Subgroups — #25013215

Order 11008.

398 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	C₂×#14583016	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582728	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136302	C₂	1	1	yes	maximal, characteristic
4	#19135341	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132691	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132672	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#359	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063332	—	2	1	no	—
9	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	#14583016	—	2	2	no	—
11	#14583016	—	2	2	no	—
12	C₂×#13063261	C₂²	1	1	yes	characteristic
13	#14582937	C₂²	1	1	yes	characteristic
14	#14582927	—	2	1	no	—
15	C₂×#13063252	—	2	1	no	—
16	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
17	#14582728	—	2	2	no	—
18	#14582703	C₄	1	1	yes	characteristic, complemented
19	#14582703	C₄	1	1	yes	characteristic, complemented
20	#14582700	C₄	1	1	yes	characteristic, complemented
21	#14582700	C₄	1	1	yes	characteristic, complemented
22	#14582660	—	2	1	no	—
23	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#359	—	2	2	no	—
27	#13063332	—	2	4	no	—
28	#13063332	—	4	2	no	—
29	C₂×#10621877	—	2	1	no	—
30	C₂×#10621877	—	4	1	no	—
31	C₂²×C₄₃⋊C₈	—	4	1	no	—
32	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
33	#13063284	—	2	1	no	—
34	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
35	C₂×C₄₃⋊C₁₆	—	2	1	no	—
36	C₂×C₄₃⋊C₁₆	—	2	1	no	—
37	C₂×C₄₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
39	#13063261	—	2	1	no	—
40	#13063261	D₄	1	2	yes	—
41	#13063259	C₂×C₄	1	1	yes	characteristic
42	#13063259	C₂×C₄	1	1	yes	characteristic
43	#13063257	—	2	1	no	—
44	Dic₄₃⋊C₈	—	4	1	no	—
45	Dic₄₃⋊C₈	—	4	1	no	—
46	#13063252	—	2	4	no	—
47	C₈×Dic₄₃	—	2	1	no	—
48	C₈×Dic₄₃	—	2	1	no	—
49	C₂×#41×C₄₃	—	2	1	no	—
50	#135×C₄₃	—	2	1	no	—
51	#135×C₄₃	—	2	1	no	—
52	#135×C₄₃	D₄	1	2	yes	—
53	#135×C₄₃	Q₈	1	2	yes	—
54	C₂×Q₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
55	C₂×SD₁₆×C₄₃	—	2	1	no	—
56	C₂×D₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₄₃	—	2	1	no	—
58	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₄₃	—	2	1	no	—
60	C₂×C₁₆×C₄₃	—	2	1	no	—
61	C₂×C₁₆×C₄₃	—	2	1	no	—
62	#108×C₄₃	—	2	1	no	—
63	#108×C₄₃	—	2	1	no	—
64	#41×C₄₃	—	2	2	no	—
65	#41×C₄₃	—	2	2	no	—
66	#41×C₄₃	—	4	1	no	—
67	#41×C₄₃	—	4	1	no	—
68	C₂×Q₈×C₄₃	—	2	1	no	—
69	C₂×D₄×C₄₃	—	2	1	no	—
70	C₂×D₄×C₄₃	—	4	1	no	—
71	C₂²×C₄×C₄₃	—	4	1	no	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₄₃	—	2	1	no	—
74	Q₁₆×C₄₃	—	2	1	no	—
75	Q₁₆×C₄₃	C₄⋊C₄	1	2	yes	—
76	SD₁₆×C₄₃	—	2	2	no	—
77	SD₁₆×C₄₃	—	2	2	no	—
78	D₈×C₄₃	—	2	1	no	—
79	D₈×C₄₃	C₄⋊C₄	1	2	yes	—
80	M₁₆×C₄₃	—	4	1	no	—
81	M₁₆×C₄₃	—	4	1	no	—
82	C₂×C₈×C₄₃	—	4	1	no	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
89	C₂²⋊Dic₄₃	—	4	1	no	—
90	C₄⋊Dic₄₃	—	2	1	no	—
91	C₄⋊Dic₄₃	—	2	1	no	—
92	C₄×Dic₄₃	—	2	1	no	—
93	C₄×Dic₄₃	—	2	1	no	—
94	C₄×Dic₄₃	—	4	1	no	—
95	#10621877	—	2	2	no	—
96	#10621877	—	4	4	no	—
97	#10621877	—	4	1	no	—
98	C₂×C₄₃⋊C₈	—	2	1	no	—
99	C₂×C₄₃⋊C₈	—	2	1	no	—
100	C₂×C₄₃⋊C₈	—	4	4	no	—
101	C₂×C₄₃⋊C₈	—	4	1	no	—
102	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
