Subgroups — #25001340

Order 11008.

402 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	#19135388	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	#19132479	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132429	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#13063247	—	2	1	no	—
11	C₂×#13063245	—	2	1	no	—
12	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
13	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
15	#14582769	C₂²	1	1	yes	characteristic
16	#14582662	—	2	1	no	—
17	#14582661	—	2	1	no	—
18	#14582661	—	2	1	no	—
19	#14582661	—	2	1	no	—
20	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#384	C₂²	1	1	yes	characteristic
23	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
24	C₂²×C₄×D₄₃	—	2	1	no	—
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
27	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
28	C₄×D₁₇₂	—	2	1	no	—
29	C₄×D₁₇₂	—	2	1	no	—
30	C₄×D₁₇₂	—	2	1	no	—
31	C₄×D₁₇₂	—	2	1	no	—
32	C₂×C₄⋊Dic₄₃	—	2	1	no	—
33	C₂×C₄⋊Dic₄₃	—	2	1	no	—
34	C₂×C₄⋊Dic₄₃	—	2	1	no	—
35	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
36	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
37	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
38	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
39	#13063247	—	4	2	no	—
40	#13063245	—	4	2	no	—
41	(C₂×D₄₃)⋊C₈	—	4	2	no	—
42	(C₂×D₄₃)⋊C₈	—	4	2	no	—
43	C₄×C₄₃⋊C₈	—	2	1	no	—
44	C₄×C₄₃⋊C₈	—	2	1	no	—
45	C₂²×C₈×C₄₃	—	2	1	no	—
46	C₂²×C₈×C₄₃	—	2	1	no	—
47	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
50	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₄₃	—	2	1	no	—
52	C₄⋊C₈×C₄₃	—	2	1	no	—
53	C₄⋊C₈×C₄₃	—	2	1	no	—
54	C₄⋊C₈×C₄₃	—	2	1	no	—
55	C₂³×D₄₃	—	2	1	no	—
56	C₂²×C₄×C₄₃	—	4	1	no	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
60	C₂²×Dic₄₃	—	2	1	no	—
61	C₂²×Dic₄₃	—	4	1	no	—
62	C₂×D₁₇₂	—	2	1	no	—
63	C₂×D₁₇₂	—	2	1	no	—
64	C₂×D₁₇₂	—	2	1	no	—
65	C₂×D₁₇₂	—	2	1	no	—
66	C₂×D₁₇₂	—	2	1	no	—
67	C₂×D₁₇₂	Q₁₆	1	2	yes	—
68	C₂×C₄×D₄₃	—	4	1	no	—
69	C₂×C₄×D₄₃	—	4	1	no	—
70	C₂×C₄×D₄₃	—	4	1	no	—
71	C₂×C₄×D₄₃	—	4	1	no	—
72	C₂×C₄×D₄₃	—	4	1	no	—
73	C₂×C₄×D₄₃	—	4	1	no	—
74	C₂×C₈×C₄₃	—	4	1	no	—
75	C₂×C₈×C₄₃	—	4	1	no	—
76	C₂×C₈×C₄₃	—	4	1	no	—
77	C₂×C₈×C₄₃	—	4	1	no	—
78	C₂×C₈×C₄₃	—	4	1	no	—
79	C₂×C₈×C₄₃	—	4	1	no	—
80	C₄⋊C₄×C₄₃	—	4	1	no	—
81	C₄⋊C₄×C₄₃	—	4	1	no	—
82	C₄⋊C₄×C₄₃	—	4	1	no	—
83	C₄⋊C₄×C₄₃	—	4	1	no	—
84	C₄⋊C₄×C₄₃	—	8	1	no	—
85	C₄²×C₄₃	#41	1	1	yes	characteristic
86	C₄²×C₄₃	#41	1	1	yes	characteristic
87	C₄²×C₄₃	#41	1	1	yes	characteristic
88	C₄²×C₄₃	#41	1	1	yes	characteristic
89	(C₂×D₄₃)⋊C₄	—	4	1	no	—
90	(C₂×D₄₃)⋊C₄	—	4	1	no	—
91	C₄⋊Dic₄₃	—	2	1	no	—
92	C₄⋊Dic₄₃	—	2	1	no	—
93	C₄⋊Dic₄₃	—	4	1	no	—
94	C₄⋊Dic₄₃	—	4	1	no	—
