Subgroups — #25251934

Order 11008.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145081	C₂	1	1	yes	maximal, characteristic
2	#19136838	C₂	1	2	yes	maximal, complemented
3	#19136834	C₂	1	2	yes	maximal, complemented
4	#19136829	C₂	1	2	yes	maximal, complemented
5	#19136820	C₂	1	2	yes	maximal, complemented
6	#19136310	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136305	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136300	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14582699	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136192	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063353	—	2	1	no	—
13	C₂×#13063347	C₂²	1	1	yes	characteristic
14	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
15	#14583468	—	2	1	no	—
16	C₂×#13063265	C₂²	1	1	yes	characteristic
17	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	#14583288	—	2	2	no	—
19	#14583283	C₂²	1	2	yes	—
20	#14583276	C₂²	1	2	yes	—
21	#14582702	C₂²	1	1	yes	characteristic, complemented
22	#14582702	C₂²	1	1	yes	characteristic, complemented
23	#14582702	C₂²	1	2	yes	complemented
24	#14582699	C₂²	1	2	yes	—
25	#14582699	C₂²	1	2	yes	—
26	#14582699	C₂²	1	2	yes	—
27	#14582699	C₂²	1	2	yes	—
28	#14582699	C₂²	1	1	yes	characteristic, complemented
29	#14582699	C₂²	1	1	yes	characteristic, complemented
30	#14582699	C₂²	1	2	yes	complemented
31	#14582675	C₂²	1	2	yes	complemented
32	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	complemented
33	C₄₃×#544	—	2	1	no	—
34	C₄₃×#501	—	2	2	no	—
35	C₄₃×#500	C₂²	1	2	yes	complemented
36	C₄₃×#499	C₂²	1	2	yes	complemented
37	C₄₃×#428	—	2	1	no	—
38	C₄₃×#427	C₂²	1	1	yes	characteristic
39	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
40	C₄₃×#405	C₂²	1	1	yes	characteristic
41	C₂×#10621898	—	2	1	no	—
42	C₂²×D₁₇₂	—	2	1	no	—
43	#13063353	—	4	4	no	—
44	#13063347	—	2	2	no	—
45	#13063347	—	2	2	no	—
46	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
47	C₂×#10621873	—	2	1	no	—
48	C₂×#10621873	—	2	2	no	—
49	C₂×D₃₄₄	—	2	2	no	—
50	C₂×D₃₄₄	—	2	2	no	—
51	C₂×D₃₄₄	—	2	1	no	—
52	C₂×D₃₄₄	—	2	2	no	—
53	C₂×D₃₄₄	C₂³	1	2	yes	—
54	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
55	#13063322	—	2	2	no	—
56	#13063313	—	2	2	no	—
57	#13063300	—	4	1	no	—
58	#13063300	—	4	1	no	—
59	#13063295	—	4	2	no	—
60	#13063265	—	2	2	no	—
61	C₂×C₄₃⋊C₁₆	—	2	2	no	—
62	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
65	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₄₃	—	2	2	no	—
68	C₄⋊Q₈×C₄₃	—	4	1	no	—
69	#125×C₄₃	—	2	2	no	—
70	#125×C₄₃	—	4	1	no	—
71	C₂²⋊Q₈×C₄₃	—	4	2	no	—
72	#117×C₄₃	—	2	1	no	—
73	#117×C₄₃	D₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
75	#107×C₄₃	—	2	2	no	—
76	#107×C₄₃	—	2	1	no	—
77	#107×C₄₃	C₂³	1	1	yes	characteristic
78	#107×C₄₃	C₂³	1	2	yes	—
79	#107×C₄₃	C₂³	1	1	yes	characteristic
80	#106×C₄₃	—	2	2	no	—
81	#106×C₄₃	—	2	1	no	—
82	#97×C₄₃	D₄	1	2	yes	—
83	C₄×C₈×C₄₃	C₂³	1	2	yes	—
84	C₂³×D₄₃	—	4	1	no	—
85	C₂×Q₈×C₄₃	—	4	2	no	—
