Subgroups — #25079699

Order 11008.

662 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145725	C₂	1	1	yes	maximal, characteristic
2	#19145522	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583297	C₂	1	1	yes	maximal, characteristic, complemented
4	#19143308	C₂	1	1	yes	maximal, characteristic
5	#19134095	C₂	1	4	yes	maximal, complemented
6	#19133971	C₂	1	4	yes	maximal, complemented
7	#19133279	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133275	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063331	—	2	1	no	—
11	#14583449	C₂²	1	1	yes	characteristic
12	C₂×#13063325	C₂²	1	1	yes	characteristic
13	C₂×#13063325	C₂²	1	1	yes	characteristic
14	#14583297	—	2	1	no	—
15	#14583297	—	2	1	no	—
16	#14583297	—	2	1	no	—
17	#14583297	C₂²	1	2	yes	—
18	#14583297	C₂²	1	2	yes	—
19	#14583272	—	2	1	no	—
20	#14583250	—	2	1	no	—
21	#14583246	—	2	1	no	—
22	#14583218	—	2	2	no	—
23	#14583190	C₂²	1	4	yes	—
24	#14583105	C₂²	1	4	yes	—
25	#14583040	—	2	2	no	—
26	#14582978	—	2	1	no	—
27	#14582978	C₂²	1	2	yes	—
28	C₈×D₁₇₂	—	2	1	no	—
29	C₈×D₁₇₂	C₂²	1	2	yes	—
30	C₂×#13063257	—	2	1	no	—
31	C₂×#13063257	—	2	1	no	—
32	#14582923	C₂²	1	1	yes	characteristic
33	#14582865	—	2	1	no	—
34	#14582647	C₂²	1	4	yes	—
35	#14582646	C₂²	1	4	yes	—
36	C₄₃×#431	C₂²	1	1	yes	characteristic
37	C₄₃×#431	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#335	C₂²	1	2	yes	—
40	C₄₃×#334	C₂²	1	2	yes	—
41	C₂×#10621898	—	2	1	no	—
42	C₂²×Dic₈₆	—	2	1	no	—
43	C₂²×Dic₈₆	—	2	1	no	—
44	C₂×#10621873	—	2	4	no	—
45	#13063331	—	4	4	no	—
46	C₂×#10621877	—	2	1	no	—
47	C₂×#10621872	—	2	2	no	—
48	C₂×C₈×D₄₃	—	2	2	no	—
49	#13063325	—	2	1	no	—
50	#13063325	—	2	1	no	—
51	#13063325	—	2	1	no	—
52	#13063325	—	2	1	no	—
53	#13063325	—	2	1	no	—
54	#13063325	—	2	1	no	—
55	#13063325	D₄	1	1	yes	characteristic
56	#13063325	C₂³	1	2	yes	—
57	#13063325	C₂³	1	2	yes	—
58	#13063322	D₄	1	1	yes	characteristic
59	#13063313	—	2	1	no	—
60	#13063299	—	2	1	no	—
61	#13063299	D₄	1	2	yes	—
62	C₄×D₁₇₂	—	2	1	no	—
63	C₄×D₁₇₂	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₄₃	—	2	1	no	—
65	C₂×C₄⋊Dic₄₃	—	2	1	no	—
66	C₂×C₄⋊Dic₄₃	—	2	1	no	—
67	C₂×C₄⋊Dic₄₃	—	2	1	no	—
68	#13063257	—	2	2	no	—
69	#13063257	—	2	2	no	—
70	#13063257	—	2	2	no	—
71	#13063257	—	2	2	no	—
72	#13063257	—	2	4	no	—
73	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
74	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
75	#13063250	—	2	4	no	—
76	#13063247	—	2	4	no	—
77	(C₂×D₄₃)⋊C₈	—	2	2	no	—
78	(C₂×D₄₃)⋊C₈	—	2	2	no	—
79	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
83	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
84	C₂²⋊C₈×C₄₃	—	2	1	no	—
85	C₂²⋊C₈×C₄₃	—	2	1	no	—
86	C₄×C₈×C₄₃	—	2	1	no	—
87	C₄×C₈×C₄₃	—	2	1	no	—
88	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
90	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₄₃	—	2	2	no	—
96	C₂²×Dic₄₃	—	4	1	no	—
97	C₂²×Dic₄₃	—	4	1	no	—
