Subgroups — #25013483

Order 11008.

474 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147697	C₂	1	1	yes	maximal, characteristic, complemented
2	#19143305	C₂	1	1	yes	maximal, characteristic
3	#19136223	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136215	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132676	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132676	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#2117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063331	C₂²	1	1	yes	characteristic
9	#14583445	C₄	1	1	yes	characteristic
10	#14583317	—	2	2	no	—
11	#14583315	—	2	2	no	—
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	M₃₂×D₄₃	—	2	2	no	—
15	#14582975	—	2	2	no	—
16	C₈×C₂²⋊D₄₃	—	2	2	no	—
17	#14582922	C₄	1	1	yes	characteristic
18	#14582867	C₄	1	1	yes	characteristic
19	#14582866	C₄	1	1	yes	characteristic
20	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
21	#14582683	—	2	2	no	—
22	#14582680	—	2	2	no	—
23	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#350	—	2	2	no	—
27	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
28	#13063331	—	2	1	no	—
29	#13063331	—	2	1	no	—
30	#13063331	—	2	1	no	—
31	#13063331	—	2	1	no	—
32	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
33	C₂×#10621872	D₄	1	2	yes	—
34	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₄₃	D₄	1	2	yes	—
36	#13063323	C₈	1	1	yes	characteristic
37	#13063321	C₈	1	1	yes	characteristic
38	#13063314	C₈	1	1	yes	characteristic
39	#13063312	C₈	1	1	yes	characteristic
40	#13063297	—	2	2	no	—
41	C₄×C₂²⋊D₄₃	—	2	2	no	—
42	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
43	#13063265	—	2	1	no	—
44	#13063265	—	2	1	no	—
45	#13063265	—	2	1	no	—
46	#13063265	D₄	1	2	yes	complemented
47	#13063264	—	2	4	no	—
48	C₂×C₄₃⋊C₁₆	—	2	1	no	—
49	C₂×C₄₃⋊C₁₆	—	2	1	no	—
50	C₁₆×D₄₃	—	2	4	no	—
51	Dic₄₃⋊C₈	—	2	2	no	—
52	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
53	(C₂×D₄₃)⋊C₈	—	2	2	no	—
54	C₈×Dic₄₃	—	2	1	no	—
55	C₈×Dic₄₃	—	2	1	no	—
56	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
57	#131×C₄₃	—	2	1	no	—
58	#131×C₄₃	—	2	1	no	—
59	#131×C₄₃	—	2	1	no	—
60	#131×C₄₃	—	2	1	no	—
61	C₂×M₁₆×C₄₃	—	2	1	no	—
62	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₃	—	2	1	no	—
64	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	M₃₂×C₄₃	—	2	1	no	—
66	M₃₂×C₄₃	—	2	1	no	—
67	M₃₂×C₄₃	—	2	1	no	—
68	M₃₂×C₄₃	D₄	1	2	yes	complemented
69	C₂×C₁₆×C₄₃	—	2	1	no	—
70	C₂×C₁₆×C₄₃	—	2	1	no	—
71	#41×C₄₃	—	2	1	no	—
72	#41×C₄₃	—	2	1	no	—
73	#41×C₄₃	—	2	1	no	—
74	#41×C₄₃	—	2	1	no	—
75	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
76	C₂×D₄×C₄₃	—	2	1	no	—
77	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	—	2	1	no	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₄₃	—	2	2	no	—
