Subgroups — #25080640

Order 11008.

706 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145174	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144686	C₂	1	2	yes	maximal, complemented
3	#19134128	C₂	1	4	yes	maximal, complemented
4	#19134118	C₂	1	2	yes	maximal, complemented
5	#19134074	C₂	1	2	yes	maximal, complemented
6	#19133358	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133288	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583449	C₂²	1	2	yes	—
10	#14583304	C₂²	1	1	yes	characteristic, complemented
11	#14583276	C₂²	1	1	yes	characteristic, complemented
12	#14583246	C₂²	1	2	yes	complemented
13	#14583217	—	2	2	no	—
14	#14583189	C₂²	1	4	yes	—
15	#14583135	—	2	1	no	—
16	#14583135	—	2	1	no	—
17	#14583094	C₂²	1	4	yes	—
18	#14583033	—	2	2	no	—
19	#14583009	—	2	4	no	—
20	C₄×D₃₄₄	—	2	1	no	—
21	C₄×D₃₄₄	—	2	1	no	—
22	#14582956	C₂²	1	2	yes	—
23	#14582652	C₂²	1	2	yes	complemented
24	#14582650	C₂²	1	2	yes	complemented
25	#14582647	C₂²	1	4	yes	complemented
26	#14582642	C₂²	1	4	yes	complemented
27	C₂²×M₁₆×C₄₃	—	2	1	no	—
28	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#436	—	2	1	no	—
30	C₄₃×#436	—	2	1	no	—
31	C₄₃×#436	C₂²	1	2	yes	—
32	C₄₃×#436	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₄₃	—	2	1	no	—
35	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#10621898	—	2	2	no	—
37	C₂×D₃₄₄	—	2	1	no	—
38	C₂×D₃₄₄	—	2	1	no	—
39	C₂×#10621882	—	2	4	no	—
40	#13063325	C₂³	1	2	yes	—
41	#13063322	C₂³	1	2	yes	—
42	#13063313	—	2	2	no	—
43	#13063299	—	2	2	no	—
44	#13063299	—	2	2	no	—
45	C₄×D₁₇₂	—	2	2	no	—
46	C₄×D₁₇₂	—	2	2	no	—
47	#13063259	—	2	1	no	—
48	#13063259	—	2	1	no	—
49	#13063256	—	2	4	no	—
50	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
51	#13063247	—	2	2	no	—
52	#13063247	—	2	2	no	—
53	#13063246	—	2	4	no	—
54	#13063245	—	2	4	no	—
55	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
56	#13063236	—	2	2	no	—
57	C₂²×D₄×C₄₃	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₄₃	—	2	1	no	—
59	C₂×M₁₆×C₄₃	—	2	2	no	—
60	C₂×M₁₆×C₄₃	—	2	2	no	—
61	C₂×M₁₆×C₄₃	—	4	4	no	—
62	C₂²×C₈×C₄₃	—	2	1	no	—
63	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
65	C₄×D₄×C₄₃	—	2	1	no	—
66	C₄×D₄×C₄₃	—	2	1	no	—
67	C₄×D₄×C₄₃	D₄	1	2	yes	—
68	C₄×D₄×C₄₃	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
71	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₄₃	—	2	1	no	—
73	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
74	C₂²⋊C₈×C₄₃	—	2	2	no	—
75	C₂²⋊C₈×C₄₃	—	2	2	no	—
76	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
80	C₂⁴×C₄₃	—	2	1	no	—
81	C₂×D₄×C₄₃	—	2	1	no	—
82	C₂×D₄×C₄₃	—	4	2	no	—
83	C₂×D₄×C₄₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₃	—	2	2	no	—
85	C₂²×C₄×C₄₃	—	2	2	no	—
86	C₂²×C₄×C₄₃	—	2	1	no	—
87	C₂²×C₄×C₄₃	—	4	2	no	—
88	C₂²×C₄×C₄₃	—	4	2	no	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₃	—	4	2	no	—
93	#10621898	—	8	4	no	—
94	C₂×D₁₇₂	—	2	2	no	—
95	C₂×D₁₇₂	—	2	2	no	—
96	C₂×D₁₇₂	—	2	2	no	—
97	C₂×C₄×D₄₃	—	4	2	no	—
98	C₂×Dic₈₆	—	2	2	no	—
99	M₁₆×C₄₃	—	4	4	no	—
100	M₁₆×C₄₃	—	4	4	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	4	1	no	—
104	C₂×C₈×C₄₃	—	4	2	no	—
105	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₄⋊C₄×C₄₃	—	2	1	no	—
110	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₄₃	—	2	2	no	—
112	C₂²⋊C₄×C₄₃	—	2	2	no	—
113	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₃	#41	1	1	yes	characteristic
116	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
117	#10621882	—	4	8	no	—
118	(C₂×D₄₃)⋊C₄	—	4	2	no	—
119	(C₂×D₄₃)⋊C₄	—	4	2	no	—
120	C₄⋊Dic₄₃	—	2	2	no	—
