Subgroups — #25251858

Order 11008.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145212	C₂	1	1	yes	maximal, characteristic, complemented
2	#19143499	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136815	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136810	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136767	C₂	1	2	yes	maximal, complemented
6	#19136665	C₂	1	2	yes	maximal, complemented
7	#19136548	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136543	C₂	1	1	yes	maximal, characteristic
9	#19136495	C₂	1	2	yes	maximal, complemented
10	#19136488	C₂	1	2	yes	maximal, complemented
11	C₄₃×#2166	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583699	C₂²	1	1	yes	characteristic
13	C₂×#13063331	—	2	1	no	—
14	#14583520	—	2	1	no	—
15	#14583342	C₂²	1	2	yes	—
16	#14583337	C₂²	1	2	yes	—
17	#14583297	C₂²	1	1	yes	characteristic, complemented
18	#14583283	C₂²	1	1	yes	characteristic, complemented
19	#14583259	—	2	1	no	—
20	#14583157	C₂²	1	1	yes	characteristic
21	#14583152	C₂²	1	1	yes	characteristic
22	#14583128	—	2	1	no	—
23	#14583022	—	2	1	no	—
24	#14583022	C₂²	1	2	yes	—
25	#14583017	—	2	1	no	—
26	#14583017	C₂²	1	2	yes	—
27	#14583012	C₂²	1	1	yes	characteristic, complemented
28	#14583007	C₂²	1	1	yes	characteristic, complemented
29	C₄×#10621873	—	2	1	no	—
30	#14582948	C₂²	1	1	yes	characteristic
31	#14582917	—	2	1	no	—
32	#14582722	C₂²	1	2	yes	complemented
33	#14582722	C₂²	1	2	yes	complemented
34	#14582716	C₂²	1	2	yes	complemented
35	#14582713	C₂²	1	2	yes	complemented
36	#14582710	C₂²	1	2	yes	—
37	#14582710	C₂²	1	2	yes	—
38	C₄₃×#405	C₂²	1	1	yes	characteristic
39	C₄₃×#364	C₂²	1	2	yes	complemented
40	C₄₃×#349	C₂²	1	1	yes	characteristic
41	C₄₃×#349	C₂²	1	1	yes	characteristic
42	C₄₃×#347	C₂²	1	2	yes	complemented
43	C₂×#10621898	—	2	1	no	—
44	#13063358	—	2	2	no	—
45	#13063353	—	2	2	no	—
46	#13063347	—	2	1	no	—
47	#13063347	C₂³	1	2	yes	—
48	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
49	C₂×#10621873	—	2	1	no	—
50	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
51	#13063331	—	2	2	no	—
52	#13063331	—	2	2	no	—
53	#13063331	—	4	1	no	—
54	#13063331	—	4	1	no	—
55	C₂×#10621877	—	2	1	no	—
56	C₂×#10621872	—	4	1	no	—
57	C₂×C₈×D₄₃	—	4	1	no	—
58	#13063325	—	2	1	no	—
59	#13063320	—	4	1	no	—
60	#13063313	—	2	1	no	—
61	#13063311	—	4	1	no	—
62	#13063307	—	4	1	no	—
63	#13063304	—	2	1	no	—
64	#13063303	—	4	1	no	—
65	#13063299	—	2	1	no	—
66	C₄×Dic₈₆	—	2	1	no	—
67	C₄×D₁₇₂	—	2	1	no	—