81	C₂×C₂²⋊D₄₃	M₁₆	1	2	yes	—
82	C₂²×Dic₄₃	—	2	1	no	—
83	#10621898	—	2	1	no	—
84	#10621898	—	2	1	no	—
85	#10621898	—	2	1	no	—
86	#10621898	—	2	1	no	—
87	C₂×D₁₇₂	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
89	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
90	M₁₆×C₄₃	—	2	1	no	—
91	M₁₆×C₄₃	—	2	1	no	—
92	M₁₆×C₄₃	—	4	1	no	—
93	M₁₆×C₄₃	—	4	1	no	—
94	C₂×C₈×C₄₃	—	2	1	no	—
95	C₂×C₈×C₄₃	—	4	1	no	—
96	C₂×C₈×C₄₃	—	4	1	no	—
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
103	C₂²⋊Dic₄₃	—	2	2	no	—
104	(C₂×D₄₃)⋊C₄	—	2	2	no	—
105	Dic₄₃⋊C₄	—	2	2	no	—
106	C₄×Dic₄₃	—	2	1	no	—
107	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
108	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
109	#10621877	—	2	1	no	—
110	#10621877	—	2	1	no	—
111	C₂×C₄₃⋊C₈	—	2	1	no	—
112	C₂×C₄₃⋊C₈	—	2	1	no	—
113	C₂×C₄₃⋊C₈	—	2	1	no	—
114	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
115	#10621872	—	2	2	no	—
116	#10621872	—	2	2	no	—
117	C₈×D₄₃	—	2	2	no	—
118	C₈×D₄₃	—	2	2	no	—
119	C₁₆×C₄₃	—	2	2	no	—
120	C₁₆×C₄₃	—	4	1	no	—
121	C₄₃⋊C₁₆	—	2	2	no	—
122	C₄₃⋊C₁₆	—	4	1	no	—
123	C₂³×C₄₃	—	2	1	no	—
124	C₂³×C₄₃	C₂×M₁₆	1	1	yes	characteristic
125	C₂²×D₄₃	C₂²⋊C₈	1	2	yes	—
126	Q₈×C₄₃	—	2	1	no	—
127	Q₈×C₄₃	—	2	1	no	—
128	D₄×C₄₃	—	2	1	no	—
129	D₄×C₄₃	—	2	1	no	—
130	D₄×C₄₃	—	4	1	no	—
131	D₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	2	1	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
141	C₂²⋊D₄₃	—	2	2	no	—
142	C₂²⋊D₄₃	—	2	2	no	—
143	C₂²⋊D₄₃	—	4	4	no	—
144	C₂×Dic₄₃	—	2	1	no	—
145	C₂×Dic₄₃	—	2	1	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	C₂²⋊C₈	1	2	yes	—
148	D₁₇₂	—	2	1	no	—
149	D₁₇₂	—	2	1	no	—
150	C₄×D₄₃	—	2	2	no	—
151	C₄×D₄₃	—	2	2	no	—
152	Dic₈₆	—	2	1	no	—
153	Dic₈₆	—	2	1	no	—
154	C₈×C₄₃	—	4	1	no	—
155	C₈×C₄₃	#120	1	1	yes	characteristic
156	C₈×C₄₃	#121	1	1	yes	characteristic
157	C₈×C₄₃	#121	1	1	yes	characteristic
158	C₈×C₄₃	#123	1	1	yes	characteristic
159	C₄₃⋊C₈	—	2	2	no	—
160	C₄₃⋊C₈	—	4	1	no	—
161	#12982	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	—	4	1	no	—
163	C₂²×C₄₃	—	4	2	no	—
164	C₂²×C₄₃	—	4	1	no	—
165	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
166	C₂²×C₄₃	#435	1	1	yes	characteristic
167	C₂²×C₄₃	#435	1	1	yes	characteristic
168	C₂×D₄₃	—	2	2	no	—
169	C₂×D₄₃	—	4	2	no	—
170	C₄×C₄₃	—	4	1	no	—
171	C₄×C₄₃	#386	1	1	yes	characteristic
172	C₄×C₄₃	#408	1	1	yes	characteristic
173	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
174	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
175	Dic₄₃	—	2	2	no	—
176	Dic₄₃	—	2	2	no	—
177	Dic₄₃	—	4	1	no	—
178	#3407	—	43	1	no	—
179	#2931	—	43	1	no	—
180	#2117	—	43	1	no	—
181	#2117	D₄₃	1	1	yes	characteristic, complemented
182	#2115	—	43	1	no	—
183	#1376	—	43	1	no	—
184	#1376	—	43	1	no	—
185	C₂×C₄₃	—	2	1	no	—
