Subgroups — #25965978

Order 11008.

2134 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19148421	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144995	C₂	1	3	yes	maximal, complemented
3	#19144969	C₂	1	3	yes	maximal, complemented
4	#19144949	C₂	1	12	yes	maximal, complemented
5	#19144940	C₂	1	3	yes	maximal, complemented
6	C₂×#14583263	C₂	1	3	yes	maximal, complemented
7	#19143121	C₂	1	4	yes	maximal, complemented
8	#19143023	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#3498	C₂	1	1	yes	maximal, characteristic
10	C₂×#13063332	C₂²	1	2	yes	—
11	#14583625	—	2	2	no	—
12	#14583608	—	2	6	no	—
13	#14583527	C₂²	1	6	yes	complemented
14	#14583456	C₂²	1	3	yes	complemented
15	C₂×#13063317	C₂²	1	3	yes	complemented
16	#14583382	C₂²	1	2	yes	—
17	#14583267	C₂²	1	3	yes	complemented
18	#14583267	C₂²	1	1	yes	characteristic, complemented
19	#14583263	C₂²	1	3	yes	complemented
20	#14583263	C₂²	1	6	yes	complemented
21	#14583263	C₂²	1	3	yes	complemented
22	#14583263	C₂²	1	12	yes	complemented
23	#14583258	C₂²	1	12	yes	complemented
24	#14583253	C₂²	1	12	yes	complemented
25	#14583245	C₂²	1	12	yes	complemented
26	#14582904	C₂²	1	12	yes	complemented
27	#14582900	C₂²	1	12	yes	complemented
28	#14582895	C₂²	1	12	yes	complemented
29	C₂×#13063246	C₂²	1	12	yes	complemented
30	#14582855	C₂²	1	6	yes	complemented
31	C₄×#10621877	C₂²	1	3	yes	complemented
32	C₂×#13063236	C₂²	1	3	yes	complemented
33	C₂×H(5,𝔽₂)×C₄₃	C₂²	1	2	yes	—
34	C₄₃×#550	—	2	2	no	—
35	D₄²×C₄₃	—	2	6	no	—
36	C₄₃×#535	C₂²	1	6	yes	—
37	C₄₃×#532	C₂²	1	3	yes	—
38	C₂×#127×C₄₃	C₂²	1	3	yes	—
39	C₄×#41×C₄₃	C₂²	1	1	yes	characteristic
40	#13063332	—	2	8	no	—
41	C₂×#10621877	C₂³	1	6	yes	—
42	C₂²×C₄₃⋊C₈	C₂³	1	6	yes	—
43	#13063325	C₂³	1	6	yes	—
44	#13063324	—	2	6	no	—
45	#13063317	—	2	6	no	—
46	#13063317	C₂³	1	3	yes	—
47	#13063317	C₂³	1	6	yes	—
48	#13063317	C₂³	1	1	yes	characteristic
49	#13063316	—	2	6	no	—
50	#13063304	—	2	6	no	—
51	#13063302	—	2	12	no	—
52	Q₈×Dic₄₃	—	2	2	no	—
53	C₄×Dic₈₆	—	2	6	no	—
54	D₄×Dic₄₃	—	2	6	no	—
55	#13063284	C₂³	1	6	yes	—
56	C₂×C₄⋊Dic₄₃	C₂³	1	6	yes	—
57	C₂×Dic₄₃⋊C₄	—	2	6	no	—
58	#13063249	C₂³	1	4	yes	—
59	#13063249	C₂³	1	12	yes	—
60	#13063246	C₂³	1	12	yes	—
61	#13063246	C₂³	1	24	yes	—
62	#13063246	C₂³	1	12	yes	—
