Subgroups — #25079692

Order 11008.

746 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	#19145905	C₂	1	2	yes	maximal, complemented
2	#19145023	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143308	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134061	C₂	1	4	yes	maximal, complemented
5	#19133961	C₂	1	2	yes	maximal, complemented
6	#19133951	C₂	1	2	yes	maximal, complemented
7	#19133270	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063347	—	2	1	no	—
10	C₂×#13063347	—	2	1	no	—
11	C₂×#13063331	—	2	1	no	—
12	#14583449	C₂²	1	2	yes	—
13	#14583449	C₂²	1	1	yes	characteristic
14	#14583283	—	2	1	no	—
15	#14583283	—	2	1	no	—
16	#14583279	C₂²	1	1	yes	characteristic
17	#14583276	C₂²	1	1	yes	characteristic
18	#14583272	C₂²	1	2	yes	—
19	#14583180	—	2	2	no	—
20	#14583180	C₂²	1	4	yes	—
21	#14583135	—	2	4	no	—
22	#14583095	—	2	2	no	—
23	#14583095	C₂²	1	4	yes	—
24	#14582978	—	2	1	no	—
25	#14582978	C₂²	1	2	yes	—
26	C₈×D₁₇₂	—	2	1	no	—
27	C₈×D₁₇₂	C₂²	1	2	yes	—
28	#14582923	C₂²	1	1	yes	characteristic
29	#14582865	—	2	1	no	—
30	#14582644	C₂²	1	4	yes	—
31	#14582642	C₂²	1	4	yes	—
32	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#335	C₂²	1	2	yes	—
35	C₄₃×#335	C₂²	1	2	yes	—
36	C₂×#10621898	—	2	1	no	—
37	C₂×#10621898	—	2	2	no	—
38	#13063347	—	4	4	no	—
39	#13063347	—	4	4	no	—
40	C₂×Dic₁₇₂	—	2	1	no	—
41	C₂×Dic₁₇₂	—	2	1	no	—
42	C₂×#10621873	—	2	2	no	—
43	C₂×#10621873	—	2	2	no	—
44	C₂×D₃₄₄	—	2	1	no	—
45	C₂×D₃₄₄	—	2	1	no	—
46	C₂×D₃₄₄	—	2	4	no	—
47	#13063331	—	4	4	no	—
48	C₂×#10621877	—	2	1	no	—
49	C₂×#10621872	—	2	2	no	—
50	C₂×C₈×D₄₃	—	2	2	no	—
51	#13063325	D₄	1	1	yes	characteristic
52	#13063325	C₂³	1	2	yes	—
53	#13063322	D₄	1	1	yes	characteristic
54	#13063322	C₂³	1	2	yes	—
55	#13063313	—	2	1	no	—
56	#13063313	—	2	2	no	—
57	#13063299	—	2	1	no	—
58	#13063299	—	2	2	no	—
59	#13063299	—	2	2	no	—
60	#13063299	D₄	1	2	yes	—
61	C₄×D₁₇₂	—	2	1	no	—
62	C₄×D₁₇₂	—	2	2	no	—
63	C₄×D₁₇₂	—	2	2	no	—
64	C₄×D₁₇₂	C₂³	1	2	yes	—
65	#13063259	—	2	4	no	—
66	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
67	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
68	#13063247	—	2	4	no	—
69	#13063247	—	2	4	no	—
70	(C₂×D₄₃)⋊C₈	—	2	2	no	—
71	(C₂×D₄₃)⋊C₈	—	2	2	no	—
72	C₂²×C₈×C₄₃	—	2	2	no	—
73	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₃	—	2	2	no	—
77	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
78	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
79	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
80	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
85	C₂×C₂²⋊D₄₃	—	2	2	no	—
86	C₂×C₂²⋊D₄₃	—	4	2	no	—
87	#10621898	—	4	2	no	—
88	#10621898	—	4	2	no	—
89	#10621898	—	4	4	no	—
90	#10621898	—	4	4	no	—
91	C₂×D₁₇₂	—	2	1	no	—
92	C₂×D₁₇₂	—	2	1	no	—
93	C₂×D₁₇₂	—	2	2	no	—
94	C₂×D₁₇₂	—	2	2	no	—
95	C₂×D₁₇₂	—	2	2	no	—
96	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
