Subgroups — #25101016

Order 11008.

730 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143397	C₂	1	1	yes	maximal, characteristic
2	#19143240	C₂	1	2	yes	maximal, complemented
3	#19135486	C₂	1	1	yes	maximal, characteristic
4	#19135481	C₂	1	1	yes	maximal, characteristic
5	#19135346	C₂	1	1	yes	maximal, characteristic, complemented
6	#19135341	C₂	1	1	yes	maximal, characteristic, complemented
7	#19135304	C₂	1	2	yes	maximal, complemented
8	#19135297	C₂	1	2	yes	maximal, complemented
9	#19134408	C₂	1	2	yes	maximal, complemented
10	#19134283	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1778	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	1	yes	characteristic
13	C₄×#10621898	C₂²	1	2	yes	—
14	#14583012	C₄	1	2	yes	complemented
15	#14583007	C₄	1	2	yes	complemented
16	#14583001	C₄	1	4	yes	complemented
17	#14582961	—	2	4	no	—
18	#14582950	—	2	4	no	—
19	#14582937	C₂²	1	1	yes	characteristic
20	#14582927	C₂²	1	2	yes	—
21	#14582927	C₂²	1	2	yes	—
22	C₂×#13063251	C₂²	1	2	yes	—
23	C₂×#13063251	C₂²	1	2	yes	—
24	#14582896	C₂²	1	2	yes	—
25	#14582867	C₂²	1	2	yes	—
26	#14582867	C₂²	1	2	yes	—
27	C₂×#13063235	C₂²	1	1	yes	characteristic
28	C₂×#13063235	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
32	#14582745	C₂²	1	1	yes	characteristic
33	#14582660	C₂²	1	2	yes	—
34	#14582660	C₂²	1	2	yes	—
35	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₄₃	C₂²	1	2	yes	—
38	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#341	C₂²	1	2	yes	—
40	C₂×#10621898	C₂³	1	2	yes	—
41	#13063347	—	2	2	no	—
42	#13063347	—	2	2	no	—
43	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
44	C₂×#10621873	C₂×C₄	1	2	yes	—
45	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
48	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
49	C₄×D₁₇₂	C₂×C₄	1	4	yes	—
50	C₄×C₂²⋊D₄₃	—	2	4	no	—
51	#13063284	C₂³	1	2	yes	—
52	#13063282	—	2	2	no	—
53	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
55	C₄²×D₄₃	—	2	2	no	—
56	#13063259	C₂×C₄	1	2	yes	—
57	#13063257	C₂×C₄	1	2	yes	—
58	Dic₄₃⋊C₈	—	2	4	no	—
59	Dic₄₃⋊C₈	—	2	4	no	—
60	#13063251	—	2	4	no	—
61	#13063251	—	2	4	no	—
62	#13063250	C₂×C₄	1	4	yes	—
63	#13063247	C₂×C₄	1	4	yes	—
64	(C₂×D₄₃)⋊C₈	—	2	4	no	—
65	(C₂×D₄₃)⋊C₈	—	2	4	no	—
66	#13063235	—	2	2	no	—
67	#13063235	—	2	2	no	—
68	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
69	C₈×Dic₄₃	—	2	2	no	—
70	C₈×Dic₄₃	—	2	2	no	—
71	C₄₃⋊#95	—	2	2	no	—
72	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
73	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
74	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
77	#108×C₄₃	—	2	2	no	—
78	#108×C₄₃	—	2	2	no	—
79	#97×C₄₃	—	2	2	no	—
80	#97×C₄₃	C₂×C₄	1	2	yes	—
81	#97×C₄₃	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₄₃	—	2	2	no	—
