×D₄×C₄₃	—	4	2	no	—
73	C₂×D₄×C₄₃	—	4	2	no	—
74	C₂×D₄×C₄₃	—	4	2	no	—
75	C₂×D₄×C₄₃	—	4	2	no	—
76	C₂²×C₄×C₄₃	—	4	1	no	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₄₃	—	4	1	no	—
81	C₂×D₁₇₂	—	2	2	no	—
82	C₂×D₁₇₂	—	2	2	no	—
83	C₂×D₁₇₂	—	2	2	no	—
84	C₂×D₁₇₂	—	4	1	no	—
85	C₂×D₁₇₂	—	4	1	no	—
86	C₂×D₁₇₂	—	4	1	no	—
87	C₂×D₁₇₂	—	4	2	no	—
88	C₂×D₁₇₂	—	4	2	no	—
89	C₂×D₁₇₂	—	4	2	no	—
90	C₂×D₁₇₂	—	4	2	no	—
91	C₂×D₁₇₂	—	4	2	no	—
92	C₂×D₁₇₂	—	4	2	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	D₈	1	4	yes	complemented
96	C₄⋊C₄×C₄₃	—	4	2	no	—
97	C₄⋊C₄×C₄₃	—	4	2	no	—
98	C₄⋊C₄×C₄₃	—	4	2	no	—
99	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₃	#41	1	1	yes	characteristic
102	C₄²×C₄₃	#41	1	1	yes	characteristic
103	#10621882	—	4	16	no	—
104	C₄⋊Dic₄₃	—	4	2	no	—
105	C₄⋊Dic₄₃	—	4	2	no	—
106	C₄⋊Dic₄₃	—	4	2	no	—
107	C₂×C₄₃⋊C₈	—	2	2	no	—
108	C₂×C₄₃⋊C₈	—	2	2	no	—
109	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	4	yes	—
110	C₂×C₄₃⋊C₈	D₈	1	4	yes	—
111	C₂×C₄₃⋊C₈	D₈	1	4	yes	complemented
112	C₂×C₄₃⋊C₈	SD₁₆	1	4	yes	—
113	C₂³×C₄₃	—	4	2	no	—
114	C₂³×C₄₃	—	8	1	no	—
115	C₂³×C₄₃	—	8	1	no	—
116	C₂³×C₄₃	—	8	1	no	—
117	C₂³×C₄₃	—	8	1	no	—
118	C₂³×C₄₃	—	8	1	no	—
119	C₂³×C₄₃	—	8	1	no	—
120	C₂³×C₄₃	#127	1	1	yes	characteristic
121	C₂²×D₄₃	—	4	2	no	—
122	C₂²×D₄₃	—	8	1	no	—
123	C₂²×D₄₃	—	8	1	no	—
124	C₂²×D₄₃	—	8	1	no	—
125	C₂²×D₄₃	—	8	1	no	—
126	C₂²×D₄₃	—	8	1	no	—
127	C₂²×D₄₃	—	8	1	no	—
128	D₄×C₄₃	—	4	4	no	—
129	D₄×C₄₃	—	4	4	no	—
130	D₄×C₄₃	—	4	4	no	—
131	D₄×C₄₃	—	8	2	no	—
132	D₄×C₄₃	—	8	2	no	—
133	D₄×C₄₃	—	8	2	no	—
134	C₂×C₄×C₄₃	—	8	1	no	—
135	C₂×C₄×C₄₃	—	8	1	no	—
136	C₂×C₄×C₄₃	—	8	1	no	—
137	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
147	C₂×C₄×C₄₃	#135	1	2	yes	—
148	C₂×C₄×C₄₃	#135	1	2	yes	—
149	C₂×Dic₄₃	—	8	1	no	—
150	C₂×Dic₄₃	—	8	1	no	—
151	C₂×Dic₄₃	—	8	1	no	—
152	D₁₇₂	—	4	4	no	—
153	D₁₇₂	—	4	4	no	—
154	D₁₇₂	—	4	4	no	—
155	D₁₇₂	—	8	2	no	—
156	D₁₇₂	—	8	2	no	—
157	D₁₇₂	—	8	2	no	—
158	C₈×C₄₃	—	2	4	no	—
159	C₄₃⋊C₈	—	2	4	no	—
160	C₄₃⋊C₈	#102	1	8	yes	complemented
161	#9824	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	—	8	2	no	—
163	C₂²×C₄₃	—	8	2	no	—
164	C₂²×C₄₃	—	8	2	no	—
165	C₂²×C₄₃	—	16	1	no	—
166	C₂²×C₄₃	—	16	1	no	—
167	C₂²×C₄₃	—	16	1	no	—
168	C₂²×C₄₃	—	16	1	no	—
169	C₂²×C₄₃	#390	1	1	yes	characteristic
