Subgroups — #25013960

Order 11008.

578 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	C₂³×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14582700	C₂	1	2	yes	maximal, complemented
3	#19134624	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132686	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₃×#466	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₁₆	—	2	4	no	—
7	C₂²×C₄₃⋊C₁₆	—	2	2	no	—
8	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	C₂×#13063259	C₂²	1	2	yes	—
10	#14582759	—	2	1	no	—
11	#14582700	—	2	4	no	—
12	#14582662	—	2	1	no	—
13	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×#121×C₄₃	—	2	1	no	—
16	C₄₃×#466	—	2	2	no	—
17	C₄₃×#466	C₄	1	4	yes	complemented
18	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×#102×C₄₃	—	2	1	no	—
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	C₂×C₂²⋊Dic₄₃	—	4	1	no	—
22	C₂×C₄₃⋊C₁₆	—	2	8	no	—
23	C₂×C₄₃⋊C₁₆	—	2	2	no	—
24	C₂×C₄₃⋊C₁₆	—	2	2	no	—
25	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	complemented
26	#13063259	—	2	2	no	—
27	#13063259	—	2	2	no	—
28	C₄₃⋊#95	—	4	1	no	—
29	C₂²×D₄×C₄₃	—	2	1	no	—
30	C₂²×D₄×C₄₃	—	4	1	no	—
31	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
32	C₂×D₈×C₄₃	—	2	2	no	—
33	C₂×D₈×C₄₃	—	2	1	no	—
34	C₂×D₈×C₄₃	—	2	2	no	—
35	C₂×D₈×C₄₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₄₃	—	2	2	no	—
37	C₂²×C₈×C₄₃	—	2	2	no	—
38	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₃	D₄	1	2	yes	—
41	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
42	#121×C₄₃	—	2	4	no	—
43	#121×C₄₃	—	4	2	no	—
44	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₄₃	—	4	1	no	—
46	#107×C₄₃	—	2	1	no	—
47	#107×C₄₃	C₂×C₄	1	2	yes	—
48	#102×C₄₃	—	2	4	no	—
49	C₂⁴×C₄₃	—	4	1	no	—
50	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
51	C₂×D₄×C₄₃	—	2	2	no	—
52	C₂×D₄×C₄₃	—	4	1	no	—
53	C₂×D₄×C₄₃	—	4	4	no	—
54	C₂×D₄×C₄₃	—	4	2	no	—
55	C₂×D₄×C₄₃	—	4	2	no	—
56	C₂×D₄×C₄₃	—	8	2	no	—
57	C₂×D₄×C₄₃	—	8	2	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	4	1	no	—
61	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₄₃	—	4	2	no	—
66	D₈×C₄₃	—	2	4	no	—
67	D₈×C₄₃	—	4	1	no	—
68	D₈×C₄₃	—	4	1	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	4	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	#41	1	2	yes	—
78	C₄⋊C₄×C₄₃	—	2	2	no	—
79	C₄⋊C₄×C₄₃	—	4	1	no	—
80	C₂²⋊C₄×C₄₃	—	4	4	no	—
81	C₂²⋊Dic₄₃	—	8	2	no	—
82	C₄⋊Dic₄₃	—	4	2	no	—
83	C₄⋊Dic₄₃	—	4	2	no	—
84	C₄₃⋊C₁₆	—	2	4	no	—
85	C₂³×C₄₃	—	2	2	no	—
86	C₂³×C₄₃	—	2	2	no	—
87	C₂³×C₄₃	—	4	2	no	—
88	C₂³×C₄₃	—	8	2	no	—
89	C₂³×C₄₃	—	8	1	no	—
90	C₂³×C₄₃	—	8	1	no	—
91	C₂³×C₄₃	—	8	2	no	—
92	C₂³×C₄₃	#102	1	2	yes	—
93	C₂³×C₄₃	D₁₆	1	2	yes	—
94	C₂³×C₄₃	SD₃₂	1	2	yes	—
95	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
96	D₄×C₄₃	—	4	2	no	—
97	D₄×C₄₃	—	4	2	no	—
98	D₄×C₄₃	—	8	8	no	—
99	D₄×C₄₃	—	8	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	—	8	1	no	—
106	C₂×C₄×C₄₃	—	8	1	no	—
107	C₂×C₄×C₄₃	#102	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#121	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#135	1	2	yes	—
112	C₂×Dic₄₃	—	8	2	no	—
113	C₂×Dic₄₃	—	8	2	no	—
114	C₂×Dic₄₃	—	8	2	no	—
115	C₈×C₄₃	—	2	2	no	—
116	C₈×C₄₃	C₄×D₄	1	2	yes	—
117	C₈×C₄₃	#120	1	1	yes	characteristic
118	C₈×C₄₃	#123	1	1	yes	characteristic
119	#13059	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	—	2	2	no	—
121	C₂²×C₄₃	—	2	2	no	—
122	C₂²×C₄₃	—	2	4	no	—
123	C₂²×C₄₃	—	2	4	no	—
124	C₂²×C₄₃	—	8	2	no	—
125	C₂²×C₄₃	—	8	2	no	—
126	C₂²×C₄₃	—	8	2	no	—
127	C₂²×C₄₃	—	16	2	no	—
128	C₂²×C₄₃	—	16	2	no	—
129	C₂²×C₄₃	#357	1	4	yes	complemented
130	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
131	C₂²×C₄₃	#466	1	2	yes	—
132	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
133	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
134	C₂²×C₄₃	#508	1	2	yes	—