97	C₂×C₄×D₄₃	—	2	2	no	—
98	C₂×C₄×D₄₃	—	4	2	no	—
99	C₂×Dic₈₆	—	2	1	no	—
100	C₂×Dic₈₆	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	4	no	—
103	C₂×C₈×C₄₃	—	8	2	no	—
104	C₂×C₈×C₄₃	D₈	1	4	yes	complemented
105	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₄₃	#41	1	1	yes	characteristic
112	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
113	(C₂×D₄₃)⋊C₄	—	2	2	no	—
114	(C₂×D₄₃)⋊C₄	—	2	2	no	—
115	(C₂×D₄₃)⋊C₄	—	4	2	no	—
116	(C₂×D₄₃)⋊C₄	—	4	2	no	—
117	C₄⋊Dic₄₃	—	2	1	no	—
118	C₄⋊Dic₄₃	—	2	2	no	—
119	C₄⋊Dic₄₃	—	2	2	no	—
120	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
121	#10621877	—	4	2	no	—
122	C₂×C₄₃⋊C₈	—	2	2	no	—
123	Dic₁₇₂	—	4	2	no	—
124	Dic₁₇₂	—	4	2	no	—
125	D₃₄₄	—	4	2	no	—
126	D₃₄₄	—	4	2	no	—
127	D₃₄₄	—	8	4	no	—
128	#10621873	—	4	4	no	—
129	#10621873	—	4	4	no	—
130	#10621872	—	4	4	no	—
131	C₈×D₄₃	—	4	4	no	—
132	C₂³×C₄₃	#121	1	1	yes	characteristic
133	C₂²×D₄₃	—	2	2	no	—
134	C₂²×D₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	—	2	1	no	—
136	C₂×C₄×C₄₃	—	2	1	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
146	C₂²⋊D₄₃	—	4	4	no	—
147	C₂²⋊D₄₃	—	8	4	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	D₁₇₂	—	2	2	no	—
151	D₁₇₂	—	2	2	no	—
152	D₁₇₂	—	4	2	no	—
153	D₁₇₂	—	4	2	no	—
154	D₁₇₂	—	4	2	no	—
155	D₁₇₂	—	4	2	no	—
156	D₁₇₂	—	4	2	no	—
157	D₁₇₂	—	4	2	no	—
158	D₁₇₂	—	8	2	no	—
159	C₄×D₄₃	—	4	2	no	—
160	C₄×D₄₃	—	4	2	no	—
161	C₄×D₄₃	—	8	4	no	—
162	Dic₈₆	—	4	2	no	—
163	Dic₈₆	—	4	2	no	—
164	Dic₈₆	—	4	2	no	—
165	C₈×C₄₃	—	2	1	no	—
166	C₈×C₄₃	—	2	1	no	—
167	C₈×C₄₃	—	4	4	no	—
168	C₈×C₄₃	C₂×D₈	1	2	yes	—
169	C₈×C₄₃	C₂×D₈	1	2	yes	—
170	C₄₃⋊C₈	—	4	2	no	—
171	#19190	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	2	1	no	—
173	C₂²×C₄₃	—	2	1	no	—
174	C₂²×C₄₃	#459	1	2	yes	—
175	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
176	C₂×D₄₃	—	4	2	no	—
177	C₂×D₄₃	—	4	2	no	—
178	C₂×D₄₃	—	4	2	no	—
179	C₂×D₄₃	—	8	2	no	—
180	C₂×D₄₃	—	8	2	no	—
181	C₂×D₄₃	—	8	2	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	2	1	no	—
184	C₄×C₄₃	—	2	1	no	—
185	C₄×C₄₃	#459	1	2	yes	—
186	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
187	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₄×C₄₃	#576	1	1	yes	characteristic
189	C₄×C₄₃	#578	1	1	yes	characteristic
190	Dic₄₃	—	4	2	no	—
191	Dic₄₃	—	8	2	no	—
192	#3245	—	43	2	no	—
193	#3147	—	43	1	no	—
194	#2931	—	43	1	no	—
195	#1685	—	43	4	no	—
196	#1669	—	43	2	no	—
197	#1667	—	43	2	no	—
198	C₂×#335	D₄₃	1	1	yes	characteristic, complemented
199	#1547	—	43	2	no	—
200	C₂×C₄₃	—	2	2	no	—
201	C₂×C₄₃	C₂×#459	1	1	yes	characteristic
202	C₂×C₄₃	#3061	1	1	yes	characteristic
203	C₂×C₄₃	#3067	1	1	yes	characteristic
204	D₄₃	—	8	2	no	—
205	D₄₃	—	16	2	no	—
206	C₂×#135	—	86	1	no	—
207	C₂×#135	—	86	1	no	—
208	C₂×#131	—	86	1	no	—
209	#532	—	43	1	no	—
210	#532	—	43	2	no	—
211	#495	—	86	1	no	—
212	#495	—	86	1	no	—
213	#494	—	43	1	no	—
