Subgroups — #25249179

Order 11008.

910 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583304	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583302	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583295	C₂	1	1	yes	maximal, characteristic
4	#19142991	C₂	1	1	yes	maximal, characteristic
5	#19142972	C₂	1	1	yes	maximal, characteristic
6	#19134760	C₂	1	8	yes	maximal
7	C₄×#13063235	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₃×#498	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063325	C₂²	1	1	yes	characteristic
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₂×#13063323	—	2	2	no	—
12	#14583304	—	2	4	no	—
13	#14583302	—	2	4	no	—
14	#14583295	—	2	4	no	—
15	C₂×#13063259	C₂²	1	4	yes	—
16	C₂×#13063258	C₂²	1	4	yes	—
17	C₂×#13063256	C₂²	1	4	yes	—
18	C₂×#13063235	C₂²	1	4	yes	—
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	#14582806	—	2	2	no	—
21	#14582804	—	2	2	no	—
22	#14582768	—	2	2	no	—
23	#14582768	C₂²	1	4	yes	—
24	#14582764	C₂²	1	4	yes	—
25	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#498	—	2	4	no	—
29	C₂×#106×C₄₃	C₂²	1	4	yes	—
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₂²×Dic₈₆	—	2	1	no	—
32	C₂²×Dic₈₆	—	2	1	no	—
33	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
34	#13063325	—	2	2	no	—
35	#13063325	—	2	2	no	—
36	#13063325	—	2	2	no	—
37	#13063325	—	2	2	no	—
38	#13063323	—	4	8	no	—
39	C₂×C₄⋊Dic₄₃	—	2	1	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
43	C₂×Dic₄₃⋊C₄	—	2	8	no	—
44	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
46	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
47	#13063259	—	2	8	no	—
48	#13063258	—	2	8	no	—
49	#13063256	—	2	8	no	—
50	#13063235	Q₈	1	16	yes	complemented
51	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
52	C₄₃⋊#95	—	2	8	no	—
53	C₂²×Q₈×C₄₃	—	2	1	no	—
54	C₂²×Q₈×C₄₃	—	2	1	no	—
55	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₄₃	—	2	2	no	—
57	C₄⋊Q₈×C₄₃	—	2	2	no	—
58	C₄⋊Q₈×C₄₃	—	2	2	no	—
59	C₄⋊Q₈×C₄₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
65	#106×C₄₃	—	2	8	no	—
66	C₄×C₈×C₄₃	D₄	1	4	yes	—
67	C₂×Q₈×C₄₃	—	4	2	no	—
68	C₂×Q₈×C₄₃	—	4	2	no	—
69	C₂×Q₈×C₄₃	—	4	4	no	—
70	C₂×Q₈×C₄₃	—	4	4	no	—
71	C₂²×C₄×C₄₃	—	2	2	no	—
72	C₂²×C₄×C₄₃	—	2	2	no	—
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₃	—	2	2	no	—
77	C₂²×Dic₄₃	—	2	2	no	—
78	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
79	C₂×Dic₈₆	—	4	2	no	—
80	C₂×Dic₈₆	—	4	2	no	—
81	C₂×Dic₈₆	—	4	4	no	—
82	C₂×Dic₈₆	—	4	4	no	—
83	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₄₃	C₂×Q₈	1	8	yes	—
85	C₄⋊C₄×C₄₃	—	2	2	no	—
86	C₄⋊C₄×C₄₃	—	2	2	no	—
87	C₄⋊C₄×C₄₃	—	2	2	no	—
88	C₄⋊C₄×C₄₃	—	2	2	no	—
89	C₄⋊C₄×C₄₃	—	4	2	no	—
90	C₄⋊C₄×C₄₃	—	4	2	no	—
91	C₄²×C₄₃	C₂×D₄	1	2	yes	—
92	C₄²×C₄₃	C₂×D₄	1	2	yes	—
93	C₄⋊Dic₄₃	—	2	2	no	—
94	C₄⋊Dic₄₃	—	2	2	no	—
95	C₄⋊Dic₄₃	—	2	2	no	—
96	C₄⋊Dic₄₃	—	2	2	no	—
97	C₄⋊Dic₄₃	—	4	2	no	—
98	C₄⋊Dic₄₃	—	4	2	no	—