68	Dic₃₄₄	—	2	2	no	—
69	#13063270	—	2	2	no	—
70	#13063270	—	2	2	no	—
71	D₆₈₈	—	2	2	no	—
72	#13063261	—	2	1	no	—
73	#13063261	C₂³	1	2	yes	—
74	#13063257	—	2	1	no	—
75	#13063257	C₂³	1	1	yes	characteristic
76	#13063257	C₂³	1	1	yes	characteristic
77	#13063253	—	2	2	no	—
78	#13063253	—	2	2	no	—
79	#13063250	—	2	1	no	—
80	#13063250	—	2	1	no	—
81	#13063247	—	2	1	no	—
82	#13063247	—	2	1	no	—
83	C₂×M₁₆×C₄₃	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
85	#117×C₄₃	D₄	1	1	yes	characteristic
86	M₃₂×C₄₃	C₂³	1	2	yes	—
87	M₃₂×C₄₃	C₂³	1	2	yes	—
88	#97×C₄₃	D₄	1	1	yes	characteristic
89	#97×C₄₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
91	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₄₃	—	4	1	no	—
94	C₂×C₂²⋊D₄₃	—	4	1	no	—
95	#10621898	—	2	2	no	—
96	#10621898	—	2	2	no	—
97	#10621898	—	4	1	no	—
98	#10621898	—	4	1	no	—
99	C₂×D₁₇₂	—	2	1	no	—
100	C₂×D₁₇₂	—	2	1	no	—
101	C₂×C₄×D₄₃	—	4	1	no	—
102	C₂×C₄×D₄₃	—	4	1	no	—
103	C₂×Dic₈₆	—	2	1	no	—
104	C₂×Dic₈₆	—	2	1	no	—
105	C₂×Dic₈₆	—	4	1	no	—
106	M₁₆×C₄₃	—	2	2	no	—
107	C₂×C₈×C₄₃	—	2	1	no	—
108	C₂×C₈×C₄₃	—	2	1	no	—
109	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₄₃	—	2	1	no	—
114	C₂²⋊C₄×C₄₃	—	2	1	no	—
115	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂²⋊Dic₄₃	—	4	1	no	—
118	(C₂×D₄₃)⋊C₄	—	4	1	no	—
119	(C₂×D₄₃)⋊C₄	—	4	1	no	—
120	(C₂×D₄₃)⋊C₄	—	4	1	no	—
121	(C₂×D₄₃)⋊C₄	—	4	1	no	—
122	C₄⋊Dic₄₃	—	2	1	no	—
123	C₄⋊Dic₄₃	—	2	1	no	—
124	C₄⋊Dic₄₃	—	4	1	no	—
125	Dic₄₃⋊C₄	—	4	1	no	—
126	Dic₄₃⋊C₄	—	4	1	no	—
127	C₄×Dic₄₃	—	4	1	no	—
128	#10621877	—	2	2	no	—
129	#10621877	—	4	1	no	—
130	C₂×C₄₃⋊C₈	—	4	1	no	—
131	Dic₁₇₂	—	2	1	no	—
132	Dic₁₇₂	—	2	2	no	—
133	Dic₁₇₂	#41	1	2	yes	—
134	D₃₄₄	—	2	1	no	—
135	D₃₄₄	—	2	2	no	—
136	D₃₄₄	#41	1	2	yes	—
137	#10621873	—	2	2	no	—
138	#10621873	—	2	2	no	—
139	#10621873	—	4	1	no	—
140	#10621873	—	4	1	no	—
141	#10621872	—	4	2	no	—
142	#10621872	—	4	2	no	—
143	C₈×D₄₃	—	4	2	no	—
144	C₈×D₄₃	—	4	2	no	—
145	C₁₆×C₄₃	—	2	2	no	—
146	C₁₆×C₄₃	—	2	2	no	—
147	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂²×D₄₃	—	4	1	no	—
149	C₂²×D₄₃	—	4	1	no	—
150	C₂×C₄×C₄₃	—	2	1	no	—
151	C₂×C₄×C₄₃	—	2	1	no	—
152	C₂×C₄×C₄₃	—	2	1	no	—
153	C₂×C₄×C₄₃	—	2	1	no	—
154	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
156	C₂²⋊D₄₃	—	4	2	no	—
157	C₂²⋊D₄₃	—	4	2	no	—
158	C₂²⋊D₄₃	—	16	1	no	—
159	C₂×Dic₄₃	—	4	1	no	—
160	C₂×Dic₄₃	—	4	1	no	—
161	C₂×Dic₄₃	—	4	1	no	—
162	C₂×Dic₄₃	—	4	1	no	—
163	D₁₇₂	—	2	2	no	—
164	D₁₇₂	—	4	1	no	—
165	D₁₇₂	—	4	1	no	—
