Subgroups — #24989700

Order 11008.

494 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582976	C₂	1	1	yes	maximal, characteristic
2	#19134722	C₂	1	1	yes	maximal, characteristic, complemented
3	#19134596	C₂	1	1	yes	maximal, characteristic
4	#19134592	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582651	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132479	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621872	—	2	1	no	—
9	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
10	#14582976	—	2	2	no	—
11	#14582976	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#13063247	—	2	1	no	—
14	C₂×#13063245	—	2	1	no	—
15	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	#14582769	C₂²	1	1	yes	characteristic
18	#14582662	—	2	1	no	—
19	#14582661	—	2	1	no	—
20	#14582661	—	2	1	no	—
21	#14582661	—	2	1	no	—
22	#14582651	—	2	2	no	—
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#384	C₂²	1	1	yes	characteristic
26	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₃	—	2	1	no	—
28	C₂×#10621872	—	2	4	no	—
29	C₂×#10621872	—	4	2	no	—
30	C₂×#10621872	—	4	2	no	—
31	C₂²×C₄₃⋊C₈	—	2	1	no	—
32	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
34	C₄×D₁₇₂	—	2	1	no	—
35	C₄×D₁₇₂	—	2	1	no	—
36	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
37	C₄×D₁₇₂	Q₈	1	2	yes	—
38	C₂×C₄⋊Dic₄₃	—	2	1	no	—
39	C₂×C₄⋊Dic₄₃	—	2	1	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
42	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
43	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
45	#13063247	—	4	2	no	—
46	#13063245	—	4	2	no	—
47	(C₂×D₄₃)⋊C₈	—	2	4	no	—
48	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
50	C₂²×C₈×C₄₃	—	2	1	no	—
51	C₂²×C₈×C₄₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	—	2	1	no	—
57	C₄⋊C₈×C₄₃	—	2	1	no	—
58	C₄⋊C₈×C₄₃	—	2	1	no	—
59	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
60	C₂³×D₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	—	4	1	no	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
65	C₂²×Dic₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	—	4	1	no	—
67	C₂×D₁₇₂	—	2	1	no	—
68	C₂×D₁₇₂	—	2	1	no	—
69	C₂×D₁₇₂	—	2	1	no	—
70	C₂×D₁₇₂	—	2	1	no	—
71	C₂×D₁₇₂	—	2	1	no	—
72	C₂×D₁₇₂	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₄₃	—	2	2	no	—
74	C₂×C₄×D₄₃	—	2	2	no	—
75	C₂×C₄×D₄₃	—	4	1	no	—
76	C₂×C₄×D₄₃	—	4	1	no	—
77	C₂×C₄×D₄₃	—	4	1	no	—
78	C₂×C₄×D₄₃	—	4	1	no	—
79	C₂×C₈×C₄₃	—	2	2	no	—
80	C₂×C₈×C₄₃	—	4	1	no	—
81	C₂×C₈×C₄₃	—	4	1	no	—
82	C₂×C₈×C₄₃	—	4	1	no	—
83	C₂×C₈×C₄₃	—	4	1	no	—
84	C₂×C₈×C₄₃	—	4	1	no	—
85	C₄⋊C₄×C₄₃	—	4	1	no	—
86	C₄⋊C₄×C₄₃	—	4	1	no	—
87	C₄⋊C₄×C₄₃	—	4	1	no	—
88	C₄⋊C₄×C₄₃	—	4	1	no	—
89	C₄⋊C₄×C₄₃	—	8	1	no	—
90	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
93	C₄²×C₄₃	#41	1	1	yes	characteristic
94	(C₂×D₄₃)⋊C₄	—	2	2	no	—
95	(C₂×D₄₃)⋊C₄	—	2	2	no	—
