Subgroups — #25155482

Order 11008.

674 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145127	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144448	C₂	1	2	yes	maximal
3	#19143187	C₂	1	1	yes	maximal, characteristic
4	#19136100	C₂	1	2	yes	maximal, complemented
5	#19135944	C₂	1	2	yes	maximal, complemented
6	#19135376	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134829	C₂	1	2	yes	maximal
8	#19134759	C₂	1	1	yes	maximal, characteristic, complemented
9	#19134651	C₂	1	1	yes	maximal, characteristic
10	#19134432	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1945	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063331	C₂²	1	1	yes	characteristic
13	#14583527	C₂²	1	2	yes	—
14	#14583379	C₂²	1	2	yes	—
15	#14583307	—	2	2	no	—
16	#14583299	—	2	1	no	—
17	#14583296	—	2	1	no	—
18	#14583207	—	2	2	no	—
19	#14583200	—	2	2	no	—
20	#14583195	—	2	2	no	—
21	#14583180	—	2	2	no	—
22	#14582942	C₂²	1	2	yes	—
23	#14582942	C₂²	1	2	yes	—
24	#14582941	C₂²	1	2	yes	—
25	#14582940	—	2	2	no	—
26	#14582934	—	2	1	no	—
27	C₂×#13063259	C₂²	1	2	yes	—
28	C₂×#13063257	C₂²	1	1	yes	characteristic
29	C₂×#13063257	C₂²	1	1	yes	characteristic
30	#106×D₄₃	—	2	1	no	—
31	C₂×#13063253	C₂²	1	1	yes	characteristic
32	#14582901	C₂²	1	2	yes	—
33	#14582896	C₂²	1	1	yes	characteristic
34	#14582854	C₂²	1	1	yes	characteristic
35	#14582765	C₂²	1	2	yes	—
36	#14582667	C₂²	1	2	yes	—
37	#14582667	C₂²	1	2	yes	—
38	#14582664	C₂²	1	1	yes	characteristic
39	#14582659	C₂²	1	1	yes	characteristic
40	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
41	C₄₃×#427	C₂²	1	1	yes	characteristic
42	C₄₃×#424	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
44	C₄₃×#339	C₂²	1	2	yes	—
45	C₂×#10621898	C₂³	1	1	yes	characteristic
46	#13063331	—	2	2	no	—
47	#13063331	Q₈	1	4	yes	complemented
48	C₂×#10621877	C₂³	1	1	yes	characteristic
49	C₂×#10621877	C₂³	1	1	yes	characteristic
50	C₂×#10621872	—	2	1	no	—
51	C₂×C₈×D₄₃	—	2	1	no	—
52	#13063325	—	2	2	no	—
53	#13063324	—	2	2	no	—
54	#13063317	—	2	2	no	—
55	#13063316	—	2	2	no	—
56	#13063302	—	4	2	no	—
57	C₄×Dic₈₆	—	2	2	no	—
58	C₄×D₁₇₂	—	2	2	no	—
59	(C₂²⋊D₄₃)⋊C₄	—	4	2	no	—
60	#13063284	C₂³	1	1	yes	characteristic
61	#13063283	—	2	2	no	—
62	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
63	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
64	C₄⋊C₄×D₄₃	—	2	2	no	—
65	#13063259	—	2	2	no	—
66	#13063259	—	2	2	no	—
67	#13063259	—	2	2	no	—
68	#13063258	—	2	2	no	—
69	#13063258	—	2	2	no	—
70	#13063257	—	2	1	no	—
71	#13063257	—	2	1	no	—
72	#13063257	—	2	1	no	—
73	#13063257	—	2	1	no	—
74	#13063257	—	2	2	no	—
75	#13063256	—	2	2	no	—
76	#13063256	—	2	2	no	—
77	#13063253	Q₈	1	4	yes	complemented
78	#13063250	—	2	2	no	—
79	#13063250	D₄	1	2	yes	—
80	#13063247	—	2	2	no	—
81	#13063247	D₄	1	2	yes	—
82	#13063235	—	2	1	no	—
83	C₈×Dic₄₃	—	2	1	no	—
84	C₄₃⋊#95	—	4	2	no	—
85	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
86	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	#117×C₄₃	C₂³	1	2	yes	—
89	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
90	#108×C₄₃	—	2	2	no	—
91	#107×C₄₃	D₄	1	2	yes	—
