Subgroups — #25079664

Order 11008.

842 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145513	C₂	1	2	yes	maximal, complemented
2	#19143308	C₂	1	1	yes	maximal, characteristic, complemented
3	#19134071	C₂	1	4	yes	maximal, complemented
4	#19133947	C₂	1	4	yes	maximal, complemented
5	#19133343	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133263	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063331	—	2	1	no	—
9	#14583449	C₂²	1	1	yes	characteristic
10	#14583275	—	2	2	no	—
11	#14583245	—	2	2	no	—
12	#14583091	C₂²	1	8	yes	complemented
13	#14583033	—	2	4	no	—
14	#14582978	—	2	1	no	—
15	#14582978	C₂²	1	2	yes	complemented
16	C₈×D₁₇₂	—	2	1	no	—
17	C₈×D₁₇₂	C₂²	1	2	yes	complemented
18	#14582952	C₂²	1	2	yes	—
19	#14582923	C₂²	1	1	yes	characteristic
20	#14582865	—	2	1	no	—
21	#14582652	C₂²	1	2	yes	complemented
22	#14582650	C₂²	1	2	yes	complemented
23	#14582643	C₂²	1	8	yes	complemented
24	C₂²×D₈×C₄₃	—	2	1	no	—
25	C₂²×D₈×C₄₃	—	2	1	no	—
26	C₂×#127×C₄₃	C₂²	1	2	yes	—
27	C₄₃×#493	—	2	1	no	—
28	C₄₃×#493	—	2	1	no	—
29	C₄₃×#493	C₂²	1	2	yes	—
30	C₄₃×#493	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×#10621898	—	2	1	no	—
33	C₂×#10621882	—	2	8	no	—
34	#13063331	—	4	4	no	—
35	C₂×#10621877	—	2	1	no	—
36	C₂×#10621872	—	2	2	no	—
37	C₂×C₈×D₄₃	—	2	2	no	—
38	#13063325	D₄	1	1	yes	characteristic
39	#13063322	D₄	1	1	yes	characteristic
40	#13063313	—	2	1	no	—
41	#13063299	—	2	1	no	—
42	#13063299	D₄	1	2	yes	—
43	C₄×D₁₇₂	—	2	1	no	—
44	C₄×D₁₇₂	C₂³	1	2	yes	—
45	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
47	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
48	#13063246	—	2	8	no	—
49	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
50	(C₂×D₄₃)⋊C₈	—	2	2	no	—
51	(C₂×D₄₃)⋊C₈	—	2	2	no	—
52	C₄×C₄₃⋊C₈	—	2	2	no	—
53	C₂²×D₄×C₄₃	—	2	2	no	—
54	C₂²×D₄×C₄₃	—	2	2	no	—
55	C₂²×D₄×C₄₃	—	2	2	no	—
56	C₂×D₈×C₄₃	—	2	2	no	—
57	C₂×D₈×C₄₃	—	2	2	no	—
58	C₂×D₈×C₄₃	—	2	2	no	—
59	C₂×D₈×C₄₃	—	2	2	no	—
60	C₂×D₈×C₄₃	—	4	4	no	—
61	C₂×D₈×C₄₃	—	4	4	no	—
62	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
64	#127×C₄₃	—	2	2	no	—
65	#127×C₄₃	—	2	2	no	—
66	#127×C₄₃	—	2	2	no	—
67	#127×C₄₃	C₂³	1	4	yes	—
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂⁴×C₄₃	—	4	2	no	—
74	C₂×D₄×C₄₃	—	2	4	no	—
75	C₂×D₄×C₄₃	—	2	4	no	—
76	C₂×D₄×C₄₃	—	2	4	no	—
77	C₂×D₄×C₄₃	—	4	2	no	—
78	C₂×D₄×C₄₃	—	4	2	no	—
79	C₂×D₄×C₄₃	—	4	2	no	—
80	C₂×D₄×C₄₃	—	4	4	no	—
81	C₂×D₄×C₄₃	—	4	4	no	—
82	C₂×D₄×C₄₃	—	8	4	no	—
83	C₂×D₄×C₄₃	—	8	4	no	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
87	C₂×C₂²⋊D₄₃	—	2	2	no	—
88	#10621898	—	4	2	no	—
