Subgroups — #24994924

Order 11008.

466 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135522	C₂	1	1	yes	maximal, characteristic
2	#19135401	C₂	1	1	yes	maximal, characteristic
3	#19135353	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582642	C₂	1	1	yes	maximal, characteristic
5	#19132479	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132430	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063322	C₂²	1	1	yes	characteristic
9	C₂×#13063258	—	2	1	no	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#13063247	—	2	1	no	—
13	C₂×#13063247	—	2	1	no	—
14	C₂×#13063245	—	2	1	no	—
15	C₂×#13063245	—	2	1	no	—
16	#14582769	C₂²	1	1	yes	characteristic
17	#14582661	—	2	1	no	—
18	#14582642	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₃	—	2	1	no	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#336	—	2	1	no	—
24	C₂²×D₁₇₂	—	2	1	no	—
25	C₂²×D₁₇₂	—	2	1	no	—
26	C₂²×D₁₇₂	—	2	1	no	—
27	C₂²×C₄₃⋊C₈	—	2	1	no	—
28	C₂²×C₄₃⋊C₈	—	2	1	no	—
29	#13063322	—	2	1	no	—
30	#13063322	—	2	1	no	—
31	#13063322	—	2	1	no	—
32	#13063322	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₄₃	—	2	1	no	—
34	C₂×C₄⋊Dic₄₃	—	2	1	no	—
35	C₂×C₄⋊Dic₄₃	—	2	1	no	—
36	#13063258	—	4	2	no	—
37	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
38	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
39	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
40	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
41	#13063247	—	4	2	no	—
42	#13063247	—	4	2	no	—
43	#13063245	—	4	2	no	—
44	#13063245	—	4	2	no	—
45	C₂²×C₈×C₄₃	—	2	1	no	—
46	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	—	2	1	no	—
51	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₄₃	—	2	1	no	—
53	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₄₃	—	2	1	no	—
55	C₄×C₈×C₄₃	—	2	1	no	—
56	C₄×C₈×C₄₃	—	4	2	no	—
57	#95×C₄₃	—	2	1	no	—
58	C₂³×D₄₃	—	4	1	no	—
59	C₂²×C₄×C₄₃	—	2	1	no	—
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂²×Dic₄₃	—	4	1	no	—
65	C₂×D₁₇₂	—	2	2	no	—
66	C₂×D₁₇₂	—	2	2	no	—
67	C₂×D₁₇₂	—	2	2	no	—
68	C₂×D₁₇₂	—	4	1	no	—
69	C₂×D₁₇₂	—	4	1	no	—
70	C₂×D₁₇₂	—	4	1	no	—
71	C₂×D₁₇₂	—	4	1	no	—
72	C₂×D₁₇₂	—	4	1	no	—
73	C₂×D₁₇₂	—	4	1	no	—
74	C₂×D₁₇₂	—	4	1	no	—
75	C₂×D₁₇₂	—	4	1	no	—
76	C₂×D₁₇₂	—	4	1	no	—
77	C₂×D₁₇₂	—	4	1	no	—
78	C₂×D₁₇₂	—	4	1	no	—
79	C₂×D₁₇₂	—	8	1	no	—
80	C₂×D₁₇₂	—	8	1	no	—
81	C₂×C₈×C₄₃	—	2	1	no	—
82	C₂×C₈×C₄₃	—	2	1	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	—	2	1	no	—
86	C₂×C₈×C₄₃	—	2	1	no	—
87	C₂×C₈×C₄₃	—	4	1	no	—
88	C₂×C₈×C₄₃	—	4	1	no	—
89	C₂×C₈×C₄₃	D₈	1	2	yes	—
90	C₄⋊C₄×C₄₃	—	2	1	no	—
91	C₄⋊C₄×C₄₃	—	2	1	no	—
92	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
93	C₄⋊C₄×C₄₃	SD₁₆	1	2	yes	—
94	C₄²×C₄₃	—	4	1	no	—
95	C₄²×C₄₃	—	4	1	no	—
96	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₄₃	#41	1	1	yes	characteristic
98	C₄²×C₄₃	#41	1	1	yes	characteristic
