Subgroups — #24992007

Order 11008.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143618	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583001	C₂	1	1	yes	maximal, characteristic
3	#19132989	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132987	C₂	1	1	yes	maximal, characteristic
5	#19132918	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132426	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1281	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621873	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
11	#14583001	—	2	2	no	—
12	#14583001	—	2	2	no	—
13	C₈×D₁₇₂	—	2	2	no	—
14	#14582964	—	2	2	no	—
15	#14582952	C₂²	1	1	yes	characteristic
16	#14582952	C₂²	1	1	yes	characteristic
17	C₂×#13063259	C₄	1	1	yes	characteristic
18	C₂×#13063250	C₄	1	1	yes	characteristic
19	C₂×#13063247	C₄	1	1	yes	characteristic
20	C₂²⋊C₈×D₄₃	—	2	2	no	—
21	#14582629	—	2	2	no	—
22	#14582628	—	2	2	no	—
23	#14582624	—	2	2	no	—
24	C₄₃×#431	C₂²	1	1	yes	characteristic
25	C₄₃×#431	C₂²	1	1	yes	characteristic
26	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₁₇₂	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₄₃	—	2	1	no	—
30	C₂×#10621873	—	2	2	no	—
31	C₂×#10621873	—	2	2	no	—
32	C₂×#10621873	C₈	1	4	yes	complemented
33	C₂×C₈×D₄₃	—	2	4	no	—
34	C₄×Dic₈₆	—	2	1	no	—
35	C₄×Dic₈₆	—	2	1	no	—
36	C₄×Dic₈₆	—	2	1	no	—
37	C₄×Dic₈₆	—	2	1	no	—
38	C₄×D₁₇₂	—	2	1	no	—
39	C₄×D₁₇₂	—	2	1	no	—
40	C₄×D₁₇₂	D₄	1	2	yes	—
41	C₄×D₁₇₂	D₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₄₃⋊C₄	—	2	1	no	—
45	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
46	C₂×C₄×Dic₄₃	—	2	1	no	—
47	#13063259	—	2	2	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
50	#13063250	—	2	2	no	—
51	#13063247	—	2	2	no	—
52	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
53	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
54	(C₂×D₄₃)⋊C₈	—	2	4	no	—
55	C₄×C₄₃⋊C₈	—	2	1	no	—
56	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
57	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₃	—	2	1	no	—
61	C₄⋊C₈×C₄₃	—	2	1	no	—
62	C₂²⋊C₈×C₄₃	—	2	1	no	—
63	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
64	#97×C₄₃	—	2	1	no	—
65	#97×C₄₃	—	2	1	no	—
66	C₄×C₈×C₄₃	—	2	1	no	—
67	C₄×C₈×C₄₃	—	2	1	no	—
68	C₂³×D₄₃	—	2	1	no	—
69	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
72	C₂²×Dic₄₃	—	2	1	no	—
73	C₂²×Dic₄₃	—	2	1	no	—
74	C₂²×Dic₄₃	—	2	1	no	—
75	C₂×D₁₇₂	—	2	1	no	—
76	C₂×D₁₇₂	—	2	1	no	—
77	C₂×D₁₇₂	—	2	1	no	—
78	C₂×D₁₇₂	C₂²⋊C₄	1	2	yes	—
79	C₂×D₁₇₂	C₂×C₈	1	2	yes	—
80	C₂×D₁₇₂	M₁₆	1	2	yes	—
81	C₂×C₄×D₄₃	—	2	2	no	—
82	C₂×C₄×D₄₃	—	2	2	no	—
83	C₂×C₄×D₄₃	—	8	2	no	—
84	C₂×Dic₈₆	—	2	1	no	—