63	#13063236	C₂³	1	1	yes	characteristic
64	#13063236	C₂³	1	3	yes	—
65	#13063236	C₂³	1	6	yes	—
66	C₄×C₄₃⋊C₈	C₂³	1	6	yes	—
67	H(5,𝔽₂)×C₄₃	—	2	16	no	—
68	C₂×#41×C₄₃	—	2	2	no	—
69	C₂×#41×C₄₃	C₂³	1	6	yes	—
70	C₂×#41×C₄₃	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₄₃	—	2	6	no	—
72	C₂²×D₄×C₄₃	—	2	6	no	—
73	C₂²×D₄×C₄₃	C₂³	1	6	yes	—
74	C₄⋊Q₈×C₄₃	C₂³	1	1	yes	characteristic
75	#127×C₄₃	—	2	6	no	—
76	#127×C₄₃	C₂³	1	3	yes	—
77	#127×C₄₃	C₂³	1	6	yes	—
78	#124×C₄₃	C₂³	1	6	yes	—
79	#121×C₄₃	—	2	6	no	—
80	#121×C₄₃	—	2	12	no	—
81	#120×C₄₃	—	2	12	no	—
82	C₄×Q₈×C₄₃	D₄	1	2	yes	—
83	C₄×D₄×C₄₃	—	2	6	no	—
84	C₄×D₄×C₄₃	D₄	1	6	yes	—
85	#117×C₄₃	C₂³	1	3	yes	—
86	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
87	C₂⁴×C₄₃	—	2	6	no	—
88	#41×C₄₃	—	2	24	no	—
89	#41×C₄₃	—	4	8	no	—
90	#41×C₄₃	#41	1	8	yes	—
91	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×Q₈×C₄₃	C₂⁴	1	2	yes	—
93	C₂×D₄×C₄₃	—	2	6	no	—
94	C₂×D₄×C₄₃	—	2	6	no	—
95	C₂×D₄×C₄₃	—	2	12	no	—
96	C₂×D₄×C₄₃	—	2	6	no	—
97	C₂×D₄×C₄₃	—	2	24	no	—
98	C₂×D₄×C₄₃	—	4	24	no	—
99	C₂×D₄×C₄₃	—	8	12	no	—
100	C₂×D₄×C₄₃	C₂×D₄	1	3	yes	—
101	C₂×D₄×C₄₃	C₂⁴	1	6	yes	—
102	C₂²×C₄×C₄₃	—	2	6	no	—
103	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
104	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
105	C₂²×Dic₄₃	—	2	6	no	—
106	C₂×Dic₈₆	—	2	6	no	—
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	C₂×D₄	1	6	yes	—
109	C₂²⋊C₄×C₄₃	—	2	12	no	—
110	C₂²⋊C₄×C₄₃	C₂×D₄	1	6	yes	—
111	C₄²×C₄₃	C₂×D₄	1	6	yes	—
112	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
113	C₄²×C₄₃	C₂⁴	1	3	yes	—
114	C₂²⋊Dic₄₃	—	2	6	no	—
115	C₄⋊Dic₄₃	—	2	6	no	—
116	C₄⋊Dic₄₃	—	2	6	no	—
117	C₄⋊Dic₄₃	C₂⁴	1	6	yes	—
118	C₄⋊Dic₄₃	C₂⁴	1	2	yes	—
119	Dic₄₃⋊C₄	—	2	6	no	—
120	Dic₄₃⋊C₄	—	2	12	no	—
121	Dic₄₃⋊C₄	—	2	2	no	—
122	C₄×Dic₄₃	—	2	6	no	—
123	#10621877	—	2	12	no	—
124	C₂×C₄₃⋊C₈	—	2	12	no	—
125	C₂×C₄₃⋊C₈	C₂⁴	1	6	yes	—
126	C₂×C₄₃⋊C₈	C₂⁴	1	2	yes	—
127	C₂³×C₄₃	—	2	6	no	—
128	C₂³×C₄₃	—	4	12	no	—
129	C₂³×C₄₃	—	8	12	no	—
130	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
131	Q₈×C₄₃	—	2	4	no	—
132	Q₈×C₄₃	#124	1	4	yes	—
133	D₄×C₄₃	—	2	12	no	—
134	D₄×C₄₃	—	4	12	no	—
135	D₄×C₄₃	—	4	12	no	—
136	D₄×C₄₃	—	4	12	no	—
137	D₄×C₄₃	—	8	12	no	—