104	C₁₆×C₄₃	—	4	1	no	—
105	C₄₃⋊C₁₆	—	2	1	no	—
106	C₄₃⋊C₁₆	—	2	1	no	—
107	C₂³×C₄₃	—	4	1	no	—
108	C₂³×C₄₃	#135	1	1	yes	characteristic
109	Q₈×C₄₃	—	2	2	no	—
110	Q₈×C₄₃	—	4	1	no	—
111	D₄×C₄₃	—	2	2	no	—
112	D₄×C₄₃	—	4	2	no	—
113	D₄×C₄₃	—	4	2	no	—
114	D₄×C₄₃	—	4	1	no	—
115	C₂×C₄×C₄₃	—	4	1	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
124	C₂×Dic₄₃	—	2	2	no	—
125	C₂×Dic₄₃	—	4	1	no	—
126	C₂×Dic₄₃	—	4	1	no	—
127	C₂×Dic₄₃	—	4	1	no	—
128	C₂×Dic₄₃	—	4	1	no	—
129	C₈×C₄₃	—	8	1	no	—
130	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
132	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
133	C₈×C₄₃	#123	1	1	yes	characteristic
134	C₄₃⋊C₈	—	2	1	no	—
135	C₄₃⋊C₈	—	2	1	no	—
136	C₄₃⋊C₈	—	4	2	no	—
137	#12939	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	—	2	1	no	—
139	C₂²×C₄₃	—	2	1	no	—
140	C₂²×C₄₃	—	4	2	no	—
141	C₂²×C₄₃	—	8	1	no	—
142	C₂²×C₄₃	—	8	1	no	—
143	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
144	C₂²×C₄₃	#452	1	1	yes	characteristic
145	C₂²×C₄₃	#478	1	1	yes	characteristic
146	C₄×C₄₃	—	4	2	no	—
147	C₄×C₄₃	#385	1	1	yes	characteristic
148	C₄×C₄₃	#440	1	1	yes	characteristic
149	C₄×C₄₃	#447	1	1	yes	characteristic
150	C₄×C₄₃	#474	1	1	yes	characteristic
151	Dic₄₃	—	4	2	no	—
152	Dic₄₃	—	8	1	no	—
153	Dic₄₃	—	8	1	no	—
154	C₂×#443	—	43	1	no	—
155	C₂×#368	—	43	1	no	—
156	C₂×#359	D₄₃	1	1	yes	characteristic
157	#2138	—	43	1	no	—
158	#1945	—	43	1	no	—
159	#1382	—	43	1	no	—
160	#1374	—	43	1	no	—
161	C₂×C₄₃	—	2	1	no	—
162	C₂×C₄₃	—	2	1	no	—
163	C₂×C₄₃	—	8	2	no	—
164	C₂×C₄₃	#1862	1	1	yes	characteristic
165	C₂×C₄₃	#2178	1	1	yes	characteristic
166	C₂×C₄₃	#2179	1	1	yes	characteristic
167	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
168	C₂×#131	—	86	1	no	—
169	C₂²×C₁₆	—	43	1	no	—
170	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
171	#443	—	86	2	no	—
172	#443	—	86	2	no	—
173	C₂×#108	—	43	1	no	—
174	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
175	#427	—	43	1	no	—
176	#424	—	86	1	no	—
177	C₂×#104	—	86	1	no	—
178	C₂×C₄×C₈	—	43	1	no	—
179	#368	—	86	2	no	—
180	#359	—	2	2	no	—
181	#358	—	43	1	no	—
182	#358	—	43	1	no	—
183	#357	—	43	1	no	—
184	#357	—	43	1	no	—
185	#339	—	86	1	no	—
186	C₄₃	#12939	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	2	1	no	—
188	#135	—	2	1	no	—
189	#135	—	2	1	no	—
190	#135	Dic₈₆	1	2	yes	—
191	#135	C₂²⋊D₄₃	1	2	yes	—
192	C₂×Q₁₆	C₄×D₄₃	1	1	yes	characteristic
193	C₂×SD₁₆	—	2	1	no	—
194	C₂×D₈	C₄×D₄₃	1	1	yes	characteristic
195	#131	—	86	4	no	—
196	#131	—	172	2	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	86	1	no	—
199	C₂×M₁₆	—	172	1	no	—
200	C₂²×C₈	—	43	1	no	—
201	C₂²×C₈	—	172	1	no	—
202	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
203	#117	—	86	1	no	—
204	C₂×C₄²	—	43	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	43	1	no	—
209	C₂×C₁₆	—	43	1	no	—
210	C₂×C₁₆	—	86	1	no	—
211	C₂×C₁₆	—	86	1	no	—
212	#108	—	2	1	no	—
213	#108	—	2	1	no	—
214	#108	—	43	2	no	—
215	#108	—	86	1	no	—
216	#107	—	43	1	no	—
217	#107	—	43	1	no	—
218	#106	—	86	1	no	—
219	C₄⋊C₈	—	172	1	no	—
220	C₄⋊C₈	—	172	1	no	—
221	#104	—	86	4	no	—
222	C₄×C₈	—	86	1	no	—
