Subgroups — #25253430

Order 11008.

622 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19146987	C₂	1	1	yes	maximal, characteristic, complemented
2	#19146270	C₂	1	1	yes	maximal, characteristic
3	#19136779	C₂	1	2	yes	maximal, complemented
4	#19136739	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136690	C₂	1	2	yes	maximal, complemented
6	#19136632	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136525	C₂	1	2	yes	maximal, complemented
8	C₄×#13063270	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₆×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#357×D₄₃	C₂	1	2	yes	maximal, complemented
11	C₄₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₄₃	C₂²	1	2	yes	—
13	#14583610	—	2	1	no	—
14	#14583534	—	2	1	no	—
15	C₂×#13063270	C₂²	1	1	yes	characteristic
16	C₂×#13063269	C₂²	1	2	yes	complemented
17	C₂×C₁₆×D₄₃	C₂²	1	2	yes	complemented
18	#14583236	—	2	2	no	—
19	#14583191	—	2	1	no	—
20	#14583141	C₂²	1	2	yes	complemented
21	#14583132	C₂²	1	2	yes	—
22	#14583091	—	2	1	no	—
23	#14583031	—	2	2	no	—
24	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic, complemented
25	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
26	C₈×D₁₇₂	C₂²	1	1	yes	characteristic
27	#107×D₄₃	C₂²	1	2	yes	complemented
28	#14582727	C₂²	1	1	yes	characteristic, complemented
29	#14582726	C₂²	1	2	yes	complemented
30	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
31	#14582704	C₂²	1	1	yes	characteristic, complemented
32	#14582701	C₂²	1	1	yes	characteristic
33	#14582700	C₂²	1	2	yes	complemented
34	#14582699	C₂²	1	2	yes	—
35	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	complemented
36	C₄₃×#500	C₂²	1	1	yes	characteristic, complemented
37	C₄₃×#493	C₂²	1	1	yes	characteristic
38	C₄₃×#357	C₂²	1	2	yes	complemented
39	C₄₃×#357	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
41	C₂×D₄×D₄₃	—	2	2	no	—
42	C₂×D₄×D₄₃	—	4	1	no	—
43	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
44	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
45	C₂×#10621882	—	2	2	no	—
46	D₈×D₄₃	—	2	4	no	—
47	D₈×D₄₃	—	4	2	no	—
48	C₂×C₈×D₄₃	C₂³	1	2	yes	—
49	#13063316	—	4	1	no	—
50	#13063305	—	2	2	no	—
51	#13063301	—	4	2	no	—
52	#13063296	—	2	2	no	—
53	#13063296	—	4	1	no	—
54	#13063293	—	4	2	no	—
55	C₄×Dic₈₆	—	2	1	no	—
56	C₄×D₁₇₂	—	2	1	no	—
57	C₄×D₁₇₂	D₄	1	1	yes	characteristic
58	#13063287	—	4	1	no	—
59	C₄⋊C₄×D₄₃	—	2	2	no	—
60	#13063270	—	2	2	no	—
61	#13063269	—	4	2	no	—
62	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
63	C₁₆×D₄₃	—	4	2	no	—
64	#13063259	C₂³	1	1	yes	characteristic
65	#13063259	C₂³	1	1	yes	characteristic
66	#13063258	—	2	2	no	—
67	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
68	#13063250	—	2	1	no	—
69	#13063248	—	2	2	no	—
70	#13063247	—	2	1	no	—
71	#13063246	—	2	2	no	—
72	(C₂×D₄₃)⋊C₈	D₄	1	2	yes	—
73	C₂×D₈×C₄₃	—	2	1	no	—
74	C₂×D₈×C₄₃	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₄₃	—	2	1	no	—
76	#127×C₄₃	—	2	1	no	—
77	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
