Subgroups — #25013474

Order 11008.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583014	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582684	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582683	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136208	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132676	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132672	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063345	C₄	1	1	yes	characteristic
9	C₂×#13063331	C₂²	1	1	yes	characteristic
10	C₂×#13063265	C₂²	1	1	yes	characteristic
11	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14583014	—	2	2	no	—
13	#14583014	—	2	2	no	—
14	C₂×#13063260	C₄	1	1	yes	characteristic
15	C₂×#13063252	C₄	1	1	yes	characteristic
16	C₂×#13063251	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
18	#14582684	—	2	2	no	—
19	#14582683	—	2	2	no	—
20	(C₂×D₄₃)⋊C₁₆	—	2	1	no	—
21	(C₂×D₄₃)⋊C₁₆	—	2	1	no	—
22	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#504	—	2	2	no	—
24	C₄₃×#504	—	2	2	no	—
25	C₂×M₃₂×C₄₃	—	2	1	no	—
26	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₃	—	2	1	no	—
28	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
30	#13063345	—	2	2	no	—
31	#13063345	—	2	2	no	—
32	C₂×#10621885	C₈	1	1	yes	characteristic
33	C₂×#10621883	C₈	1	1	yes	characteristic
34	C₂×#10621884	C₈	1	1	yes	characteristic
35	C₂×#10621882	C₈	1	1	yes	characteristic
36	#13063331	—	2	1	no	—
37	#13063331	—	2	1	no	—
38	#13063331	—	2	1	no	—
39	#13063331	—	2	1	no	—
40	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
41	C₂×#10621872	—	2	1	no	—
42	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₄₃	—	2	1	no	—
44	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
45	#13063265	—	2	1	no	—
46	#13063265	—	2	1	no	—
47	C₂×C₄₃⋊C₁₆	—	2	1	no	—
48	C₂×C₄₃⋊C₁₆	—	2	1	no	—
49	C₂×C₄₃⋊C₁₆	—	2	1	no	—
50	C₂×C₄₃⋊C₁₆	—	2	1	no	—
51	#13063260	—	2	2	no	—
52	#13063252	—	2	2	no	—
53	#13063251	—	2	2	no	—
54	C₈×Dic₄₃	—	2	1	no	—
55	C₈×Dic₄₃	—	2	1	no	—
56	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
57	#131×C₄₃	—	2	1	no	—
58	#131×C₄₃	—	2	1	no	—
59	#131×C₄₃	D₄	1	2	yes	—
60	#131×C₄₃	D₄	1	2	yes	—
61	C₂×M₁₆×C₄₃	—	2	1	no	—
62	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₃	—	2	1	no	—
64	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	M₃₂×C₄₃	—	2	4	no	—
66	M₃₂×C₄₃	—	2	1	no	—
67	M₃₂×C₄₃	D₄	1	2	yes	—
68	C₂×C₁₆×C₄₃	—	2	4	no	—
69	C₂×C₁₆×C₄₃	—	2	1	no	—
70	C₂×C₁₆×C₄₃	—	2	1	no	—
71	C₂×C₁₆×C₄₃	—	2	1	no	—
72	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
73	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
74	#41×C₄₃	—	2	1	no	—
75	#41×C₄₃	—	2	1	no	—
76	#41×C₄₃	C₂²⋊C₄	1	2	yes	—
77	#41×C₄₃	M₁₆	1	2	yes	—
78	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
79	C₂×D₄×C₄₃	—	2	1	no	—
80	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	—	2	1	no	—