98	#10621898	—	4	2	no	—
99	#10621898	—	4	2	no	—
100	C₂×D₁₇₂	—	2	1	no	—
101	C₂×D₁₇₂	—	2	1	no	—
102	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
103	C₂×C₄×D₄₃	—	2	2	no	—
104	C₂×Dic₈₆	—	2	1	no	—
105	C₂×Dic₈₆	—	2	2	no	—
106	C₂×Dic₈₆	—	2	2	no	—
107	C₂×Dic₈₆	—	4	1	no	—
108	C₂×Dic₈₆	—	4	1	no	—
109	C₂×Dic₈₆	—	8	2	no	—
110	C₂×Dic₈₆	—	8	2	no	—
111	C₂×C₈×C₄₃	—	2	2	no	—
112	C₂×C₈×C₄₃	—	2	2	no	—
113	C₂×C₈×C₄₃	—	2	2	no	—
114	C₂×C₈×C₄₃	SD₁₆	1	4	yes	complemented
115	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₄₃	#41	1	1	yes	characteristic
122	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
123	(C₂×D₄₃)⋊C₄	—	2	2	no	—
124	(C₂×D₄₃)⋊C₄	—	2	2	no	—
125	C₄⋊Dic₄₃	—	2	1	no	—
126	C₄⋊Dic₄₃	—	2	2	no	—
127	C₄⋊Dic₄₃	—	2	2	no	—
128	C₄⋊Dic₄₃	—	2	2	no	—
129	C₄⋊Dic₄₃	—	2	2	no	—
130	C₄⋊Dic₄₃	—	4	1	no	—
131	C₄⋊Dic₄₃	—	4	1	no	—
132	C₄⋊Dic₄₃	—	4	1	no	—
133	C₄⋊Dic₄₃	—	4	1	no	—
134	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
135	#10621877	—	4	2	no	—
136	C₂×C₄₃⋊C₈	—	2	2	no	—
137	#10621873	—	8	4	no	—
138	#10621872	—	4	4	no	—
139	C₈×D₄₃	—	4	4	no	—
140	C₂³×C₄₃	#121	1	1	yes	characteristic
141	C₂²×D₄₃	—	2	2	no	—
142	C₂×C₄×C₄₃	—	2	2	no	—
143	C₂×C₄×C₄₃	—	2	2	no	—
144	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
148	C₂×C₄×C₄₃	#137	1	2	yes	—
149	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₄₃	—	4	4	no	—
153	C₂×Dic₄₃	—	2	2	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	C₂×Dic₄₃	—	8	2	no	—
157	C₂×Dic₄₃	—	8	2	no	—
158	D₁₇₂	—	2	4	no	—
159	D₁₇₂	—	4	2	no	—
160	D₁₇₂	—	4	2	no	—
161	C₄×D₄₃	—	4	2	no	—
162	C₄×D₄₃	—	4	2	no	—
163	Dic₈₆	—	4	2	no	—
164	Dic₈₆	—	4	4	no	—
165	Dic₈₆	—	8	2	no	—
166	Dic₈₆	—	8	2	no	—
167	Dic₈₆	—	8	2	no	—
168	C₈×C₄₃	—	2	1	no	—
169	C₈×C₄₃	—	2	1	no	—
170	C₈×C₄₃	—	4	2	no	—
171	C₈×C₄₃	—	4	2	no	—
172	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
173	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
174	C₄₃⋊C₈	—	4	2	no	—
175	#19191	—	43	1	no	maximal, Sylow
176	C₂²×C₄₃	—	2	1	no	—
177	C₂²×C₄₃	—	2	1	no	—
178	C₂²×C₄₃	#461	1	2	yes	—
179	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
180	C₂×D₄₃	—	4	2	no	—
181	C₂×D₄₃	—	4	2	no	—
182	C₂×D₄₃	—	4	2	no	—
183	C₄×C₄₃	—	2	1	no	—
184	C₄×C₄₃	—	2	1	no	—
185	C₄×C₄₃	#461	1	2	yes	—
186	C₄×C₄₃	#471	1	2	yes	—
187	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
188	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
189	C₄×C₄₃	#577	1	2	yes	—
190	Dic₄₃	—	4	2	no	—
191	Dic₄₃	—	8	2	no	—
192	Dic₄₃	—	8	2	no	—
193	#3227	—	43	1	no	—
194	#3206	—	43	1	no	—
195	C₂×#498	—	43	1	no	—
196	#2931	—	43	1	no	—
197	#1691	—	43	4	no	—
198	#1671	—	43	4	no	—
199	#1564	D₄₃	1	1	yes	characteristic, complemented
200	#1549	—	43	1	no	—
201	#1548	—	43	1	no	—
202	C₂×C₄₃	—	2	2	no	—
203	C₂×C₄₃	C₂×#461	1	1	yes	characteristic
204	C₂×C₄₃	#3057	1	1	yes	characteristic
205	C₂×C₄₃	C₂×#471	1	1	yes	characteristic
206	D₄₃	—	8	2	no	—
207	C₂×#131	—	86	1	no	—
208	#532	—	43	1	no	—
