Subgroups — #25105122

Order 11008.

1238 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583011	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×Dic₄₃	C₂	1	2	yes	maximal, complemented
3	#19137321	C₂	1	2	yes	maximal, complemented
4	#19135472	C₂	1	2	yes	maximal, complemented
5	#19135462	C₂	1	2	yes	maximal, complemented
6	#19135420	C₂	1	2	yes	maximal, complemented
7	#19134759	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134281	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic
10	C₂×D₄×Dic₄₃	C₂²	1	4	yes	—
11	#14583011	—	2	8	no	—
12	D₈×Dic₄₃	—	2	8	no	—
13	C₂×#13063259	C₂²	1	2	yes	—
14	C₂×#13063257	C₂²	1	2	yes	—
15	C₂×#13063246	C₂²	1	4	yes	—
16	C₂×#13063246	C₂²	1	4	yes	—
17	C₂×#13063236	C₂²	1	1	yes	characteristic
18	C₂×#13063235	C₂²	1	1	yes	characteristic
19	C₂×#13063235	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
22	#14582802	C₂²	1	4	yes	—
23	#14582779	—	2	8	no	—
24	#14582765	C₂²	1	2	yes	—
25	#14582745	C₂²	1	1	yes	characteristic
26	C₂²×D₈×C₄₃	C₂²	1	2	yes	—
27	C₂²×D₈×C₄₃	C₂²	1	2	yes	—
28	C₂×#127×C₄₃	C₂²	1	2	yes	—
29	C₄₃×#493	C₄	1	8	yes	complemented
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₂³×Dic₄₃	—	2	4	no	—
32	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
34	D₄×Dic₄₃	—	2	8	no	—
35	D₄×Dic₄₃	—	2	8	no	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
37	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
38	C₂×C₂²⋊Dic₄₃	—	2	8	no	—
39	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
41	#13063259	—	2	4	no	—
42	#13063257	—	2	4	no	—
43	#13063246	—	2	8	no	—
44	#13063246	—	2	8	no	—
45	#13063236	—	2	2	no	—
46	#13063235	—	2	2	no	—
47	#13063235	D₄	1	4	yes	complemented
48	C₈×Dic₄₃	—	2	2	no	—
49	C₈×Dic₄₃	D₄	1	4	yes	complemented
50	C₄₃⋊#95	—	2	4	no	—
51	C₄₃⋊#95	D₄	1	4	yes	—
52	C₂²×D₄×C₄₃	—	2	4	no	—
53	C₂²×D₄×C₄₃	C₂³	1	4	yes	—
54	C₂×D₈×C₄₃	—	2	4	no	—
55	C₂×D₈×C₄₃	—	2	4	no	—
56	C₂×D₈×C₄₃	—	2	4	no	—
57	C₂×D₈×C₄₃	—	2	4	no	—
58	C₂×D₈×C₄₃	C₂×C₄	1	8	yes	complemented
59	C₂×D₈×C₄₃	C₂×C₄	1	8	yes	complemented
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	#127×C₄₃	—	2	4	no	—
63	#127×C₄₃	C₂×C₄	1	8	yes	complemented
64	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
66	C₂⁴×C₄₃	—	2	4	no	—
67	C₂×D₄×C₄₃	—	2	16	no	—
68	C₂×D₄×C₄₃	—	2	4	no	—
69	C₂×D₄×C₄₃	—	2	4	no	—
70	C₂×D₄×C₄₃	—	2	4	no	—
71	C₂×D₄×C₄₃	—	2	4	no	—
72	C₂×D₄×C₄₃	—	2	4	no	—
73	C₂×D₄×C₄₃	—	8	8	no	—
74	C₂×D₄×C₄₃	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₄₃	—	2	4	no	—
78	C₂²×Dic₄₃	—	4	8	no	—
79	C₂²×Dic₄₃	—	8	8	no	—
80	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
81	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
82	D₈×C₄₃	—	2	16	no	—
83	D₈×C₄₃	—	2	16	no	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₃	#41	1	2	yes	—
89	C₂×C₈×C₄₃	#41	1	2	yes	—
90	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₃	#41	1	2	yes	—
92	C₂²⋊Dic₄₃	—	4	8	no	—
93	C₂²⋊Dic₄₃	—	8	8	no	—
94	C₄⋊Dic₄₃	—	2	4	no	—
95	C₄⋊Dic₄₃	—	2	4	no	—
96	C₄×Dic₄₃	—	2	1	no	—
97	C₄×Dic₄₃	—	2	1	no	—
98	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
99	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
100	C₂×C₄₃⋊C₈	—	2	1	no	—
101	C₂×C₄₃⋊C₈	—	2	1	no	—
102	C₂×C₄₃⋊C₈	—	2	1	no	—
103	C₂×C₄₃⋊C₈	—	2	1	no	—
104	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
105	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
106	C₂³×C₄₃	—	2	8	no	—
