Subgroups — #25005428

Order 11008.

634 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	C₂²×#13063259	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582667	C₂	1	2	yes	maximal
3	C₂×#14582664	C₂	1	1	yes	maximal, characteristic
4	#19134316	C₂	1	2	yes	maximal
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	C₂×#13063259	—	2	2	no	—
9	C₂×#13063259	—	2	2	no	—
10	C₂×#13063259	C₄	1	4	yes	complemented
11	#14582667	—	2	4	no	—
12	#14582664	—	2	2	no	—
13	#14582661	—	2	4	no	—
14	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#336	—	2	2	no	—
18	C₂³×Dic₄₃	—	2	2	no	—
19	C₂×#10621877	—	2	2	no	—
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	Q₈	1	4	yes	—
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	—	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	C₂×C₄	1	4	yes	—
32	#13063259	—	2	8	no	—
33	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₄₃	—	2	1	no	—
35	C₂×M₁₆×C₄₃	—	2	2	no	—
36	C₂×M₁₆×C₄₃	—	2	2	no	—
37	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
38	C₂²×C₈×C₄₃	—	2	2	no	—
39	C₂²×C₈×C₄₃	—	2	2	no	—
40	C₂²×C₈×C₄₃	—	2	1	no	—
41	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₄₃	D₄	1	2	yes	—
43	C₂²×C₈×C₄₃	D₄	1	2	yes	—
44	#117×C₄₃	—	2	2	no	—
45	#117×C₄₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
48	C₂×C₄²×C₄₃	—	2	1	no	—
49	C₂⁴×C₄₃	#41	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	#41	1	2	yes	—
59	C₂²×C₄×C₄₃	#41	1	2	yes	—
60	C₂²×C₄×C₄₃	#41	1	2	yes	—
61	C₂²×Dic₄₃	—	2	4	no	—
62	C₂²×Dic₄₃	—	4	2	no	—
63	C₂²×Dic₄₃	—	4	2	no	—
64	C₂²×Dic₄₃	—	8	4	no	—
65	M₁₆×C₄₃	—	2	4	no	—
66	M₁₆×C₄₃	—	4	2	no	—
67	C₂×C₈×C₄₃	—	2	4	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	4	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	4	1	no	—
73	C₂×C₈×C₄₃	—	4	1	no	—
74	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
75	C₄⋊C₄×C₄₃	—	2	4	no	—
76	C₂²⋊C₄×C₄₃	—	4	2	no	—
77	C₄²×C₄₃	—	4	1	no	—
78	C₄²×C₄₃	—	4	1	no	—
79	C₄⋊Dic₄₃	—	2	4	no	—
80	C₄⋊Dic₄₃	—	2	4	no	—
81	C₄⋊Dic₄₃	—	4	2	no	—
82	C₄⋊Dic₄₃	—	4	2	no	—
83	#10621877	—	2	4	no	—
84	#10621877	—	2	4	no	—
85	#10621877	—	4	2	no	—
86	#10621877	—	4	2	no	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	4	2	no	—
89	C₂×C₄₃⋊C₈	—	4	2	no	—
90	C₂³×C₄₃	—	2	2	no	—
91	C₂³×C₄₃	—	2	2	no	—
92	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₄₃	#121	1	1	yes	characteristic
94	C₂³×C₄₃	#124	1	1	yes	characteristic
95	C₂³×C₄₃	#125	1	2	yes	—
96	C₂³×C₄₃	#126	1	2	yes	—
97	C₂×C₄×C₄₃	—	4	2	no	—
98	C₂×C₄×C₄₃	—	4	2	no	—
99	C₂×C₄×C₄₃	—	4	2	no	—
100	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#117	1	2	yes	—
102	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#123	1	2	yes	—
111	C₂×C₄×C₄₃	#123	1	2	yes	—
112	C₂×C₄×C₄₃	#123	1	2	yes	—
113	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#125	1	2	yes	—
115	C₂×C₄×C₄₃	#126	1	2	yes	—
116	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×Dic₄₃	—	4	4	no	—
119	C₂×Dic₄₃	—	8	4	no	—
120	C₂×Dic₄₃	—	8	4	no	—
121	C₂×Dic₄₃	—	8	4	no	—
122	C₈×C₄₃	—	2	4	no	—
123	C₈×C₄₃	—	4	2	no	—
124	C₄₃⋊C₈	—	4	4	no	—
125	#11365	—	43	1	no	maximal, Sylow
126	C₂²×C₄₃	—	2	4	no	—
127	C₂²×C₄₃	—	2	4	no	—
128	C₂²×C₄₃	—	2	1	no	—
129	C₂²×C₄₃	—	2	1	no	—
130	C₂²×C₄₃	—	2	1	no	—
131	C₂²×C₄₃	—	2	1	no	—
132	C₂²×C₄₃	#387	1	2	yes	—
133	C₂²×C₄₃	#390	1	1	yes	characteristic
134	C₂²×C₄₃	#397	1	1	yes	characteristic
135	C₂²×C₄₃	#398	1	1	yes	characteristic
136	C₂²×C₄₃	#400	1	2	yes	—
137	C₂²×C₄₃	#472	1	2	yes	—
138	C₂²×C₄₃	#473	1	2	yes	—
139	C₄×C₄₃	—	8	2	no	—
140	C₄×C₄₃	#381	1	1	yes	characteristic
141	C₄×C₄₃	#387	1	2	yes	—
142	C₄×C₄₃	#391	1	1	yes	characteristic
143	C₄×C₄₃	#397	1	1	yes	characteristic
144	C₄×C₄₃	#398	1	1	yes	characteristic
145	C₄×C₄₃	#399	1	2	yes	—
146	Dic₄₃	—	8	4	no	—
147	C₂²×#107	—	43	1	no	—
148	C₂×#343	—	43	2	no	—
149	C₂×#341	—	43	1	no	—
150	#1736	—	43	2	no	—
151	#1729	D₄₃	1	1	yes	characteristic
152	C₂×C₄₃	—	2	2	no	—
153	C₂×C₄₃	—	2	2	no	—
