Subgroups — #25079805

Order 11008.

774 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	#19145538	C₂	1	2	yes	maximal, complemented
2	C₂×#14583276	C₂	1	1	yes	maximal, characteristic
3	#19143529	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134080	C₂	1	4	yes	maximal, complemented
5	#19133956	C₂	1	4	yes	maximal, complemented
6	#19133262	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₄₄	—	2	1	no	—
9	C₂²×D₃₄₄	—	2	1	no	—
10	#14583456	C₂²	1	1	yes	characteristic
11	C₂×#13063322	C₂²	1	2	yes	—
12	#14583276	—	2	1	no	—
13	#14583276	—	2	1	no	—
14	#14583276	C₂²	1	2	yes	—
15	#14583276	C₂²	1	2	yes	—
16	#14583272	—	2	2	no	—
17	#14583246	—	2	2	no	—
18	#14583100	C₂²	1	8	yes	complemented
19	#14583052	—	2	4	no	—
20	#14583029	—	2	1	no	—
21	#14583029	C₂²	1	2	yes	complemented
22	C₈×Dic₈₆	—	2	1	no	—
23	C₈×Dic₈₆	C₂²	1	2	yes	complemented
24	#14582925	—	2	1	no	—
25	#14582923	C₂²	1	1	yes	characteristic
26	#14582854	—	2	1	no	—
27	#14582642	C₂²	1	8	yes	complemented
28	C₄₃×#431	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₄₃×#335	C₂²	1	2	yes	complemented
31	C₄₃×#334	C₂²	1	2	yes	complemented
32	C₂²×D₁₇₂	—	2	2	no	—
33	C₂²×D₁₇₂	—	2	2	no	—
34	C₂²×D₁₇₂	—	2	2	no	—
35	C₂×#10621873	—	2	8	no	—
36	C₂×D₃₄₄	—	2	2	no	—
37	C₂×D₃₄₄	—	2	2	no	—
38	C₂×D₃₄₄	—	2	2	no	—
39	C₂×D₃₄₄	—	2	2	no	—
40	C₂×D₃₄₄	—	4	4	no	—
41	C₂×D₃₄₄	—	4	4	no	—
42	C₂×#10621877	—	2	1	no	—
43	#13063325	D₄	1	1	yes	characteristic
44	#13063325	D₄	1	1	yes	characteristic
45	#13063322	—	2	2	no	—
46	#13063322	—	2	2	no	—
47	#13063322	—	2	2	no	—
48	#13063322	C₂³	1	4	yes	—
49	#13063317	—	2	1	no	—
50	#13063304	—	2	1	no	—
51	#13063304	D₄	1	2	yes	—
52	C₄×Dic₈₆	—	2	1	no	—
53	C₄×Dic₈₆	C₂³	1	2	yes	—
54	#13063284	—	2	1	no	—
55	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
56	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
57	Dic₄₃⋊C₈	—	2	2	no	—
58	Dic₄₃⋊C₈	—	2	2	no	—
59	#13063247	—	2	8	no	—
60	#13063235	—	2	2	no	—
61	C₈×Dic₄₃	—	2	2	no	—
62	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
66	C₂²⋊C₈×C₄₃	—	2	2	no	—
67	C₄×C₈×C₄₃	—	2	2	no	—
68	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂³×D₄₃	—	4	2	no	—
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
76	C₂×D₁₇₂	—	2	4	no	—
77	C₂×D₁₇₂	—	2	4	no	—
78	C₂×D₁₇₂	—	2	4	no	—
79	C₂×D₁₇₂	—	4	2	no	—
80	C₂×D₁₇₂	—	4	2	no	—
81	C₂×D₁₇₂	—	4	2	no	—
82	C₂×D₁₇₂	—	4	4	no	—
83	C₂×D₁₇₂	—	4	4	no	—
84	C₂×D₁₇₂	—	8	4	no	—
85	C₂×D₁₇₂	—	8	4	no	—
86	C₂×Dic₈₆	—	2	1	no	—
87	C₂×Dic₈₆	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	4	no	—
90	C₂×C₈×C₄₃	SD₁₆	1	4	yes	complemented
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₃	#41	1	1	yes	characteristic
98	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
99	C₂²⋊Dic₄₃	—	2	2	no	—
100	C₄⋊Dic₄₃	—	2	1	no	—
101	C₄⋊Dic₄₃	—	2	1	no	—
102	C₄⋊Dic₄₃	—	2	1	no	—
103	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
104	Dic₄₃⋊C₄	—	2	2	no	—
105	Dic₄₃⋊C₄	—	2	2	no	—
106	C₄×Dic₄₃	—	2	2	no	—
107	#10621877	—	4	2	no	—
108	C₂×C₄₃⋊C₈	—	2	2	no	—
109	D₃₄₄	—	4	4	no	—
110	D₃₄₄	—	4	4	no	—
111	D₃₄₄	—	4	4	no	—
112	D₃₄₄	—	4	4	no	—
113	#10621873	—	8	8	no	—
114	C₂³×C₄₃	#121	1	1	yes	characteristic
115	C₂²×D₄₃	—	4	4	no	—
116	C₂²×D₄₃	—	8	4	no	—
117	C₂²×D₄₃	—	8	4	no	—
118	C₂²×D₄₃	—	8	4	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	—	2	2	no	—
131	D₁₇₂	—	4	4	no	—
132	D₁₇₂	—	4	4	no	—
