Subgroups — #24987308

Order 11008.

554 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063252	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132876	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132432	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1476	C₂	1	1	yes	maximal, characteristic
8	Q₈×C₄₃⋊C₈	C₄	1	2	yes	complemented
9	D₄×C₄₃⋊C₈	—	2	2	no	—
10	D₄×C₄₃⋊C₈	C₄	1	2	yes	complemented
11	#14582952	—	2	1	no	—
12	#14582952	C₂²	1	1	yes	characteristic
13	C₂×#13063252	C₄	1	2	yes	—
14	C₄×#10621877	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₃	—	2	2	no	—
18	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
19	#14582658	C₄	1	2	yes	—
20	#14582621	—	2	1	no	—
21	C₈×C₄₃⋊C₈	—	2	1	no	—
22	C₄×#41×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#431	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×Q₈⋊C₈	C₄	1	2	yes	complemented
26	C₄₃×D₄⋊C₈	C₄	1	2	yes	complemented
27	C₂×#10621877	C₂×C₄	1	2	yes	—
28	C₂²×C₄₃⋊C₈	—	2	2	no	—
29	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
30	C₂×C₄×Dic₄₃	—	2	1	no	—
31	C₂×C₄×Dic₄₃	—	2	1	no	—
32	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
33	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
34	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
35	#13063252	C₈	1	8	yes	complemented
36	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
37	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
38	#13063236	—	2	1	no	—
39	C₄×C₄₃⋊C₈	—	2	1	no	—
40	C₄×C₄₃⋊C₈	—	2	1	no	—
41	C₄×C₄₃⋊C₈	—	2	2	no	—
42	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
44	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
45	C₈×Dic₄₃	—	2	8	no	—
46	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	—	2	1	no	—
48	C₄×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₄×D₄×C₄₃	—	2	1	no	—
51	C₄×D₄×C₄₃	—	2	1	no	—
52	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
54	#117×C₄₃	—	2	1	no	—
55	#117×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	—	2	1	no	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
59	C₂²⋊C₈×C₄₃	—	2	1	no	—
60	C₄×C₈×C₄₃	—	2	1	no	—
61	C₄×C₈×C₄₃	—	2	1	no	—
62	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
63	#41×C₄₃	—	2	2	no	—
64	#41×C₄₃	C₂×C₈	1	4	yes	—
65	C₂×Q₈×C₄₃	C₄²	1	1	yes	characteristic
66	C₂×D₄×C₄₃	—	2	1	no	—
67	C₂×D₄×C₄₃	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	—	2	1	no	—
69	C₂²×C₄×C₄₃	—	2	1	no	—
70	C₂²×C₄×C₄₃	—	2	1	no	—
71	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
74	C₂²×Dic₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	—	2	4	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	2	no	—
78	C₄⋊C₄×C₄₃	—	2	1	no	—
79	C₄⋊C₄×C₄₃	—	2	1	no	—
80	C₄⋊C₄×C₄₃	C₄²	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	C₄²	1	1	yes	characteristic
82	C₂²⋊C₄×C₄₃	—	2	1	no	—
83	C₂²⋊C₄×C₄₃	—	2	1	no	—
84	C₂²⋊C₄×C₄₃	—	2	1	no	—
85	C₄²×C₄₃	—	2	1	no	—
86	C₄²×C₄₃	—	2	1	no	—
87	C₄²×C₄₃	—	2	1	no	—
88	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₃	#41	1	1	yes	characteristic
92	C₄×Dic₄₃	—	2	4	no	—
93	C₄×Dic₄₃	—	2	4	no	—
94	C₄×Dic₄₃	—	2	2	no	—
95	C₄×Dic₄₃	C₂×C₈	1	4	yes	complemented
96	#10621877	—	2	2	no	—
97	#10621877	C₂×C₈	1	4	yes	complemented
98	C₂×C₄₃⋊C₈	—	2	2	no	—
99	C₂×C₄₃⋊C₈	—	2	4	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	—	4	4	no	—
102	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
103	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
104	C₂³×C₄₃	—	2	1	no	—
105	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
106	Q₈×C₄₃	—	2	1	no	—
107	Q₈×C₄₃	C₄×C₈	1	2	yes	complemented
108	D₄×C₄₃	—	2	4	no	—
109	D₄×C₄₃	—	2	1	no	—
110	D₄×C₄₃	C₄×C₈	1	2	yes	complemented
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₄₃	#131	1	2	yes	—
127	C₂×Dic₄₃	—	2	4	no	—
128	C₂×Dic₄₃	—	2	4	no	—
129	C₂×Dic₄₃	—	2	4	no	—
130	C₈×C₄₃	—	2	4	no	—
131	C₈×C₄₃	—	4	2	no	—
132	C₄₃⋊C₈	—	2	2	no	—
133	C₄₃⋊C₈	—	2	4	no	—
134	C₄₃⋊C₈	—	4	2	no	—
135	C₄₃⋊C₈	#104	1	4	yes	complemented
136	C₈×#104	—	43	1	no	maximal, Sylow
137	C₂²×C₄₃	—	2	1	no	—
138	C₂²×C₄₃	—	2	1	no	—
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	4	1	no	—
141	C₂²×C₄₃	—	4	1	no	—
142	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄₃	C₈×D₄	1	2	yes	—
144	C₄×C₄₃	—	2	2	no	—
145	C₄×C₄₃	—	2	1	no	—
146	C₄×C₄₃	—	2	1	no	—
147	C₄×C₄₃	—	4	1	no	—
148	C₄×C₄₃	—	4	1	no	—
149	C₄×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
150	C₄×C₄₃	#405	1	1	yes	characteristic
151	C₄×C₄₃	C₂×#104	1	2	yes	—
152	C₄×C₄₃	C₈×D₄	1	2	yes	—
153	C₄×C₄₃	#443	1	2	yes	—