86	C₂²×C₄×C₄₃	—	4	1	no	—
87	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₄₃	—	4	1	no	—
89	#10621898	—	4	2	no	—
90	#10621898	—	4	2	no	—
91	C₂×D₁₇₂	—	2	2	no	—
92	C₂×D₁₇₂	—	2	1	no	—
93	C₂×D₁₇₂	—	4	1	no	—
94	C₂×D₁₇₂	—	4	2	no	—
95	C₂×D₁₇₂	—	4	2	no	—
96	C₂×D₁₇₂	—	4	2	no	—
97	C₂×D₁₇₂	—	4	2	no	—
98	C₂×C₄×D₄₃	—	4	1	no	—
99	C₂×Dic₈₆	—	2	1	no	—
100	M₁₆×C₄₃	SD₁₆	1	4	yes	—
101	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₃	#41	1	2	yes	—
104	C₂×C₈×C₄₃	#41	1	2	yes	—
105	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₃	—	2	1	no	—
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	—	2	1	no	—
109	C₄⋊C₄×C₄₃	—	4	1	no	—
110	C₄⋊C₄×C₄₃	—	4	2	no	—
111	C₄⋊C₄×C₄₃	—	4	2	no	—
112	C₄⋊C₄×C₄₃	—	4	2	no	—
113	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
115	C₂²⋊C₄×C₄₃	—	4	1	no	—
116	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
117	C₄²×C₄₃	—	4	1	no	—
118	C₄²×C₄₃	C₂×D₄	1	2	yes	—
119	(C₂×D₄₃)⋊C₄	—	4	2	no	—
120	(C₂×D₄₃)⋊C₄	—	4	2	no	—
121	Dic₁₇₂	—	2	1	no	—
122	Dic₁₇₂	—	2	1	no	—
123	D₃₄₄	—	2	2	no	—
124	D₃₄₄	—	2	2	no	—
125	D₃₄₄	—	2	1	no	—
126	D₃₄₄	—	2	1	no	—
127	D₃₄₄	—	4	2	no	—
128	D₃₄₄	—	4	4	no	—
129	#10621873	—	4	2	no	—
130	#10621873	—	4	4	no	—
131	C₄₃⋊C₁₆	—	2	1	no	—
132	C₄₃⋊C₁₆	—	2	1	no	—
133	C₄₃⋊C₁₆	—	2	2	no	—
134	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₄₃	—	4	2	no	—
136	C₂²×D₄₃	—	4	1	no	—
137	C₂²×D₄₃	—	8	2	no	—
138	C₂²×D₄₃	—	16	1	no	—
139	C₂²×D₄₃	—	16	1	no	—
140	Q₈×C₄₃	—	16	2	no	—
141	C₂×C₄×C₄₃	—	4	1	no	—
142	C₂×C₄×C₄₃	—	4	1	no	—
143	C₂×C₄×C₄₃	—	4	2	no	—
144	C₂×C₄×C₄₃	—	16	1	no	—
145	C₂×C₄×C₄₃	#124	1	2	yes	—
146	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₄₃	—	8	2	no	—
152	C₂×Dic₄₃	—	4	1	no	—
153	D₁₇₂	—	4	1	no	—
154	D₁₇₂	—	4	2	no	—
155	D₁₇₂	—	4	2	no	—
156	D₁₇₂	—	4	1	no	—
157	D₁₇₂	—	8	2	no	—
158	D₁₇₂	—	16	2	no	—
159	D₁₇₂	—	16	2	no	—
160	C₄×D₄₃	—	8	2	no	—
161	Dic₈₆	—	4	1	no	—
162	Dic₈₆	—	4	1	no	—
163	C₈×C₄₃	#124	1	2	yes	—
164	C₈×C₄₃	C₂×SD₁₆	1	4	yes	—
165	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
166	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
167	#33702	—	43	1	no	maximal, Sylow
168	C₂²×C₄₃	—	2	1	no	—
169	C₂²×C₄₃	—	2	1	no	—
170	C₂²×C₄₃	#486	1	2	yes	—
171	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
172	C₂×D₄₃	—	8	2	no	—
173	C₂×D₄₃	—	8	1	no	—
174	C₂×D₄₃	—	8	2	no	—
175	C₂×D₄₃	—	8	1	no	—
176	C₂×D₄₃	—	8	2	no	—
177	C₂×D₄₃	—	8	1	no	—
178	C₂×D₄₃	—	16	4	no	—
179	C₄×C₄₃	—	2	2	no	—
180	C₄×C₄₃	—	2	2	no	—
181	C₄×C₄₃	—	8	2	no	—
182	C₄×C₄₃	—	8	1	no	—
183	C₄×C₄₃	—	8	1	no	—
184	C₄×C₄₃	#486	1	2	yes	—
185	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
186	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
187	Dic₄₃	—	8	1	no	—
188	#3164	D₄₃	1	1	yes	characteristic, complemented
189	#3155	—	43	1	no	—
190	#2244	—	43	2	no	—
191	#2243	—	43	2	no	—
192	#2242	—	43	2	no	—
193	#2240	—	43	2	no	—
194	#2140	—	43	1	no	—
195	#2139	—	43	1	no	—
196	#2138	—	43	1	no	—
197	C₂×#357	—	43	1	no	—
198	#2108	—	43	1	no	—
