Subgroups — #25067072

Order 11008.

558 conjugacy classes of subgroups.

The subgroup lattice has 364 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	#19143692	C₂	1	1	yes	maximal, characteristic
3	#19143678	C₂	1	1	yes	maximal, characteristic
4	#19133503	C₂	1	2	yes	maximal, complemented
5	#19133491	C₂	1	2	yes	maximal, complemented
6	#19133465	C₂	1	2	yes	maximal, complemented
7	#19133444	C₂	1	2	yes	maximal, complemented
8	#19133041	C₂	1	1	yes	maximal, characteristic
9	#19133015	C₂	1	1	yes	maximal, characteristic
10	#19132890	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1572	C₂	1	1	yes	maximal, characteristic
12	#14583456	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
14	#14583235	C₄	1	2	yes	—
15	#14583195	C₄	1	2	yes	—
16	#14583110	C₄	1	2	yes	—
17	#14583057	C₄	1	2	yes	—
18	#14583029	—	2	2	no	—
19	#14583027	—	2	2	no	—
20	#14583026	C₂²	1	2	yes	—
21	#14583026	C₂²	1	2	yes	—
22	#14583026	C₂²	1	2	yes	—
23	#14583023	C₂²	1	2	yes	—
24	C₄×Dic₁₇₂	C₂²	1	2	yes	—
25	C₄×Dic₁₇₂	C₂²	1	2	yes	—
26	#14582957	—	2	2	no	—
27	#14582953	—	2	2	no	—
28	#14582952	C₂²	1	1	yes	characteristic
29	C₄×#10621877	C₂²	1	1	yes	characteristic
30	#14582629	C₂²	1	2	yes	—
31	#14582629	C₂²	1	2	yes	—
32	#14582629	C₂²	1	2	yes	—
33	#14582629	C₂²	1	2	yes	—
34	#14582621	C₂²	1	1	yes	characteristic
35	#14582621	C₂²	1	1	yes	characteristic
36	#14582620	C₂²	1	1	yes	characteristic
37	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₃×#432	C₂²	1	1	yes	characteristic
39	C₄₃×#431	C₂²	1	1	yes	characteristic
40	C₄₃×#431	C₂²	1	1	yes	characteristic
41	C₄₃×#335	C₂²	1	2	yes	—
42	C₄₃×#334	C₂²	1	2	yes	—
43	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
45	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
46	C₂×#10621877	—	2	1	no	—
47	#13063325	C₂×C₄	1	1	yes	characteristic
48	#13063325	C₂×C₄	1	1	yes	characteristic
49	#13063317	—	2	1	no	—
50	#13063304	—	2	1	no	—
51	#13063304	C₂×C₄	1	2	yes	—
52	C₄×Dic₈₆	—	2	1	no	—
53	C₄×Dic₈₆	—	2	1	no	—
54	C₄×Dic₈₆	—	2	1	no	—
55	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
56	C₄×Dic₈₆	C₂³	1	2	yes	—
57	C₄×Dic₈₆	C₂³	1	2	yes	—
58	#13063284	—	2	1	no	—
59	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
60	C₂×Dic₄₃⋊C₄	—	2	1	no	—
61	C₂×C₄×Dic₄₃	—	2	1	no	—
62	#13063259	C₂×C₄	1	2	yes	—
63	#13063259	C₂×C₄	1	2	yes	—
64	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
65	Dic₄₃⋊C₈	—	2	2	no	—
66	Dic₄₃⋊C₈	—	2	2	no	—
67	Dic₄₃⋊C₈	—	2	2	no	—
68	Dic₄₃⋊C₈	—	2	2	no	—
69	#13063250	C₂×C₄	1	2	yes	—
70	#13063250	C₂×C₄	1	2	yes	—
71	#13063250	C₂×C₄	1	2	yes	—
72	#13063250	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
74	#13063236	—	2	1	no	—
75	#13063235	—	2	2	no	—
76	#13063235	—	2	2	no	—
77	#13063235	—	2	2	no	—
78	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₄₃	—	2	2	no	—
83	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
85	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
86	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
87	C₂²⋊C₈×C₄₃	—	2	1	no	—
88	C₂²⋊C₈×C₄₃	—	2	1	no	—
89	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
90	#97×C₄₃	—	2	1	no	—
91	C₄×C₈×C₄₃	—	2	1	no	—
92	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
97	C₂²×Dic₄₃	—	2	1	no	—
98	C₂²×Dic₄₃	—	2	1	no	—
99	C₂×Dic₈₆	—	2	1	no	—
100	C₂×Dic₈₆	—	2	1	no	—
101	C₂×Dic₈₆	—	4	2	no	—
102	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
103	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₄₃	—	2	2	no	—
105	C₂×C₈×C₄₃	—	2	2	no	—