214	#493	—	43	1	no	—
215	#492	—	43	2	no	—
216	#474	—	43	4	no	—
217	#474	—	86	2	no	—
218	#466	—	86	4	no	—
219	#459	—	43	4	no	—
220	#459	—	86	2	no	—
221	#436	—	43	2	no	—
222	#436	—	86	1	no	—
223	C₈×D₄	—	43	2	no	—
224	C₈×D₄	—	86	1	no	—
225	#423	—	43	1	no	—
226	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
227	#408	—	86	1	no	—
228	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
229	#335	C₂×D₄₃	1	2	yes	—
230	#335	C₂×D₄₃	1	2	yes	—
231	#332	—	43	4	no	—
232	#331	—	43	4	no	—
233	C₄₃	#19190	1	1	yes	characteristic, complemented, Sylow
234	C₂×#41	—	86	1	no	—
235	C₂×#41	—	86	2	no	—
236	#135	—	172	4	no	—
237	#135	—	172	4	no	—
238	C₂×Q₁₆	—	86	1	no	—
239	C₂×Q₁₆	—	86	1	no	—
240	C₂×SD₁₆	—	86	2	no	—
241	C₂×SD₁₆	—	86	2	no	—
242	C₂×D₈	—	86	4	no	—
243	C₂×D₈	—	86	1	no	—
244	C₂×D₈	—	86	1	no	—
245	#131	—	172	4	no	—
246	C₂×M₁₆	—	86	1	no	—
247	C₂×M₁₆	—	86	2	no	—
248	C₂²×C₈	—	2	2	no	—
249	C₂²×C₈	—	86	2	no	—
250	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
251	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
252	C₄⋊Q₈	—	43	1	no	—
253	C₄⋊Q₈	—	43	2	no	—
254	#127	—	43	1	no	—
255	#127	—	43	2	no	—
256	#124	—	86	2	no	—
257	#124	—	86	1	no	—
258	#121	—	43	2	no	—
259	#121	—	86	1	no	—
260	#121	—	86	2	no	—
261	#121	—	86	2	no	—
262	C₄×D₄	—	43	2	no	—
263	C₄×D₄	—	86	1	no	—
264	C₄×D₄	—	86	2	no	—
265	C₄×D₄	—	86	2	no	—
266	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
267	#107	—	86	4	no	—
268	C₄⋊C₈	—	2	2	no	—
269	C₄⋊C₈	—	43	2	no	—
270	C₄⋊C₈	—	86	1	no	—
271	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
272	#102	—	86	4	no	—
273	#102	—	86	4	no	—
274	C₂²⋊C₈	—	86	2	no	—
275	C₂²⋊C₈	—	86	2	no	—
276	C₄×C₈	D₁₇₂	1	1	yes	characteristic
277	C₄×C₈	D₁₇₂	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
280	#41	—	172	4	no	—
281	#41	—	172	4	no	—
282	#41	—	172	2	no	—
283	#41	—	172	2	no	—
284	C₂×Q₈	—	86	1	no	—
285	C₂×Q₈	—	86	2	no	—
286	C₂×D₄	—	43	2	no	—
287	C₂×D₄	—	86	2	no	—
288	C₂×D₄	—	86	1	no	—
289	C₂×D₄	—	86	1	no	—
290	C₂×D₄	—	86	2	no	—
291	C₂×D₄	—	86	2	no	—
292	C₂×D₄	—	86	2	no	—
293	C₂×D₄	—	172	2	no	—
294	C₂²×C₄	—	86	2	no	—
295	C₂²×C₄	—	172	2	no	—
296	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
297	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
298	C₂²×C₄	#10621902	1	1	yes	characteristic
299	Q₁₆	—	172	2	no	—
300	Q₁₆	—	172	2	no	—
301	SD₁₆	—	172	4	no	—
302	SD₁₆	—	172	4	no	—
303	D₈	—	344	4	no	—
304	D₈	—	172	2	no	—
305	D₈	—	172	2	no	—
306	M₁₆	—	172	2	no	—
307	M₁₆	—	172	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	4	no	—
310	C₂×C₈	—	8	2	no	—
311	C₂×C₈	—	86	2	no	—
312	C₂×C₈	—	172	4	no	—
313	C₂×C₈	D₃₄₄	1	4	yes	complemented
314	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
315	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
316	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
317	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
318	C₄⋊C₄	—	43	2	no	—
