Subgroups — #25013524

Order 11008.

470 conjugacy classes of subgroups.

The subgroup lattice has 366 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	#19143300	C₂	1	1	yes	maximal, characteristic
3	#19136215	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582680	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582680	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132681	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063331	C₂²	1	1	yes	characteristic
9	C₂²×#10621872	C₂²	1	1	yes	characteristic
10	#14583411	C₄	1	1	yes	characteristic
11	#14583317	—	2	2	no	—
12	#14583315	—	2	2	no	—
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	C₂×#13063265	C₂²	1	1	yes	characteristic
15	M₃₂×D₄₃	—	2	2	no	—
16	#14582978	—	2	2	no	—
17	#14582972	—	2	2	no	—
18	#14582926	C₄	1	1	yes	characteristic
19	#14582867	C₄	1	1	yes	characteristic
20	#14582862	C₄	1	1	yes	characteristic
21	#14582680	—	2	2	no	—
22	#14582680	—	2	2	no	—
23	#14582680	—	2	2	no	—
24	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#405	C₂²	1	1	yes	characteristic
27	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
29	#13063331	—	2	1	no	—
30	#13063331	—	2	1	no	—
31	#13063331	—	2	1	no	—
32	#13063331	—	2	1	no	—
33	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
34	C₂×#10621872	—	2	1	no	—
35	C₂×#10621872	—	2	1	no	—
36	C₂×#10621872	—	2	1	no	—
37	C₂×#10621872	—	2	1	no	—
38	C₂×#10621872	—	2	1	no	—
39	C₂×#10621872	—	2	1	no	—
40	C₂×#10621872	D₄	1	2	yes	—
41	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₄₃	D₄	1	2	yes	—
43	#13063309	C₈	1	2	yes	—
44	#13063305	—	2	1	no	—
45	#13063300	—	2	1	no	—
46	#13063295	C₈	1	2	yes	—
47	C₄×D₁₇₂	—	2	2	no	—
48	#13063265	—	2	1	no	—
49	#13063265	—	2	1	no	—
50	#13063265	—	2	1	no	—
51	#13063265	D₄	1	2	yes	complemented
52	#13063264	—	2	4	no	—
53	C₂×C₄₃⋊C₁₆	—	2	1	no	—
54	C₂×C₄₃⋊C₁₆	—	2	1	no	—
55	C₁₆×D₄₃	—	2	4	no	—
56	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
57	(C₂×D₄₃)⋊C₈	—	2	2	no	—
58	(C₂×D₄₃)⋊C₈	—	2	2	no	—
59	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	#117×C₄₃	C₂×C₄	1	1	yes	characteristic
62	M₃₂×C₄₃	—	2	1	no	—
63	M₃₂×C₄₃	—	2	1	no	—
64	M₃₂×C₄₃	—	2	1	no	—
65	M₃₂×C₄₃	D₄	1	2	yes	complemented
66	C₂×C₁₆×C₄₃	—	2	1	no	—
67	C₂×C₁₆×C₄₃	—	2	1	no	—
68	#97×C₄₃	—	2	1	no	—
69	C₄×C₈×C₄₃	—	2	1	no	—
70	C₂³×D₄₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊D₄₃	C₂×C₈	1	2	yes	—
73	C₂²×Dic₄₃	C₂×C₈	1	1	yes	characteristic
74	#10621898	—	2	1	no	—
75	#10621898	—	2	1	no	—
76	#10621898	—	2	1	no	—
77	#10621898	—	2	1	no	—
78	C₂×D₁₇₂	—	2	2	no	—
79	C₂×D₁₇₂	M₁₆	1	1	yes	characteristic
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×C₄×D₄₃	—	2	1	no	—
82	C₂×C₄×D₄₃	—	2	1	no	—
83	C₂×C₄×D₄₃	—	2	1	no	—
84	C₂×C₄×D₄₃	—	2	1	no	—