99	Dic₄₃⋊C₄	—	4	16	no	—
100	C₄×Dic₄₃	—	2	4	no	—
101	C₄×Dic₄₃	—	2	4	no	—
102	C₄×Dic₄₃	C₂×Q₈	1	8	yes	—
103	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
104	C₂×C₄₃⋊C₈	C₂×Q₈	1	8	yes	—
105	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₄₃	—	8	4	no	—
107	Q₈×C₄₃	—	8	4	no	—
108	C₂×C₄×C₄₃	—	4	2	no	—
109	C₂×C₄×C₄₃	—	4	2	no	—
110	C₂×C₄×C₄₃	—	4	4	no	—
111	C₂×C₄×C₄₃	—	4	4	no	—
112	C₂×C₄×C₄₃	#127	1	2	yes	—
113	C₂×C₄×C₄₃	#127	1	2	yes	—
114	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
115	C₂×C₄×C₄₃	#137	1	4	yes	—
116	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
119	C₂×Dic₄₃	—	2	4	no	—
120	C₂×Dic₄₃	—	2	4	no	—
121	C₂×Dic₄₃	—	4	2	no	—
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	—	4	4	no	—
124	C₂×Dic₄₃	—	4	4	no	—
125	C₂×Dic₄₃	C₄⋊Q₈	1	8	yes	—
126	Dic₈₆	—	8	4	no	—
127	Dic₈₆	—	8	4	no	—
128	C₈×C₄₃	C₄⋊Q₈	1	8	yes	complemented
129	C₄₃⋊C₈	C₄⋊Q₈	1	8	yes	complemented
130	#33314	—	43	1	no	maximal, Sylow
131	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
132	C₂²×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
133	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
134	C₂²×C₄₃	C₂×#137	1	2	yes	—
135	C₄×C₄₃	—	8	2	no	—
136	C₄×C₄₃	—	8	2	no	—
137	C₄×C₄₃	#496	1	4	yes	—
138	C₄×C₄₃	#497	1	4	yes	—
139	C₄×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
140	C₄×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
141	Dic₄₃	—	2	8	no	—
142	Dic₄₃	—	8	2	no	—
143	Dic₄₃	—	8	2	no	—
144	C₂×#500	—	43	1	no	—
145	C₂×#500	—	43	1	no	—
146	C₂×#498	—	43	1	no	—
147	C₂×#498	D₄₃	1	1	yes	characteristic
148	#2868	—	43	1	no	—
149	#2862	—	43	1	no	—
150	#1847	—	43	8	no	—
151	C₄×#97	—	43	1	no	—
152	C₂×C₄₃	#2868	1	1	yes	characteristic
153	C₂×C₄₃	C₂×#496	1	1	yes	characteristic
154	C₂×C₄₃	C₂×#497	1	1	yes	characteristic
155	C₂×C₄₃	#3165	1	4	yes	—
156	C₂×C₄⋊Q₈	—	43	1	no	—
157	C₂×C₄⋊Q₈	—	43	1	no	—
158	C₂×C₄⋊Q₈	—	86	2	no	—
159	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
160	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
161	#500	—	86	4	no	—
162	#500	—	86	4	no	—
163	#498	—	2	4	no	—
164	#498	—	86	4	no	—
165	C₂×#107	—	43	4	no	—
166	C₂×#107	—	43	4	no	—
167	C₂×#106	—	43	4	no	—
168	C₂×#106	C₂×D₄₃	1	4	yes	—
169	C₂×#97	—	43	4	no	—
170	C₂×C₄×C₈	—	43	1	no	—
171	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
172	#391	—	86	2	no	—
173	#391	—	86	2	no	—
174	#384	—	43	4	no	—
175	#384	—	86	2	no	—
176	#382	—	43	4	no	—
177	C₄³	—	43	1	no	—
178	C₄₃	#33314	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	—	2	1	no	—
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×Q₈	—	86	1	no	—
182	C₂²×Q₈	—	86	1	no	—
183	C₂²×C₈	—	43	2	no	—
184	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	86	2	no	—
190	C₄⋊Q₈	—	86	2	no	—
191	C₄⋊Q₈	—	86	2	no	—
192	C₄⋊Q₈	—	86	2	no	—
193	C₄⋊Q₈	—	172	8	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	43	2	no	—
197	C₂×C₄⋊C₄	—	43	2	no	—
198	C₂×C₄⋊C₄	—	86	8	no	—
