Subgroups — #25252949

Order 11008.

490 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	#19145181	C₂	1	1	yes	maximal, characteristic, complemented
2	#19145165	C₂	1	1	yes	maximal, characteristic
3	#19136760	C₂	1	2	yes	maximal, complemented
4	#19136748	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136743	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136708	C₂	1	2	yes	maximal
7	#19136703	C₂	1	2	yes	maximal, complemented
8	#19136658	C₂	1	2	yes	maximal, complemented
9	#19136646	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136641	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583456	—	2	1	no	—
13	#14583433	—	2	1	no	—
14	C₂×Dic₃₄₄	C₂²	1	2	yes	—
15	C₂×#13063270	C₂²	1	2	yes	—
16	#14583304	C₂²	1	1	yes	characteristic
17	#14583301	C₂²	1	1	yes	characteristic
18	#14583283	C₂²	1	1	yes	characteristic
19	#14583279	C₂²	1	1	yes	characteristic
20	#14583254	—	2	1	no	—
21	#14583250	—	2	1	no	—
22	#14583140	—	2	1	no	—
23	#14583140	C₂²	1	1	yes	characteristic
24	#14583140	C₂²	1	2	yes	—
25	#14583135	C₂²	1	1	yes	characteristic
26	#14583128	—	2	1	no	—
27	C₄×Dic₁₇₂	—	2	1	no	—
28	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₇₂	C₂²	1	2	yes	—
30	C₄×#10621873	—	2	1	no	—
31	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
32	#14582727	C₂²	1	2	yes	—
33	#14582725	C₂²	1	2	yes	—
34	#14582704	C₂²	1	2	yes	—
35	#14582704	C₂²	1	2	yes	—
36	#14582704	C₂²	1	2	yes	—
37	#14582701	C₂²	1	2	yes	—
38	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
43	C₂×Dic₁₇₂	—	2	1	no	—
44	C₂×Dic₁₇₂	C₂³	1	2	yes	—
45	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
46	C₂×#10621873	—	2	1	no	—
47	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
48	#13063325	—	2	1	no	—
49	#13063325	—	2	1	no	—
50	#13063319	—	4	1	no	—
51	#13063317	—	2	1	no	—
52	#13063317	—	4	1	no	—
53	#13063313	—	2	1	no	—
54	#13063308	—	4	1	no	—
55	#13063304	—	2	1	no	—
56	#13063304	—	2	2	no	—
57	#13063304	—	4	1	no	—
58	#13063299	—	2	1	no	—
59	C₄×Dic₈₆	—	2	1	no	—
60	C₄×Dic₈₆	—	2	2	no	—
61	C₄×Dic₈₆	—	4	1	no	—
62	C₄×D₁₇₂	—	2	1	no	—
63	C₄×C₂²⋊D₄₃	—	4	1	no	—
64	#13063284	—	4	1	no	—
65	#13063282	—	4	1	no	—
66	Dic₃₄₄	—	4	2	no	—
67	#13063270	—	4	2	no	—
68	#13063259	—	2	1	no	—
69	#13063259	C₂³	1	1	yes	characteristic
70	#13063259	C₂³	1	1	yes	characteristic
71	#13063259	C₂³	1	2	yes	—
72	#13063257	—	2	1	no	—
73	#13063250	—	2	1	no	—
74	#13063250	—	2	2	no	—
75	#13063250	—	2	1	no	—
76	#13063250	—	2	2	no	—
77	#13063247	—	2	1	no	—
78	#13063247	—	2	1	no	—
79	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
82	M₃₂×C₄₃	—	4	2	no	—
83	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
84	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
85	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
86	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