85	C₂×C₄×D₄₃	—	2	1	no	—
86	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
87	C₂×Dic₈₆	M₁₆	1	1	yes	characteristic
88	M₁₆×C₄₃	—	2	1	no	—
89	M₁₆×C₄₃	—	2	1	no	—
90	C₂×C₈×C₄₃	—	2	1	no	—
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
97	C₄⋊C₄×C₄₃	—	2	1	no	—
98	C₂²⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
99	C₄²×C₄₃	—	2	1	no	—
100	(C₂×D₄₃)⋊C₄	—	2	2	no	—
101	(C₂×D₄₃)⋊C₄	—	2	2	no	—
102	C₄⋊Dic₄₃	—	2	2	no	—
103	#10621877	—	2	1	no	—
104	#10621877	—	2	1	no	—
105	C₂×C₄₃⋊C₈	—	2	1	no	—
106	C₂×C₄₃⋊C₈	—	2	1	no	—
107	C₂×C₄₃⋊C₈	—	2	1	no	—
108	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
109	#10621872	—	2	2	no	—
110	#10621872	—	2	2	no	—
111	#10621872	—	4	1	no	—
112	#10621872	—	4	1	no	—
113	#10621872	—	4	1	no	—
114	#10621872	—	4	1	no	—
115	C₈×D₄₃	—	2	2	no	—
116	C₈×D₄₃	—	2	2	no	—
117	C₁₆×C₄₃	—	2	2	no	—
118	C₁₆×C₄₃	—	4	1	no	—
119	C₄₃⋊C₁₆	—	2	2	no	—
120	C₄₃⋊C₁₆	—	4	1	no	—
121	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
122	C₂²×D₄₃	—	2	1	no	—
123	C₂²×D₄₃	—	2	1	no	—
124	C₂²×D₄₃	—	2	1	no	—
125	C₂²×D₄₃	—	8	1	no	—
126	C₂²×D₄₃	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₄×D₄	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	C₂²⋊D₄₃	—	2	2	no	—
135	C₂²⋊D₄₃	—	2	2	no	—
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	2	1	no	—
138	C₂×Dic₄₃	—	2	1	no	—
139	C₂×Dic₄₃	C₂²⋊C₈	1	2	yes	—
140	D₁₇₂	—	2	1	no	—
141	D₁₇₂	—	2	1	no	—
142	D₁₇₂	—	8	2	no	—
143	C₄×D₄₃	—	2	2	no	—
144	C₄×D₄₃	—	2	2	no	—
145	C₄×D₄₃	—	4	1	no	—
146	C₄×D₄₃	—	4	1	no	—
147	C₄×D₄₃	—	4	1	no	—
148	C₄×D₄₃	—	4	1	no	—
149	Dic₈₆	—	2	1	no	—
150	Dic₈₆	—	2	1	no	—
151	C₈×C₄₃	—	4	1	no	—
152	C₈×C₄₃	#120	1	1	yes	characteristic
153	C₈×C₄₃	#121	1	1	yes	characteristic
154	C₈×C₄₃	#121	1	1	yes	characteristic
155	C₈×C₄₃	#123	1	1	yes	characteristic
156	C₄₃⋊C₈	—	2	2	no	—
157	C₄₃⋊C₈	—	4	1	no	—
158	#12989	—	43	1	no	maximal, Sylow
159	C₂²×C₄₃	—	4	1	no	—
160	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
162	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
163	C₂×D₄₃	—	2	2	no	—
164	C₂×D₄₃	—	4	2	no	—
165	C₂×D₄₃	—	4	1	no	—
166	C₂×D₄₃	—	4	1	no	—
167	C₂×D₄₃	—	4	1	no	—
168	C₂×D₄₃	—	8	1	no	—
169	C₂×D₄₃	—	8	1	no	—
170	C₄×C₄₃	—	4	2	no	—
171	C₄×C₄₃	#386	1	1	yes	characteristic
172	C₄×C₄₃	#408	1	1	yes	characteristic
173	C₄×C₄₃	#435	1	1	yes	characteristic
174	C₄×C₄₃	#435	1	1	yes	characteristic
175	Dic₄₃	—	2	2	no	—
176	Dic₄₃	—	4	1	no	—
177	#3407	—	43	1	no	—
178	#2930	—	43	1	no	—
179	#2115	—	43	1	no	—
180	C₂×#349	—	43	1	no	—
181	C₂×#349	—	43	1	no	—
182	#1378	—	43	1	no	—
183	#1378	D₄₃	1	1	yes	characteristic, complemented
184	C₂×C₄₃	—	2	1	no	—
185	C₂×C₄₃	—	2	1	no	—
186	C₂×C₄₃	—	2	1	no	—
187	C₂×C₄₃	#1869	1	1	yes	characteristic
188	D₄₃	—	4	2	no	—
189	D₄₃	—	8	1	no	—
190	C₂×#131	—	43	1	no	—