106	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
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	C₂²⋊Dic₄₃	—	2	2	no	—
113	C₄⋊Dic₄₃	—	2	1	no	—
114	C₄⋊Dic₄₃	—	2	1	no	—
115	C₄⋊Dic₄₃	—	2	1	no	—
116	C₄⋊Dic₄₃	—	2	1	no	—
117	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
118	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
119	Dic₄₃⋊C₄	—	2	2	no	—
120	Dic₄₃⋊C₄	—	2	2	no	—
121	Dic₄₃⋊C₄	—	2	2	no	—
122	Dic₄₃⋊C₄	—	2	2	no	—
123	C₄×Dic₄₃	—	2	2	no	—
124	C₄×Dic₄₃	—	2	2	no	—
125	C₄×Dic₄₃	—	2	2	no	—
126	#10621877	—	4	2	no	—
127	C₂×C₄₃⋊C₈	—	2	1	no	—
128	C₂×C₄₃⋊C₈	—	2	1	no	—
129	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
130	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
131	Dic₁₇₂	—	4	2	no	—
132	Dic₁₇₂	—	4	2	no	—
133	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₄₃	—	2	2	no	—
144	C₂×Dic₄₃	—	2	2	no	—
145	C₂×Dic₄₃	—	2	2	no	—
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	8	1	no	—
149	Dic₈₆	—	2	2	no	—
150	Dic₈₆	—	2	2	no	—
151	Dic₈₆	—	2	2	no	—
152	Dic₈₆	—	2	2	no	—
153	Dic₈₆	—	4	2	no	—
154	Dic₈₆	—	8	1	no	—
155	C₈×C₄₃	—	2	2	no	—
156	C₈×C₄₃	—	2	2	no	—
157	C₈×C₄₃	—	4	2	no	—
158	C₈×C₄₃	—	4	2	no	—
159	C₄₃⋊C₈	—	2	2	no	—
160	C₄₃⋊C₈	—	2	2	no	—
161	C₄₃⋊C₈	—	2	2	no	—
162	C₄₃⋊C₈	—	4	1	no	—
163	#17953	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	—	2	1	no	—
167	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
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	#568	1	2	yes	—
173	C₄×C₄₃	#578	1	2	yes	—
174	Dic₄₃	—	4	2	no	—
175	Dic₄₃	—	4	2	no	—
176	Dic₄₃	—	4	2	no	—
177	Dic₄₃	—	4	2	no	—
178	#3104	—	43	1	no	—
179	#2985	—	43	1	no	—
180	#2983	—	43	1	no	—
181	#1593	—	43	2	no	—
182	#1591	—	43	2	no	—
183	#1587	—	43	2	no	—
184	#1584	—	43	2	no	—
185	#1572	D₄₃	1	1	yes	characteristic
186	#1491	—	43	1	no	—
187	#1485	—	43	1	no	—
188	#1453	—	43	1	no	—
189	C₂×C₄₃	—	4	1	no	—
190	C₂×C₄₃	#2930	1	1	yes	characteristic
191	C₂×C₄₃	#2950	1	1	yes	characteristic
192	C₂×C₄₃	#2956	1	1	yes	characteristic
193	#533	—	43	1	no	—
194	C₂×C₄×Q₈	—	43	1	no	—
195	#484	—	43	2	no	—
196	#477	—	43	2	no	—
197	#462	—	43	2	no	—
198	#451	—	43	2	no	—
199	#446	—	43	2	no	—
200	#446	—	43	2	no	—
201	#446	—	43	2	no	—
202	#446	—	86	2	no	—
203	#446	—	86	2	no	—
204	C₈×Q₈	—	43	2	no	—
205	C₄×Q₁₆	—	43	2	no	—
206	C₄×Q₁₆	—	43	2	no	—
207	#433	—	86	2	no	—
208	#432	—	86	2	no	—
209	#432	C₂×D₄₃	1	1	yes	characteristic
210	#431	—	43	1	no	—
211	#431	C₂×D₄₃	1	1	yes	characteristic
212	#431	C₂×D₄₃	1	1	yes	characteristic
213	C₄×M₁₆	—	43	1	no	—
214	#335	C₂×D₄₃	1	2	yes	—
215	#334	C₂×D₄₃	1	2	yes	—
216	Q₈⋊C₈	—	43	2	no	—
217	Q₈⋊C₈	—	43	2	no	—
218	Q₈⋊C₈	—	43	2	no	—
219	Q₈⋊C₈	—	43	2	no	—
220	#322	—	43	1	no	—
221	#322	—	43	1	no	—
222	#322	—	43	1	no	—
223	C₈²	—	43	1	no	—
224	C₄₃	#17953	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	—	43	1	no	—
226	C₂×Q₁₆	—	43	2	no	—
227	C₂×Q₁₆	—	43	2	no	—
228	C₂×M₁₆	—	86	1	no	—
229	C₄⋊Q₈	—	43	1	no	—
230	C₄⋊Q₈	—	43	1	no	—
231	#125	—	86	1	no	—
232	C₂²⋊Q₈	—	43	2	no	—
233	C₂²⋊Q₈	—	86	1	no	—
234	C₄×Q₈	—	43	2	no	—
235	C₄×Q₈	—	43	2	no	—
236	C₄×Q₈	—	43	2	no	—
237	C₄×Q₈	—	86	1	no	—
238	C₄×Q₈	—	86	1	no	—
239	C₄×Q₈	—	86	1	no	—
240	#117	—	86	1	no	—
241	C₂×C₄⋊C₄	—	43	1	no	—
242	C₂×C₄⋊C₄	—	86	1	no	—
243	C₂×C₄²	—	86	1	no	—
244	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
245	#107	—	43	2	no	—
246	#107	—	43	2	no	—
247	C₄⋊C₈	—	43	2	no	—
248	C₄⋊C₈	—	86	2	no	—
249	C₄⋊C₈	—	86	2	no	—
250	C₄⋊C₈	—	86	2	no	—
251	C₄⋊C₈	—	86	2	no	—
