Subgroups — #25066527

Order 11008.

610 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144635	C₂	1	1	yes	maximal, characteristic
2	#19143529	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
4	#19133468	C₂	1	4	yes	maximal, complemented
5	#19133415	C₂	1	4	yes	maximal, complemented
6	#19133366	C₂	1	1	yes	maximal, characteristic
7	#19133025	C₂	1	1	yes	maximal, characteristic
8	#19132999	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1452	C₂	1	1	yes	maximal, characteristic
10	#14583456	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
12	#14583235	C₄	1	2	yes	—
13	#14583195	C₄	1	2	yes	—
14	#14583110	C₄	1	2	yes	—
15	#14583057	C₄	1	2	yes	—
16	#14583029	—	2	1	no	—
17	#14583029	C₂²	1	2	yes	—
18	C₈×Dic₈₆	—	2	1	no	—
19	C₈×Dic₈₆	—	2	1	no	—
20	C₈×Dic₈₆	—	2	1	no	—
21	C₈×Dic₈₆	C₂²	1	2	yes	—
22	C₈×Dic₈₆	C₂²	1	2	yes	—
23	C₈×Dic₈₆	C₂²	1	2	yes	—
24	C₄×Dic₁₇₂	C₂²	1	4	yes	—
25	#14582925	—	2	1	no	—
26	#14582923	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×Dic₄₃⋊C₈	—	2	1	no	—
29	#14582854	—	2	1	no	—
30	C₂×C₈×Dic₄₃	—	2	1	no	—
31	#14582650	C₂²	1	4	yes	—
32	#14582631	C₂²	1	4	yes	—
33	#14582631	C₂²	1	4	yes	—
34	C₄₃×#431	C₂²	1	1	yes	characteristic
35	C₄₃×#431	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₄₃×#322	C₂²	1	2	yes	—
38	C₄₃×#322	C₂²	1	2	yes	—
39	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
40	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
41	C₂×#10621877	—	2	1	no	—
42	C₂²×C₄₃⋊C₈	—	2	1	no	—
43	#13063325	C₂×C₄	1	1	yes	characteristic
44	#13063325	C₂×C₄	1	1	yes	characteristic
45	#13063317	—	2	1	no	—
46	#13063304	—	2	1	no	—
47	#13063304	C₂×C₄	1	2	yes	—
48	C₄×Dic₈₆	—	2	1	no	—
49	C₄×Dic₈₆	—	2	1	no	—
50	C₄×Dic₈₆	—	2	1	no	—
51	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
52	C₄×Dic₈₆	C₂³	1	2	yes	—
53	C₄×Dic₈₆	C₂³	1	2	yes	—
54	#13063284	—	2	1	no	—
55	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
56	C₂×Dic₄₃⋊C₄	—	2	1	no	—
57	C₂×C₄×Dic₄₃	—	2	1	no	—
58	#13063259	C₂×C₄	1	4	yes	—
59	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
60	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
61	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
62	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
63	Dic₄₃⋊C₈	—	2	2	no	—
64	Dic₄₃⋊C₈	—	2	2	no	—
65	Dic₄₃⋊C₈	—	2	2	no	—
66	Dic₄₃⋊C₈	—	2	2	no	—
67	#13063250	C₂×C₄	1	4	yes	—
68	#13063250	C₂×C₄	1	4	yes	—
69	#13063235	—	2	2	no	—
70	C₈×Dic₄₃	—	2	2	no	—
71	C₈×Dic₄₃	—	2	2	no	—
72	C₈×Dic₄₃	—	2	2	no	—
73	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₃	—	2	1	no	—
77	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
78	C₂²⋊C₈×C₄₃	—	2	1	no	—
79	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
80	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
81	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
82	C₄×C₈×C₄₃	C₂³	1	2	yes	—
83	C₄×C₈×C₄₃	C₂³	1	2	yes	—
84	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
89	C₂²×Dic₄₃	—	2	1	no	—
90	C₂²×Dic₄₃	—	2	1	no	—
91	C₂×Dic₈₆	—	2	1	no	—
92	C₂×Dic₈₆	—	2	4	no	—
93	C₂×Dic₈₆	—	2	1	no	—
94	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
95	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
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	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₃	#41	1	1	yes	characteristic
107	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	2	1	no	—
110	C₄⋊Dic₄₃	—	2	1	no	—
111	C₄⋊Dic₄₃	—	2	1	no	—
112	C₄⋊Dic₄₃	—	2	1	no	—
113	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
