Subgroups — #25002315

Order 11008.

734 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582664	C₂	1	1	yes	maximal, characteristic
2	#19134312	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582659	C₂	1	2	yes	maximal
4	#19134291	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂×#13063284	C₂²	1	2	yes	—
8	#14582664	C₄	1	4	yes	complemented
9	#14582662	C₄	1	4	yes	complemented
10	#14582659	—	2	4	no	—
11	#14582653	—	2	4	no	—
12	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₃	—	2	4	no	—
16	C₂×C₄×C₈×C₄₃	C₄	1	4	yes	complemented
17	C₂×#10621877	—	2	2	no	—
18	C₂×#10621877	—	2	2	no	—
19	C₂×#10621877	—	2	2	no	—
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	C₂×C₄	1	4	yes	—
23	C₂²×C₄₃⋊C₈	—	2	2	no	—
24	#13063284	—	2	2	no	—
25	#13063284	—	2	2	no	—
26	#13063284	—	2	2	no	—
27	#13063284	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
30	C₂×C₄×Dic₄₃	—	2	2	no	—
31	C₂³×C₄×C₄₃	—	2	1	no	—
32	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₄₃	—	2	2	no	—
34	C₂²×C₈×C₄₃	—	2	2	no	—
35	C₂²×C₈×C₄₃	—	2	2	no	—
36	C₂²×C₈×C₄₃	—	2	2	no	—
37	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₄₃	D₄	1	2	yes	—
42	C₂²×C₈×C₄₃	Q₈	1	2	yes	—
43	C₂×C₄²×C₄₃	—	2	2	no	—
44	C₂×C₄²×C₄₃	—	2	1	no	—
45	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₃	D₄	1	4	yes	—
47	C₄×C₈×C₄₃	—	2	8	no	—
48	C₂⁴×C₄₃	#41	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	—	2	8	no	—
50	C₂²×C₄×C₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	—	2	4	no	—
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	#41	1	2	yes	—
60	C₂²×C₄×C₄₃	#41	1	2	yes	—
61	C₂²×C₄×C₄₃	#41	1	2	yes	—
62	C₂²×Dic₄₃	—	2	4	no	—
63	C₂×C₈×C₄₃	—	2	2	no	—
64	C₂×C₈×C₄₃	—	2	2	no	—
65	C₂×C₈×C₄₃	—	2	2	no	—
66	C₂×C₈×C₄₃	—	2	2	no	—
67	C₂×C₈×C₄₃	—	2	4	no	—
68	C₂×C₈×C₄₃	—	2	4	no	—
69	C₂×C₈×C₄₃	—	2	4	no	—
70	C₂×C₈×C₄₃	—	2	4	no	—
71	C₂×C₈×C₄₃	C₄²	1	4	yes	—
72	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
73	C₄²×C₄₃	—	2	2	no	—
74	C₄²×C₄₃	—	2	4	no	—
75	C₄²×C₄₃	C₂²⋊C₄	1	4	yes	—
76	C₂²⋊Dic₄₃	—	4	2	no	—
77	C₂²⋊Dic₄₃	—	4	2	no	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	C₄⋊Dic₄₃	—	2	4	no	—
80	C₄×Dic₄₃	—	4	2	no	—
81	C₄×Dic₄₃	—	4	2	no	—
82	#10621877	—	2	4	no	—
83	#10621877	—	2	4	no	—
84	#10621877	—	4	2	no	—
85	#10621877	—	4	2	no	—
86	C₂×C₄₃⋊C₈	—	2	4	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂×C₄₃⋊C₈	—	4	2	no	—
89	C₂³×C₄₃	—	2	4	no	—
90	C₂³×C₄₃	#117	1	1	yes	characteristic
91	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
92	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
93	C₂³×C₄₃	#123	1	2	yes	—
94	C₂³×C₄₃	#126	1	2	yes	—
95	C₂×C₄×C₄₃	—	2	16	no	—
96	C₂×C₄×C₄₃	—	2	8	no	—
97	C₂×C₄×C₄₃	—	2	4	no	—
98	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#117	1	2	yes	—
