Subgroups — #25097166

Order 11008.

810 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582853	C₂	1	1	yes	maximal, characteristic
2	#19137205	C₂	1	2	yes	maximal
3	#19134755	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134728	C₂	1	2	yes	maximal, complemented
5	#19134625	C₂	1	2	yes	maximal
6	#19134621	C₂	1	2	yes	maximal
7	#19134495	C₂	1	2	yes	maximal, complemented
8	#107×Dic₄₃	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10621872	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
12	C₂×#13063282	C₂²	1	1	yes	characteristic
13	C₂×#13063259	C₂²	1	2	yes	—
14	C₂×#13063259	C₂²	1	2	yes	—
15	#14582853	C₄	1	8	yes	complemented
16	C₂×#13063236	C₂²	1	1	yes	characteristic
17	C₂×#13063235	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
19	#14582791	—	2	4	no	—
20	#14582786	—	2	4	no	—
21	#14582769	C₂²	1	2	yes	—
22	#14582756	C₂²	1	4	yes	—
23	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
24	#14582661	C₂²	1	4	yes	—
25	#14582661	C₂²	1	4	yes	—
26	C₂×#107×C₄₃	C₂²	1	2	yes	—
27	C₂×#106×C₄₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#382	C₂²	1	2	yes	—
30	C₂²×C₄×D₄₃	C₂³	1	1	yes	characteristic
31	C₂×#10621872	—	2	2	no	—
32	C₂×#10621872	—	2	2	no	—
33	C₂×#10621872	C₂×C₄	1	4	yes	—
34	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₄₃	—	2	2	no	—
37	C₂×C₈×D₄₃	—	2	2	no	—
38	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
39	#13063282	C₂×C₄	1	4	yes	—
40	#13063282	Q₈	1	4	yes	—
41	C₂×C₄⋊Dic₄₃	—	2	4	no	—
42	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
43	C₂×Dic₄₃⋊C₄	D₄	1	2	yes	—
44	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
45	C₂×(C₂×D₄₃)⋊C₄	D₄	1	2	yes	—
46	C₂×C₄×Dic₄₃	—	2	4	no	—
47	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
48	#13063259	—	2	4	no	—
49	#13063259	—	2	4	no	—
50	#13063236	C₂×C₄	1	4	yes	—
51	#13063235	C₂×C₄	1	4	yes	—
52	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
53	C₄₃⋊#95	—	2	4	no	—
54	C₄₃⋊#95	—	2	4	no	—
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	#107×C₄₃	—	2	4	no	—
60	#106×C₄₃	—	2	4	no	—
61	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
62	#95×C₄₃	—	2	4	no	—
63	C₂³×D₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	4	no	—
68	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	—	2	1	no	—
74	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₃	#41	1	2	yes	—
79	C₂×C₈×C₄₃	#41	1	2	yes	—
80	C₂×C₈×C₄₃	#41	1	2	yes	—
81	C₂×C₈×C₄₃	#41	1	2	yes	—
82	C₄⋊C₄×C₄₃	—	2	4	no	—
83	C₄⋊C₄×C₄₃	—	2	4	no	—
84	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
86	(C₂×D₄₃)⋊C₄	—	8	4	no	—
87	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	8	yes	—
88	C₄⋊Dic₄₃	—	2	4	no	—
89	C₄⋊Dic₄₃	—	2	4	no	—
90	C₄⋊Dic₄₃	—	8	4	no	—
91	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
92	C₄×Dic₄₃	—	8	4	no	—
93	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
94	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
95	C₂×C₄₃⋊C₈	—	2	1	no	—
96	C₂×C₄₃⋊C₈	—	2	1	no	—
97	C₂×C₄₃⋊C₈	—	2	1	no	—
98	C₂×C₄₃⋊C₈	—	2	1	no	—
99	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
101	#10621872	—	2	8	no	—
102	C₈×D₄₃	—	2	8	no	—
103	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
104	C₂²×D₄₃	—	2	1	no	—
105	C₂²×D₄₃	—	2	1	no	—