85	C₂×Dic₈₆	—	2	1	no	—
86	C₂×Dic₈₆	—	2	1	no	—
87	C₂×Dic₈₆	—	2	1	no	—
88	C₂×Dic₈₆	—	2	1	no	—
89	C₂×Dic₈₆	C₂×C₈	1	2	yes	complemented
90	C₂×C₈×C₄₃	—	2	1	no	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	2	1	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	1	no	—
95	C₂×C₈×C₄₃	—	2	1	no	—
96	C₂×C₈×C₄₃	—	4	1	no	—
97	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	complemented
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₃	#41	1	1	yes	characteristic
102	(C₂×D₄₃)⋊C₄	—	2	2	no	—
103	(C₂×D₄₃)⋊C₄	—	2	2	no	—
104	C₄⋊Dic₄₃	—	2	1	no	—
105	C₄⋊Dic₄₃	—	2	1	no	—
106	C₄⋊Dic₄₃	—	2	1	no	—
107	C₄⋊Dic₄₃	C₂²⋊C₄	1	2	yes	—
108	Dic₄₃⋊C₄	—	4	1	no	—
109	Dic₄₃⋊C₄	—	4	1	no	—
110	C₄×Dic₄₃	—	4	1	no	—
111	C₄×Dic₄₃	—	4	1	no	—
112	C₂×C₄₃⋊C₈	—	2	2	no	—
113	C₂×C₄₃⋊C₈	—	4	1	no	—
114	#10621873	—	2	8	no	—
115	C₈×D₄₃	—	8	4	no	—
116	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₄₃	—	2	2	no	—
118	C₂²×D₄₃	—	2	2	no	—
119	C₂²×D₄₃	—	2	2	no	—
120	C₂²×D₄₃	—	8	1	no	—
121	C₂²×D₄₃	—	8	1	no	—
122	C₂×C₄×C₄₃	—	4	1	no	—
123	C₂×C₄×C₄₃	—	4	1	no	—
124	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
134	C₂×Dic₄₃	—	2	2	no	—
135	C₂×Dic₄₃	—	2	2	no	—
136	C₂×Dic₄₃	—	4	1	no	—
137	C₂×Dic₄₃	—	4	1	no	—
138	C₂×Dic₄₃	—	4	1	no	—
139	C₂×Dic₄₃	—	8	1	no	—
140	C₂×Dic₄₃	—	8	1	no	—
141	D₁₇₂	—	2	2	no	—
142	D₁₇₂	—	2	2	no	—
143	D₁₇₂	—	2	2	no	—
144	D₁₇₂	C₂²⋊C₈	1	4	yes	—
145	C₄×D₄₃	—	4	4	no	—
146	C₄×D₄₃	—	8	2	no	—
147	Dic₈₆	—	2	2	no	—
148	Dic₈₆	—	2	2	no	—
149	Dic₈₆	—	4	1	no	—
150	Dic₈₆	—	4	1	no	—
151	C₈×C₄₃	—	2	2	no	—
152	C₈×C₄₃	—	4	2	no	—
153	C₈×C₄₃	—	4	1	no	—
154	C₈×C₄₃	—	8	1	no	—
155	C₄₃⋊C₈	—	4	2	no	—
156	C₄₃⋊C₈	—	8	1	no	—
157	#9087	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	#386	1	1	yes	characteristic
161	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
162	C₂²×C₄₃	#435	1	1	yes	characteristic
163	C₂×D₄₃	—	2	4	no	—
164	C₂×D₄₃	—	4	2	no	—
165	C₂×D₄₃	—	4	2	no	—
166	C₂×D₄₃	—	8	2	no	—
167	C₂×D₄₃	—	8	2	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
173	C₄×C₄₃	#408	1	1	yes	characteristic
174	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
175	C₄×C₄₃	#435	1	1	yes	characteristic
176	Dic₄₃	—	4	2	no	—
177	Dic₄₃	—	4	2	no	—
178	Dic₄₃	—	8	1	no	—
179	#2972	—	43	1	no	—
180	C₂×#440	—	43	1	no	—
181	#1479	—	43	1	no	—
182	#1478	—	43	1	no	—
183	#1461	—	43	1	no	—
184	#1281	D₄₃	1	1	yes	characteristic, complemented
185	#1275	—	43	1	no	—
186	C₂×C₄₃	—	2	1	no	—
187	C₂×C₄₃	—	2	1	no	—
188	C₂×C₄₃	#1869	1	1	yes	characteristic
189	C₂×C₄₃	#1902	1	1	yes	characteristic
190	C₂×C₄₃	#1904	1	1	yes	characteristic
191	D₄₃	—	4	4	no	—
192	C₂²×SD₁₆	—	43	1	no	—
193	C₂×C₄×Q₈	—	43	1	no	—
194	C₂×C₄×D₄	—	43	1	no	—