99	C₄²×C₄₃	#41	1	1	yes	characteristic
100	C₄⋊Dic₄₃	—	4	1	no	—
101	C₄⋊Dic₄₃	—	4	1	no	—
102	C₄⋊Dic₄₃	—	4	1	no	—
103	C₄⋊Dic₄₃	—	4	1	no	—
104	C₄⋊Dic₄₃	—	8	1	no	—
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	C₂×C₄₃⋊C₈	—	4	1	no	—
107	C₂×C₄₃⋊C₈	—	4	1	no	—
108	C₂×C₄₃⋊C₈	—	4	1	no	—
109	C₂×C₄₃⋊C₈	—	4	1	no	—
110	C₂×C₄₃⋊C₈	—	4	1	no	—
111	C₂³×C₄₃	#121	1	1	yes	characteristic
112	C₂²×D₄₃	—	4	2	no	—
113	C₂²×D₄₃	—	8	1	no	—
114	C₂²×D₄₃	—	8	1	no	—
115	C₂²×D₄₃	—	8	1	no	—
116	C₂²×D₄₃	—	8	1	no	—
117	C₂²×D₄₃	—	8	1	no	—
118	C₂²×D₄₃	—	8	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	4	1	no	—
124	C₂×C₄×C₄₃	—	4	1	no	—
125	C₂×C₄×C₄₃	—	4	1	no	—
126	C₂×C₄×C₄₃	—	4	1	no	—
127	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#126	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
139	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
141	C₂×Dic₄₃	—	8	1	no	—
142	C₂×Dic₄₃	—	8	1	no	—
143	C₂×Dic₄₃	—	8	1	no	—
144	D₁₇₂	—	4	2	no	—
145	D₁₇₂	—	4	2	no	—
146	D₁₇₂	—	4	2	no	—
147	D₁₇₂	—	4	2	no	—
148	D₁₇₂	—	8	2	no	—
149	D₁₇₂	—	8	2	no	—
150	D₁₇₂	—	8	1	no	—
151	D₁₇₂	—	8	1	no	—
152	D₁₇₂	—	8	1	no	—
153	D₁₇₂	—	8	1	no	—
154	C₈×C₄₃	—	2	2	no	—
155	C₈×C₄₃	—	4	2	no	—
156	C₈×C₄₃	—	4	1	no	—
157	C₄₃⋊C₈	—	4	2	no	—
158	C₄₃⋊C₈	—	8	1	no	—
159	#9563	—	43	1	no	maximal, Sylow
160	C₂²×C₄₃	#388	1	1	yes	characteristic
161	C₂²×C₄₃	#459	1	1	yes	characteristic
162	C₂²×C₄₃	#464	1	1	yes	characteristic
163	C₂²×C₄₃	#465	1	1	yes	characteristic
164	C₂²×C₄₃	#466	1	1	yes	characteristic
165	C₂²×C₄₃	#471	1	1	yes	characteristic
166	C₂²×C₄₃	#472	1	1	yes	characteristic
167	C₂×D₄₃	—	8	2	no	—
168	C₂×D₄₃	—	8	2	no	—
169	C₂×D₄₃	—	8	2	no	—
170	C₂×D₄₃	—	16	1	no	—
171	C₂×D₄₃	—	16	1	no	—
172	C₂×D₄₃	—	16	1	no	—
173	C₂×D₄₃	—	16	1	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	4	2	no	—
177	C₄×C₄₃	—	4	2	no	—
178	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
179	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
180	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
181	C₄×C₄₃	#440	1	1	yes	characteristic
182	C₄×C₄₃	#480	1	1	yes	characteristic
183	C₄×C₄₃	#482	1	1	yes	characteristic
184	C₄×C₄₃	#486	1	1	yes	characteristic
185	C₄×C₄₃	#488	1	1	yes	characteristic
186	Dic₄₃	—	16	1	no	—
187	#1982	—	43	1	no	—
188	#1957	—	43	1	no	—
189	#1947	—	43	1	no	—
190	C₈×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
191	C₂×#331	—	43	1	no	—
192	#1296	—	43	1	no	—
193	#1277	—	43	1	no	—
194	C₂×C₄₃	#1665	1	1	yes	characteristic
195	C₂×C₄₃	#1667	1	1	yes	characteristic
196	C₂×C₄₃	#1678	1	1	yes	characteristic
197	C₂×C₄₃	#1682	1	1	yes	characteristic
198	C₂×C₄₃	#1925	1	1	yes	characteristic
199	C₂×C₄₃	#1931	1	1	yes	characteristic
200	C₂×C₄₃	#1934	1	1	yes	characteristic
201	D₄₃	—	16	2	no	—
202	C₂×#127	—	43	1	no	—
203	C₂×#107	—	86	1	no	—
204	C₂×C₄⋊C₈	—	43	1	no	—
205	C₂×C₄⋊C₈	—	43	1	no	—
