Subgroups — #25253398

Order 11008.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147744	C₂	1	1	yes	maximal, characteristic, complemented
2	#19147736	C₂	1	1	yes	maximal, characteristic, complemented
3	#19146384	C₂	1	2	yes	maximal, complemented
4	#19136857	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136716	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136667	C₂	1	2	yes	maximal, complemented
7	#19136564	C₂	1	2	yes	maximal, complemented
8	#19136562	C₂	1	2	yes	maximal, complemented
9	#19136475	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136430	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2222	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583712	C₂²	1	2	yes	complemented
13	#14583708	—	2	2	no	—
14	#14583704	C₂²	1	2	yes	—
15	#14583341	—	2	2	no	—
16	#14583340	C₂²	1	2	yes	complemented
17	#14583337	—	2	2	no	—
18	#14583337	C₂²	1	2	yes	—
19	#14583332	—	2	1	no	—
20	#14583331	—	2	1	no	—
21	#14583331	C₂²	1	1	yes	characteristic
22	Q₃₂×D₄₃	—	2	1	no	—
23	C₂×#13063268	C₂²	1	2	yes	complemented
24	C₂×D₆₈₈	—	2	1	no	—
25	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
26	#14583315	C₂²	1	1	yes	characteristic, complemented
27	#14583284	C₂²	1	2	yes	complemented
28	#14583170	C₂²	1	2	yes	complemented
29	#14583165	—	2	2	no	—
30	#14583075	—	2	2	no	—
31	#14583074	—	2	2	no	—
32	#14583015	C₂²	1	2	yes	complemented
33	#14583014	C₂²	1	1	yes	characteristic
34	#14582719	C₂²	1	1	yes	characteristic, complemented
35	#14582711	C₂²	1	2	yes	complemented
36	#14582705	C₂²	1	2	yes	complemented
37	#14582684	C₂²	1	1	yes	characteristic, complemented
38	#14582683	C₂²	1	1	yes	characteristic
39	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
40	C₂×Q₃₂×C₄₃	C₂²	1	1	yes	characteristic, complemented
41	C₄₃×#504	C₂²	1	1	yes	characteristic
42	C₄₃×#473	C₂²	1	2	yes	—
43	C₄₃×#368	C₂²	1	1	yes	characteristic, complemented
44	C₄₃×#362	C₂²	1	2	yes	complemented
45	#13063373	—	2	2	no	—
46	#13063371	—	2	2	no	—
47	#13063357	—	4	2	no	—
48	#13063357	—	4	2	no	—
49	#13063356	—	2	2	no	—
50	#13063353	—	2	2	no	—
51	#13063353	—	2	2	no	—
52	#13063353	—	4	2	no	—
53	#13063352	—	4	2	no	—
54	#13063348	—	2	2	no	—
55	#13063348	—	4	2	no	—
56	#13063347	—	2	2	no	—
57	#13063347	C₂³	1	2	yes	—
58	#13063345	D₄	1	1	yes	characteristic
59	Q₁₆×D₄₃	—	2	2	no	—
60	C₂×#10621884	—	2	2	no	—
61	SD₁₆×D₄₃	—	4	2	no	—
62	C₂×D₃₄₄	—	2	2	no	—
63	C₂×D₃₄₄	C₂³	1	2	yes	—
64	#13063333	—	2	2	no	—
65	#13063331	C₂³	1	1	yes	characteristic
66	#13063314	—	2	2	no	—
67	Dic₃₄₄	D₄	1	1	yes	characteristic
68	#13063271	—	2	2	no	—
69	#13063270	—	2	2	no	—
70	#13063270	D₄	1	2	yes	—
71	#13063268	—	2	2	no	—