199	C₂×C₄⋊C₄	—	86	1	no	—
200	C₂×C₄⋊C₄	—	86	1	no	—
201	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
202	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
203	C₂×C₄²	—	43	2	no	—
204	C₂×C₄²	—	43	2	no	—
205	C₂×C₄²	—	43	2	no	—
206	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
207	#107	—	86	8	no	—
208	#107	—	86	8	no	—
209	#106	—	2	8	no	—
210	#106	—	86	8	no	—
211	#97	—	43	16	no	—
212	C₄×C₈	—	43	4	no	—
213	C₄×C₈	C₂²⋊D₄₃	1	4	yes	—
214	#95	—	86	8	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	172	2	no	—
220	C₂×Q₈	—	172	2	no	—
221	C₂×Q₈	—	172	4	no	—
222	C₂×Q₈	—	172	4	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	43	4	no	—
226	C₂²×C₄	—	86	2	no	—
227	C₂²×C₄	—	86	2	no	—
228	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
230	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
231	C₂×C₈	—	43	4	no	—
232	C₂×C₈	—	43	8	no	—
233	C₂×C₈	Q₈×D₄₃	1	8	yes	—
234	C₂×C₈	C₂×C₂²⋊D₄₃	1	4	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	86	2	no	—
242	C₄⋊C₄	—	86	2	no	—
243	C₄⋊C₄	—	86	2	no	—
244	C₄⋊C₄	—	86	2	no	—
245	C₄⋊C₄	—	172	16	no	—
246	C₄⋊C₄	—	172	2	no	—
247	C₄⋊C₄	—	172	2	no	—
248	C₄²	—	43	8	no	—
249	C₄²	—	86	4	no	—
250	C₄²	—	86	4	no	—
251	C₄²	D₄×D₄₃	1	2	yes	—
252	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
253	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
254	Q₈	—	8	4	no	—
255	Q₈	—	8	4	no	—
256	Q₈	—	344	4	no	—
257	Q₈	—	344	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	43	8	no	—
263	C₂×C₄	—	86	4	no	—
264	C₂×C₄	—	86	4	no	—
265	C₂×C₄	—	172	4	no	—
266	C₂×C₄	—	172	4	no	—
267	C₂×C₄	—	172	2	no	—
268	C₂×C₄	—	172	2	no	—
269	C₂×C₄	#13063321	1	2	yes	—
270	C₂×C₄	#13063321	1	2	yes	—
271	C₂×C₄	#13063352	1	4	yes	—
272	C₂×C₄	#13063354	1	4	yes	—
273	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
275	C₂×C₄	C₂×Q₈×D₄₃	1	4	yes	—
276	C₈	—	43	8	no	—
277	C₈	#13063323	1	8	yes	complemented
278	C₂²	C₂×#13063321	1	1	yes	characteristic
279	C₂²	C₄⋊Q₈×D₄₃	1	2	yes	—
280	C₂²	C₂×#13063352	1	2	yes	—
281	C₂²	C₂×#13063354	1	2	yes	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	86	8	no	—
285	C₄	—	344	2	no	—
286	C₄	—	344	2	no	—
287	C₄	#14583285	1	4	yes	—
288	C₄	#14583290	1	4	yes	—
289	C₄	C₄⋊Q₈×D₄₃	1	2	yes	—
290	C₄	C₂×#13063323	1	2	yes	—
291	C₂	#19142991	1	1	yes	characteristic
292	C₂	C₂×#14583285	1	1	yes	characteristic
293	C₂	C₂×#14583290	1	1	yes	characteristic
294	C₂	#19145136	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#498	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₂×#498	D₄₃
hypocenter	C₄₃	#33314
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
50. #13063235	Q₈	semidirect
128. C₈×C₄₃	C₄⋊Q₈	semidirect
129. C₄₃⋊C₈	C₄⋊Q₈	semidirect
178. C₄₃	#33314	semidirect
277. C₈	#13063323	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583304	2
C₂×#14583302	2
C₂×#14583295	2
#19142991	2
#19142972	2
#19134760	2
C₄×#13063235	2
C₂×C₄₃×#498	2
#33314	43