89	#10621898	—	4	2	no	—
90	C₂×D₁₇₂	—	2	1	no	—
91	C₂×D₁₇₂	—	2	1	no	—
92	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
93	C₂×C₄×D₄₃	—	2	2	no	—
94	C₂×Dic₈₆	—	2	1	no	—
95	D₈×C₄₃	—	4	4	no	—
96	D₈×C₄₃	—	4	4	no	—
97	D₈×C₄₃	—	4	4	no	—
98	D₈×C₄₃	—	4	4	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	D₈	1	4	yes	complemented
101	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₃	#41	1	1	yes	characteristic
108	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
109	#10621882	—	8	8	no	—
110	(C₂×D₄₃)⋊C₄	—	2	2	no	—
111	(C₂×D₄₃)⋊C₄	—	2	2	no	—
112	C₄⋊Dic₄₃	—	2	1	no	—
113	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
114	#10621877	—	4	2	no	—
115	C₂×C₄₃⋊C₈	—	2	4	no	—
116	C₂×C₄₃⋊C₈	—	2	2	no	—
117	#10621872	—	4	4	no	—
118	C₈×D₄₃	—	4	4	no	—
119	C₂³×C₄₃	—	4	4	no	—
120	C₂³×C₄₃	—	8	4	no	—
121	C₂³×C₄₃	—	8	4	no	—
122	C₂³×C₄₃	—	8	4	no	—
123	C₂³×C₄₃	#121	1	1	yes	characteristic
124	C₂²×D₄₃	—	2	2	no	—
125	D₄×C₄₃	—	4	4	no	—
126	D₄×C₄₃	—	4	4	no	—
127	D₄×C₄₃	—	4	8	no	—
128	D₄×C₄₃	—	8	4	no	—
129	D₄×C₄₃	—	8	4	no	—
130	D₄×C₄₃	—	8	4	no	—
131	D₄×C₄₃	—	8	4	no	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
139	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
142	C₂²⋊D₄₃	—	4	4	no	—
143	C₂×Dic₄₃	—	2	2	no	—
144	D₁₇₂	—	2	4	no	—
145	D₁₇₂	—	4	2	no	—
146	D₁₇₂	—	4	2	no	—
147	C₄×D₄₃	—	4	2	no	—
148	C₄×D₄₃	—	4	2	no	—
149	Dic₈₆	—	4	2	no	—
150	C₈×C₄₃	—	2	1	no	—
151	C₈×C₄₃	—	2	1	no	—
152	C₈×C₄₃	C₂×D₈	1	2	yes	—
153	C₈×C₄₃	C₂×D₈	1	2	yes	—
154	C₄₃⋊C₈	—	4	2	no	—
155	C₄₃⋊C₈	—	4	4	no	—
156	#19187	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	—	2	1	no	—
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	8	4	no	—
160	C₂²×C₄₃	—	8	4	no	—
161	C₂²×C₄₃	—	8	4	no	—
162	C₂²×C₄₃	—	16	8	no	—
163	C₂²×C₄₃	#459	1	2	yes	—
164	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
165	C₂×D₄₃	—	4	2	no	—
166	C₂×D₄₃	—	4	2	no	—
167	C₂×D₄₃	—	4	2	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	#459	1	2	yes	—
171	C₄×C₄₃	#472	1	2	yes	—
172	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₄×C₄₃	#576	1	2	yes	—
175	Dic₄₃	—	4	2	no	—
176	#3205	—	43	2	no	—
177	C₂×#493	D₄₃	1	1	yes	characteristic, complemented
178	#2931	—	43	1	no	—
179	#1687	—	43	4	no	—
180	#1667	—	43	4	no	—
181	#1564	—	43	1	no	—
182	#1545	—	43	2	no	—
183	C₂×C₄₃	—	2	2	no	—
184	C₂×C₄₃	—	16	4	no	—
185	C₂×C₄₃	C₂×#459	1	1	yes	characteristic
186	C₂×C₄₃	#3058	1	1	yes	characteristic
187	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
188	D₄₃	—	8	2	no	—
189	C₂²×D₈	—	2	1	no	—
190	C₂²×D₈	—	2	1	no	—
191	C₂×#131	—	86	1	no	—
192	#532	—	43	1	no	—