95	C₄⋊Dic₄₃	—	4	1	no	—
96	C₄⋊Dic₄₃	—	4	1	no	—
97	C₄⋊Dic₄₃	—	8	1	no	—
98	C₂×C₄₃⋊C₈	—	2	1	no	—
99	C₂×C₄₃⋊C₈	—	2	1	no	—
100	C₂×C₄₃⋊C₈	—	2	1	no	—
101	C₂×C₄₃⋊C₈	—	2	1	no	—
102	C₂×C₄₃⋊C₈	—	2	1	no	—
103	C₂×C₄₃⋊C₈	—	2	1	no	—
104	C₂×C₄₃⋊C₈	—	4	1	no	—
105	C₂×C₄₃⋊C₈	—	4	1	no	—
106	C₂×C₄₃⋊C₈	—	4	1	no	—
107	C₂³×C₄₃	#121	1	1	yes	characteristic
108	C₂²×D₄₃	—	2	2	no	—
109	C₂²×D₄₃	—	4	1	no	—
110	C₂²×D₄₃	—	4	1	no	—
111	C₂²×D₄₃	—	4	1	no	—
112	C₂²×D₄₃	—	4	1	no	—
113	C₂²×D₄₃	—	4	1	no	—
114	C₂²×D₄₃	—	4	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	8	1	no	—
118	C₂×C₄×C₄₃	—	8	1	no	—
119	C₂×C₄×C₄₃	—	8	1	no	—
120	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#126	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#126	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
130	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×Dic₄₃	—	4	1	no	—
135	C₂×Dic₄₃	—	4	1	no	—
136	C₂×Dic₄₃	—	4	1	no	—
137	C₂×Dic₄₃	—	8	1	no	—
138	C₂×Dic₄₃	—	8	1	no	—
139	C₂×Dic₄₃	—	8	1	no	—
140	D₁₇₂	—	2	2	no	—
141	D₁₇₂	—	2	2	no	—
142	D₁₇₂	—	4	1	no	—
143	D₁₇₂	—	4	1	no	—
144	C₄×D₄₃	—	8	1	no	—
145	C₄×D₄₃	—	8	1	no	—
146	C₄×D₄₃	—	8	1	no	—
147	C₄×D₄₃	—	8	1	no	—
148	C₈×C₄₃	—	8	1	no	—
149	C₈×C₄₃	—	8	1	no	—
150	C₄₃⋊C₈	—	4	1	no	—
151	C₄₃⋊C₈	—	4	1	no	—
152	C₄₃⋊C₈	—	8	1	no	—
153	#10622	—	43	1	no	maximal, Sylow
154	C₂²×C₄₃	#397	1	1	yes	characteristic
155	C₂²×C₄₃	#463	1	1	yes	characteristic
156	C₂²×C₄₃	#463	1	1	yes	characteristic
157	C₂²×C₄₃	#467	1	1	yes	characteristic
158	C₂²×C₄₃	#469	1	1	yes	characteristic
159	C₂²×C₄₃	#471	1	1	yes	characteristic
160	C₂²×C₄₃	#471	1	1	yes	characteristic
161	C₂×D₄₃	—	4	2	no	—
162	C₂×D₄₃	—	4	2	no	—
163	C₂×D₄₃	—	4	2	no	—
164	C₂×D₄₃	—	8	1	no	—
165	C₂×D₄₃	—	8	1	no	—
166	C₂×D₄₃	—	8	1	no	—
167	C₂×D₄₃	—	8	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	16	1	no	—
171	C₄×C₄₃	#482	1	1	yes	characteristic
172	C₄×C₄₃	#482	1	1	yes	characteristic
173	C₄×C₄₃	#482	1	1	yes	characteristic
174	C₄×C₄₃	#482	1	1	yes	characteristic
175	C₄×C₄₃	#484	1	1	yes	characteristic
176	C₄×C₄₃	#484	1	1	yes	characteristic
177	C₄×C₄₃	#488	1	1	yes	characteristic
178	C₄×C₄₃	#490	1	1	yes	characteristic
179	Dic₄₃	—	8	1	no	—
180	Dic₄₃	—	16	1	no	—
181	#1954	—	43	1	no	—
182	#1834	—	43	1	no	—
183	#1802	—	43	1	no	—
184	#1802	—	43	1	no	—
185	#1296	—	43	1	no	—
186	#1296	D₄₃	1	1	yes	characteristic, complemented
187	#1277	—	43	1	no	—
188	C₂×C₄₃	#1683	1	1	yes	characteristic
189	C₂×C₄₃	#1683	1	1	yes	characteristic
190	C₂×C₄₃	#1701	1	1	yes	characteristic
191	C₂×C₄₃	#1701	1	1	yes	characteristic
192	C₂×C₄₃	#2073	1	1	yes	characteristic
193	C₂×C₄₃	#2076	1	1	yes	characteristic
194	C₂×C₄₃	#2079	1	1	yes	characteristic
195	D₄₃	—	8	2	no	—
196	C₂×C₄×D₄	—	43	1	no	—
197	C₂×C₄⋊C₈	—	43	1	no	—