79	#107×C₄₃	—	2	1	no	—
80	#107×C₄₃	C₂³	1	2	yes	—
81	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂³×D₄₃	—	4	2	no	—
83	C₂×Q₈×C₄₃	—	4	1	no	—
84	C₂×D₄×C₄₃	—	2	2	no	—
85	C₂×D₄×C₄₃	—	4	1	no	—
86	C₂×D₄×C₄₃	—	4	1	no	—
87	C₂×C₂²⋊D₄₃	—	4	2	no	—
88	C₂×C₂²⋊D₄₃	—	4	2	no	—
89	D₄×D₄₃	—	4	4	no	—
90	D₄×D₄₃	—	8	2	no	—
91	D₄×D₄₃	—	8	4	no	—
92	C₂×D₁₇₂	—	2	1	no	—
93	C₂×D₁₇₂	D₈	1	1	yes	characteristic
94	C₂×C₄×D₄₃	—	4	1	no	—
95	C₂×C₄×D₄₃	—	4	2	no	—
96	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
97	C₂×Dic₈₆	—	2	1	no	—
98	D₈×C₄₃	—	2	2	no	—
99	D₈×C₄₃	—	2	2	no	—
100	D₈×C₄₃	—	8	1	no	—
101	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	4	1	no	—
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₄₃	—	4	2	no	—
107	#10621882	—	4	4	no	—
108	(C₂×D₄₃)⋊C₄	—	4	1	no	—
109	(C₂×D₄₃)⋊C₄	—	4	2	no	—
110	(C₂×D₄₃)⋊C₄	D₈	1	2	yes	—
111	C₄⋊Dic₄₃	—	2	1	no	—
112	C₄⋊Dic₄₃	—	2	1	no	—
113	C₄⋊Dic₄₃	—	4	2	no	—
114	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
115	Dic₄₃⋊C₄	—	4	1	no	—
116	Dic₄₃⋊C₄	—	4	2	no	—
117	C₄×Dic₄₃	—	4	1	no	—
118	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
119	Dic₁₇₂	—	2	1	no	—
120	Dic₁₇₂	—	2	1	no	—
121	D₃₄₄	—	2	1	no	—
122	D₃₄₄	#41	1	2	yes	—
123	C₈×D₄₃	—	2	2	no	—
124	C₈×D₄₃	—	2	2	no	—
125	C₁₆×C₄₃	—	2	1	no	—
126	C₁₆×C₄₃	—	2	1	no	—
127	C₄₃⋊C₁₆	—	2	2	no	—
128	C₂³×C₄₃	—	4	2	no	—
129	C₂²×D₄₃	—	4	1	no	—
130	C₂²×D₄₃	—	8	4	no	—
131	C₂²×D₄₃	—	8	4	no	—
132	C₂²×D₄₃	—	16	2	no	—
133	C₂²×D₄₃	C₂×D₈	1	2	yes	—
134	Q₈×C₄₃	—	16	1	no	—
135	D₄×C₄₃	—	4	2	no	—
136	D₄×C₄₃	—	4	2	no	—
137	D₄×C₄₃	—	8	2	no	—
138	D₄×C₄₃	—	16	1	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂²⋊D₄₃	—	8	4	no	—
144	C₂²⋊D₄₃	—	8	4	no	—
145	C₂×Dic₄₃	—	4	1	no	—
146	C₂×Dic₄₃	—	4	1	no	—
147	C₂×Dic₄₃	—	4	1	no	—
148	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
149	D₁₇₂	—	2	2	no	—
150	D₁₇₂	—	4	1	no	—
151	D₁₇₂	SD₃₂	1	4	yes	complemented
152	C₄×D₄₃	—	2	2	no	—
153	C₄×D₄₃	—	2	2	no	—
154	C₄×D₄₃	—	8	2	no	—
155	C₄×D₄₃	—	16	2	no	—
156	Dic₈₆	—	4	1	no	—
157	Dic₈₆	—	4	1	no	—
158	C₈×C₄₃	—	2	1	no	—
159	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
160	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
161	C₄₃⋊C₈	—	2	2	no	—
162	#33862	—	43	1	no	maximal, Sylow
163	C₂²×C₄₃	—	8	2	no	—
164	C₂²×C₄₃	—	8	2	no	—
165	C₂²×C₄₃	—	8	2	no	—
166	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₄₃	—	2	2	no	—
168	C₂×D₄₃	—	2	2	no	—
169	C₂×D₄₃	—	4	2	no	—
170	C₂×D₄₃	—	8	1	no	—
171	C₂×D₄₃	—	8	1	no	—
172	C₂×D₄₃	—	16	8	no	—
173	C₂×D₄₃	C₂×SD₃₂	1	4	yes	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	8	2	no	—
176	C₄×C₄₃	C₂×SD₃₂	1	2	yes	—
177	C₄×C₄₃	#546	1	1	yes	characteristic
178	C₄×C₄₃	#577	1	1	yes	characteristic
179	Dic₄₃	—	2	2	no	—
180	Dic₄₃	—	8	1	no	—
181	Dic₄₃	—	8	1	no	—
182	Dic₄₃	—	8	1	no	—
183	#3338	—	43	1	no	—
184	D₄×D₈	—	43	1	no	—
185	#2240	D₄₃	1	1	yes	characteristic, complemented
186	#2233	—	43	2	no	—
187	#2226	—	43	1	no	—
188	#2217	—	43	2	no	—
189	#2207	—	43	1	no	—
190	#2187	—	43	2	no	—
