Subgroups — #25251455

Order 11008.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144251	C₂	1	1	yes	maximal, characteristic
2	#19143192	C₂	1	1	yes	maximal, characteristic
3	#19136806	C₂	1	2	yes	maximal, complemented
4	#19136805	C₂	1	2	yes	maximal, complemented
5	#19136799	C₂	1	2	yes	maximal, complemented
6	#19136794	C₂	1	2	yes	maximal, complemented
7	#19136384	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136314	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14582704	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136208	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
13	C₂×#13063331	C₂²	1	1	yes	characteristic
14	#14583475	—	2	1	no	—
15	#14583379	—	2	1	no	—
16	C₂×#13063265	C₂²	1	1	yes	characteristic
17	#14583158	—	2	2	no	—
18	#14583141	C₂²	1	2	yes	—
19	#14583141	C₂²	1	2	yes	—
20	#14582942	—	2	1	no	—
21	#14582938	—	2	2	no	—
22	#14582936	C₂²	1	2	yes	—
23	C₂×#13063259	C₂²	1	1	yes	characteristic
24	#107×D₄₃	C₂²	1	2	yes	—
25	#14582907	—	2	1	no	—
26	#14582726	C₂²	1	2	yes	—
27	#14582724	C₂²	1	2	yes	—
28	#14582704	C₂²	1	2	yes	—
29	#14582704	C₂²	1	2	yes	—
30	#14582702	C₂²	1	2	yes	—
31	#14582702	C₂²	1	2	yes	—
32	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
33	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
34	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
35	C₄₃×#427	C₂²	1	1	yes	characteristic
36	C₄₃×#427	C₂²	1	1	yes	characteristic
37	C₄₃×#367	C₂²	1	1	yes	characteristic
38	C₄₃×#367	C₂²	1	1	yes	characteristic
39	C₄₃×#366	C₂²	1	1	yes	characteristic
40	C₄₃×#366	C₂²	1	1	yes	characteristic
41	C₂×#10621898	D₄	1	1	yes	characteristic
42	C₂²×Dic₈₆	—	2	1	no	—
43	C₂×Dic₁₇₂	—	2	2	no	—
44	C₂×Dic₁₇₂	—	2	2	no	—
45	C₂×Dic₁₇₂	—	2	1	no	—
46	C₂×Dic₁₇₂	C₂³	1	2	yes	—
47	#13063331	—	2	2	no	—
48	#13063331	—	2	2	no	—
49	C₂×#10621877	D₄	1	1	yes	characteristic
50	C₂×#10621872	D₄	1	2	yes	—
51	C₂×C₈×D₄₃	C₂³	1	2	yes	—
52	#13063325	—	4	1	no	—
53	#13063313	—	4	1	no	—
54	#13063309	—	4	2	no	—
55	#13063305	—	2	2	no	—
56	#13063305	—	2	2	no	—
57	C₄×Dic₈₆	—	4	1	no	—
58	C₄×D₁₇₂	—	4	1	no	—
59	(C₂²⋊D₄₃)⋊C₄	—	4	2	no	—
60	#13063283	—	2	2	no	—
61	C₂×C₄⋊Dic₄₃	—	2	1	no	—
62	C₄⋊C₄×D₄₃	—	2	2	no	—
63	#13063265	—	2	2	no	—
64	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
65	#13063259	—	2	1	no	—
66	#13063259	C₂³	1	2	yes	—
67	#13063258	—	2	2	no	—
68	#13063256	—	2	2	no	—
69	#13063248	—	2	2	no	—
70	#13063248	—	2	2	no	—
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	#117×C₄₃	—	2	1	no	—
73	#117×C₄₃	—	2	1	no	—
74	C₂×C₁₆×C₄₃	—	2	2	no	—
75	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
77	#107×C₄₃	C₂³	1	1	yes	characteristic
78	#107×C₄₃	C₂³	1	1	yes	characteristic
79	#107×C₄₃	C₂³	1	1	yes	characteristic