133	D₁₇₂	—	4	8	no	—
134	D₁₇₂	—	8	4	no	—
135	D₁₇₂	—	8	4	no	—
136	D₁₇₂	—	8	4	no	—
137	D₁₇₂	—	8	4	no	—
138	Dic₈₆	—	2	4	no	—
139	Dic₈₆	—	4	2	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	—	4	4	no	—
143	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
144	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
145	C₄₃⋊C₈	—	4	2	no	—
146	#19203	—	43	1	no	maximal, Sylow
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	—	2	1	no	—
149	C₂²×C₄₃	#460	1	2	yes	—
150	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
151	C₂×D₄₃	—	8	4	no	—
152	C₂×D₄₃	—	8	4	no	—
153	C₂×D₄₃	—	8	4	no	—
154	C₂×D₄₃	—	16	8	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	#460	1	2	yes	—
158	C₄×C₄₃	#472	1	2	yes	—
159	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
160	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₄×C₄₃	#576	1	2	yes	—
162	Dic₄₃	—	4	2	no	—
163	Dic₄₃	—	4	2	no	—
164	#3208	—	43	2	no	—
165	C₂×#493	—	43	1	no	—
166	#2960	—	43	1	no	—
167	#1688	—	43	4	no	—
168	#1668	—	43	4	no	—
169	#1564	D₄₃	1	1	yes	characteristic, complemented
170	#1545	—	43	2	no	—
171	C₂×C₄₃	—	2	2	no	—
172	C₂×C₄₃	C₂×#460	1	1	yes	characteristic
173	C₂×C₄₃	#3058	1	1	yes	characteristic
174	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
175	D₄₃	—	16	4	no	—
176	C₂²×D₈	—	86	1	no	—
177	C₂²×D₈	—	86	1	no	—
178	#533	—	43	1	no	—
179	C₂×#127	—	43	2	no	—
180	#493	—	43	2	no	—
181	#493	—	43	2	no	—
182	#493	—	86	1	no	—
183	#493	—	86	1	no	—
184	#492	—	86	2	no	—
185	#486	—	86	2	no	—
186	#460	—	43	8	no	—
187	#450	—	86	4	no	—
188	#446	—	43	2	no	—
189	#446	—	86	1	no	—
190	C₈×Q₈	—	43	2	no	—
191	C₈×Q₈	—	86	1	no	—
192	#431	C₂×D₄₃	1	2	yes	—
193	#424	—	86	1	no	—
194	#423	—	43	1	no	—
195	#405	—	86	1	no	—
196	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
197	#335	C₂×D₄₃	1	2	yes	complemented
198	#334	C₂×D₄₃	1	2	yes	complemented
199	#331	—	43	8	no	—
200	C₄₃	#19203	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₄	—	86	2	no	—
202	C₂²×D₄	—	86	2	no	—
203	C₂²×D₄	—	86	2	no	—
204	C₂×SD₁₆	—	86	8	no	—
205	C₂×D₈	—	86	2	no	—
206	C₂×D₈	—	86	2	no	—
207	C₂×D₈	—	86	2	no	—
208	C₂×D₈	—	86	2	no	—
209	C₂×D₈	—	172	4	no	—
210	C₂×D₈	—	172	4	no	—
211	C₂×M₁₆	—	86	1	no	—
212	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
213	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
214	C₄⋊Q₈	—	43	1	no	—
215	C₄⋊Q₈	—	43	1	no	—
216	#127	—	43	4	no	—
217	#127	—	86	2	no	—
218	#127	—	86	2	no	—
219	#127	—	86	2	no	—
220	#125	—	86	1	no	—
221	C₂²⋊Q₈	—	43	2	no	—
222	C₂²⋊Q₈	—	86	1	no	—
223	C₄×Q₈	—	43	2	no	—
224	C₄×Q₈	—	86	1	no	—
225	#117	—	86	1	no	—
226	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
227	C₄⋊C₈	—	43	2	no	—
228	C₄⋊C₈	—	86	2	no	—
229	C₄⋊C₈	—	86	2	no	—
230	C₄⋊C₈	—	86	1	no	—
231	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
232	#102	—	86	8	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	#97	—	86	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	86	2	no	—
237	C₄×C₈	D₁₇₂	1	1	yes	characteristic
238	C₄×C₈	D₁₇₂	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	C₂⁴	—	172	2	no	—
242	C₂×Q₈	—	43	2	no	—
243	C₂×Q₈	—	86	1	no	—
244	C₂×D₄	—	86	4	no	—
245	C₂×D₄	—	86	4	no	—
246	C₂×D₄	—	86	4	no	—
247	C₂×D₄	—	344	4	no	—
248	C₂×D₄	—	344	4	no	—
249	C₂×D₄	—	172	2	no	—
250	C₂×D₄	—	172	2	no	—
251	C₂×D₄	—	172	2	no	—
252	C₂×D₄	—	172	4	no	—