121	C₄⋊Dic₄₃	—	2	2	no	—
122	C₂×C₄₃⋊C₈	—	2	4	no	—
123	D₃₄₄	—	4	2	no	—
124	D₃₄₄	—	4	2	no	—
125	C₂³×C₄₃	—	2	2	no	—
126	C₂³×C₄₃	—	4	2	no	—
127	C₂³×C₄₃	—	4	2	no	—
128	C₂³×C₄₃	—	4	2	no	—
129	C₂³×C₄₃	#121	1	1	yes	characteristic
130	C₂²×D₄₃	—	4	2	no	—
131	D₄×C₄₃	—	2	4	no	—
132	D₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	2	1	no	—
135	C₂×C₄×C₄₃	—	2	1	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	4	4	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₄₃	—	8	4	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	D₁₇₂	—	4	2	no	—
151	D₁₇₂	—	4	2	no	—
152	D₁₇₂	—	4	4	no	—
153	D₁₇₂	—	8	2	no	—
154	C₄×D₄₃	—	8	4	no	—
155	Dic₈₆	—	8	2	no	—
156	C₈×C₄₃	—	2	1	no	—
157	C₈×C₄₃	—	2	1	no	—
158	C₈×C₄₃	—	2	1	no	—
159	C₈×C₄₃	—	4	2	no	—
160	C₈×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
161	C₄₃⋊C₈	—	4	4	no	—
162	#19283	—	43	1	no	maximal, Sylow
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	—	4	2	no	—
166	C₂²×C₄₃	—	4	2	no	—
167	C₂²×C₄₃	—	4	2	no	—
168	C₂²×C₄₃	—	8	4	no	—
169	C₂²×C₄₃	#472	1	2	yes	—
170	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
171	C₂×D₄₃	—	8	2	no	—
172	C₂×D₄₃	—	8	2	no	—
173	C₂×D₄₃	—	8	2	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	2	1	no	—
177	C₄×C₄₃	—	2	1	no	—
178	C₄×C₄₃	—	4	2	no	—
179	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₄×C₄₃	#575	1	1	yes	characteristic
181	C₄×C₄₃	#575	1	2	yes	—
182	Dic₄₃	—	8	2	no	—
183	#3169	—	43	1	no	—
184	#3109	—	43	2	no	—
185	C₂×#436	D₄₃	1	1	yes	characteristic, complemented
186	#1697	—	43	4	no	—
187	#1695	—	43	2	no	—
188	#1687	—	43	2	no	—
189	#1568	—	43	1	no	—
190	#1551	—	43	2	no	—
191	C₂×C₄₃	—	2	2	no	—
192	C₂×C₄₃	—	8	2	no	—
193	C₂×C₄₃	#3036	1	1	yes	characteristic
194	C₂×C₄₃	#3054	1	1	yes	characteristic
195	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
196	D₄₃	—	16	2	no	—
197	C₂²×M₁₆	—	2	1	no	—
198	#532	—	43	2	no	—
199	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
200	#500	—	43	1	no	—
201	#493	—	43	1	no	—
202	#486	—	43	2	no	—
203	#481	—	86	2	no	—
204	#476	—	43	4	no	—
205	#466	—	86	1	no	—
206	#466	—	86	1	no	—
207	#459	—	43	4	no	—
208	#447	—	86	2	no	—
209	#442	—	86	4	no	—
210	C₄×D₈	—	86	1	no	—
211	C₄×D₈	—	86	1	no	—
212	#436	—	2	1	no	—
213	#436	—	2	1	no	—
214	#436	C₂×D₄₃	1	2	yes	—
215	#436	C₂×D₄₃	1	2	yes	—
216	#432	—	43	2	no	—
217	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
218	C₂×C₂²⋊C₈	—	2	1	no	—
219	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
220	#335	—	43	2	no	—
221	#334	—	43	2	no	—
222	#333	—	43	4	no	—
223	#331	—	43	4	no	—
224	C₄₃	#19283	1	1	yes	characteristic, complemented, Sylow
225	C₂×#41	—	86	2	no	—
226	C₂²×D₄	D₁₇₂	1	1	yes	characteristic
227	C₂³×C₄	—	2	1	no	—
228	C₂×D₈	—	86	4	no	—
229	C₂×D₈	—	86	1	no	—
230	C₂×D₈	—	86	1	no	—
231	C₂×M₁₆	—	2	2	no	—
232	C₂×M₁₆	—	2	2	no	—
233	C₂×M₁₆	—	4	4	no	—
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
236	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
237	C₄⋊Q₈	—	43	2	no	—
238	#127	—	43	2	no	—
239	#124	—	86	2	no	—
240	#121	—	86	2	no	—
241	#121	—	86	2	no	—
242	C₄×D₄	—	2	1	no	—
243	C₄×D₄	—	2	1	no	—
244	C₄×D₄	—	86	2	no	—
245	C₄×D₄	—	86	2	no	—
246	C₄×D₄	D₁₇₂	1	2	yes	—
247	C₄×D₄	C₂²×D₄₃	1	2	yes	—
248	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
249	C₂×C₂²⋊C₄	—	2	1	no	—
250	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	#107	—	86	1	no	—
252	#107	—	86	1	no	—
253	#106	—	86	4	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	—	43	4	no	—
256	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
257	#102	—	86	4	no	—
258	#102	—	86	2	no	—
259	#102	—	86	2	no	—