83	C₂²×C₄×C₄₃	#41	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₂×C₂²⋊D₄₃	—	2	2	no	—
86	C₂²×Dic₄₃	—	2	1	no	—
87	C₂²×Dic₄₃	—	2	1	no	—
88	#10621898	—	2	2	no	—
89	#10621898	—	2	2	no	—
90	#10621898	—	2	2	no	—
91	#10621898	—	2	2	no	—
92	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₄₃	—	2	4	no	—
94	C₂×C₄×D₄₃	—	2	2	no	—
95	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
96	M₁₆×C₄₃	—	2	2	no	—
97	M₁₆×C₄₃	—	2	2	no	—
98	M₁₆×C₄₃	—	2	2	no	—
99	M₁₆×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	—	2	1	no	—
101	C₂×C₈×C₄₃	—	2	1	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	2	2	no	—
104	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₄₃	#41	1	2	yes	—
108	C₂×C₈×C₄₃	#41	1	2	yes	—
109	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
110	C₄²×C₄₃	C₂×D₄	1	2	yes	—
111	C₂²⋊Dic₄₃	—	2	2	no	—
112	(C₂×D₄₃)⋊C₄	—	2	4	no	—
113	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
114	Dic₄₃⋊C₄	—	2	4	no	—
115	C₄×Dic₄₃	—	2	4	no	—
116	C₄×Dic₄₃	—	2	2	no	—
117	C₄×Dic₄₃	—	2	1	no	—
118	C₄×Dic₄₃	—	2	1	no	—
119	C₄×Dic₄₃	#41	1	2	yes	—
120	C₄×Dic₄₃	#41	1	2	yes	—
121	C₂×C₄₃⋊C₈	—	2	1	no	—
122	C₂×C₄₃⋊C₈	—	2	1	no	—
123	C₂×C₄₃⋊C₈	—	2	1	no	—
124	C₂×C₄₃⋊C₈	—	2	1	no	—
125	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
126	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
127	Dic₁₇₂	—	2	2	no	—
128	D₃₄₄	—	2	2	no	—
129	#10621873	—	2	4	no	—
130	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
131	C₂²×D₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
134	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
140	C₂²⋊D₄₃	—	4	2	no	—
141	C₂²⋊D₄₃	—	4	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	—	2	4	no	—
144	C₂×Dic₄₃	—	2	2	no	—
145	C₂×Dic₄₃	—	2	2	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	D₁₇₂	—	2	2	no	—
149	D₁₇₂	—	2	2	no	—
150	C₄×D₄₃	—	4	8	no	—
151	C₄×D₄₃	—	4	2	no	—
152	C₄×D₄₃	—	4	2	no	—
153	Dic₈₆	—	2	2	no	—
154	Dic₈₆	—	2	2	no	—
155	C₈×C₄₃	—	2	2	no	—
156	C₈×C₄₃	—	4	2	no	—
157	C₈×C₄₃	—	4	2	no	—
158	C₈×C₄₃	#117	1	2	yes	—
159	C₈×C₄₃	#117	1	2	yes	—
160	C₄₃⋊C₈	—	2	2	no	—
161	C₄₃⋊C₈	—	2	2	no	—
162	#21296	—	43	1	no	maximal, Sylow
163	C₂²×C₄₃	—	2	2	no	—
164	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₄₃	#529	1	1	yes	characteristic
166	C₂²×C₄₃	#529	1	1	yes	characteristic
167	C₂×D₄₃	—	4	4	no	—
168	C₂×D₄₃	—	4	2	no	—
169	C₄×C₄₃	—	2	4	no	—
170	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
171	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
172	C₄×C₄₃	#532	1	1	yes	characteristic
173	C₄×C₄₃	#532	1	1	yes	characteristic
174	Dic₄₃	—	4	2	no	—
175	Dic₄₃	—	4	2	no	—
176	Dic₄₃	—	4	4	no	—
177	Dic₄₃	—	4	2	no	—
178	#2943	—	43	1	no	—
179	#2922	—	43	2	no	—
180	#1975	—	43	1	no	—
181	#1974	—	43	1	no	—
182	#1946	—	43	1	no	—
183	#1945	—	43	1	no	—