96	C₄⋊Dic₄₃	—	2	1	no	—
97	C₄⋊Dic₄₃	—	4	1	no	—
98	C₄⋊Dic₄₃	—	4	1	no	—
99	C₄⋊Dic₄₃	—	4	1	no	—
100	C₄⋊Dic₄₃	—	4	1	no	—
101	C₄⋊Dic₄₃	—	8	1	no	—
102	C₄⋊Dic₄₃	C₄⋊C₄	1	2	yes	—
103	C₂×C₄₃⋊C₈	—	2	1	no	—
104	C₂×C₄₃⋊C₈	—	2	1	no	—
105	C₂×C₄₃⋊C₈	—	2	1	no	—
106	C₂×C₄₃⋊C₈	—	2	1	no	—
107	C₂×C₄₃⋊C₈	—	4	1	no	—
108	C₂×C₄₃⋊C₈	—	4	1	no	—
109	C₂×C₄₃⋊C₈	—	4	1	no	—
110	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
111	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
112	#10621872	—	4	8	no	—
113	C₂³×C₄₃	#121	1	1	yes	characteristic
114	C₂²×D₄₃	—	2	2	no	—
115	C₂²×D₄₃	—	4	1	no	—
116	C₂²×D₄₃	—	4	1	no	—
117	C₂²×D₄₃	—	4	1	no	—
118	C₂²×D₄₃	—	4	1	no	—
119	C₂²×D₄₃	—	4	1	no	—
120	C₂²×D₄₃	—	4	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	8	1	no	—
124	C₂×C₄×C₄₃	—	8	1	no	—
125	C₂×C₄×C₄₃	—	8	1	no	—
126	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
134	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
140	C₂×Dic₄₃	—	2	2	no	—
141	C₂×Dic₄₃	—	4	1	no	—
142	C₂×Dic₄₃	—	4	1	no	—
143	C₂×Dic₄₃	—	8	1	no	—
144	C₂×Dic₄₃	—	8	1	no	—
145	C₂×Dic₄₃	—	8	1	no	—
146	D₁₇₂	—	2	2	no	—
147	D₁₇₂	—	2	2	no	—
148	D₁₇₂	—	4	1	no	—
149	D₁₇₂	—	4	1	no	—
150	C₄×D₄₃	—	4	2	no	—
151	C₄×D₄₃	—	4	2	no	—
152	C₄×D₄₃	—	4	2	no	—
153	C₄×D₄₃	—	4	2	no	—
154	C₈×C₄₃	—	4	2	no	—
155	C₈×C₄₃	—	8	1	no	—
156	C₄₃⋊C₈	—	2	2	no	—
157	C₄₃⋊C₈	—	2	2	no	—
158	C₄₃⋊C₈	—	8	1	no	—
159	#8654	—	43	1	no	maximal, Sylow
160	C₂²×C₄₃	#380	1	1	yes	characteristic
161	C₂²×C₄₃	#463	1	1	yes	characteristic
162	C₂²×C₄₃	#463	1	1	yes	characteristic
163	C₂²×C₄₃	#468	1	1	yes	characteristic
164	C₂²×C₄₃	#468	1	1	yes	characteristic
165	C₂²×C₄₃	#472	1	1	yes	characteristic
166	C₂²×C₄₃	#473	1	1	yes	characteristic
167	C₂×D₄₃	—	4	2	no	—
168	C₂×D₄₃	—	4	2	no	—
169	C₂×D₄₃	—	4	2	no	—
170	C₂×D₄₃	—	8	1	no	—
171	C₂×D₄₃	—	8	1	no	—
172	C₂×D₄₃	—	8	1	no	—
173	C₂×D₄₃	—	8	1	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	16	1	no	—
177	C₄×C₄₃	#416	1	1	yes	characteristic
178	C₄×C₄₃	#418	1	1	yes	characteristic
179	C₄×C₄₃	#428	1	1	yes	characteristic
180	C₄×C₄₃	#428	1	1	yes	characteristic
181	C₄×C₄₃	#469	1	1	yes	characteristic
182	C₄×C₄₃	#470	1	1	yes	characteristic
183	C₄×C₄₃	#478	1	1	yes	characteristic
184	C₄×C₄₃	#478	1	1	yes	characteristic
185	Dic₄₃	—	4	2	no	—
186	Dic₄₃	—	16	1	no	—
187	C₂×#436	—	43	1	no	—
188	#1834	—	43	1	no	—
189	#1802	—	43	1	no	—
190	#1802	—	43	1	no	—
191	C₂×#335	—	43	1	no	—
192	#1296	—	43	1	no	—
193	#1296	D₄₃	1	1	yes	characteristic, complemented
194	C₂×C₄₃	#1697	1	1	yes	characteristic
195	C₂×C₄₃	#1697	1	1	yes	characteristic
196	C₂×C₄₃	#1699	1	1	yes	characteristic
197	C₂×C₄₃	#1699	1	1	yes	characteristic
198	C₂×C₄₃	#1804	1	1	yes	characteristic
199	C₂×C₄₃	#1809	1	1	yes	characteristic
200	C₂×C₄₃	#1816	1	1	yes	characteristic
201	D₄₃	—	8	2	no	—
202	C₂²×M₁₆	—	86	1	no	—
203	C₂×C₄×D₄	—	43	1	no	—
204	#436	—	43	4	no	—
