Subgroups — #25097066

Order 11008.

954 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582946	C₂	1	2	yes	maximal
2	C₂×#108×D₄₃	C₂	1	2	yes	maximal
3	#19143157	C₂	1	2	yes	maximal
4	C₂×#14582856	C₂	1	1	yes	maximal, characteristic
5	#19134770	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134768	C₂	1	2	yes	maximal, complemented
7	#19134766	C₂	1	2	yes	maximal, complemented
8	#19134765	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1849	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10621872	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₄₃	C₂²	1	4	yes	—
12	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
13	C₂×#13063282	C₂²	1	1	yes	characteristic
14	#14582946	—	2	8	no	—
15	C₂×#13063261	C₂²	1	2	yes	—
16	C₂×#13063261	C₂²	1	2	yes	—
17	C₂×#13063260	C₂²	1	4	yes	—
18	C₂×#13063260	C₂²	1	4	yes	—
19	#108×D₄₃	—	2	8	no	—
20	#14582924	—	2	8	no	—
21	#14582923	C₂²	1	2	yes	—
22	#14582922	C₂²	1	4	yes	—
23	#14582856	C₄	1	8	yes	—
24	C₂×#13063235	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₃	C₂²	1	2	yes	—
28	C₂×#108×C₄₃	C₂²	1	2	yes	—
29	C₄₃×#423	C₂²	1	2	yes	—
30	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄×D₄₃	C₂³	1	1	yes	characteristic
32	C₂×#10621877	C₂³	1	4	yes	—
33	C₂×#10621872	—	2	4	no	—
34	C₂×#10621872	C₂×C₄	1	4	yes	—
35	C₂×#10621872	D₄	1	4	yes	—
36	C₂×#10621872	Q₈	1	4	yes	—
37	M₁₆×D₄₃	—	2	8	no	—
38	M₁₆×D₄₃	—	2	8	no	—
39	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂×C₈×D₄₃	—	2	4	no	—
42	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
43	C₂×C₈×D₄₃	D₄	1	4	yes	—
44	C₂×C₈×D₄₃	Q₈	1	4	yes	—
45	#13063282	C₂×C₄	1	4	yes	—
46	#13063282	D₄	1	4	yes	—
47	C₂×Dic₄₃⋊C₄	Q₈	1	2	yes	—
48	C₂×(C₂×D₄₃)⋊C₄	Q₈	1	2	yes	—
49	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
50	#13063261	—	2	4	no	—
51	#13063261	—	2	4	no	—
52	#13063260	—	2	8	no	—
53	#13063260	—	2	8	no	—
54	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
55	Dic₄₃⋊C₈	—	2	8	no	—
56	#13063235	C₂×C₄	1	4	yes	—
57	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
58	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
59	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	#108×C₄₃	—	2	4	no	—
64	#108×C₄₃	—	2	4	no	—
65	C₄⋊C₈×C₄₃	—	2	4	no	—
66	#97×C₄₃	C₂×C₄	1	4	yes	—
67	C₂³×D₄₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₃	C₂×Q₈	1	2	yes	—
71	C₂²×Dic₄₃	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
76	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
77	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
78	M₁₆×C₄₃	—	2	4	no	—
79	M₁₆×C₄₃	—	2	4	no	—
80	C₂×C₈×C₄₃	—	2	4	no	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
87	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₃	C₂×D₄	1	2	yes	—
89	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	8	yes	—
90	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
91	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
92	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
93	#10621877	—	2	4	no	—
94	#10621877	—	2	4	no	—
95	C₂×C₄₃⋊C₈	—	2	4	no	—
96	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
100	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
