Subgroups — #25088114

Order 11008.

672 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19146943	C₂	1	1	yes	maximal, characteristic, complemented
2	#19146902	C₂	1	2	yes	maximal, complemented
3	#19146317	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134192	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134143	C₂	1	2	yes	maximal, complemented
6	#19133984	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133960	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133882	C₂	1	2	yes	maximal, complemented
9	#19133881	C₂	1	2	yes	maximal, complemented
10	#19133481	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	—	2	1	no	—
13	#14583554	C₂²	1	2	yes	—
14	#14583553	C₂²	1	2	yes	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583297	C₂²	1	2	yes	—
17	#14583279	C₂²	1	1	yes	characteristic
18	#14583250	C₂²	1	2	yes	—
19	#14583236	—	2	2	no	—
20	#14583213	—	2	2	no	—
21	#14583180	C₂²	1	2	yes	—
22	#14583179	C₂²	1	2	yes	—
23	#14583144	—	2	2	no	—
24	#14583140	—	2	1	no	—
25	#14583135	—	2	1	no	—
26	#14583105	C₂²	1	2	yes	—
27	#14583104	C₂²	1	2	yes	—
28	#14583063	—	2	2	no	—
29	#14583041	—	2	2	no	—
30	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
31	#14582978	C₂²	1	2	yes	—
32	C₂×#13063259	—	2	1	no	—
33	#14582897	—	2	2	no	—
34	#14582652	C₂²	1	1	yes	characteristic
35	#14582646	C₂²	1	2	yes	—
36	#14582644	C₂²	1	2	yes	—
37	#14582628	C₂²	1	2	yes	—
38	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
41	C₄₃×#335	C₂²	1	1	yes	characteristic
42	C₄₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₄₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₄₃	C₂²	1	1	yes	characteristic
45	C₂×#10621898	—	2	2	no	—
46	C₂×#10621900	—	2	2	no	—
47	#13063347	—	4	4	no	—
48	C₂×Dic₁₇₂	—	2	1	no	—
49	C₂×Dic₁₇₂	—	2	1	no	—
50	C₂×#10621873	—	2	2	no	—
51	C₂×#10621883	—	2	2	no	—
52	C₂×D₃₄₄	D₄	1	1	yes	characteristic
53	C₂×#10621872	—	2	2	no	—
54	#13063325	C₂³	1	2	yes	—
55	#13063325	C₂³	1	2	yes	—
56	#13063309	—	4	2	no	—
57	#13063307	—	4	2	no	—
58	#13063305	—	2	2	no	—
59	#13063304	—	2	2	no	—
60	#13063299	—	2	2	no	—
61	#13063296	—	2	2	no	—
62	C₄×D₁₇₂	C₂³	1	2	yes	—
63	C₂×C₄⋊Dic₄₃	—	2	2	no	—
64	C₄⋊C₄×D₄₃	—	2	2	no	—
65	#13063259	—	2	1	no	—
66	#13063259	—	2	1	no	—
67	#13063259	—	2	2	no	—
68	#13063259	D₄	1	1	yes	characteristic
69	#13063258	—	2	2	no	—
70	#13063257	—	4	2	no	—
71	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
72	#13063250	—	2	2	no	—
73	#13063250	—	2	2	no	—
74	#13063250	—	2	2	no	—
75	#13063248	—	2	2	no	—
76	#13063247	—	2	2	no	—
77	#13063247	D₄	1	2	yes	—
78	#13063246	—	2	2	no	—
79	(C₂×D₄₃)⋊C₈	—	2	2	no	—
80	C₂×D₈×C₄₃	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₄₃	—	2	1	no	—
82	C₂²×C₈×C₄₃	—	2	1	no	—
83	C₄×D₄×C₄₃	C₂³	1	1	yes	characteristic
84	C₄×D₄×C₄₃	C₂³	1	1	yes	characteristic
85	#107×C₄₃	D₄	1	1	yes	characteristic
86	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
88	#102×C₄₃	D₄	1	1	yes	characteristic
89	#102×C₄₃	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₄₃	—	2	1	no	—
91	C₂²⋊C₈×C₄₃	—	2	1	no	—