260	#102	—	86	4	no	—
261	C₂²⋊C₈	—	2	2	no	—
262	C₂²⋊C₈	—	2	2	no	—
263	C₂²⋊C₈	—	86	2	no	—
264	#97	—	86	2	no	—
265	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
266	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
269	C₂⁴	—	2	1	no	—
270	#41	—	344	4	no	—
271	C₂×Q₈	—	86	2	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	86	2	no	—
275	C₂×D₄	—	86	2	no	—
276	C₂×D₄	—	86	2	no	—
277	C₂×D₄	—	172	2	no	—
278	C₂×D₄	C₂×D₁₇₂	1	2	yes	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	—	172	2	no	—
285	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
286	C₂²×C₄	#10621898	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
288	D₈	—	172	2	no	—
289	D₈	—	172	8	no	—
290	D₈	—	172	2	no	—
291	M₁₆	—	4	4	no	—
292	M₁₆	—	4	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	4	1	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	86	4	no	—
298	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
299	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
300	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
301	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	86	2	no	—
304	C₄⋊C₄	—	86	2	no	—
305	C₄⋊C₄	C₂×D₁₇₂	1	2	yes	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	172	2	no	—
309	C₂²⋊C₄	—	172	2	no	—
310	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
311	C₄²	#10621898	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
313	C₄²	C₂³×D₄₃	1	1	yes	characteristic
314	C₂³	—	2	2	no	—
315	C₂³	—	4	2	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	172	2	no	—
319	C₂³	#13063299	1	1	yes	characteristic
320	Q₈	—	344	2	no	—
321	D₄	—	2	4	no	—
322	D₄	—	4	2	no	—
323	D₄	—	344	2	no	—
324	D₄	—	344	4	no	—
325	D₄	—	172	2	no	—
326	D₄	—	172	2	no	—
327	D₄	—	172	4	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	344	4	no	—
338	C₂×C₄	—	172	2	no	—
339	C₂×C₄	#13063299	1	1	yes	characteristic
340	C₂×C₄	#13063299	1	1	yes	characteristic
341	C₂×C₄	#13063299	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
344	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	4	2	no	—
349	C₈	—	172	4	no	—
350	C₈	#13063351	1	2	yes	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	8	4	no	—
357	C₂²	—	344	2	no	—
358	C₂²	—	344	2	no	—
359	C₂²	—	344	2	no	—
360	C₂²	#14583169	1	2	yes	—
361	C₂²	C₂×#13063299	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	344	2	no	—
368	C₄	C₂×#13063351	1	1	yes	characteristic
369	C₄	#14583697	1	1	yes	characteristic
370	C₄	#14583699	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	8	2	no	—
373	C₂	—	688	2	no	—
374	C₂	#19144059	1	1	yes	characteristic
375	C₂	#19144204	1	1	yes	characteristic
376	C₂	C₂×#14583169	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063299
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#436	C₂
socle	C₂²×C₄₃	C₂×#121
hypercenter	C₂×#436	D₄₃
hypocenter	C₄₃	#19283
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #19145174	C₂	semidirect
2. #19144686	C₂	semidirect
3. #19134128	C₂	semidirect
4. #19134118	C₂	semidirect
5. #19134074	C₂	semidirect
6. #19133358	C₂	semidirect
7. #19133288	C₂	semidirect
8. C₂×C₄₃×#436	C₂	semidirect
10. #14583304	C₂²	semidirect
11. #14583276	C₂²	semidirect
12. #14583246	C₂²	semidirect
23. #14582652	C₂²	semidirect
24. #14582650	C₂²	semidirect
25. #14582647	C₂²	semidirect
26. #14582642	C₂²	semidirect
185. C₂×#436	D₄₃	semidirect
224. C₄₃	#19283	semidirect

Maximal subgroups

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

subgroup	index
#19145174	2
#19144686	2
#19134128	2
#19134118	2
#19134074	2
#19133358	2
#19133288	2
C₂×C₄₃×#436	2
#19283	43