Subgroups — #25005243

Order 11008.

722 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582942	C₂	1	2	yes	maximal
2	C₂²×#13063257	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582667	C₂	1	2	yes	maximal
4	C₂×#14582664	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
8	#14582942	—	2	4	no	—
9	#14582942	—	2	4	no	—
10	C₂×#13063258	—	2	2	no	—
11	C₂×#13063257	—	2	4	no	—
12	C₂×#13063257	—	2	2	no	—
13	C₂×#13063256	—	2	2	no	—
14	#14582667	—	2	4	no	—
15	#14582664	C₄	1	4	yes	complemented
16	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#336	—	2	2	no	—
20	C₂³×Dic₄₃	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	C₂×C₄	1	4	yes	—
25	C₂×#10621877	D₄	1	4	yes	—
26	C₂×#10621877	Q₈	1	4	yes	—
27	C₂²×C₄₃⋊C₈	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	—	2	2	no	—
29	C₂×C₄⋊Dic₄₃	—	2	2	no	—
30	C₂×C₄⋊Dic₄₃	—	2	2	no	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄⋊Dic₄₃	D₄	1	4	yes	—
34	#13063258	—	2	8	no	—
35	#13063257	—	2	8	no	—
36	#13063256	—	2	8	no	—
37	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₄₃	—	2	1	no	—
39	C₂×M₁₆×C₄₃	—	2	2	no	—
40	C₂×M₁₆×C₄₃	—	2	2	no	—
41	C₂×M₁₆×C₄₃	Q₈	1	2	yes	—
42	C₂²×C₈×C₄₃	—	2	2	no	—
43	C₂²×C₈×C₄₃	—	2	2	no	—
44	C₂²×C₈×C₄₃	—	2	1	no	—
45	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₃	D₄	1	2	yes	—
47	C₂²×C₈×C₄₃	Q₈	1	2	yes	—
48	#117×C₄₃	—	2	2	no	—
49	#117×C₄₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
52	C₂×C₄²×C₄₃	—	2	1	no	—
53	C₂⁴×C₄₃	#41	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	2	yes	—
65	C₂²×Dic₄₃	—	2	4	no	—
66	C₂²×Dic₄₃	—	4	2	no	—
67	C₂²×Dic₄₃	—	4	2	no	—
68	C₂²×Dic₄₃	—	8	4	no	—
69	M₁₆×C₄₃	—	2	4	no	—
70	M₁₆×C₄₃	—	4	2	no	—
71	C₂×C₈×C₄₃	—	2	4	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	—	4	1	no	—
77	C₂×C₈×C₄₃	—	4	1	no	—
78	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
79	C₄⋊C₄×C₄₃	—	2	2	no	—
80	C₄⋊C₄×C₄₃	—	2	2	no	—
81	C₂²⋊C₄×C₄₃	—	4	2	no	—
82	C₄²×C₄₃	—	4	1	no	—
83	C₄²×C₄₃	—	4	1	no	—
84	C₄⋊Dic₄₃	—	2	4	no	—
85	C₄⋊Dic₄₃	—	2	4	no	—
86	C₄⋊Dic₄₃	—	4	2	no	—
87	C₄⋊Dic₄₃	—	4	2	no	—
88	#10621877	—	2	4	no	—
89	#10621877	—	2	4	no	—
90	#10621877	—	2	4	no	—
91	#10621877	C₄⋊C₄	1	8	yes	—
92	C₂×C₄₃⋊C₈	—	2	4	no	—
93	C₂×C₄₃⋊C₈	—	2	4	no	—
94	C₂×C₄₃⋊C₈	—	2	4	no	—
95	C₂³×C₄₃	—	2	4	no	—
96	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
97	C₂³×C₄₃	#121	1	2	yes	—
98	C₂³×C₄₃	#121	1	1	yes	characteristic
99	C₂³×C₄₃	#125	1	2	yes	—
100	C₂³×C₄₃	#125	1	1	yes	characteristic
101	C₂×C₄×C₄₃	—	4	2	no	—
102	C₂×C₄×C₄₃	—	4	4	no	—
103	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#120	1	2	yes	—
109	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#123	1	2	yes	—