80	#107×C₄₃	C₂³	1	1	yes	characteristic
81	#106×C₄₃	—	2	1	no	—
82	#106×C₄₃	—	2	1	no	—
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₄₃	D₈	1	2	yes	—
85	C₂²×Dic₄₃	—	4	1	no	—
86	C₂²×Dic₄₃	—	4	1	no	—
87	#10621898	—	2	2	no	—
88	#10621898	—	2	2	no	—
89	C₂×D₁₇₂	D₈	1	1	yes	characteristic
90	C₂×C₄×D₄₃	—	4	2	no	—
91	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
92	C₂×Dic₈₆	—	2	2	no	—
93	C₂×Dic₈₆	—	4	2	no	—
94	C₂×Dic₈₆	—	4	2	no	—
95	C₂×Dic₈₆	—	4	1	no	—
96	C₂×Dic₈₆	D₈	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	#41	1	2	yes	—
99	C₂×C₈×C₄₃	#41	1	2	yes	—
100	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₄₃	—	2	1	no	—
102	C₄⋊C₄×C₄₃	—	2	1	no	—
103	C₄⋊C₄×C₄₃	—	2	1	no	—
104	C₄⋊C₄×C₄₃	—	2	1	no	—
105	C₂²⋊C₄×C₄₃	—	4	1	no	—
106	C₂²⋊C₄×C₄₃	—	4	1	no	—
107	C₄²×C₄₃	—	4	1	no	—
108	C₄²×C₄₃	—	4	1	no	—
109	(C₂×D₄₃)⋊C₄	—	4	2	no	—
110	(C₂×D₄₃)⋊C₄	—	4	2	no	—
111	(C₂×D₄₃)⋊C₄	—	4	2	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	C₄⋊Dic₄₃	—	4	1	no	—
114	C₄⋊Dic₄₃	—	4	2	no	—
115	C₄⋊Dic₄₃	—	4	2	no	—
116	Dic₄₃⋊C₄	—	4	2	no	—
117	C₄×Dic₄₃	—	4	2	no	—
118	#10621877	—	2	2	no	—
119	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
120	Dic₁₇₂	—	2	2	no	—
121	Dic₁₇₂	—	2	2	no	—
122	Dic₁₇₂	—	4	2	no	—
123	#10621872	—	2	4	no	—
124	C₈×D₄₃	—	2	2	no	—
125	C₈×D₄₃	—	2	2	no	—
126	C₁₆×C₄₃	—	2	1	no	—
127	C₁₆×C₄₃	—	2	1	no	—
128	C₄₃⋊C₁₆	—	2	2	no	—
129	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
130	C₂²×D₄₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	4	1	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	#123	1	2	yes	—
136	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#137	1	2	yes	—
138	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₄₃	—	2	4	no	—
140	C₂×Dic₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	C₂×Dic₄₃	—	8	2	no	—
143	C₂×Dic₄₃	—	16	1	no	—
144	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
145	D₁₇₂	—	2	2	no	—
146	C₄×D₄₃	—	2	2	no	—
147	C₄×D₄₃	—	2	2	no	—
148	C₄×D₄₃	—	16	2	no	—
149	Dic₈₆	—	2	2	no	—
150	Dic₈₆	—	4	2	no	—
151	Dic₈₆	—	4	2	no	—
152	Dic₈₆	—	8	2	no	—
153	C₈×C₄₃	#123	1	2	yes	—
154	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
155	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
156	C₄₃⋊C₈	—	2	2	no	—
157	#33651	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	#480	1	2	yes	—
161	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₄₃	—	2	2	no	—
163	C₂×D₄₃	—	2	2	no	—
164	C₂×D₄₃	—	2	2	no	—
165	C₄×C₄₃	—	8	1	no	—
166	C₄×C₄₃	—	8	1	no	—
167	C₄×C₄₃	—	8	1	no	—
168	C₄×C₄₃	—	8	1	no	—
169	C₄×C₄₃	#480	1	2	yes	—
170	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
171	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
172	Dic₄₃	—	2	2	no	—
173	Dic₄₃	—	8	2	no	—