72	#13063268	—	4	2	no	—
73	D₆₈₈	—	2	2	no	—
74	D₆₈₈	—	2	1	no	—
75	D₆₈₈	—	2	1	no	—
76	D₆₈₈	D₄	1	1	yes	characteristic
77	#13063265	C₂³	1	1	yes	characteristic
78	#13063264	—	2	1	no	—
79	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
80	C₁₆×D₄₃	—	2	1	no	—
81	#13063260	D₄	1	1	yes	characteristic
82	#13063252	D₄	1	1	yes	characteristic
83	#13063251	—	2	2	no	—
84	#13063251	D₄	1	1	yes	characteristic
85	#13063241	—	2	2	no	—
86	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
87	#137×C₄₃	—	2	2	no	—
88	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
89	#131×C₄₃	C₂³	1	1	yes	characteristic
90	Q₃₂×C₄₃	—	2	1	no	—
91	Q₃₂×C₄₃	—	2	1	no	—
92	M₃₂×C₄₃	—	2	1	no	—
93	M₃₂×C₄₃	C₂³	1	1	yes	characteristic
94	C₂×C₁₆×C₄₃	—	2	1	no	—
95	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
96	#108×C₄₃	C₂³	1	2	yes	—
97	#101×C₄₃	—	2	2	no	—
98	#41×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Q₈×C₄₃	—	2	2	no	—
100	#10621902	—	4	2	no	—
101	#10621902	—	4	2	no	—
102	#10621902	—	4	2	no	—
103	Q₈×D₄₃	—	4	2	no	—
104	Q₈×D₄₃	—	4	2	no	—
105	D₄×D₄₃	—	4	2	no	—
106	D₄×D₄₃	—	8	2	no	—
107	#10621898	—	2	2	no	—
108	#10621898	—	4	2	no	—
109	#10621898	—	8	2	no	—
110	#10621898	C₂×D₄	1	1	yes	characteristic
111	C₂×D₁₇₂	—	2	2	no	—
112	C₂×D₁₇₂	—	4	2	no	—
113	Q₁₆×C₄₃	—	2	2	no	—
114	Q₁₆×C₄₃	—	2	2	no	—
115	Q₁₆×C₄₃	—	4	2	no	—
116	SD₁₆×C₄₃	—	4	2	no	—
117	M₁₆×C₄₃	—	2	1	no	—
118	M₁₆×C₄₃	—	2	2	no	—
119	M₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₄₃	—	2	1	no	—
121	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
122	#10621885	—	4	2	no	—
123	#10621885	D₈	1	1	yes	characteristic
124	#10621884	—	4	2	no	—
125	#10621884	—	4	2	no	—
126	#10621884	—	4	2	no	—
127	#10621884	D₈	1	1	yes	characteristic
128	#10621883	D₈	1	1	yes	characteristic
129	#10621882	D₈	1	1	yes	characteristic
130	(C₂×D₄₃)⋊C₄	—	8	2	no	—
131	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
132	#10621877	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
134	Dic₁₇₂	C₂×D₄	1	2	yes	—
135	D₃₄₄	—	2	2	no	—
136	D₃₄₄	—	2	2	no	—
137	D₃₄₄	—	4	2	no	—
138	D₃₄₄	—	4	2	no	—
139	D₃₄₄	C₂×D₄	1	2	yes	—
140	#10621873	—	2	2	no	—
141	#10621873	—	4	2	no	—
142	#10621873	—	4	2	no	—
143	#10621872	—	2	1	no	—
144	#10621872	—	4	2	no	—
145	C₈×D₄₃	—	2	1	no	—
146	C₈×D₄₃	—	4	2	no	—
147	C₁₆×C₄₃	—	2	1	no	—
148	C₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
149	C₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
150	C₄₃⋊C₁₆	—	2	1	no	—
151	C₄₃⋊C₁₆	—	2	1	no	—
152	C₂²×D₄₃	—	4	2	no	—
153	C₂²×D₄₃	—	8	2	no	—
154	Q₈×C₄₃	—	4	2	no	—
155	Q₈×C₄₃	—	4	2	no	—
156	Q₈×C₄₃	C₂×D₈	1	1	yes	characteristic
157	D₄×C₄₃	—	2	1	no	—
