Subgroups — #25251492

Order 11008.

556 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144200	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582946	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136813	C₂	1	4	yes	maximal, complemented
4	#19136808	C₂	1	4	yes	maximal, complemented
5	#19136369	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136368	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136344	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14582680	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#2148	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10621872	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₄₃	—	2	1	no	—
12	#14583406	—	2	1	no	—
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	#14583154	C₂²	1	2	yes	—
15	#14583149	C₂²	1	2	yes	—
16	#14583142	—	2	2	no	—
17	#14582946	—	2	2	no	—
18	#14582946	C₂²	1	4	yes	—
19	C₂×#13063261	C₂²	1	1	yes	characteristic
20	C₂×#13063260	—	2	1	no	—
21	#14582937	C₂²	1	1	yes	characteristic
22	#14582895	—	2	1	no	—
23	#14582723	C₂²	1	4	yes	complemented
24	#14582722	C₂²	1	4	yes	complemented
25	#14582715	C₂²	1	2	yes	complemented
26	#14582712	C₂²	1	2	yes	complemented
27	#14582680	C₂²	1	4	yes	—
28	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#362	C₂²	1	2	yes	complemented
32	C₄₃×#361	C₂²	1	2	yes	complemented
33	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
34	C₂×#10621877	—	2	1	no	—
35	C₂×#10621872	—	2	1	no	—
36	C₂×#10621872	—	2	1	no	—
37	C₂×#10621872	—	2	1	no	—
38	C₂×#10621872	—	2	1	no	—
39	C₂×#10621872	—	4	1	no	—
40	C₂×#10621872	D₄	1	2	yes	—
41	C₂×#10621872	C₂³	1	2	yes	—
42	M₁₆×D₄₃	—	2	4	no	—
43	M₁₆×D₄₃	—	4	2	no	—
44	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
45	C₂×C₈×D₄₃	—	4	1	no	—
46	#13063306	—	4	1	no	—
47	#13063301	—	2	2	no	—
48	#13063296	—	2	2	no	—
49	#13063293	—	4	1	no	—
50	C₄×C₂²⋊D₄₃	—	4	2	no	—
51	#13063284	—	2	1	no	—
52	#13063265	—	2	1	no	—
53	#13063265	C₂³	1	2	yes	—
54	C₂×C₄₃⋊C₁₆	—	2	1	no	—
55	#13063261	—	2	1	no	—
56	#13063261	C₂³	1	2	yes	—
57	#13063260	—	2	4	no	—
58	#13063259	—	2	1	no	—
59	#13063257	C₂³	1	2	yes	—
60	#13063249	—	2	2	no	—
61	#13063246	—	2	2	no	—
62	C₂×#41×C₄₃	—	2	1	no	—
63	#135×C₄₃	—	2	1	no	—
64	#135×C₄₃	—	2	1	no	—
65	#135×C₄₃	—	2	1	no	—
66	#135×C₄₃	—	2	1	no	—
67	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×SD₁₆×C₄₃	—	2	1	no	—
69	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×M₁₆×C₄₃	—	2	1	no	—
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	M₃₂×C₄₃	—	2	1	no	—
73	M₃₂×C₄₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₄₃	—	2	1	no	—
75	#108×C₄₃	—	2	1	no	—
76	#108×C₄₃	C₂³	1	2	yes	—
77	C₂³×D₄₃	SD₁₆	1	1	yes	characteristic
78	#41×C₄₃	—	4	1	no	—
79	#41×C₄₃	—	4	1	no	—
80	#41×C₄₃	—	4	1	no	—
81	#41×C₄₃	—	4	1	no	—
82	C₂×Q₈×C₄₃	—	2	1	no	—
83	C₂×D₄×C₄₃	—	2	1	no	—
84	C₂×D₄×C₄₃	—	4	1	no	—
85	C₂²×C₄×C₄₃	—	4	1	no	—
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₄₃	—	4	2	no	—
88	C₂²×Dic₄₃	SD₁₆	1	1	yes	characteristic
89	C₂×C₄×D₄₃	—	2	1	no	—
90	C₂×C₄×D₄₃	—	2	1	no	—
