Subgroups — #25079861

Order 11008.

570 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145937	C₂	1	2	yes	maximal
2	#19143529	C₂	1	1	yes	maximal, characteristic
3	#19134098	C₂	1	4	yes	maximal, complemented
4	#19133996	C₂	1	2	yes	maximal, complemented
5	#19133986	C₂	1	2	yes	maximal, complemented
6	C₂×#14582652	C₂	1	1	yes	maximal, characteristic
7	#19133285	C₂	1	2	yes	maximal
8	C₄₃×#3160	C₂	1	1	yes	maximal, characteristic
9	#14583456	C₂²	1	1	yes	characteristic
10	#14583194	—	2	2	no	—
11	#14583191	C₂²	1	4	yes	—
12	#14583138	—	2	4	no	—
13	#14583109	—	2	2	no	—
14	#14583106	C₂²	1	4	yes	—
15	#14583029	—	2	1	no	—
16	#14583029	C₂²	1	2	yes	—
17	C₈×Dic₈₆	—	2	1	no	—
18	C₈×Dic₈₆	C₂²	1	2	yes	—
19	#14582925	—	2	1	no	—
20	#14582923	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
22	#14582854	—	2	1	no	—
23	#14582652	C₂²	1	2	yes	—
24	#14582652	C₂²	1	2	yes	—
25	#14582648	C₂²	1	4	yes	—
26	#14582645	C₂²	1	4	yes	—
27	C₄₃×#533	C₂²	1	2	yes	—
28	C₄₃×#500	C₂²	1	1	yes	characteristic
29	C₄₃×#500	C₂²	1	1	yes	characteristic
30	C₄₃×#499	—	2	1	no	—
31	C₄₃×#499	—	2	1	no	—
32	C₄₃×#498	C₂²	1	2	yes	—
33	C₄₃×#427	—	2	1	no	—
34	C₄₃×#427	—	2	1	no	—
35	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#10621885	—	2	4	no	—
37	C₂×#10621877	—	2	1	no	—
38	C₂²×C₄₃⋊C₈	—	2	2	no	—
39	#13063325	D₄	1	1	yes	characteristic
40	#13063325	D₄	1	1	yes	characteristic
41	#13063317	—	2	1	no	—
42	#13063304	—	2	1	no	—
43	#13063304	D₄	1	2	yes	—
44	C₄×Dic₈₆	—	2	1	no	—
45	C₄×Dic₈₆	C₂³	1	2	yes	—
46	#13063284	—	2	1	no	—
47	#13063258	—	2	4	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
49	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
50	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
51	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
52	Dic₄₃⋊C₈	—	2	2	no	—
53	Dic₄₃⋊C₈	—	2	2	no	—
54	#13063249	—	2	4	no	—
55	#13063248	—	2	4	no	—
56	#13063235	—	2	2	no	—
57	C₈×Dic₄₃	—	2	2	no	—
58	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
62	#125×C₄₃	—	2	2	no	—
63	C₂²⋊Q₈×C₄₃	—	2	2	no	—
64	C₂²⋊Q₈×C₄₃	—	2	2	no	—
65	C₄×Q₈×C₄₃	—	2	2	no	—
66	C₄×Q₈×C₄₃	—	2	2	no	—
67	#117×C₄₃	—	2	2	no	—
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	#107×C₄₃	—	2	1	no	—
70	#107×C₄₃	—	2	1	no	—
71	#107×C₄₃	—	2	1	no	—
72	#107×C₄₃	—	2	1	no	—
73	#106×C₄₃	—	2	2	no	—
74	#106×C₄₃	—	2	2	no	—
75	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₃	—	2	2	no	—
80	C₂×Q₈×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
84	C₂×Dic₈₆	—	2	1	no	—
85	C₂×Dic₈₆	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	Q₁₆	1	4	yes	complemented
88	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	2	2	no	—
94	C₄⋊C₄×C₄₃	—	2	2	no	—
95	C₄⋊C₄×C₄₃	—	2	2	no	—
96	C₄⋊C₄×C₄₃	—	4	2	no	—
97	C₄⋊C₄×C₄₃	—	4	2	no	—
98	C₂²⋊C₄×C₄₃	—	4	2	no	—
99	C₄²×C₄₃	—	4	2	no	—
100	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₃	#41	1	1	yes	characteristic
103	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊Dic₄₃	—	2	2	no	—
105	#10621885	—	8	4	no	—
106	C₄⋊Dic₄₃	—	2	1	no	—
107	C₄⋊Dic₄₃	—	2	1	no	—
108	C₄⋊Dic₄₃	—	2	1	no	—
