Subgroups — #25088177

Order 11008.

602 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147584	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₃₄₄	C₂	1	1	yes	maximal, characteristic
3	#19146880	C₂	1	2	yes	maximal, complemented
4	#19134192	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134143	C₂	1	2	yes	maximal, complemented
6	#19133989	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133965	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133867	C₂	1	2	yes	maximal, complemented
9	#19133861	C₂	1	2	yes	maximal, complemented
10	#19133481	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583622	C₂²	1	2	yes	—
13	Q₈×D₁₇₂	C₂²	1	2	yes	—
14	#14583304	—	2	1	no	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583297	C₂²	1	2	yes	—
17	#14583283	—	2	1	no	—
18	#14583279	C₂²	1	1	yes	characteristic
19	#14583250	C₂²	1	2	yes	—
20	#14583242	—	2	2	no	—
21	#14583184	C₂²	1	2	yes	—
22	#14583180	—	2	2	no	—
23	#14583180	C₂²	1	2	yes	—
24	#14583156	—	2	2	no	—
25	#14583122	—	2	2	no	—
26	#14583109	C₂²	1	2	yes	—
27	#14583105	C₂²	1	2	yes	—
28	#14583053	—	2	2	no	—
29	C₄×Dic₁₇₂	—	2	1	no	—
30	C₄×D₃₄₄	—	2	1	no	—
31	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
32	#14582978	C₂²	1	2	yes	—
33	#14582906	—	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	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₄₃×#335	C₂²	1	1	yes	characteristic
42	C₄₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₄₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₄₃	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₄₃	—	2	2	no	—
46	C₂×Dic₁₇₂	—	2	1	no	—
47	C₂×Dic₁₇₂	—	2	1	no	—
48	C₂×#10621885	—	2	2	no	—
49	C₂×#10621873	—	2	2	no	—
50	C₂×D₃₄₄	D₄	1	1	yes	characteristic
51	C₂×#10621872	—	2	2	no	—
52	#13063325	—	2	2	no	—
53	#13063325	C₂³	1	2	yes	—
54	#13063325	C₂³	1	2	yes	—
55	#13063320	—	4	2	no	—
56	#13063313	—	2	2	no	—
57	#13063305	—	2	2	no	—
58	#13063305	—	4	2	no	—
59	#13063301	—	2	2	no	—
60	C₄×Dic₈₆	—	2	2	no	—
61	C₄×D₁₇₂	—	2	2	no	—
62	C₄×D₁₇₂	C₂³	1	2	yes	—
63	#13063283	—	2	2	no	—
64	#13063259	—	2	1	no	—
65	#13063259	—	2	1	no	—
66	#13063259	—	2	2	no	—
67	#13063259	D₄	1	1	yes	characteristic
68	#13063257	—	4	2	no	—
69	#13063256	—	2	2	no	—
70	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
71	#13063250	—	2	2	no	—
72	#13063250	—	2	2	no	—
73	#13063250	—	2	2	no	—
74	#13063249	—	2	2	no	—
75	#13063248	—	2	2	no	—
76	#13063247	—	2	2	no	—
77	#13063247	D₄	1	2	yes	—
78	(C₂×D₄₃)⋊C₈	—	2	2	no	—
79	C₂×Q₁₆×C₄₃	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₄×Q₈×C₄₃	C₂³	1	1	yes	characteristic
82	#107×C₄₃	D₄	1	1	yes	characteristic
83	C₄⋊C₈×C₄₃	—	2	1	no	—
84	C₄⋊C₈×C₄₃	—	2	1	no	—
85	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
87	#103×C₄₃	D₄	1	1	yes	characteristic
