Subgroups — #25087905

Order 11008.

690 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#19146942	C₂	1	1	yes	maximal, characteristic
3	#19146236	C₂	1	2	yes	maximal, complemented
4	#19134212	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134083	C₂	1	2	yes	maximal, complemented
6	#19133985	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133842	C₂	1	2	yes	maximal, complemented
8	#19133831	C₂	1	2	yes	maximal, complemented
9	#19133479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10621884	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1674	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063357	—	2	2	no	—
13	Q₈×D₁₇₂	C₂²	1	2	yes	—
14	#14583552	C₂²	1	2	yes	—
15	#14583304	—	2	1	no	—
16	#14583304	C₂²	1	1	yes	characteristic
17	#14583302	C₂²	1	2	yes	—
18	#14583278	C₂²	1	2	yes	—
19	#14583214	—	2	2	no	—
20	#14583180	—	2	2	no	—
21	#14583180	C₂²	1	2	yes	—
22	#14583176	C₂²	1	2	yes	—
23	#14583156	—	2	2	no	—
24	#14583136	—	2	1	no	—
25	#14583105	C₂²	1	2	yes	—
26	#14583064	—	2	2	no	—
27	#14583053	—	2	2	no	—
28	C₄×#10621884	C₂²	1	1	yes	characteristic
29	C₄×D₃₄₄	—	2	1	no	—
30	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
31	#14582976	C₂²	1	2	yes	—
32	C₈×D₁₇₂	C₂²	1	1	yes	characteristic
33	#107×D₄₃	—	2	1	no	—
34	#14582649	C₂²	1	1	yes	characteristic
35	#14582646	C₂²	1	2	yes	—
36	Q₈⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
37	#14582626	C₂²	1	2	yes	—
38	#14582626	C₂²	1	1	yes	characteristic
39	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₄₃×#500	C₂²	1	1	yes	characteristic
41	C₄₃×#462	C₂²	1	2	yes	—
42	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
43	C₄₃×#335	C₂²	1	1	yes	characteristic
44	C₄₃×#332	C₂²	1	2	yes	—
45	C₂×#10621902	—	2	2	no	—
46	C₂×Q₈×D₄₃	—	2	2	no	—
47	#13063357	—	4	8	no	—
48	C₂×#10621885	—	2	2	no	—
49	C₂×#10621885	—	2	2	no	—
50	C₂×#10621873	—	2	2	no	—
51	C₂×#10621884	D₄	1	1	yes	characteristic
52	C₂×D₃₄₄	D₄	1	1	yes	characteristic
53	C₂×#10621872	—	2	2	no	—
54	C₂×C₈×D₄₃	—	2	1	no	—
55	#13063325	—	2	2	no	—
56	#13063325	C₂³	1	2	yes	—
57	#13063310	—	4	2	no	—
58	#13063305	—	2	2	no	—
59	#13063305	—	4	2	no	—
60	#13063301	—	2	2	no	—
61	#13063300	—	2	2	no	—
62	C₄×D₁₇₂	—	2	2	no	—
63	C₄×D₁₇₂	C₂³	1	1	yes	characteristic
64	C₄×D₁₇₂	C₂³	1	2	yes	—
65	C₄⋊C₄×D₄₃	—	2	2	no	—
66	#13063259	—	2	1	no	—
67	#13063259	—	2	1	no	—
68	#13063259	—	2	2	no	—
69	#13063259	D₄	1	1	yes	characteristic
70	#13063258	—	2	2	no	—
71	#13063258	—	4	2	no	—
72	#13063256	D₄	1	1	yes	characteristic
73	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
74	#13063250	—	2	2	no	—
75	#13063249	D₄	1	1	yes	characteristic
76	#13063247	—	2	2	no	—
77	#13063247	D₄	1	2	yes	—
78	#13063245	—	2	2	no	—
79	#13063245	D₄	1	1	yes	characteristic
80	(C₂×D₄₃)⋊C₈	—	2	1	no	—
81	(C₂×D₄₃)⋊C₈	—	2	2	no	—
82	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
84	C₂×Q₁₆×C₄₃	—	2	2	no	—
85	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
86	C₄×Q₈×C₄₃	C₂³	1	1	yes	characteristic
87	#107×C₄₃	—	2	1	no	—
88	#107×C₄₃	—	2	1	no	—
89	C₄⋊C₈×C₄₃	—	2	1	no	—
90	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