92	#106×C₄₃	D₄	1	1	yes	characteristic
93	#106×C₄₃	D₄	1	1	yes	characteristic
94	C₄⋊C₈×C₄₃	—	2	2	no	—
95	C₄⋊C₈×C₄₃	—	2	2	no	—
96	C₄×C₈×C₄₃	—	2	2	no	—
97	C₂²×C₄×C₄₃	—	2	1	no	—
98	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
99	C₂×C₂²⋊D₄₃	—	2	1	no	—
100	C₂²×Dic₄₃	—	2	2	no	—
101	C₂²×Dic₄₃	—	2	2	no	—
102	#10621898	—	2	1	no	—
103	#10621898	—	2	1	no	—
104	#10621898	C₂×Q₈	1	2	yes	—
105	#10621898	C₂×Q₈	1	2	yes	—
106	C₂×D₁₇₂	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₄₃	—	2	1	no	—
108	C₂×C₄×D₄₃	—	4	2	no	—
109	C₂×Dic₈₆	—	4	2	no	—
110	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
111	M₁₆×C₄₃	—	2	2	no	—
112	M₁₆×C₄₃	—	2	2	no	—
113	C₂×C₈×C₄₃	—	2	1	no	—
114	C₂×C₈×C₄₃	—	2	1	no	—
115	C₂×C₈×C₄₃	—	2	2	no	—
116	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
120	C₂×C₈×C₄₃	#41	1	2	yes	—
121	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
122	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
123	C₂²⋊C₄×C₄₃	—	2	2	no	—
124	C₄²×C₄₃	—	2	2	no	—
125	C₄²×C₄₃	—	2	1	no	—
126	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
127	C₂²⋊Dic₄₃	—	2	1	no	—
128	(C₂×D₄₃)⋊C₄	—	4	2	no	—
129	C₄⋊Dic₄₃	—	2	2	no	—
130	C₄⋊Dic₄₃	—	2	2	no	—
131	C₄⋊Dic₄₃	—	2	2	no	—
132	C₄⋊Dic₄₃	—	2	2	no	—
133	C₄⋊Dic₄₃	—	4	2	no	—
134	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
135	Dic₄₃⋊C₄	—	4	2	no	—
136	Dic₄₃⋊C₄	—	4	2	no	—
137	Dic₄₃⋊C₄	—	4	2	no	—
138	C₄×Dic₄₃	—	2	1	no	—
139	C₄×Dic₄₃	—	4	2	no	—
140	#10621877	—	2	1	no	—
141	#10621877	—	2	1	no	—
142	#10621877	C₂×Q₈	1	2	yes	—
143	#10621877	C₂×Q₈	1	2	yes	—
144	C₂×C₄₃⋊C₈	—	2	1	no	—
145	C₂×C₄₃⋊C₈	—	2	1	no	—
146	#10621872	—	2	4	no	—
147	C₈×D₄₃	—	2	4	no	—
148	C₂³×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂²×D₄₃	—	2	1	no	—
150	C₂×C₄×C₄₃	—	2	2	no	—
151	C₂×C₄×C₄₃	—	2	2	no	—
152	C₂×C₄×C₄₃	—	2	2	no	—
153	C₂×C₄×C₄₃	—	4	2	no	—
154	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
160	C₂²⋊D₄₃	—	2	2	no	—
161	C₂²⋊D₄₃	—	2	2	no	—
162	C₂×Dic₄₃	—	2	1	no	—
163	C₂×Dic₄₃	—	2	2	no	—
164	C₂×Dic₄₃	—	4	2	no	—
165	C₂×Dic₄₃	—	4	2	no	—
166	C₂×Dic₄₃	—	8	2	no	—
167	C₂×Dic₄₃	—	8	2	no	—
168	D₁₇₂	—	2	1	no	—
169	D₁₇₂	C₄⋊Q₈	1	2	yes	—
170	C₄×D₄₃	—	2	2	no	—
171	C₄×D₄₃	—	2	2	no	—
172	C₄×D₄₃	—	16	2	no	—
173	Dic₈₆	—	2	1	no	—
174	Dic₈₆	—	16	2	no	—
175	Dic₈₆	C₄⋊Q₈	1	2	yes	—
176	C₈×C₄₃	—	2	2	no	—
177	C₈×C₄₃	—	4	2	no	—
178	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
179	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
180	C₄₃⋊C₈	—	2	2	no	—
181	C₄₃⋊C₈	—	2	2	no	—
182	#26080	—	43	1	no	maximal, Sylow
183	C₂²×C₄₃	—	2	2	no	—
184	C₂²×C₄₃	#544	1	1	yes	characteristic
185	C₂²×C₄₃	D₄²	1	1	yes	characteristic
186	C₂²×C₄₃	#554	1	1	yes	characteristic
187	C₂×D₄₃	—	2	2	no	—
188	C₂×D₄₃	—	4	2	no	—
189	C₄×C₄₃	—	4	2	no	—
190	C₄×C₄₃	—	4	2	no	—
191	C₄×C₄₃	—	4	2	no	—
192	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
193	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
194	C₄×C₄₃	#547	1	1	yes	characteristic
195	C₄×C₄₃	D₄×Q₈	1	1	yes	characteristic
196	Dic₄₃	—	2	2	no	—
197	Dic₄₃	—	4	2	no	—
198	Dic₄₃	—	8	2	no	—
199	Dic₄₃	—	8	2	no	—
200	#3163	—	43	1	no	—
201	#3083	—	43	2	no	—
202	#2912	—	43	1	no	—