88	#103×C₄₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₄₃	—	2	1	no	—
90	C₄×C₈×C₄₃	—	2	1	no	—
91	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	Q₈×D₄₃	—	4	4	no	—
97	Q₈×D₄₃	—	8	2	no	—
98	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
99	C₂×C₄×D₄₃	—	2	2	no	—
100	C₂×C₄×D₄₃	—	4	2	no	—
101	C₂×Dic₈₆	—	2	2	no	—
102	C₂×Dic₈₆	—	2	2	no	—
103	C₂×Dic₈₆	—	4	2	no	—
104	Q₁₆×C₄₃	—	2	2	no	—
105	C₂×C₈×C₄₃	—	2	1	no	—
106	C₂×C₈×C₄₃	—	2	1	no	—
107	C₂×C₈×C₄₃	—	2	1	no	—
108	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₄₃	—	2	1	no	—
113	C₄⋊C₄×C₄₃	—	2	1	no	—
114	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₃	—	2	1	no	—
117	C₄²×C₄₃	—	2	1	no	—
118	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
119	#10621885	—	8	2	no	—
120	(C₂×D₄₃)⋊C₄	—	2	2	no	—
121	(C₂×D₄₃)⋊C₄	—	4	2	no	—
122	(C₂×D₄₃)⋊C₄	—	4	2	no	—
123	(C₂×D₄₃)⋊C₄	—	4	2	no	—
124	C₄⋊Dic₄₃	—	2	2	no	—
125	C₄⋊Dic₄₃	—	2	2	no	—
126	C₄⋊Dic₄₃	—	2	2	no	—
127	C₄⋊Dic₄₃	—	2	2	no	—
128	C₄⋊Dic₄₃	—	4	2	no	—
129	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
130	Dic₄₃⋊C₄	—	4	2	no	—
131	C₄×Dic₄₃	—	4	2	no	—
132	C₂×C₄₃⋊C₈	—	2	2	no	—
133	Dic₁₇₂	—	4	2	no	—
134	Dic₁₇₂	—	8	1	no	—
135	D₃₄₄	—	2	2	no	—
136	#10621873	—	8	2	no	—
137	#10621872	—	8	2	no	—
138	C₂²×D₄₃	—	2	2	no	—
139	Q₈×C₄₃	—	2	1	no	—
140	Q₈×C₄₃	—	2	1	no	—
141	Q₈×C₄₃	—	2	1	no	—
142	Q₈×C₄₃	#135	1	2	yes	—
143	C₂×C₄×C₄₃	—	2	1	no	—
144	C₂×C₄×C₄₃	—	2	1	no	—
145	C₂×C₄×C₄₃	—	2	1	no	—
146	C₂×C₄×C₄₃	—	2	1	no	—
147	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×Dic₄₃	—	2	2	no	—
151	C₂×Dic₄₃	—	4	2	no	—
152	C₂×Dic₄₃	—	4	2	no	—
153	D₁₇₂	—	2	2	no	—
154	D₁₇₂	—	2	2	no	—
155	C₄×D₄₃	—	4	2	no	—
156	C₄×D₄₃	—	4	2	no	—
157	C₄×D₄₃	—	8	4	no	—
158	Dic₈₆	—	4	2	no	—
159	Dic₈₆	—	4	2	no	—
160	Dic₈₆	—	4	2	no	—
161	Dic₈₆	—	4	2	no	—
162	Dic₈₆	—	8	4	no	—
163	C₈×C₄₃	—	2	2	no	—
164	C₈×C₄₃	—	2	1	no	—
165	C₈×C₄₃	—	2	1	no	—
166	C₈×C₄₃	—	2	1	no	—
167	C₈×C₄₃	—	4	1	no	—
168	C₈×C₄₃	—	4	1	no	—
169	C₈×C₄₃	#137	1	2	yes	—
170	C₄₃⋊C₈	—	4	2	no	—
171	#19907	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	D₄²	1	1	yes	characteristic
173	C₂×D₄₃	—	4	2	no	—
174	C₂×D₄₃	—	4	2	no	—
175	C₂×D₄₃	—	4	2	no	—
176	C₄×C₄₃	—	2	2	no	—
177	C₄×C₄₃	—	2	1	no	—
178	C₄×C₄₃	—	4	1	no	—
179	C₄×C₄₃	—	4	1	no	—
180	C₄×C₄₃	—	4	1	no	—
181	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
182	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
183	C₄×C₄₃	#576	1	1	yes	characteristic
184	C₄×C₄₃	#577	1	1	yes	characteristic
185	Dic₄₃	—	4	2	no	—
186	Dic₄₃	—	8	2	no	—
187	Dic₄₃	—	8	2	no	—
188	#3393	—	43	1	no	—
189	Q₈×D₈	—	43	1	no	—
190	#3328	—	43	2	no	—
191	#1707	—	43	1	no	—
192	#1699	—	43	2	no	—
193	#1674	—	43	1	no	—
194	#1670	—	43	1	no	—
195	#1652	—	43	2	no	—
196	#1651	—	43	2	no	—
197	#1589	—	43	1	no	—
198	C₈×Q₁₆	D₄₃	1	1	yes	characteristic, complemented