191	C₄×SD₃₂	—	43	1	no	—
192	D₄×C₁₆	—	43	1	no	—
193	C₂×#357	—	43	2	no	—
194	C₂×C₄₃	—	16	2	no	—
195	C₂×C₄₃	#3294	1	1	yes	characteristic
196	C₂×C₄₃	C₂²×SD₃₂	1	1	yes	characteristic
197	C₂×C₄₃	#3415	1	1	yes	characteristic
198	D₄₃	—	2	4	no	—
199	D₄₃	—	8	2	no	—
200	C₂²×D₈	—	43	2	no	—
201	#554	—	86	1	no	—
202	D₄²	—	86	1	no	—
203	C₂×SD₃₂	—	43	2	no	—
204	C₂×SD₃₂	—	43	1	no	—
205	C₂²×C₁₆	—	43	2	no	—
206	#500	C₂×D₄₃	1	1	yes	characteristic, complemented
207	#493	C₂×D₄₃	1	1	yes	characteristic
208	#484	—	86	2	no	—
209	#477	—	86	1	no	—
210	#467	—	43	2	no	—
211	#466	—	43	2	no	—
212	#459	—	86	1	no	—
213	#447	—	86	2	no	—
214	C₄×Q₁₆	—	43	1	no	—
215	C₄×D₈	—	43	1	no	—
216	C₈×D₄	—	43	1	no	—
217	C₂×#107	—	43	2	no	—
218	#367	—	43	1	no	—
219	#367	—	43	2	no	—
220	C₄⋊C₁₆	—	43	1	no	—
221	#358	—	43	1	no	—
222	#357	—	43	2	no	—
223	#357	—	43	1	no	—
224	#357	—	43	2	no	—
225	#357	C₂×D₄₃	1	2	yes	complemented
226	#357	C₂×D₄₃	1	2	yes	complemented
227	C₂²⋊C₁₆	—	43	2	no	—
228	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
229	C₄₃	#33862	1	1	yes	characteristic, complemented, Sylow
230	C₂²×D₄	—	86	2	no	—
231	C₂²×D₄	—	172	1	no	—
232	C₂×Q₁₆	—	43	1	no	—
233	C₂×D₈	—	2	1	no	—
234	C₂×D₈	—	43	1	no	—
235	C₂×D₈	—	86	2	no	—
236	C₂×D₈	—	86	4	no	—
237	C₂×D₈	—	172	2	no	—
238	C₂×D₈	C₂²×D₄₃	1	2	yes	—
239	C₂²×C₈	—	43	2	no	—
240	C₄⋊Q₈	—	2	1	no	—
241	#127	—	2	1	no	—
242	#125	—	172	1	no	—
243	C₂²⋊Q₈	—	86	2	no	—
244	C₂²⋊Q₈	—	172	2	no	—
245	#121	—	86	2	no	—
246	#121	—	172	1	no	—
247	#120	—	172	2	no	—
248	C₄×Q₈	—	86	1	no	—
249	C₄×D₄	—	43	1	no	—
250	C₄×D₄	—	86	1	no	—
251	C₄×D₄	—	172	1	no	—
252	C₂×C₄⋊C₄	—	86	2	no	—
253	SD₃₂	—	86	2	no	—
254	SD₃₂	—	172	2	no	—
255	C₂×C₁₆	—	43	2	no	—
256	C₂×C₁₆	—	172	2	no	—
257	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
258	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
259	#107	—	2	1	no	—
260	#107	—	43	1	no	—
261	#107	—	43	1	no	—
262	#107	—	86	2	no	—
263	#107	C₂²×D₄₃	1	2	yes	—
264	C₄⋊C₈	—	43	1	no	—
265	#103	—	86	2	no	—
266	#103	—	86	1	no	—
267	#102	—	86	2	no	—
268	#102	—	86	1	no	—
269	C₂²⋊C₈	—	43	2	no	—
270	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
271	C₂⁴	—	172	2	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	86	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	43	1	no	—
278	C₂×D₄	—	86	1	no	—
279	C₂×D₄	—	344	2	no	—
280	C₂×D₄	—	344	4	no	—
281	C₂×D₄	—	172	2	no	—
282	C₂×D₄	—	172	4	no	—
283	C₂×D₄	—	172	2	no	—
284	C₂²×C₄	—	43	2	no	—
285	C₂²×C₄	—	172	2	no	—
286	C₂²×C₄	—	172	1	no	—
287	Q₁₆	—	86	1	no	—
288	Q₁₆	—	86	1	no	—
289	D₈	—	2	2	no	—
290	D₈	—	2	2	no	—
291	D₈	—	8	1	no	—
292	D₈	—	43	2	no	—
293	D₈	—	86	1	no	—
294	D₈	—	172	4	no	—
295	C₂×C₈	—	43	2	no	—
296	C₂×C₈	—	86	2	no	—
297	C₂×C₈	—	86	2	no	—
298	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	43	1	no	—
303	C₄⋊C₄	—	86	1	no	—
304	C₄⋊C₄	—	86	1	no	—
305	C₄⋊C₄	—	172	2	no	—
306	C₄⋊C₄	—	172	2	no	—
307	C₄⋊C₄	—	172	1	no	—
308	C₂²⋊C₄	—	43	2	no	—
309	C₂²⋊C₄	—	172	2	no	—
310	C₂²⋊C₄	—	172	2	no	—