203	#2089	—	43	2	no	—
204	#2061	—	43	2	no	—
205	#1951	—	43	1	no	—
206	#1945	D₄₃	1	1	yes	characteristic, complemented
207	#1860	—	43	2	no	—
208	#1846	—	43	1	no	—
209	#1815	—	43	1	no	—
210	#1764	—	43	1	no	—
211	C₂×C₄₃	—	2	1	no	—
212	C₂×C₄₃	—	2	1	no	—
213	C₂×C₄₃	#2440	1	1	yes	characteristic
214	C₂×C₄₃	#3290	1	2	yes	—
215	D₄₃	—	4	2	no	—
216	C₂×#131	—	43	1	no	—
217	#544	—	43	2	no	—
218	#520	—	43	2	no	—
219	#501	—	86	2	no	—
220	#499	—	86	1	no	—
221	#498	—	86	1	no	—
222	#479	—	86	2	no	—
223	#478	—	86	2	no	—
224	#477	—	86	2	no	—
225	#474	—	86	2	no	—
226	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
227	#428	—	43	2	no	—
228	#428	—	43	2	no	—
229	#428	—	43	2	no	—
230	#428	—	86	2	no	—
231	#427	—	86	1	no	—
232	#427	C₂×D₄₃	1	1	yes	characteristic
233	C₂×#107	—	43	2	no	—
234	C₂×#106	—	43	1	no	—
235	C₂×#106	—	43	1	no	—
236	C₂×#106	—	86	1	no	—
237	#424	C₂×D₄₃	1	2	yes	—
238	C₂×#104	—	43	1	no	—
239	#417	—	43	2	no	—
240	#416	—	43	1	no	—
241	#405	—	43	1	no	—
242	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
243	#382	—	43	2	no	—
244	#343	—	43	2	no	—
245	#343	—	43	2	no	—
246	#341	—	43	1	no	—
247	#339	—	43	1	no	—
248	#339	C₂×D₄₃	1	2	yes	—
249	C₄₃	#26080	1	1	yes	characteristic, complemented, Sylow
250	C₂×#41	—	43	1	no	—
251	#131	—	43	4	no	—
252	#131	—	86	2	no	—
253	C₂×M₁₆	—	43	1	no	—
254	C₂×M₁₆	—	43	1	no	—
255	C₂×M₁₆	—	86	1	no	—
256	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
257	C₂²×C₈	—	86	1	no	—
258	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
259	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
260	C₄⋊Q₈	—	86	2	no	—
261	C₄⋊Q₈	—	86	2	no	—
262	#125	—	86	2	no	—
263	#125	—	86	2	no	—
264	C₂²⋊Q₈	—	172	2	no	—
265	C₄×Q₈	—	86	2	no	—
266	C₄×D₄	—	86	2	no	—
267	C₄×D₄	—	172	2	no	—
268	#117	—	43	1	no	—
269	#117	—	86	2	no	—
270	#117	C₂²×D₄₃	1	2	yes	—
271	C₂×C₄⋊C₄	—	43	2	no	—
272	C₂×C₄⋊C₄	—	43	2	no	—
273	C₂×C₄⋊C₄	—	86	2	no	—
274	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
275	#108	—	2	2	no	—
276	#107	—	86	2	no	—
277	#107	—	86	2	no	—
278	#107	—	86	2	no	—
279	#107	—	86	2	no	—
280	#107	—	86	2	no	—
281	#107	D₁₇₂	1	2	yes	—
282	#106	—	86	2	no	—
283	#106	—	86	1	no	—
284	#106	—	86	1	no	—
285	#106	—	86	1	no	—
286	#106	—	86	1	no	—
287	#106	—	86	2	no	—
288	#106	—	86	2	no	—
289	#106	D₁₇₂	1	1	yes	characteristic
290	#106	D₁₇₂	1	1	yes	characteristic
291	C₄⋊C₈	—	2	2	no	—
292	C₄⋊C₈	—	2	2	no	—
293	#104	—	43	4	no	—
294	#103	—	43	2	no	—
295	#103	—	86	2	no	—
296	#102	—	43	2	no	—
297	#102	—	86	2	no	—
298	#97	—	86	1	no	—
299	C₄×C₈	—	2	2	no	—
300	C₄×C₈	—	86	1	no	—
301	#95	—	172	2	no	—
302	#41	—	43	2	no	—
303	#41	—	43	2	no	—
304	#41	—	86	1	no	—
305	#41	—	86	1	no	—
306	C₂×Q₈	—	43	1	no	—
307	C₂×Q₈	—	172	2	no	—
308	C₂×D₄	—	43	1	no	—
309	C₂×D₄	—	86	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	86	1	no	—
312	C₂²×C₄	—	86	2	no	—
313	C₂²×C₄	—	86	2	no	—
314	C₂²×C₄	—	172	2	no	—
315	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
316	M₁₆	—	2	2	no	—
317	M₁₆	—	2	2	no	—
318	M₁₆	—	43	2	no	—
319	M₁₆	—	43	2	no	—
320	M₁₆	—	86	1	no	—
321	M₁₆	—	86	1	no	—
322	M₁₆	—	86	4	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	86	1	no	—