174	Dic₄₃	—	8	2	no	—
175	#3059	—	43	1	no	—
176	#2913	—	43	1	no	—
177	#2237	—	43	2	no	—
178	#2237	—	43	2	no	—
179	#2236	—	43	2	no	—
180	#2235	—	43	2	no	—
181	#2159	—	43	1	no	—
182	#2158	D₄₃	1	1	yes	characteristic, complemented
183	#2141	—	43	1	no	—
184	C₂×#358	—	43	1	no	—
185	#2113	—	43	1	no	—
186	C₂×C₄₃	—	2	2	no	—
187	C₂×C₄₃	C₂×#480	1	1	yes	characteristic
188	C₂×C₄₃	#3416	1	1	yes	characteristic
189	C₂×C₄₃	#3417	1	1	yes	characteristic
190	D₄₃	—	4	2	no	—
191	C₂²×Q₁₆	—	43	1	no	—
192	C₂×#131	—	43	1	no	—
193	#536	—	86	1	no	—
194	#520	—	86	1	no	—
195	C₂×M₃₂	—	43	1	no	—
196	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
197	#470	—	86	2	no	—
198	#467	—	43	2	no	—
199	#467	—	43	2	no	—
200	#428	—	86	2	no	—
201	#428	—	86	1	no	—
202	#427	—	43	2	no	—
203	#427	C₂×D₄₃	1	1	yes	characteristic
204	#427	C₂×D₄₃	1	1	yes	characteristic
205	C₂×#107	—	43	1	no	—
206	C₂×#107	—	43	2	no	—
207	#419	—	86	1	no	—
208	#367	—	43	2	no	—
209	#367	C₂×D₄₃	1	1	yes	characteristic
210	#367	C₂×D₄₃	1	1	yes	characteristic
211	#366	—	43	2	no	—
212	#366	C₂×D₄₃	1	1	yes	characteristic
213	#366	C₂×D₄₃	1	1	yes	characteristic
214	#358	—	43	2	no	—
215	#358	—	43	2	no	—
216	#358	—	43	2	no	—
217	#358	—	43	2	no	—
218	C₂²⋊C₁₆	—	43	2	no	—
219	C₂²⋊C₁₆	—	43	2	no	—
220	C₄₃	#33651	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	43	1	no	—
222	C₂²×Q₈	—	86	1	no	—
223	C₂×Q₁₆	—	43	2	no	—
224	C₂×Q₁₆	—	86	2	no	—
225	C₂×Q₁₆	—	86	1	no	—
226	C₂×Q₁₆	—	86	2	no	—
227	#131	—	86	2	no	—
228	#131	—	86	2	no	—
229	C₂×M₁₆	—	43	1	no	—
230	C₂×M₁₆	—	43	2	no	—
231	C₂²×C₈	—	43	2	no	—
232	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
233	C₄⋊Q₈	—	172	1	no	—
234	#124	—	172	1	no	—
235	#123	—	172	2	no	—
236	C₂²⋊Q₈	—	86	2	no	—
237	C₂²⋊Q₈	—	86	2	no	—
238	C₄×Q₈	—	172	1	no	—
239	C₄×D₄	—	172	1	no	—
240	C₄×D₄	—	172	2	no	—
241	#117	—	2	1	no	—
242	#117	—	2	1	no	—
243	#117	—	86	2	no	—
244	C₂×C₄⋊C₄	—	86	2	no	—
245	C₂×C₄⋊C₄	—	86	1	no	—
246	M₃₂	—	86	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	43	2	no	—
249	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
251	#107	—	43	2	no	—
252	#107	—	86	2	no	—
253	#107	—	86	1	no	—
254	#107	C₂²×D₄₃	1	1	yes	characteristic
255	#107	C₂²×D₄₃	1	1	yes	characteristic
256	#107	C₂²×D₄₃	1	1	yes	characteristic
257	#107	C₂²×D₄₃	1	1	yes	characteristic
258	#106	—	2	1	no	—
259	#106	—	2	1	no	—
260	#106	—	86	2	no	—
261	#103	—	86	2	no	—
262	#103	—	86	2	no	—
263	#41	—	86	2	no	—
264	#41	—	86	2	no	—
265	C₂×Q₈	—	43	1	no	—
266	C₂×Q₈	—	86	2	no	—
267	C₂×Q₈	—	172	2	no	—
268	C₂×Q₈	—	172	2	no	—
269	C₂×Q₈	—	172	1	no	—
270	C₂×D₄	—	43	2	no	—
271	C₂×D₄	—	43	1	no	—
272	C₂²×C₄	—	43	2	no	—
273	C₂²×C₄	—	172	2	no	—
274	C₂²×C₄	—	172	1	no	—
275	C₂²×C₄	—	172	1	no	—
