Subgroups — #24989457

Order 11008.

462 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582856	C₂	1	1	yes	maximal, characteristic
2	#19133341	C₂	1	2	yes	maximal, complemented
3	#19132907	C₂	1	2	yes	maximal
4	#19132426	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1275	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10621872	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₄₃	C₄	1	1	yes	characteristic
8	C₂×#13063282	C₂²	1	1	yes	characteristic
9	#14582952	C₂²	1	2	yes	—
10	#14582856	—	2	2	no	—
11	#14582856	—	2	2	no	—
12	C₂×#13063235	C₄	1	1	yes	characteristic
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₄₃	C₄	1	1	yes	characteristic
15	#14582651	—	2	2	no	—
16	#14582649	—	2	2	no	—
17	#14582624	—	2	2	no	—
18	#14582622	—	2	2	no	—
19	C₄₃×#431	C₂²	1	2	yes	—
20	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#10621872	—	2	2	no	—
23	C₂×#10621872	—	2	2	no	—
24	C₂×#10621872	—	2	2	no	—
25	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₄₃	—	2	1	no	—
28	C₂×C₈×D₄₃	—	2	1	no	—
29	C₂×C₈×D₄₃	—	2	2	no	—
30	C₂×C₈×D₄₃	C₈	1	4	yes	complemented
31	#13063282	—	2	2	no	—
32	#13063282	—	2	2	no	—
33	C₂×Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
34	C₂×Dic₄₃⋊C₄	Q₈	1	1	yes	characteristic
35	C₂×(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
36	C₂×(C₂×D₄₃)⋊C₄	Q₈	1	1	yes	characteristic
37	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
39	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
40	#13063235	—	2	2	no	—
41	C₄×C₄₃⋊C₈	—	2	2	no	—
42	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
43	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
44	C₈×Dic₄₃	—	2	1	no	—
45	C₈×Dic₄₃	—	2	1	no	—
46	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₃	—	2	2	no	—
50	C₂²⋊C₈×C₄₃	—	2	2	no	—
51	#97×C₄₃	—	2	2	no	—
52	C₄×C₈×C₄₃	—	2	2	no	—
53	C₂³×D₄₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×D₄₃	—	2	1	no	—
61	C₂×C₄×D₄₃	—	2	1	no	—
62	C₂×C₄×D₄₃	—	2	1	no	—
63	C₂×C₄×D₄₃	—	2	1	no	—
64	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₄₃	M₁₆	1	2	yes	—
66	C₂×C₈×C₄₃	—	2	1	no	—
67	C₂×C₈×C₄₃	—	2	1	no	—
68	C₂×C₈×C₄₃	—	2	1	no	—
69	C₂×C₈×C₄₃	—	2	1	no	—
70	C₂×C₈×C₄₃	—	4	2	no	—
71	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
73	C₄²×C₄₃	C₂×D₄	1	2	yes	—
74	C₄²×C₄₃	#41	1	2	yes	—
75	(C₂×D₄₃)⋊C₄	—	2	2	no	—
76	(C₂×D₄₃)⋊C₄	—	2	2	no	—
77	Dic₄₃⋊C₄	—	2	2	no	—
78	Dic₄₃⋊C₄	—	2	2	no	—
79	C₄×Dic₄₃	—	2	1	no	—
80	C₄×Dic₄₃	—	2	1	no	—
81	C₂×C₄₃⋊C₈	—	2	1	no	—
82	C₂×C₄₃⋊C₈	—	2	1	no	—
83	C₂×C₄₃⋊C₈	—	4	2	no	—
84	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
88	C₂×C₄₃⋊C₈	M₁₆	1	2	yes	—
89	#10621872	—	2	8	no	—
90	C₈×D₄₃	—	2	4	no	—
91	C₈×D₄₃	—	2	4	no	—
92	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×D₄₃	—	2	1	no	—
94	C₂²×D₄₃	—	2	1	no	—
95	C₂²×D₄₃	—	4	1	no	—
96	C₂²×D₄₃	—	4	1	no	—
97	C₂²×D₄₃	C₄⋊C₈	1	2	yes	—
98	C₂×C₄×C₄₃	—	4	1	no	—
99	C₂×C₄×C₄₃	—	4	1	no	—
100	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₄₃	#121	1	2	yes	—