116	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×Dic₄₃	—	4	4	no	—
122	C₂×Dic₄₃	—	8	4	no	—
123	C₂×Dic₄₃	—	8	4	no	—
124	C₂×Dic₄₃	—	8	4	no	—
125	C₈×C₄₃	—	2	4	no	—
126	C₈×C₄₃	—	4	2	no	—
127	C₄₃⋊C₈	—	2	8	no	—
128	#11330	—	43	1	no	maximal, Sylow
129	C₂²×C₄₃	—	2	4	no	—
130	C₂²×C₄₃	—	2	2	no	—
131	C₂²×C₄₃	—	2	2	no	—
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	#389	1	2	yes	—
135	C₂²×C₄₃	#389	1	1	yes	characteristic
136	C₂²×C₄₃	#392	1	1	yes	characteristic
137	C₂²×C₄₃	#398	1	1	yes	characteristic
138	C₂²×C₄₃	#398	1	2	yes	—
139	C₂²×C₄₃	#471	1	4	yes	—
140	C₄×C₄₃	—	8	2	no	—
141	C₄×C₄₃	#384	1	1	yes	characteristic
142	C₄×C₄₃	#385	1	1	yes	characteristic
143	C₄×C₄₃	#385	1	2	yes	—
144	C₄×C₄₃	#395	1	1	yes	characteristic
145	C₄×C₄₃	#395	1	2	yes	—
146	C₄×C₄₃	#399	1	1	yes	characteristic
147	Dic₄₃	—	8	4	no	—
148	C₂×#428	—	43	2	no	—
149	C₂²×#106	—	43	1	no	—
150	C₂×#343	—	43	2	no	—
151	C₂×#341	—	43	1	no	—
152	#1729	D₄₃	1	1	yes	characteristic
153	C₂×C₄₃	—	2	2	no	—
154	C₂×C₄₃	—	2	2	no	—
155	C₂×C₄₃	#1444	1	1	yes	characteristic
156	C₂×C₄₃	#1951	1	2	yes	—
157	C₂×C₄₃	#2019	1	2	yes	—
158	C₂×C₄₃	#2090	1	2	yes	—
159	C₂²×M₁₆	—	43	2	no	—
160	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
161	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
162	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
163	C₂²×C₄⋊C₄	—	43	2	no	—
164	#428	—	86	4	no	—
165	#428	—	86	4	no	—
166	C₂×#107	—	86	2	no	—
167	C₂×#106	—	86	4	no	—
168	C₂×#106	—	86	2	no	—
169	C₂×#106	—	86	2	no	—
170	#343	—	86	4	no	—
171	#341	—	43	4	no	—
172	#336	—	2	2	no	—
173	C₄₃	#11330	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	—	86	2	no	—
175	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	43	4	no	—
180	C₂×M₁₆	—	43	4	no	—
181	C₂×M₁₆	—	43	4	no	—
182	C₂×M₁₆	—	86	2	no	—
183	C₂×M₁₆	—	86	2	no	—
184	C₂×M₁₆	—	86	2	no	—
185	C₂×M₁₆	Dic₈₆	1	2	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	86	2	no	—
190	C₂²×C₈	Dic₈₆	1	2	yes	—
191	C₂²×C₈	C₄×D₄₃	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	C₂×C₄⋊C₄	—	43	4	no	—
196	C₂×C₄⋊C₄	—	86	2	no	—
197	C₂×C₄⋊C₄	—	86	2	no	—
198	C₂×C₄⋊C₄	—	86	2	no	—
199	C₂×C₄⋊C₄	—	86	2	no	—
200	C₂×C₄⋊C₄	—	86	2	no	—
201	C₂×C₄⋊C₄	D₁₇₂	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	C₂×C₄²	—	2	1	no	—
204	#107	—	86	8	no	—
205	#106	—	86	8	no	—
206	#106	—	86	8	no	—
207	C₂⁴	#10621898	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	86	4	no	—
210	C₂²×C₄	—	344	4	no	—
211	C₂²×C₄	—	172	2	no	—
212	C₂²×C₄	—	172	2	no	—
213	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
214	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
217	C₂²×C₄	#10621898	1	1	yes	characteristic
218	C₂²×C₄	D₄×D₄₃	1	2	yes	—
219	C₂²×C₄	D₄×D₄₃	1	2	yes	—