114	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
115	Dic₄₃⋊C₄	—	2	2	no	—
116	Dic₄₃⋊C₄	—	2	2	no	—
117	Dic₄₃⋊C₄	—	2	2	no	—
118	Dic₄₃⋊C₄	—	2	2	no	—
119	C₄×Dic₄₃	—	2	2	no	—
120	C₄×Dic₄₃	—	2	2	no	—
121	C₄×Dic₄₃	—	2	2	no	—
122	#10621877	—	4	2	no	—
123	C₂×C₄₃⋊C₈	—	2	2	no	—
124	C₂×C₄₃⋊C₈	—	2	2	no	—
125	C₂×C₄₃⋊C₈	—	4	2	no	—
126	Dic₁₇₂	—	2	8	no	—
127	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	4	1	no	—
130	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#137	1	2	yes	—
134	C₂×C₄×C₄₃	#137	1	2	yes	—
135	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₄₃	—	2	4	no	—
139	C₂×Dic₄₃	—	2	2	no	—
140	C₂×Dic₄₃	—	2	2	no	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	—	4	2	no	—
144	Dic₈₆	—	2	4	no	—
145	Dic₈₆	—	2	2	no	—
146	Dic₈₆	—	2	4	no	—
147	Dic₈₆	—	4	2	no	—
148	Dic₈₆	#131	1	4	yes	—
149	C₈×C₄₃	—	2	4	no	—
150	C₈×C₄₃	—	2	1	no	—
151	C₈×C₄₃	—	2	1	no	—
152	C₈×C₄₃	—	2	1	no	—
153	C₈×C₄₃	—	2	1	no	—
154	C₈×C₄₃	—	4	2	no	—
155	C₄₃⋊C₈	—	4	2	no	—
156	C₄₃⋊C₈	—	4	2	no	—
157	#17906	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	#441	1	2	yes	—
161	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₄×C₄₃	—	2	1	no	—
163	C₄×C₄₃	—	2	1	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	#441	1	2	yes	—
166	C₄×C₄₃	C₂×#131	1	2	yes	—
167	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
168	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
169	Dic₄₃	—	2	4	no	—
170	Dic₄₃	—	4	2	no	—
171	Dic₄₃	—	4	2	no	—
172	Dic₄₃	—	4	2	no	—
173	#3104	—	43	1	no	—
174	#2960	—	43	1	no	—
175	C₂×C₈×Q₈	—	43	1	no	—
176	#1587	—	43	4	no	—
177	#1579	—	43	4	no	—
178	#1570	—	43	1	no	—
179	#1487	—	43	1	no	—
180	#1481	—	43	1	no	—
181	#1452	D₄₃	1	1	yes	characteristic
182	C₂×C₄₃	—	2	2	no	—
183	C₂×C₄₃	#2929	1	1	yes	characteristic
184	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
185	C₂×C₄₃	#2955	1	1	yes	characteristic
186	#533	—	43	1	no	—
187	C₂×C₄×Q₈	—	43	1	no	—
188	#484	—	43	2	no	—
189	#477	—	43	2	no	—
190	#462	—	43	2	no	—
191	#451	—	43	2	no	—
192	#446	—	43	2	no	—
193	#446	—	86	1	no	—
194	C₈×Q₈	—	43	2	no	—
195	C₈×Q₈	—	43	2	no	—
196	C₈×Q₈	—	43	2	no	—
197	C₈×Q₈	—	86	1	no	—
198	C₈×Q₈	—	86	1	no	—
199	C₈×Q₈	—	86	1	no	—
200	C₄×Q₁₆	—	43	4	no	—
201	#431	C₂×D₄₃	1	1	yes	characteristic
202	#431	C₂×D₄₃	1	1	yes	characteristic
203	#424	—	86	1	no	—
204	#423	—	43	1	no	—
205	C₂×C₄⋊C₈	—	43	1	no	—
206	C₂×C₄⋊C₈	—	86	1	no	—
207	#405	—	86	1	no	—
208	C₂×C₄×C₈	—	86	1	no	—
209	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
210	#334	—	43	4	no	—
211	Q₈⋊C₈	—	43	4	no	—
212	Q₈⋊C₈	—	43	4	no	—
213	#322	C₂×D₄₃	1	2	yes	—
214	#322	C₂×D₄₃	1	2	yes	—
215	C₄₃	#17906	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	—	43	1	no	—
217	C₂×Q₁₆	—	43	4	no	—
218	C₂×M₁₆	—	86	1	no	—
219	C₂²×C₈	—	86	1	no	—
220	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
221	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
222	C₄⋊Q₈	—	43	1	no	—
223	C₄⋊Q₈	—	43	1	no	—
224	#125	—	86	1	no	—
225	C₂²⋊Q₈	—	43	2	no	—
226	C₂²⋊Q₈	—	86	1	no	—
227	C₄×Q₈	—	43	2	no	—
228	C₄×Q₈	—	43	2	no	—
229	C₄×Q₈	—	43	2	no	—
230	C₄×Q₈	—	86	1	no	—
231	C₄×Q₈	—	86	1	no	—
232	C₄×Q₈	—	86	1	no	—
233	#117	—	86	1	no	—
234	C₂×C₄⋊C₄	—	43	1	no	—
235	C₂×C₄⋊C₄	—	86	1	no	—
236	C₂×C₄²	—	86	1	no	—
237	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
238	#107	—	43	4	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	43	2	no	—
241	C₄⋊C₈	—	43	2	no	—
242	C₄⋊C₈	—	86	2	no	—
243	C₄⋊C₈	—	86	2	no	—
244	C₄⋊C₈	—	86	2	no	—
245	C₄⋊C₈	—	86	2	no	—
246	C₄⋊C₈	—	86	1	no	—
247	C₄⋊C₈	—	86	1	no	—