82	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊D₄₃	—	2	1	no	—
84	C₂²×Dic₄₃	—	2	1	no	—
85	#10621898	—	2	1	no	—
86	#10621898	—	2	1	no	—
87	#10621898	—	2	1	no	—
88	#10621898	—	2	1	no	—
89	C₂×D₁₇₂	C₂×C₈	1	1	yes	characteristic
90	C₂×C₄×D₄₃	—	2	1	no	—
91	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
92	M₁₆×C₄₃	—	2	1	no	—
93	M₁₆×C₄₃	—	2	2	no	—
94	M₁₆×C₄₃	—	2	2	no	—
95	M₁₆×C₄₃	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₄₃	—	2	1	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
105	#10621885	—	2	2	no	—
106	#10621884	—	2	2	no	—
107	#10621883	—	2	2	no	—
108	#10621882	—	2	2	no	—
109	C₄×Dic₄₃	—	2	1	no	—
110	C₄×Dic₄₃	—	2	1	no	—
111	#10621877	—	2	1	no	—
112	#10621877	—	2	1	no	—
113	C₂×C₄₃⋊C₈	—	2	1	no	—
114	C₂×C₄₃⋊C₈	—	2	1	no	—
115	C₂×C₄₃⋊C₈	—	2	1	no	—
116	C₂×C₄₃⋊C₈	C₂×C₈	1	1	yes	characteristic
117	C₂×C₄₃⋊C₈	C₂×C₈	1	1	yes	characteristic
118	#10621872	—	4	1	no	—
119	#10621872	—	4	1	no	—
120	C₈×D₄₃	—	4	1	no	—
121	C₈×D₄₃	—	4	1	no	—
122	C₁₆×C₄₃	—	2	1	no	—
123	C₁₆×C₄₃	—	2	2	no	—
124	C₁₆×C₄₃	—	2	1	no	—
125	C₄₃⋊C₁₆	—	4	1	no	—
126	C₄₃⋊C₁₆	—	4	1	no	—
127	C₂³×C₄₃	—	2	1	no	—
128	C₂³×C₄₃	#131	1	1	yes	characteristic
129	C₂²×D₄₃	—	2	1	no	—
130	Q₈×C₄₃	—	2	1	no	—
131	Q₈×C₄₃	C₂²⋊C₈	1	2	yes	—
132	D₄×C₄₃	—	2	2	no	—
133	D₄×C₄₃	—	2	2	no	—
134	D₄×C₄₃	—	2	1	no	—
135	D₄×C₄₃	C₂²⋊C₈	1	2	yes	—
136	C₂×C₄×C₄₃	—	2	1	no	—
137	C₂×C₄×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
145	C₂²⋊D₄₃	—	4	1	no	—
146	C₂²⋊D₄₃	—	4	1	no	—
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	1	no	—
149	C₂×Dic₄₃	—	4	1	no	—
150	C₂×Dic₄₃	—	4	1	no	—
151	D₁₇₂	—	2	1	no	—
152	D₁₇₂	—	2	1	no	—
153	C₄×D₄₃	—	4	1	no	—
154	C₄×D₄₃	—	4	1	no	—
155	Dic₈₆	—	2	1	no	—
156	Dic₈₆	—	2	1	no	—
157	C₈×C₄₃	—	2	2	no	—
158	C₈×C₄₃	#120	1	1	yes	characteristic
159	C₈×C₄₃	#121	1	1	yes	characteristic
160	C₈×C₄₃	#121	1	1	yes	characteristic
161	C₈×C₄₃	#123	1	1	yes	characteristic
162	C₄₃⋊C₈	—	2	1	no	—
163	C₄₃⋊C₈	—	2	1	no	—
164	C₄₃⋊C₈	—	4	1	no	—
165	#12980	—	43	1	no	maximal, Sylow
166	C₂²×C₄₃	—	2	1	no	—
167	C₂²×C₄₃	—	2	1	no	—
168	C₂²×C₄₃	—	2	2	no	—
169	C₂²×C₄₃	—	4	1	no	—
170	C₂²×C₄₃	—	4	1	no	—
171	C₂²×C₄₃	#408	1	1	yes	characteristic
172	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
173	C₂²×C₄₃	#435	1	1	yes	characteristic
174	C₂×D₄₃	—	4	1	no	—
175	C₂×D₄₃	—	4	1	no	—
176	C₂×D₄₃	—	4	1	no	—
177	C₄×C₄₃	—	2	2	no	—
178	C₄×C₄₃	#386	1	1	yes	characteristic
179	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
180	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
181	C₄×C₄₃	#435	1	1	yes	characteristic
182	Dic₄₃	—	4	2	no	—
183	Dic₄₃	—	4	1	no	—
184	C₂×#504	D₄₃	1	1	yes	characteristic, complemented
185	C₂×#443	—	43	1	no	—
186	C₂×#350	—	43	1	no	—
187	C₂×#350	—	43	1	no	—
188	#2113	—	43	1	no	—
189	#1376	—	43	1	no	—
190	#1374	—	43	1	no	—
191	C₂×C₄₃	—	2	1	no	—
192	C₂×C₄₃	—	2	1	no	—
193	C₂×C₄₃	—	4	2	no	—
194	C₂×C₄₃	#1869	1	1	yes	characteristic
195	C₂×C₄₃	#2170	1	1	yes	characteristic