184	#1938	—	43	2	no	—
185	#1937	—	43	2	no	—
186	#1778	D₄₃	1	1	yes	characteristic, complemented
187	#1759	—	43	2	no	—
188	#1726	—	43	1	no	—
189	C₂×C₄₃	—	2	1	no	—
190	C₂×C₄₃	—	2	1	no	—
191	C₂×C₄₃	#2293	1	1	yes	characteristic
192	C₂×C₄₃	#2955	1	2	yes	—
193	D₄₃	—	8	2	no	—
194	C₂×#135	—	43	1	no	—
195	C₄×#41	—	43	2	no	—
196	#442	—	43	2	no	—
197	#441	—	43	2	no	—
198	#440	—	43	4	no	—
199	#433	—	86	4	no	—
200	#431	—	86	4	no	—
201	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
202	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
203	#427	—	43	1	no	—
204	#424	—	43	2	no	—
205	#424	—	43	2	no	—
206	C₂×#104	—	43	2	no	—
207	C₂×#104	—	43	2	no	—
208	#416	—	43	2	no	—
209	#408	—	43	2	no	—
210	#408	—	43	2	no	—
211	C₄×M₁₆	C₂×D₄₃	1	2	yes	—
212	C₂×#97	—	43	1	no	—
213	C₂×#97	—	43	1	no	—
214	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
215	C₂×C₄×C₈	—	43	1	no	—
216	C₂×C₄×C₈	—	43	1	no	—
217	C₂×C₄×C₈	—	43	1	no	—
218	#376	—	43	1	no	—
219	#341	C₂×D₄₃	1	2	yes	—
220	#339	—	43	2	no	—
221	#339	—	43	2	no	—
222	C₄₃	#21296	1	1	yes	characteristic, complemented, Sylow
223	C₂×#41	—	43	2	no	—
224	#135	—	86	2	no	—
225	#135	—	86	2	no	—
226	C₂×Q₁₆	—	43	1	no	—
227	C₂×SD₁₆	—	43	2	no	—
228	C₂×D₈	—	43	1	no	—
229	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
230	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
231	C₂²×C₈	—	43	1	no	—
232	C₂²×C₈	—	43	1	no	—
233	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
235	C₄×Q₈	—	43	4	no	—
236	C₄×D₄	—	43	4	no	—
237	C₄×D₄	—	86	4	no	—
238	#117	—	43	2	no	—
239	#117	—	86	2	no	—
240	C₂×C₄²	—	43	1	no	—
241	C₂×C₄²	—	43	1	no	—
242	C₂×C₄²	—	86	2	no	—
243	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
244	#108	—	2	2	no	—
245	#108	—	2	2	no	—
246	#107	—	43	2	no	—
247	#106	—	43	2	no	—
248	C₄⋊C₈	—	86	4	no	—
249	C₄⋊C₈	—	86	4	no	—
250	#104	—	86	4	no	—
251	#104	—	86	4	no	—
252	#103	—	43	4	no	—
253	#102	—	43	4	no	—
254	C₂²⋊C₈	—	86	4	no	—
255	C₂²⋊C₈	—	86	4	no	—
256	#97	—	2	2	no	—
257	#97	—	86	2	no	—
258	#97	—	86	2	no	—
259	#97	C₄×D₄₃	1	2	yes	—
260	#97	C₄×D₄₃	1	2	yes	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	43	4	no	—
263	C₄×C₈	—	86	2	no	—
264	C₄×C₈	—	86	2	no	—
265	#95	—	86	2	no	—
266	#41	—	86	2	no	—
267	#41	—	86	2	no	—
268	#41	—	86	2	no	—
269	#41	—	86	2	no	—
270	C₂×Q₈	—	43	2	no	—
271	C₂×D₄	—	43	2	no	—
272	C₂×D₄	—	86	2	no	—
273	C₂²×C₄	—	86	1	no	—
274	C₂²×C₄	—	86	1	no	—
275	C₂²×C₄	—	86	4	no	—
276	C₂²×C₄	—	86	2	no	—
277	C₂²×C₄	#10621902	1	2	yes	—
278	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
279	Q₁₆	—	86	2	no	—
280	SD₁₆	—	86	4	no	—
281	D₈	—	86	2	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	43	2	no	—
291	C₂×C₈	—	43	2	no	—
292	C₂×C₈	—	86	1	no	—
293	C₂×C₈	—	86	1	no	—
294	C₂×C₈	—	86	1	no	—
295	C₂×C₈	—	86	1	no	—
