Subgroups — #25013485

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	#19143308	C₂	1	1	yes	maximal, characteristic
3	#19136224	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136222	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136215	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132677	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063331	C₂²	1	1	yes	characteristic
9	C₂×#13063331	C₂²	1	1	yes	characteristic
10	#14583449	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	C₈×D₁₇₂	—	2	2	no	—
18	#14582923	C₄	1	1	yes	characteristic
19	#14582865	C₄	1	1	yes	characteristic
20	#14582865	C₄	1	1	yes	characteristic
21	#14582684	—	2	2	no	—
22	#14582682	—	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	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
28	C₂×#10621898	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	#13063331	—	2	1	no	—
34	#13063331	—	2	1	no	—
35	#13063331	—	2	1	no	—
36	#13063331	—	2	1	no	—
37	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
38	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
39	C₂×#10621872	—	2	1	no	—
40	C₂×#10621872	D₄	1	2	yes	—
41	C₂×C₈×D₄₃	—	2	1	no	—
42	C₂×C₈×D₄₃	D₄	1	2	yes	—
43	#13063325	C₈	1	1	yes	characteristic
44	#13063322	C₈	1	1	yes	characteristic
45	#13063313	C₈	1	1	yes	characteristic
46	#13063313	C₈	1	1	yes	characteristic
47	#13063299	—	2	2	no	—
48	C₄×D₁₇₂	—	2	2	no	—
49	#13063265	—	2	1	no	—
50	#13063265	—	2	1	no	—
51	#13063265	—	2	1	no	—
52	#13063265	D₄	1	2	yes	complemented
53	#13063264	—	2	4	no	—
54	C₂×C₄₃⋊C₁₆	—	2	1	no	—
55	C₂×C₄₃⋊C₁₆	—	2	1	no	—
56	C₁₆×D₄₃	—	2	4	no	—
57	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
58	(C₂×D₄₃)⋊C₈	—	2	2	no	—
59	(C₂×D₄₃)⋊C₈	—	2	2	no	—
60	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
63	M₃₂×C₄₃	—	2	1	no	—
64	M₃₂×C₄₃	—	2	1	no	—
65	M₃₂×C₄₃	—	2	1	no	—
66	M₃₂×C₄₃	D₄	1	2	yes	complemented
67	C₂×C₁₆×C₄₃	—	2	1	no	—
68	C₂×C₁₆×C₄₃	—	2	1	no	—
69	C₄×C₈×C₄₃	—	2	1	no	—
70	C₄×C₈×C₄₃	—	2	1	no	—
71	C₂²×C₄×C₄₃	—	2	1	no	—
72	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊D₄₃	—	2	1	no	—
74	C₂×C₂²⋊D₄₃	M₁₆	1	2	yes	—
75	#10621898	—	2	1	no	—
76	#10621898	—	2	1	no	—
77	#10621898	—	2	1	no	—
78	#10621898	—	2	1	no	—
79	#10621898	—	2	1	no	—
80	#10621898	—	2	1	no	—
81	#10621898	—	2	1	no	—
82	#10621898	—	2	1	no	—
83	C₂×D₁₇₂	—	2	2	no	—
84	C₂×D₁₇₂	C₂×C₈	1	1	yes	characteristic
85	C₂×D₁₇₂	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₄₃	—	2	1	no	—
87	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
88	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
89	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
99	C₄²×C₄₃	—	2	1	no	—
100	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
101	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
102	(C₂×D₄₃)⋊C₄	—	2	2	no	—
103	(C₂×D₄₃)⋊C₄	—	2	2	no	—
104	C₄⋊Dic₄₃	—	2	2	no	—
105	#10621877	—	2	1	no	—