209	C₂×C₄⋊Q₈	—	43	1	no	—
210	C₂×C₄⋊Q₈	—	43	1	no	—
211	#498	—	43	2	no	—
212	#498	—	43	2	no	—
213	#498	—	86	1	no	—
214	#498	—	86	1	no	—
215	#498	—	86	1	no	—
216	#492	—	86	1	no	—
217	#487	—	86	1	no	—
218	#486	—	86	1	no	—
219	#481	—	86	2	no	—
220	#476	—	43	4	no	—
221	#461	—	43	4	no	—
222	#448	—	86	2	no	—
223	#436	—	43	2	no	—
224	#436	—	86	1	no	—
225	C₈×D₄	—	43	2	no	—
226	C₈×D₄	—	86	1	no	—
227	#431	C₂×D₄₃	1	1	yes	characteristic
228	#431	C₂×D₄₃	1	1	yes	characteristic
229	C₂×#106	—	86	1	no	—
230	C₂×#106	—	86	1	no	—
231	#423	—	43	1	no	—
232	#408	—	86	1	no	—
233	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
234	#335	C₂×D₄₃	1	2	yes	—
235	#334	C₂×D₄₃	1	2	yes	—
236	#333	—	43	4	no	—
237	#332	—	43	4	no	—
238	C₄₃	#19191	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	86	1	no	—
240	C₂²×Q₈	—	86	1	no	—
241	C₂²×Q₈	—	86	1	no	—
242	C₂×SD₁₆	—	86	4	no	—
243	#131	—	172	4	no	—
244	C₂×M₁₆	—	86	1	no	—
245	C₂×M₁₆	—	86	2	no	—
246	C₂²×C₈	—	86	2	no	—
247	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
248	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
249	C₄⋊Q₈	—	43	1	no	—
250	C₄⋊Q₈	—	43	2	no	—
251	C₄⋊Q₈	—	43	2	no	—
252	C₄⋊Q₈	—	86	1	no	—
253	C₄⋊Q₈	—	86	1	no	—
254	C₄⋊Q₈	—	86	1	no	—
255	C₄⋊Q₈	—	86	1	no	—
256	C₄⋊Q₈	—	86	1	no	—
257	C₄⋊Q₈	—	86	1	no	—
258	#127	—	43	1	no	—
259	#124	—	86	1	no	—
260	#121	—	43	2	no	—
261	#121	—	86	1	no	—
262	C₄×D₄	—	43	2	no	—
263	C₄×D₄	—	86	1	no	—
264	C₂×C₄⋊C₄	—	86	1	no	—
265	C₂×C₄⋊C₄	—	86	1	no	—
266	C₂×C₄⋊C₄	—	86	1	no	—
267	C₂×C₄⋊C₄	—	86	1	no	—
268	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
269	#106	—	86	4	no	—
270	#106	—	86	2	no	—
271	#106	—	86	2	no	—
272	#106	—	86	2	no	—
273	#106	—	86	2	no	—
274	C₄⋊C₈	—	43	2	no	—
275	C₄⋊C₈	—	86	1	no	—
276	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
277	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
278	#103	—	86	4	no	—
279	#102	—	86	4	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	86	2	no	—
283	C₂²⋊C₈	—	86	2	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	D₁₇₂	1	1	yes	characteristic
287	C₄×C₈	D₁₇₂	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
290	#41	—	172	2	no	—
291	#41	—	172	2	no	—
292	C₂×Q₈	—	86	1	no	—
293	C₂×Q₈	—	86	2	no	—
294	C₂×Q₈	—	86	2	no	—
295	C₂×Q₈	—	344	2	no	—
296	C₂×Q₈	—	344	2	no	—
297	C₂×Q₈	—	172	1	no	—
298	C₂×Q₈	—	172	1	no	—
299	C₂×D₄	—	43	2	no	—
300	C₂×D₄	—	86	2	no	—
301	C₂×D₄	—	86	1	no	—
302	C₂×D₄	—	86	1	no	—
303	C₂²×C₄	—	86	2	no	—
304	C₂²×C₄	—	172	1	no	—
305	C₂²×C₄	—	172	1	no	—
306	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
307	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
308	C₂²×C₄	#10621902	1	1	yes	characteristic
309	SD₁₆	—	344	4	no	—
310	M₁₆	—	172	2	no	—
311	M₁₆	—	172	4	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	86	2	no	—
316	C₂×C₈	—	172	4	no	—
317	C₂×C₈	#10621873	1	4	yes	complemented
318	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