170	C₂²×C₄₃	#492	1	1	yes	characteristic
171	C₂²×C₄₃	#493	1	1	yes	characteristic
172	C₂²×C₄₃	#493	1	1	yes	characteristic
173	C₂²×C₄₃	#493	1	1	yes	characteristic
174	C₂²×C₄₃	#495	1	1	yes	characteristic
175	C₂²×C₄₃	#495	1	1	yes	characteristic
176	C₂×D₄₃	—	8	2	no	—
177	C₂×D₄₃	—	8	2	no	—
178	C₂×D₄₃	—	8	2	no	—
179	C₂×D₄₃	—	16	1	no	—
180	C₂×D₄₃	—	16	1	no	—
181	C₂×D₄₃	—	16	1	no	—
182	C₂×D₄₃	—	16	1	no	—
183	C₄×C₄₃	—	16	1	no	—
184	C₄×C₄₃	C₂×#102	1	2	yes	—
185	C₄×C₄₃	C₄×D₈	1	2	yes	—
186	C₄×C₄₃	C₄×D₈	1	2	yes	—
187	C₄×C₄₃	#466	1	2	yes	—
188	C₄×C₄₃	#466	1	2	yes	—
189	C₄×C₄₃	#486	1	2	yes	—
190	Dic₄₃	—	16	1	no	—
191	C₂×#493	—	43	1	no	—
192	#1957	—	43	1	no	—
193	#1957	—	43	1	no	—
194	#1957	—	43	1	no	—
195	#1957	D₄₃	1	1	yes	characteristic, complemented
196	#1845	—	43	1	no	—
197	C₂×C₈²	—	43	1	no	—
198	C₂×C₄₃	—	16	2	no	—
199	C₂×C₄₃	#1706	1	1	yes	characteristic
200	C₂×C₄₃	#1706	1	1	yes	characteristic
201	C₂×C₄₃	#1709	1	1	yes	characteristic
202	C₂×C₄₃	#1712	1	1	yes	characteristic
203	C₂×C₄₃	#1957	1	1	yes	characteristic
204	C₂×C₄₃	#1970	1	1	yes	characteristic
205	C₂×C₄₃	#1970	1	1	yes	characteristic
206	D₄₃	—	16	2	no	—
207	C₂²×D₈	—	86	2	no	—
208	C₂×#127	—	43	1	no	—
209	C₂×#127	C₂×D₄₃	1	1	yes	characteristic
210	#493	—	43	4	no	—
211	#493	—	86	2	no	—
212	C₂×#107	—	86	2	no	—
213	C₂×#102	—	2	2	no	—
214	C₂×#102	—	86	2	no	—
215	C₂×#102	—	86	2	no	—
216	C₂×#102	—	86	2	no	—
217	C₂×C₄×C₈	—	43	1	no	—
218	C₂×C₄×C₈	—	43	1	no	—
219	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
220	#384	—	43	1	no	—
221	#384	C₂×D₄₃	1	1	yes	characteristic
222	C₈²	—	86	2	no	—
223	C₄₃	#9824	1	1	yes	characteristic, complemented, Sylow
224	C₂²×D₄	—	2	1	no	—
225	C₂²×D₄	—	2	1	no	—
226	C₂²×D₄	—	2	1	no	—
227	C₂²×D₄	—	86	1	no	—
228	C₂²×D₄	—	86	1	no	—
229	C₂²×D₄	—	86	1	no	—
230	C₂×D₈	—	86	8	no	—
231	C₂×D₈	—	172	4	no	—
232	C₂×D₈	—	172	4	no	—
233	C₂²×C₈	—	43	2	no	—
234	C₂²×C₈	—	43	2	no	—
235	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
236	#127	—	2	1	no	—
237	#127	—	2	1	no	—
238	#127	—	2	1	no	—
239	#127	—	43	2	no	—
240	#127	—	86	1	no	—
241	#127	—	86	1	no	—
242	#127	—	86	1	no	—
243	#127	C₄×D₄₃	1	2	yes	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	86	1	no	—
248	C₂×C₄⋊C₄	—	86	1	no	—
249	C₂×C₄⋊C₄	—	86	1	no	—
250	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	#107	—	172	4	no	—