106	C₂²×D₄₃	—	2	2	no	—
107	C₂²×D₄₃	—	2	2	no	—
108	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	4	4	no	—
112	C₂×C₄×C₄₃	—	4	4	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	2	1	no	—
123	C₂×Dic₄₃	—	2	2	no	—
124	C₂×Dic₄₃	—	2	2	no	—
125	C₂×Dic₄₃	—	4	4	no	—
126	C₂×Dic₄₃	—	4	4	no	—
127	C₂×Dic₄₃	—	4	4	no	—
128	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
129	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₄×D₄₃	—	2	2	no	—
131	C₄×D₄₃	—	2	2	no	—
132	C₄×D₄₃	—	2	2	no	—
133	C₄×D₄₃	—	2	2	no	—
134	C₈×C₄₃	#117	1	4	yes	—
135	C₈×C₄₃	#117	1	4	yes	—
136	C₄₃⋊C₈	—	2	2	no	—
137	C₄₃⋊C₈	—	2	2	no	—
138	#20821	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₄₃	#532	1	1	yes	characteristic
141	C₂²×C₄₃	#532	1	1	yes	characteristic
142	C₂²×C₄₃	#576	1	2	yes	—
143	C₂²×C₄₃	#578	1	2	yes	—
144	C₂×D₄₃	—	2	4	no	—
145	C₂×D₄₃	—	2	2	no	—
146	C₂×D₄₃	—	4	2	no	—
147	C₂×D₄₃	—	4	1	no	—
148	C₂×D₄₃	—	4	1	no	—
149	C₄×C₄₃	—	2	4	no	—
150	C₄×C₄₃	—	8	4	no	—
151	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
152	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
153	C₄×C₄₃	#533	1	1	yes	characteristic
154	C₄×C₄₃	#533	1	1	yes	characteristic
155	Dic₄₃	—	2	4	no	—
156	Dic₄₃	—	2	2	no	—
157	Dic₄₃	—	8	4	no	—
158	C₂×#405	—	43	1	no	—
159	#2314	—	43	2	no	—
160	#1846	D₄₃	1	1	yes	characteristic, complemented
161	#1845	—	43	1	no	—
162	#1836	—	43	2	no	—
163	#1809	—	43	2	no	—
164	#1808	—	43	2	no	—
165	#1776	—	43	2	no	—
166	C₄×#107	—	43	2	no	—
167	C₂×C₄₃	#2285	1	1	yes	characteristic
168	C₂×C₄₃	#2913	1	2	yes	—
169	C₂×C₄₃	#3175	1	2	yes	—
170	C₂×C₄₃	#3176	1	1	yes	characteristic
171	C₂×C₄₃	#3177	1	1	yes	characteristic
172	D₄₃	—	4	2	no	—
173	C₂²×M₁₆	—	43	1	no	—
174	C₂³×C₈	—	43	1	no	—
175	C₂×#117	—	43	1	no	—
176	C₂×#107	—	43	2	no	—
177	C₂×#107	—	43	2	no	—
178	C₂×#107	C₂×D₄₃	1	2	yes	—
179	C₂×#106	C₂×D₄₃	1	2	yes	—
180	#405	—	43	8	no	—
181	C₂×#97	—	43	1	no	—
182	C₂×#97	—	43	1	no	—
183	C₂×C₄×C₈	—	43	1	no	—
184	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
185	#388	—	86	4	no	—
186	#387	—	86	4	no	—
187	#384	—	43	2	no	—
188	#382	C₂×D₄₃	1	2	yes	—
189	#380	—	43	4	no	—
190	C₄×C₄⋊C₄	—	43	4	no	—
191	#340	—	43	4	no	—
192	#340	—	43	4	no	—
193	C₄₃	#20821	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₄	—	43	1	no	—
195	C₂×M₁₆	—	43	4	no	—
196	C₂×M₁₆	—	86	2	no	—
197	C₂×M₁₆	—	86	2	no	—
198	C₂²×C₈	—	43	1	no	—
199	C₂²×C₈	—	43	1	no	—
200	C₂²×C₈	—	43	4	no	—
201	C₂²×C₈	—	86	2	no	—
202	C₂²×C₈	—	86	2	no	—
203	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
205	#117	—	43	4	no	—
206	#117	—	43	4	no	—
207	C₂×C₄⋊C₄	—	43	2	no	—
208	C₂×C₄⋊C₄	—	43	4	no	—
209	C₂×C₄⋊C₄	—	86	4	no	—
210	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
211	C₂×C₂²⋊C₄	—	43	2	no	—
212	C₂×C₂²⋊C₄	—	86	4	no	—
213	C₂×C₄²	—	43	1	no	—
214	C₂×C₄²	—	86	4	no	—
215	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
216	#107	—	2	4	no	—
217	#107	—	86	4	no	—
218	#107	—	86	4	no	—
219	#106	—	2	4	no	—
220	#97	—	43	4	no	—
221	#97	—	43	4	no	—
222	C₄×C₈	—	43	4	no	—
223	C₄×C₈	C₄×D₄₃	1	4	yes	—
224	#95	—	2	4	no	—
225	#95	—	86	4	no	—
226	#95	—	86	4	no	—
227	C₂⁴	—	43	1	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	43	1	no	—
230	C₂²×C₄	—	43	2	no	—
231	C₂²×C₄	—	43	2	no	—
232	C₂²×C₄	—	43	2	no	—
