Subgroups — #25249903

Order 11008.

682 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583329	C₂	1	1	yes	maximal, characteristic
2	#135×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143366	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136468	C₂	1	2	yes	maximal, complemented
5	#19136463	C₂	1	2	yes	maximal, complemented
6	#19136462	C₂	1	2	yes	maximal, complemented
7	#19136456	C₂	1	2	yes	maximal, complemented
8	#19136377	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136303	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14582674	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2138	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	1	yes	characteristic
13	#41×Dic₄₃	—	2	1	no	—
14	#14583367	—	2	1	no	—
15	#14583329	C₄	1	8	yes	complemented
16	C₂×#13063271	C₂²	1	1	yes	characteristic
17	C₂×#13063269	C₂²	1	1	yes	characteristic
18	C₂×#13063268	C₂²	1	1	yes	characteristic
19	C₂×#13063266	C₂²	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	Q₁₆×Dic₄₃	C₂²	1	2	yes	—
22	#14582993	C₂²	1	2	yes	—
23	SD₁₆×Dic₄₃	—	2	2	no	—
24	#14582986	—	2	2	no	—
25	D₈×Dic₄₃	C₂²	1	2	yes	—
26	#14582981	C₂²	1	2	yes	—
27	#14582937	C₂²	1	1	yes	characteristic
28	#14582895	—	2	1	no	—
29	#14582893	—	2	1	no	—
30	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
31	#14582726	C₂²	1	2	yes	—
32	#14582724	C₂²	1	2	yes	—
33	#14582704	C₂²	1	2	yes	—
34	#14582701	C₂²	1	2	yes	—
35	#14582674	C₂²	1	2	yes	—
36	#14582674	C₂²	1	2	yes	—
37	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
38	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#427	C₂²	1	1	yes	characteristic
40	C₄₃×#358	C₂²	1	2	yes	—
41	C₄₃×#357	C₂²	1	2	yes	—
42	C₂×#10621898	—	2	1	no	—
43	#13063347	—	2	2	no	—
44	#13063347	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
46	C₂×#10621873	—	2	1	no	—
47	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
48	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
49	Q₈×Dic₄₃	—	2	2	no	—
50	#13063290	—	2	2	no	—
51	D₄×Dic₄₃	—	2	2	no	—
52	D₄×Dic₄₃	—	4	2	no	—
53	#13063287	—	2	2	no	—
54	(C₂²⋊D₄₃)⋊C₄	—	4	2	no	—
55	#13063284	—	2	1	no	—
56	#13063284	—	4	1	no	—
57	#13063282	—	4	1	no	—
58	C₂×C₄×Dic₄₃	—	4	1	no	—
59	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
60	C₄²×D₄₃	—	4	1	no	—
61	#13063271	C₂×C₄	1	4	yes	complemented
62	#13063269	C₂×C₄	1	4	yes	complemented
63	#13063268	C₂×C₄	1	4	yes	complemented
64	#13063266	C₂×C₄	1	4	yes	complemented
65	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
66	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
68	#13063259	C₂³	1	2	yes	—
69	#13063257	—	2	1	no	—
70	#13063249	—	2	2	no	—
71	#13063248	—	2	2	no	—
72	#13063246	—	2	2	no	—
73	#13063245	—	2	2	no	—
74	C₈×Dic₄₃	D₄	1	2	yes	—
75	C₈×Dic₄₃	C₂³	1	2	yes	—
76	C₂×#41×C₄₃	—	2	1	no	—
77	#135×C₄₃	—	2	2	no	—
78	#135×C₄₃	C₂×C₄	1	4	yes	complemented
79	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
80	C₂×SD₁₆×C₄₃	—	2	1	no	—
81	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
83	#117×C₄₃	—	2	1	no	—
84	C₂×C₁₆×C₄₃	—	2	2	no	—
85	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
86	#107×C₄₃	C₂³	1	2	yes	—