101	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
102	#10621872	—	4	8	no	—
103	#10621872	C₄⋊C₄	1	16	yes	—
104	C₈×D₄₃	—	4	8	no	—
105	C₈×D₄₃	C₄⋊C₄	1	16	yes	—
106	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×D₄₃	—	4	2	no	—
108	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂²×D₄₃	C₄⋊Q₈	1	2	yes	—
110	C₂²×D₄₃	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	—	4	2	no	—
112	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×Dic₄₃	—	4	2	no	—
120	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×Dic₄₃	C₄⋊Q₈	1	2	yes	—
123	C₂×Dic₄₃	C₄⋊Q₈	1	2	yes	—
124	C₄×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₄×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
126	C₄×D₄₃	#127	1	4	yes	—
127	C₄×D₄₃	C₄⋊Q₈	1	4	yes	—
128	C₈×C₄₃	—	4	4	no	—
129	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
130	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
131	C₄₃⋊C₈	—	4	4	no	—
132	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
133	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
134	#20807	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₂×D₄₃	—	2	4	no	—
140	C₂×D₄₃	—	4	2	no	—
141	C₂×D₄₃	—	4	2	no	—
142	C₂×D₄₃	#384	1	4	yes	—
143	C₄×C₄₃	—	2	4	no	—
144	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
147	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	Dic₄₃	—	2	4	no	—
149	Dic₄₃	#384	1	4	yes	—
150	C₂×#430	—	43	2	no	—
151	C₂²×#108	—	43	2	no	—
152	#2905	—	43	2	no	—
153	C₂×#405	—	43	1	no	—
154	#1849	—	43	2	no	—
155	#1849	—	43	1	no	—
156	#1849	D₄₃	1	1	yes	characteristic, complemented
157	#1848	—	43	2	no	—
158	#1848	—	43	2	no	—
159	C₂×C₄₃	—	2	1	no	—
160	C₂×C₄₃	—	2	1	no	—
161	C₂×C₄₃	C₂×#384	1	1	yes	characteristic
162	C₂×C₄₃	#2916	1	2	yes	—
163	D₄₃	—	2	4	no	—
164	C₂²×M₁₆	—	43	1	no	—
165	C₂²×M₁₆	—	43	4	no	—
166	C₂³×C₈	—	43	1	no	—
167	C₂×#117	—	43	1	no	—
168	#430	—	86	8	no	—
169	C₂×#108	—	43	4	no	—
170	C₂×#108	—	43	4	no	—
171	C₂×#108	—	43	2	no	—
172	C₂×#108	—	43	2	no	—
173	C₂×#108	—	86	8	no	—
174	C₂×#108	C₂×D₄₃	1	2	yes	—
175	C₂×#108	C₂×D₄₃	1	2	yes	—
176	#424	—	86	8	no	—
177	#423	—	43	4	no	—
178	#423	—	43	2	no	—
179	#423	C₂×D₄₃	1	2	yes	—
180	#405	—	43	8	no	—
181	C₂×#97	—	43	1	no	—
182	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
183	C₂×C₄×C₈	—	43	1	no	—
184	C₂×C₄×C₈	—	43	1	no	—
185	C₄₃	#20807	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₄	—	43	1	no	—
187	C₂×M₁₆	—	43	4	no	—
188	C₂×M₁₆	—	43	4	no	—
189	C₂×M₁₆	—	43	4	no	—
190	C₂×M₁₆	—	43	4	no	—
191	C₂×M₁₆	—	86	4	no	—
192	C₂×M₁₆	—	86	8	no	—
193	C₂×M₁₆	—	86	8	no	—
194	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
195	C₂²×C₈	—	43	1	no	—
196	C₂²×C₈	—	43	1	no	—
197	C₂²×C₈	—	43	4	no	—
198	C₂²×C₈	—	43	4	no	—
199	C₂²×C₈	—	43	4	no	—
200	C₂²×C₈	—	86	4	no	—
201	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
203	#117	—	43	4	no	—
204	#117	—	43	4	no	—
205	C₂×C₄⋊C₄	—	43	2	no	—
206	C₂×C₂²⋊C₄	—	43	2	no	—
207	C₂×C₄²	—	43	1	no	—
208	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
209	#108	—	2	4	no	—
210	#108	—	2	4	no	—
211	#108	—	86	8	no	—
212	#108	—	86	8	no	—
213	#108	—	86	4	no	—
214	#108	—	86	4	no	—