206	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
207	C₂×#102	—	86	1	no	—
208	C₂×#102	—	86	1	no	—
209	C₂×#102	—	86	1	no	—
210	C₂×#102	—	86	1	no	—
211	C₂×C₄×C₈	—	2	1	no	—
212	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
213	#384	—	43	1	no	—
214	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
215	#340	—	86	1	no	—
216	#336	—	2	1	no	—
217	#331	—	43	4	no	—
218	C₄₃	#9563	1	1	yes	characteristic, complemented, Sylow
219	C₂²×D₄	—	86	1	no	—
220	C₂²×D₄	—	86	1	no	—
221	C₂²×D₄	—	86	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	86	1	no	—
224	C₂²×C₈	—	86	1	no	—
225	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
226	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
227	#127	—	43	2	no	—
228	#127	—	86	1	no	—
229	#127	—	86	1	no	—
230	#127	—	86	1	no	—
231	C₂×C₄⋊C₄	—	86	1	no	—
232	C₂×C₄⋊C₄	—	86	1	no	—
233	C₂×C₄⋊C₄	—	86	1	no	—
234	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
235	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
236	C₂×C₄²	—	2	1	no	—
237	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
238	#107	—	172	2	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	43	2	no	—
241	C₄⋊C₈	—	86	1	no	—
242	C₄⋊C₈	—	86	1	no	—
243	C₄⋊C₈	—	86	1	no	—
244	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
245	#102	—	172	2	no	—
246	#102	—	172	2	no	—
247	#102	—	172	2	no	—
248	#102	—	172	2	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	—	4	2	no	—
252	#95	—	2	1	no	—
253	C₂⁴	—	172	1	no	—
254	C₂×D₄	—	86	2	no	—
255	C₂×D₄	—	86	2	no	—
256	C₂×D₄	—	86	2	no	—
257	C₂×D₄	—	344	1	no	—
258	C₂×D₄	—	344	1	no	—
259	C₂×D₄	—	172	1	no	—
260	C₂×D₄	—	172	1	no	—
261	C₂×D₄	—	172	1	no	—
262	C₂×D₄	—	172	1	no	—
263	C₂×D₄	—	172	1	no	—
264	C₂×D₄	—	172	1	no	—
265	C₂×D₄	—	172	1	no	—
266	C₂×D₄	—	172	1	no	—
267	C₂×D₄	—	172	1	no	—
268	C₂×D₄	—	172	1	no	—
269	C₂×D₄	—	172	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	172	1	no	—
273	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
274	C₂²×C₄	#10621898	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	86	2	no	—
285	C₂×C₈	—	172	1	no	—
286	C₂×C₈	—	172	1	no	—
287	C₂×C₈	—	172	1	no	—
288	C₂×C₈	—	172	1	no	—
289	C₂×C₈	—	172	1	no	—
290	C₂×C₈	#10621882	1	2	yes	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	344	1	no	—
294	C₄⋊C₄	—	172	1	no	—
295	C₄⋊C₄	—	172	1	no	—
296	C₄⋊C₄	—	172	1	no	—
297	C₄⋊C₄	—	172	1	no	—
298	C₄⋊C₄	#10621873	1	2	yes	—
299	C₄⋊C₄	D₃₄₄	1	2	yes	—
300	C₄²	—	4	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
303	C₄²	#10621898	1	1	yes	characteristic
304	C₄²	#10621898	1	1	yes	characteristic
305	C₄²	#10621898	1	1	yes	characteristic
306	C₂³	—	344	1	no	—
307	C₂³	—	344	1	no	—
308	C₂³	—	344	1	no	—
309	C₂³	—	344	1	no	—
310	C₂³	—	344	1	no	—
311	C₂³	—	344	1	no	—
312	C₂³	—	172	2	no	—
313	C₂³	#13063299	1	1	yes	characteristic
314	D₄	—	344	2	no	—
315	D₄	—	344	2	no	—
