Subgroups — #25066294

Order 11008.

626 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143678	C₂	1	1	yes	maximal, characteristic
2	#19143675	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
4	#19133465	C₂	1	4	yes	maximal, complemented
5	#19133444	C₂	1	4	yes	maximal, complemented
6	#19132989	C₂	1	1	yes	maximal, characteristic
7	#19132989	C₂	1	1	yes	maximal, characteristic
8	#19132887	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₇₂	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
13	#14583029	—	2	2	no	—
14	#14583026	C₂²	1	4	yes	—
15	C₈×Dic₈₆	—	2	2	no	—
16	#14583023	C₂²	1	4	yes	—
17	C₄×Dic₁₇₂	C₄	1	2	yes	—
18	C₄×Dic₁₇₂	C₄	1	2	yes	—
19	C₄×Dic₁₇₂	C₂²	1	2	yes	—
20	C₄×Dic₁₇₂	C₂²	1	2	yes	—
21	#14582953	—	2	2	no	—
22	#14582952	C₂²	1	1	yes	characteristic
23	#14582952	C₂²	1	1	yes	characteristic
24	#14582949	—	2	2	no	—
25	C₂×#13063259	C₄	1	1	yes	characteristic
26	C₂×#13063250	C₄	1	1	yes	characteristic
27	C₂×#13063250	C₄	1	1	yes	characteristic
28	#14582629	C₂²	1	4	yes	—
29	#14582629	C₂²	1	4	yes	—
30	#14582621	C₂²	1	2	yes	—
31	#14582621	C₂²	1	2	yes	—
32	C₄₃×#431	C₂²	1	1	yes	characteristic
33	C₄₃×#431	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
35	C₄₃×#335	C₂²	1	2	yes	—
36	C₄₃×#334	C₂²	1	2	yes	—
37	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
39	C₂×Dic₁₇₂	—	2	4	no	—
40	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
41	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
42	C₄×Dic₈₆	—	2	1	no	—
43	C₄×Dic₈₆	—	2	1	no	—
44	C₄×Dic₈₆	—	2	1	no	—
45	C₄×Dic₈₆	—	2	1	no	—
46	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
47	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
48	C₄×Dic₈₆	C₂³	1	2	yes	—
49	C₄×Dic₈₆	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
52	C₂×Dic₄₃⋊C₄	—	2	1	no	—
53	C₂×Dic₄₃⋊C₄	—	2	1	no	—
54	C₂×C₄×Dic₄₃	—	2	1	no	—
55	C₂×C₄×Dic₄₃	—	2	1	no	—
56	#13063259	C₂×C₄	1	2	yes	—
57	#13063259	C₂×C₄	1	2	yes	—
58	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
59	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
60	Dic₄₃⋊C₈	—	2	4	no	—
61	Dic₄₃⋊C₈	—	2	4	no	—
62	#13063250	C₂×C₄	1	2	yes	—
63	#13063250	C₂×C₄	1	2	yes	—
64	#13063250	C₂×C₄	1	2	yes	—
65	#13063250	C₂×C₄	1	2	yes	—
66	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
67	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
68	#13063235	—	2	4	no	—
69	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
70	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
71	C₈×Dic₄₃	—	2	4	no	—
72	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
76	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
77	C₂²⋊C₈×C₄₃	—	2	1	no	—
78	C₂²⋊C₈×C₄₃	—	2	1	no	—
79	C₄×C₈×C₄₃	—	2	1	no	—
80	C₄×C₈×C₄₃	—	2	1	no	—
81	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
88	C₂²×Dic₄₃	—	2	1	no	—
89	C₂²×Dic₄₃	—	2	1	no	—
90	C₂²×Dic₄₃	—	2	1	no	—
91	C₂²×Dic₄₃	—	2	1	no	—
92	C₂×Dic₈₆	—	2	1	no	—
93	C₂×Dic₈₆	—	2	2	no	—
94	C₂×Dic₈₆	—	2	2	no	—
95	C₂×Dic₈₆	—	2	2	no	—
96	C₂×Dic₈₆	—	2	1	no	—
97	C₂×Dic₈₆	M₁₆	1	4	yes	—
98	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
99	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	2	2	no	—
104	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₃	#41	1	1	yes	characteristic
111	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
112	C₄⋊Dic₄₃	—	2	1	no	—
113	C₄⋊Dic₄₃	—	2	1	no	—