220	C₂²×C₄	#10621900	1	2	yes	—
221	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	43	8	no	—
226	M₁₆	—	86	4	no	—
227	M₁₆	—	86	4	no	—
228	M₁₆	—	86	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	86	4	no	—
237	C₂×C₈	—	86	4	no	—
238	C₂×C₈	—	86	4	no	—
239	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	86	4	no	—
243	C₄⋊C₄	—	86	4	no	—
244	C₄⋊C₄	—	172	2	no	—
245	C₄⋊C₄	—	172	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	1	no	—
249	C₂³	—	2	4	no	—
250	C₂³	C₄×Dic₈₆	1	1	yes	characteristic
251	C₂³	#13063298	1	2	yes	—
252	C₂³	#13063299	1	1	yes	characteristic
253	C₂³	#13063315	1	2	yes	—
254	C₂³	#13063317	1	1	yes	characteristic
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	344	4	no	—
258	C₂×C₄	—	344	4	no	—
259	C₂×C₄	—	344	4	no	—
260	C₂×C₄	—	172	4	no	—
261	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
262	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
263	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
264	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
265	C₂×C₄	#13063293	1	1	yes	characteristic
266	C₂×C₄	#13063295	1	2	yes	—
267	C₂×C₄	#13063302	1	2	yes	—
268	C₂×C₄	#13063302	1	2	yes	—
269	C₂×C₄	#13063303	1	2	yes	—
270	C₂×C₄	#13063304	1	1	yes	characteristic
271	C₂×C₄	#13063305	1	2	yes	—
272	C₂×C₄	#13063306	1	1	yes	characteristic
273	C₂×C₄	#13063307	1	2	yes	—
274	C₂×C₄	#13063321	1	1	yes	characteristic
275	C₂×C₄	#13063323	1	1	yes	characteristic
276	C₂×C₄	#13063324	1	2	yes	—
277	C₂×C₄	#13063324	1	2	yes	—
278	C₂×C₄	#13063325	1	1	yes	characteristic
279	C₈	—	2	4	no	—
280	C₈	—	4	2	no	—
281	C₈	—	86	8	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	#14582796	1	2	yes	—
288	C₂²	#14582797	1	1	yes	characteristic
289	C₂²	#14582808	1	1	yes	characteristic
290	C₂²	#14582829	1	1	yes	characteristic
291	C₂²	#14582833	1	2	yes	—
292	C₂²	#14583162	1	4	yes	—
293	C₄	—	8	2	no	—
294	C₄	—	344	4	no	—
295	C₄	#14582768	1	1	yes	characteristic
296	C₄	#14582771	1	1	yes	characteristic
297	C₄	#14582773	1	2	yes	—
298	C₄	#14582819	1	1	yes	characteristic
299	C₄	#14582820	1	2	yes	—
300	C₄	#14582835	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	2	2	no	—
303	C₂	#19132859	1	1	yes	characteristic
304	C₂	#19135375	1	2	yes	—
305	C₂	#19135710	1	2	yes	—
306	C₂	#19136105	1	2	yes	—

	subgroup	G/H
center	C₂³	#13063299
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1729	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1729	D₄₃
hypocenter	C₄₃	#11330
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
15. #14582664	C₄	semidirect
173. C₄₃	#11330	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582942	2
C₂²×#13063257	2
C₂×#14582667	2
C₂×#14582664	2
C₄₃×#1729	2
#11330	43