92	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×D₄×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₃	—	2	1	no	—
98	C₂²×C₄×C₄₃	—	2	1	no	—
99	C₂×C₂²⋊D₄₃	—	4	2	no	—
100	C₂×C₂²⋊D₄₃	—	4	2	no	—
101	C₂²×Dic₄₃	—	4	2	no	—
102	#10621900	—	4	4	no	—
103	#10621900	—	8	2	no	—
104	#10621898	—	4	4	no	—
105	#10621898	—	8	2	no	—
106	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
107	C₂×C₄×D₄₃	—	2	2	no	—
108	C₂×C₄×D₄₃	—	4	2	no	—
109	C₂×Dic₈₆	—	2	2	no	—
110	C₂×Dic₈₆	—	2	2	no	—
111	D₈×C₄₃	—	2	2	no	—
112	C₂×C₈×C₄₃	—	2	1	no	—
113	C₂×C₈×C₄₃	—	2	1	no	—
114	C₂×C₈×C₄₃	—	2	1	no	—
115	C₂×C₈×C₄₃	—	4	2	no	—
116	C₂×C₈×C₄₃	—	4	2	no	—
117	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂²⋊C₄×C₄₃	—	2	1	no	—
124	C₂²⋊C₄×C₄₃	—	2	1	no	—
125	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
126	C₂²⋊Dic₄₃	—	4	2	no	—
127	#10621883	—	8	2	no	—
128	(C₂×D₄₃)⋊C₄	—	2	2	no	—
129	(C₂×D₄₃)⋊C₄	—	4	2	no	—
130	C₄⋊Dic₄₃	—	2	2	no	—
131	C₄⋊Dic₄₃	—	2	2	no	—
132	C₄⋊Dic₄₃	—	2	2	no	—
133	C₄⋊Dic₄₃	—	2	2	no	—
134	C₄⋊Dic₄₃	—	4	2	no	—
135	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
136	Dic₄₃⋊C₄	—	4	2	no	—
137	C₂×C₄₃⋊C₈	—	2	2	no	—
138	Dic₁₇₂	—	4	2	no	—
139	Dic₁₇₂	—	8	1	no	—
140	D₃₄₄	—	2	2	no	—
141	#10621873	—	4	4	no	—
142	#10621872	—	8	2	no	—
143	C₂³×C₄₃	—	2	1	no	—
144	C₂³×C₄₃	—	2	1	no	—
145	C₂²×D₄₃	—	2	2	no	—
146	D₄×C₄₃	—	2	1	no	—
147	D₄×C₄₃	—	2	1	no	—
148	D₄×C₄₃	—	2	1	no	—
149	D₄×C₄₃	#135	1	2	yes	complemented
150	C₂×C₄×C₄₃	—	2	1	no	—
151	C₂×C₄×C₄₃	—	2	1	no	—
152	C₂×C₄×C₄₃	—	4	1	no	—
153	C₂×C₄×C₄₃	—	4	1	no	—
154	C₂×C₄×C₄₃	—	4	1	no	—
155	C₂×C₄×C₄₃	—	4	1	no	—
156	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂²⋊D₄₃	—	8	4	no	—
160	C₂²⋊D₄₃	—	8	4	no	—
161	C₂×Dic₄₃	—	2	2	no	—
162	C₂×Dic₄₃	—	4	2	no	—
163	C₂×Dic₄₃	—	4	2	no	—
164	C₂×Dic₄₃	—	8	4	no	—
165	D₁₇₂	—	2	2	no	—
166	D₁₇₂	—	2	2	no	—
167	C₄×D₄₃	—	4	2	no	—
168	C₄×D₄₃	—	4	2	no	—
169	C₄×D₄₃	—	8	4	no	—
170	Dic₈₆	—	4	2	no	—
171	Dic₈₆	—	4	2	no	—
172	Dic₈₆	—	4	2	no	—
173	Dic₈₆	—	4	2	no	—
174	C₈×C₄₃	—	2	2	no	—
175	C₈×C₄₃	—	2	1	no	—
176	C₈×C₄₃	—	2	1	no	—
177	C₈×C₄₃	—	2	1	no	—
178	C₈×C₄₃	—	4	1	no	—
179	C₈×C₄₃	—	4	1	no	—
180	C₈×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
181	C₄₃⋊C₈	—	4	2	no	—
182	#19902	—	43	1	no	maximal, Sylow
183	C₂²×C₄₃	—	2	2	no	—
184	C₂²×C₄₃	—	4	1	no	—
185	C₂²×C₄₃	—	4	1	no	—
186	C₂²×C₄₃	—	4	1	no	—
187	C₂²×C₄₃	—	4	1	no	—
188	C₂²×C₄₃	—	4	1	no	—
189	C₂²×C₄₃	D₄²	1	1	yes	characteristic
190	C₂×D₄₃	—	4	2	no	—
191	C₂×D₄₃	—	4	2	no	—
192	C₂×D₄₃	—	4	2	no	—
193	C₄×C₄₃	—	2	1	no	—
194	C₄×C₄₃	—	4	1	no	—
195	C₄×C₄₃	—	4	1	no	—
196	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
197	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₄×C₄₃	#576	1	1	yes	characteristic
199	C₄×C₄₃	#577	1	1	yes	characteristic
200	Dic₄₃	—	4	2	no	—
201	Dic₄₃	—	8	2	no	—
202	Dic₄₃	—	8	2	no	—
203	#3334	—	43	1	no	—
204	#3330	—	43	2	no	—
205	#3283	—	43	1	no	—
206	#1707	—	43	1	no	—
207	#1699	—	43	2	no	—
208	#1673	—	43	1	no	—
209	#1669	—	43	1	no	—
210	#1654	—	43	2	no	—
211	#1654	—	43	2	no	—
212	#1589	—	43	1	no	—
213	C₈×D₈	D₄₃	1	1	yes	characteristic, complemented
214	C₂×C₄₃	—	4	2	no	—