87	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₃	—	4	1	no	—
93	C₂×D₁₇₂	—	2	1	no	—
94	C₂×C₄×D₄₃	—	4	1	no	—
95	C₂×Dic₈₆	—	2	1	no	—
96	C₂×Dic₈₆	—	2	2	no	—
97	C₂×Dic₈₆	—	4	1	no	—
98	C₂×C₈×C₄₃	—	2	1	no	—
99	C₂×C₈×C₄₃	—	2	1	no	—
100	C₂×C₈×C₄₃	—	4	1	no	—
101	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
105	C₄²×C₄₃	D₈	1	1	yes	characteristic
106	C₄²×C₄₃	D₈	1	1	yes	characteristic
107	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂²⋊Dic₄₃	—	4	2	no	—
110	C₂²⋊Dic₄₃	—	4	1	no	—
111	(C₂×D₄₃)⋊C₄	—	4	1	no	—
112	(C₂×D₄₃)⋊C₄	—	4	1	no	—
113	(C₂×D₄₃)⋊C₄	—	4	1	no	—
114	(C₂×D₄₃)⋊C₄	—	4	1	no	—
115	C₄⋊Dic₄₃	—	2	1	no	—
116	C₄⋊Dic₄₃	—	2	1	no	—
117	C₄⋊Dic₄₃	—	2	2	no	—
118	C₄⋊Dic₄₃	—	4	1	no	—
119	C₄⋊Dic₄₃	—	4	1	no	—
120	C₄⋊Dic₄₃	—	4	1	no	—
121	Dic₄₃⋊C₄	—	4	1	no	—
122	Dic₄₃⋊C₄	—	4	1	no	—
123	Dic₄₃⋊C₄	—	4	1	no	—
124	Dic₄₃⋊C₄	—	4	2	no	—
125	Dic₄₃⋊C₄	—	4	1	no	—
126	Dic₄₃⋊C₄	—	4	2	no	—
127	C₄×Dic₄₃	—	4	1	no	—
128	C₄×Dic₄₃	—	4	2	no	—
129	Dic₁₇₂	—	2	2	no	—
130	Dic₁₇₂	—	2	1	no	—
131	Dic₁₇₂	—	2	2	no	—
132	Dic₁₇₂	—	2	1	no	—
133	Dic₁₇₂	—	4	2	no	—
134	D₃₄₄	—	2	1	no	—
135	D₃₄₄	—	2	1	no	—
136	#10621873	—	4	2	no	—
137	C₁₆×C₄₃	—	2	2	no	—
138	C₁₆×C₄₃	—	2	2	no	—
139	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
140	C₂²×D₄₃	—	4	1	no	—
141	C₂×C₄×C₄₃	—	2	1	no	—
142	C₂×C₄×C₄₃	—	2	1	no	—
143	C₂×C₄×C₄₃	—	2	2	no	—
144	C₂×C₄×C₄₃	—	4	1	no	—
145	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₄₃	—	8	2	no	—
152	C₂×Dic₄₃	—	4	1	no	—
153	C₂×Dic₄₃	—	4	2	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	4	1	no	—
156	C₂×Dic₄₃	—	4	1	no	—
157	D₁₇₂	—	4	1	no	—
158	D₁₇₂	—	4	1	no	—
159	C₄×D₄₃	—	8	2	no	—
160	Dic₈₆	—	4	2	no	—
161	Dic₈₆	—	4	1	no	—
162	Dic₈₆	—	4	1	no	—
163	Dic₈₆	—	4	2	no	—
164	Dic₈₆	—	8	2	no	—
165	C₈×C₄₃	—	2	1	no	—
166	C₈×C₄₃	—	2	1	no	—
167	C₈×C₄₃	—	2	1	no	—
168	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
169	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
170	#33818	—	43	1	no	maximal, Sylow
171	C₂²×C₄₃	—	2	1	no	—
172	C₂²×C₄₃	—	2	1	no	—
173	C₂²×C₄₃	—	2	1	no	—
174	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₄₃	—	8	1	no	—
176	C₂×D₄₃	—	8	1	no	—
177	C₂×D₄₃	—	8	1	no	—
178	C₄×C₄₃	—	2	1	no	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	2	1	no	—
181	C₄×C₄₃	—	2	1	no	—
182	C₄×C₄₃	#510	1	2	yes	—
183	C₄×C₄₃	#537	1	1	yes	characteristic
184	C₄×C₄₃	#575	1	1	yes	characteristic
185	Dic₄₃	—	8	1	no	—
186	Dic₄₃	—	8	2	no	—
187	Dic₄₃	—	8	1	no	—
188	Dic₄₃	—	8	1	no	—
189	Dic₄₃	—	8	2	no	—
190	#3170	—	43	1	no	—
191	#3168	—	43	1	no	—
192	#2229	—	43	2	no	—
193	#2227	—	43	1	no	—
194	#2226	—	43	1	no	—
195	#2220	—	43	2	no	—
196	#2219	—	43	2	no	—
197	#2211	—	43	2	no	—
198	#2209	—	43	1	no	—
199	#2208	—	43	1	no	—
200	#2150	D₄₃	1	1	yes	characteristic, complemented
201	C₂×C₄₃	—	4	1	no	—
202	C₂×C₄₃	#3245	1	1	yes	characteristic
203	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
