Subgroups — #25096806

Order 11008.

894 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137218	C₂	1	2	yes	maximal
2	#19134729	C₂	1	4	yes	maximal, complemented
3	#19134726	C₂	1	2	yes	maximal, complemented
4	#19134723	C₂	1	2	yes	maximal, complemented
5	#19134315	C₂	1	2	yes	maximal
6	#19134312	C₂	1	1	yes	maximal, characteristic
7	C₂×#14582662	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₃×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	#14582798	—	2	2	no	—
12	#14582798	—	2	2	no	—
13	#14582774	—	2	4	no	—
14	#14582769	C₂²	1	4	yes	—
15	#14582765	C₂²	1	4	yes	—
16	#14582662	C₂²	1	1	yes	characteristic
17	#14582662	C₂²	1	2	yes	—
18	#14582662	C₂²	1	4	yes	—
19	#14582662	C₂²	1	1	yes	characteristic
20	#14582662	C₂²	1	4	yes	—
21	#14582662	C₂²	1	2	yes	—
22	#14582662	C₂²	1	2	yes	—
23	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#405	C₄	1	8	yes	complemented
27	C₂×#97×C₄₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
29	C₂³×Dic₄₃	—	2	2	no	—
30	#13063284	—	2	4	no	—
31	#13063284	—	2	4	no	—
32	C₂×C₄⋊Dic₄₃	—	2	4	no	—
33	C₂×C₄⋊Dic₄₃	—	2	4	no	—
34	C₂×C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄⋊Dic₄₃	—	2	4	no	—
36	C₂×C₄⋊Dic₄₃	—	2	4	no	—
37	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
40	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
41	C₂×C₄×Dic₄₃	—	2	2	no	—
42	C₄₃⋊#95	—	2	4	no	—
43	C₄₃⋊#95	—	2	4	no	—
44	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₄₃	—	2	4	no	—
46	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₄₃	—	2	4	no	—
48	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
52	#117×C₄₃	C₂×C₄	1	4	yes	—
53	#117×C₄₃	Q₈	1	4	yes	—
54	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
58	#97×C₄₃	C₂×C₄	1	8	yes	complemented
59	C₄×C₈×C₄₃	C₂×C₄	1	8	yes	complemented
60	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₃	#41	1	4	yes	—
67	C₂²×C₄×C₄₃	#41	1	4	yes	—
68	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	2	2	no	—
70	C₂²×Dic₄₃	—	2	4	no	—
71	C₂²×Dic₄₃	—	2	2	no	—
72	C₂²×Dic₄₃	—	4	4	no	—
73	C₂²×Dic₄₃	—	8	2	no	—
74	C₂²×Dic₄₃	—	8	2	no	—
75	M₁₆×C₄₃	—	2	8	no	—
76	C₂×C₈×C₄₃	—	2	8	no	—
77	C₂×C₈×C₄₃	—	2	2	no	—
78	C₂×C₈×C₄₃	—	2	2	no	—
79	C₂×C₈×C₄₃	—	2	4	no	—
80	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₄₃	C₄⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₄₃	C₄⋊C₄	1	4	yes	—
85	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	8	yes	—
86	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
87	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
88	C₂²⋊Dic₄₃	—	4	4	no	—
89	C₄⋊Dic₄₃	—	2	4	no	—
90	C₄⋊Dic₄₃	—	2	4	no	—
91	C₄⋊Dic₄₃	—	2	2	no	—
92	C₄⋊Dic₄₃	—	2	2	no	—
93	C₄⋊Dic₄₃	—	2	2	no	—
94	C₄⋊Dic₄₃	—	2	2	no	—
95	C₄⋊Dic₄₃	—	4	4	no	—
96	C₄⋊Dic₄₃	—	8	4	no	—
97	C₄⋊Dic₄₃	—	8	4	no	—
98	C₄×Dic₄₃	—	4	4	no	—
99	C₂³×C₄₃	—	2	1	no	—
100	C₂³×C₄₃	—	2	2	no	—
101	C₂³×C₄₃	—	2	1	no	—
102	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₄₃	#124	1	4	yes	—
104	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₃	#117	1	4	yes	—
112	C₂×C₄×C₄₃	#117	1	4	yes	—
113	C₂×C₄×C₄₃	#124	1	4	yes	—
114	C₂×C₄×C₄₃	#125	1	8	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	2	yes	—
118	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