319	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
320	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
321	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
322	C₄⋊C₄	—	43	2	no	—
323	C₄⋊C₄	—	86	1	no	—
324	C₄⋊C₄	—	86	2	no	—
325	C₄⋊C₄	—	86	2	no	—
326	C₄⋊C₄	—	86	2	no	—
327	C₄⋊C₄	—	86	2	no	—
328	C₄⋊C₄	—	172	1	no	—
329	C₄⋊C₄	—	172	1	no	—
330	C₄⋊C₄	—	172	1	no	—
331	C₄⋊C₄	—	172	1	no	—
332	C₂²⋊C₄	—	86	2	no	—
333	C₂²⋊C₄	—	86	2	no	—
334	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
335	C₄²	D₄×D₄₃	1	1	yes	characteristic
336	C₄²	#10621902	1	1	yes	characteristic
337	C₄²	C₂³×D₄₃	1	1	yes	characteristic
338	C₂³	—	86	2	no	—
339	C₂³	#13063300	1	1	yes	characteristic
340	Q₈	—	344	2	no	—
341	Q₈	—	344	2	no	—
342	Q₈	—	344	2	no	—
343	Q₈	—	172	4	no	—
344	Q₈	—	172	2	no	—
345	D₄	—	86	4	no	—
346	D₄	—	172	4	no	—
347	D₄	—	172	2	no	—
348	D₄	—	172	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	86	2	no	—
352	C₂×C₄	—	344	2	no	—
353	C₂×C₄	—	344	2	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	—	172	2	no	—
356	C₂×C₄	—	172	2	no	—
357	C₂×C₄	—	172	2	no	—
358	C₂×C₄	#13063300	1	1	yes	characteristic
359	C₂×C₄	#13063300	1	1	yes	characteristic
360	C₂×C₄	#13063300	1	1	yes	characteristic
361	C₂×C₄	C₂×#10621873	1	2	yes	—
362	C₂×C₄	#13063358	1	2	yes	—
363	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
364	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
365	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	172	2	no	—
371	C₈	C₂×#10621873	1	2	yes	—
372	C₈	C₂×#10621873	1	2	yes	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	172	2	no	—
376	C₂²	—	172	2	no	—
377	C₂²	—	172	2	no	—
378	C₂²	#14583105	1	2	yes	—
379	C₂²	C₂×#13063300	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	344	2	no	—
383	C₄	—	344	2	no	—
384	C₄	—	172	2	no	—
385	C₄	#14583105	1	2	yes	—
386	C₄	#14583163	1	2	yes	—
387	C₄	C₂²×#10621873	1	1	yes	characteristic
388	C₄	C₂×#13063358	1	1	yes	characteristic
389	C₄	#14583706	1	2	yes	—
390	C₂	—	2	2	no	—
391	C₂	—	344	2	no	—
392	C₂	C₂×#14583105	1	1	yes	characteristic
393	C₂	#19144235	1	1	yes	characteristic
394	C₂	C₂×#14583163	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063300
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1564	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	#14583449	C₂²
hypercenter	#1564	D₄₃
hypocenter	C₄₃	#19191
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. C₂×#14583297	C₂	semidirect
5. #19134095	C₂	semidirect
6. #19133971	C₂	semidirect
7. #19133279	C₂	semidirect
8. #19133275	C₂	semidirect
9. C₄₃×#1564	C₂	semidirect
114. C₂×C₈×C₄₃	SD₁₆	semidirect
199. #1564	D₄₃	semidirect
238. C₄₃	#19191	semidirect
317. C₂×C₈	#10621873	semidirect

Maximal subgroups

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

subgroup	index
#19145725	2
#19145522	2
C₂×#14583297	2
#19143308	2
#19134095	2
#19133971	2
#19133279	2
#19133275	2
C₄₃×#1564	2
#19191	43