103	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×Dic₄₃	—	2	1	no	—
109	C₂×Dic₄₃	—	2	1	no	—
110	C₂×Dic₄₃	—	2	1	no	—
111	C₂×Dic₄₃	—	2	1	no	—
112	C₂×Dic₄₃	—	4	1	no	—
113	C₂×Dic₄₃	—	4	1	no	—
114	C₂×Dic₄₃	C₄⋊C₈	1	2	yes	—
115	C₄×D₄₃	—	2	2	no	—
116	C₄×D₄₃	—	2	2	no	—
117	C₄×D₄₃	—	2	2	no	—
118	C₄×D₄₃	—	2	2	no	—
119	C₈×C₄₃	—	2	2	no	—
120	C₈×C₄₃	—	2	2	no	—
121	C₈×C₄₃	—	8	2	no	—
122	C₄₃⋊C₈	—	2	2	no	—
123	C₄₃⋊C₈	—	8	2	no	—
124	C₄₃⋊C₈	C₂²⋊C₈	1	2	yes	—
125	C₄₃⋊C₈	C₂²⋊C₈	1	2	yes	—
126	#8601	—	43	1	no	maximal, Sylow
127	C₂²×C₄₃	—	2	1	no	—
128	C₂²×C₄₃	—	2	1	no	—
129	C₂²×C₄₃	#380	1	1	yes	characteristic
130	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂²×C₄₃	#423	1	1	yes	characteristic
132	C₂×D₄₃	—	2	2	no	—
133	C₂×D₄₃	—	4	2	no	—
134	C₂×D₄₃	—	4	1	no	—
135	C₂×D₄₃	—	4	1	no	—
136	C₂×D₄₃	—	4	2	no	—
137	C₄×C₄₃	—	2	2	no	—
138	C₄×C₄₃	—	2	2	no	—
139	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₄×C₄₃	#407	1	1	yes	characteristic
141	C₄×C₄₃	#431	1	1	yes	characteristic
142	C₄×C₄₃	#432	1	1	yes	characteristic
143	Dic₄₃	—	2	2	no	—
144	Dic₄₃	—	4	1	no	—
145	Dic₄₃	—	4	1	no	—
146	C₂×#405	—	43	1	no	—
147	#1564	—	43	2	no	—
148	#1458	—	43	2	no	—
149	#1275	—	43	1	no	—
150	#1275	D₄₃	1	1	yes	characteristic, complemented
151	C₂×C₄₃	—	2	1	no	—
152	C₂×C₄₃	—	2	1	no	—
153	C₂×C₄₃	#1797	1	1	yes	characteristic
154	C₂×C₄₃	#1812	1	1	yes	characteristic
155	C₂×C₄₃	#1813	1	1	yes	characteristic
156	D₄₃	—	4	2	no	—
157	C₂²×M₁₆	—	43	1	no	—
158	C₂³×C₈	—	43	1	no	—
159	C₂×#117	—	43	1	no	—
160	#431	—	43	2	no	—
161	#431	C₂×D₄₃	1	2	yes	—
162	#405	—	86	2	no	—
163	#405	—	86	2	no	—
164	C₂×#97	—	43	1	no	—
165	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
166	C₂×C₄×C₈	—	43	1	no	—
167	C₂×C₄×C₈	—	43	1	no	—
168	#335	—	86	2	no	—
169	#334	—	86	2	no	—
170	#324	—	86	2	no	—
171	#323	—	86	2	no	—
172	C₄₃	#8601	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	43	1	no	—
174	C₂×M₁₆	—	86	2	no	—
175	C₂×M₁₆	—	86	2	no	—
176	C₂×M₁₆	—	86	2	no	—
177	C₂²×C₈	—	43	1	no	—
178	C₂²×C₈	—	43	1	no	—
179	C₂²×C₈	—	43	4	no	—
180	C₂²×C₈	—	86	1	no	—
181	C₂²×C₈	—	86	1	no	—
182	C₂²×C₈	—	86	2	no	—
183	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
184	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
185	#117	—	86	2	no	—
186	#117	—	86	2	no	—
187	C₂×C₄⋊C₄	—	43	1	no	—
188	C₂×C₄⋊C₄	—	43	1	no	—
189	C₂×C₂²⋊C₄	—	43	1	no	—
190	C₂×C₂²⋊C₄	—	43	1	no	—
191	C₂×C₄²	—	43	1	no	—
192	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	86	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	—	86	2	no	—
197	#97	—	2	2	no	—
198	#97	—	86	2	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	—	43	2	no	—
201	C₄×C₈	—	43	2	no	—
202	C₄×C₈	—	86	1	no	—
203	C₄×C₈	—	86	1	no	—
204	C₄×C₈	—	86	2	no	—
205	C₂⁴	—	43	1	no	—
206	C₂²×C₄	—	43	1	no	—
207	C₂²×C₄	—	43	2	no	—
208	C₂²×C₄	—	43	2	no	—
209	C₂²×C₄	—	43	1	no	—
210	C₂²×C₄	—	43	1	no	—
211	C₂²×C₄	—	86	1	no	—
212	C₂²×C₄	—	86	1	no	—
213	C₂²×C₄	—	86	1	no	—