198	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
199	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
200	C₂×#102	—	86	1	no	—
201	C₂×#102	—	86	1	no	—
202	C₂×C₂²⋊C₈	—	86	1	no	—
203	C₂×C₂²⋊C₈	—	86	1	no	—
204	C₂×C₄×C₈	—	43	1	no	—
205	#384	—	43	1	no	—
206	#384	C₂×D₄₃	1	1	yes	characteristic
207	#340	—	86	1	no	—
208	#340	—	86	1	no	—
209	#340	—	86	1	no	—
210	#340	—	86	1	no	—
211	C₄₃	#10622	1	1	yes	characteristic, complemented, Sylow
212	C₂²×D₄	—	43	1	no	—
213	C₂³×C₄	—	86	1	no	—
214	C₂²×C₈	—	2	1	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	43	1	no	—
217	C₂²×C₈	—	43	1	no	—
218	C₂²×C₈	—	86	1	no	—
219	C₄×D₄	—	86	1	no	—
220	C₄×D₄	—	86	1	no	—
221	C₄×D₄	—	86	1	no	—
222	C₄×D₄	—	86	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	43	1	no	—
227	C₂×C₄⋊C₄	—	86	1	no	—
228	C₂×C₄⋊C₄	—	86	1	no	—
229	C₂×C₄⋊C₄	—	86	1	no	—
230	C₂×C₂²⋊C₄	—	86	1	no	—
231	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	—	2	1	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	86	1	no	—
237	C₄⋊C₈	—	86	1	no	—
238	#102	—	172	2	no	—
239	#102	—	172	2	no	—
240	C₂²⋊C₈	—	172	2	no	—
241	C₂²⋊C₈	—	172	2	no	—
242	C₄×C₈	—	86	1	no	—
243	C₄×C₈	—	86	1	no	—
244	C₂⁴	—	86	1	no	—
245	C₂×D₄	—	43	2	no	—
246	C₂×D₄	—	86	1	no	—
247	C₂×D₄	—	86	1	no	—
248	C₂×D₄	—	86	1	no	—
249	C₂×D₄	—	86	1	no	—
250	C₂×D₄	—	86	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	86	1	no	—
253	C₂²×C₄	—	172	1	no	—
254	C₂²×C₄	—	172	1	no	—
255	C₂²×C₄	—	172	1	no	—
256	C₂²×C₄	—	172	1	no	—
257	C₂²×C₄	—	172	1	no	—
258	C₂²×C₄	—	172	1	no	—
259	C₂²×C₄	—	172	1	no	—
260	C₂²×C₄	#10621898	1	1	yes	characteristic
261	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
262	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	86	1	no	—
270	C₂×C₈	—	86	1	no	—
271	C₂×C₈	—	86	1	no	—
272	C₂×C₈	—	86	1	no	—
273	C₂×C₈	—	86	1	no	—
274	C₂×C₈	—	86	1	no	—
275	C₂×C₈	—	172	1	no	—
276	C₂×C₈	—	172	1	no	—
277	C₂×C₈	—	172	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	8	1	no	—
283	C₄⋊C₄	—	86	1	no	—
284	C₄⋊C₄	—	86	1	no	—
285	C₄⋊C₄	—	344	1	no	—
286	C₄⋊C₄	—	172	1	no	—
287	C₄⋊C₄	—	172	1	no	—
288	C₄⋊C₄	—	172	1	no	—
289	C₄⋊C₄	—	172	1	no	—
290	C₂²⋊C₄	—	172	1	no	—
291	C₂²⋊C₄	—	172	1	no	—
292	C₄²	#10621898	1	1	yes	characteristic
293	C₄²	#10621898	1	1	yes	characteristic
294	C₄²	#10621900	1	1	yes	characteristic
295	C₄²	#10621902	1	1	yes	characteristic
296	C₂³	—	86	2	no	—
297	C₂³	—	172	1	no	—
298	C₂³	—	172	1	no	—
299	C₂³	—	172	1	no	—
300	C₂³	—	172	1	no	—
301	C₂³	—	172	1	no	—
302	C₂³	—	172	1	no	—
303	C₂³	#13063298	1	1	yes	characteristic
304	D₄	—	86	2	no	—
305	D₄	—	86	2	no	—
306	D₄	—	172	1	no	—
307	D₄	—	172	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	8	1	no	—
313	C₂×C₄	—	344	1	no	—