106	#10621877	—	2	1	no	—
107	#10621877	—	2	1	no	—
108	#10621877	—	2	1	no	—
109	C₂×C₄₃⋊C₈	—	2	1	no	—
110	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
111	#10621872	—	2	2	no	—
112	#10621872	—	2	2	no	—
113	#10621872	—	4	1	no	—
114	#10621872	—	4	1	no	—
115	C₈×D₄₃	—	2	2	no	—
116	C₈×D₄₃	—	2	2	no	—
117	C₈×D₄₃	—	4	1	no	—
118	C₈×D₄₃	—	4	1	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	C₂×M₁₆	1	1	yes	characteristic
124	C₂²×D₄₃	—	2	1	no	—
125	C₂²×D₄₃	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	4	2	no	—
129	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
135	C₂²⋊D₄₃	—	2	2	no	—
136	C₂²⋊D₄₃	—	2	2	no	—
137	C₂²⋊D₄₃	—	4	1	no	—
138	C₂²⋊D₄₃	—	4	1	no	—
139	C₂×Dic₄₃	—	2	1	no	—
140	C₂×Dic₄₃	C₂²⋊C₈	1	2	yes	—
141	D₁₇₂	—	2	1	no	—
142	D₁₇₂	—	2	1	no	—
143	D₁₇₂	—	2	1	no	—
144	D₁₇₂	—	2	1	no	—
145	D₁₇₂	—	4	4	no	—
146	C₄×D₄₃	—	2	2	no	—
147	C₄×D₄₃	—	2	2	no	—
148	C₄×D₄₃	—	4	1	no	—
149	C₄×D₄₃	—	4	1	no	—
150	Dic₈₆	—	2	1	no	—
151	Dic₈₆	—	2	1	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	C₂×C₂²⋊C₈	1	1	yes	characteristic
164	C₂²×C₄₃	#435	1	1	yes	characteristic
165	C₂²×C₄₃	#435	1	1	yes	characteristic
166	C₂×D₄₃	—	2	2	no	—
167	C₂×D₄₃	—	4	2	no	—
168	C₂×D₄₃	—	4	2	no	—
169	C₂×D₄₃	—	4	1	no	—
170	C₄×C₄₃	—	2	2	no	—
171	C₄×C₄₃	—	4	1	no	—
172	C₄×C₄₃	#386	1	1	yes	characteristic
173	C₄×C₄₃	#408	1	1	yes	characteristic
174	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
175	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
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	—	43	1	no	—
182	#2115	—	43	1	no	—
183	#1376	—	43	1	no	—
184	#1376	D₄₃	1	1	yes	characteristic, complemented
185	C₂×C₄₃	—	2	1	no	—
186	C₂×C₄₃	—	2	1	no	—
187	C₂×C₄₃	—	2	1	no	—
188	C₂×C₄₃	#1869	1	1	yes	characteristic
189	D₄₃	—	4	2	no	—
190	D₄₃	—	8	1	no	—
191	C₂×#131	—	43	1	no	—
192	C₂×#131	—	43	1	no	—
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	C₂×D₄₃	1	1	yes	characteristic
207	#350	—	86	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	—	43	1	no	—
213	#131	—	86	1	no	—
214	#131	—	86	1	no	—
215	#131	—	86	1	no	—
216	#131	—	86	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	—	43	1	no	—
222	C₂×M₁₆	—	43	2	no	—
223	C₂×M₁₆	—	43	1	no	—
224	C₂×M₁₆	—	86	1	no	—
225	C₂²×C₈	—	43	2	no	—
226	C₂²×C₈	—	86	1	no	—
227	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
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	C₄×D₄₃	1	1	yes	characteristic
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	D₁₇₂	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	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	#41	—	86	1	no	—
256	#41	—	86	1	no	—
257	#41	—	86	1	no	—
258	#41	—	86	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	—	43	1	no	—
265	C₂×D₄	—	43	2	no	—
266	C₂×D₄	—	43	1	no	—
267	C₂×D₄	—	43	1	no	—
268	C₂×D₄	—	86	2	no	—
269	C₂×D₄	—	86	1	no	—
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	—	86	1	no	—