166	D₁₇₂	—	4	1	no	—
167	C₄×D₄₃	—	4	2	no	—
168	C₄×D₄₃	—	4	2	no	—
169	C₄×D₄₃	—	16	1	no	—
170	Dic₈₆	—	2	2	no	—
171	Dic₈₆	—	4	1	no	—
172	Dic₈₆	—	4	1	no	—
173	Dic₈₆	—	4	1	no	—
174	Dic₈₆	—	16	1	no	—
175	C₈×C₄₃	—	2	1	no	—
176	C₈×C₄₃	—	2	1	no	—
177	C₈×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₈×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₈×C₄₃	C₂×#41	1	2	yes	—
180	C₄₃⋊C₈	—	4	2	no	—
181	#33691	—	43	1	no	maximal, Sylow
182	C₂²×C₄₃	—	2	1	no	—
183	C₂²×C₄₃	—	4	1	no	—
184	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₂×D₄₃	—	4	2	no	—
186	C₂×D₄₃	—	8	2	no	—
187	C₂×D₄₃	—	8	1	no	—
188	C₂×D₄₃	—	8	1	no	—
189	C₂×D₄₃	—	8	1	no	—
190	C₄×C₄₃	—	2	1	no	—
191	C₄×C₄₃	—	4	1	no	—
192	C₄×C₄₃	—	4	1	no	—
193	C₄×C₄₃	#525	1	1	yes	characteristic
194	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
195	Dic₄₃	—	4	2	no	—
196	Dic₄₃	—	8	1	no	—
197	Dic₄₃	—	8	1	no	—
198	Dic₄₃	—	8	1	no	—
199	#3173	—	43	1	no	—
200	#2955	—	43	1	no	—
201	#2239	—	43	1	no	—
202	#2238	—	43	1	no	—
203	#2230	—	43	2	no	—
204	#2212	—	43	2	no	—
205	#2191	—	43	1	no	—
206	#2190	—	43	1	no	—
207	AGL(1,ℤ/16)	—	43	2	no	—
208	#2180	—	43	2	no	—
209	#2166	D₄₃	1	1	yes	characteristic, complemented
210	C₂×C₄₃	—	2	1	no	—
211	C₂×C₄₃	—	4	1	no	—
212	C₂×C₄₃	#3109	1	1	yes	characteristic
213	D₄₃	—	8	2	no	—
214	D₄₃	—	16	1	no	—
215	#575	—	43	1	no	—
216	C₂×#131	—	86	1	no	—
217	#542	—	86	1	no	—
218	#510	—	43	2	no	—
219	#509	—	43	2	no	—
220	#498	—	43	1	no	—
221	#495	—	43	1	no	—
222	#489	—	86	1	no	—
223	#470	—	43	1	no	—
224	#469	—	43	1	no	—
225	#465	—	86	1	no	—
226	#444	—	43	2	no	—
227	#444	—	86	1	no	—
228	#443	—	43	2	no	—
229	#443	—	86	1	no	—
230	#442	—	43	1	no	—
231	#441	—	43	1	no	—
232	C₄×SD₁₆	—	86	1	no	—
233	#430	—	43	1	no	—
234	#421	—	86	1	no	—
235	#405	C₂×D₄₃	1	1	yes	characteristic
236	#365	—	43	2	no	—
237	#365	—	43	2	no	—
238	#364	C₂×D₄₃	1	2	yes	complemented
239	#362	—	43	2	no	—
240	#361	—	43	2	no	—
241	#360	—	43	2	no	—
242	#360	—	43	2	no	—
243	#349	C₂×D₄₃	1	1	yes	characteristic
244	#349	C₂×D₄₃	1	1	yes	characteristic
245	#347	C₂×D₄₃	1	2	yes	complemented
246	C₄₃	#33691	1	1	yes	characteristic, complemented, Sylow
247	C₂×#41	—	86	1	no	—
248	#137	—	86	2	no	—
249	AGL(1,ℤ/8)	—	86	2	no	—
250	#135	—	43	2	no	—
251	#135	—	86	1	no	—
252	C₂×Q₁₆	—	43	1	no	—
253	C₂×SD₁₆	—	86	1	no	—
254	C₂×D₈	—	43	1	no	—
255	#131	—	86	2	no	—
256	#131	—	86	2	no	—
257	#131	—	172	1	no	—
258	#131	—	172	1	no	—
259	C₂×M₁₆	—	86	1	no	—
260	C₂×M₁₆	—	172	1	no	—