119	C₂×Dic₄₃	—	4	2	no	—
120	C₂×Dic₄₃	—	4	2	no	—
121	C₂×Dic₄₃	—	4	4	no	—
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	—	4	4	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	C₂×Dic₄₃	—	4	4	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	4	2	no	—
128	C₂×Dic₄₃	—	8	8	no	—
129	C₈×C₄₃	—	2	2	no	—
130	C₈×C₄₃	—	2	2	no	—
131	C₈×C₄₃	—	2	4	no	—
132	#20763	—	43	1	no	maximal, Sylow
133	C₂²×C₄₃	—	2	4	no	—
134	C₂²×C₄₃	—	2	4	no	—
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	#382	1	4	yes	—
138	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₃	C₂×#124	1	2	yes	—
140	C₂²×C₄₃	#576	1	1	yes	characteristic
141	C₂²×C₄₃	#577	1	2	yes	—
142	C₂²×C₄₃	#578	1	1	yes	characteristic
143	C₄×C₄₃	—	2	4	no	—
144	C₄×C₄₃	—	2	4	no	—
145	C₄×C₄₃	#382	1	4	yes	—
146	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
147	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
148	C₄×C₄₃	C₂×#125	1	2	yes	—
149	Dic₄₃	—	8	2	no	—
150	Dic₄₃	—	8	4	no	—
151	Dic₄₃	—	8	2	no	—
152	C₂×#405	D₄₃	1	1	yes	characteristic
153	#2317	—	43	2	no	—
154	#1836	—	43	4	no	—
155	#1835	—	43	2	no	—
156	#1834	—	43	2	no	—
157	#1736	—	43	2	no	—
158	#1735	—	43	1	no	—
159	C₂×#340	—	43	1	no	—
160	C₂×C₄₃	—	2	4	no	—
161	C₂×C₄₃	C₂×#382	1	1	yes	characteristic
162	C₂×C₄₃	#2912	1	1	yes	characteristic
163	C₂×C₄₃	#2913	1	1	yes	characteristic
164	C₂×C₄₃	#3136	1	2	yes	—
165	C₂×C₄₃	#3137	1	2	yes	—
166	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
167	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
168	C₂×#117	—	43	2	no	—
169	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
170	C₂²×C₄⋊C₄	—	43	2	no	—
171	#405	Dic₄₃	1	8	yes	complemented
172	C₂×#97	C₂×D₄₃	1	2	yes	—
173	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
174	#389	—	86	2	no	—
175	#389	—	86	2	no	—
176	#385	—	86	4	no	—
177	#384	—	43	4	no	—
178	#382	—	43	4	no	—
179	#340	—	43	1	no	—
180	#340	—	43	2	no	—
181	#340	—	43	4	no	—
182	#340	—	43	1	no	—
183	#340	—	43	4	no	—
184	#340	—	43	2	no	—
185	#340	—	43	2	no	—
186	C₄₃	#20763	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₄	—	86	2	no	—
188	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	C₂×Dic₄₃	1	4	yes	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
193	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
196	#117	—	86	4	no	—
197	#117	—	86	4	no	—
198	#117	Dic₈₆	1	4	yes	—
199	#117	C₂×Dic₄₃	1	4	yes	—
200	C₂×C₄⋊C₄	—	43	4	no	—
201	C₂×C₄⋊C₄	—	43	2	no	—
202	C₂×C₄⋊C₄	—	43	2	no	—
203	C₂×C₄⋊C₄	—	86	4	no	—
204	C₂×C₄⋊C₄	—	86	4	no	—
205	C₂×C₄⋊C₄	—	86	4	no	—
206	C₂×C₄⋊C₄	—	86	4	no	—
207	C₂×C₄⋊C₄	—	86	2	no	—
208	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
209	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	86	2	no	—
211	C₂×C₂²⋊C₄	D₁₇₂	1	2	yes	—
212	C₂×C₄²	—	86	2	no	—
213	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
214	#97	C₂×Dic₄₃	1	8	yes	complemented
215	C₄×C₈	C₂×Dic₄₃	1	8	yes	complemented
216	#95	—	86	4	no	—
217	#95	—	86	4	no	—
218	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
219	C₂²×C₄	—	86	2	no	—
220	C₂²×C₄	—	86	4	no	—
221	C₂²×C₄	—	86	2	no	—
222	C₂²×C₄	—	344	2	no	—
223	C₂²×C₄	—	344	2	no	—
224	C₂²×C₄	—	172	4	no	—
225	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
