Subgroups — #25114190

Order 11008.

704 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143249	C₂	1	2	yes	maximal
2	C₂×#14582937	C₂	1	1	yes	maximal, characteristic
3	#19135343	C₂	1	2	yes	maximal, complemented
4	#19135195	C₂	1	4	yes	maximal, complemented
5	#19134695	C₂	1	2	yes	maximal
6	#19134633	C₂	1	1	yes	maximal, characteristic
7	#19134474	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	C₂²	1	2	yes	—
10	#14582957	—	2	4	no	—
11	#14582953	—	2	2	no	—
12	#14582949	—	2	2	no	—
13	C₂×#13063261	C₂²	1	2	yes	—
14	#14582937	C₄	1	1	yes	characteristic, complemented
15	#14582937	C₄	1	1	yes	characteristic, complemented
16	#14582937	C₄	1	2	yes	complemented
17	#14582937	C₂²	1	1	yes	characteristic
18	#14582937	C₂²	1	1	yes	characteristic
19	#14582937	C₂²	1	2	yes	—
20	C₂×#13063259	C₄	1	1	yes	characteristic, complemented
21	C₂×#13063259	C₄	1	1	yes	characteristic, complemented
22	C₂×#13063257	C₄	1	2	yes	complemented
23	#14582925	C₂²	1	4	yes	—
24	#14582863	C₂²	1	2	yes	—
25	#14582854	C₂²	1	4	yes	—
26	#14582664	C₂²	1	2	yes	—
27	#14582659	C₂²	1	4	yes	—
28	#14582659	C₂²	1	4	yes	—
29	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
32	C₄×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
33	C₄₃×#336	C₂²	1	2	yes	—
34	C₂×#10621877	C₂³	1	4	yes	—
35	#13063284	—	2	2	no	—
36	#13063284	—	2	2	no	—
37	#13063284	C₂×C₄	1	2	yes	—
38	#13063284	C₂×C₄	1	2	yes	—
39	#13063284	C₂³	1	2	yes	—
40	#13063284	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
44	C₂×C₄×Dic₄₃	—	2	2	no	—
45	#13063261	—	2	4	no	—
46	#13063259	C₂×C₄	1	2	yes	—
47	#13063259	C₂×C₄	1	2	yes	—
48	#13063259	C₂×C₄	1	1	yes	characteristic
49	#13063259	C₂×C₄	1	1	yes	characteristic
50	#13063259	C₂×C₄	1	1	yes	characteristic
51	#13063259	C₂×C₄	1	1	yes	characteristic
52	#13063257	C₂×C₄	1	2	yes	—
53	#13063257	C₂×C₄	1	2	yes	—
54	#13063257	C₂×C₄	1	2	yes	—
55	#13063257	C₂×C₄	1	2	yes	—
56	Dic₄₃⋊C₈	—	2	4	no	—
57	Dic₄₃⋊C₈	—	2	4	no	—
58	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
59	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
60	#13063235	—	2	4	no	—
61	C₈×Dic₄₃	—	2	4	no	—
62	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₃	—	2	4	no	—
64	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
70	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
72	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
73	C₂²⋊C₈×C₄₃	D₄	1	1	yes	characteristic
74	C₂²⋊C₈×C₄₃	D₄	1	1	yes	characteristic
75	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
76	C₄×C₈×C₄₃	—	2	4	no	—
77	#95×C₄₃	—	2	1	no	—
78	C₂⁴×C₄₃	#41	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	—	2	2	no	—
80	C₂²×C₄×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	—	2	2	no	—
82	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²×Dic₄₃	—	2	2	no	—
89	C₂²×Dic₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	2	2	no	—
91	C₂×C₈×C₄₃	—	2	2	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₃	#41	1	4	yes	—
104	C₂²⋊C₄×C₄₃	—	2	4	no	—
105	C₄²×C₄₃	—	2	2	no	—
106	C₄²×C₄₃	—	2	2	no	—
107	C₂²⋊Dic₄₃	—	2	2	no	—
108	C₂²⋊Dic₄₃	—	2	2	no	—
109	C₂²⋊Dic₄₃	—	2	2	no	—
110	C₂²⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
114	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
115	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
116	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
117	C₄×Dic₄₃	—	2	2	no	—
118	C₄×Dic₄₃	—	2	2	no	—
119	C₄×Dic₄₃	—	2	2	no	—
120	C₄×Dic₄₃	—	2	2	no	—