158	D₄×C₄₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₄₃	—	2	1	no	—
160	C₂×C₄×C₄₃	—	4	2	no	—
161	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
162	C₂²⋊D₄₃	—	2	1	no	—
163	C₂²⋊D₄₃	—	4	2	no	—
164	C₂²⋊D₄₃	—	8	2	no	—
165	C₂×Dic₄₃	—	2	1	no	—
166	D₁₇₂	—	2	2	no	—
167	D₁₇₂	—	4	2	no	—
168	D₁₇₂	—	4	2	no	—
169	D₁₇₂	—	8	2	no	—
170	D₁₇₂	—	8	2	no	—
171	D₁₇₂	C₂×D₈	1	1	yes	characteristic
172	C₄×D₄₃	—	2	1	no	—
173	C₄×D₄₃	—	4	2	no	—
174	C₄×D₄₃	—	8	2	no	—
175	C₄×D₄₃	—	8	2	no	—
176	C₄×D₄₃	—	8	2	no	—
177	Dic₈₆	—	2	2	no	—
178	Dic₈₆	—	8	2	no	—
179	Dic₈₆	C₂×D₈	1	1	yes	characteristic
180	C₈×C₄₃	—	2	1	no	—
181	C₈×C₄₃	—	4	2	no	—
182	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
183	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
184	C₄₃⋊C₈	—	2	1	no	—
185	C₄₃⋊C₈	C₂×D₈	1	1	yes	characteristic
186	C₄₃⋊C₈	C₂×D₈	1	1	yes	characteristic
187	#33859	—	43	1	no	maximal, Sylow
188	C₂²×C₄₃	—	2	1	no	—
189	C₂²×C₄₃	#576	1	1	yes	characteristic
190	C₂×D₄₃	—	2	1	no	—
191	C₂×D₄₃	—	4	2	no	—
192	C₂×D₄₃	—	8	2	no	—
193	C₂×D₄₃	—	16	2	no	—
194	C₂×D₄₃	—	32	2	no	—
195	C₄×C₄₃	—	2	1	no	—
196	C₄×C₄₃	—	8	2	no	—
197	C₄×C₄₃	D₄²	1	1	yes	characteristic
198	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
199	Dic₄₃	—	2	1	no	—
200	Dic₄₃	—	4	1	no	—
201	Dic₄₃	—	4	2	no	—
202	#3417	—	43	1	no	—
203	#3415	—	43	1	no	—
204	#3290	—	43	2	no	—
205	#2249	—	43	1	no	—
206	#2222	D₄₃	1	1	yes	characteristic, complemented
207	#2221	—	43	1	no	—
208	#2212	—	43	2	no	—
209	#2194	—	43	2	no	—
210	#2193	—	43	2	no	—
211	#2178	—	43	1	no	—
212	#2170	—	43	1	no	—
213	C₂×C₄₃	—	2	1	no	—
214	C₂×C₄₃	—	4	1	no	—
215	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
216	D₄₃	—	4	1	no	—
217	D₄₃	—	8	2	no	—
218	D₄₃	—	16	2	no	—
219	#578	—	43	2	no	—
220	#577	—	86	2	no	—
221	#576	—	43	2	no	—
222	#510	—	43	2	no	—
223	#510	—	86	2	no	—
224	#509	—	43	2	no	—
225	#509	—	86	2	no	—
226	#508	—	43	1	no	—
227	#508	—	86	1	no	—
228	#508	—	86	1	no	—
229	C₂×Q₃₂	—	86	1	no	—
230	C₂×Q₃₂	C₂×D₄₃	1	1	yes	characteristic, complemented
231	C₂×SD₃₂	—	43	2	no	—
232	C₂×D₁₆	—	43	1	no	—
233	C₂×D₁₆	—	86	1	no	—
234	#504	—	43	1	no	—
235	#504	C₂×D₄₃	1	1	yes	characteristic
236	#495	—	43	2	no	—
237	#473	C₂×D₄₃	1	2	yes	—
238	#472	—	43	2	no	—
239	#471	—	86	2	no	—
240	#455	—	86	2	no	—
241	#454	—	86	2	no	—
242	#443	—	43	1	no	—
243	#443	—	43	2	no	—
244	#368	C₂×D₄₃	1	1	yes	characteristic, complemented
245	#364	—	43	1	no	—
246	#362	C₂×D₄₃	1	2	yes	complemented
247	#361	—	43	2	no	—
248	#359	—	43	2	no	—
249	#350	—	43	1	no	—
250	#350	—	43	1	no	—
251	C₄×C₁₆	—	43	1	no	—
252	C₄₃	#33859	1	1	yes	characteristic, complemented, Sylow
253	2₋¹⁺⁴	—	86	2	no	—