91	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
92	#103×C₄₃	—	2	2	no	—
93	C₄×C₈×C₄₃	—	2	1	no	—
94	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₄₃	—	2	2	no	—
96	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	#10621902	—	4	4	no	—
98	#10621902	—	8	2	no	—
99	Q₈×D₄₃	—	4	4	no	—
100	Q₈×D₄₃	—	8	2	no	—
101	C₂×D₁₇₂	—	4	2	no	—
102	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
103	C₂×D₁₇₂	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₄₃	—	2	1	no	—
105	C₂×C₄×D₄₃	—	2	2	no	—
106	C₂×C₄×D₄₃	—	4	2	no	—
107	C₂×C₄×D₄₃	—	4	2	no	—
108	C₂×Dic₈₆	—	2	2	no	—
109	C₂×Dic₈₆	—	4	2	no	—
110	Q₁₆×C₄₃	—	4	4	no	—
111	C₂×C₈×C₄₃	—	2	1	no	—
112	C₂×C₈×C₄₃	—	2	2	no	—
113	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₄₃	—	2	1	no	—
116	C₄⋊C₄×C₄₃	—	2	2	no	—
117	C₄⋊C₄×C₄₃	—	2	2	no	—
118	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₃	—	2	1	no	—
120	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
121	#10621885	—	4	4	no	—
122	#10621885	—	8	2	no	—
123	#10621884	D₈	1	4	yes	complemented
124	(C₂×D₄₃)⋊C₄	—	2	2	no	—
125	(C₂×D₄₃)⋊C₄	—	2	1	no	—
126	(C₂×D₄₃)⋊C₄	—	4	2	no	—
127	(C₂×D₄₃)⋊C₄	—	4	2	no	—
128	C₄⋊Dic₄₃	—	2	2	no	—
129	C₄⋊Dic₄₃	—	2	2	no	—
130	C₄⋊Dic₄₃	—	4	2	no	—
131	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
132	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
133	Dic₄₃⋊C₄	—	4	2	no	—
134	C₂×C₄₃⋊C₈	—	2	1	no	—
135	C₂×C₄₃⋊C₈	—	2	1	no	—
136	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
138	D₃₄₄	—	2	2	no	—
139	#10621873	—	4	4	no	—
140	#10621872	—	4	4	no	—
141	C₈×D₄₃	—	4	2	no	—
142	C₂²×D₄₃	—	2	1	no	—
143	C₂²×D₄₃	—	2	2	no	—
144	Q₈×C₄₃	—	2	1	no	—
145	Q₈×C₄₃	—	4	2	no	—
146	Q₈×C₄₃	—	4	2	no	—
147	Q₈×C₄₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₄₃	—	2	1	no	—
149	C₂×C₄×C₄₃	—	2	1	no	—
150	C₂×C₄×C₄₃	—	4	2	no	—
151	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×Dic₄₃	—	2	1	no	—
155	C₂×Dic₄₃	—	2	2	no	—
156	C₂×Dic₄₃	—	4	2	no	—
157	D₁₇₂	—	2	2	no	—
158	D₁₇₂	—	2	1	no	—
159	D₁₇₂	—	2	2	no	—
160	D₁₇₂	—	8	4	no	—
161	D₁₇₂	C₂×D₈	1	2	yes	—
162	C₄×D₄₃	—	4	2	no	—
163	C₄×D₄₃	—	4	1	no	—
164	C₄×D₄₃	—	4	2	no	—
165	C₄×D₄₃	—	4	1	no	—
166	C₄×D₄₃	—	8	4	no	—
167	C₄×D₄₃	—	8	4	no	—
168	Dic₈₆	—	4	2	no	—
169	Dic₈₆	—	4	2	no	—
170	Dic₈₆	—	8	4	no	—
171	C₈×C₄₃	—	2	1	no	—
172	C₈×C₄₃	—	4	2	no	—
173	C₈×C₄₃	—	4	1	no	—
174	C₈×C₄₃	#137	1	2	yes	complemented
175	C₄₃⋊C₈	—	2	2	no	—
176	C₄₃⋊C₈	—	2	1	no	—
177	C₄₃⋊C₈	—	4	1	no	—
178	C₄₃⋊C₈	C₂×D₈	1	2	yes	—
179	#19887	—	43	1	no	maximal, Sylow
180	C₂²×C₄₃	D₄²	1	1	yes	characteristic
181	C₂×D₄₃	—	2	2	no	—
182	C₂×D₄₃	—	4	1	no	—
183	C₂×D₄₃	—	4	1	no	—
184	C₂×D₄₃	—	4	2	no	—
185	C₂×D₄₃	—	4	2	no	—
186	C₂×D₄₃	—	4	2	no	—
187	C₄×C₄₃	—	2	2	no	—
188	C₄×C₄₃	—	2	1	no	—
189	C₄×C₄₃	—	4	1	no	—
190	C₄×C₄₃	—	8	2	no	—
191	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
193	C₄×C₄₃	#576	1	1	yes	characteristic
194	C₄×C₄₃	#578	1	1	yes	characteristic
195	Dic₄₃	—	4	1	no	—
196	Dic₄₃	—	4	2	no	—
197	Dic₄₃	—	8	2	no	—
198	Q₈×D₈	—	43	1	no	—
199	#3334	—	43	1	no	—
200	#3276	—	43	2	no	—
201	#1713	—	43	1	no	—