327	C₂×C₈	—	86	1	no	—
328	C₂×C₈	—	86	4	no	—
329	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
330	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
331	C₂×C₈	D₄×D₄₃	1	2	yes	—
332	C₂×C₈	#10621900	1	2	yes	—
333	C₂×C₈	Q₈×D₄₃	1	2	yes	—
334	C₄⋊C₄	—	43	2	no	—
335	C₄⋊C₄	—	86	2	no	—
336	C₄⋊C₄	—	86	2	no	—
337	C₄⋊C₄	—	86	2	no	—
338	C₄⋊C₄	—	86	2	no	—
339	C₄⋊C₄	—	172	2	no	—
340	C₄⋊C₄	—	172	2	no	—
341	C₄⋊C₄	—	172	2	no	—
342	C₄⋊C₄	—	172	2	no	—
343	C₄⋊C₄	C₂×D₁₇₂	1	2	yes	—
344	C₄⋊C₄	C₂×D₁₇₂	1	2	yes	—
345	C₂²⋊C₄	—	2	2	no	—
346	C₂²⋊C₄	—	86	1	no	—
347	C₂²⋊C₄	—	172	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	2	1	no	—
350	C₄²	—	86	1	no	—
351	C₄²	—	172	2	no	—
352	C₄²	Q₈×D₄₃	1	2	yes	—
353	C₂³	—	86	1	no	—
354	C₂³	#13063371	1	1	yes	characteristic
355	Q₈	—	43	2	no	—
356	Q₈	—	86	1	no	—
357	Q₈	—	688	2	no	—
358	D₄	—	43	2	no	—
359	D₄	—	86	2	no	—
360	D₄	—	86	2	no	—
361	D₄	—	86	1	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	86	1	no	—
367	C₂×C₄	—	86	2	no	—
368	C₂×C₄	—	86	2	no	—
369	C₂×C₄	—	86	2	no	—
370	C₂×C₄	—	344	2	no	—
371	C₂×C₄	—	344	2	no	—
372	C₂×C₄	—	172	2	no	—
373	C₂×C₄	—	172	2	no	—
374	C₂×C₄	—	688	2	no	—
375	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
377	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
378	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
379	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
380	C₂×C₄	#13063372	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	4	2	no	—
383	C₈	—	86	2	no	—
384	C₈	—	86	2	no	—
385	C₈	#13063305	1	2	yes	—
386	C₈	#13063305	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	86	2	no	—
389	C₂²	—	172	2	no	—
390	C₂²	#14583529	1	1	yes	characteristic
391	C₂²	D₄×D₁₇₂	1	1	yes	characteristic
392	C₂²	#14583607	1	1	yes	characteristic
393	C₄	—	4	2	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	2	no	—
396	C₄	—	86	2	no	—
397	C₄	—	344	2	no	—
398	C₄	—	344	2	no	—
399	C₄	—	172	2	no	—
400	C₄	C₂×#13063305	1	1	yes	characteristic
401	C₄	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
402	C₄	#14583549	1	1	yes	characteristic
403	C₄	Q₈×D₁₇₂	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	172	2	no	—
407	C₂	#19138295	1	1	yes	characteristic
408	C₂	#19146382	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₇₂
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1945	C₂
socle	C₂²×C₄₃	D₄²
prime radical	C₂×#10621898	C₂³
hypercenter	#1945	D₄₃
hypocenter	C₄₃	#26080
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1945

Complements

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

N	H ≅ G/N	type
1. #19145127	C₂	semidirect
4. #19136100	C₂	semidirect
5. #19135944	C₂	semidirect
6. #19135376	C₂	semidirect
8. #19134759	C₂	semidirect
11. C₄₃×#1945	C₂	semidirect
47. #13063331	Q₈	semidirect
77. #13063253	Q₈	semidirect
206. #1945	D₄₃	semidirect
249. C₄₃	#26080	semidirect

Maximal subgroups

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

subgroup	index
#19145127	2
#19144448	2
#19143187	2
#19136100	2
#19135944	2
#19135376	2
#19134829	2
#19134759	2
#19134651	2
#19134432	2
C₄₃×#1945	2
#26080	43