Subgroups — #25079683

Order 11008.

746 conjugacy classes of subgroups.

The subgroup lattice has 390 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	#19134085	C₂	1	2	yes	maximal, complemented
5	#19134075	C₂	1	2	yes	maximal, complemented
6	#19133937	C₂	1	4	yes	maximal, complemented
7	#19133270	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₃×#334	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×#334	C₂²	1	2	yes	—
35	C₄₃×#334	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	2	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
145	C₂²⋊D₄₃	—	4	4	no	—
146	C₂²⋊D₄₃	—	8	4	no	—
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	D₁₇₂	—	2	4	no	—
150	D₁₇₂	—	4	2	no	—
151	D₁₇₂	—	4	2	no	—
152	D₁₇₂	—	4	2	no	—
153	D₁₇₂	—	4	2	no	—
154	D₁₇₂	—	4	2	no	—
155	D₁₇₂	—	4	2	no	—
156	D₁₇₂	—	8	2	no	—
157	C₄×D₄₃	—	4	2	no	—
158	C₄×D₄₃	—	4	2	no	—
159	C₄×D₄₃	—	8	4	no	—
160	Dic₈₆	—	4	2	no	—
161	Dic₈₆	—	4	2	no	—
162	Dic₈₆	—	4	2	no	—
163	C₈×C₄₃	—	2	1	no	—
164	C₈×C₄₃	—	2	1	no	—
165	C₈×C₄₃	—	4	4	no	—
166	C₈×C₄₃	C₂×D₈	1	2	yes	—
167	C₈×C₄₃	C₂×D₈	1	2	yes	—
168	C₄₃⋊C₈	—	4	2	no	—
169	#19189	—	43	1	no	maximal, Sylow
170	C₂²×C₄₃	—	2	1	no	—
171	C₂²×C₄₃	—	2	1	no	—
172	C₂²×C₄₃	#459	1	2	yes	—
173	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
174	C₂×D₄₃	—	4	2	no	—
175	C₂×D₄₃	—	4	2	no	—
176	C₂×D₄₃	—	4	2	no	—
177	C₂×D₄₃	—	8	2	no	—
178	C₂×D₄₃	—	8	2	no	—
179	C₂×D₄₃	—	8	2	no	—
180	C₄×C₄₃	—	2	1	no	—
181	C₄×C₄₃	—	2	1	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	#459	1	2	yes	—
184	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₄₃	#577	1	2	yes	—
187	Dic₄₃	—	4	2	no	—
188	Dic₄₃	—	8	2	no	—
189	#3245	—	43	2	no	—
190	#3147	—	43	1	no	—
191	#2931	—	43	1	no	—
192	#1689	—	43	2	no	—
193	#1687	—	43	2	no	—
194	#1665	—	43	4	no	—
195	C₂×#334	D₄₃	1	1	yes	characteristic, complemented
196	#1547	—	43	2	no	—
197	C₂×C₄₃	—	2	2	no	—
198	C₂×C₄₃	C₂×#459	1	1	yes	characteristic
199	C₂×C₄₃	#3060	1	1	yes	characteristic
200	C₂×C₄₃	#3067	1	1	yes	characteristic
201	D₄₃	—	8	2	no	—
202	D₄₃	—	16	2	no	—
203	C₂×#135	—	86	1	no	—
204	C₂×#135	—	86	1	no	—
205	C₂×#131	—	86	1	no	—
206	#532	—	43	1	no	—
207	#532	—	43	2	no	—
208	#495	—	86	1	no	—
209	#495	—	86	1	no	—
210	#494	—	43	1	no	—
211	#493	—	43	1	no	—
212	#492	—	43	2	no	—
213	#474	—	43	4	no	—
214	#474	—	86	2	no	—
215	#466	—	86	4	no	—
216	#459	—	43	4	no	—
217	#459	—	86	2	no	—
218	#436	—	43	2	no	—
219	#436	—	86	1	no	—
220	C₈×D₄	—	43	2	no	—
221	C₈×D₄	—	86	1	no	—
222	#423	—	43	1	no	—
223	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
224	#408	—	86	1	no	—
225	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
226	#334	C₂×D₄₃	1	2	yes	—
227	#334	C₂×D₄₃	1	2	yes	—
228	#332	—	43	4	no	—
229	#331	—	43	4	no	—
230	C₄₃	#19189	1	1	yes	characteristic, complemented, Sylow
231	C₂×#41	—	86	1	no	—
232	C₂×#41	—	86	2	no	—
233	#135	—	172	4	no	—
234	#135	—	172	4	no	—
235	C₂×Q₁₆	—	86	1	no	—
236	C₂×Q₁₆	—	86	1	no	—
237	C₂×SD₁₆	—	86	2	no	—
238	C₂×SD₁₆	—	86	2	no	—
239	C₂×D₈	—	86	4	no	—
240	C₂×D₈	—	86	1	no	—
241	C₂×D₈	—	86	1	no	—
242	#131	—	172	4	no	—
243	C₂×M₁₆	—	86	1	no	—
244	C₂×M₁₆	—	86	2	no	—
245	C₂²×C₈	—	2	2	no	—
246	C₂²×C₈	—	86	2	no	—
247	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
248	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
249	C₄⋊Q₈	—	43	1	no	—
250	C₄⋊Q₈	—	43	2	no	—
251	#127	—	43	1	no	—
252	#127	—	43	2	no	—
253	#124	—	86	2	no	—
254	#124	—	86	1	no	—
255	#121	—	43	2	no	—
256	#121	—	86	1	no	—
257	#121	—	86	2	no	—
258	#121	—	86	2	no	—
259	C₄×D₄	—	43	2	no	—
260	C₄×D₄	—	86	1	no	—
261	C₄×D₄	—	86	2	no	—
262	C₄×D₄	—	86	2	no	—
263	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
264	#107	—	86	4	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	43	2	no	—
267	C₄⋊C₈	—	86	1	no	—
268	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
269	#102	—	86	4	no	—