252	#102	—	4	4	no	—
253	#102	—	172	4	no	—
254	#102	—	172	4	no	—
255	#102	—	172	4	no	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	43	2	no	—
259	C₄×C₈	—	43	2	no	—
260	C₄×C₈	—	86	1	no	—
261	C₄×C₈	—	86	1	no	—
262	C₂⁴	—	4	1	no	—
263	C₂⁴	—	172	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	—	4	2	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	4	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	86	2	no	—
277	C₂×D₄	—	86	2	no	—
278	C₂×D₄	—	86	2	no	—
279	C₂×D₄	—	172	1	no	—
280	C₂×D₄	—	172	1	no	—
281	C₂×D₄	—	172	1	no	—
282	C₂×D₄	—	172	2	no	—
283	C₂×D₄	—	172	2	no	—
284	C₂×D₄	—	172	2	no	—
285	C₂×D₄	—	172	2	no	—
286	C₂×D₄	—	172	2	no	—
287	C₂×D₄	—	172	2	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	—	172	1	no	—
290	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
293	D₈	—	172	16	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	43	4	no	—
297	C₂×C₈	—	43	4	no	—
298	C₂×C₈	—	43	4	no	—
299	C₂×C₈	—	43	4	no	—
300	C₂×C₈	—	86	2	no	—
301	C₂×C₈	—	86	2	no	—
302	C₂×C₈	#10621882	1	4	yes	complemented
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	172	2	no	—
307	C₄⋊C₄	—	172	2	no	—
308	C₄⋊C₄	—	172	2	no	—
309	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
310	C₄²	#10621898	1	1	yes	characteristic
311	C₄²	D₄×D₄₃	1	1	yes	characteristic
312	C₄²	#10621900	1	1	yes	characteristic
313	C₂³	—	4	2	no	—
314	C₂³	—	8	1	no	—
315	C₂³	—	8	1	no	—
316	C₂³	—	8	1	no	—
317	C₂³	—	8	1	no	—
318	C₂³	—	8	1	no	—
319	C₂³	—	8	1	no	—
320	C₂³	—	344	1	no	—
321	C₂³	—	344	1	no	—
322	C₂³	—	344	1	no	—
323	C₂³	—	344	1	no	—
324	C₂³	—	344	1	no	—
325	C₂³	—	344	1	no	—
326	C₂³	—	172	2	no	—
327	C₂³	#13063321	1	1	yes	characteristic
328	D₄	—	4	4	no	—
329	D₄	—	4	4	no	—
330	D₄	—	4	4	no	—
331	D₄	—	8	2	no	—
332	D₄	—	8	2	no	—
333	D₄	—	8	2	no	—
334	D₄	—	344	2	no	—
335	D₄	—	344	2	no	—
336	D₄	—	344	2	no	—
337	D₄	—	172	4	no	—
338	D₄	—	172	4	no	—
339	D₄	—	172	4	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	8	1	no	—
342	C₂×C₄	—	8	1	no	—
343	C₂×C₄	—	344	1	no	—
344	C₂×C₄	—	344	1	no	—
345	C₂×C₄	—	344	1	no	—
346	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
347	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
348	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
349	C₂×C₄	#13063297	1	1	yes	characteristic