205	#436	—	86	2	no	—
206	C₂×C₄⋊C₈	—	43	1	no	—
207	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
208	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
209	C₂×#102	—	86	1	no	—
210	C₂×#102	—	86	1	no	—
211	C₂×C₂²⋊C₈	—	86	1	no	—
212	C₂×C₄×C₈	—	43	1	no	—
213	#384	—	43	1	no	—
214	#384	C₂×D₄₃	1	1	yes	characteristic
215	#340	—	86	1	no	—
216	#340	—	86	1	no	—
217	#340	—	86	1	no	—
218	#340	—	86	1	no	—
219	#335	—	86	2	no	—
220	C₄₃	#8654	1	1	yes	characteristic, complemented, Sylow
221	C₂²×D₄	—	43	1	no	—
222	C₂³×C₄	—	86	1	no	—
223	C₂×M₁₆	—	86	4	no	—
224	C₂×M₁₆	—	172	2	no	—
225	C₂×M₁₆	—	172	2	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	—	2	1	no	—
228	C₂²×C₈	—	43	1	no	—
229	C₂²×C₈	—	43	1	no	—
230	C₂²×C₈	—	86	1	no	—
231	C₄×D₄	—	43	2	no	—
232	C₄×D₄	—	43	2	no	—
233	C₄×D₄	—	86	1	no	—
234	C₄×D₄	—	86	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	43	1	no	—
239	C₂×C₄⋊C₄	—	86	1	no	—
240	C₂×C₄⋊C₄	—	86	1	no	—
241	C₂×C₄⋊C₄	—	86	1	no	—
242	C₂×C₂²⋊C₄	—	86	1	no	—
243	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	86	1	no	—
248	C₄⋊C₈	—	86	1	no	—
249	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
250	#102	—	172	2	no	—
251	#102	—	172	2	no	—
252	C₂²⋊C₈	—	86	4	no	—
253	C₄×C₈	—	43	2	no	—
254	C₄×C₈	—	43	2	no	—
255	C₂⁴	—	86	1	no	—
256	C₂×D₄	—	43	2	no	—
257	C₂×D₄	—	86	1	no	—
258	C₂×D₄	—	86	1	no	—
259	C₂×D₄	—	86	1	no	—
260	C₂×D₄	—	86	1	no	—
261	C₂×D₄	—	86	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	86	1	no	—
264	C₂²×C₄	—	86	2	no	—
265	C₂²×C₄	—	86	2	no	—
266	C₂²×C₄	—	172	1	no	—
267	C₂²×C₄	—	172	1	no	—
268	C₂²×C₄	—	172	1	no	—
269	C₂²×C₄	—	172	1	no	—
270	C₂²×C₄	—	172	1	no	—
271	C₂²×C₄	#10621898	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
273	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
274	M₁₆	—	172	8	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	43	2	no	—
282	C₂×C₈	—	43	2	no	—
283	C₂×C₈	—	86	1	no	—
284	C₂×C₈	—	86	1	no	—
285	C₂×C₈	—	86	1	no	—
286	C₂×C₈	—	86	1	no	—
287	C₂×C₈	—	172	1	no	—
288	C₂×C₈	—	172	1	no	—
289	C₂×C₈	—	172	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	8	1	no	—
295	C₄⋊C₄	—	43	2	no	—
296	C₄⋊C₄	—	86	1	no	—
297	C₄⋊C₄	—	344	1	no	—
298	C₄⋊C₄	—	172	1	no	—
299	C₄⋊C₄	—	172	1	no	—
300	C₄⋊C₄	—	172	1	no	—
301	C₄⋊C₄	—	172	1	no	—
302	C₂²⋊C₄	—	86	2	no	—
303	C₂²⋊C₄	—	86	2	no	—
304	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
305	C₄²	#10621898	1	1	yes	characteristic
306	C₄²	D₄×D₄₃	1	1	yes	characteristic
307	C₄²	Q₈×D₄₃	1	1	yes	characteristic
308	C₂³	—	86	2	no	—
309	C₂³	—	172	1	no	—
310	C₂³	—	172	1	no	—
311	C₂³	—	172	1	no	—
312	C₂³	—	172	1	no	—
313	C₂³	—	172	1	no	—
314	C₂³	—	172	1	no	—
315	C₂³	#13063298	1	1	yes	characteristic
316	D₄	—	86	2	no	—
317	D₄	—	86	2	no	—
318	D₄	—	172	1	no	—
319	D₄	—	172	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	8	1	no	—