186	C₂×C₄₃	—	2	1	no	—
187	C₂×C₄₃	—	2	1	no	—
188	C₂×C₄₃	—	8	1	no	—
189	C₂×C₄₃	#1869	1	1	yes	characteristic
190	D₄₃	—	4	2	no	—
191	C₂×#131	—	43	1	no	—
192	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
193	#532	—	43	1	no	—
194	#504	—	86	2	no	—
195	#504	—	86	2	no	—
196	C₂×M₃₂	—	43	1	no	—
197	C₂×M₃₂	—	43	1	no	—
198	C₂×M₃₂	—	86	2	no	—
199	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
200	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
201	#436	—	86	2	no	—
202	C₈×D₄	—	86	2	no	—
203	#423	—	43	1	no	—
204	#408	—	43	1	no	—
205	#408	—	43	1	no	—
206	C₂×C₄×C₈	—	43	1	no	—
207	#350	—	2	2	no	—
208	#350	—	86	2	no	—
209	#349	—	86	2	no	—
210	C₄₃	#12982	1	1	yes	characteristic, complemented, Sylow
211	C₂×#41	—	43	1	no	—
212	C₂×#41	C₄×D₄₃	1	1	yes	characteristic
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	#131	—	2	1	no	—
216	#131	—	2	1	no	—
217	#131	—	86	1	no	—
218	#131	—	86	1	no	—
219	#131	—	86	1	no	—
220	#131	—	86	1	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	—	43	1	no	—
223	C₂×M₁₆	—	43	2	no	—
224	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	43	2	no	—
227	C₂²×C₈	—	43	1	no	—
228	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
229	C₄⋊Q₈	—	43	1	no	—
230	#127	—	43	1	no	—
231	#124	—	43	1	no	—
232	#124	—	43	1	no	—
233	#121	—	86	2	no	—
234	C₄×D₄	—	86	2	no	—
235	C₂×C₄²	—	43	1	no	—
236	M₃₂	—	2	1	no	—
237	M₃₂	—	2	1	no	—
238	M₃₂	—	2	1	no	—
239	M₃₂	—	43	2	no	—
240	M₃₂	—	86	4	no	—
241	M₃₂	—	86	1	no	—
242	M₃₂	—	86	1	no	—
243	M₃₂	—	86	1	no	—
244	M₃₂	C₂²⋊D₄₃	1	2	yes	complemented
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	—	86	4	no	—
248	C₂×C₁₆	—	86	1	no	—
249	C₂×C₁₆	—	86	1	no	—
250	C₄⋊C₈	—	86	2	no	—
251	C₂²⋊C₈	—	86	2	no	—
252	C₂²⋊C₈	—	86	2	no	—
253	C₄×C₈	—	86	1	no	—
254	C₄×C₈	—	86	1	no	—
255	#41	—	2	1	no	—
256	#41	—	2	1	no	—
257	#41	—	2	1	no	—
258	#41	—	2	1	no	—
259	#41	—	86	1	no	—
260	#41	—	86	1	no	—
261	#41	—	86	1	no	—
262	#41	—	86	1	no	—
263	C₂×Q₈	—	43	1	no	—
264	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	43	2	no	—
267	C₂×D₄	—	43	1	no	—
268	C₂×D₄	—	86	2	no	—
269	C₂×D₄	C₈×D₄₃	1	1	yes	characteristic
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	43	2	no	—
272	C₂²×C₄	—	86	1	no	—
273	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
274	M₁₆	—	2	1	no	—
275	M₁₆	—	2	1	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	86	1	no	—
279	M₁₆	—	86	1	no	—
280	M₁₆	—	86	2	no	—
281	M₁₆	—	86	2	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	43	2	no	—
286	C₂×C₈	—	86	2	no	—
287	C₂×C₈	—	86	2	no	—
288	C₂×C₈	—	86	1	no	—
289	C₂×C₈	—	86	1	no	—
290	C₂×C₈	—	86	1	no	—
291	C₂×C₈	#10621898	1	1	yes	characteristic
292	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
293	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