91	C₂×C₄×D₄₃	—	2	1	no	—
92	C₂×C₄×D₄₃	—	2	1	no	—
93	C₂×C₄×D₄₃	SD₁₆	1	2	yes	—
94	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
95	Q₁₆×C₄₃	—	2	1	no	—
96	Q₁₆×C₄₃	—	2	1	no	—
97	SD₁₆×C₄₃	—	4	1	no	—
98	SD₁₆×C₄₃	—	4	1	no	—
99	D₈×C₄₃	—	2	1	no	—
100	D₈×C₄₃	—	2	1	no	—
101	M₁₆×C₄₃	—	2	2	no	—
102	M₁₆×C₄₃	—	4	1	no	—
103	C₂×C₈×C₄₃	—	4	1	no	—
104	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
106	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
107	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
108	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
109	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
110	C₂²⋊Dic₄₃	—	4	2	no	—
111	C₂²⋊Dic₄₃	—	4	1	no	—
112	(C₂×D₄₃)⋊C₄	—	4	2	no	—
113	C₄⋊Dic₄₃	—	2	2	no	—
114	Dic₄₃⋊C₄	—	4	2	no	—
115	C₄×Dic₄₃	—	4	1	no	—
116	#10621877	—	2	2	no	—
117	#10621877	—	4	1	no	—
118	C₂×C₄₃⋊C₈	—	2	1	no	—
119	C₂×C₄₃⋊C₈	—	2	1	no	—
120	C₂×C₄₃⋊C₈	—	4	1	no	—
121	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
122	#10621872	—	2	2	no	—
123	#10621872	—	2	2	no	—
124	#10621872	—	2	2	no	—
125	#10621872	—	2	2	no	—
126	#10621872	—	4	4	no	—
127	C₈×D₄₃	—	4	4	no	—
128	C₁₆×C₄₃	—	2	2	no	—
129	C₄₃⋊C₁₆	—	2	2	no	—
130	C₂³×C₄₃	—	4	1	no	—
131	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂²×D₄₃	—	2	1	no	—
133	C₂²×D₄₃	—	2	1	no	—
134	C₂²×D₄₃	—	8	1	no	—
135	C₂²×D₄₃	C₂×SD₁₆	1	2	yes	—
136	Q₈×C₄₃	—	4	1	no	—
137	Q₈×C₄₃	—	4	1	no	—
138	D₄×C₄₃	—	4	1	no	—
139	D₄×C₄₃	—	4	1	no	—
140	D₄×C₄₃	—	8	1	no	—
141	D₄×C₄₃	—	8	1	no	—
142	C₂×C₄×C₄₃	—	4	1	no	—
143	C₂×C₄×C₄₃	—	8	1	no	—
144	C₂×C₄×C₄₃	—	8	1	no	—
145	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
146	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₄₃	—	8	4	no	—
152	C₂×Dic₄₃	—	2	1	no	—
153	C₂×Dic₄₃	—	2	1	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	C₂×SD₁₆	1	2	yes	—
156	C₄×D₄₃	—	2	2	no	—
157	C₄×D₄₃	—	2	2	no	—
158	C₄×D₄₃	—	2	2	no	—
159	C₄×D₄₃	—	2	2	no	—
160	C₈×C₄₃	—	4	2	no	—
161	C₈×C₄₃	#123	1	1	yes	characteristic
162	C₈×C₄₃	#123	1	1	yes	characteristic
163	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
164	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
165	C₄₃⋊C₈	—	2	2	no	—
166	C₄₃⋊C₈	—	4	2	no	—
167	#33656	—	43	1	no	maximal, Sylow
168	C₂²×C₄₃	—	4	1	no	—
169	C₂²×C₄₃	—	8	2	no	—
170	C₂²×C₄₃	—	8	1	no	—
171	C₂²×C₄₃	#481	1	1	yes	characteristic
172	C₂²×C₄₃	#481	1	1	yes	characteristic
173	C₂²×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
174	C₂×D₄₃	—	2	2	no	—
175	C₂×D₄₃	—	4	2	no	—
176	C₂×D₄₃	—	4	2	no	—
177	C₂×D₄₃	—	8	1	no	—
178	C₄×C₄₃	—	8	1	no	—
179	C₄×C₄₃	#481	1	1	yes	characteristic
180	C₄×C₄₃	#481	1	1	yes	characteristic
181	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
182	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	Dic₄₃	—	2	2	no	—
184	Dic₄₃	—	8	2	no	—
185	#3054	—	43	1	no	—
186	C₂×#430	—	43	1	no	—
187	#2239	—	43	4	no	—
188	#2238	—	43	4	no	—
189	#2155	—	43	1	no	—
190	#2155	—	43	1	no	—
191	#2148	—	43	1	no	—
192	#2148	D₄₃	1	1	yes	characteristic, complemented
193	C₂×#349	—	43	1	no	—
194	C₂×C₄₃	—	2	1	no	—