261	C₂×M₁₆	C₂²×D₄₃	1	1	yes	characteristic
262	C₂²×C₈	—	172	1	no	—
263	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
264	C₄⋊Q₈	—	86	1	no	—
265	#126	—	172	1	no	—
266	#124	—	86	1	no	—
267	#124	—	172	1	no	—
268	#123	—	172	1	no	—
269	C₂²⋊Q₈	—	86	1	no	—
270	C₂²⋊Q₈	—	172	1	no	—
271	#121	—	86	1	no	—
272	C₄×Q₈	—	86	1	no	—
273	C₄×D₄	—	86	1	no	—
274	#117	D₁₇₂	1	1	yes	characteristic
275	Q₃₂	—	86	2	no	—
276	SD₃₂	—	86	2	no	—
277	SD₃₂	—	86	2	no	—
278	D₁₆	—	86	2	no	—
279	M₃₂	C₂²×D₄₃	1	2	yes	—
280	M₃₂	C₂²×D₄₃	1	2	yes	—
281	#108	—	43	2	no	—
282	#108	—	86	1	no	—
283	#106	—	43	1	no	—
284	#106	—	43	1	no	—
285	#106	—	86	1	no	—
286	#104	—	86	2	no	—
287	#104	—	86	2	no	—
288	#103	—	86	1	no	—
289	#103	—	86	1	no	—
290	#102	—	86	1	no	—
291	#102	—	86	1	no	—
292	#97	D₁₇₂	1	1	yes	characteristic
293	#97	C₂²×D₄₃	1	1	yes	characteristic
294	C₄×C₈	D₁₇₂	1	1	yes	characteristic
295	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
296	#41	—	86	2	no	—
297	#41	—	86	2	no	—
298	#41	—	172	1	no	—
299	#41	—	172	1	no	—
300	C₂×Q₈	—	86	1	no	—
301	C₂×Q₈	—	86	1	no	—
302	C₂×Q₈	—	172	1	no	—
303	C₂×D₄	—	86	1	no	—
304	C₂×D₄	—	86	1	no	—
305	C₂×D₄	—	172	1	no	—
306	C₂×D₄	—	172	1	no	—
307	C₂²×C₄	—	172	1	no	—
308	C₂²×C₄	—	172	1	no	—
309	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
310	Q₁₆	—	43	2	no	—
311	Q₁₆	—	86	1	no	—
312	Q₁₆	—	86	2	no	—
313	SD₁₆	—	86	2	no	—
314	SD₁₆	—	86	2	no	—
315	SD₁₆	—	172	1	no	—
316	SD₁₆	—	172	1	no	—
317	D₈	—	43	2	no	—
318	D₈	—	86	1	no	—
319	D₈	—	86	2	no	—
320	M₁₆	—	2	2	no	—
321	M₁₆	—	86	2	no	—
322	M₁₆	—	172	2	no	—
323	M₁₆	—	172	1	no	—
324	M₁₆	—	172	2	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	172	2	no	—
328	C₂×C₈	—	172	1	no	—
329	C₂×C₈	—	172	2	no	—
330	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
331	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
332	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
333	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	86	1	no	—
336	C₄⋊C₄	—	86	1	no	—
337	C₄⋊C₄	—	172	1	no	—
338	C₄⋊C₄	—	172	1	no	—
339	C₄⋊C₄	—	172	1	no	—
340	C₂²⋊C₄	—	2	1	no	—
341	C₂²⋊C₄	—	172	1	no	—
342	C₂²⋊C₄	—	172	1	no	—
343	C₂²⋊C₄	—	172	1	no	—
344	C₂²⋊C₄	—	172	1	no	—
345	C₂²⋊C₄	—	172	1	no	—
346	C₄²	—	172	1	no	—
347	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
348	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
349	C₁₆	—	2	2	no	—
350	C₁₆	—	2	2	no	—
351	C₂³	—	172	1	no	—
352	C₂³	—	172	1	no	—
353	C₂³	#13063353	1	1	yes	characteristic
354	Q₈	—	86	2	no	—
355	Q₈	—	172	1	no	—