193	C₂×#127	C₂×D₄₃	1	2	yes	—
194	#493	—	2	1	no	—
195	#493	—	2	1	no	—
196	#493	—	86	2	no	—
197	#493	C₂×D₄₃	1	2	yes	—
198	#493	C₂×D₄₃	1	2	yes	—
199	#486	—	86	2	no	—
200	#459	—	43	8	no	—
201	#447	—	86	4	no	—
202	#436	—	43	2	no	—
203	#436	—	86	1	no	—
204	C₈×D₄	—	43	2	no	—
205	C₈×D₄	—	86	1	no	—
206	#431	—	43	2	no	—
207	#423	—	43	1	no	—
208	#408	—	86	1	no	—
209	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
210	#335	—	43	2	no	—
211	#334	—	43	2	no	—
212	#331	—	43	8	no	—
213	C₄₃	#19187	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	86	1	no	—
215	C₂²×D₄	—	2	2	no	—
216	C₂²×D₄	—	2	2	no	—
217	C₂²×D₄	—	2	2	no	—
218	C₂×D₈	—	2	2	no	—
219	C₂×D₈	—	2	2	no	—
220	C₂×D₈	—	2	2	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	—	4	4	no	—
223	C₂×D₈	—	4	4	no	—
224	C₂×D₈	—	86	8	no	—
225	#131	—	172	4	no	—
226	C₂×M₁₆	—	86	1	no	—
227	C₂×M₁₆	—	86	2	no	—
228	C₂²×C₈	—	86	2	no	—
229	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
230	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
231	C₄⋊Q₈	—	43	1	no	—
232	#127	—	2	2	no	—
233	#127	—	2	2	no	—
234	#127	—	2	2	no	—
235	#127	—	43	1	no	—
236	#127	C₂²×D₄₃	1	4	yes	—
237	#124	—	86	1	no	—
238	#121	—	43	2	no	—
239	#121	—	86	1	no	—
240	C₄×D₄	—	43	2	no	—
241	C₄×D₄	—	86	1	no	—
242	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
243	C₄⋊C₈	—	43	2	no	—
244	C₄⋊C₈	—	43	4	no	—
245	C₄⋊C₈	—	86	1	no	—
246	#102	—	86	8	no	—
247	C₂²⋊C₈	—	86	2	no	—
248	C₂²⋊C₈	—	86	2	no	—
249	C₂²⋊C₈	—	86	2	no	—
250	C₄×C₈	—	86	2	no	—
251	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
252	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
255	C₂⁴	—	4	2	no	—
256	#41	—	172	2	no	—
257	#41	—	172	2	no	—
258	C₂×Q₈	—	86	1	no	—
259	C₂×D₄	—	2	4	no	—
260	C₂×D₄	—	2	4	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	4	no	—
266	C₂×D₄	—	4	4	no	—
267	C₂×D₄	—	8	4	no	—
268	C₂×D₄	—	8	4	no	—
269	C₂×D₄	—	43	2	no	—
270	C₂×D₄	—	86	2	no	—
271	C₂×D₄	—	86	1	no	—
272	C₂×D₄	—	86	1	no	—
273	C₂²×C₄	—	86	2	no	—
274	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
275	C₂²×C₄	#10621900	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
277	D₈	—	4	4	no	—
278	D₈	—	4	4	no	—
279	D₈	—	4	4	no	—
280	D₈	—	4	4	no	—
281	D₈	—	344	8	no	—
282	M₁₆	—	172	2	no	—
283	M₁₆	—	172	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	86	4	no	—
286	C₂×C₈	—	86	2	no	—
287	C₂×C₈	—	172	4	no	—
288	C₂×C₈	#10621882	1	4	yes	complemented
289	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
293	C₄⋊C₄	—	43	2	no	—
294	C₄⋊C₄	—	86	1	no	—
295	C₂²⋊C₄	—	86	2	no	—
296	C₂²⋊C₄	—	86	2	no	—
297	C₄²	D₄×D₄₃	1	1	yes	characteristic