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	—	86	8	no	—
217	C₄⋊C₈	—	86	4	no	—
218	#97	—	43	4	no	—
219	#97	C₄×D₄₃	1	4	yes	—
220	C₄×C₈	—	43	4	no	—
221	C₄×C₈	—	43	4	no	—
222	C₂⁴	—	43	1	no	—
223	C₂²×C₄	—	43	1	no	—
224	C₂²×C₄	—	43	2	no	—
225	C₂²×C₄	—	43	2	no	—
226	C₂²×C₄	—	43	2	no	—
227	C₂²×C₄	—	43	2	no	—
228	C₂²×C₄	—	43	2	no	—
229	C₂²×C₄	—	43	2	no	—
230	C₂²×C₄	—	43	2	no	—
231	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
232	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	43	16	no	—
236	M₁₆	—	86	4	no	—
237	M₁₆	—	86	4	no	—
238	M₁₆	—	172	8	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	43	2	no	—
241	C₂×C₈	—	43	2	no	—
242	C₂×C₈	—	43	2	no	—
243	C₂×C₈	—	43	2	no	—
244	C₂×C₈	—	43	2	no	—
245	C₂×C₈	—	43	2	no	—
246	C₂×C₈	—	43	16	no	—
247	C₂×C₈	—	86	4	no	—
248	C₂×C₈	—	172	8	no	—
249	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
251	C₂×C₈	D₄×D₄₃	1	2	yes	—
252	C₂×C₈	D₄×D₄₃	1	2	yes	—
253	C₂×C₈	Q₈×D₄₃	1	2	yes	—
254	C₂×C₈	Q₈×D₄₃	1	2	yes	—
255	C₄⋊C₄	—	43	8	no	—
256	C₂²⋊C₄	—	43	8	no	—
257	C₄²	—	43	2	no	—
258	C₄²	—	43	2	no	—
259	C₄²	C₂×C₄×D₄₃	1	2	yes	—
260	C₄²	D₄×D₄₃	1	2	yes	—
261	C₂³	—	43	2	no	—
262	C₂³	—	43	2	no	—
263	C₂³	—	43	2	no	—
264	C₂³	—	172	2	no	—
265	C₂³	C₂×Q₈×D₄₃	1	1	yes	characteristic
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	43	2	no	—
268	C₂×C₄	—	43	2	no	—
269	C₂×C₄	—	43	2	no	—
270	C₂×C₄	—	43	4	no	—
271	C₂×C₄	—	43	4	no	—
272	C₂×C₄	—	43	4	no	—
273	C₂×C₄	—	43	4	no	—
274	C₂×C₄	—	43	4	no	—
275	C₂×C₄	—	172	2	no	—
276	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
277	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
281	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
282	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
283	C₈	—	4	4	no	—
284	C₈	—	43	4	no	—
285	C₈	—	43	4	no	—
286	C₈	—	172	4	no	—
287	C₈	C₄⋊C₄×D₄₃	1	4	yes	—
288	C₈	C₄⋊C₄×D₄₃	1	4	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	43	4	no	—
291	C₂²	—	86	4	no	—
292	C₂²	—	172	2	no	—
293	C₂²	—	172	2	no	—
294	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
295	C₂²	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
296	C₂²	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
297	C₄	—	2	4	no	—
298	C₄	—	43	4	no	—
299	C₄	—	86	4	no	—
300	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
301	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
302	C₄	#127×D₄₃	1	1	yes	characteristic
303	C₄	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	86	4	no	—
307	C₂	#384×D₄₃	1	1	yes	characteristic
308	C₂	#19143205	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1849	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1849	D₄₃
hypocenter	C₄₃	#20807
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

Complements

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

N	H ≅ G/N	type
5. #19134770	C₂	semidirect
6. #19134768	C₂	semidirect
7. #19134766	C₂	semidirect
8. #19134765	C₂	semidirect
9. C₄₃×#1849	C₂	semidirect
156. #1849	D₄₃	semidirect
185. C₄₃	#20807	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582946	2
C₂×#108×D₄₃	2
#19143157	2
C₂×#14582856	2
#19134770	2
#19134768	2
#19134766	2
#19134765	2
C₄₃×#1849	2
#20807	43