226	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
227	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
228	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
229	C₂²×C₄	#10621898	1	4	yes	—
230	C₂²×C₄	#10621898	1	4	yes	—
231	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
232	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
233	M₁₆	—	2	8	no	—
234	C₂×C₈	—	2	8	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
239	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
240	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
241	C₄⋊C₄	—	86	4	no	—
242	C₄⋊C₄	—	86	4	no	—
243	C₄⋊C₄	—	86	2	no	—
244	C₄⋊C₄	—	86	2	no	—
245	C₄⋊C₄	—	86	2	no	—
246	C₄⋊C₄	—	86	2	no	—
247	C₄⋊C₄	—	344	4	no	—
248	C₄⋊C₄	—	344	4	no	—
249	C₄⋊C₄	—	172	4	no	—
250	C₄⋊C₄	C₄⋊Dic₄₃	1	4	yes	—
251	C₄⋊C₄	C₄⋊Dic₄₃	1	4	yes	—
252	C₂²⋊C₄	—	172	4	no	—
253	C₂²⋊C₄	C₄⋊Dic₄₃	1	8	yes	—
254	C₄²	—	172	4	no	—
255	C₄²	C₂×Dic₈₆	1	4	yes	—
256	C₄²	C₂²×Dic₄₃	1	4	yes	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	1	no	—
260	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
261	C₂³	#13063313	1	4	yes	—
262	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	344	8	no	—
267	C₂×C₄	—	172	2	no	—
268	C₂×C₄	—	172	2	no	—
269	C₂×C₄	—	172	4	no	—
270	C₂×C₄	—	172	2	no	—
271	C₂×C₄	—	172	4	no	—
272	C₂×C₄	—	172	2	no	—
273	C₂×C₄	—	172	4	no	—
274	C₂×C₄	—	172	2	no	—
275	C₂×C₄	—	172	2	no	—
276	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
277	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
278	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
279	C₂×C₄	#13063284	1	4	yes	—
280	C₂×C₄	#13063284	1	4	yes	—
281	C₂×C₄	#13063313	1	4	yes	—
282	C₂×C₄	#13063317	1	8	yes	—
283	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
285	C₂×C₄	C₂×#10621898	1	2	yes	—
286	C₂×C₄	C₂×#10621898	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	#14582765	1	4	yes	—
295	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
296	C₂²	C₂×#13063313	1	2	yes	—
297	C₂²	#14583704	1	1	yes	characteristic
298	C₂²	#14583711	1	2	yes	—
299	C₂²	#14583716	1	1	yes	characteristic
300	C₄	—	2	4	no	—
301	C₄	—	2	4	no	—
302	C₄	—	344	2	no	—
303	C₄	—	344	4	no	—
304	C₄	—	344	2	no	—
305	C₄	#14582765	1	4	yes	—
306	C₄	C₂×#13063284	1	1	yes	characteristic
307	C₄	C₂×#13063284	1	1	yes	characteristic
308	C₄	C₂×#13063317	1	2	yes	—
309	C₂	—	2	4	no	—
310	C₂	C₂×#14582765	1	1	yes	characteristic
311	C₂	#19143190	1	1	yes	characteristic
312	C₂	#19143195	1	1	yes	characteristic
313	C₂	#19144951	1	2	yes	—
314	C₂	#19144962	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#405	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#405	D₄₃
hypocenter	C₄₃	#20763
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #19134729	C₂	semidirect
3. #19134726	C₂	semidirect
4. #19134723	C₂	semidirect
26. C₄₃×#405	C₄	semidirect
58. #97×C₄₃	C₂×C₄	semidirect
59. C₄×C₈×C₄₃	C₂×C₄	semidirect
171. #405	Dic₄₃	semidirect
186. C₄₃	#20763	semidirect
214. #97	C₂×Dic₄₃	semidirect
215. C₄×C₈	C₂×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19137218	2
#19134729	2
#19134726	2
#19134723	2
#19134315	2
#19134312	2
C₂×#14582662	2
C₂×C₄₃×#405	2
#20763	43