253	C₂×D₄	—	172	4	no	—
254	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
256	C₂²×C₄	#10621902	1	1	yes	characteristic
257	SD₁₆	—	344	8	no	—
258	D₈	—	172	4	no	—
259	D₈	—	172	4	no	—
260	D₈	—	172	4	no	—
261	D₈	—	172	4	no	—
262	M₁₆	—	172	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	86	2	no	—
266	C₂×C₈	#10621873	1	4	yes	complemented
267	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
268	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
269	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
270	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
271	C₄⋊C₄	—	43	2	no	—
272	C₄⋊C₄	—	86	1	no	—
273	C₄⋊C₄	—	86	1	no	—
274	C₄⋊C₄	—	86	2	no	—
275	C₄⋊C₄	—	86	1	no	—
276	C₄⋊C₄	—	86	2	no	—
277	C₂²⋊C₄	—	86	2	no	—
278	C₄²	—	86	2	no	—
279	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
280	C₄²	D₄×D₄₃	1	1	yes	characteristic
281	C₄²	#10621902	1	1	yes	characteristic
282	C₄²	C₂³×D₄₃	1	1	yes	characteristic
283	C₂³	—	344	4	no	—
284	C₂³	—	344	4	no	—
285	C₂³	—	344	4	no	—
286	C₂³	—	172	4	no	—
287	C₂³	#13063300	1	1	yes	characteristic
288	Q₈	—	86	4	no	—
289	Q₈	—	172	2	no	—
290	D₄	—	344	4	no	—
291	D₄	—	344	4	no	—
292	D₄	—	344	4	no	—
293	D₄	—	344	4	no	—
294	D₄	—	172	4	no	—
295	D₄	—	172	4	no	—
296	D₄	—	172	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	86	2	no	—
300	C₂×C₄	—	86	2	no	—
301	C₂×C₄	#13063300	1	1	yes	characteristic
302	C₂×C₄	#13063300	1	1	yes	characteristic
303	C₂×C₄	#13063300	1	1	yes	characteristic
304	C₂×C₄	C₂×#10621873	1	2	yes	—
305	C₂×C₄	#13063353	1	2	yes	—
306	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
308	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	4	no	—
312	C₈	—	172	2	no	—
313	C₈	C₂×#10621873	1	2	yes	—
314	C₈	C₂×#10621873	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	344	4	no	—
318	C₂²	—	344	4	no	—
319	C₂²	—	344	4	no	—
320	C₂²	—	688	8	no	—
321	C₂²	#14583100	1	2	yes	—
322	C₂²	C₂×#13063300	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	172	2	no	—
326	C₄	—	172	2	no	—
327	C₄	#14583100	1	2	yes	—
328	C₄	#14583170	1	2	yes	—
329	C₄	C₂²×#10621873	1	1	yes	characteristic
330	C₄	C₂×#13063353	1	1	yes	characteristic
331	C₄	#14583701	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	688	4	no	—
334	C₂	C₂×#14583100	1	1	yes	characteristic
335	C₂	#19144244	1	1	yes	characteristic
336	C₂	C₂×#14583170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063300
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1564	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂×#14583276	C₂
hypercenter	#1564	D₄₃
hypocenter	C₄₃	#19203
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #19145538	C₂	semidirect
3. #19143529	C₂	semidirect
4. #19134080	C₂	semidirect
5. #19133956	C₂	semidirect
6. #19133262	C₂	semidirect
7. C₄₃×#1564	C₂	semidirect
18. #14583100	C₂²	semidirect
21. #14583029	C₂²	semidirect
23. C₈×Dic₈₆	C₂²	semidirect
27. #14582642	C₂²	semidirect
30. C₄₃×#335	C₂²	semidirect
31. C₄₃×#334	C₂²	semidirect
90. C₂×C₈×C₄₃	SD₁₆	semidirect
169. #1564	D₄₃	semidirect
197. #335	C₂×D₄₃	semidirect
198. #334	C₂×D₄₃	semidirect
200. C₄₃	#19203	semidirect
266. C₂×C₈	#10621873	semidirect

Maximal subgroups

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

subgroup	index
#19145538	2
C₂×#14583276	2
#19143529	2
#19134080	2
#19133956	2
#19133262	2
C₄₃×#1564	2
#19203	43