270	#102	—	86	4	no	—
271	C₂²⋊C₈	—	86	2	no	—
272	C₂²⋊C₈	—	86	2	no	—
273	C₄×C₈	D₁₇₂	1	1	yes	characteristic
274	C₄×C₈	D₁₇₂	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
277	#41	—	172	4	no	—
278	#41	—	172	4	no	—
279	#41	—	172	2	no	—
280	#41	—	172	2	no	—
281	C₂×Q₈	—	86	1	no	—
282	C₂×Q₈	—	86	2	no	—
283	C₂×D₄	—	43	2	no	—
284	C₂×D₄	—	86	2	no	—
285	C₂×D₄	—	86	1	no	—
286	C₂×D₄	—	86	1	no	—
287	C₂×D₄	—	86	2	no	—
288	C₂×D₄	—	86	2	no	—
289	C₂×D₄	—	86	2	no	—
290	C₂×D₄	—	172	2	no	—
291	C₂²×C₄	—	86	2	no	—
292	C₂²×C₄	—	172	2	no	—
293	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
294	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
295	C₂²×C₄	#10621902	1	1	yes	characteristic
296	Q₁₆	—	172	2	no	—
297	Q₁₆	—	172	2	no	—
298	SD₁₆	—	172	4	no	—
299	SD₁₆	—	172	4	no	—
300	D₈	—	344	4	no	—
301	D₈	—	172	2	no	—
302	D₈	—	172	2	no	—
303	M₁₆	—	172	2	no	—
304	M₁₆	—	172	4	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	4	no	—
307	C₂×C₈	—	8	2	no	—
308	C₂×C₈	—	86	2	no	—
309	C₂×C₈	—	172	4	no	—
310	C₂×C₈	D₃₄₄	1	4	yes	complemented
311	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
312	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
313	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
314	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
315	C₄⋊C₄	—	43	2	no	—
316	C₄⋊C₄	—	86	1	no	—
317	C₄⋊C₄	—	86	2	no	—
318	C₄⋊C₄	—	86	2	no	—
319	C₂²⋊C₄	—	86	2	no	—
320	C₂²⋊C₄	—	86	2	no	—
321	C₂²⋊C₄	—	172	2	no	—
322	C₂²⋊C₄	—	172	2	no	—
323	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
324	C₄²	D₄×D₄₃	1	1	yes	characteristic
325	C₄²	#10621902	1	1	yes	characteristic
326	C₄²	C₂³×D₄₃	1	1	yes	characteristic
327	C₂³	—	86	2	no	—
328	C₂³	—	172	2	no	—
329	C₂³	#13063300	1	1	yes	characteristic
330	Q₈	—	172	2	no	—
331	Q₈	—	172	2	no	—
332	Q₈	—	172	2	no	—
333	D₄	—	86	4	no	—
334	D₄	—	344	2	no	—
335	D₄	—	344	4	no	—
336	D₄	—	172	2	no	—
337	D₄	—	172	4	no	—
338	D₄	—	172	2	no	—
339	D₄	—	172	2	no	—
340	D₄	—	172	2	no	—
341	D₄	—	172	2	no	—
342	D₄	—	172	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	86	2	no	—
346	C₂×C₄	—	344	4	no	—
347	C₂×C₄	—	172	2	no	—
348	C₂×C₄	—	172	2	no	—
349	C₂×C₄	—	172	2	no	—
350	C₂×C₄	#13063300	1	1	yes	characteristic
351	C₂×C₄	#13063300	1	1	yes	characteristic
352	C₂×C₄	#13063300	1	1	yes	characteristic
353	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
354	C₂×C₄	#13063353	1	2	yes	—
355	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
357	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	4	4	no	—
361	C₈	—	172	2	no	—
362	C₈	C₂×D₃₄₄	1	2	yes	—
363	C₈	C₂×D₃₄₄	1	2	yes	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	344	2	no	—
367	C₂²	—	344	2	no	—
368	C₂²	—	344	2	no	—
369	C₂²	—	172	2	no	—
370	C₂²	—	172	2	no	—
371	C₂²	—	172	2	no	—
372	C₂²	#14583095	1	2	yes	—
373	C₂²	C₂×#13063300	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	344	2	no	—
378	C₄	—	172	2	no	—
379	C₄	#14583095	1	2	yes	—
380	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
381	C₄	C₂×#13063353	1	1	yes	characteristic
382	C₄	#14583705	1	2	yes	—
383	C₂	—	2	2	no	—
384	C₂	—	344	2	no	—
385	C₂	—	688	2	no	—
386	C₂	C₂×#14583095	1	1	yes	characteristic
387	C₂	#19144258	1	1	yes	characteristic
388	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×#334	C₂
socle	C₂²×C₄₃	C₂×#121
hypercenter	C₂×#334	D₄₃
hypocenter	C₄₃	#19189
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#334

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. #19134085	C₂	semidirect
5. #19134075	C₂	semidirect
6. #19133937	C₂	semidirect
7. #19133270	C₂	semidirect
8. C₂×C₄₃×#334	C₂	semidirect
104. C₂×C₈×C₄₃	D₈	semidirect
195. C₂×#334	D₄₃	semidirect
230. C₄₃	#19189	semidirect
310. 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
#19134085	2
#19134075	2
#19133937	2
#19133270	2
C₂×C₄₃×#334	2
#19189	43