114	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
115	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
116	Dic₄₃⋊C₄	—	2	2	no	—
117	Dic₄₃⋊C₄	—	2	2	no	—
118	Dic₄₃⋊C₄	—	2	2	no	—
119	Dic₄₃⋊C₄	—	2	2	no	—
120	C₄×Dic₄₃	—	2	2	no	—
121	C₄×Dic₄₃	—	2	2	no	—
122	C₄×Dic₄₃	—	2	2	no	—
123	C₄×Dic₄₃	—	2	2	no	—
124	C₂×C₄₃⋊C₈	—	2	2	no	—
125	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
126	Dic₁₇₂	—	2	4	no	—
127	Dic₁₇₂	—	2	4	no	—
128	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	4	1	no	—
131	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₄₃	#131	1	2	yes	—
136	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₄₃	—	2	2	no	—
140	C₂×Dic₄₃	—	2	4	no	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	—	2	2	no	—
144	C₂×Dic₄₃	—	4	2	no	—
145	C₂×Dic₄₃	—	8	1	no	—
146	C₂×Dic₄₃	—	8	1	no	—
147	Dic₈₆	—	2	2	no	—
148	Dic₈₆	—	2	4	no	—
149	Dic₈₆	—	2	2	no	—
150	Dic₈₆	—	2	2	no	—
151	Dic₈₆	—	4	2	no	—
152	Dic₈₆	C₂×M₁₆	1	4	yes	—
153	C₈×C₄₃	—	2	1	no	—
154	C₈×C₄₃	—	2	1	no	—
155	C₈×C₄₃	—	2	1	no	—
156	C₈×C₄₃	—	2	1	no	—
157	C₈×C₄₃	—	4	2	no	—
158	C₈×C₄₃	—	4	2	no	—
159	C₄₃⋊C₈	—	2	4	no	—
160	C₄₃⋊C₈	—	4	2	no	—
161	#17880	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	#435	1	2	yes	—
165	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	#435	1	2	yes	—
170	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
171	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
172	C₄×C₄₃	#578	1	2	yes	—
173	Dic₄₃	—	2	4	no	—
174	Dic₄₃	—	4	2	no	—
175	Dic₄₃	—	4	2	no	—
176	Dic₄₃	—	4	2	no	—
177	#2983	—	43	1	no	—
178	#2982	—	43	1	no	—
179	C₂×C₄×Q₁₆	—	43	1	no	—
180	#1587	—	43	4	no	—
181	#1584	—	43	4	no	—
182	#1564	D₄₃	1	1	yes	characteristic
183	#1479	—	43	1	no	—
184	#1479	—	43	1	no	—
185	#1452	—	43	1	no	—
186	C₂×C₄₃	—	2	2	no	—
187	C₂×C₄₃	C₂×#435	1	1	yes	characteristic
188	C₂×C₄₃	#2950	1	1	yes	characteristic
189	C₂×C₄₃	#2957	1	1	yes	characteristic
190	C₂²×Q₁₆	—	43	1	no	—
191	C₂×C₄×Q₈	—	43	1	no	—
192	C₂×C₄×Q₈	—	43	1	no	—
193	#446	—	43	4	no	—
194	#446	—	86	2	no	—
195	C₈×Q₈	—	43	4	no	—
196	C₈×Q₈	—	86	2	no	—
197	C₄×Q₁₆	—	43	2	no	—
198	C₄×Q₁₆	—	43	2	no	—
199	C₄×Q₁₆	—	43	2	no	—
200	C₄×Q₁₆	—	43	2	no	—
201	#432	—	86	2	no	—
202	#431	—	43	1	no	—
203	#431	—	43	1	no	—
204	#431	—	86	2	no	—
205	#431	C₂×D₄₃	1	1	yes	characteristic
206	#431	C₂×D₄₃	1	1	yes	characteristic
207	C₂×#107	—	43	1	no	—
208	C₂×#103	—	43	1	no	—
209	C₂×#103	—	43	1	no	—
210	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
211	#335	C₂×D₄₃	1	2	yes	—
212	#334	C₂×D₄₃	1	2	yes	—
213	Q₈⋊C₈	—	43	4	no	—
214	Q₈⋊C₈	—	43	4	no	—
215	#322	—	43	2	no	—
216	#322	—	43	2	no	—
217	C₄₃	#17880	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	—	43	1	no	—
219	C₂²×Q₈	—	43	1	no	—
220	C₂×Q₁₆	—	43	2	no	—
221	C₂×Q₁₆	—	43	2	no	—
222	C₂×Q₁₆	—	86	4	no	—
223	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
224	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
225	C₄×Q₈	—	43	2	no	—
226	C₄×Q₈	—	43	2	no	—
227	C₄×Q₈	—	43	2	no	—
228	C₄×Q₈	—	43	2	no	—
229	C₄×Q₈	—	86	1	no	—
230	C₄×Q₈	—	86	1	no	—
231	C₄×Q₈	—	86	1	no	—
232	C₄×Q₈	—	86	1	no	—
233	C₂×C₄⋊C₄	—	43	1	no	—
234	C₂×C₄⋊C₄	—	43	1	no	—
235	C₂×C₄⋊C₄	—	86	1	no	—
236	C₂×C₄⋊C₄	—	86	1	no	—
237	C₂×C₄²	—	86	1	no	—
238	C₂×C₄²	—	86	1	no	—
239	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
240	#107	—	43	2	no	—
241	#107	—	43	2	no	—
242	C₄⋊C₈	—	43	2	no	—
243	C₄⋊C₈	—	86	4	no	—