195	C₂×C₄₃	—	2	1	no	—
196	C₂×C₄₃	—	2	1	no	—
197	C₂×C₄₃	—	16	1	no	—
198	C₂×C₄₃	C₂×#481	1	1	yes	characteristic
199	D₄₃	—	4	2	no	—
200	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
201	C₂²×M₁₆	—	43	1	no	—
202	C₂²×M₁₆	—	86	1	no	—
203	#525	—	86	1	no	—
204	C₂×M₃₂	—	43	1	no	—
205	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
206	#470	—	43	2	no	—
207	#469	—	43	2	no	—
208	#468	—	86	2	no	—
209	#430	—	43	4	no	—
210	#430	—	86	2	no	—
211	C₂×#108	—	43	1	no	—
212	C₂×#108	—	86	1	no	—
213	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
214	#427	—	43	1	no	—
215	#416	—	86	1	no	—
216	#365	—	43	4	no	—
217	#365	—	43	4	no	—
218	#362	—	43	2	no	—
219	#362	C₂×D₄₃	1	2	yes	complemented
220	#361	—	43	2	no	—
221	#361	C₂×D₄₃	1	2	yes	complemented
222	#349	—	43	4	no	—
223	C₄₃	#33656	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	—	2	1	no	—
225	C₂³×C₄	—	43	1	no	—
226	#135	—	2	1	no	—
227	#135	—	2	1	no	—
228	#135	—	2	1	no	—
229	#135	—	2	1	no	—
230	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
231	C₂×SD₁₆	—	2	1	no	—
232	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
233	C₂×M₁₆	—	2	1	no	—
234	C₂×M₁₆	—	43	2	no	—
235	C₂×M₁₆	—	43	2	no	—
236	C₂×M₁₆	—	86	1	no	—
237	C₂×M₁₆	—	86	1	no	—
238	C₂×M₁₆	—	86	1	no	—
239	C₂×M₁₆	—	86	1	no	—
240	C₂×M₁₆	—	86	4	no	—
241	C₂×M₁₆	—	86	1	no	—
242	C₂×M₁₆	—	172	1	no	—
243	C₂×M₁₆	—	172	2	no	—
244	C₂²×C₈	—	43	1	no	—
245	C₂²×C₈	—	172	1	no	—
246	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
247	#123	—	172	1	no	—
248	C₂²⋊Q₈	—	86	2	no	—
249	#121	—	86	2	no	—
250	#120	—	172	1	no	—
251	C₄×D₄	—	172	2	no	—
252	#117	—	86	1	no	—
253	M₃₂	—	2	1	no	—
254	M₃₂	—	43	2	no	—
255	M₃₂	—	86	1	no	—
256	M₃₂	C₂²×D₄₃	1	2	yes	—
257	C₂×C₁₆	—	2	1	no	—
258	C₂×C₁₆	—	86	1	no	—
259	#108	—	2	1	no	—
260	#108	—	43	2	no	—
261	#108	—	86	4	no	—
262	#108	—	86	1	no	—
263	#108	C₂²×D₄₃	1	2	yes	—
264	#107	—	86	1	no	—
265	#106	—	43	2	no	—
266	#103	—	86	2	no	—
267	#102	—	86	2	no	—
268	C₂⁴	—	43	1	no	—
269	#41	—	4	1	no	—
270	#41	—	4	1	no	—
271	#41	—	4	1	no	—
272	#41	—	4	1	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	172	2	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	43	1	no	—
279	C₂²×C₄	—	43	2	no	—
280	C₂²×C₄	—	43	2	no	—
281	C₂²×C₄	—	86	1	no	—
282	C₂²×C₄	—	86	1	no	—
283	C₂²×C₄	—	86	1	no	—
284	C₂²×C₄	—	86	1	no	—
285	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
286	Q₁₆	—	2	1	no	—
287	Q₁₆	—	2	1	no	—
288	SD₁₆	—	4	1	no	—
289	SD₁₆	—	4	1	no	—
290	D₈	—	2	1	no	—
291	D₈	—	2	1	no	—
292	M₁₆	—	2	2	no	—
293	M₁₆	—	4	1	no	—
294	M₁₆	—	86	2	no	—
295	M₁₆	—	86	2	no	—
296	M₁₆	—	86	2	no	—
297	M₁₆	—	86	2	no	—
298	M₁₆	—	86	2	no	—
299	M₁₆	—	172	4	no	—
300	M₁₆	—	172	1	no	—
301	C₂×C₈	—	4	1	no	—
302	C₂×C₈	—	43	2	no	—
303	C₂×C₈	—	86	1	no	—
304	C₂×C₈	—	86	1	no	—
305	C₂×C₈	—	172	4	no	—
306	C₂×C₈	—	172	1	no	—
307	C₂×C₈	#10621898	1	1	yes	characteristic
308	C₂×C₈	#10621898	1	1	yes	characteristic
309	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
310	C₂×C₈	#10621900	1	1	yes	characteristic
311	C₂×C₈	#10621900	1	1	yes	characteristic