87	#106×C₄₃	—	2	1	no	—
88	#41×C₄₃	—	2	4	no	—
89	#41×C₄₃	—	4	2	no	—
90	C₂×Q₈×C₄₃	—	2	1	no	—
91	C₂×D₄×C₄₃	—	2	1	no	—
92	C₂×D₄×C₄₃	—	4	1	no	—
93	C₂²×C₄×C₄₃	—	4	1	no	—
94	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₂²⋊D₄₃	—	4	1	no	—
96	C₂²×Dic₄₃	—	4	2	no	—
97	C₂²×Dic₄₃	D₈	1	2	yes	—
98	#10621898	—	2	4	no	—
99	#10621898	—	4	2	no	—
100	C₂×D₁₇₂	—	2	1	no	—
101	C₂×C₄×D₄₃	—	4	1	no	—
102	C₂×C₄×D₄₃	—	4	2	no	—
103	C₂×Dic₈₆	—	2	1	no	—
104	Q₁₆×C₄₃	—	2	1	no	—
105	Q₁₆×C₄₃	C₂²×C₄	1	2	yes	—
106	SD₁₆×C₄₃	—	2	4	no	—
107	D₈×C₄₃	—	2	1	no	—
108	D₈×C₄₃	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₃	#41	1	2	yes	—
112	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₄₃	—	2	2	no	—
114	C₂²⋊C₄×C₄₃	—	4	1	no	—
115	C₄²×C₄₃	—	4	1	no	—
116	C₂²⋊Dic₄₃	—	4	2	no	—
117	C₂²⋊Dic₄₃	—	4	1	no	—
118	(C₂×D₄₃)⋊C₄	—	4	2	no	—
119	C₄⋊Dic₄₃	—	2	2	no	—
120	C₄⋊Dic₄₃	—	4	2	no	—
121	Dic₄₃⋊C₄	—	4	2	no	—
122	C₄×Dic₄₃	—	4	1	no	—
123	C₄×Dic₄₃	—	4	2	no	—
124	C₄×Dic₄₃	—	4	2	no	—
125	C₄×Dic₄₃	D₈	1	1	yes	characteristic
126	C₄×Dic₄₃	D₈	1	1	yes	characteristic
127	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
128	C₂×C₄₃⋊C₈	—	2	2	no	—
129	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
130	Dic₁₇₂	—	2	1	no	—
131	Dic₁₇₂	C₂²×C₄	1	2	yes	—
132	D₃₄₄	—	2	1	no	—
133	D₃₄₄	C₂²×C₄	1	2	yes	—
134	#10621873	—	2	4	no	—
135	C₁₆×C₄₃	—	2	2	no	—
136	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
137	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
138	C₂³×C₄₃	—	4	1	no	—
139	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
140	C₂²×D₄₃	—	4	1	no	—
141	Q₈×C₄₃	—	2	2	no	—
142	Q₈×C₄₃	—	4	1	no	—
143	D₄×C₄₃	—	2	2	no	—
144	D₄×C₄₃	—	4	4	no	—
145	D₄×C₄₃	—	4	1	no	—
146	C₂×C₄×C₄₃	—	4	1	no	—
147	C₂×C₄×C₄₃	—	4	4	no	—
148	C₂×C₄×C₄₃	—	4	2	no	—
149	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₄₃	#135	1	2	yes	—
152	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
153	C₂²⋊D₄₃	—	4	4	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	4	1	no	—
156	C₂×Dic₄₃	—	4	2	no	—
157	C₂×Dic₄₃	—	8	4	no	—
158	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
159	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
160	D₁₇₂	—	2	2	no	—
161	D₁₇₂	—	4	1	no	—
162	C₄×D₄₃	—	4	4	no	—
163	C₄×D₄₃	—	8	4	no	—
164	Dic₈₆	—	2	2	no	—
165	Dic₈₆	—	4	1	no	—
166	C₈×C₄₃	C₄×D₄	1	2	yes	—
167	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
168	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
169	C₄₃⋊C₈	—	2	2	no	—
170	#33451	—	43	1	no	maximal, Sylow
171	C₂²×C₄₃	—	2	2	no	—
172	C₂²×C₄₃	—	4	2	no	—
173	C₂²×C₄₃	—	8	2	no	—
174	C₂²×C₄₃	C₄×D₈	1	2	yes	—
175	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
176	C₂×D₄₃	—	4	2	no	—
177	C₂×D₄₃	—	8	2	no	—
178	C₄×C₄₃	—	4	2	no	—
179	C₄×C₄₃	—	8	2	no	—
180	C₄×C₄₃	C₄×D₈	1	2	yes	—
181	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
182	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
183	Dic₄₃	—	2	2	no	—
184	Dic₄₃	—	2	2	no	—
185	Dic₄₃	—	4	2	no	—
186	Dic₄₃	—	8	2	no	—
187	C₂×#508	—	43	1	no	—
188	C₄×#135	—	43	1	no	—
189	C₄×#135	—	43	1	no	—
190	#2177	—	43	2	no	—
191	#2176	—	43	2	no	—
192	#2175	—	43	2	no	—