204	C₂×C₄₃	#3414	1	1	yes	characteristic
205	D₄₃	—	16	1	no	—
206	#533	—	86	1	no	—
207	#529	—	86	1	no	—
208	C₂×Q₃₂	—	43	2	no	—
209	C₂×SD₃₂	—	43	2	no	—
210	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
211	#500	—	43	1	no	—
212	#499	—	43	1	no	—
213	#495	—	43	1	no	—
214	#494	—	43	1	no	—
215	#488	—	86	1	no	—
216	#487	—	86	1	no	—
217	#467	—	43	1	no	—
218	#467	—	43	2	no	—
219	#467	—	86	1	no	—
220	#466	—	43	1	no	—
221	#465	—	86	1	no	—
222	C₄×Q₁₆	—	43	1	no	—
223	C₄×Q₁₆	—	43	2	no	—
224	C₄×Q₁₆	—	86	1	no	—
225	C₄×SD₁₆	—	86	1	no	—
226	C₄×D₈	—	43	1	no	—
227	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
228	#367	—	43	2	no	—
229	#366	—	43	2	no	—
230	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
231	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
232	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
233	#358	—	43	2	no	—
234	#358	—	43	2	no	—
235	#358	—	43	2	no	—
236	#357	—	43	2	no	—
237	C₄₃	#33818	1	1	yes	characteristic, complemented, Sylow
238	C₂×Q₁₆	—	43	2	no	—
239	C₂×Q₁₆	—	43	1	no	—
240	C₂×Q₁₆	—	86	1	no	—
241	C₂×SD₁₆	—	86	1	no	—
242	C₂×D₈	—	43	1	no	—
243	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
244	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
245	C₄⋊Q₈	—	86	1	no	—
246	C₄⋊Q₈	—	86	1	no	—
247	#126	—	172	1	no	—
248	#125	—	86	1	no	—
249	#125	—	172	1	no	—
250	#124	—	86	1	no	—
251	#123	—	172	1	no	—
252	C₂²⋊Q₈	—	86	1	no	—
253	C₂²⋊Q₈	—	86	2	no	—
254	C₂²⋊Q₈	—	172	1	no	—
255	#121	—	86	1	no	—
256	C₄×Q₈	—	86	1	no	—
257	C₄×Q₈	—	86	2	no	—
258	C₄×Q₈	—	172	1	no	—
259	C₄×D₄	—	86	1	no	—
260	C₄×D₄	—	172	1	no	—
261	#117	—	172	1	no	—
262	#117	—	172	1	no	—
263	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
264	Q₃₂	—	172	2	no	—
265	SD₃₂	—	172	2	no	—
266	M₃₂	—	4	2	no	—
267	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
268	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
269	#107	—	43	2	no	—
270	#107	—	43	1	no	—
271	#107	—	43	1	no	—
272	#107	—	86	1	no	—
273	#106	—	86	1	no	—
274	#103	—	86	1	no	—
275	#103	—	86	1	no	—
276	#103	—	86	2	no	—
277	#103	—	86	2	no	—
278	#102	—	86	1	no	—
279	#102	—	86	1	no	—
280	C₄×C₈	D₁₇₂	1	1	yes	characteristic
281	C₄×C₈	D₁₇₂	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
284	C₂×Q₈	—	86	1	no	—
285	C₂×Q₈	—	86	2	no	—
286	C₂×Q₈	—	172	1	no	—
287	C₂×D₄	—	86	1	no	—
288	C₂×D₄	—	172	1	no	—
289	C₂²×C₄	—	172	1	no	—
290	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
291	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
293	Q₁₆	—	86	2	no	—
294	Q₁₆	—	86	1	no	—
295	Q₁₆	—	86	2	no	—
296	Q₁₆	—	86	1	no	—
297	Q₁₆	—	172	2	no	—
298	SD₁₆	—	172	2	no	—
299	D₈	—	86	1	no	—
300	D₈	—	86	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
305	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
306	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