233	C₂²×C₄	—	86	1	no	—
234	C₂²×C₄	—	86	1	no	—
235	C₂²×C₄	—	86	1	no	—
236	C₂²×C₄	—	86	1	no	—
237	C₂²×C₄	—	86	4	no	—
238	C₂²×C₄	D₄×D₄₃	1	2	yes	—
239	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
240	M₁₆	—	86	8	no	—
241	C₂×C₈	—	43	2	no	—
242	C₂×C₈	—	43	2	no	—
243	C₂×C₈	—	86	1	no	—
244	C₂×C₈	—	86	1	no	—
245	C₂×C₈	—	86	1	no	—
246	C₂×C₈	—	86	1	no	—
247	C₂×C₈	—	86	8	no	—
248	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
250	C₂×C₈	#10621900	1	2	yes	—
251	C₂×C₈	#10621900	1	2	yes	—
252	C₂×C₈	#10621902	1	2	yes	—
253	C₂×C₈	#10621902	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	43	8	no	—
257	C₄⋊C₄	—	86	4	no	—
258	C₄⋊C₄	—	86	4	no	—
259	C₄⋊C₄	—	344	4	no	—
260	C₂²⋊C₄	—	43	8	no	—
261	C₂²⋊C₄	—	344	4	no	—
262	C₄²	—	43	2	no	—
263	C₄²	—	43	2	no	—
264	C₄²	—	344	4	no	—
265	C₄²	C₂×C₄×D₄₃	1	2	yes	—
266	C₄²	Q₈×D₄₃	1	2	yes	—
267	C₂³	—	43	2	no	—
268	C₂³	—	86	2	no	—
269	C₂³	—	86	1	no	—
270	C₂³	—	86	1	no	—
271	C₂³	—	86	2	no	—
272	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	43	4	no	—
279	C₂×C₄	—	43	2	no	—
280	C₂×C₄	—	86	2	no	—
281	C₂×C₄	—	86	1	no	—
282	C₂×C₄	—	86	1	no	—
283	C₂×C₄	—	86	2	no	—
284	C₂×C₄	—	86	2	no	—
285	C₂×C₄	—	86	2	no	—
286	C₂×C₄	—	86	2	no	—
287	C₂×C₄	—	86	2	no	—
288	C₂×C₄	—	172	4	no	—
289	C₂×C₄	—	172	4	no	—
290	C₂×C₄	—	172	4	no	—
291	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
292	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
293	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
294	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
295	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
296	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
297	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
298	C₈	—	86	2	no	—
299	C₈	—	86	2	no	—
300	C₈	#13063283	1	4	yes	—
301	C₈	#13063283	1	4	yes	—
302	C₂²	—	86	4	no	—
303	C₂²	—	86	2	no	—
304	C₂²	—	172	2	no	—
305	C₂²	—	172	1	no	—
306	C₂²	—	172	1	no	—
307	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
308	C₂²	#14583447	1	1	yes	characteristic
309	C₂²	#14583450	1	1	yes	characteristic
310	C₂²	#14583701	1	2	yes	—
311	C₂²	#14583713	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	8	4	no	—
314	C₄	—	86	4	no	—
315	C₄	—	86	2	no	—
316	C₄	—	344	4	no	—
317	C₄	C₂×#13063283	1	1	yes	characteristic
318	C₄	C₂×#13063283	1	1	yes	characteristic
319	C₄	#14583454	1	1	yes	characteristic
320	C₄	#14583454	1	1	yes	characteristic
321	C₂	—	172	2	no	—
322	C₂	#19137000	1	1	yes	characteristic
323	C₂	#19143192	1	2	yes	—
324	C₂	#19145231	1	2	yes	—
325	C₂	#19145246	1	1	yes	characteristic
326	C₂	#19145255	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1846	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂³×D₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1846	D₄₃
hypocenter	C₄₃	#20821
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
3. #19134755	C₂	semidirect
4. #19134728	C₂	semidirect
7. #19134495	C₂	semidirect
8. #107×Dic₄₃	C₂	semidirect
9. C₄₃×#1846	C₂	semidirect
15. #14582853	C₄	semidirect
160. #1846	D₄₃	semidirect
193. C₄₃	#20821	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582853	2
#19137205	2
#19134755	2
#19134728	2
#19134625	2
#19134621	2
#19134495	2
#107×Dic₄₃	2
C₄₃×#1846	2
#20821	43