296	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
297	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
299	C₂×C₈	#10621898	1	2	yes	—
300	C₂×C₈	#10621898	1	2	yes	—
301	C₄⋊C₄	—	43	4	no	—
302	C₄⋊C₄	—	86	4	no	—
303	C₂²⋊C₄	—	86	2	no	—
304	C₂²⋊C₄	—	86	4	no	—
305	C₄²	—	43	2	no	—
306	C₄²	—	43	2	no	—
307	C₄²	—	86	4	no	—
308	C₄²	—	86	2	no	—
309	C₄²	—	86	1	no	—
310	C₄²	—	86	1	no	—
311	C₄²	C₂×C₄×D₄₃	1	2	yes	—
312	C₄²	D₄×D₄₃	1	2	yes	—
313	C₂³	—	86	2	no	—
314	C₂³	C₂×#10621902	1	1	yes	characteristic
315	Q₈	—	86	2	no	—
316	Q₈	—	86	2	no	—
317	D₄	—	86	2	no	—
318	D₄	—	86	2	no	—
319	D₄	—	172	2	no	—
320	D₄	—	172	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	86	4	no	—
323	C₂×C₄	—	86	2	no	—
324	C₂×C₄	—	86	2	no	—
325	C₂×C₄	—	86	2	no	—
326	C₂×C₄	—	172	8	no	—
327	C₂×C₄	—	172	2	no	—
328	C₂×C₄	—	172	2	no	—
329	C₂×C₄	—	172	2	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	#13063287	1	4	yes	—
332	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
334	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
335	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
336	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
337	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	86	2	no	—
342	C₈	—	86	2	no	—
343	C₈	#13063282	1	2	yes	—
344	C₈	#13063282	1	2	yes	—
345	C₂²	—	2	2	no	—
346	C₂²	—	172	2	no	—
347	C₂²	—	172	4	no	—
348	C₂²	C₂×#13063287	1	1	yes	characteristic
349	C₂²	#14583436	1	1	yes	characteristic
350	C₂²	#14583436	1	1	yes	characteristic
351	C₄	—	2	4	no	—
352	C₄	—	172	2	no	—
353	C₄	—	172	2	no	—
354	C₄	—	172	2	no	—
355	C₄	—	172	4	no	—
356	C₄	C₂×#13063282	1	1	yes	characteristic
357	C₄	#14583367	1	1	yes	characteristic
358	C₄	#14583446	1	1	yes	characteristic
359	C₄	#14583446	1	1	yes	characteristic
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	344	2	no	—
363	C₂	#19137056	1	1	yes	characteristic
364	C₂	#19143497	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1778	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂×#13063347	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1778	D₄₃
hypocenter	C₄₃	#21296
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
2. #19143240	C₂	semidirect
5. #19135346	C₂	semidirect
6. #19135341	C₂	semidirect
7. #19135304	C₂	semidirect
8. #19135297	C₂	semidirect
9. #19134408	C₂	semidirect
10. #19134283	C₂	semidirect
11. C₄₃×#1778	C₂	semidirect
14. #14583012	C₄	semidirect
15. #14583007	C₄	semidirect
16. #14583001	C₄	semidirect
186. #1778	D₄₃	semidirect
222. C₄₃	#21296	semidirect

Maximal subgroups

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

subgroup	index
#19143397	2
#19143240	2
#19135486	2
#19135481	2
#19135346	2
#19135341	2
#19135304	2
#19135297	2
#19134408	2
#19134283	2
C₄₃×#1778	2
#21296	43