154	C₂×C₄₃	#1445	1	1	yes	characteristic
155	C₂×C₄₃	#1978	1	1	yes	characteristic
156	C₂×C₄₃	#1979	1	1	yes	characteristic
157	C₂×C₄₃	#2078	1	2	yes	—
158	C₂×C₄₃	#2089	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	C₂×#107	—	43	4	no	—
165	C₂×#107	—	86	2	no	—
166	C₂×#107	—	86	2	no	—
167	#343	—	86	4	no	—
168	#341	—	86	2	no	—
169	#340	—	86	4	no	—
170	#336	—	2	2	no	—
171	C₄₃	#11365	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	—	86	2	no	—
173	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	43	4	no	—
178	C₂×M₁₆	—	86	2	no	—
179	C₂×M₁₆	—	86	2	no	—
180	C₂×M₁₆	—	86	2	no	—
181	C₂×M₁₆	—	86	2	no	—
182	C₂×M₁₆	—	86	2	no	—
183	C₂×M₁₆	D₁₇₂	1	2	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	86	2	no	—
188	C₂²×C₈	C₄×D₄₃	1	2	yes	—
189	C₂²×C₈	D₁₇₂	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	C₂×C₄⋊C₄	—	43	4	no	—
194	C₂×C₄⋊C₄	—	86	2	no	—
195	C₂×C₄⋊C₄	—	86	2	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	D₁₇₂	1	2	yes	—
200	C₂×C₂²⋊C₄	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	#107	—	86	8	no	—
203	C₂⁴	#10621898	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	86	4	no	—
206	C₂²×C₄	—	344	4	no	—
207	C₂²×C₄	—	172	2	no	—
208	C₂²×C₄	—	172	2	no	—
209	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
213	C₂²×C₄	#10621898	1	1	yes	characteristic
214	C₂²×C₄	#10621900	1	2	yes	—
215	C₂²×C₄	#10621900	1	2	yes	—
216	C₂²×C₄	#10621900	1	2	yes	—
217	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	86	4	no	—
222	M₁₆	—	86	4	no	—
223	M₁₆	—	172	2	no	—
224	M₁₆	—	172	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	86	4	no	—
233	C₂×C₈	—	172	2	no	—
234	C₂×C₈	—	172	2	no	—
235	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	86	4	no	—
238	C₄⋊C₄	—	86	4	no	—
239	C₄⋊C₄	—	172	2	no	—
240	C₄⋊C₄	—	172	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
247	C₂³	#13063299	1	1	yes	characteristic
248	C₂³	#13063313	1	1	yes	characteristic
249	C₂³	#13063315	1	2	yes	—
250	C₂³	#13063318	1	2	yes	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	344	4	no	—
255	C₂×C₄	—	344	4	no	—
256	C₂×C₄	—	344	4	no	—
257	C₂×C₄	—	172	4	no	—
258	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	#13063281	1	2	yes	—
260	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
261	C₂×C₄	#13063290	1	2	yes	—
262	C₂×C₄	#13063299	1	1	yes	characteristic
263	C₂×C₄	#13063305	1	2	yes	—
264	C₂×C₄	#13063305	1	2	yes	—
265	C₂×C₄	#13063305	1	2	yes	—
266	C₂×C₄	#13063306	1	1	yes	characteristic
267	C₂×C₄	#13063306	1	1	yes	characteristic
268	C₂×C₄	#13063309	1	2	yes	—
269	C₂×C₄	#13063309	1	2	yes	—
270	C₂×C₄	#13063309	1	2	yes	—
271	C₂×C₄	#13063312	1	1	yes	characteristic
272	C₂×C₄	#13063315	1	2	yes	—
273	C₂×C₄	#13063318	1	2	yes	—
274	C₂×C₄	#13063322	1	1	yes	characteristic
275	C₂×C₄	#13063323	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	4	2	no	—
278	C₈	—	172	4	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	#14582784	1	2	yes	—
286	C₂²	#14582801	1	1	yes	characteristic
287	C₂²	#14582824	1	1	yes	characteristic
288	C₂²	#14582829	1	1	yes	characteristic
289	C₂²	#14582839	1	2	yes	—
290	C₂²	#14583169	1	2	yes	—
291	C₂²	#14583174	1	2	yes	—
292	C₄	—	8	2	no	—
293	C₄	—	344	4	no	—
294	C₄	#14582760	1	1	yes	characteristic
295	C₄	#14582784	1	2	yes	—
296	C₄	#14582804	1	1	yes	characteristic
297	C₄	#14582824	1	1	yes	characteristic
298	C₄	#14582829	1	1	yes	characteristic
299	C₄	#14582836	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	#19132862	1	1	yes	characteristic
303	C₂	#19135505	1	1	yes	characteristic
304	C₂	#19135510	1	1	yes	characteristic
305	C₂	#19136035	1	2	yes	—
306	C₂	#19136100	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	#11365
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
10. C₂×#13063259	C₄	semidirect
171. C₄₃	#11365	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063259	2
C₂×#14582667	2
C₂×#14582664	2
#19134316	2
C₄₃×#1729	2
#11365	43