252	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
253	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
254	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
255	#103	—	43	2	no	—
256	#103	—	43	2	no	—
257	#103	—	43	2	no	—
258	#103	—	43	2	no	—
259	C₂²⋊C₈	—	2	1	no	—
260	C₂²⋊C₈	—	2	1	no	—
261	C₂²⋊C₈	—	43	2	no	—
262	C₂²⋊C₈	C₄×D₄₃	1	2	yes	—
263	#97	—	2	1	no	—
264	#97	—	86	2	no	—
265	#97	—	86	2	no	—
266	#97	—	86	2	no	—
267	#97	—	86	1	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	43	1	no	—
270	C₄×C₈	—	43	1	no	—
271	C₄×C₈	—	43	1	no	—
272	C₄×C₈	—	43	1	no	—
273	C₄×C₈	—	86	2	no	—
274	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
276	C₂×Q₈	—	43	2	no	—
277	C₂×Q₈	—	43	2	no	—
278	C₂×Q₈	—	86	1	no	—
279	C₂×Q₈	—	86	1	no	—
280	C₂×Q₈	—	172	2	no	—
281	C₂²×C₄	—	86	1	no	—
282	C₂²×C₄	—	86	1	no	—
283	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
285	C₂²×C₄	#10621902	1	1	yes	characteristic
286	Q₁₆	—	172	2	no	—
287	Q₁₆	—	172	2	no	—
288	M₁₆	—	172	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	43	2	no	—
292	C₂×C₈	—	43	2	no	—
293	C₂×C₈	—	86	1	no	—
294	C₂×C₈	—	86	1	no	—
295	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
296	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
297	C₄⋊C₄	—	43	2	no	—
298	C₄⋊C₄	—	43	2	no	—
299	C₄⋊C₄	—	86	1	no	—
300	C₄⋊C₄	—	86	2	no	—
301	C₄⋊C₄	—	86	1	no	—
302	C₄⋊C₄	—	86	1	no	—
303	C₄⋊C₄	—	86	2	no	—
304	C₄⋊C₄	—	86	1	no	—
305	C₄⋊C₄	—	86	2	no	—
306	C₄⋊C₄	—	86	2	no	—
307	C₂²⋊C₄	—	86	2	no	—
308	C₄²	—	86	2	no	—
309	C₄²	—	86	2	no	—
310	C₄²	—	86	2	no	—
311	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
312	C₄²	D₄×D₄₃	1	1	yes	characteristic
313	C₄²	#10621902	1	1	yes	characteristic
314	C₄²	C₂³×D₄₃	1	1	yes	characteristic
315	C₂³	#13063287	1	1	yes	characteristic
316	Q₈	—	86	2	no	—
317	Q₈	—	86	2	no	—
318	Q₈	—	86	2	no	—
319	Q₈	—	86	2	no	—
320	Q₈	—	344	1	no	—
321	Q₈	—	172	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	86	2	no	—
326	C₂×C₄	—	86	2	no	—
327	C₂×C₄	—	86	2	no	—
328	C₂×C₄	—	86	2	no	—
329	C₂×C₄	—	344	1	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	#13063287	1	1	yes	characteristic
332	C₂×C₄	#13063287	1	1	yes	characteristic
333	C₂×C₄	#13063287	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
336	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
337	C₈	—	2	2	no	—
338	C₈	—	2	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	86	2	no	—
342	C₈	—	86	2	no	—
343	C₈	—	86	2	no	—
344	C₈	—	172	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	C₂×#13063287	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	172	2	no	—
354	C₄	—	172	2	no	—
355	C₄	—	172	2	no	—
356	C₄	—	172	2	no	—
357	C₄	#14583664	1	2	yes	—
358	C₄	#14583713	1	2	yes	—
359	C₂	—	4	1	no	—
360	C₂	#19143297	1	1	yes	characteristic
361	C₂	#19143462	1	1	yes	characteristic
362	C₂	#19143504	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×#1572	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1572	D₄₃
hypocenter	C₄₃	#17953
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1572

Complements

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

N	H ≅ G/N	type
4. #19133503	C₂	semidirect
5. #19133491	C₂	semidirect
6. #19133465	C₂	semidirect
7. #19133444	C₂	semidirect
224. C₄₃	#17953	semidirect

Maximal subgroups

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

subgroup	index
#19144635	2
#19143692	2
#19143678	2
#19133503	2
#19133491	2
#19133465	2
#19133444	2
#19133041	2
#19133015	2
#19132890	2
C₄₃×#1572	2
#17953	43