138	D₄×C₄₃	#124	1	12	yes	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	4	6	no	—
141	C₂×C₄×C₄₃	—	4	12	no	—
142	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
143	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
144	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
145	C₂×C₄×C₄₃	C₂×#41	1	12	yes	—
146	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	6	no	—
149	C₂×Dic₄₃	—	8	6	no	—
150	Dic₈₆	—	8	6	no	—
151	C₄₃⋊C₈	—	2	2	no	—
152	C₄₃⋊C₈	—	2	6	no	—
153	#60626	—	43	1	no	maximal, Sylow
154	C₂²×C₄₃	—	2	6	no	—
155	C₂²×C₄₃	—	4	12	no	—
156	C₂²×C₄₃	—	4	6	no	—
157	C₂²×C₄₃	—	16	12	no	—
158	C₂²×C₄₃	C₂×#124	1	6	yes	—
159	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
160	C₄×C₄₃	—	2	6	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	4	2	no	—
163	C₄×C₄₃	C₂×#124	1	6	yes	—
164	C₄×C₄₃	C₂²×#41	1	2	yes	—
165	Dic₄₃	—	4	2	no	—
166	Dic₄₃	—	4	6	no	—
167	#3504	—	43	1	no	—
168	#3498	D₄₃	1	1	yes	characteristic
169	#3142	—	43	3	no	—
170	#3138	—	43	3	no	—
171	#3136	—	43	12	no	—
172	#3135	—	43	3	no	—
173	C₂×#490	—	43	3	no	—
174	#2897	—	43	4	no	—
175	#2875	—	43	1	no	—
176	C₂×C₄₃	—	2	6	no	—
177	C₂×C₄₃	—	8	6	no	—
178	C₂×C₄₃	C₂²×#124	1	1	yes	characteristic
179	C₂×C₄₃	#3585	1	2	yes	—
180	C₂×H(5,𝔽₂)	C₂×D₄₃	1	2	yes	—
181	C₂×#131	—	43	2	no	—
182	#557	—	86	2	no	—
183	#554	—	86	6	no	—
184	#550	—	2	2	no	—
185	D₄²	—	2	6	no	—
186	#544	—	43	6	no	—
187	#535	C₂×D₄₃	1	6	yes	—
188	#533	—	43	3	no	—
189	#532	C₂×D₄₃	1	3	yes	—
190	C₂×#127	C₂×D₄₃	1	3	yes	—
191	C₂×#125	—	43	3	no	—
192	#520	—	43	2	no	—
193	C₄×#41	C₂×D₄₃	1	1	yes	characteristic
194	#491	—	43	1	no	—
195	#491	—	43	3	no	—
196	#490	—	43	3	no	—
197	#490	—	43	12	no	—
198	#490	—	43	6	no	—
199	#490	—	43	3	no	—
200	#489	—	43	12	no	—
201	#488	—	43	12	no	—
202	#486	—	43	12	no	—
203	#418	—	43	12	no	—
204	#417	—	43	12	no	—
205	#416	—	43	12	no	—
206	C₂×#102	—	43	12	no	—
207	#405	—	43	6	no	—
208	C₄×M₁₆	—	43	3	no	—
209	C₂×#97	—	43	3	no	—
210	C₄₃	#60626	1	1	yes	characteristic, complemented, Sylow
211	H(5,𝔽₂)	—	2	16	no	—
212	C₂×#41	—	2	2	no	—
213	C₂×#41	C₂²×D₄₃	1	6	yes	—
214	C₂×#41	C₂²×D₄₃	1	1	yes	characteristic
215	C₂²×D₄	—	2	6	no	—
216	C₂²×D₄	—	2	6	no	—
217	C₂²×D₄	C₂²×D₄₃	1	6	yes	—
218	#131	—	86	8	no	—
219	C₂×M₁₆	—	43	6	no	—
220	C₂²×C₈	—	43	6	no	—