196	C₂×C₄₃	#2171	1	1	yes	characteristic
197	D₄₃	—	8	1	no	—
198	C₂×#135	—	43	1	no	—
199	C₂×#131	—	43	1	no	—
200	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
201	#504	—	2	2	no	—
202	#504	—	2	2	no	—
203	C₂×M₃₂	—	2	1	no	—
204	C₂×M₃₂	—	43	1	no	—
205	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
206	C₂²×C₁₆	—	2	1	no	—
207	C₂²×C₁₆	—	43	1	no	—
208	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
209	#443	—	86	2	no	—
210	#443	—	86	2	no	—
211	C₂×#108	—	43	1	no	—
212	C₂×#104	—	43	1	no	—
213	C₂×#104	—	43	1	no	—
214	C₂×C₄×C₈	—	43	1	no	—
215	#350	—	86	2	no	—
216	#350	—	86	2	no	—
217	C₂²⋊C₁₆	—	86	1	no	—
218	C₂²⋊C₁₆	—	86	1	no	—
219	C₄₃	#12980	1	1	yes	characteristic, complemented, Sylow
220	C₂×#41	—	43	1	no	—
221	C₂×#41	C₄×D₄₃	1	1	yes	characteristic
222	#135	—	86	2	no	—
223	#135	—	86	2	no	—
224	C₂×Q₁₆	—	43	1	no	—
225	C₂×SD₁₆	—	43	1	no	—
226	C₂×SD₁₆	—	43	1	no	—
227	C₂×D₈	—	43	1	no	—
228	#131	—	2	1	no	—
229	#131	—	2	1	no	—
230	#131	—	86	1	no	—
231	#131	—	86	1	no	—
232	#131	—	86	1	no	—
233	#131	—	86	1	no	—
234	#131	D₁₇₂	1	2	yes	—
235	#131	C₂²⋊D₄₃	1	2	yes	—
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	43	1	no	—
238	C₂×M₁₆	—	86	1	no	—
239	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
240	C₂²×C₈	—	2	1	no	—
241	C₂²×C₈	—	43	1	no	—
242	C₂²×C₈	—	86	1	no	—
243	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
244	C₂×C₄²	—	43	1	no	—
245	M₃₂	—	2	4	no	—
246	M₃₂	—	2	1	no	—
247	M₃₂	—	86	1	no	—
248	M₃₂	—	86	1	no	—
249	M₃₂	C₂²⋊D₄₃	1	2	yes	—
250	C₂×C₁₆	—	2	4	no	—
251	C₂×C₁₆	—	2	1	no	—
252	C₂×C₁₆	—	2	1	no	—
253	C₂×C₁₆	—	2	1	no	—
254	C₂×C₁₆	—	86	1	no	—
255	C₂×C₁₆	—	86	1	no	—
256	C₂×C₁₆	—	86	1	no	—
257	C₂×C₁₆	—	86	1	no	—
258	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
259	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
260	#108	—	86	2	no	—
261	#104	—	86	2	no	—
262	#104	—	86	2	no	—
263	C₄×C₈	—	86	1	no	—
264	C₄×C₈	—	86	1	no	—
265	#41	—	2	1	no	—
266	#41	—	2	1	no	—
267	#41	—	86	1	no	—
268	#41	—	86	1	no	—
269	#41	—	86	1	no	—
270	#41	—	86	1	no	—
271	#41	#10621872	1	2	yes	—
272	#41	(C₂×D₄₃)⋊C₄	1	2	yes	—
273	C₂×Q₈	—	43	1	no	—
274	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
275	C₂×D₄	—	2	1	no	—
276	C₂×D₄	—	43	1	no	—
277	C₂×D₄	—	86	1	no	—
278	C₂×D₄	C₈×D₄₃	1	1	yes	characteristic
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	86	1	no	—
281	C₂²×C₄	—	86	1	no	—
282	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
283	Q₁₆	—	86	2	no	—
284	SD₁₆	—	86	2	no	—
285	SD₁₆	—	86	2	no	—
286	D₈	—	86	2	no	—
287	M₁₆	—	2	1	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	86	1	no	—
291	M₁₆	—	86	1	no	—
292	M₁₆	—	172	1	no	—
293	M₁₆	—	172	1	no	—
294	M₁₆	(C₂×D₄₃)⋊C₄	1	2	yes	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	43	1	no	—
299	C₂×C₈	—	43	1	no	—
300	C₂×C₈	—	86	1	no	—
301	C₂×C₈	—	86	1	no	—
302	C₂×C₈	—	86	1	no	—
303	C₂×C₈	—	172	1	no	—