121	#10621877	—	2	4	no	—
122	#10621877	—	2	4	no	—
123	C₂×C₄₃⋊C₈	—	2	4	no	—
124	C₂³×C₄₃	—	2	2	no	—
125	C₂³×C₄₃	—	2	2	no	—
126	C₂³×C₄₃	—	4	1	no	—
127	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
128	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
129	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	1	no	—
132	C₂×C₄×C₄₃	—	2	1	no	—
133	C₂×C₄×C₄₃	—	2	1	no	—
134	C₂×C₄×C₄₃	—	2	1	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	4	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
145	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	—	2	2	no	—
150	C₂×Dic₄₃	—	4	4	no	—
151	C₂×Dic₄₃	—	8	2	no	—
152	C₈×C₄₃	—	2	1	no	—
153	C₈×C₄₃	—	2	1	no	—
154	C₈×C₄₃	—	2	2	no	—
155	C₈×C₄₃	—	2	2	no	—
156	C₈×C₄₃	#117	1	2	yes	—
157	C₈×C₄₃	#117	1	2	yes	—
158	C₄₃⋊C₈	—	4	4	no	—
159	#22606	—	43	1	no	maximal, Sylow
160	C₂²×C₄₃	—	2	1	no	—
161	C₂²×C₄₃	—	2	1	no	—
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	—	2	1	no	—
167	C₂²×C₄₃	—	4	4	no	—
168	C₂²×C₄₃	#443	1	2	yes	—
169	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₄₃	#552	1	2	yes	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	—	2	1	no	—
173	C₄×C₄₃	—	4	4	no	—
174	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
175	C₄×C₄₃	#519	1	1	yes	characteristic
176	C₄×C₄₃	#547	1	2	yes	—
177	Dic₄₃	—	4	2	no	—
178	Dic₄₃	—	4	2	no	—
179	Dic₄₃	—	4	2	no	—
180	Dic₄₃	—	4	2	no	—
181	#2924	—	43	2	no	—
182	C₂×#427	—	43	1	no	—
183	#1945	—	43	2	no	—
184	#1922	—	43	4	no	—
185	#1826	—	43	2	no	—
186	#1810	—	43	1	no	—
187	#1772	—	43	2	no	—
188	C₈×C₂²⋊C₄	D₄₃	1	1	yes	characteristic
189	C₂×C₄₃	—	2	2	no	—
190	C₂×C₄₃	—	2	2	no	—
191	C₂×C₄₃	—	4	1	no	—
192	C₂×C₄₃	#2344	1	1	yes	characteristic
193	C₂×C₄₃	C₂×#443	1	1	yes	characteristic
194	C₂×C₄₃	#2957	1	1	yes	characteristic
195	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
196	C₂×#117	—	43	2	no	—
197	#433	—	86	4	no	—
198	#432	—	86	2	no	—
199	#431	—	86	2	no	—
200	C₂×#108	—	43	2	no	—
201	#427	—	43	2	no	—
202	#427	—	43	1	no	—
203	#427	—	43	1	no	—
204	#427	—	43	1	no	—
205	#427	—	43	1	no	—
206	#427	—	43	2	no	—
207	C₂×#107	—	43	1	no	—
208	C₂×#107	—	43	1	no	—
209	C₂×#106	—	43	2	no	—
210	#424	—	43	4	no	—
211	#407	—	43	2	no	—
212	C₂×C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
213	#405	—	43	4	no	—
214	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
215	C₄×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
216	#341	—	43	2	no	—
217	#339	—	43	4	no	—
218	#339	—	43	4	no	—
219	#336	C₂×D₄₃	1	2	yes	—
220	C₄₃	#22606	1	1	yes	characteristic, complemented, Sylow
221	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
222	C₂×M₁₆	—	43	4	no	—
223	C₂²×C₈	—	2	4	no	—
224	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
225	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
226	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
227	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
230	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
231	#117	—	43	2	no	—
232	#117	—	43	2	no	—
233	#117	—	43	2	no	—
234	#117	—	43	2	no	—
235	#117	—	86	2	no	—
236	#117	—	86	2	no	—
237	C₂×C₄⋊C₄	—	43	2	no	—
238	C₂×C₄⋊C₄	—	43	2	no	—
239	C₂×C₂²⋊C₄	—	2	1	no	—
240	C₂×C₂²⋊C₄	—	86	2	no	—
241	C₂×C₄²	—	86	2	no	—
242	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
243	#108	—	86	4	no	—
244	#107	—	43	2	no	—
245	#107	—	43	2	no	—
246	#107	—	43	1	no	—
247	#107	—	43	1	no	—
248	#107	—	43	1	no	—
249	#107	—	43	1	no	—
250	#106	—	43	2	no	—