202	#1689	—	43	2	no	—
203	#1674	D₄₃	1	1	yes	characteristic, complemented
204	#1673	—	43	1	no	—
205	#1649	—	43	2	no	—
206	#1647	—	43	2	no	—
207	#1589	—	43	1	no	—
208	C₈×SD₁₆	—	43	1	no	—
209	C₂×C₄₃	#3276	1	1	yes	characteristic
210	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
211	C₂×C₄₃	#3293	1	1	yes	characteristic
212	D₄₃	—	4	2	no	—
213	D₄₃	—	8	2	no	—
214	C₂×#137	—	86	2	no	—
215	D₄×Q₈	—	43	2	no	—
216	#547	—	43	2	no	—
217	#500	—	43	2	no	—
218	#500	—	43	1	no	—
219	#500	—	86	1	no	—
220	#500	C₂×D₄₃	1	1	yes	characteristic
221	#494	—	43	2	no	—
222	#480	—	86	2	no	—
223	#474	—	43	2	no	—
224	#474	—	43	2	no	—
225	#474	—	86	2	no	—
226	#470	—	86	2	no	—
227	#466	—	86	1	no	—
228	#462	C₂×D₄₃	1	2	yes	—
229	#461	—	43	2	no	—
230	#452	—	86	2	no	—
231	#451	—	86	2	no	—
232	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
233	C₄×SD₁₆	—	43	1	no	—
234	C₄×D₈	—	43	1	no	—
235	C₄×D₈	—	86	1	no	—
236	#436	—	43	2	no	—
237	C₈×D₄	—	43	1	no	—
238	C₂×#107	—	86	1	no	—
239	#335	C₂×D₄₃	1	1	yes	characteristic
240	#334	—	43	1	no	—
241	#332	—	43	2	no	—
242	#332	C₂×D₄₃	1	2	yes	—
243	Q₈⋊C₈	—	43	1	no	—
244	D₄⋊C₈	—	43	1	no	—
245	D₄⋊C₈	—	43	2	no	—
246	C₈²	—	43	1	no	—
247	C₄₃	#19887	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	86	2	no	—
249	C₂²×Q₈	—	86	2	no	—
250	#137	—	172	8	no	—
251	C₂×Q₁₆	—	2	2	no	—
252	C₂×Q₁₆	—	86	2	no	—
253	C₂×Q₁₆	—	86	2	no	—
254	C₂×SD₁₆	—	43	1	no	—
255	C₂×SD₁₆	—	86	2	no	—
256	C₂×D₈	—	43	1	no	—
257	C₂×M₁₆	—	86	2	no	—
258	C₂²×C₈	—	86	1	no	—
259	C₄⋊Q₈	—	43	2	no	—
260	C₄⋊Q₈	—	86	2	no	—
261	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
262	#123	—	172	2	no	—
263	C₂²⋊Q₈	—	86	2	no	—
264	C₂²⋊Q₈	—	86	2	no	—
265	C₂²⋊Q₈	—	172	2	no	—
266	#121	—	86	2	no	—
267	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
268	C₄×D₄	—	43	1	no	—
269	C₄×D₄	—	43	2	no	—
270	C₄×D₄	—	86	2	no	—
271	C₂×C₄⋊C₄	—	86	2	no	—
272	#107	—	2	1	no	—
273	#107	—	2	1	no	—
274	#107	—	43	1	no	—
275	#107	—	86	2	no	—
276	#107	—	86	2	no	—
277	#107	—	86	1	no	—
278	#107	—	86	1	no	—
279	#107	—	172	2	no	—
280	#106	—	43	1	no	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	43	1	no	—
283	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
284	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
285	#103	—	2	2	no	—
286	#103	—	43	1	no	—
287	#103	—	86	2	no	—
288	#102	—	43	2	no	—
289	#102	—	43	1	no	—
290	#102	—	86	2	no	—
291	#102	—	86	2	no	—
292	C₂²⋊C₈	—	86	2	no	—
293	C₂²⋊C₈	—	86	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	—	43	1	no	—
296	C₄×C₈	—	43	1	no	—
297	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
298	#41	—	344	2	no	—
299	#41	—	172	4	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	86	2	no	—
302	C₂×Q₈	—	344	2	no	—
303	C₂×Q₈	—	172	2	no	—
304	C₂×Q₈	—	172	4	no	—
305	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
306	C₂×D₄	—	43	1	no	—
307	C₂×D₄	—	43	2	no	—
308	C₂×D₄	—	172	2	no	—
309	C₂²×C₄	—	86	1	no	—
310	C₂²×C₄	—	86	2	no	—
311	C₂²×C₄	—	172	2	no	—
312	C₂²×C₄	—	172	2	no	—
313	Q₁₆	—	4	4	no	—
314	Q₁₆	—	344	2	no	—
315	Q₁₆	—	172	4	no	—
316	SD₁₆	—	43	4	no	—
317	SD₁₆	—	172	4	no	—