311	C₂²⋊C₄	—	172	1	no	—
312	C₄²	—	172	1	no	—
313	C₄²	D₄×D₄₃	1	1	yes	characteristic
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	86	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	43	2	no	—
319	C₂³	—	344	4	no	—
320	C₂³	—	344	4	no	—
321	C₂³	—	172	1	no	—
322	C₂³	—	688	2	no	—
323	Q₈	—	16	1	no	—
324	Q₈	—	172	1	no	—
325	Q₈	—	172	1	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	16	1	no	—
330	D₄	—	43	4	no	—
331	D₄	—	86	2	no	—
332	D₄	—	344	4	no	—
333	D₄	—	344	4	no	—
334	D₄	—	172	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	43	2	no	—
337	C₂×C₄	—	86	2	no	—
338	C₂×C₄	—	86	2	no	—
339	C₂×C₄	—	344	2	no	—
340	C₂×C₄	—	172	1	no	—
341	C₂×C₄	—	172	1	no	—
342	C₂×C₄	—	172	1	no	—
343	C₂×C₄	—	688	2	no	—
344	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
345	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	86	2	no	—
349	C₈	#41×D₄₃	1	1	yes	characteristic
350	C₈	#13063371	1	1	yes	characteristic
351	C₂²	—	8	2	no	—
352	C₂²	—	8	2	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	43	4	no	—
355	C₂²	—	86	2	no	—
356	C₂²	—	86	2	no	—
357	C₂²	—	344	1	no	—
358	C₂²	—	344	1	no	—
359	C₂²	—	172	2	no	—
360	C₂²	—	688	8	no	—
361	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	8	2	no	—
364	C₄	—	86	2	no	—
365	C₄	—	344	1	no	—
366	C₄	—	344	1	no	—
367	C₄	—	344	1	no	—
368	C₄	SD₃₂×D₄₃	1	2	yes	—
369	C₄	#14583542	1	1	yes	characteristic
370	C₄	#14583705	1	1	yes	characteristic
371	C₂	—	16	2	no	—
372	C₂	—	86	4	no	—
373	C₂	—	344	2	no	—
374	C₂	#19146407	1	1	yes	characteristic
375	C₂	C₂×SD₃₂×D₄₃	1	1	yes	characteristic
376	C₂	#19147732	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2240	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19146270	C₂
hypercenter	#2240	D₄₃
hypocenter	C₄₃	#33862
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #19146987	C₂	semidirect
3. #19136779	C₂	semidirect
4. #19136739	C₂	semidirect
5. #19136690	C₂	semidirect
6. #19136632	C₂	semidirect
7. #19136525	C₂	semidirect
8. C₄×#13063270	C₂	semidirect
9. C₁₆×D₁₇₂	C₂	semidirect
10. #357×D₄₃	C₂	semidirect
11. C₄₃×#2240	C₂	semidirect
16. C₂×#13063269	C₂²	semidirect
17. C₂×C₁₆×D₄₃	C₂²	semidirect
20. #14583141	C₂²	semidirect
24. C₄×Dic₁₇₂	C₂²	semidirect
27. #107×D₄₃	C₂²	semidirect
28. #14582727	C₂²	semidirect
29. #14582726	C₂²	semidirect
30. C₄⋊(C₄₃⋊C₁₆)	C₂²	semidirect
31. #14582704	C₂²	semidirect
33. #14582700	C₂²	semidirect
35. (C₂×D₄₃)⋊C₁₆	C₂²	semidirect
36. C₄₃×#500	C₂²	semidirect
38. C₄₃×#357	C₂²	semidirect
39. C₄₃×#357	C₂²	semidirect
40. C₄×C₁₆×C₄₃	C₂²	semidirect
151. D₁₇₂	SD₃₂	semidirect
185. #2240	D₄₃	semidirect
206. #500	C₂×D₄₃	semidirect
225. #357	C₂×D₄₃	semidirect
226. #357	C₂×D₄₃	semidirect
228. C₄×C₁₆	C₂×D₄₃	semidirect
229. C₄₃	#33862	semidirect

Maximal subgroups

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

subgroup	index
#19146987	2
#19146270	2
#19136779	2
#19136739	2
#19136690	2
#19136632	2
#19136525	2
C₄×#13063270	2
C₁₆×D₁₇₂	2
#357×D₄₃	2
C₄₃×#2240	2
#33862	43