199	C₂×C₄₃	#3278	1	1	yes	characteristic
200	C₂×C₄₃	#3280	1	1	yes	characteristic
201	C₂×C₄₃	#3293	1	1	yes	characteristic
202	D₄₃	—	8	2	no	—
203	#556	—	43	2	no	—
204	D₄×Q₈	—	43	2	no	—
205	#500	—	43	1	no	—
206	#500	—	86	1	no	—
207	#498	—	43	2	no	—
208	#495	—	86	1	no	—
209	#494	—	43	1	no	—
210	#487	—	43	2	no	—
211	#485	—	86	2	no	—
212	#475	—	43	2	no	—
213	#474	—	43	2	no	—
214	#474	—	86	2	no	—
215	#470	—	86	2	no	—
216	#464	—	86	2	no	—
217	#462	—	43	2	no	—
218	#461	—	43	2	no	—
219	#451	—	86	2	no	—
220	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
221	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
222	C₄×Q₁₆	—	86	1	no	—
223	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
224	C₄×D₈	—	43	1	no	—
225	C₄×D₈	—	86	1	no	—
226	#436	—	43	2	no	—
227	#419	—	86	2	no	—
228	#335	—	43	1	no	—
229	#335	C₂×D₄₃	1	1	yes	characteristic
230	#332	—	43	2	no	—
231	#332	—	43	2	no	—
232	Q₈⋊C₈	C₂×D₄₃	1	1	yes	characteristic
233	Q₈⋊C₈	C₂×D₄₃	1	1	yes	characteristic
234	D₄⋊C₈	—	43	2	no	—
235	C₈²	C₂×D₄₃	1	1	yes	characteristic
236	C₄₃	#19907	1	1	yes	characteristic, complemented, Sylow
237	C₂²×Q₈	—	86	2	no	—
238	C₂×Q₁₆	—	86	2	no	—
239	C₂×Q₁₆	—	86	1	no	—
240	C₂×Q₁₆	—	86	1	no	—
241	C₂×Q₁₆	D₁₇₂	1	1	yes	characteristic
242	C₂×SD₁₆	—	86	2	no	—
243	C₂×D₈	—	43	1	no	—
244	C₂×M₁₆	—	86	2	no	—
245	C₄⋊Q₈	—	43	2	no	—
246	C₄⋊Q₈	—	43	2	no	—
247	C₄⋊Q₈	—	86	2	no	—
248	#126	—	172	2	no	—
249	#124	—	86	2	no	—
250	C₂²⋊Q₈	—	86	2	no	—
251	C₂²⋊Q₈	—	86	2	no	—
252	C₂²⋊Q₈	—	172	2	no	—
253	C₄×Q₈	—	86	2	no	—
254	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
255	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
256	C₄×D₄	—	43	2	no	—
257	C₄×D₄	—	86	2	no	—
258	#117	—	86	2	no	—
259	#107	—	43	1	no	—
260	#107	—	86	2	no	—
261	#107	—	86	1	no	—
262	#107	—	86	1	no	—
263	#107	D₁₇₂	1	1	yes	characteristic
264	#106	—	86	2	no	—
265	#106	—	172	2	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	43	2	no	—
269	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
270	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
271	#103	—	86	2	no	—
272	#103	—	86	2	no	—
273	#103	—	86	2	no	—
274	#103	—	86	2	no	—
275	#103	—	86	2	no	—
276	#103	D₁₇₂	1	1	yes	characteristic
277	#103	D₁₇₂	1	1	yes	characteristic
278	#102	—	43	2	no	—
279	#102	—	86	2	no	—
280	C₂²⋊C₈	—	86	2	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	2	1	no	—
283	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
286	C₂×Q₈	—	86	2	no	—
287	C₂×Q₈	—	86	2	no	—
288	C₂×Q₈	—	344	2	no	—
289	C₂×Q₈	—	172	2	no	—
290	C₂×Q₈	—	172	4	no	—
291	C₂×Q₈	C₂×D₁₇₂	1	1	yes	characteristic
292	C₂×Q₈	C₂×D₁₇₂	1	1	yes	characteristic
293	C₂×D₄	—	43	2	no	—
294	C₂²×C₄	—	86	2	no	—
295	C₂²×C₄	—	172	2	no	—
296	Q₁₆	—	2	2	no	—
297	Q₁₆	—	344	1	no	—
298	Q₁₆	—	344	2	no	—