323	C₂×C₄	—	8	1	no	—
324	C₂×C₄	—	8	1	no	—
325	C₂×C₄	—	86	2	no	—
326	C₂×C₄	—	344	1	no	—
327	C₂×C₄	—	344	1	no	—
328	C₂×C₄	—	344	1	no	—
329	C₂×C₄	—	172	1	no	—
330	C₂×C₄	—	172	1	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	—	172	2	no	—
333	C₂×C₄	—	172	2	no	—
334	C₂×C₄	—	172	2	no	—
335	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
336	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
337	C₂×C₄	#13063282	1	1	yes	characteristic
338	C₂×C₄	#13063298	1	1	yes	characteristic
339	C₂×C₄	#13063303	1	1	yes	characteristic
340	C₂×C₄	#13063303	1	1	yes	characteristic
341	C₂×C₄	#13063346	1	1	yes	characteristic
342	C₂×C₄	#13063348	1	1	yes	characteristic
343	C₂×C₄	#13063349	1	1	yes	characteristic
344	C₂×C₄	#13063349	1	1	yes	characteristic
345	C₂×C₄	#13063352	1	1	yes	characteristic
346	C₂×C₄	#13063352	1	1	yes	characteristic
347	C₂×C₄	#13063354	1	1	yes	characteristic
348	C₂×C₄	#13063357	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	8	1	no	—
351	C₈	—	86	2	no	—
352	C₈	—	86	2	no	—
353	C₈	—	344	1	no	—
354	C₂²	—	344	1	no	—
355	C₂²	—	344	1	no	—
356	C₂²	—	344	1	no	—
357	C₂²	—	344	1	no	—
358	C₂²	—	172	2	no	—
359	C₂²	—	172	2	no	—
360	C₂²	—	172	2	no	—
361	C₂²	#14582758	1	1	yes	characteristic
362	C₂²	#14583113	1	1	yes	characteristic
363	C₂²	#14583117	1	1	yes	characteristic
364	C₂²	#14583144	1	1	yes	characteristic
365	C₂²	#14583148	1	1	yes	characteristic
366	C₂²	#14583168	1	1	yes	characteristic
367	C₂²	#14583173	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	16	1	no	—
371	C₄	—	172	2	no	—
372	C₄	—	688	1	no	—
373	C₄	#14582894	1	1	yes	characteristic
374	C₄	#14582902	1	1	yes	characteristic
375	C₄	#14582938	1	1	yes	characteristic
376	C₄	#14582940	1	1	yes	characteristic
377	C₄	#14583151	1	1	yes	characteristic
378	C₄	#14583156	1	1	yes	characteristic
379	C₄	#14583198	1	1	yes	characteristic
380	C₄	#14583202	1	1	yes	characteristic
381	C₂	—	344	2	no	—
382	C₂	#19134127	1	1	yes	characteristic
383	C₂	#19134131	1	1	yes	characteristic
384	C₂	#19134141	1	1	yes	characteristic
385	C₂	#19134145	1	1	yes	characteristic
386	C₂	#19134604	1	1	yes	characteristic
387	C₂	#19134627	1	1	yes	characteristic
388	C₂	#19134658	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063298
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1296	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂²×D₁₇₂	D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1296	D₄₃
hypocenter	C₄₃	#8654
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #19134722	C₂	semidirect
5. C₂×#14582651	C₂	semidirect
6. #19132479	C₂	semidirect
7. C₄₃×#1296	C₂	semidirect
11. #14582976	C₄	semidirect
193. #1296	D₄₃	semidirect
220. C₄₃	#8654	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582976	2
#19134722	2
#19134596	2
#19134592	2
C₂×#14582651	2
#19132479	2
C₄₃×#1296	2
#8654	43