102	C₂×C₄×C₄₃	#117	1	2	yes	—
103	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#121	1	2	yes	—
111	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₃	#123	1	2	yes	—
113	C₂×C₄×C₄₃	#124	1	2	yes	—
114	C₂×C₄×C₄₃	#125	1	2	yes	—
115	C₂×C₄×C₄₃	#126	1	2	yes	—
116	C₂×Dic₄₃	—	4	4	no	—
117	C₂×Dic₄₃	—	4	8	no	—
118	C₈×C₄₃	—	2	4	no	—
119	C₈×C₄₃	—	2	4	no	—
120	C₄₃⋊C₈	—	4	4	no	—
121	#10818	—	43	1	no	maximal, Sylow
122	C₂²×C₄₃	—	2	4	no	—
123	C₂²×C₄₃	—	2	4	no	—
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₃	#381	1	1	yes	characteristic
128	C₂²×C₄₃	#383	1	1	yes	characteristic
129	C₂²×C₄₃	#387	1	2	yes	—
130	C₂²×C₄₃	#388	1	2	yes	—
131	C₂²×C₄₃	#443	1	4	yes	—
132	C₄×C₄₃	—	2	8	no	—
133	C₄×C₄₃	#376	1	1	yes	characteristic
134	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₄₃	#380	1	1	yes	characteristic
136	C₄×C₄₃	#382	1	1	yes	characteristic
137	C₄×C₄₃	#387	1	2	yes	—
138	C₄×C₄₃	#388	1	2	yes	—
139	Dic₄₃	—	8	4	no	—
140	C₂²×C₄×C₈	D₄₃	1	1	yes	characteristic
141	C₂×#341	—	43	1	no	—
142	#1735	—	43	1	no	—
143	C₂×#339	—	43	2	no	—
144	#1729	—	43	2	no	—
145	C₂×C₄₃	—	2	1	no	—
146	C₂×C₄₃	—	2	1	no	—
147	C₂×C₄₃	—	2	2	no	—
148	C₂×C₄₃	#1434	1	1	yes	characteristic
149	C₂×C₄₃	#1772	1	2	yes	—
150	C₂×C₄₃	#1937	1	4	yes	—
151	C₂²×M₁₆	—	43	2	no	—
152	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
153	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
154	C₂×#117	—	43	2	no	—
155	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
156	C₂×C₄×C₈	—	2	4	no	—
157	C₂×C₄×C₈	Dic₄₃	1	4	yes	complemented
158	#341	—	43	4	no	—
159	#340	—	43	4	no	—
160	#339	—	86	4	no	—
161	#336	—	86	4	no	—
162	C₄₃	#10818	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	—	2	1	no	—
164	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
165	C₂×M₁₆	—	43	4	no	—
166	C₂×M₁₆	—	86	2	no	—
167	C₂×M₁₆	—	86	2	no	—
168	C₂×M₁₆	—	86	2	no	—
169	C₂×M₁₆	—	86	2	no	—
170	C₂×M₁₆	—	86	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	86	2	no	—
176	C₂²×C₈	Dic₈₆	1	2	yes	—
177	C₂²×C₈	C₄×D₄₃	1	2	yes	—
178	C₂²×C₈	C₄×D₄₃	1	2	yes	—
179	C₂²×C₈	D₁₇₂	1	2	yes	—
180	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
181	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
182	#117	—	86	2	no	—
183	#117	—	86	2	no	—
184	#117	—	86	2	no	—
185	#117	—	86	2	no	—
186	C₂×C₄⋊C₄	—	43	4	no	—
187	C₂×C₂²⋊C₄	—	86	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	—	86	2	no	—
191	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₄₃	1	4	yes	—
193	C₄×C₈	—	2	8	no	—
194	C₂⁴	#10621898	1	1	yes	characteristic
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	86	4	no	—
199	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
203	C₂²×C₄	#10621898	1	2	yes	—
204	C₂²×C₄	#10621898	1	2	yes	—
205	C₂²×C₄	#10621898	1	2	yes	—