275	M₁₆	—	86	1	no	—
276	M₁₆	—	86	2	no	—
277	M₁₆	—	86	2	no	—
278	M₁₆	—	86	1	no	—
279	M₁₆	—	86	1	no	—
280	M₁₆	—	172	1	no	—
281	M₁₆	—	172	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	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	—	172	1	no	—
290	C₂×C₈	—	172	1	no	—
291	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
292	C₂×C₈	#10621898	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	D₄×D₄₃	1	1	yes	characteristic
296	C₂×C₈	#10621902	1	1	yes	characteristic
297	C₄⋊C₄	—	86	2	no	—
298	C₂²⋊C₄	—	86	2	no	—
299	C₂²⋊C₄	—	86	2	no	—
300	C₄²	—	2	1	no	—
301	C₄²	C₈×D₄₃	1	1	yes	characteristic
302	C₄²	C₈×D₄₃	1	1	yes	characteristic
303	C₁₆	—	2	2	no	—
304	C₁₆	—	4	1	no	—
305	C₁₆	—	86	2	no	—
306	C₁₆	—	172	1	no	—
307	C₂³	—	43	2	no	—
308	C₂³	—	86	1	no	—
309	C₂³	M₁₆×D₄₃	1	1	yes	characteristic
310	Q₈	—	86	1	no	—
311	Q₈	—	86	1	no	—
312	Q₈	—	86	1	no	—
313	Q₈	—	86	1	no	—
314	D₄	—	86	2	no	—
315	D₄	—	86	2	no	—
316	D₄	—	86	1	no	—
317	D₄	—	86	1	no	—
318	D₄	—	86	1	no	—
319	D₄	—	86	1	no	—
320	D₄	—	172	4	no	—
321	D₄	—	172	1	no	—
322	D₄	—	172	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	43	2	no	—
327	C₂×C₄	—	86	2	no	—
328	C₂×C₄	—	86	2	no	—
329	C₂×C₄	—	86	1	no	—
330	C₂×C₄	—	172	1	no	—
331	C₂×C₄	—	172	1	no	—
332	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
333	C₂×C₄	#13063287	1	1	yes	characteristic
334	C₂×C₄	C₄×D₁₇₂	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	#13063295	1	1	yes	characteristic
342	C₈	#13063298	1	1	yes	characteristic
343	C₈	#13063300	1	1	yes	characteristic
344	C₈	#13063310	1	1	yes	characteristic
345	C₂²	—	4	1	no	—
346	C₂²	—	86	2	no	—
347	C₂²	—	172	2	no	—
348	C₂²	—	172	2	no	—
349	C₂²	—	172	1	no	—
350	C₂²	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
351	C₂²	#14582969	1	1	yes	characteristic
352	C₂²	#14582972	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	86	2	no	—
356	C₄	—	172	1	no	—
357	C₄	#14582777	1	1	yes	characteristic
358	C₄	#14582864	1	1	yes	characteristic
359	C₄	#14582964	1	1	yes	characteristic
360	C₄	C₈×D₁₇₂	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	344	1	no	—
365	C₂	—	172	2	no	—
366	C₂	#19134886	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13063295
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1376	C₂
socle	C₂×C₄₃	#1869
prime radical	#14583449	C₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1376	D₄₃
hypocenter	C₄₃	#12982
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
1. #19147697	C₂	semidirect
3. #19136224	C₂	semidirect
4. #19136222	C₂	semidirect
5. #19136215	C₂	semidirect
6. #19132677	C₂	semidirect
7. C₄₃×#1376	C₂	semidirect
52. #13063265	D₄	semidirect
66. M₃₂×C₄₃	D₄	semidirect
184. #1376	D₄₃	semidirect
210. C₄₃	#12982	semidirect
244. 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
#19143308	2
#19136224	2
#19136222	2
#19136215	2
#19132677	2
C₄₃×#1376	2
#12982	43