Subgroups — #24987680

Order 11008.

462 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143308	C₂	1	1	yes	maximal, characteristic
2	#19133245	C₂	1	1	yes	maximal, characteristic
3	#19133089	C₂	1	1	yes	maximal, characteristic
4	#19132887	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132498	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063331	—	2	1	no	—
9	#14583449	C₂²	1	1	yes	characteristic
10	#14582978	—	2	1	no	—
11	#14582978	C₄	1	2	yes	complemented
12	C₈×D₁₇₂	—	2	1	no	—
13	C₈×D₁₇₂	C₄	1	2	yes	complemented
14	#14582952	C₂²	1	1	yes	characteristic
15	#14582952	C₂²	1	1	yes	characteristic
16	#14582923	C₂²	1	1	yes	characteristic
17	#14582865	—	2	1	no	—
18	#14582646	C₄	1	2	yes	complemented
19	#14582642	C₄	1	2	yes	complemented
20	#14582632	C₄	1	4	yes	—
21	#14582621	—	2	1	no	—
22	#14582621	—	2	1	no	—
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#97×C₄₃	—	2	1	no	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#336	—	2	1	no	—
28	C₂×#10621898	—	2	1	no	—
29	#13063331	—	4	2	no	—
30	#13063331	—	4	2	no	—
31	C₂×#10621877	—	2	1	no	—
32	C₂×#10621872	—	2	2	no	—
33	C₂×C₈×D₄₃	—	2	2	no	—
34	#13063325	C₂×C₄	1	1	yes	characteristic
35	#13063322	C₂×C₄	1	1	yes	characteristic
36	#13063313	—	2	1	no	—
37	#13063299	—	2	1	no	—
38	#13063299	C₂×C₄	1	2	yes	—
39	C₄×D₁₇₂	—	2	1	no	—
40	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
41	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
42	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
44	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
45	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
46	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
47	(C₂×D₄₃)⋊C₈	—	2	2	no	—
48	(C₂×D₄₃)⋊C₈	—	2	2	no	—
49	C₄×C₄₃⋊C₈	—	2	1	no	—
50	C₄×C₄₃⋊C₈	—	2	1	no	—
51	C₂²×C₈×C₄₃	—	2	1	no	—
52	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
55	C₂×C₄²×C₄₃	—	2	1	no	—
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₃	—	2	1	no	—
58	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
59	#97×C₄₃	—	4	2	no	—
60	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
64	#95×C₄₃	C₂×C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₃	—	2	2	no	—
70	#10621898	—	4	1	no	—
71	#10621898	—	4	1	no	—
72	#10621898	—	4	1	no	—
73	#10621898	—	4	1	no	—
74	C₂×D₁₇₂	—	2	1	no	—
75	C₂×D₁₇₂	—	2	1	no	—
76	C₂×D₁₇₂	C₄²	1	2	yes	—
77	C₂×C₄×D₄₃	—	2	2	no	—
78	C₂×Dic₈₆	—	2	1	no	—
79	C₂×C₈×C₄₃	—	2	1	no	—
80	C₂×C₈×C₄₃	—	2	1	no	—
81	C₂×C₈×C₄₃	—	2	1	no	—
82	C₂×C₈×C₄₃	—	2	1	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	—	4	1	no	—
85	C₂×C₈×C₄₃	—	4	1	no	—
86	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	—	4	2	no	—
91	C₄²×C₄₃	—	4	1	no	—
92	C₄²×C₄₃	—	4	1	no	—
93	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₄₃	#41	1	1	yes	characteristic
97	(C₂×D₄₃)⋊C₄	—	2	2	no	—
98	(C₂×D₄₃)⋊C₄	—	2	2	no	—
99	C₄⋊Dic₄₃	—	2	1	no	—
100	C₄⋊Dic₄₃	C₄²	1	2	yes	—
101	#10621877	—	4	1	no	—
102	#10621877	—	4	1	no	—
103	C₂×C₄₃⋊C₈	—	2	2	no	—
104	C₂×C₄₃⋊C₈	—	2	2	no	—
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	#10621872	—	4	4	no	—
107	C₈×D₄₃	—	4	4	no	—
108	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