276	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
277	Q₁₆	—	86	2	no	—
278	Q₁₆	—	86	2	no	—
279	Q₁₆	—	172	2	no	—
280	M₁₆	—	86	2	no	—
281	M₁₆	—	86	4	no	—
282	C₂×C₈	—	43	2	no	—
283	C₂×C₈	—	86	2	no	—
284	C₂×C₈	—	86	2	no	—
285	C₂×C₈	#10621898	1	2	yes	—
286	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
287	C₂×C₈	#10621902	1	2	yes	—
288	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	86	2	no	—
294	C₄⋊C₄	—	172	2	no	—
295	C₄⋊C₄	—	172	2	no	—
296	C₄⋊C₄	—	172	2	no	—
297	C₄⋊C₄	—	172	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	172	2	no	—
301	C₂²⋊C₄	—	172	2	no	—
302	C₂²⋊C₄	—	172	2	no	—
303	C₄²	—	4	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	172	2	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	86	2	no	—
309	C₂³	—	43	2	no	—
310	C₂³	D₈×D₄₃	1	1	yes	characteristic
311	Q₈	—	86	2	no	—
312	Q₈	—	344	2	no	—
313	Q₈	—	172	2	no	—
314	Q₈	—	172	2	no	—
315	D₄	—	86	4	no	—
316	D₄	—	86	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	43	2	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	86	2	no	—
324	C₂×C₄	—	344	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	—	688	2	no	—
328	C₂×C₄	—	688	1	no	—
329	C₂×C₄	#13063310	1	2	yes	—
330	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
331	C₂×C₄	#13063357	1	2	yes	—
332	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
333	C₈	—	86	2	no	—
334	C₈	#13063310	1	2	yes	—
335	C₈	C₂×#10621898	1	1	yes	characteristic
336	C₈	C₂×#10621902	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	86	2	no	—
340	C₂²	—	86	2	no	—
341	C₂²	—	86	2	no	—
342	C₂²	#14583214	1	2	yes	—
343	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	86	2	no	—
349	C₄	—	344	2	no	—
350	C₄	—	344	2	no	—
351	C₄	#14583214	1	2	yes	—
352	C₄	C₂×#13063310	1	1	yes	characteristic
353	C₄	C₂×#13063357	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	172	2	no	—
356	C₂	C₂×#14583214	1	1	yes	characteristic
357	C₂	#19147738	1	1	yes	characteristic
358	C₂	#19147745	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×#2158	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#10621898	D₄
hypercenter	#2158	D₄₃
hypocenter	C₄₃	#33651
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #19136806	C₂	semidirect
4. #19136805	C₂	semidirect
5. #19136799	C₂	semidirect
6. #19136794	C₂	semidirect
7. #19136384	C₂	semidirect
8. #19136314	C₂	semidirect
9. C₂×#14582704	C₂	semidirect
11. C₄₃×#2158	C₂	semidirect
182. #2158	D₄₃	semidirect
220. C₄₃	#33651	semidirect

Maximal subgroups

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

subgroup	index
#19144251	2
#19143192	2
#19136806	2
#19136805	2
#19136799	2
#19136794	2
#19136384	2
#19136314	2
C₂×#14582704	2
#19136208	2
C₄₃×#2158	2
#33651	43