350	C₂×C₄	#13063298	1	1	yes	characteristic
351	C₂×C₄	#13063311	1	1	yes	characteristic
352	C₂×C₄	D₈×D₄₃	1	2	yes	—
353	C₂×C₄	D₈×D₄₃	1	2	yes	—
354	C₂×C₄	C₂×#10621882	1	2	yes	—
355	C₂×C₄	SD₁₆×D₄₃	1	2	yes	—
356	C₂×C₄	#13063345	1	2	yes	—
357	C₂×C₄	#13063346	1	2	yes	—
358	C₈	—	2	4	no	—
359	C₈	—	43	8	no	—
360	C₈	—	86	4	no	—
361	C₂²	—	8	2	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	16	1	no	—
365	C₂²	—	16	1	no	—
366	C₂²	—	16	1	no	—
367	C₂²	—	16	1	no	—
368	C₂²	—	344	2	no	—
369	C₂²	—	344	2	no	—
370	C₂²	—	344	2	no	—
371	C₂²	—	688	1	no	—
372	C₂²	—	688	1	no	—
373	C₂²	—	688	1	no	—
374	C₂²	—	688	1	no	—
375	C₂²	#14582799	1	1	yes	characteristic
376	C₂²	#14583270	1	1	yes	characteristic
377	C₂²	#14583274	1	1	yes	characteristic
378	C₂²	#14583274	1	1	yes	characteristic
379	C₂²	#14583275	1	1	yes	characteristic
380	C₂²	#14583280	1	1	yes	characteristic
381	C₂²	#14583281	1	1	yes	characteristic
382	C₄	—	16	1	no	—
383	C₄	—	688	1	no	—
384	C₄	#102×D₄₃	1	2	yes	—
385	C₄	#14582979	1	2	yes	—
386	C₄	C₄×#10621882	1	2	yes	—
387	C₄	#14583133	1	2	yes	—
388	C₄	#14583134	1	2	yes	—
389	C₄	#14583244	1	2	yes	—
390	C₂	—	16	2	no	—
391	C₂	—	688	2	no	—
392	C₂	#19134185	1	1	yes	characteristic
393	C₂	#19134186	1	1	yes	characteristic
394	C₂	#19134199	1	1	yes	characteristic
395	C₂	#19134209	1	1	yes	characteristic
396	C₂	#19135399	1	1	yes	characteristic
397	C₂	#19135458	1	1	yes	characteristic
398	C₂	#19135460	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063321
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1957	C₂
socle	C₂³×C₄₃	#127
prime radical	C₂×#14583275	C₂
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1957	D₄₃
hypocenter	C₄₃	#9824
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

Complements

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

N	H ≅ G/N	type
2. #19135402	C₂	semidirect
3. #19135401	C₂	semidirect
4. #19135400	C₂	semidirect
5. #19134754	C₂	semidirect
6. C₂×C₈×C₄₃⋊C₈	C₂	semidirect
7. C₄₃×#1957	C₂	semidirect
11. #14583275	C₄	semidirect
95. C₂×C₈×C₄₃	D₈	semidirect
111. C₂×C₄₃⋊C₈	D₈	semidirect
160. C₄₃⋊C₈	#102	semidirect
195. #1957	D₄₃	semidirect
223. C₄₃	#9824	semidirect
302. C₂×C₈	#10621882	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583275	2
#19135402	2
#19135401	2
#19135400	2
#19134754	2
C₂×C₈×C₄₃⋊C₈	2
C₄₃×#1957	2
#9824	43