191	C₂²×M₁₆	—	43	1	no	—
192	#525	—	43	1	no	—
193	#504	—	86	2	no	—
194	#504	—	86	2	no	—
195	C₂×M₃₂	—	43	1	no	—
196	C₂×M₃₂	—	43	1	no	—
197	C₂×M₃₂	—	86	2	no	—
198	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
199	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
200	#436	—	86	2	no	—
201	#435	—	86	2	no	—
202	#424	—	43	1	no	—
203	#408	—	43	1	no	—
204	#407	—	43	1	no	—
205	#405	C₂×D₄₃	1	1	yes	characteristic
206	#349	—	86	2	no	—
207	#349	—	86	2	no	—
208	#349	—	86	2	no	—
209	C₄₃	#12989	1	1	yes	characteristic, complemented, Sylow
210	C₂×#41	—	43	1	no	—
211	C₂³×C₄	—	43	1	no	—
212	#131	—	86	1	no	—
213	#131	—	86	1	no	—
214	#131	—	86	1	no	—
215	#131	—	86	1	no	—
216	C₂×M₁₆	—	43	1	no	—
217	C₂×M₁₆	—	43	2	no	—
218	C₂×M₁₆	—	86	1	no	—
219	C₂×M₁₆	—	86	1	no	—
220	C₂×M₁₆	—	86	1	no	—
221	C₂×M₁₆	—	86	1	no	—
222	C₂×M₁₆	—	86	1	no	—
223	C₂×M₁₆	—	86	1	no	—
224	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
225	C₂²×C₈	—	43	2	no	—
226	C₂²×C₈	—	43	1	no	—
227	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
228	#123	—	43	2	no	—
229	C₂²⋊Q₈	—	86	1	no	—
230	#121	—	86	1	no	—
231	#120	—	43	2	no	—
232	C₄×D₄	—	86	2	no	—
233	#117	C₄×D₄₃	1	1	yes	characteristic
234	M₃₂	—	2	1	no	—
235	M₃₂	—	2	1	no	—
236	M₃₂	—	2	1	no	—
237	M₃₂	—	43	2	no	—
238	M₃₂	—	86	4	no	—
239	M₃₂	—	86	1	no	—
240	M₃₂	—	86	1	no	—
241	M₃₂	—	86	1	no	—
242	M₃₂	D₁₇₂	1	2	yes	complemented
243	C₂×C₁₆	—	2	1	no	—
244	C₂×C₁₆	—	2	1	no	—
245	C₂×C₁₆	—	86	4	no	—
246	C₂×C₁₆	—	86	1	no	—
247	C₂×C₁₆	—	86	1	no	—
248	C₄⋊C₈	—	86	2	no	—
249	C₂²⋊C₈	—	86	2	no	—
250	C₂²⋊C₈	—	86	2	no	—
251	#97	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₂⁴	—	43	1	no	—
254	#41	—	86	1	no	—
255	#41	—	86	1	no	—
256	#41	—	86	1	no	—
257	#41	—	86	1	no	—
258	C₂×Q₈	—	43	1	no	—
259	C₂×D₄	—	43	2	no	—
260	C₂×D₄	—	43	1	no	—
261	C₂×D₄	—	86	2	no	—
262	C₂²×C₄	—	43	2	no	—
263	C₂²×C₄	—	43	1	no	—
264	C₂²×C₄	—	86	1	no	—
265	C₂²×C₄	—	86	1	no	—
266	C₂²×C₄	—	86	1	no	—
267	C₂²×C₄	—	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	M₁₆	—	2	1	no	—
272	M₁₆	—	2	1	no	—
273	M₁₆	—	86	1	no	—
274	M₁₆	—	86	1	no	—
275	M₁₆	—	86	2	no	—
276	M₁₆	—	86	2	no	—
277	M₁₆	—	172	1	no	—
278	M₁₆	—	172	1	no	—
279	M₁₆	—	172	1	no	—
280	M₁₆	—	172	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	43	2	no	—
283	C₂×C₈	—	86	2	no	—
284	C₂×C₈	—	86	2	no	—
285	C₂×C₈	—	86	1	no	—
286	C₂×C₈	—	86	1	no	—
287	C₂×C₈	—	86	1	no	—
288	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
289	C₂×C₈	#10621898	1	1	yes	characteristic
290	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
291	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
292	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
293	C₂×C₈	#10621902	1	1	yes	characteristic
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	86	2	no	—