195	#440	—	86	2	no	—
196	#440	—	86	2	no	—
197	C₈×D₄	—	86	2	no	—
198	C₈×D₄	—	86	2	no	—
199	#431	—	43	1	no	—
200	#431	—	43	1	no	—
201	#431	C₂×D₄₃	1	1	yes	characteristic
202	#431	C₂×D₄₃	1	1	yes	characteristic
203	C₂×#107	—	43	1	no	—
204	C₂×#103	—	43	1	no	—
205	C₂×#102	—	43	1	no	—
206	C₂×C₂²⋊C₈	—	86	2	no	—
207	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
208	Q₈⋊C₈	—	86	2	no	—
209	D₄⋊C₈	—	86	2	no	—
210	#324	—	86	2	no	—
211	C₄₃	#9087	1	1	yes	characteristic, complemented, Sylow
212	C₂²×Q₈	—	43	1	no	—
213	C₂²×D₄	—	43	1	no	—
214	C₂³×C₄	—	86	1	no	—
215	C₂×SD₁₆	—	43	4	no	—
216	C₂×SD₁₆	—	86	2	no	—
217	C₂×SD₁₆	—	86	2	no	—
218	C₂²×C₈	—	86	4	no	—
219	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
220	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
221	C₄×Q₈	—	86	1	no	—
222	C₄×Q₈	—	86	1	no	—
223	C₄×Q₈	—	86	1	no	—
224	C₄×Q₈	—	86	1	no	—
225	C₄×D₄	—	43	2	no	—
226	C₄×D₄	—	43	2	no	—
227	C₄×D₄	—	86	1	no	—
228	C₄×D₄	—	86	1	no	—
229	C₂×C₄⋊C₄	—	43	1	no	—
230	C₂×C₄⋊C₄	—	43	1	no	—
231	C₂×C₄⋊C₄	—	86	1	no	—
232	C₂×C₂²⋊C₄	—	86	1	no	—
233	C₂×C₄²	—	86	1	no	—
234	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
235	#107	—	86	2	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	86	1	no	—
239	C₄⋊C₈	—	86	1	no	—
240	#103	—	86	2	no	—
241	#102	—	86	2	no	—
242	C₂²⋊C₈	—	2	1	no	—
243	C₂²⋊C₈	—	86	4	no	—
244	C₂²⋊C₈	—	86	1	no	—
245	C₂²⋊C₈	—	86	1	no	—
246	C₂²⋊C₈	D₁₇₂	1	2	yes	—
247	#97	—	2	1	no	—
248	#97	—	2	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	—	43	2	no	—
252	C₄×C₈	—	86	1	no	—
253	C₂⁴	—	86	1	no	—
254	C₂×Q₈	—	43	2	no	—
255	C₂×Q₈	—	86	1	no	—
256	C₂×Q₈	—	86	1	no	—
257	C₂×Q₈	—	86	1	no	—
258	C₂×Q₈	—	86	1	no	—
259	C₂×Q₈	—	86	1	no	—
260	C₂×D₄	—	43	2	no	—
261	C₂×D₄	—	43	2	no	—
262	C₂×D₄	—	43	2	no	—
263	C₂×D₄	—	86	1	no	—
264	C₂×D₄	—	86	1	no	—
265	C₂×D₄	—	86	1	no	—
266	C₂²×C₄	—	86	1	no	—
267	C₂²×C₄	—	86	1	no	—
268	C₂²×C₄	—	86	2	no	—
269	C₂²×C₄	—	86	2	no	—
270	C₂²×C₄	—	86	1	no	—
271	C₂²×C₄	—	344	2	no	—
272	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
274	C₂²×C₄	#10621898	1	1	yes	characteristic
275	SD₁₆	—	86	8	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	86	2	no	—
284	C₂×C₈	—	344	4	no	—
285	C₂×C₈	—	172	1	no	—
286	C₂×C₈	C₈×D₄₃	1	2	yes	complemented
287	C₄⋊C₄	—	43	2	no	—
288	C₄⋊C₄	—	86	1	no	—
289	C₄⋊C₄	—	86	1	no	—
290	C₄⋊C₄	—	86	1	no	—
291	C₄⋊C₄	—	172	1	no	—
292	C₄⋊C₄	—	172	1	no	—
293	C₂²⋊C₄	—	86	2	no	—
294	C₂²⋊C₄	—	86	2	no	—
295	C₄²	—	172	1	no	—
296	C₄²	—	172	1	no	—
297	C₄²	D₄×D₄₃	1	1	yes	characteristic
298	C₄²	D₄×D₄₃	1	1	yes	characteristic
299	C₄²	D₄×D₄₃	1	1	yes	characteristic
300	C₄²	#10621902	1	1	yes	characteristic
301	C₂³	—	86	2	no	—
