Subgroups — #25249686

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	#19136399	C₂	1	4	yes	maximal, complemented
3	#19136391	C₂	1	4	yes	maximal, complemented
4	#19136355	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136353	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136209	C₂	1	1	yes	maximal, characteristic
7	M₃₂×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#2113	C₂	1	1	yes	maximal, characteristic
10	C₂×#13063332	C₄	1	1	yes	characteristic
11	#41×Dic₄₃	C₄	1	1	yes	characteristic
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	#14582857	C₄	1	2	yes	—
15	#14582857	C₄	1	1	yes	characteristic
16	#14582854	C₂²	1	1	yes	characteristic
17	#14582854	C₂²	1	2	yes	—
18	M₁₆×Dic₄₃	C₄	1	1	yes	characteristic
19	M₁₆×Dic₄₃	C₄	1	2	yes	—
20	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
21	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
22	Dic₄₃⋊C₁₆	C₂²	1	4	yes	—
23	Dic₄₃⋊C₁₆	C₂²	1	4	yes	—
24	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
25	#14582674	C₂²	1	2	yes	—
26	C₁₆×Dic₄₃	C₂²	1	2	yes	—
27	C₁₆×Dic₄₃	C₂²	1	2	yes	—
28	C₁₆×Dic₄₃	C₂²	1	2	yes	—
29	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
31	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
32	C₄₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
33	#13063332	—	2	4	no	—
34	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
35	C₂×#10621877	C₂×C₄	1	2	yes	—
36	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	Q₈×Dic₄₃	C₈	1	2	yes	—
39	D₄×Dic₄₃	C₈	1	4	yes	—
40	D₄×Dic₄₃	C₈	1	2	yes	—
41	#13063284	C₂×C₄	1	2	yes	—
42	#13063284	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
45	#13063265	—	2	2	no	—
46	C₂×C₄₃⋊C₁₆	—	2	2	no	—
47	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
49	#13063235	C₂×C₄	1	2	yes	—
50	#13063235	C₂×C₄	1	2	yes	—
51	#13063235	C₂×C₄	1	2	yes	—
52	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
57	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
58	C₈×Dic₄₃	C₂³	1	2	yes	—
59	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
60	#131×C₄₃	—	2	2	no	—
61	#131×C₄₃	—	2	2	no	—
62	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
64	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
65	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
66	M₃₂×C₄₃	—	2	2	no	—
67	C₂×C₁₆×C₄₃	—	2	2	no	—
68	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
69	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
70	#41×C₄₃	—	2	2	no	—
71	#41×C₄₃	—	2	2	no	—
72	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
73	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
74	C₂×D₄×C₄₃	C₂×C₈	1	2	yes	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×Dic₄₃	C₂×C₈	1	4	yes	—
78	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
79	M₁₆×C₄₃	—	2	2	no	—
80	M₁₆×C₄₃	—	2	4	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	#41	1	2	yes	—
86	C₂×C₈×C₄₃	#41	1	2	yes	—
87	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₄₃	C₂×C₈	1	4	yes	—
89	C₂²⋊Dic₄₃	C₂×C₈	1	2	yes	—
90	C₄⋊Dic₄₃	C₂×C₈	1	4	yes	—
91	C₄⋊Dic₄₃	C₂×C₈	1	2	yes	—
92	C₄×Dic₄₃	C₂×C₈	1	4	yes	—
93	C₄×Dic₄₃	C₂×C₈	1	2	yes	—
94	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
95	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
97	#10621877	—	2	2	no	—
98	#10621877	—	2	4	no	—
99	C₂×C₄₃⋊C₈	—	2	4	no	—
100	C₂×C₄₃⋊C₈	M₁₆	1	4	yes	—
101	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
104	C₁₆×C₄₃	—	2	2	no	—
105	C₁₆×C₄₃	—	2	2	no	—
106	C₄₃⋊C₁₆	—	2	2	no	—
107	C₄₃⋊C₁₆	—	2	2	no	—
108	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
109	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
110	Q₈×C₄₃	—	2	2	no	—
111	D₄×C₄₃	—	2	4	no	—
112	D₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	#117	1	2	yes	—
116	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×Dic₄₃	—	2	4	no	—
121	C₂×Dic₄₃	—	2	8	no	—
122	C₂×Dic₄₃	C₂²×C₈	1	4	yes	—
123	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
124	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
125	C₈×C₄₃	—	2	2	no	—
126	C₈×C₄₃	#117	1	2	yes	—
127	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
128	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
129	C₄₃⋊C₈	—	2	2	no	—
130	C₄₃⋊C₈	C₂×M₁₆	1	2	yes	—
131	C₄₃⋊C₈	C₂×M₁₆	1	2	yes	—
132	#33420	—	43	1	no	maximal, Sylow
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	—	2	4	no	—
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	#431	1	2	yes	—
137	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
138	C₄×C₄₃	—	2	2	no	—
139	C₄×C₄₃	#431	1	2	yes	—
140	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