199	C₂×C₄₃	—	2	2	no	—
200	C₂×C₄₃	C₂×#486	1	1	yes	characteristic
201	C₂×C₄₃	#3415	1	1	yes	characteristic
202	C₂×C₄₃	#3416	1	1	yes	characteristic
203	D₄₃	—	16	1	no	—
204	D₄₃	—	16	2	no	—
205	C₂×AGL(1,ℤ/8)	—	86	1	no	—
206	C₂×#135	—	43	1	no	—
207	C₂²×D₈	—	43	1	no	—
208	#544	—	2	1	no	—
209	#535	—	86	1	no	—
210	C₂×M₃₂	—	43	1	no	—
211	C₂²×C₁₆	—	43	1	no	—
212	#501	—	2	2	no	—
213	#500	C₂×D₄₃	1	2	yes	complemented
214	#499	C₂×D₄₃	1	2	yes	complemented
215	#496	—	86	2	no	—
216	#495	—	43	2	no	—
217	#493	—	43	2	no	—
218	#428	—	2	1	no	—
219	#427	C₂×D₄₃	1	1	yes	characteristic
220	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
221	#405	C₂×D₄₃	1	1	yes	characteristic
222	#358	—	43	1	no	—
223	#358	—	43	1	no	—
224	#358	—	43	2	no	—
225	#357	—	43	2	no	—
226	#357	—	43	2	no	—
227	#357	—	43	1	no	—
228	#357	—	43	1	no	—
229	#357	—	43	2	no	—
230	#357	—	43	2	no	—
231	#357	—	43	2	no	—
232	#346	—	43	2	no	—
233	C₄×C₁₆	—	43	2	no	—
234	C₄₃	#33702	1	1	yes	characteristic, complemented, Sylow
235	C₂×#41	—	86	1	no	—
236	C₂²×D₄	—	86	1	no	—
237	AGL(1,ℤ/8)	—	172	4	no	—
238	#135	—	86	2	no	—
239	#135	—	86	2	no	—
240	C₂×Q₁₆	—	43	1	no	—
241	C₂×SD₁₆	—	86	1	no	—
242	C₂×SD₁₆	—	86	2	no	—
243	C₂×D₈	—	43	2	no	—
244	C₂×D₈	—	43	1	no	—
245	C₂×D₈	—	86	2	no	—
246	C₂×D₈	—	86	2	no	—
247	C₂×D₈	—	86	1	no	—
248	C₂×D₈	—	86	2	no	—
249	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
251	C₄⋊Q₈	—	2	2	no	—
252	C₄⋊Q₈	—	4	1	no	—
253	#127	—	86	2	no	—
254	#125	—	2	2	no	—
255	#125	—	4	1	no	—
256	#124	—	86	2	no	—
257	C₂²⋊Q₈	—	4	2	no	—
258	#121	—	172	1	no	—
259	#121	—	172	1	no	—
260	#120	—	172	2	no	—
261	#117	—	2	1	no	—
262	#117	C₂²⋊D₄₃	1	1	yes	characteristic
263	C₂×C₄⋊C₄	—	2	1	no	—
264	M₃₂	—	86	2	no	—
265	C₂×C₁₆	—	43	1	no	—
266	C₂×C₁₆	—	43	2	no	—
267	C₂×C₁₆	—	43	1	no	—
268	C₂×C₁₆	—	86	2	no	—
269	#107	—	2	1	no	—
270	#107	—	2	2	no	—
271	#107	C₂²×D₄₃	1	2	yes	—
272	#107	C₂²×D₄₃	1	1	yes	characteristic
273	#107	C₂²×D₄₃	1	1	yes	characteristic
274	#106	—	2	1	no	—
275	#106	—	2	2	no	—
276	#97	C₂²⋊D₄₃	1	2	yes	—
277	C₄×C₈	C₂²×D₄₃	1	2	yes	—
278	C₂⁴	—	172	1	no	—
279	#41	—	172	2	no	—
280	#41	—	172	2	no	—
281	C₂×Q₈	—	4	2	no	—
282	C₂×Q₈	—	86	1	no	—
283	C₂×D₄	—	86	2	no	—
284	C₂×D₄	—	86	1	no	—
285	C₂×D₄	—	172	1	no	—
286	C₂×D₄	—	172	2	no	—
287	C₂×D₄	—	172	1	no	—
288	C₂×D₄	—	172	2	no	—
289	C₂×D₄	—	172	2	no	—
290	C₂×D₄	—	172	2	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	172	1	no	—
293	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
294	Q₁₆	—	86	1	no	—
295	Q₁₆	—	86	1	no	—
296	SD₁₆	—	172	2	no	—
297	SD₁₆	—	172	4	no	—
298	D₈	—	86	2	no	—
299	D₈	—	86	2	no	—
300	D₈	—	86	1	no	—
301	D₈	—	86	1	no	—
302	D₈	—	172	2	no	—
303	D₈	—	172	4	no	—
304	M₁₆	#10621884	1	4	yes	—
305	C₂×C₈	#10621902	1	2	yes	—
306	C₂×C₈	#10621902	1	2	yes	—
307	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
308	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
309	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	#10621882	1	1	yes	characteristic