109	C₂²×D₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	4	1	no	—
112	C₂×C₄×C₄₃	—	4	2	no	—
113	C₂×C₄×C₄₃	—	4	2	no	—
114	C₂×C₄×C₄₃	—	4	2	no	—
115	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
125	C₂²⋊D₄₃	—	4	2	no	—
126	C₂²⋊D₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	2	2	no	—
128	D₁₇₂	—	4	2	no	—
129	D₁₇₂	—	4	1	no	—
130	D₁₇₂	—	4	1	no	—
131	D₁₇₂	—	4	2	no	—
132	C₄×D₄₃	—	4	2	no	—
133	C₄×D₄₃	—	4	2	no	—
134	Dic₈₆	—	4	1	no	—
135	Dic₈₆	—	4	1	no	—
136	C₈×C₄₃	—	2	1	no	—
137	C₈×C₄₃	—	2	1	no	—
138	C₈×C₄₃	—	2	1	no	—
139	C₈×C₄₃	—	2	1	no	—
140	C₈×C₄₃	—	4	2	no	—
141	C₄₃⋊C₈	—	4	2	no	—
142	C₄₃⋊C₈	—	4	2	no	—
143	C₄₃⋊C₈	—	4	2	no	—
144	#8253	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₃	#440	1	1	yes	characteristic
149	C₂²×C₄₃	#442	1	1	yes	characteristic
150	C₂×D₄₃	—	4	2	no	—
151	C₂×D₄₃	—	4	2	no	—
152	C₂×D₄₃	—	4	2	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	8	2	no	—
157	C₄×C₄₃	#417	1	1	yes	characteristic
158	C₄×C₄₃	#418	1	1	yes	characteristic
159	C₄×C₄₃	#440	1	1	yes	characteristic
160	C₄×C₄₃	#442	1	1	yes	characteristic
161	C₄×C₄₃	#444	1	2	yes	—
162	Dic₄₃	—	4	2	no	—
163	#2931	—	43	1	no	—
164	#1770	D₄₃	1	1	yes	characteristic, complemented
165	#1540	—	43	1	no	—
166	#1502	—	43	1	no	—
167	#1452	—	43	1	no	—
168	#1303	—	43	1	no	—
169	#1301	—	43	1	no	—
170	C₂×C₄₃	—	2	1	no	—
171	C₂×C₄₃	—	2	1	no	—
172	C₂×C₄₃	#1757	1	1	yes	characteristic
173	C₂×C₄₃	#1762	1	1	yes	characteristic
174	C₂×C₄₃	#1765	1	1	yes	characteristic
175	D₄₃	—	8	2	no	—
176	C₂×#131	—	86	1	no	—
177	#532	—	43	1	no	—
178	#436	—	43	2	no	—
179	#436	—	86	1	no	—
180	C₈×D₄	—	43	2	no	—
181	C₈×D₄	—	86	1	no	—
182	#431	—	43	1	no	—
183	#431	—	43	1	no	—
184	#423	—	43	1	no	—
185	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
186	#408	—	86	1	no	—
187	C₂×#97	—	2	1	no	—
188	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
189	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
190	#336	—	2	1	no	—
191	#332	—	43	2	no	—
192	#331	—	43	2	no	—
193	#327	—	43	4	no	—
194	#322	—	86	1	no	—
195	#322	—	86	1	no	—
196	C₄₃	#8253	1	1	yes	characteristic, complemented, Sylow
197	C₂×#41	—	86	1	no	—
198	#131	—	172	2	no	—
199	#131	—	172	2	no	—
200	C₂×M₁₆	—	86	1	no	—
201	C₂×M₁₆	—	86	2	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	86	2	no	—
204	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
206	C₄⋊Q₈	—	43	1	no	—
207	#127	—	43	1	no	—
208	#124	—	86	1	no	—
209	#121	—	43	2	no	—
210	#121	—	86	1	no	—
211	C₄×D₄	—	43	2	no	—
212	C₄×D₄	—	86	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	43	2	no	—
218	C₄⋊C₈	—	43	2	no	—
219	C₄⋊C₈	—	86	1	no	—
220	C₄⋊C₈	—	86	1	no	—
221	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
222	C₂²⋊C₈	—	43	2	no	—
223	C₂²⋊C₈	—	86	1	no	—
224	C₂²⋊C₈	—	86	2	no	—
225	C₂²⋊C₈	—	86	2	no	—
226	#97	—	4	2	no	—
227	C₄×C₈	—	86	1	no	—
228	C₄×C₈	—	86	1	no	—
229	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
230	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
231	C₄×C₈	D₁₇₂	1	1	yes	characteristic
232	C₄×C₈	D₁₇₂	1	1	yes	characteristic
233	#95	C₄×D₄₃	1	2	yes	—
234	#41	—	172	1	no	—
235	#41	—	172	1	no	—
236	#41	—	172	1	no	—
237	#41	—	172	1	no	—
238	C₂×Q₈	—	86	1	no	—
239	C₂×D₄	—	43	2	no	—
240	C₂×D₄	—	86	2	no	—