223	C₄×C₈	—	86	1	no	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	#41	—	4	1	no	—
227	#41	—	4	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	86	1	no	—
233	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
234	Q₁₆	—	2	1	no	—
235	Q₁₆	Dic₄₃⋊C₄	1	2	yes	—
236	SD₁₆	—	2	2	no	—
237	SD₁₆	—	2	2	no	—
238	D₈	—	2	1	no	—
239	D₈	Dic₄₃⋊C₄	1	2	yes	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	86	2	no	—
243	M₁₆	—	172	4	no	—
244	M₁₆	—	172	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	43	1	no	—
247	C₂×C₈	—	43	1	no	—
248	C₂×C₈	—	86	1	no	—
249	C₂×C₈	—	86	1	no	—
250	C₂×C₈	—	172	4	no	—
251	C₂×C₈	—	172	1	no	—
252	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
254	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
255	C₂×C₈	#10621900	1	1	yes	characteristic
256	C₂×C₈	#10621900	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
258	C₄⋊C₄	—	86	1	no	—
259	C₄⋊C₄	—	86	1	no	—
260	C₂²⋊C₄	—	172	1	no	—
261	C₄²	—	86	1	no	—
262	C₄²	—	86	1	no	—
263	C₄²	—	172	1	no	—
264	C₁₆	—	4	1	no	—
265	C₁₆	—	86	1	no	—
266	C₁₆	—	86	1	no	—
267	C₂³	—	4	1	no	—
268	C₂³	#13063346	1	1	yes	characteristic
269	Q₈	—	2	2	no	—
270	Q₈	—	4	1	no	—
271	D₄	—	2	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	86	2	no	—
279	C₂×C₄	—	172	1	no	—
280	C₂×C₄	—	172	1	no	—
281	C₂×C₄	—	172	1	no	—
282	C₂×C₄	—	172	1	no	—
283	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#13063293	1	1	yes	characteristic
285	C₂×C₄	#13063302	1	1	yes	characteristic
286	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
287	C₂×C₄	#13063349	1	1	yes	characteristic
288	C₂×C₄	#13063354	1	1	yes	characteristic
289	C₈	—	8	1	no	—
290	C₈	—	86	1	no	—
291	C₈	—	86	1	no	—
292	C₈	—	172	2	no	—
293	C₈	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
294	C₈	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
295	C₈	#13063302	1	1	yes	characteristic
296	C₈	#13063306	1	1	yes	characteristic
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	1	no	—
302	C₂²	#14582979	1	1	yes	characteristic
303	C₂²	#14583058	1	1	yes	characteristic
304	C₂²	#14583201	1	1	yes	characteristic
305	C₄	—	4	2	no	—
306	C₄	—	344	1	no	—
307	C₄	—	344	1	no	—
308	C₄	—	172	2	no	—
309	C₄	#14582771	1	1	yes	characteristic
310	C₄	#14582996	1	1	yes	characteristic
311	C₄	#14583031	1	1	yes	characteristic
312	C₄	#14583177	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	8	2	no	—
316	C₂	#19134840	1	1	yes	characteristic
317	C₂	#19136473	1	1	yes	characteristic
318	C₂	#19136478	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#359	C₂
socle	C₂²×C₄₃	C₄×D₈
prime radical	C₂×#135×C₄₃	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#359	D₄₃
hypocenter	C₄₃	#12939
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#359

Complements

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

N	H ≅ G/N	type
2. C₂×#14582728	C₂	semidirect
4. #19135341	C₂	semidirect
5. #19132691	C₂	semidirect
6. #19132672	C₂	semidirect
18. #14582703	C₄	semidirect
19. #14582703	C₄	semidirect
20. #14582700	C₄	semidirect
21. #14582700	C₄	semidirect
186. C₄₃	#12939	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583016	2
C₂×#14582728	2
#19136302	2
#19135341	2
#19132691	2
#19132672	2
C₂×C₄₃×#359	2
#12939	43