154	Dic₄₃	—	2	8	no	—
155	C₈×#41	—	43	1	no	—
156	C₄×#104	—	43	1	no	—
157	C₄²×C₈	—	43	1	no	—
158	#1476	D₄₃	1	1	yes	characteristic
159	C₈×M₁₆	—	43	1	no	—
160	#1278	—	43	1	no	—
161	#1276	—	43	1	no	—
162	C₂×C₄₃	—	2	2	no	—
163	C₂×C₄₃	—	4	2	no	—
164	C₂×C₄₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
166	C₂×C₄₃	#1764	1	1	yes	characteristic
167	C₄×#41	C₂×D₄₃	1	1	yes	characteristic
168	C₈×Q₈	—	43	2	no	—
169	C₈×D₄	—	43	2	no	—
170	C₈×D₄	—	86	2	no	—
171	#431	—	43	1	no	—
172	#431	—	86	1	no	—
173	#431	C₂×D₄₃	1	1	yes	characteristic
174	C₂×#104	—	43	2	no	—
175	C₄×M₁₆	—	43	1	no	—
176	C₂×C₄×C₈	—	43	1	no	—
177	C₂×C₄×C₈	—	86	1	no	—
178	C₂×C₄×C₈	—	86	2	no	—
179	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
180	C₄³	—	43	1	no	—
181	#339	—	43	2	no	—
182	Q₈⋊C₈	Dic₄₃	1	2	yes	complemented
183	D₄⋊C₈	Dic₄₃	1	2	yes	complemented
184	#322	—	86	1	no	—
185	C₈²	—	86	1	no	—
186	C₄₃	C₈×#104	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	C₄×D₄₃	1	1	yes	characteristic
188	C₂×M₁₆	—	43	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	43	2	no	—
191	C₂²×C₈	—	86	2	no	—
192	C₄×Q₈	C₄×D₄₃	1	1	yes	characteristic
193	C₄×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	C₄×D₄₃	1	1	yes	characteristic
197	C₄×D₄	C₂×Dic₄₃	1	1	yes	characteristic
198	#117	—	2	1	no	—
199	#117	C₄×D₄₃	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	43	2	no	—
202	C₂×C₄²	—	86	1	no	—
203	C₂×C₄²	—	86	1	no	—
204	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
205	C₄⋊C₈	—	43	2	no	—
206	C₄⋊C₈	—	86	2	no	—
207	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
208	#104	—	43	8	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	86	2	no	—
211	C₂²⋊C₈	—	86	1	no	—
212	#97	—	86	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	43	2	no	—
216	C₄×C₈	—	43	1	no	—
217	C₄×C₈	—	43	1	no	—
218	C₄×C₈	—	86	2	no	—
219	C₄×C₈	—	86	8	no	—
220	C₄×C₈	—	86	1	no	—
221	C₄×C₈	—	86	1	no	—
222	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
223	#41	—	2	2	no	—
224	#41	C₈×D₄₃	1	4	yes	—
225	C₂×Q₈	C₄×Dic₄₃	1	1	yes	characteristic
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	C₄×Dic₄₃	1	1	yes	characteristic
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	86	2	no	—
232	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
234	C₂²×C₄	#10621900	1	1	yes	characteristic
235	M₁₆	—	43	4	no	—
236	M₁₆	—	86	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	43	2	no	—
241	C₂×C₈	—	43	2	no	—
242	C₂×C₈	—	86	4	no	—
243	C₂×C₈	—	86	2	no	—
244	C₂×C₈	—	86	2	no	—
245	C₂×C₈	—	172	4	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	C₄×Dic₄₃	1	1	yes	characteristic
249	C₄⋊C₄	C₄×Dic₄₃	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	43	4	no	—
257	C₄²	—	86	4	no	—
258	C₄²	—	86	4	no	—
259	C₄²	—	86	2	no	—
260	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
261	C₄²	D₄×D₄₃	1	1	yes	characteristic
262	C₄²	#10621900	1	1	yes	characteristic
263	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	C₂³	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
266	Q₈	—	2	1	no	—
267	Q₈	C₈×Dic₄₃	1	2	yes	complemented
268	D₄	—	2	4	no	—
269	D₄	—	2	1	no	—
270	D₄	C₈×Dic₄₃	1	2	yes	complemented
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	86	4	no	—
280	C₂×C₄	—	86	4	no	—
281	C₂×C₄	—	86	4	no	—
282	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
284	C₂×C₄	#13063281	1	1	yes	characteristic
285	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
287	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
288	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
289	C₂×C₄	#13063333	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	43	4	no	—
293	C₈	—	86	2	no	—
294	C₈	—	86	4	no	—
295	C₈	—	172	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
302	C₂²	(C₂²⋊D₄₃)⋊C₈	1	2	yes	—
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	86	8	no	—
309	C₄	C₂×C₈×Dic₄₃	1	1	yes	characteristic
310	C₄	#14582857	1	1	yes	characteristic
311	C₄	#104×D₄₃	1	2	yes	—
312	C₄	(C₂²⋊D₄₃)⋊C₈	1	2	yes	—
313	C₄	#14583016	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	4	2	no	—
316	C₂	#19134369	1	1	yes	characteristic
317	C₂	#104×Dic₄₃	1	1	yes	characteristic
318	C₂	#19134435	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₄₃
commutator	C₄×C₄₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1476	C₂
socle	C₂²×C₄₃	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₄₃	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1476	D₄₃
hypocenter	C₄₃	C₈×#104
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1476