107	C₂³×C₄₃	—	4	4	no	—
108	C₂³×C₄₃	—	4	4	no	—
109	C₂³×C₄₃	—	4	8	no	—
110	C₂³×C₄₃	—	4	8	no	—
111	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
112	D₄×C₄₃	—	2	8	no	—
113	D₄×C₄₃	—	2	8	no	—
114	D₄×C₄₃	—	4	4	no	—
115	D₄×C₄₃	—	4	4	no	—
116	D₄×C₄₃	—	8	16	no	—
117	C₂×C₄×C₄₃	—	2	4	no	—
118	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
119	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₄₃	—	2	4	no	—
126	C₂×Dic₄₃	—	2	2	no	—
127	C₂×Dic₄₃	—	2	4	no	—
128	C₂×Dic₄₃	—	4	4	no	—
129	C₂×Dic₄₃	—	4	8	no	—
130	C₂×Dic₄₃	—	8	16	no	—
131	C₂×Dic₄₃	#127	1	4	yes	—
132	C₈×C₄₃	C₄×D₄	1	4	yes	—
133	C₈×C₄₃	C₄×D₄	1	4	yes	—
134	C₄₃⋊C₈	—	2	2	no	—
135	C₄₃⋊C₈	—	2	2	no	—
136	#21697	—	43	1	no	maximal, Sylow
137	C₂²×C₄₃	—	4	8	no	—
138	C₂²×C₄₃	—	4	16	no	—
139	C₂²×C₄₃	—	8	4	no	—
140	C₂²×C₄₃	—	8	4	no	—
141	C₂²×C₄₃	—	8	8	no	—
142	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
144	C₂²×C₄₃	#532	1	1	yes	characteristic
145	C₂²×C₄₃	#576	1	4	yes	—
146	C₄×C₄₃	—	2	4	no	—
147	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₄×C₄₃	#532	1	1	yes	characteristic
150	C₄×C₄₃	#532	1	1	yes	characteristic
151	Dic₄₃	—	2	4	no	—
152	Dic₄₃	—	4	2	no	—
153	Dic₄₃	—	8	4	no	—
154	C₂×#493	D₄₃	1	1	yes	characteristic
155	C₂×#442	—	43	2	no	—
156	C₂×C₄×D₈	—	43	2	no	—
157	#2328	—	43	2	no	—
158	#1972	—	43	2	no	—
159	#1970	—	43	2	no	—
160	#1961	—	43	2	no	—
161	#1846	—	43	1	no	—
162	#1726	—	43	1	no	—
163	C₂×C₄₃	—	8	8	no	—
164	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
165	C₂×C₄₃	#2948	1	2	yes	—
166	C₂×C₄₃	#3155	1	2	yes	—
167	C₂×C₄₃	#3176	1	2	yes	—
168	C₂²×D₈	C₂×D₄₃	1	2	yes	—
169	C₂²×D₈	C₂×D₄₃	1	2	yes	—
170	C₂×#127	C₂×D₄₃	1	2	yes	—
171	C₂×C₄×D₄	—	43	4	no	—
172	#493	Dic₄₃	1	8	yes	complemented
173	#442	—	86	8	no	—
174	C₄×D₈	—	86	8	no	—
175	C₂×#107	—	43	2	no	—
176	C₂×#106	—	43	2	no	—
177	C₂×#102	—	43	4	no	—
178	C₂×#102	—	43	4	no	—
179	C₂×#97	—	43	1	no	—
180	C₂×#97	—	43	1	no	—
181	C₂×#97	—	43	1	no	—
182	C₂×C₄×C₈	—	43	1	no	—
183	C₂×C₄×C₈	—	43	1	no	—
184	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
185	#390	—	43	4	no	—
186	#386	—	86	8	no	—
187	#382	—	43	2	no	—
188	#376	—	43	1	no	—
189	C₄₃	#21697	1	1	yes	characteristic, complemented, Sylow
190	C₂²×D₄	—	2	4	no	—
191	C₂²×D₄	C₂²×D₄₃	1	4	yes	—
192	C₂³×C₄	—	86	4	no	—
193	C₂×D₈	—	2	4	no	—
194	C₂×D₈	—	2	4	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	2	4	no	—
197	C₂×D₈	C₂×Dic₄₃	1	8	yes	complemented
198	C₂×D₈	C₂×Dic₄₃	1	8	yes	complemented
199	C₂²×C₈	—	43	1	no	—
200	C₂²×C₈	—	43	1	no	—
201	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
203	#127	—	2	4	no	—
204	#127	C₂×Dic₄₃	1	8	yes	complemented
205	C₄×D₄	—	86	8	no	—
206	C₄×D₄	—	86	8	no	—
207	C₂×C₄⋊C₄	—	43	4	no	—
208	C₂×C₂²⋊C₄	—	86	8	no	—
209	C₂×C₂²⋊C₄	—	86	4	no	—
210	C₂×C₄²	—	43	1	no	—
211	C₂×C₄²	—	43	1	no	—
212	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
213	#107	—	86	4	no	—
214	#106	—	86	4	no	—
215	#102	—	86	8	no	—
216	#102	—	86	8	no	—
217	#97	—	43	4	no	—
218	#97	—	86	2	no	—
219	#97	—	86	2	no	—
220	C₄×C₈	—	43	4	no	—
221	C₄×C₈	—	86	2	no	—
222	C₄×C₈	C₂×Dic₄₃	1	4	yes	complemented
223	#95	—	43	4	no	—
224	#95	—	86	4	no	—
225	C₂⁴	—	2	4	no	—
226	C₂×D₄	—	2	16	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	8	8	no	—
233	C₂×D₄	C₂²×Dic₄₃	1	8	yes	—
234	C₂²×C₄	—	43	2	no	—
235	C₂²×C₄	—	43	2	no	—
236	C₂²×C₄	—	86	4	no	—
237	C₂²×C₄	—	344	8	no	—