141	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
142	Dic₄₃	—	2	2	no	—
143	Dic₄₃	—	2	4	no	—
144	Dic₄₃	#504	1	4	yes	—
145	C₄×#131	—	43	1	no	—
146	#2164	—	43	4	no	—
147	#2162	—	43	4	no	—
148	#2151	—	43	1	no	—
149	#2151	—	43	1	no	—
150	#2113	—	43	1	no	—
151	#2113	D₄₃	1	1	yes	characteristic
152	C₄×M₃₂	—	43	1	no	—
153	C₂×C₄×C₁₆	—	43	1	no	—
154	C₂×C₄₃	—	2	2	no	—
155	C₂×C₄₃	—	4	2	no	—
156	C₂×C₄₃	C₂×#431	1	1	yes	characteristic
157	C₂×C₄₃	C₂×#504	1	2	yes	—
158	C₂×#131	—	43	1	no	—
159	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
160	C₄×#41	—	43	1	no	—
161	C₂×M₃₂	—	43	1	no	—
162	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
163	C₂²×C₁₆	—	43	1	no	—
164	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
165	#405	—	43	1	no	—
166	#405	—	43	2	no	—
167	#405	—	43	1	no	—
168	#405	—	43	2	no	—
169	C₄×M₁₆	—	43	2	no	—
170	C₄×M₁₆	—	43	1	no	—
171	C₂×C₄×C₈	—	43	1	no	—
172	C₂×C₄×C₈	—	43	2	no	—
173	C₄⋊C₁₆	—	43	4	no	—
174	C₄⋊C₁₆	—	43	4	no	—
175	C₂²⋊C₁₆	—	43	4	no	—
176	C₂²⋊C₁₆	C₂×D₄₃	1	4	yes	—
177	#346	—	43	2	no	—
178	C₄×C₁₆	—	43	2	no	—
179	C₄×C₁₆	—	43	2	no	—
180	C₄×C₁₆	—	43	2	no	—
181	C₄₃	#33420	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	C₄×D₄₃	1	1	yes	characteristic
183	#131	—	2	2	no	—
184	#131	—	2	2	no	—
185	#131	—	86	4	no	—
186	C₂×M₁₆	—	43	2	no	—
187	C₂×M₁₆	—	43	1	no	—
188	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
189	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
190	C₂²×C₈	—	43	1	no	—
191	C₂²×C₈	—	43	2	no	—
192	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
193	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
194	C₄×Q₈	—	43	2	no	—
195	C₄×D₄	—	43	4	no	—
196	C₄×D₄	—	43	2	no	—
197	#117	—	43	2	no	—
198	#117	—	43	1	no	—
199	C₂×C₄²	—	43	2	no	—
200	C₂×C₄²	—	43	1	no	—
201	M₃₂	—	2	2	no	—
202	M₃₂	—	86	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	43	2	no	—
205	C₂×C₁₆	—	43	2	no	—
206	C₂×C₁₆	—	86	2	no	—
207	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
208	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
209	#97	—	43	2	no	—
210	#97	—	43	2	no	—
211	#97	—	43	2	no	—
212	C₄×C₈	—	43	1	no	—
213	C₄×C₈	—	43	2	no	—
214	C₄×C₈	—	43	2	no	—
215	C₄×C₈	—	43	1	no	—
216	C₄×C₈	—	43	1	no	—
217	C₄×C₈	—	43	1	no	—
218	C₄×C₈	—	43	2	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
222	C₂×D₄	C₈×D₄₃	1	2	yes	—
223	C₂×D₄	C₈×D₄₃	1	1	yes	characteristic
224	C₂²×C₄	—	43	4	no	—
225	C₂²×C₄	—	43	2	no	—
226	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	86	4	no	—
231	M₁₆	—	86	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	43	4	no	—
235	C₂×C₈	—	43	1	no	—
236	C₂×C₈	—	43	1	no	—
237	C₂×C₈	—	43	2	no	—
238	C₂×C₈	—	86	4	no	—
239	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
241	C₂×C₈	#10621900	1	2	yes	—
242	C₂×C₈	#10621900	1	2	yes	—
243	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
244	C₄⋊C₄	—	43	4	no	—
245	C₄⋊C₄	—	43	2	no	—
246	C₂²⋊C₄	—	43	4	no	—
247	C₂²⋊C₄	—	43	2	no	—
248	C₄²	—	43	4	no	—
249	C₄²	—	43	2	no	—
250	C₄²	—	43	1	no	—
251	C₄²	—	43	1	no	—
252	C₄²	—	43	2	no	—
253	C₁₆	—	2	2	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	86	2	no	—
256	C₁₆	—	86	2	no	—
257	C₂³	C₂×C₈×D₄₃	1	2	yes	—
258	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
259	Q₈	—	2	2	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	43	2	no	—
265	C₂×C₄	—	43	4	no	—
266	C₂×C₄	—	43	2	no	—
267	C₂×C₄	—	86	4	no	—
268	C₂×C₄	—	86	8	no	—
269	C₂×C₄	#13063281	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	43	2	no	—
276	C₈	—	43	2	no	—
277	C₈	—	86	2	no	—
278	C₈	#13063281	1	2	yes	—
279	C₈	C₂×#10621900	1	1	yes	characteristic
280	C₈	C₂×#10621900	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	#14582949	1	2	yes	—
285	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	43	4	no	—
288	C₄	—	86	2	no	—
289	C₄	—	86	4	no	—
290	C₄	#14582949	1	2	yes	—
291	C₄	C₂×#13063281	1	1	yes	characteristic
292	C₄	C₂×M₁₆×D₄₃	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	C₂×#14582949	1	1	yes	characteristic
296	C₂	#504×D₄₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	C₄₃×#2113	C₂
prime radical	C₂×#41×C₄₃	C₂×C₄
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2113	D₄₃
hypocenter	C₄₃	#33420
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
2. #19136399	C₂	semidirect
3. #19136391	C₂	semidirect
4. #19136355	C₂	semidirect
5. #19136353	C₂	semidirect
7. M₃₂×Dic₄₃	C₂	semidirect
8. C₂×C₁₆×Dic₄₃	C₂	semidirect
181. C₄₃	#33420	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₄₃	2
#19136399	2
#19136391	2
#19136355	2
#19136353	2
#19136209	2
M₃₂×Dic₄₃	2
C₂×C₁₆×Dic₄₃	2
C₄₃×#2113	2
#33420	43