251	#106	—	43	2	no	—
252	#106	—	43	2	no	—
253	#106	—	43	2	no	—
254	C₄⋊C₈	—	86	4	no	—
255	C₄⋊C₈	—	86	4	no	—
256	C₂²⋊C₈	—	86	2	no	—
257	C₂²⋊C₈	—	86	2	no	—
258	C₂²⋊C₈	D₁₇₂	1	1	yes	characteristic
259	C₂²⋊C₈	D₁₇₂	1	1	yes	characteristic
260	C₂²⋊C₈	D₁₇₂	1	2	yes	—
261	#97	—	86	4	no	—
262	C₄×C₈	—	2	4	no	—
263	C₄×C₈	—	86	4	no	—
264	#95	—	2	1	no	—
265	C₂⁴	#10621898	1	1	yes	characteristic
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	86	2	no	—
270	C₂²×C₄	—	86	2	no	—
271	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
275	C₂²×C₄	#10621898	1	1	yes	characteristic
276	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
277	M₁₆	—	86	4	no	—
278	M₁₆	—	86	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	86	4	no	—
287	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
291	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
292	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
293	C₂×C₈	#10621900	1	4	yes	—
294	C₄⋊C₄	—	43	2	no	—
295	C₄⋊C₄	—	43	2	no	—
296	C₄⋊C₄	—	43	2	no	—
297	C₄⋊C₄	—	43	2	no	—
298	C₄⋊C₄	—	86	2	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₂²⋊C₄	—	2	4	no	—
301	C₂²⋊C₄	—	86	2	no	—
302	C₂²⋊C₄	—	86	2	no	—
303	C₂²⋊C₄	—	86	2	no	—
304	C₂²⋊C₄	—	86	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	86	2	no	—
308	C₄²	—	86	2	no	—
309	C₄²	—	86	2	no	—
310	C₄²	—	86	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	—	2	2	no	—
313	C₂³	—	4	1	no	—
314	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
315	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
316	C₂³	C₂×#10621898	1	1	yes	characteristic
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	86	2	no	—
328	C₂×C₄	—	86	2	no	—
329	C₂×C₄	—	86	2	no	—
330	C₂×C₄	—	86	2	no	—
331	C₂×C₄	—	344	2	no	—
332	C₂×C₄	—	172	4	no	—
333	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
334	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
336	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
337	C₂×C₄	C₂×#10621900	1	2	yes	—
338	C₂×C₄	#13063372	1	2	yes	—
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	172	4	no	—
344	C₈	#13063281	1	2	yes	—
345	C₈	#13063281	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	4	4	no	—
354	C₂²	#14583017	1	2	yes	—
355	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
356	C₂²	#14583589	1	2	yes	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	4	no	—
360	C₄	—	172	2	no	—
361	C₄	—	172	2	no	—
362	C₄	—	172	2	no	—
363	C₄	—	172	2	no	—
364	C₄	C₂×#13063281	1	1	yes	characteristic
365	C₄	#14583376	1	1	yes	characteristic
366	C₄	#14583553	1	2	yes	—
367	C₂	—	2	2	no	—
368	C₂	—	2	2	no	—
369	C₂	—	4	1	no	—
370	C₂	#19137462	1	1	yes	characteristic
371	C₂	C₂×#14583017	1	1	yes	characteristic
372	C₂	#19143513	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₈×C₂²⋊C₄×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₈×C₂²⋊C₄	D₄₃
hypocenter	C₄₃	#22606
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. #19135343	C₂	semidirect
4. #19135195	C₂	semidirect
7. #19134474	C₂	semidirect
14. #14582937	C₄	semidirect
15. #14582937	C₄	semidirect
16. #14582937	C₄	semidirect
20. C₂×#13063259	C₄	semidirect
21. C₂×#13063259	C₄	semidirect
22. C₂×#13063257	C₄	semidirect
220. C₄₃	#22606	semidirect

Maximal subgroups

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

subgroup	index
#19143249	2
C₂×#14582937	2
#19135343	2
#19135195	2
#19134695	2
#19134633	2
#19134474	2
C₈×C₂²⋊C₄×C₄₃	2
#22606	43