215	C₂×C₄₃	—	8	1	no	—
216	C₂×C₄₃	#3272	1	1	yes	characteristic
217	C₂×C₄₃	#3280	1	1	yes	characteristic
218	C₂×C₄₃	#3293	1	1	yes	characteristic
219	D₄₃	—	8	2	no	—
220	C₂×#135	—	86	1	no	—
221	#547	—	43	2	no	—
222	#547	—	43	2	no	—
223	#500	—	43	1	no	—
224	#498	—	43	2	no	—
225	#494	—	43	1	no	—
226	#487	—	43	2	no	—
227	#484	—	86	2	no	—
228	#480	—	86	2	no	—
229	#474	—	43	2	no	—
230	#474	—	43	2	no	—
231	#468	—	86	2	no	—
232	#467	—	86	1	no	—
233	#466	—	86	1	no	—
234	#461	—	43	2	no	—
235	#461	—	43	2	no	—
236	#452	—	86	2	no	—
237	#449	—	86	2	no	—
238	C₄×D₈	—	43	1	no	—
239	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
240	#436	—	43	2	no	—
241	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
242	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
243	C₂×#107	—	86	1	no	—
244	#417	—	86	2	no	—
245	#335	—	43	1	no	—
246	#335	C₂×D₄₃	1	1	yes	characteristic
247	#332	—	43	2	no	—
248	#332	—	43	2	no	—
249	D₄⋊C₈	—	43	2	no	—
250	D₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
251	D₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
252	C₈²	C₂×D₄₃	1	1	yes	characteristic
253	C₄₃	#19902	1	1	yes	characteristic, complemented, Sylow
254	C₂×#41	—	86	2	no	—
255	C₂×#41	—	86	2	no	—
256	#135	—	172	4	no	—
257	C₂×Q₁₆	—	86	1	no	—
258	C₂×Q₁₆	—	86	1	no	—
259	C₂×SD₁₆	—	86	2	no	—
260	C₂×SD₁₆	—	86	2	no	—
261	C₂×D₈	—	43	1	no	—
262	C₂×D₈	D₁₇₂	1	1	yes	characteristic
263	C₂×M₁₆	—	86	2	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₄⋊Q₈	—	43	2	no	—
267	C₄⋊Q₈	—	43	2	no	—
268	#123	—	172	2	no	—
269	#123	—	172	2	no	—
270	C₂²⋊Q₈	—	86	2	no	—
271	C₂²⋊Q₈	—	86	2	no	—
272	#121	—	86	2	no	—
273	#121	—	86	2	no	—
274	C₄×D₄	—	43	2	no	—
275	C₄×D₄	C₂²×D₄₃	1	1	yes	characteristic
276	C₄×D₄	C₂²×D₄₃	1	1	yes	characteristic
277	C₂×C₄⋊C₄	—	86	2	no	—
278	C₂×C₄⋊C₄	—	86	2	no	—
279	#107	—	43	1	no	—
280	#107	—	86	2	no	—
281	#107	—	86	1	no	—
282	#107	—	86	1	no	—
283	#107	—	86	2	no	—
284	#107	D₁₇₂	1	1	yes	characteristic
285	#106	—	172	2	no	—
286	C₄⋊C₈	—	43	2	no	—
287	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
288	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
289	#103	—	86	2	no	—
290	#103	—	86	2	no	—
291	#103	—	86	2	no	—
292	#103	—	86	2	no	—
293	#102	—	43	2	no	—
294	#102	—	86	2	no	—
295	#102	—	86	2	no	—
296	#102	D₁₇₂	1	1	yes	characteristic
297	#102	D₁₇₂	1	1	yes	characteristic
298	C₂²⋊C₈	—	2	1	no	—
299	C₂²⋊C₈	—	2	1	no	—
300	C₂²⋊C₈	—	86	2	no	—
301	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
302	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
303	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
304	#41	—	344	2	no	—
305	#41	—	344	2	no	—
306	#41	—	172	4	no	—
307	#41	—	172	4	no	—
308	C₂×Q₈	—	86	2	no	—
309	C₂×Q₈	—	86	2	no	—
310	C₂×D₄	—	43	2	no	—
311	C₂×D₄	—	172	2	no	—
312	C₂×D₄	—	172	2	no	—
313	C₂×D₄	C₂×D₁₇₂	1	1	yes	characteristic
314	C₂×D₄	C₂×D₁₇₂	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	86	2	no	—
318	C₂²×C₄	—	172	2	no	—
319	C₂²×C₄	—	172	2	no	—
320	Q₁₆	—	344	1	no	—
321	Q₁₆	—	172	2	no	—
322	SD₁₆	—	344	2	no	—
323	SD₁₆	—	172	4	no	—
324	D₈	—	2	2	no	—