308	C₄⋊C₄	—	86	1	no	—
309	C₄⋊C₄	—	86	1	no	—
310	C₄⋊C₄	—	86	2	no	—
311	C₄⋊C₄	—	172	1	no	—
312	C₄⋊C₄	—	172	1	no	—
313	C₄⋊C₄	—	172	1	no	—
314	C₄⋊C₄	—	172	1	no	—
315	C₄⋊C₄	—	172	1	no	—
316	C₄⋊C₄	—	172	2	no	—
317	C₄⋊C₄	—	172	1	no	—
318	C₄⋊C₄	—	172	2	no	—
319	C₄⋊C₄	—	172	1	no	—
320	C₂²⋊C₄	—	172	2	no	—
321	C₂²⋊C₄	—	172	1	no	—
322	C₂²⋊C₄	—	172	1	no	—
323	C₂²⋊C₄	—	172	1	no	—
324	C₂²⋊C₄	—	172	1	no	—
325	C₂²⋊C₄	—	172	1	no	—
326	C₄²	—	172	1	no	—
327	C₄²	—	172	2	no	—
328	C₄²	D₃₄₄	1	1	yes	characteristic
329	C₄²	D₃₄₄	1	1	yes	characteristic
330	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
331	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
332	C₁₆	—	2	2	no	—
333	C₁₆	—	2	2	no	—
334	C₂³	—	172	1	no	—
335	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
336	Q₈	—	344	2	no	—
337	Q₈	—	172	1	no	—
338	Q₈	—	172	2	no	—
339	Q₈	—	172	2	no	—
340	Q₈	—	172	1	no	—
341	D₄	—	344	2	no	—
342	D₄	—	172	1	no	—
343	D₄	—	172	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	344	2	no	—
349	C₂×C₄	—	172	1	no	—
350	C₂×C₄	—	172	1	no	—
351	C₂×C₄	—	172	2	no	—
352	C₂×C₄	—	172	2	no	—
353	C₂×C₄	—	172	1	no	—
354	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
355	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
356	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
357	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
358	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
359	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	#13063370	1	1	yes	characteristic
364	C₈	#13063373	1	1	yes	characteristic
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	344	1	no	—
369	C₂²	—	344	1	no	—
370	C₂²	—	344	1	no	—
371	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	344	1	no	—
377	C₄	—	344	2	no	—
378	C₄	—	344	1	no	—
379	C₄	—	344	1	no	—
380	C₄	—	344	2	no	—
381	C₄	#14583342	1	2	yes	—
382	C₄	#14583480	1	1	yes	characteristic
383	C₄	#14583699	1	1	yes	characteristic
384	C₂	—	4	1	no	—
385	C₂	—	688	1	no	—
386	C₂	#19145905	1	1	yes	characteristic
387	C₂	C₂×#14583342	1	1	yes	characteristic
388	C₂	#19147731	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2150	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#14583135	C₂²
hypercenter	#2150	D₄₃
hypocenter	C₄₃	#33818
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. #19145181	C₂	semidirect
3. #19136760	C₂	semidirect
4. #19136748	C₂	semidirect
5. #19136743	C₂	semidirect
7. #19136703	C₂	semidirect
8. #19136658	C₂	semidirect
9. #19136646	C₂	semidirect
10. #19136641	C₂	semidirect
11. C₄₃×#2150	C₂	semidirect
200. #2150	D₄₃	semidirect
237. C₄₃	#33818	semidirect

Maximal subgroups

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

subgroup	index
#19145181	2
#19145165	2
#19136760	2
#19136748	2
#19136743	2
#19136708	2
#19136703	2
#19136658	2
#19136646	2
#19136641	2
C₄₃×#2150	2
#33818	43