318	D₈	—	86	2	no	—
319	M₁₆	—	172	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	43	1	no	—
323	C₂×C₈	—	43	1	no	—
324	C₂×C₈	—	86	1	no	—
325	C₂×C₈	—	86	1	no	—
326	C₂×C₈	—	172	2	no	—
327	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
328	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	43	2	no	—
333	C₄⋊C₄	—	43	1	no	—
334	C₄⋊C₄	—	86	2	no	—
335	C₄⋊C₄	—	86	2	no	—
336	C₄⋊C₄	—	172	2	no	—
337	C₄⋊C₄	—	172	2	no	—
338	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
339	C₂²⋊C₄	—	86	2	no	—
340	C₂²⋊C₄	—	86	1	no	—
341	C₂²⋊C₄	—	172	2	no	—
342	C₂²⋊C₄	—	172	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂³×D₄₃	1	1	yes	characteristic
345	C₂³	—	86	1	no	—
346	C₂³	—	86	2	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	344	4	no	—
351	Q₈	—	172	2	no	—
352	Q₈	—	172	2	no	—
353	Q₈	D₈×D₄₃	1	2	yes	—
354	D₄	—	43	2	no	—
355	D₄	—	86	2	no	—
356	D₄	—	86	1	no	—
357	D₄	—	86	2	no	—
358	D₄	—	344	4	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	86	1	no	—
363	C₂×C₄	—	86	2	no	—
364	C₂×C₄	—	344	4	no	—
365	C₂×C₄	—	344	4	no	—
366	C₂×C₄	—	172	2	no	—
367	C₂×C₄	—	172	2	no	—
368	C₂×C₄	—	172	1	no	—
369	C₂×C₄	—	172	2	no	—
370	C₂×C₄	—	172	1	no	—
371	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
372	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
373	C₂×C₄	#13063371	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	2	no	—
376	C₈	—	4	1	no	—
377	C₈	—	43	2	no	—
378	C₈	—	86	2	no	—
379	C₈	—	86	1	no	—
380	C₈	—	172	1	no	—
381	C₈	#13063357	1	2	yes	complemented
382	C₂²	—	86	2	no	—
383	C₂²	—	172	2	no	—
384	C₂²	—	172	2	no	—
385	C₂²	—	172	2	no	—
386	C₂²	—	172	1	no	—
387	C₂²	—	172	1	no	—
388	C₂²	#14583533	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	8	2	no	—
393	C₄	—	344	2	no	—
394	C₄	—	172	2	no	—
395	C₄	—	172	1	no	—
396	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
397	C₄	C₂×#13063357	1	1	yes	characteristic
398	C₄	#14583701	1	1	yes	characteristic
399	C₄	#14583715	1	1	yes	characteristic
400	C₂	—	344	2	no	—
401	C₂	—	172	2	no	—
402	C₂	#19146236	1	1	yes	characteristic
403	C₂	#19146269	1	1	yes	characteristic
404	C₂	#19146398	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14583533
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1674	C₂
socle	C₂²×C₄₃	D₄²
prime radical	#19146942	C₂
hypercenter	#1674	D₄₃
hypocenter	C₄₃	#19887
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1674

Complements

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

N	H ≅ G/N	type
1. Q₈×D₃₄₄	C₂	semidirect
3. #19146236	C₂	semidirect
4. #19134212	C₂	semidirect
5. #19134083	C₂	semidirect
6. #19133985	C₂	semidirect
7. #19133842	C₂	semidirect
8. #19133831	C₂	semidirect
9. #19133479	C₂	semidirect
10. C₈×#10621884	C₂	semidirect
11. C₄₃×#1674	C₂	semidirect
123. #10621884	D₈	semidirect
174. C₈×C₄₃	#137	semidirect
203. #1674	D₄₃	semidirect
247. C₄₃	#19887	semidirect
381. C₈	#13063357	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₃₄₄	2
#19146942	2
#19146236	2
#19134212	2
#19134083	2
#19133985	2
#19133842	2
#19133831	2
#19133479	2
C₈×#10621884	2
C₄₃×#1674	2
#19887	43