193	#2175	—	43	2	no	—
194	#2158	—	43	1	no	—
195	#2138	—	43	1	no	—
196	#2138	D₄₃	1	1	yes	characteristic, complemented
197	C₂×#346	—	43	1	no	—
198	C₂×C₄₃	—	2	2	no	—
199	C₂×C₄₃	—	8	2	no	—
200	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
201	C₂×C₄₃	#3414	1	2	yes	—
202	D₄₃	—	8	2	no	—
203	C₂×#135	—	43	1	no	—
204	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
205	C₄×#41	—	86	1	no	—
206	C₄×#41	—	86	1	no	—
207	#508	—	43	8	no	—
208	C₂×Q₃₂	—	43	1	no	—
209	C₂×SD₃₂	—	43	1	no	—
210	C₂×SD₃₂	—	43	1	no	—
211	C₂×D₁₆	—	43	1	no	—
212	C₂²×C₁₆	—	43	1	no	—
213	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
214	C₄×Q₁₆	—	43	2	no	—
215	C₄×Q₁₆	—	43	2	no	—
216	C₄×SD₁₆	—	86	2	no	—
217	C₄×SD₁₆	—	86	2	no	—
218	C₄×D₈	—	43	2	no	—
219	C₄×D₈	—	43	2	no	—
220	#427	—	43	1	no	—
221	#427	C₂×D₄₃	1	1	yes	characteristic
222	#416	—	86	1	no	—
223	#416	—	86	1	no	—
224	C₂×C₄×C₈	—	43	1	no	—
225	#367	—	43	2	no	—
226	#366	—	43	2	no	—
227	#358	—	43	2	no	—
228	#358	C₂×D₄₃	1	2	yes	—
229	#357	—	43	2	no	—
230	#357	C₂×D₄₃	1	2	yes	—
231	#346	—	43	2	no	—
232	#346	—	43	2	no	—
233	C₄₃	#33451	1	1	yes	characteristic, complemented, Sylow
234	C₂×#41	—	2	1	no	—
235	C₂×#41	—	86	1	no	—
236	#135	—	2	2	no	—
237	#135	—	43	4	no	—
238	#135	—	86	2	no	—
239	#135	C₄×D₄₃	1	4	yes	complemented
240	C₂×Q₁₆	—	43	1	no	—
241	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
242	C₂×SD₁₆	—	2	1	no	—
243	C₂×SD₁₆	—	86	1	no	—
244	C₂×D₈	—	43	1	no	—
245	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
246	C₂²×C₈	—	43	1	no	—
247	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₄×Q₈	—	86	2	no	—
249	C₄×Q₈	—	86	2	no	—
250	C₄×D₄	—	86	2	no	—
251	C₄×D₄	—	86	2	no	—
252	C₄×D₄	—	172	2	no	—
253	C₄×D₄	—	172	2	no	—
254	#117	—	2	1	no	—
255	#117	—	86	1	no	—
256	#117	—	172	1	no	—
257	#117	—	172	1	no	—
258	C₂×C₄²	—	43	1	no	—
259	C₂×C₄²	—	172	1	no	—
260	C₂×C₄²	—	172	1	no	—
261	Q₃₂	—	43	4	no	—
262	SD₃₂	—	43	4	no	—
263	SD₃₂	—	43	4	no	—
264	D₁₆	—	43	4	no	—
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	—	43	4	no	—
267	C₂×C₁₆	—	43	1	no	—
268	C₂×C₁₆	—	43	1	no	—
269	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
270	#107	—	43	2	no	—
271	#107	C₂²×D₄₃	1	2	yes	—
272	#106	—	2	1	no	—
273	#106	—	86	1	no	—
274	#103	—	86	2	no	—
275	#103	—	86	2	no	—
276	#102	—	86	2	no	—
277	#102	—	86	2	no	—
278	C₄×C₈	—	43	2	no	—
279	C₄×C₈	—	43	2	no	—
280	#41	—	2	4	no	—
281	#41	—	4	2	no	—
282	#41	—	86	4	no	—
283	#41	—	172	2	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	—	86	1	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	86	1	no	—
289	C₂×D₄	—	172	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	43	2	no	—
292	C₂²×C₄	—	172	2	no	—
293	C₂²×C₄	—	172	2	no	—
294	C₂²×C₄	—	172	1	no	—
295	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
296	Q₁₆	—	2	1	no	—
297	Q₁₆	—	43	2	no	—
298	Q₁₆	—	86	1	no	—
299	Q₁₆	C₂×C₄×D₄₃	1	2	yes	—
300	SD₁₆	—	2	4	no	—
301	SD₁₆	—	86	4	no	—
302	D₈	—	2	1	no	—
303	D₈	—	43	2	no	—
304	D₈	—	86	1	no	—
305	D₈	C₂×C₄×D₄₃	1	2	yes	—
306	C₂×C₈	—	43	2	no	—
307	C₂×C₈	—	86	2	no	—
308	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
309	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