302	C₂³	—	86	2	no	—
303	C₂³	—	86	2	no	—
304	C₂³	—	344	1	no	—
305	C₂³	—	344	1	no	—
306	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
307	Q₈	—	86	2	no	—
308	Q₈	—	86	2	no	—
309	Q₈	—	172	1	no	—
310	Q₈	—	172	1	no	—
311	D₄	—	43	4	no	—
312	D₄	—	86	2	no	—
313	D₄	—	86	2	no	—
314	D₄	—	86	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	86	2	no	—
318	C₂×C₄	—	86	2	no	—
319	C₂×C₄	—	344	2	no	—
320	C₂×C₄	—	344	1	no	—
321	C₂×C₄	—	344	1	no	—
322	C₂×C₄	—	172	4	no	—
323	C₂×C₄	—	172	1	no	—
324	C₂×C₄	—	172	1	no	—
325	C₂×C₄	—	172	1	no	—
326	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
327	C₂×C₄	#13063287	1	1	yes	characteristic
328	C₂×C₄	#13063295	1	1	yes	characteristic
329	C₂×C₄	#13063298	1	1	yes	characteristic
330	C₂×C₄	#13063300	1	1	yes	characteristic
331	C₂×C₄	#13063310	1	1	yes	characteristic
332	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
333	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
334	C₂×C₄	#13063331	1	1	yes	characteristic
335	C₂×C₄	#13063333	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	1	no	—
339	C₈	—	8	1	no	—
340	C₈	—	344	1	no	—
341	C₈	—	172	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	86	4	no	—
345	C₂²	—	344	2	no	—
346	C₂²	—	344	2	no	—
347	C₂²	—	172	2	no	—
348	C₂²	—	172	2	no	—
349	C₂²	#14582777	1	1	yes	characteristic
350	C₂²	C₈×D₁₇₂	1	1	yes	characteristic
351	C₂²	#14582972	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	344	1	no	—
357	C₄	—	172	2	no	—
358	C₄	—	172	2	no	—
359	C₄	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
360	C₄	#14582864	1	1	yes	characteristic
361	C₄	#14582964	1	1	yes	characteristic
362	C₄	#14582969	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	172	4	no	—
366	C₂	#19134886	1	1	yes	characteristic
367	C₂	#19135067	1	1	yes	characteristic
368	C₂	#19135081	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582777
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1281	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂²×D₁₇₂	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1281	D₄₃
hypocenter	C₄₃	#9087
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

Complements

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

N	H ≅ G/N	type
3. #19132989	C₂	semidirect
5. #19132918	C₂	semidirect
6. #19132426	C₂	semidirect
7. C₄₃×#1281	C₂	semidirect
32. C₂×#10621873	C₈	semidirect
89. C₂×Dic₈₆	C₂×C₈	semidirect
97. C₂×C₈×C₄₃	C₂×C₈	semidirect
184. #1281	D₄₃	semidirect
211. C₄₃	#9087	semidirect
286. C₂×C₈	C₈×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19143618	2
C₂×#14583001	2
#19132989	2
#19132987	2
#19132918	2
#19132426	2
C₄₃×#1281	2
#9087	43