221	C₄⋊Q₈	—	43	6	no	—
222	C₄⋊Q₈	—	86	6	no	—
223	C₄⋊Q₈	C₂²×D₄₃	1	1	yes	characteristic
224	#127	—	2	6	no	—
225	#127	C₂²×D₄₃	1	3	yes	—
226	#127	C₂²×D₄₃	1	6	yes	—
227	#125	—	43	3	no	—
228	#125	—	43	6	no	—
229	#125	—	43	1	no	—
230	#125	—	86	6	no	—
231	#125	—	86	6	no	—
232	#124	C₂²×D₄₃	1	6	yes	—
233	C₂²⋊Q₈	—	86	6	no	—
234	C₂²⋊Q₈	—	86	12	no	—
235	#121	—	2	6	no	—
236	#121	—	2	12	no	—
237	#120	—	2	12	no	—
238	C₄×Q₈	—	86	2	no	—
239	C₄×Q₈	—	86	6	no	—
240	C₄×Q₈	C₂²⋊D₄₃	1	2	yes	—
241	C₄×D₄	—	2	6	no	—
242	C₄×D₄	—	86	6	no	—
243	C₄×D₄	C₂²⋊D₄₃	1	6	yes	—
244	#117	—	43	6	no	—
245	#117	C₂²×D₄₃	1	3	yes	—
246	C₂×C₄⋊C₄	—	43	6	no	—
247	C₂×C₄⋊C₄	—	86	6	no	—
248	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
249	#103	—	43	4	no	—
250	#103	—	43	12	no	—
251	#102	—	43	12	no	—
252	#102	—	43	24	no	—
253	#102	—	43	12	no	—
254	#97	—	43	1	no	—
255	#97	—	43	3	no	—
256	#97	—	43	6	no	—
257	C₄×C₈	—	43	6	no	—
258	C₂⁴	—	2	6	no	—
259	#41	—	2	24	no	—
260	#41	—	4	8	no	—
261	#41	#10621900	1	8	yes	—
262	C₂×Q₈	—	86	6	no	—
263	C₂×Q₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₂×Q₈	C₂³×D₄₃	1	2	yes	—
265	C₂×D₄	—	2	6	no	—
266	C₂×D₄	—	2	6	no	—
267	C₂×D₄	—	2	6	no	—
268	C₂×D₄	—	2	12	no	—
269	C₂×D₄	—	2	24	no	—
270	C₂×D₄	—	4	24	no	—
271	C₂×D₄	—	8	12	no	—
272	C₂×D₄	C₂×C₂²⋊D₄₃	1	3	yes	—
273	C₂×D₄	C₂³×D₄₃	1	6	yes	—
274	C₂²×C₄	—	2	6	no	—
275	C₂²×C₄	—	86	6	no	—
276	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
277	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
278	M₁₆	—	86	12	no	—
279	C₂×C₈	—	43	6	no	—
280	C₂×C₈	—	43	2	no	—
281	C₂×C₈	—	86	12	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	43	6	no	—
284	C₄⋊C₄	—	43	2	no	—
285	C₄⋊C₄	—	86	6	no	—
286	C₄⋊C₄	—	86	6	no	—
287	C₄⋊C₄	—	86	12	no	—
288	C₄⋊C₄	—	86	2	no	—
289	C₄⋊C₄	—	86	6	no	—
290	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
291	C₂²⋊C₄	—	2	12	no	—
292	C₂²⋊C₄	—	86	6	no	—
293	C₂²⋊C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
294	C₄²	—	86	6	no	—
295	C₄²	C₂×C₂²⋊D₄₃	1	6	yes	—
296	C₄²	C₂³×D₄₃	1	3	yes	—
297	C₄²	C₂³×D₄₃	1	1	yes	characteristic
298	C₂³	—	2	6	no	—
299	C₂³	—	4	12	no	—
300	C₂³	—	8	12	no	—
301	C₂³	C₂²×C₂²⋊D₄₃	1	3	yes	—
302	Q₈	—	2	4	no	—
303	Q₈	—	344	6	no	—
304	Q₈	#13063312	1	4	yes	—
305	D₄	—	2	12	no	—
306	D₄	—	4	12	no	—