214	C₂²×C₄	—	86	1	no	—
215	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
217	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
218	M₁₆	—	86	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	43	1	no	—
225	C₂×C₈	—	43	1	no	—
226	C₂×C₈	—	43	2	no	—
227	C₂×C₈	—	43	2	no	—
228	C₂×C₈	—	43	2	no	—
229	C₂×C₈	—	86	4	no	—
230	C₂×C₈	—	86	4	no	—
231	C₂×C₈	—	86	1	no	—
232	C₂×C₈	—	86	1	no	—
233	C₂×C₈	—	172	2	no	—
234	C₂×C₈	C₈×D₄₃	1	2	yes	—
235	C₂×C₈	#10621872	1	2	yes	—
236	C₄⋊C₄	—	86	2	no	—
237	C₄⋊C₄	—	86	2	no	—
238	C₂²⋊C₄	—	86	2	no	—
239	C₂²⋊C₄	—	86	2	no	—
240	C₄²	—	86	1	no	—
241	C₄²	—	86	1	no	—
242	C₄²	#10621898	1	2	yes	—
243	C₄²	D₄×D₄₃	1	2	yes	—
244	C₂³	—	43	2	no	—
245	C₂³	—	86	1	no	—
246	C₂³	—	86	1	no	—
247	C₂³	—	172	1	no	—
248	C₂³	—	172	1	no	—
249	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	43	2	no	—
253	C₂×C₄	—	86	1	no	—
254	C₂×C₄	—	86	1	no	—
255	C₂×C₄	—	86	2	no	—
256	C₂×C₄	—	86	2	no	—
257	C₂×C₄	—	86	2	no	—
258	C₂×C₄	—	86	2	no	—
259	C₂×C₄	—	86	1	no	—
260	C₂×C₄	—	86	1	no	—
261	C₂×C₄	—	172	1	no	—
262	C₂×C₄	—	172	1	no	—
263	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
264	C₂×C₄	#13063282	1	1	yes	characteristic
265	C₂×C₄	#13063298	1	2	yes	—
266	C₂×C₄	#13063303	1	2	yes	—
267	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
268	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
269	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
270	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	2	no	—
273	C₈	—	8	2	no	—
274	C₈	—	43	2	no	—
275	C₈	—	43	2	no	—
276	C₈	—	86	2	no	—
277	C₈	—	344	2	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	86	2	no	—
281	C₂²	—	172	2	no	—
282	C₂²	—	172	2	no	—
283	C₂²	—	172	1	no	—
284	C₂²	—	172	1	no	—
285	C₂²	#14582758	1	1	yes	characteristic
286	C₂²	C₄⋊C₈×D₄₃	1	1	yes	characteristic
287	C₂²	#14582921	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	86	2	no	—
291	C₄	—	172	1	no	—
292	C₄	—	172	1	no	—
293	C₄	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
294	C₄	#14582861	1	1	yes	characteristic
295	C₄	#14582950	1	1	yes	characteristic
296	C₄	#14582954	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	172	2	no	—
300	C₂	#19134574	1	1	yes	characteristic
301	C₂	#19134640	1	1	yes	characteristic
302	C₂	#19134644	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063282
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1275	C₂
socle	C₂²×C₄₃	#380
prime radical	C₂³×D₄₃	C₄⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1275	D₄₃
hypocenter	C₄₃	#8601
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1275

Complements

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

N	H ≅ G/N	type
2. #19133341	C₂	semidirect
4. #19132426	C₂	semidirect
5. C₄₃×#1275	C₂	semidirect
30. C₂×C₈×D₄₃	C₈	semidirect
150. #1275	D₄₃	semidirect
172. C₄₃	#8601	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582856	2
#19133341	2
#19132907	2
#19132426	2
C₄₃×#1275	2
#8601	43