312	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
313	C₄⋊C₄	—	86	2	no	—
314	C₄⋊C₄	—	172	2	no	—
315	C₂²⋊C₄	—	172	2	no	—
316	C₂²⋊C₄	—	172	2	no	—
317	C₂²⋊C₄	—	172	1	no	—
318	C₄²	—	172	1	no	—
319	C₁₆	—	2	2	no	—
320	C₁₆	—	86	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	43	2	no	—
323	C₂³	—	86	1	no	—
324	C₂³	—	86	1	no	—
325	C₂³	—	344	1	no	—
326	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
327	Q₈	—	4	1	no	—
328	Q₈	—	4	1	no	—
329	D₄	—	4	1	no	—
330	D₄	—	4	1	no	—
331	D₄	—	8	1	no	—
332	D₄	—	8	1	no	—
333	D₄	—	344	4	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	8	1	no	—
337	C₂×C₄	—	43	2	no	—
338	C₂×C₄	—	86	1	no	—
339	C₂×C₄	—	86	1	no	—
340	C₂×C₄	—	86	2	no	—
341	C₂×C₄	—	86	2	no	—
342	C₂×C₄	—	86	2	no	—
343	C₂×C₄	—	86	2	no	—
344	C₂×C₄	—	172	2	no	—
345	C₂×C₄	#13063307	1	1	yes	characteristic
346	C₂×C₄	#13063307	1	1	yes	characteristic
347	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
348	C₂×C₄	#13063349	1	1	yes	characteristic
349	C₂×C₄	#13063349	1	1	yes	characteristic
350	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
351	C₈	—	4	2	no	—
352	C₈	—	86	2	no	—
353	C₈	—	172	2	no	—
354	C₈	#13063307	1	1	yes	characteristic
355	C₈	#13063307	1	1	yes	characteristic
356	C₈	C₂×#10621900	1	1	yes	characteristic
357	C₈	C₂×#10621898	1	1	yes	characteristic
358	C₂²	—	4	1	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	8	1	no	—
361	C₂²	—	86	2	no	—
362	C₂²	—	344	1	no	—
363	C₂²	—	172	2	no	—
364	C₂²	—	172	2	no	—
365	C₂²	#14583216	1	1	yes	characteristic
366	C₂²	#14583216	1	1	yes	characteristic
367	C₂²	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
368	C₄	—	8	1	no	—
369	C₄	—	86	2	no	—
370	C₄	—	344	2	no	—
371	C₄	#14583216	1	1	yes	characteristic
372	C₄	#14583216	1	1	yes	characteristic
373	C₄	C₂×#13063307	1	1	yes	characteristic
374	C₄	C₂×#13063349	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	16	1	no	—
379	C₂	—	172	2	no	—
380	C₂	C₂×#14583216	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14583216
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2148	C₂
socle	C₂×C₄₃	C₂×#481
prime radical	#19144200	C₂
hypercenter	#2148	D₄₃
hypocenter	C₄₃	#33656
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2148

Complements

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

N	H ≅ G/N	type
2. C₂×#14582946	C₂	semidirect
3. #19136813	C₂	semidirect
4. #19136808	C₂	semidirect
5. #19136369	C₂	semidirect
6. #19136368	C₂	semidirect
7. #19136344	C₂	semidirect
8. C₂×#14582680	C₂	semidirect
9. C₄₃×#2148	C₂	semidirect
23. #14582723	C₂²	semidirect
24. #14582722	C₂²	semidirect
25. #14582715	C₂²	semidirect
26. #14582712	C₂²	semidirect
31. C₄₃×#362	C₂²	semidirect
32. C₄₃×#361	C₂²	semidirect
192. #2148	D₄₃	semidirect
219. #362	C₂×D₄₃	semidirect
221. #361	C₂×D₄₃	semidirect
223. C₄₃	#33656	semidirect

Maximal subgroups

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

subgroup	index
#19144200	2
C₂×#14582946	2
#19136813	2
#19136808	2
#19136369	2
#19136368	2
#19136344	2
C₂×#14582680	2
C₄₃×#2148	2
#33656	43