298	C₄²	#10621900	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
300	C₄²	C₂³×D₄₃	1	1	yes	characteristic
301	C₂³	—	4	4	no	—
302	C₂³	—	8	4	no	—
303	C₂³	—	8	4	no	—
304	C₂³	—	8	4	no	—
305	C₂³	—	86	2	no	—
306	C₂³	#13063296	1	1	yes	characteristic
307	Q₈	—	172	2	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	8	no	—
311	D₄	—	8	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	4	no	—
314	D₄	—	8	4	no	—
315	D₄	—	86	4	no	—
316	D₄	—	172	4	no	—
317	D₄	—	172	2	no	—
318	D₄	—	172	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	86	2	no	—
322	C₂×C₄	—	172	2	no	—
323	C₂×C₄	—	172	2	no	—
324	C₂×C₄	#13063296	1	1	yes	characteristic
325	C₂×C₄	#13063296	1	1	yes	characteristic
326	C₂×C₄	#13063296	1	1	yes	characteristic
327	C₂×C₄	C₂×#10621882	1	2	yes	—
328	C₂×C₄	#13063351	1	2	yes	—
329	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
331	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	172	2	no	—
335	C₈	—	172	4	no	—
336	C₈	C₂×#10621882	1	2	yes	—
337	C₈	C₂×#10621882	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	8	4	no	—
341	C₂²	—	8	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	16	8	no	—
344	C₂²	—	172	2	no	—
345	C₂²	—	172	2	no	—
346	C₂²	—	172	2	no	—
347	C₂²	#14583091	1	2	yes	—
348	C₂²	C₂×#13063296	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	172	2	no	—
352	C₄	#14583091	1	2	yes	—
353	C₄	#14583166	1	2	yes	—
354	C₄	C₂²×#10621882	1	1	yes	characteristic
355	C₄	C₂×#13063351	1	1	yes	characteristic
356	C₄	#14583700	1	2	yes	—
357	C₂	—	2	2	no	—
358	C₂	—	16	4	no	—
359	C₂	—	344	2	no	—
360	C₂	C₂×#14583091	1	1	yes	characteristic
361	C₂	#19144239	1	1	yes	characteristic
362	C₂	C₂×#14583166	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063296
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#493	C₂
socle	C₂²×C₄₃	C₂×#121
hypercenter	C₂×#493	D₄₃
hypocenter	C₄₃	#19187
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #19145513	C₂	semidirect
2. #19143308	C₂	semidirect
3. #19134071	C₂	semidirect
4. #19133947	C₂	semidirect
5. #19133343	C₂	semidirect
6. #19133263	C₂	semidirect
7. C₂×C₄₃×#493	C₂	semidirect
12. #14583091	C₂²	semidirect
15. #14582978	C₂²	semidirect
17. C₈×D₁₇₂	C₂²	semidirect
21. #14582652	C₂²	semidirect
22. #14582650	C₂²	semidirect
23. #14582643	C₂²	semidirect
100. C₂×C₈×C₄₃	D₈	semidirect
177. C₂×#493	D₄₃	semidirect
213. C₄₃	#19187	semidirect
288. C₂×C₈	#10621882	semidirect

Maximal subgroups

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

subgroup	index
#19145513	2
#19143308	2
#19134071	2
#19133947	2
#19133343	2
#19133263	2
C₂×C₄₃×#493	2
#19187	43