238	C₂²×C₄	—	172	8	no	—
239	C₂²×C₄	D₄×D₄₃	1	2	yes	—
240	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
241	D₈	—	2	16	no	—
242	D₈	—	2	16	no	—
243	C₂×C₈	—	43	2	no	—
244	C₂×C₈	—	43	2	no	—
245	C₂×C₈	—	86	1	no	—
246	C₂×C₈	—	86	1	no	—
247	C₂×C₈	—	86	1	no	—
248	C₂×C₈	—	86	1	no	—
249	C₂×C₈	D₄×D₄₃	1	2	yes	—
250	C₂×C₈	D₄×D₄₃	1	2	yes	—
251	C₂×C₈	#10621900	1	2	yes	—
252	C₂×C₈	#10621900	1	2	yes	—
253	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
254	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
255	C₄⋊C₄	—	86	4	no	—
256	C₄⋊C₄	—	86	4	no	—
257	C₂²⋊C₄	—	344	8	no	—
258	C₂²⋊C₄	—	172	8	no	—
259	C₄²	—	43	2	no	—
260	C₄²	—	43	2	no	—
261	C₄²	—	86	1	no	—
262	C₄²	—	86	1	no	—
263	C₄²	#10621900	1	2	yes	—
264	C₄²	C₂²×Dic₄₃	1	2	yes	—
265	C₂³	—	2	8	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	4	8	no	—
268	C₂³	—	4	8	no	—
269	C₂³	—	4	4	no	—
270	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
271	D₄	—	2	8	no	—
272	D₄	—	2	8	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	8	16	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	43	4	no	—
278	C₂×C₄	—	86	4	no	—
279	C₂×C₄	—	86	4	no	—
280	C₂×C₄	—	86	2	no	—
281	C₂×C₄	—	344	16	no	—
282	C₂×C₄	—	172	4	no	—
283	C₂×C₄	—	172	8	no	—
284	C₂×C₄	D₄×Dic₄₃	1	4	yes	—
285	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
286	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
287	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
289	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
290	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
291	C₈	—	86	2	no	—
292	C₈	—	86	2	no	—
293	C₈	D₄×Dic₄₃	1	4	yes	—
294	C₈	D₄×Dic₄₃	1	4	yes	—
295	C₂²	—	4	16	no	—
296	C₂²	—	4	8	no	—
297	C₂²	—	8	8	no	—
298	C₂²	—	8	4	no	—
299	C₂²	—	8	4	no	—
300	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
301	C₂²	#127×D₄₃	1	1	yes	characteristic
302	C₂²	#14583447	1	1	yes	characteristic
303	C₂²	#14583700	1	4	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	86	4	no	—
306	C₄	—	344	4	no	—
307	C₄	—	172	2	no	—
308	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
309	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
310	C₄	#14583447	1	1	yes	characteristic
311	C₄	#14583447	1	1	yes	characteristic
312	C₂	—	8	8	no	—
313	C₂	#127×Dic₄₃	1	1	yes	characteristic
314	C₂	#19143447	1	2	yes	—
315	C₂	#19145075	1	2	yes	—
316	C₂	#19145242	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting, prime radical	C₂×C₄₃×#493	C₂
socle	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#493	D₄₃
hypocenter	C₄₃	#21697
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#14583011	C₂	semidirect
2. C₂×D₈×Dic₄₃	C₂	semidirect
3. #19137321	C₂	semidirect
4. #19135472	C₂	semidirect
5. #19135462	C₂	semidirect
6. #19135420	C₂	semidirect
7. #19134759	C₂	semidirect
8. #19134281	C₂	semidirect
29. C₄₃×#493	C₄	semidirect
47. #13063235	D₄	semidirect
49. C₈×Dic₄₃	D₄	semidirect
58. C₂×D₈×C₄₃	C₂×C₄	semidirect
59. C₂×D₈×C₄₃	C₂×C₄	semidirect
63. #127×C₄₃	C₂×C₄	semidirect
65. C₄×C₈×C₄₃	C₂×C₄	semidirect
172. #493	Dic₄₃	semidirect
189. C₄₃	#21697	semidirect
197. C₂×D₈	C₂×Dic₄₃	semidirect
198. C₂×D₈	C₂×Dic₄₃	semidirect
204. #127	C₂×Dic₄₃	semidirect
222. 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
C₂×#14583011	2
C₂×D₈×Dic₄₃	2
#19137321	2
#19135472	2
#19135462	2
#19135420	2
#19134759	2
#19134281	2
C₂×C₄₃×#493	2
#21697	43