244	C₄⋊C₈	—	86	4	no	—
245	C₄⋊C₈	—	86	1	no	—
246	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
247	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
248	#103	—	43	2	no	—
249	#103	—	43	2	no	—
250	#103	—	43	2	no	—
251	#103	—	43	2	no	—
252	C₂²⋊C₈	—	2	1	no	—
253	C₂²⋊C₈	—	2	1	no	—
254	C₂²⋊C₈	—	43	2	no	—
255	C₂²⋊C₈	—	86	1	no	—
256	#97	—	86	4	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	43	2	no	—
260	C₄×C₈	—	43	2	no	—
261	C₄×C₈	—	86	4	no	—
262	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
263	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
266	C₂×Q₈	—	43	4	no	—
267	C₂×Q₈	—	43	2	no	—
268	C₂×Q₈	—	43	2	no	—
269	C₂×Q₈	—	86	1	no	—
270	C₂×Q₈	—	86	1	no	—
271	C₂×Q₈	—	86	2	no	—
272	C₂×Q₈	—	86	2	no	—
273	C₂×Q₈	—	86	2	no	—
274	C₂²×C₄	—	86	1	no	—
275	C₂²×C₄	—	86	1	no	—
276	C₂²×C₄	—	86	1	no	—
277	C₂²×C₄	—	86	1	no	—
278	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
280	C₂²×C₄	#10621902	1	1	yes	characteristic
281	Q₁₆	—	86	4	no	—
282	Q₁₆	—	86	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	43	4	no	—
288	C₂×C₈	—	86	2	no	—
289	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
293	C₄⋊C₄	—	43	2	no	—
294	C₄⋊C₄	—	43	2	no	—
295	C₄⋊C₄	—	86	1	no	—
296	C₄⋊C₄	—	86	1	no	—
297	C₄⋊C₄	—	86	2	no	—
298	C₄⋊C₄	—	86	2	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₄⋊C₄	—	86	2	no	—
301	C₄²	—	86	2	no	—
302	C₄²	—	86	2	no	—
303	C₄²	—	86	2	no	—
304	C₄²	—	86	2	no	—
305	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
306	C₄²	D₄×D₄₃	1	1	yes	characteristic
307	C₄²	#10621902	1	1	yes	characteristic
308	C₄²	C₂³×D₄₃	1	1	yes	characteristic
309	C₂³	#13063287	1	1	yes	characteristic
310	Q₈	—	43	4	no	—
311	Q₈	—	86	2	no	—
312	Q₈	—	86	4	no	—
313	Q₈	—	86	2	no	—
314	Q₈	—	86	2	no	—
315	Q₈	—	172	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	86	2	no	—
319	C₂×C₄	—	86	4	no	—
320	C₂×C₄	—	86	2	no	—
321	C₂×C₄	—	86	2	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	344	1	no	—
324	C₂×C₄	—	344	1	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	#13063287	1	1	yes	characteristic
327	C₂×C₄	#13063287	1	1	yes	characteristic
328	C₂×C₄	#13063287	1	1	yes	characteristic
329	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
330	C₂×C₄	#13063331	1	2	yes	—
331	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
333	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	86	4	no	—
341	C₈	—	172	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	#14582969	1	2	yes	—
345	C₂²	C₂×#13063287	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	86	4	no	—
350	C₄	—	172	2	no	—
351	C₄	—	172	2	no	—
352	C₄	—	172	2	no	—
353	C₄	#14582969	1	2	yes	—
354	C₄	C₂×M₁₆×D₄₃	1	1	yes	characteristic
355	C₄	C₂×#13063331	1	1	yes	characteristic
356	C₄	#14583713	1	2	yes	—
357	C₂	—	2	2	no	—
358	C₂	C₂×#14582969	1	1	yes	characteristic
359	C₂	#19143462	1	1	yes	characteristic
360	C₂	#19143511	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1564	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1564	D₄₃
hypocenter	C₄₃	#17880
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
4. #19133465	C₂	semidirect
5. #19133444	C₂	semidirect
217. C₄₃	#17880	semidirect

Maximal subgroups

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

subgroup	index
#19143678	2
#19143675	2
C₂×C₄×Dic₁₇₂	2
#19133465	2
#19133444	2
#19132989	2
#19132887	2
C₄₃×#1564	2
#17880	43