109	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
110	Dic₄₃⋊C₄	—	2	2	no	—
111	Dic₄₃⋊C₄	—	2	2	no	—
112	C₄×Dic₄₃	—	2	2	no	—
113	#10621877	—	4	2	no	—
114	C₂×C₄₃⋊C₈	—	2	2	no	—
115	C₂×C₄₃⋊C₈	—	2	4	no	—
116	C₂×C₄₃⋊C₈	—	8	2	no	—
117	C₂³×C₄₃	#121	1	1	yes	characteristic
118	Q₈×C₄₃	—	4	2	no	—
119	Q₈×C₄₃	—	4	2	no	—
120	Q₈×C₄₃	—	8	2	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	4	1	no	—
124	C₂×C₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
130	C₂×C₄×C₄₃	#137	1	2	yes	—
131	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₄₃	—	2	2	no	—
135	C₂×Dic₄₃	—	2	2	no	—
136	Dic₈₆	—	2	2	no	—
137	Dic₈₆	—	2	2	no	—
138	Dic₈₆	—	4	2	no	—
139	C₈×C₄₃	—	2	1	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
142	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
143	C₄₃⋊C₈	—	4	2	no	—
144	C₄₃⋊C₈	—	4	4	no	—
145	#19211	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	#462	1	2	yes	—
149	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
150	C₄×C₄₃	—	2	1	no	—
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	8	2	no	—
154	C₄×C₄₃	—	8	2	no	—
155	C₄×C₄₃	#462	1	2	yes	—
156	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
157	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
158	C₄×C₄₃	#576	1	1	yes	characteristic
159	C₄×C₄₃	#578	1	1	yes	characteristic
160	Dic₄₃	—	4	2	no	—
161	Dic₄₃	—	4	2	no	—
162	#3249	—	43	2	no	—
163	#3160	D₄₃	1	1	yes	characteristic
164	#2960	—	43	1	no	—
165	#1692	—	43	4	no	—
166	#1676	—	43	2	no	—
167	#1674	—	43	2	no	—
168	C₂×#335	—	43	1	no	—
169	#1550	—	43	2	no	—
170	C₂×C₄₃	—	2	2	no	—
171	C₂×C₄₃	C₂×#462	1	1	yes	characteristic
172	C₂×C₄₃	#3061	1	1	yes	characteristic
173	C₂×C₄₃	#3067	1	1	yes	characteristic
174	#533	—	43	1	no	—
175	#533	C₂×D₄₃	1	2	yes	—
176	#500	C₂×D₄₃	1	1	yes	characteristic
177	#500	C₂×D₄₃	1	1	yes	characteristic
178	#499	—	2	1	no	—
179	#499	—	2	1	no	—
180	#498	C₂×D₄₃	1	2	yes	—
181	#477	—	43	4	no	—
182	#477	—	86	2	no	—
183	#467	—	86	4	no	—
184	#462	—	43	4	no	—
185	#462	—	86	2	no	—
186	#446	—	43	2	no	—
187	#446	—	86	1	no	—
188	C₈×Q₈	—	43	2	no	—
189	C₈×Q₈	—	86	1	no	—
190	#427	—	2	1	no	—
191	#427	—	2	1	no	—
192	#424	—	86	1	no	—
193	#423	—	43	1	no	—
194	C₂×C₄⋊C₈	—	43	2	no	—
195	#405	—	86	1	no	—
196	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
197	#335	—	43	2	no	—
198	#335	—	43	2	no	—
199	#333	—	43	4	no	—
200	#332	—	43	4	no	—
201	C₄₃	#19211	1	1	yes	characteristic, complemented, Sylow
202	C₂×Q₁₆	—	86	4	no	—
203	C₂×M₁₆	—	86	1	no	—
204	C₂²×C₈	—	86	2	no	—
205	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
207	C₄⋊Q₈	—	43	1	no	—
208	C₄⋊Q₈	—	43	1	no	—
209	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
210	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
211	#125	—	2	2	no	—
212	#125	—	86	1	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	43	2	no	—
216	C₂²⋊Q₈	—	86	1	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	43	2	no	—
220	C₄×Q₈	—	86	1	no	—
221	#117	—	2	2	no	—
222	#117	—	86	1	no	—
223	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
224	#107	—	2	1	no	—