248	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
249	#103	—	43	4	no	—
250	#103	—	43	4	no	—
251	C₂²⋊C₈	—	2	1	no	—
252	C₂²⋊C₈	C₄×D₄₃	1	2	yes	—
253	#97	—	86	2	no	—
254	C₄×C₈	—	86	2	no	—
255	C₄×C₈	—	86	2	no	—
256	C₄×C₈	—	86	2	no	—
257	C₄×C₈	D₁₇₂	1	1	yes	characteristic
258	C₄×C₈	D₁₇₂	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₄₃	1	2	yes	—
260	C₄×C₈	C₂²×D₄₃	1	2	yes	—
261	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
263	C₂×Q₈	—	43	2	no	—
264	C₂×Q₈	—	43	2	no	—
265	C₂×Q₈	—	86	4	no	—
266	C₂×Q₈	—	86	1	no	—
267	C₂×Q₈	—	86	1	no	—
268	C₂²×C₄	—	86	1	no	—
269	C₂²×C₄	—	86	1	no	—
270	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
272	C₂²×C₄	#10621898	1	1	yes	characteristic
273	Q₁₆	—	86	8	no	—
274	M₁₆	—	172	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	86	2	no	—
279	C₂×C₈	—	86	2	no	—
280	C₂×C₈	—	172	2	no	—
281	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
282	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
283	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
284	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
285	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
286	C₄⋊C₄	—	43	2	no	—
287	C₄⋊C₄	—	43	2	no	—
288	C₄⋊C₄	—	86	1	no	—
289	C₄⋊C₄	—	86	2	no	—
290	C₄⋊C₄	—	86	1	no	—
291	C₄⋊C₄	—	86	1	no	—
292	C₄⋊C₄	—	86	2	no	—
293	C₄⋊C₄	—	86	1	no	—
294	C₄⋊C₄	—	86	2	no	—
295	C₄⋊C₄	—	86	2	no	—
296	C₂²⋊C₄	—	86	2	no	—
297	C₄²	—	86	2	no	—
298	C₄²	—	86	2	no	—
299	C₄²	—	86	2	no	—
300	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
301	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
302	C₄²	#10621898	1	1	yes	characteristic
303	C₄²	C₂³×D₄₃	1	1	yes	characteristic
304	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
305	Q₈	—	43	4	no	—
306	Q₈	—	86	4	no	—
307	Q₈	—	86	2	no	—
308	Q₈	—	86	4	no	—
309	Q₈	—	172	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	86	4	no	—
313	C₂×C₄	—	86	2	no	—
314	C₂×C₄	—	86	2	no	—
315	C₂×C₄	—	86	2	no	—
316	C₂×C₄	—	86	2	no	—
317	C₂×C₄	—	172	2	no	—
318	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
319	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
320	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
321	C₂×C₄	#13063358	1	2	yes	—
322	C₂×C₄	#13063358	1	2	yes	—
323	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
325	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
326	C₈	—	2	4	no	—
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	—	4	2	no	—
332	C₈	—	172	2	no	—
333	C₈	—	172	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	#14583007	1	2	yes	—
337	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	86	4	no	—
342	C₄	—	172	2	no	—
343	C₄	—	172	2	no	—
344	C₄	—	172	2	no	—
345	C₄	#14583007	1	2	yes	—
346	C₄	#131×D₄₃	1	2	yes	—
347	C₄	C₂×#13063358	1	1	yes	characteristic
348	C₄	C₂×#13063358	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	#19143291	1	1	yes	characteristic
351	C₂	C₂×#14583007	1	1	yes	characteristic
352	C₂	#19143499	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1452	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1452	D₄₃
hypocenter	C₄₃	#17906
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1452

Complements

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

N	H ≅ G/N	type
4. #19133468	C₂	semidirect
5. #19133415	C₂	semidirect
215. C₄₃	#17906	semidirect

Maximal subgroups

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

subgroup	index
#19144635	2
#19143529	2
C₂×C₈×Dic₈₆	2
#19133468	2
#19133415	2
#19133366	2
#19133025	2
#19132999	2
C₄₃×#1452	2
#17906	43