319	C₄⋊C₄	—	86	1	no	—
320	C₄⋊C₄	—	86	2	no	—
321	C₄⋊C₄	—	86	2	no	—
322	C₂²⋊C₄	—	86	2	no	—
323	C₂²⋊C₄	—	86	2	no	—
324	C₂²⋊C₄	—	172	2	no	—
325	C₂²⋊C₄	—	172	2	no	—
326	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
327	C₄²	D₄×D₄₃	1	1	yes	characteristic
328	C₄²	#10621902	1	1	yes	characteristic
329	C₄²	C₂³×D₄₃	1	1	yes	characteristic
330	C₂³	—	86	2	no	—
331	C₂³	—	172	2	no	—
332	C₂³	#13063300	1	1	yes	characteristic
333	Q₈	—	172	2	no	—
334	Q₈	—	172	2	no	—
335	Q₈	—	172	2	no	—
336	D₄	—	86	2	no	—
337	D₄	—	86	2	no	—
338	D₄	—	344	2	no	—
339	D₄	—	344	4	no	—
340	D₄	—	172	2	no	—
341	D₄	—	172	4	no	—
342	D₄	—	172	2	no	—
343	D₄	—	172	2	no	—
344	D₄	—	172	2	no	—
345	D₄	—	172	2	no	—
346	D₄	—	172	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	86	2	no	—
351	C₂×C₄	—	344	4	no	—
352	C₂×C₄	—	172	2	no	—
353	C₂×C₄	—	172	2	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	#13063300	1	1	yes	characteristic
356	C₂×C₄	#13063300	1	1	yes	characteristic
357	C₂×C₄	#13063300	1	1	yes	characteristic
358	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
359	C₂×C₄	#13063353	1	2	yes	—
360	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
361	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
362	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
363	C₈	—	2	1	no	—
364	C₈	—	2	1	no	—
365	C₈	—	4	4	no	—
366	C₈	—	172	2	no	—
367	C₈	C₂×D₃₄₄	1	2	yes	—
368	C₈	C₂×D₃₄₄	1	2	yes	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	344	2	no	—
372	C₂²	—	344	2	no	—
373	C₂²	—	344	2	no	—
374	C₂²	—	172	2	no	—
375	C₂²	—	172	2	no	—
376	C₂²	—	172	2	no	—
377	C₂²	#14583095	1	2	yes	—
378	C₂²	C₂×#13063300	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	344	2	no	—
383	C₄	—	172	2	no	—
384	C₄	#14583095	1	2	yes	—
385	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
386	C₄	C₂×#13063353	1	1	yes	characteristic
387	C₄	#14583700	1	1	yes	characteristic
388	C₄	#14583712	1	1	yes	characteristic
389	C₂	—	2	2	no	—
390	C₂	—	344	2	no	—
391	C₂	—	688	2	no	—
392	C₂	C₂×#14583095	1	1	yes	characteristic
393	C₂	#19144267	1	1	yes	characteristic
394	C₂	#19144316	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063300
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#335	C₂
socle	C₂²×C₄₃	C₂×#121
hypercenter	C₂×#335	D₄₃
hypocenter	C₄₃	#19190
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
1. #19145905	C₂	semidirect
2. #19145023	C₂	semidirect
3. #19143308	C₂	semidirect
4. #19134061	C₂	semidirect
5. #19133961	C₂	semidirect
6. #19133951	C₂	semidirect
7. #19133270	C₂	semidirect
8. C₂×C₄₃×#335	C₂	semidirect
104. C₂×C₈×C₄₃	D₈	semidirect
198. C₂×#335	D₄₃	semidirect
233. C₄₃	#19190	semidirect
313. C₂×C₈	D₃₄₄	semidirect

Maximal subgroups

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

subgroup	index
#19145905	2
#19145023	2
#19143308	2
#19134061	2
#19133961	2
#19133951	2
#19133270	2
C₂×C₄₃×#335	2
#19190	43