310	C₂×C₈	#10621900	1	2	yes	—
311	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	86	2	no	—
314	C₄⋊C₄	—	172	2	no	—
315	C₄⋊C₄	—	172	2	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	172	2	no	—
318	C₂²⋊C₄	—	172	2	no	—
319	C₂²⋊C₄	—	172	1	no	—
320	C₄²	—	4	1	no	—
321	C₄²	—	43	1	no	—
322	C₄²	—	43	1	no	—
323	C₄²	—	43	2	no	—
324	C₄²	—	172	2	no	—
325	C₄²	—	172	2	no	—
326	C₄²	—	172	1	no	—
327	C₁₆	—	2	2	no	—
328	C₁₆	—	43	2	no	—
329	C₁₆	—	43	2	no	—
330	C₂³	—	4	1	no	—
331	C₂³	—	172	1	no	—
332	C₂³	D₈×D₄₃	1	1	yes	characteristic
333	Q₈	—	2	2	no	—
334	Q₈	—	4	1	no	—
335	Q₈	—	86	2	no	—
336	Q₈	—	172	1	no	—
337	D₄	—	2	2	no	—
338	D₄	—	4	4	no	—
339	D₄	—	4	1	no	—
340	D₄	—	86	2	no	—
341	D₄	—	172	4	no	—
342	D₄	—	172	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	43	2	no	—
347	C₂×C₄	—	43	2	no	—
348	C₂×C₄	—	344	4	no	—
349	C₂×C₄	—	344	4	no	—
350	C₂×C₄	—	172	2	no	—
351	C₂×C₄	—	172	2	no	—
352	C₂×C₄	—	172	1	no	—
353	C₂×C₄	—	172	4	no	—
354	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
355	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
356	C₂×C₄	#13063346	1	2	yes	—
357	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
358	C₈	—	86	2	no	—
359	C₈	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
360	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
361	C₈	C₂×#10621900	1	1	yes	characteristic
362	C₂²	—	2	2	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	344	2	no	—
366	C₂²	—	172	2	no	—
367	C₂²	#14582979	1	2	yes	—
368	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
369	C₄	—	4	2	no	—
370	C₄	—	8	2	no	—
371	C₄	—	86	2	no	—
372	C₄	—	86	2	no	—
373	C₄	—	344	2	no	—
374	C₄	—	172	2	no	—
375	C₄	#14582979	1	2	yes	—
376	C₄	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
377	C₄	C₂×#13063346	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	8	2	no	—
380	C₂	—	344	2	no	—
381	C₂	C₂×#14582979	1	1	yes	characteristic
382	C₂	#19147728	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2138	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#14583329	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2138	D₄₃
hypocenter	C₄₃	#33451
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2138

Complements

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

N	H ≅ G/N	type
2. #135×Dic₄₃	C₂	semidirect
3. #19143366	C₂	semidirect
4. #19136468	C₂	semidirect
5. #19136463	C₂	semidirect
6. #19136462	C₂	semidirect
7. #19136456	C₂	semidirect
8. #19136377	C₂	semidirect
9. #19136303	C₂	semidirect
10. C₂×#14582674	C₂	semidirect
11. C₄₃×#2138	C₂	semidirect
15. #14583329	C₄	semidirect
44. #13063347	C₂×C₄	semidirect
61. #13063271	C₂×C₄	semidirect
62. #13063269	C₂×C₄	semidirect
63. #13063268	C₂×C₄	semidirect
64. #13063266	C₂×C₄	semidirect
65. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
78. #135×C₄₃	C₂×C₄	semidirect
196. #2138	D₄₃	semidirect
233. C₄₃	#33451	semidirect
239. #135	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583329	2
#135×Dic₄₃	2
#19143366	2
#19136468	2
#19136463	2
#19136462	2
#19136456	2
#19136377	2
#19136303	2
C₂×#14582674	2
C₄₃×#2138	2
#33451	43