307	D₄	—	4	12	no	—
308	D₄	—	4	12	no	—
309	D₄	—	8	12	no	—
310	D₄	#13063312	1	12	yes	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	6	no	—
313	C₂×C₄	—	4	12	no	—
314	C₂×C₄	—	86	2	no	—
315	C₂×C₄	—	86	6	no	—
316	C₂×C₄	—	344	6	no	—
317	C₂×C₄	C₂²×C₂²⋊D₄₃	1	2	yes	—
318	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
319	C₂×C₄	C₂²×C₂²⋊D₄₃	1	3	yes	—
320	C₂×C₄	C₂×#10621900	1	12	yes	—
321	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
322	C₈	—	86	2	no	—
323	C₈	—	86	6	no	—
324	C₂²	—	2	6	no	—
325	C₂²	—	4	12	no	—
326	C₂²	—	4	6	no	—
327	C₂²	—	16	12	no	—
328	C₂²	C₂×#13063312	1	6	yes	—
329	C₂²	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	6	no	—
332	C₄	—	4	2	no	—
333	C₄	—	172	6	no	—
334	C₄	—	172	2	no	—
335	C₄	C₂×#13063312	1	6	yes	—
336	C₄	C₂²×#10621900	1	2	yes	—
337	C₂	—	2	6	no	—
338	C₂	—	8	6	no	—
339	C₂	C₂²×#13063312	1	1	yes	characteristic
340	C₂	#19149375	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting, prime radical	C₄₃×#3498	C₂
socle	C₂²×C₄₃	C₂³×D₄
hypercenter	#3498	D₄₃
hypocenter	C₄₃	#60626
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #3498

Complements

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

N	H ≅ G/N	type
1. #19148421	C₂	semidirect
2. #19144995	C₂	semidirect
3. #19144969	C₂	semidirect
4. #19144949	C₂	semidirect
5. #19144940	C₂	semidirect
6. C₂×#14583263	C₂	semidirect
7. #19143121	C₂	semidirect
8. #19143023	C₂	semidirect
13. #14583527	C₂²	semidirect
14. #14583456	C₂²	semidirect
15. C₂×#13063317	C₂²	semidirect
17. #14583267	C₂²	semidirect
18. #14583267	C₂²	semidirect
19. #14583263	C₂²	semidirect
20. #14583263	C₂²	semidirect
21. #14583263	C₂²	semidirect
22. #14583263	C₂²	semidirect
23. #14583258	C₂²	semidirect
24. #14583253	C₂²	semidirect
25. #14583245	C₂²	semidirect
26. #14582904	C₂²	semidirect
27. #14582900	C₂²	semidirect
28. #14582895	C₂²	semidirect
29. C₂×#13063246	C₂²	semidirect
30. #14582855	C₂²	semidirect
31. C₄×#10621877	C₂²	semidirect
32. C₂×#13063236	C₂²	semidirect
210. C₄₃	#60626	semidirect

Maximal subgroups

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

subgroup	index
#19148421	2
#19144995	2
#19144969	2
#19144949	2
#19144940	2
C₂×#14583263	2
#19143121	2
#19143023	2
C₄₃×#3498	2
#60626	43