296	C₂²⋊C₄	—	86	2	no	—
297	C₂²⋊C₄	—	86	2	no	—
298	C₂²⋊C₄	C₈×D₄₃	1	2	yes	—
299	C₄²	—	2	1	no	—
300	C₁₆	—	2	2	no	—
301	C₁₆	—	4	1	no	—
302	C₁₆	—	86	2	no	—
303	C₁₆	—	172	1	no	—
304	C₂³	—	43	2	no	—
305	C₂³	—	86	1	no	—
306	C₂³	—	86	1	no	—
307	C₂³	—	86	1	no	—
308	C₂³	—	344	1	no	—
309	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
310	Q₈	—	86	1	no	—
311	Q₈	—	86	1	no	—
312	D₄	—	86	2	no	—
313	D₄	—	86	2	no	—
314	D₄	—	86	1	no	—
315	D₄	—	86	1	no	—
316	D₄	—	344	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	43	2	no	—
319	C₂×C₄	—	86	2	no	—
320	C₂×C₄	—	86	2	no	—
321	C₂×C₄	—	86	1	no	—
322	C₂×C₄	—	86	1	no	—
323	C₂×C₄	—	86	1	no	—
324	C₂×C₄	—	172	1	no	—
325	C₂×C₄	—	172	1	no	—
326	C₂×C₄	—	172	1	no	—
327	C₂×C₄	—	172	1	no	—
328	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
329	C₂×C₄	#13063287	1	1	yes	characteristic
330	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
331	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
332	C₂×C₄	#13063331	1	1	yes	characteristic
333	C₂×C₄	#13063333	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	86	2	no	—
336	C₈	—	172	1	no	—
337	C₈	#13063295	1	1	yes	characteristic
338	C₈	#13063298	1	1	yes	characteristic
339	C₈	#13063300	1	1	yes	characteristic
340	C₈	#13063310	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	86	2	no	—
343	C₂²	—	344	1	no	—
344	C₂²	—	344	1	no	—
345	C₂²	—	172	1	no	—
346	C₂²	—	172	1	no	—
347	C₂²	—	172	2	no	—
348	C₂²	—	172	1	no	—
349	C₂²	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
350	C₂²	#14582964	1	1	yes	characteristic
351	C₂²	C₈×D₁₇₂	1	1	yes	characteristic
352	C₄	—	4	2	no	—
353	C₄	—	86	2	no	—
354	C₄	—	172	1	no	—
355	C₄	#14582777	1	1	yes	characteristic
356	C₄	#14582864	1	1	yes	characteristic
357	C₄	#14582969	1	1	yes	characteristic
358	C₄	#14582972	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	344	1	no	—
363	C₂	—	172	2	no	—
364	C₂	#19134886	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14582777
commutator	C₂³×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1378	C₂
socle	C₂×C₄₃	#1869
prime radical	#14583411	C₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1378	D₄₃
hypocenter	C₄₃	#12989
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1378

Complements

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

N	H ≅ G/N	type
1. #19147697	C₂	semidirect
3. #19136215	C₂	semidirect
4. C₂×#14582680	C₂	semidirect
5. C₂×#14582680	C₂	semidirect
6. #19132681	C₂	semidirect
7. C₄₃×#1378	C₂	semidirect
51. #13063265	D₄	semidirect
65. M₃₂×C₄₃	D₄	semidirect
183. #1378	D₄₃	semidirect
209. C₄₃	#12989	semidirect
242. M₃₂	D₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
#19147697	2
#19143300	2
#19136215	2
C₂×#14582680	2
#19132681	2
C₄₃×#1378	2
#12989	43