325	D₈	—	86	2	no	—
326	M₁₆	—	344	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	4	2	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	—	86	2	no	—
333	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
334	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
335	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
336	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
337	C₄⋊C₄	—	43	2	no	—
338	C₄⋊C₄	—	86	2	no	—
339	C₄⋊C₄	—	86	2	no	—
340	C₄⋊C₄	—	86	2	no	—
341	C₄⋊C₄	—	86	2	no	—
342	C₄⋊C₄	—	172	2	no	—
343	C₄⋊C₄	—	172	2	no	—
344	C₄⋊C₄	C₂×D₁₇₂	1	1	yes	characteristic
345	C₄⋊C₄	C₂×D₁₇₂	1	1	yes	characteristic
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	86	2	no	—
349	C₂²⋊C₄	—	172	2	no	—
350	C₂²⋊C₄	—	172	2	no	—
351	C₄²	C₂³×D₄₃	1	1	yes	characteristic
352	C₂³	—	2	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	86	2	no	—
355	Q₈	—	172	2	no	—
356	Q₈	—	172	2	no	—
357	Q₈	—	172	2	no	—
358	Q₈	—	172	2	no	—
359	D₄	—	2	1	no	—
360	D₄	—	2	1	no	—
361	D₄	—	2	1	no	—
362	D₄	—	86	2	no	—
363	D₄	—	86	2	no	—
364	D₄	—	344	4	no	—
365	D₄	—	344	4	no	—
366	D₄	#13063347	1	2	yes	complemented
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	86	2	no	—
374	C₂×C₄	—	344	4	no	—
375	C₂×C₄	—	344	4	no	—
376	C₂×C₄	—	172	2	no	—
377	C₂×C₄	—	172	2	no	—
378	C₂×C₄	—	172	2	no	—
379	C₂×C₄	—	172	2	no	—
380	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
382	C₂×C₄	#13063371	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	2	1	no	—
385	C₈	—	2	1	no	—
386	C₈	—	2	1	no	—
387	C₈	—	4	1	no	—
388	C₈	—	4	1	no	—
389	C₈	—	172	2	no	—
390	C₈	#13063349	1	2	yes	—
391	C₂²	—	2	2	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	172	2	no	—
398	C₂²	—	172	2	no	—
399	C₂²	—	172	2	no	—
400	C₂²	D₄×D₁₇₂	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	344	2	no	—
405	C₄	—	344	2	no	—
406	C₄	—	172	2	no	—
407	C₄	C₂×#13063347	1	1	yes	characteristic
408	C₄	C₂×#13063349	1	1	yes	characteristic
409	C₄	#14583704	1	1	yes	characteristic
410	C₄	#14583706	1	1	yes	characteristic
411	C₂	—	4	2	no	—
412	C₂	—	8	1	no	—
413	C₂	—	344	2	no	—
414	C₂	#19146196	1	1	yes	characteristic
415	C₂	#19146282	1	1	yes	characteristic
416	C₂	#19146400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₇₂
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₈×D₈×C₄₃	C₂
socle	C₂²×C₄₃	D₄²
hypercenter	C₈×D₈	D₄₃
hypocenter	C₄₃	#19902
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
1. #19146943	C₂	semidirect
2. #19146902	C₂	semidirect
3. #19146317	C₂	semidirect
4. #19134192	C₂	semidirect
5. #19134143	C₂	semidirect
6. #19133984	C₂	semidirect
7. #19133960	C₂	semidirect
8. #19133882	C₂	semidirect
9. #19133881	C₂	semidirect
10. #19133481	C₂	semidirect
11. C₈×D₈×C₄₃	C₂	semidirect
149. D₄×C₄₃	#135	semidirect
213. C₈×D₈	D₄₃	semidirect
253. C₄₃	#19902	semidirect
366. D₄	#13063347	semidirect

Maximal subgroups

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

subgroup	index
#19146943	2
#19146902	2
#19146317	2
#19134192	2
#19134143	2
#19133984	2
#19133960	2
#19133882	2
#19133881	2
#19133481	2
C₈×D₈×C₄₃	2
#19902	43