Complements

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

N	H ≅ G/N	type
3. C₄×C₈×Dic₄₃	C₂	semidirect
4. #19132876	C₂	semidirect
5. #19132432	C₂	semidirect
6. #19132427	C₂	semidirect
8. Q₈×C₄₃⋊C₈	C₄	semidirect
10. D₄×C₄₃⋊C₈	C₄	semidirect
25. C₄₃×Q₈⋊C₈	C₄	semidirect
26. C₄₃×D₄⋊C₈	C₄	semidirect
35. #13063252	C₈	semidirect
95. C₄×Dic₄₃	C₂×C₈	semidirect
97. #10621877	C₂×C₈	semidirect
107. Q₈×C₄₃	C₄×C₈	semidirect
110. D₄×C₄₃	C₄×C₈	semidirect
135. C₄₃⋊C₈	#104	semidirect
182. Q₈⋊C₈	Dic₄₃	semidirect
183. D₄⋊C₈	Dic₄₃	semidirect
186. C₄₃	C₈×#104	semidirect
267. Q₈	C₈×Dic₄₃	semidirect
270. D₄	C₈×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
#41×C₄₃⋊C₈	2
C₄×#13063252	2
C₄×C₈×Dic₄₃	2
#19132876	2
#19132432	2
#19132427	2
C₄₃×#1476	2
C₈×#104	43