314	C₂×C₄	—	344	1	no	—
315	C₂×C₄	—	344	1	no	—
316	C₂×C₄	—	344	1	no	—
317	C₂×C₄	—	344	1	no	—
318	C₂×C₄	—	344	1	no	—
319	C₂×C₄	—	344	1	no	—
320	C₂×C₄	—	172	1	no	—
321	C₂×C₄	—	172	1	no	—
322	C₂×C₄	—	172	1	no	—
323	C₂×C₄	#13063307	1	1	yes	characteristic
324	C₂×C₄	#13063310	1	1	yes	characteristic
325	C₂×C₄	#13063310	1	1	yes	characteristic
326	C₂×C₄	#13063311	1	1	yes	characteristic
327	C₂×C₄	#13063319	1	1	yes	characteristic
328	C₂×C₄	#13063320	1	1	yes	characteristic
329	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
330	C₂×C₄	#13063346	1	1	yes	characteristic
331	C₂×C₄	#13063346	1	1	yes	characteristic
332	C₂×C₄	#13063348	1	1	yes	characteristic
333	C₂×C₄	#13063349	1	1	yes	characteristic
334	C₂×C₄	#13063349	1	1	yes	characteristic
335	C₂×C₄	#13063352	1	1	yes	characteristic
336	C₂×C₄	#13063352	1	1	yes	characteristic
337	C₈	—	8	1	no	—
338	C₈	—	8	1	no	—
339	C₈	—	344	1	no	—
340	C₈	—	172	1	no	—
341	C₈	—	172	1	no	—
342	C₂²	—	344	1	no	—
343	C₂²	—	344	1	no	—
344	C₂²	—	344	1	no	—
345	C₂²	—	344	1	no	—
346	C₂²	—	172	2	no	—
347	C₂²	—	172	2	no	—
348	C₂²	—	172	2	no	—
349	C₂²	#14582828	1	1	yes	characteristic
350	C₂²	#14583113	1	1	yes	characteristic
351	C₂²	#14583117	1	1	yes	characteristic
352	C₂²	#14583139	1	1	yes	characteristic
353	C₂²	#14583151	1	1	yes	characteristic
354	C₂²	#14583161	1	1	yes	characteristic
355	C₂²	#14583165	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	16	1	no	—
359	C₄	—	344	1	no	—
360	C₄	—	688	1	no	—
361	C₄	#14583222	1	1	yes	characteristic
362	C₄	#14583225	1	1	yes	characteristic
363	C₄	#14583225	1	1	yes	characteristic
364	C₄	#14583226	1	1	yes	characteristic
365	C₄	#14583233	1	1	yes	characteristic
366	C₄	#14583236	1	1	yes	characteristic
367	C₄	#14583255	1	1	yes	characteristic
368	C₄	#14583262	1	1	yes	characteristic
369	C₂	—	344	2	no	—
370	C₂	#19134045	1	1	yes	characteristic
371	C₂	#19134049	1	1	yes	characteristic
372	C₂	#19134155	1	1	yes	characteristic
373	C₂	#19134159	1	1	yes	characteristic
374	C₂	#19136012	1	1	yes	characteristic
375	C₂	#19136027	1	1	yes	characteristic
376	C₂	#19136044	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063298
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1296	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂²×D₁₇₂	D₄
hypercenter	#1296	D₄₃
hypocenter	C₄₃	#10622
LCS	G ▷ C₂×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. #19132479	C₂	semidirect
6. #19132429	C₂	semidirect
7. C₄₃×#1296	C₂	semidirect
186. #1296	D₄₃	semidirect
211. C₄₃	#10622	semidirect

Maximal subgroups

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

subgroup	index
#19135388	2
#19134722	2
#19134596	2
#19134592	2
#19132479	2
#19132429	2
C₄₃×#1296	2
#10622	43