225	#107	—	2	1	no	—
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#107	—	86	4	no	—
229	#106	—	2	2	no	—
230	#106	—	2	2	no	—
231	C₄⋊C₈	—	43	2	no	—
232	C₄⋊C₈	—	43	4	no	—
233	C₄⋊C₈	—	86	2	no	—
234	C₄⋊C₈	—	86	2	no	—
235	C₄⋊C₈	—	86	2	no	—
236	C₄⋊C₈	—	86	1	no	—
237	#103	—	86	4	no	—
238	#103	—	86	4	no	—
239	#97	—	86	2	no	—
240	C₄×C₈	—	86	2	no	—
241	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
242	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	43	2	no	—
248	C₂×Q₈	—	86	1	no	—
249	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
250	C₂²×C₄	#10621900	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
252	Q₁₆	—	344	4	no	—
253	M₁₆	—	172	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	86	4	no	—
256	C₂×C₈	—	86	2	no	—
257	C₂×C₈	—	344	2	no	—
258	C₂×C₈	#10621885	1	4	yes	complemented
259	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	43	2	no	—
270	C₄⋊C₄	—	86	1	no	—
271	C₄⋊C₄	—	86	1	no	—
272	C₄⋊C₄	—	86	2	no	—
273	C₄⋊C₄	—	86	1	no	—
274	C₄⋊C₄	—	86	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	86	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	86	2	no	—
279	C₄²	D₄×D₄₃	1	1	yes	characteristic
280	C₄²	#10621900	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
282	C₄²	C₂³×D₄₃	1	1	yes	characteristic
283	C₂³	#13063296	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	86	2	no	—
288	Q₈	—	86	2	no	—
289	Q₈	—	172	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	86	2	no	—
296	C₂×C₄	—	86	2	no	—
297	C₂×C₄	#13063296	1	1	yes	characteristic
298	C₂×C₄	#13063296	1	1	yes	characteristic
299	C₂×C₄	#13063296	1	1	yes	characteristic
300	C₂×C₄	C₂×#10621885	1	2	yes	—
301	C₂×C₄	#13063356	1	2	yes	—
302	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
304	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	172	2	no	—
308	C₈	—	172	4	no	—
309	C₈	C₂×#10621885	1	2	yes	—
310	C₈	C₂×#10621885	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	#14583106	1	2	yes	—
314	C₂²	C₂×#13063296	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	172	2	no	—
321	C₄	—	172	2	no	—
322	C₄	#14583106	1	2	yes	—
323	C₄	C₂²×#10621885	1	1	yes	characteristic
324	C₄	C₂×#13063356	1	1	yes	characteristic
325	C₄	#14583701	1	1	yes	characteristic
326	C₄	#14583713	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	C₂×#14583106	1	1	yes	characteristic
329	C₂	#19144261	1	1	yes	characteristic
330	C₂	#19144311	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063296
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#3160	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂³×C₄₃	#121
hypercenter	#3160	D₄₃
hypocenter	C₄₃	#19211
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #19134098	C₂	semidirect
4. #19133996	C₂	semidirect
5. #19133986	C₂	semidirect
87. C₂×C₈×C₄₃	Q₁₆	semidirect
201. C₄₃	#19211	semidirect
258. C₂×C₈	#10621885	semidirect

Maximal subgroups

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

subgroup	index
#19145937	2
#19143529	2
#19134098	2
#19133996	2
#19133986	2
C₂×#14582652	2
#19133285	2
C₄₃×#3160	2
#19211	43