356	Q₈	—	172	1	no	—
357	Q₈	—	172	1	no	—
358	Q₈	—	688	1	no	—
359	D₄	—	86	2	no	—
360	D₄	—	172	1	no	—
361	D₄	—	172	2	no	—
362	D₄	—	172	2	no	—
363	D₄	—	172	1	no	—
364	D₄	—	172	1	no	—
365	D₄	—	688	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	172	2	no	—
371	C₂×C₄	—	172	1	no	—
372	C₂×C₄	—	172	1	no	—
373	C₂×C₄	—	172	1	no	—
374	C₂×C₄	—	172	1	no	—
375	C₂×C₄	—	172	2	no	—
376	C₂×C₄	—	688	1	no	—
377	C₂×C₄	#13063353	1	1	yes	characteristic
378	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
379	C₈	—	2	1	no	—
380	C₈	—	2	1	no	—
381	C₈	—	172	2	no	—
382	C₈	#13063353	1	1	yes	characteristic
383	C₈	#13063353	1	1	yes	characteristic
384	C₈	#41×D₄₃	1	2	yes	—
385	C₂²	—	2	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	344	1	no	—
388	C₂²	—	344	1	no	—
389	C₂²	—	344	2	no	—
390	C₂²	—	344	1	no	—
391	C₂²	—	172	2	no	—
392	C₂²	C₂×#13063353	1	1	yes	characteristic
393	C₄	—	2	1	no	—
394	C₄	—	4	1	no	—
395	C₄	—	4	1	no	—
396	C₄	—	344	1	no	—
397	C₄	—	344	1	no	—
398	C₄	—	344	1	no	—
399	C₄	—	172	2	no	—
400	C₄	#14583411	1	1	yes	characteristic
401	C₄	C₂×#13063353	1	1	yes	characteristic
402	C₂	—	2	1	no	—
403	C₂	—	4	1	no	—
404	C₂	—	344	2	no	—
405	C₂	—	688	1	no	—
406	C₂	#19144687	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19144687
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2166	C₂
socle	C₂×C₄₃	#3109
prime radical	#19136543	C₂
hypercenter	#2166	D₄₃
hypocenter	C₄₃	#33691
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ #2166

Complements

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

N	H ≅ G/N	type
1. #19145212	C₂	semidirect
2. #19143499	C₂	semidirect
3. #19136815	C₂	semidirect
4. #19136810	C₂	semidirect
5. #19136767	C₂	semidirect
6. #19136665	C₂	semidirect
7. #19136548	C₂	semidirect
9. #19136495	C₂	semidirect
10. #19136488	C₂	semidirect
11. C₄₃×#2166	C₂	semidirect
17. #14583297	C₂²	semidirect
18. #14583283	C₂²	semidirect
27. #14583012	C₂²	semidirect
28. #14583007	C₂²	semidirect
32. #14582722	C₂²	semidirect
33. #14582722	C₂²	semidirect
34. #14582716	C₂²	semidirect
35. #14582713	C₂²	semidirect
39. C₄₃×#364	C₂²	semidirect
42. C₄₃×#347	C₂²	semidirect
209. #2166	D₄₃	semidirect
238. #364	C₂×D₄₃	semidirect
245. #347	C₂×D₄₃	semidirect
246. C₄₃	#33691	semidirect

Maximal subgroups

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

subgroup	index
#19145212	2
#19143499	2
#19136815	2
#19136810	2
#19136767	2
#19136665	2
#19136548	2
#19136543	2
#19136495	2
#19136488	2
C₄₃×#2166	2
#33691	43