316	D₄	—	344	1	no	—
317	D₄	—	344	1	no	—
318	D₄	—	344	1	no	—
319	D₄	—	344	1	no	—
320	D₄	—	172	2	no	—
321	D₄	—	172	2	no	—
322	D₄	—	172	2	no	—
323	D₄	—	172	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	344	1	no	—
333	C₂×C₄	—	344	1	no	—
334	C₂×C₄	—	344	1	no	—
335	C₂×C₄	#13063282	1	1	yes	characteristic
336	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
337	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
338	C₂×C₄	#13063308	1	1	yes	characteristic
339	C₂×C₄	#13063313	1	1	yes	characteristic
340	C₂×C₄	#13063319	1	1	yes	characteristic
341	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
342	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
343	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
344	C₂×C₄	#13063345	1	1	yes	characteristic
345	C₂×C₄	#13063347	1	1	yes	characteristic
346	C₂×C₄	#13063347	1	1	yes	characteristic
347	C₂×C₄	#13063350	1	1	yes	characteristic
348	C₂×C₄	#13063356	1	1	yes	characteristic
349	C₈	—	2	2	no	—
350	C₈	—	4	1	no	—
351	C₈	—	4	2	no	—
352	C₈	—	344	1	no	—
353	C₈	—	172	2	no	—
354	C₂²	—	344	2	no	—
355	C₂²	—	344	2	no	—
356	C₂²	—	344	2	no	—
357	C₂²	—	688	1	no	—
358	C₂²	—	688	1	no	—
359	C₂²	—	688	1	no	—
360	C₂²	—	688	1	no	—
361	C₂²	#14582790	1	1	yes	characteristic
362	C₂²	#14583094	1	1	yes	characteristic
363	C₂²	#14583121	1	1	yes	characteristic
364	C₂²	#14583128	1	1	yes	characteristic
365	C₂²	#14583135	1	1	yes	characteristic
366	C₂²	#14583162	1	1	yes	characteristic
367	C₂²	#14583169	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	688	1	no	—
373	C₄	C₄×#10621882	1	1	yes	characteristic
374	C₄	C₄×D₃₄₄	1	1	yes	characteristic
375	C₄	C₄×#10621873	1	1	yes	characteristic
376	C₄	#14582997	1	1	yes	characteristic
377	C₄	#14583212	1	1	yes	characteristic
378	C₄	#14583223	1	1	yes	characteristic
379	C₄	#14583246	1	1	yes	characteristic
380	C₄	#14583254	1	1	yes	characteristic
381	C₂	—	688	2	no	—
382	C₂	#19133936	1	1	yes	characteristic
383	C₂	#19133950	1	1	yes	characteristic
384	C₂	#19134011	1	1	yes	characteristic
385	C₂	#19134039	1	1	yes	characteristic
386	C₂	#19135215	1	1	yes	characteristic
387	C₂	#19135257	1	1	yes	characteristic
388	C₂	#19135278	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063299
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₈×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂×#13063322	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#9563
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #19135353	C₂	semidirect
5. #19132479	C₂	semidirect
6. #19132430	C₂	semidirect
7. C₈×C₄⋊C₄×C₄₃	C₂	semidirect
18. #14582642	C₄	semidirect
190. C₈×C₄⋊C₄	D₄₃	semidirect
218. C₄₃	#9563	semidirect

Maximal subgroups

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

subgroup	index
#19135522	2
#19135401	2
#19135353	2
C₂×#14582642	2
#19132479	2
#19132430	2
C₈×C₄⋊C₄×C₄₃	2
#9563	43