304	C₂×C₈	—	172	1	no	—
305	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
306	C₂×C₈	#10621898	1	1	yes	characteristic
307	C₂×C₈	#10621898	1	1	yes	characteristic
308	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
310	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
311	C₄²	—	86	1	no	—
312	C₄²	—	86	1	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	—	2	2	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	172	1	no	—
317	C₁₆	—	172	1	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	86	1	no	—
320	C₂³	#13063331	1	1	yes	characteristic
321	Q₈	—	2	1	no	—
322	Q₈	—	86	1	no	—
323	Q₈	—	86	1	no	—
324	Q₈	(C₂×D₄₃)⋊C₈	1	2	yes	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	2	1	no	—
328	D₄	—	86	1	no	—
329	D₄	—	86	1	no	—
330	D₄	—	172	1	no	—
331	D₄	—	172	1	no	—
332	D₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	86	2	no	—
337	C₂×C₄	—	86	1	no	—
338	C₂×C₄	—	172	1	no	—
339	C₂×C₄	—	172	1	no	—
340	C₂×C₄	—	172	1	no	—
341	C₂×C₄	—	172	1	no	—
342	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
343	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
344	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
345	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
346	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
347	C₂×C₄	#13063331	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	86	1	no	—
350	C₈	—	86	1	no	—
351	C₈	—	172	1	no	—
352	C₈	#13063294	1	1	yes	characteristic
353	C₈	#13063299	1	1	yes	characteristic
354	C₈	#13063299	1	1	yes	characteristic
355	C₈	#13063308	1	1	yes	characteristic
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	172	1	no	—
362	C₂²	—	172	1	no	—
363	C₂²	—	172	1	no	—
364	C₂²	#14582865	1	1	yes	characteristic
365	C₂²	C₈×C₂²⋊D₄₃	1	1	yes	characteristic
366	C₂²	#14582970	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	172	2	no	—
369	C₄	—	172	1	no	—
370	C₄	#14582778	1	1	yes	characteristic
371	C₄	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
372	C₄	C₈×C₂²⋊D₄₃	1	1	yes	characteristic
373	C₄	#14582970	1	1	yes	characteristic
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	4	2	no	—
377	C₂	—	344	1	no	—
378	C₂	#19134887	1	1	yes	characteristic
379	C₂	#19136430	1	1	yes	characteristic
380	C₂	#19136435	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#504	C₂
socle	C₂²×C₄₃	C₈×D₄
prime radical	C₂×#13063345	C₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#504	D₄₃
hypocenter	C₄₃	#12980
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. C₂×#14582684	C₂	semidirect
3. C₂×#14582683	C₂	semidirect
4. #19136208	C₂	semidirect
5. #19132676	C₂	semidirect
6. #19132672	C₂	semidirect
7. C₂×C₄₃×#504	C₂	semidirect
184. C₂×#504	D₄₃	semidirect
219. C₄₃	#12980	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583014	2
C₂×#14582684	2
C₂×#14582683	2
#19136208	2
#19132676	2
#19132672	2
C₂×C₄₃×#504	2
#12980	43