318	C₄⋊C₄	#10621882	1	1	yes	characteristic
319	C₂²⋊C₄	—	4	1	no	—
320	C₂²⋊C₄	—	172	2	no	—
321	C₂²⋊C₄	—	172	2	no	—
322	C₂²⋊C₄	#10621882	1	2	yes	—
323	C₄²	—	4	1	no	—
324	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
325	C₁₆	—	86	1	no	—
326	C₁₆	—	86	1	no	—
327	C₁₆	—	86	2	no	—
328	C₂³	—	344	2	no	—
329	C₂³	—	172	2	no	—
330	C₂³	—	172	1	no	—
331	C₂³	—	688	1	no	—
332	C₂³	—	688	1	no	—
333	C₂³	C₂×#10621882	1	1	yes	characteristic
334	Q₈	—	16	2	no	—
335	Q₈	—	172	1	no	—
336	Q₈	—	172	1	no	—
337	D₄	—	344	2	no	—
338	D₄	—	344	2	no	—
339	D₄	—	172	1	no	—
340	D₄	—	172	2	no	—
341	D₄	—	172	2	no	—
342	D₄	—	172	1	no	—
343	D₄	—	688	2	no	—
344	D₄	—	688	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	16	1	no	—
349	C₂×C₄	—	344	2	no	—
350	C₂×C₄	—	172	1	no	—
351	C₂×C₄	#13063314	1	2	yes	—
352	C₂×C₄	C₂×#10621882	1	2	yes	—
353	C₂×C₄	C₂×#10621882	1	2	yes	—
354	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
355	C₂×C₄	C₂×#10621884	1	2	yes	—
356	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
357	C₈	#13063314	1	2	yes	—
358	C₈	C₂×#10621884	1	4	yes	—
359	C₈	C₂×#10621902	1	1	yes	characteristic
360	C₈	C₂×#10621902	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	344	1	no	—
364	C₂²	—	344	2	no	—
365	C₂²	—	344	1	no	—
366	C₂²	—	344	2	no	—
367	C₂²	—	344	1	no	—
368	C₂²	—	344	2	no	—
369	C₂²	—	688	4	no	—
370	C₂²	#14583247	1	2	yes	—
371	C₂²	C₂²×#10621882	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	8	2	no	—
375	C₄	—	8	1	no	—
376	C₄	—	8	1	no	—
377	C₄	—	344	1	no	—
378	C₄	#14583247	1	2	yes	—
379	C₄	C₂×#13063314	1	1	yes	characteristic
380	C₄	C₂²×#10621884	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	688	2	no	—
383	C₂	—	688	1	no	—
384	C₂	C₂×#14583247	1	1	yes	characteristic
385	C₂	#19147734	1	1	yes	characteristic
386	C₂	#19147739	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3164	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19145081	C₂
hypercenter	#3164	D₄₃
hypocenter	C₄₃	#33702
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #19136838	C₂	semidirect
3. #19136834	C₂	semidirect
4. #19136829	C₂	semidirect
5. #19136820	C₂	semidirect
6. #19136310	C₂	semidirect
7. #19136305	C₂	semidirect
8. #19136300	C₂	semidirect
9. C₂×#14582699	C₂	semidirect
10. #19136192	C₂	semidirect
11. C₄₃×#3164	C₂	semidirect
21. #14582702	C₂²	semidirect
22. #14582702	C₂²	semidirect
23. #14582702	C₂²	semidirect
28. #14582699	C₂²	semidirect
29. #14582699	C₂²	semidirect
30. #14582699	C₂²	semidirect
31. #14582675	C₂²	semidirect
32. C₄×C₄₃⋊C₁₆	C₂²	semidirect
35. C₄₃×#500	C₂²	semidirect
36. C₄₃×#499	C₂²	semidirect
188. #3164	D₄₃	semidirect
213. #500	C₂×D₄₃	semidirect
214. #499	C₂×D₄₃	semidirect
234. C₄₃	#33702	semidirect

Maximal subgroups

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

subgroup	index
#19145081	2
#19136838	2
#19136834	2
#19136829	2
#19136820	2
#19136310	2
#19136305	2
#19136300	2
C₂×#14582699	2
#19136192	2
C₄₃×#3164	2
#33702	43