299	Q₁₆	—	172	2	no	—
300	SD₁₆	—	344	2	no	—
301	D₈	—	86	2	no	—
302	M₁₆	—	344	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	86	2	no	—
307	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
308	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
309	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
310	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	43	2	no	—
314	C₄⋊C₄	—	86	2	no	—
315	C₄⋊C₄	—	86	2	no	—
316	C₄⋊C₄	—	86	2	no	—
317	C₄⋊C₄	—	86	2	no	—
318	C₄⋊C₄	—	172	2	no	—
319	C₄⋊C₄	—	172	2	no	—
320	C₄⋊C₄	C₂×D₁₇₂	1	1	yes	characteristic
321	C₄⋊C₄	C₂×D₁₇₂	1	1	yes	characteristic
322	C₂²⋊C₄	—	86	2	no	—
323	C₂²⋊C₄	—	172	2	no	—
324	C₂²⋊C₄	—	172	2	no	—
325	C₂²⋊C₄	—	172	2	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	2	1	no	—
328	C₄²	—	172	2	no	—
329	C₄²	C₂³×D₄₃	1	1	yes	characteristic
330	C₂³	—	86	2	no	—
331	Q₈	—	2	1	no	—
332	Q₈	—	2	1	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	344	4	no	—
335	Q₈	—	172	2	no	—
336	Q₈	—	172	2	no	—
337	Q₈	—	172	2	no	—
338	Q₈	—	172	2	no	—
339	Q₈	#13063347	1	2	yes	—
340	D₄	—	86	2	no	—
341	D₄	—	86	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	86	2	no	—
347	C₂×C₄	—	344	4	no	—
348	C₂×C₄	—	172	2	no	—
349	C₂×C₄	—	172	2	no	—
350	C₂×C₄	—	172	2	no	—
351	C₂×C₄	—	172	2	no	—
352	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
353	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
354	C₂×C₄	#13063371	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	4	1	no	—
361	C₈	—	172	2	no	—
362	C₈	#13063357	1	2	yes	—
363	C₂²	—	172	2	no	—
364	C₂²	—	172	2	no	—
365	C₂²	—	172	2	no	—
366	C₂²	D₄×D₁₇₂	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	344	2	no	—
373	C₄	—	344	2	no	—
374	C₄	—	172	2	no	—
375	C₄	C₂×#13063347	1	1	yes	characteristic
376	C₄	C₂×#13063357	1	1	yes	characteristic
377	C₄	#14583704	1	1	yes	characteristic
378	C₄	#14583706	1	1	yes	characteristic
379	C₂	—	344	2	no	—
380	C₂	#19146262	1	1	yes	characteristic
381	C₂	#19146282	1	1	yes	characteristic
382	C₂	#19146400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₇₂
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₈×Q₁₆×C₄₃	C₂
socle	C₂²×C₄₃	D₄²
prime radical	C₂×D₃₄₄	D₄
hypercenter	C₈×Q₁₆	D₄₃
hypocenter	C₄₃	#19907
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #19146880	C₂	semidirect
4. #19134192	C₂	semidirect
5. #19134143	C₂	semidirect
6. #19133989	C₂	semidirect
7. #19133965	C₂	semidirect
8. #19133867	C₂	semidirect
9. #19133861	C₂	semidirect
11. C₈×Q₁₆×C₄₃	C₂	semidirect
198. C₈×Q₁₆	D₄₃	semidirect
236. C₄₃	#19907	semidirect

Maximal subgroups

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

subgroup	index
#19147584	2
Q₈×D₃₄₄	2
#19146880	2
#19134192	2
#19134143	2
#19133989	2
#19133965	2
#19133867	2
#19133861	2
#19133481	2
C₈×Q₁₆×C₄₃	2
#19907	43