254	H(5,𝔽₂)	—	86	2	no	—
255	#137	—	2	2	no	—
256	#137	—	86	2	no	—
257	#137	—	172	2	no	—
258	#137	—	172	2	no	—
259	AGL(1,ℤ/8)	—	86	2	no	—
260	AGL(1,ℤ/8)	—	86	2	no	—
261	AGL(1,ℤ/8)	—	172	2	no	—
262	AGL(1,ℤ/8)	—	172	2	no	—
263	#135	—	43	1	no	—
264	#135	—	43	2	no	—
265	#135	—	86	2	no	—
266	#135	—	86	2	no	—
267	#135	—	172	2	no	—
268	C₂×Q₁₆	—	86	2	no	—
269	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
270	C₂×SD₁₆	—	86	2	no	—
271	C₂×SD₁₆	—	172	2	no	—
272	C₂×D₈	—	43	2	no	—
273	C₂×D₈	—	86	2	no	—
274	#131	—	43	1	no	—
275	#131	—	86	2	no	—
276	#131	C₂²×D₄₃	1	1	yes	characteristic
277	#124	—	86	2	no	—
278	Q₃₂	—	2	1	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	43	1	no	—
281	Q₃₂	—	86	2	no	—
282	SD₃₂	—	43	2	no	—
283	SD₃₂	—	86	2	no	—
284	SD₃₂	—	86	2	no	—
285	SD₃₂	—	172	2	no	—
286	D₁₆	—	43	1	no	—
287	D₁₆	—	86	2	no	—
288	D₁₆	—	86	1	no	—
289	D₁₆	—	86	1	no	—
290	M₃₂	—	2	1	no	—
291	M₃₂	—	43	1	no	—
292	M₃₂	—	86	1	no	—
293	M₃₂	C₂²×D₄₃	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	43	1	no	—
296	C₂×C₁₆	—	86	1	no	—
297	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
298	#108	—	43	1	no	—
299	#108	C₂²×D₄₃	1	2	yes	—
300	#104	—	43	1	no	—
301	#104	—	43	1	no	—
302	#104	—	86	2	no	—
303	#101	—	2	2	no	—
304	#100	—	86	2	no	—
305	C₄×C₈	—	43	1	no	—
306	#41	—	43	1	no	—
307	#41	—	86	2	no	—
308	#41	—	344	2	no	—
309	#41	—	172	2	no	—
310	#41	—	172	2	no	—
311	#41	—	172	2	no	—
312	#41	—	172	2	no	—
313	#41	D₄×D₄₃	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	172	2	no	—
316	C₂×Q₈	—	172	2	no	—
317	C₂×D₄	—	86	2	no	—
318	C₂×D₄	—	344	2	no	—
319	C₂×D₄	—	172	2	no	—
320	C₂×D₄	—	172	2	no	—
321	Q₁₆	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	43	1	no	—
325	Q₁₆	—	43	2	no	—
326	Q₁₆	—	172	2	no	—
327	SD₁₆	—	4	2	no	—
328	SD₁₆	—	43	1	no	—
329	SD₁₆	—	43	1	no	—
330	SD₁₆	—	86	2	no	—
331	SD₁₆	—	172	2	no	—
332	SD₁₆	—	172	2	no	—
333	SD₁₆	—	172	2	no	—
334	SD₁₆	—	172	2	no	—
335	SD₁₆	—	172	2	no	—
336	D₈	—	43	1	no	—
337	D₈	—	43	2	no	—
338	D₈	—	86	2	no	—
339	D₈	—	86	2	no	—
340	D₈	—	172	2	no	—
341	D₈	—	172	2	no	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	43	1	no	—
345	M₁₆	—	86	1	no	—
346	M₁₆	—	172	2	no	—
347	M₁₆	D₄×D₄₃	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	43	1	no	—
350	C₂×C₈	—	86	1	no	—
351	C₂×C₈	—	172	2	no	—
352	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
353	C₂²⋊C₄	—	344	2	no	—
354	C₄²	—	43	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	86	1	no	—
357	C₁₆	—	86	1	no	—
358	C₁₆	D₄×D₄₃	1	1	yes	characteristic
359	C₁₆	D₄×D₄₃	1	1	yes	characteristic
360	C₂³	—	344	2	no	—
361	C₂³	—	172	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	43	1	no	—