241	C₂×D₄	—	86	1	no	—
242	C₂×D₄	—	86	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	86	2	no	—
245	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
246	C₂²×C₄	#10621898	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
248	M₁₆	—	172	4	no	—
249	M₁₆	—	172	1	no	—
250	M₁₆	—	172	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	86	2	no	—
259	C₂×C₈	—	86	2	no	—
260	C₂×C₈	—	86	2	no	—
261	C₂×C₈	—	172	4	no	—
262	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
263	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
264	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
265	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	43	2	no	—
268	C₄⋊C₄	—	86	1	no	—
269	C₂²⋊C₄	—	86	2	no	—
270	C₂²⋊C₄	—	86	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
275	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
276	C₄²	#10621898	1	1	yes	characteristic
277	C₂³	—	86	2	no	—
278	C₂³	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
279	Q₈	—	172	1	no	—
280	Q₈	—	172	1	no	—
281	D₄	—	172	2	no	—
282	D₄	—	172	2	no	—
283	D₄	—	172	2	no	—
284	D₄	—	172	1	no	—
285	D₄	—	172	1	no	—
286	D₄	—	172	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	86	2	no	—
293	C₂×C₄	—	172	2	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
296	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
297	C₂×C₄	#13063282	1	1	yes	characteristic
298	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
299	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
300	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
301	C₂×C₄	#13063350	1	1	yes	characteristic
302	C₂×C₄	#13063350	1	1	yes	characteristic
303	C₂×C₄	#13063351	1	1	yes	characteristic
304	C₂×C₄	#13063356	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	—	172	2	no	—
311	C₈	—	172	2	no	—
312	C₈	—	172	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	172	2	no	—
316	C₂²	—	172	2	no	—
317	C₂²	—	172	2	no	—
318	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
319	C₂²	#14582997	1	1	yes	characteristic
320	C₂²	#14583009	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	172	2	no	—
326	C₄	#14582898	1	1	yes	characteristic
327	C₄	#14582903	1	1	yes	characteristic
328	C₄	#14582997	1	1	yes	characteristic
329	C₄	#14583009	1	1	yes	characteristic
330	C₄	#14583022	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	344	2	no	—
334	C₂	#19134396	1	1	yes	characteristic
335	C₂	#19134423	1	1	yes	characteristic
336	C₂	#19134439	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1770	C₂
socle	C₂²×C₄₃	C₄×C₂²⋊C₄
prime radical	#14583449	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1770	D₄₃
hypocenter	C₄₃	#8253
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
4. #19132887	C₂	semidirect
5. #19132498	C₂	semidirect
6. #19132492	C₂	semidirect
7. C₄₃×#1770	C₂	semidirect
11. #14582978	C₄	semidirect
13. C₈×D₁₇₂	C₄	semidirect
18. #14582646	C₄	semidirect
19. #14582642	C₄	semidirect
164. #1770	D₄₃	semidirect
196. C₄₃	#8253	semidirect

Maximal subgroups

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

subgroup	index
#19143308	2
#19133245	2
#19133089	2
#19132887	2
#19132498	2
#19132492	2
C₄₃×#1770	2
#8253	43