294	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
295	C₂×C₈	#10621900	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
297	C₄⋊C₄	—	86	2	no	—
298	C₂²⋊C₄	—	86	2	no	—
299	C₂²⋊C₄	—	86	2	no	—
300	C₄²	—	43	1	no	—
301	C₄²	—	43	1	no	—
302	C₄²	—	86	1	no	—
303	C₁₆	—	2	2	no	—
304	C₁₆	—	4	1	no	—
305	C₁₆	—	86	2	no	—
306	C₁₆	—	172	1	no	—
307	C₂³	—	2	1	no	—
308	C₂³	—	43	2	no	—
309	C₂³	M₁₆×D₄₃	1	1	yes	characteristic
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	86	1	no	—
313	Q₈	—	86	1	no	—
314	D₄	—	2	1	no	—
315	D₄	—	2	1	no	—
316	D₄	—	4	1	no	—
317	D₄	—	4	1	no	—
318	D₄	—	86	2	no	—
319	D₄	—	86	2	no	—
320	D₄	—	86	1	no	—
321	D₄	—	86	1	no	—
322	D₄	—	172	4	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	43	2	no	—
327	C₂×C₄	—	86	1	no	—
328	C₂×C₄	—	86	2	no	—
329	C₂×C₄	—	86	2	no	—
330	C₂×C₄	—	86	1	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
333	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
334	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
335	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
336	C₂×C₄	#13063331	1	1	yes	characteristic
337	C₂×C₄	#13063333	1	1	yes	characteristic
338	C₈	—	4	1	no	—
339	C₈	—	86	2	no	—
340	C₈	—	172	1	no	—
341	C₈	#13063293	1	1	yes	characteristic
342	C₈	#13063296	1	1	yes	characteristic
343	C₈	#13063298	1	1	yes	characteristic
344	C₈	#13063307	1	1	yes	characteristic
345	C₂²	—	4	1	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	86	2	no	—
349	C₂²	—	172	2	no	—
350	C₂²	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
351	C₂²	#14582968	1	1	yes	characteristic
352	C₂²	#14582973	1	1	yes	characteristic
353	C₄	—	4	1	no	—
354	C₄	—	86	2	no	—
355	C₄	—	86	2	no	—
356	C₄	—	172	1	no	—
357	C₄	#14582775	1	1	yes	characteristic
358	C₄	#14582864	1	1	yes	characteristic
359	C₄	(C₂²⋊D₄₃)⋊C₈	1	1	yes	characteristic
360	C₄	C₈×C₂²⋊D₄₃	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	8	1	no	—
365	C₂	—	172	2	no	—
366	C₂	#19134884	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13063293
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#2117	C₂
socle	C₂×C₄₃	#1869
prime radical	#14583445	C₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#2117	D₄₃
hypocenter	C₄₃	#12982
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2117

Complements

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

N	H ≅ G/N	type
1. #19147697	C₂	semidirect
3. #19136223	C₂	semidirect
4. #19136215	C₂	semidirect
5. #19132676	C₂	semidirect
6. #19132676	C₂	semidirect
7. C₄₃×#2117	C₂	semidirect
46. #13063265	D₄	semidirect
68. M₃₂×C₄₃	D₄	semidirect
181. #2117	D₄₃	semidirect
210. C₄₃	#12982	semidirect
244. M₃₂	C₂²⋊D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19147697	2
#19143305	2
#19136223	2
#19136215	2
#19132676	2
C₄₃×#2117	2
#12982	43