365	Q₈	—	86	2	no	—
366	Q₈	—	344	2	no	—
367	Q₈	D₈×D₄₃	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	43	1	no	—
370	D₄	—	86	1	no	—
371	D₄	—	86	2	no	—
372	D₄	—	344	2	no	—
373	D₄	—	344	2	no	—
374	D₄	—	344	2	no	—
375	D₄	—	172	2	no	—
376	D₄	—	172	2	no	—
377	D₄	—	172	2	no	—
378	D₄	D₈×D₄₃	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	86	1	no	—
382	C₂×C₄	—	86	1	no	—
383	C₂×C₄	—	344	2	no	—
384	C₂×C₄	—	344	2	no	—
385	C₂×C₄	—	344	2	no	—
386	C₂×C₄	—	172	2	no	—
387	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	4	2	no	—
390	C₈	—	43	1	no	—
391	C₈	—	43	1	no	—
392	C₈	—	86	1	no	—
393	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
394	C₈	#13063371	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	86	1	no	—
397	C₂²	—	344	2	no	—
398	C₂²	—	1376	2	no	—
399	C₂²	—	172	2	no	—
400	C₂²	—	688	2	no	—
401	C₂²	#14583701	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	8	2	no	—
404	C₄	—	86	1	no	—
405	C₄	—	172	1	no	—
406	C₄	—	172	2	no	—
407	C₄	#14583533	1	1	yes	characteristic
408	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	344	2	no	—
412	C₂	—	172	1	no	—
413	C₂	—	688	2	no	—
414	C₂	#19146269	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19146269
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2222	C₂
socle	C₂×C₄₃	D₄×D₈
hypercenter	#2222	D₄₃
hypocenter	C₄₃	#33859
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2222

Complements

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

N	H ≅ G/N	type
1. #19147744	C₂	semidirect
2. #19147736	C₂	semidirect
3. #19146384	C₂	semidirect
4. #19136857	C₂	semidirect
5. #19136716	C₂	semidirect
6. #19136667	C₂	semidirect
7. #19136564	C₂	semidirect
8. #19136562	C₂	semidirect
9. #19136475	C₂	semidirect
10. #19136430	C₂	semidirect
11. C₄₃×#2222	C₂	semidirect
12. #14583712	C₂²	semidirect
16. #14583340	C₂²	semidirect
23. C₂×#13063268	C₂²	semidirect
26. #14583315	C₂²	semidirect
27. #14583284	C₂²	semidirect
28. #14583170	C₂²	semidirect
32. #14583015	C₂²	semidirect
34. #14582719	C₂²	semidirect
35. #14582711	C₂²	semidirect
36. #14582705	C₂²	semidirect
37. #14582684	C₂²	semidirect
39. C₁₆×Dic₄₃	C₂²	semidirect
40. C₂×Q₃₂×C₄₃	C₂²	semidirect
43. C₄₃×#368	C₂²	semidirect
44. C₄₃×#362	C₂²	semidirect
206. #2222	D₄₃	semidirect
230. C₂×Q₃₂	C₂×D₄₃	semidirect
244. #368	C₂×D₄₃	semidirect
246. #362	C₂×D₄₃	semidirect
252. C₄₃	#33859	semidirect

Maximal subgroups

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

subgroup	index
#19147744	2
#19147736	2
#19146384	2
#19136857	2
#19136716	2
#19136667	2
#19136564	2
#19136562	2
#19136475	2
#19136430	2
C₄₃×#2222	2
#33859	43