206	C₂²×C₄	#10621898	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
209	M₁₆	—	86	4	no	—
210	M₁₆	—	86	4	no	—
211	M₁₆	—	172	2	no	—
212	M₁₆	—	172	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	86	4	no	—
222	C₂×C₈	—	172	2	no	—
223	C₂×C₈	—	172	2	no	—
224	C₂×C₈	C₄×Dic₄₃	1	4	yes	—
225	C₂×C₈	C₄⋊Dic₄₃	1	4	yes	—
226	C₄⋊C₄	—	86	4	no	—
227	C₄⋊C₄	—	86	4	no	—
228	C₂²⋊C₄	—	172	2	no	—
229	C₂²⋊C₄	—	172	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	172	2	no	—
233	C₄²	—	172	2	no	—
234	C₄²	C₂²⋊Dic₄₃	1	4	yes	—
235	C₂³	—	2	4	no	—
236	C₂³	#13063284	1	1	yes	characteristic
237	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
238	C₂³	C₄×Dic₈₆	1	1	yes	characteristic
239	C₂³	#13063308	1	2	yes	—
240	C₂³	#13063319	1	2	yes	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	172	4	no	—
245	C₂×C₄	—	172	8	no	—
246	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
249	C₂×C₄	#13063282	1	2	yes	—
250	C₂×C₄	#13063282	1	2	yes	—
251	C₂×C₄	#13063284	1	1	yes	characteristic
252	C₂×C₄	#13063284	1	1	yes	characteristic
253	C₂×C₄	#13063284	1	1	yes	characteristic
254	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
255	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
256	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
257	C₂×C₄	C₄×Dic₈₆	1	1	yes	characteristic
258	C₂×C₄	#13063299	1	2	yes	—
259	C₂×C₄	#13063304	1	2	yes	—
260	C₂×C₄	#13063308	1	2	yes	—
261	C₂×C₄	#13063313	1	2	yes	—
262	C₂×C₄	#13063317	1	2	yes	—
263	C₂×C₄	#13063319	1	2	yes	—
264	C₈	—	2	4	no	—
265	C₈	—	2	4	no	—
266	C₈	—	172	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	C₄×C₄⋊Dic₄₃	1	1	yes	characteristic
272	C₂²	#14582762	1	1	yes	characteristic
273	C₂²	#14582767	1	1	yes	characteristic
274	C₂²	#14582783	1	2	yes	—
275	C₂²	#14582790	1	2	yes	—
276	C₂²	#14583017	1	4	yes	—
277	C₄	—	2	8	no	—
278	C₄	—	344	4	no	—
279	C₄	#14582745	1	1	yes	characteristic
280	C₄	C₄×C₂²⋊Dic₄₃	1	1	yes	characteristic
281	C₄	#14582759	1	1	yes	characteristic
282	C₄	#14582765	1	1	yes	characteristic
283	C₄	#14582783	1	2	yes	—
284	C₄	#14582790	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	2	no	—
288	C₂	#19132822	1	1	yes	characteristic
289	C₂	#19134474	1	2	yes	—
290	C₂	#19135299	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂²×C₄×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₄×D₄
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂²×C₄×C₈	D₄₃
hypocenter	C₄₃	#10818
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. #14582664	C₄	semidirect
9. #14582662	C₄	semidirect
16. C₂×C₄×C₈×C₄₃	C₄	semidirect
157. C₂×C₄×C₈	Dic₄₃	semidirect
162. C₄₃	#10818	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582664	2
#19134312	2
C₂×#14582659	2
#19134291	2
C₂²×C₄×C₈×C₄₃	2
#10818	43