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	—	8	2	no	—
137	C₄×C₄₃	#386	1	1	yes	characteristic
138	C₄×C₄₃	#416	1	1	yes	characteristic
139	C₄×C₄₃	#465	1	2	yes	—
140	Dic₄₃	—	16	2	no	—
141	C₂³×C₁₆	—	43	1	no	—
142	C₂×#466	D₄₃	1	1	yes	characteristic
143	C₂×#357	—	43	2	no	—
144	#1808	—	43	1	no	—
145	#1380	—	43	2	no	—
146	C₂×C₄₃	—	2	4	no	—
147	C₂×C₄₃	—	16	2	no	—
148	C₂×C₄₃	#1893	1	1	yes	characteristic
149	C₂×C₄₃	C₂×#357	1	1	yes	characteristic
150	C₂×C₄₃	#2138	1	1	yes	characteristic
151	C₂×C₄₃	#2215	1	2	yes	—
152	C₂×C₄₃	#2217	1	2	yes	—
153	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
154	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
155	C₂×#121	—	2	1	no	—
156	C₂²×C₁₆	—	43	2	no	—
157	C₂²×C₁₆	—	86	4	no	—
158	C₂²×C₁₆	—	86	2	no	—
159	#466	—	2	2	no	—
160	#466	Dic₄₃	1	4	yes	complemented
161	C₂×#107	—	43	2	no	—
162	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
163	C₂×#102	—	2	1	no	—
164	#380	—	86	1	no	—
165	#357	—	86	4	no	—
166	#340	—	86	1	no	—
167	C₄₃	#13059	1	1	yes	characteristic, complemented, Sylow
168	C₂²×D₄	—	2	1	no	—
169	C₂²×D₄	—	4	1	no	—
170	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
171	C₂×D₈	—	2	2	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	C₂×Dic₄₃	1	2	yes	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
178	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
180	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
181	#121	—	2	4	no	—
182	#121	—	4	2	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	86	2	no	—
185	C₂×C₂²⋊C₄	—	4	1	no	—
186	C₂×C₂²⋊C₄	—	172	1	no	—
187	C₂×C₁₆	—	43	4	no	—
188	C₂×C₁₆	—	86	8	no	—
189	C₂×C₁₆	—	86	2	no	—
190	C₂×C₁₆	—	86	2	no	—
191	#107	—	2	1	no	—
192	#107	—	86	2	no	—
193	#107	—	86	2	no	—
194	#107	C₂×Dic₄₃	1	2	yes	—
195	#102	—	2	4	no	—
196	#95	—	172	1	no	—
197	C₂⁴	—	4	1	no	—
198	C₂⁴	#10621882	1	1	yes	characteristic
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	4	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	8	2	no	—
205	C₂×D₄	—	8	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	172	2	no	—
210	C₂²×C₄	#10621882	1	1	yes	characteristic
211	C₂²×C₄	#10621883	1	2	yes	—
212	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
214	D₈	—	2	4	no	—
215	D₈	—	4	1	no	—
216	D₈	—	4	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
222	C₂×C₈	D₄×D₄₃	1	2	yes	—
223	C₂×C₈	#10621900	1	2	yes	—
224	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	172	2	no	—
229	C₄⋊C₄	—	172	2	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	344	2	no	—
232	C₁₆	—	86	4	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	8	1	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	2	no	—
239	C₂³	—	8	2	no	—
240	C₂³	#13063246	1	2	yes	—
241	C₂³	#13063266	1	2	yes	—
242	C₂³	#13063269	1	2	yes	—
243	C₂³	C₂×#10621882	1	1	yes	characteristic
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	8	8	no	—
247	D₄	—	8	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	344	2	no	—
256	C₂×C₄	—	344	2	no	—
257	C₂×C₄	—	344	2	no	—
258	C₂×C₄	#13063246	1	2	yes	—
259	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
260	C₂×C₄	#13063297	1	2	yes	—
261	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
262	C₂×C₄	#13063345	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	D₄×Dic₄₃	1	2	yes	—
265	C₈	#13063293	1	1	yes	characteristic
266	C₈	#13063306	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	#14582700	1	4	yes	complemented
277	C₂²	C₂×#13063246	1	1	yes	characteristic
278	C₂²	#14583133	1	2	yes	—
279	C₂²	C₂×#13063266	1	1	yes	characteristic
280	C₂²	C₂×#13063269	1	1	yes	characteristic
281	C₂²	#14583329	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	688	2	no	—
285	C₄	#14582779	1	1	yes	characteristic
286	C₄	#14582895	1	1	yes	characteristic
287	C₄	#14583126	1	2	yes	—
288	C₂	—	2	4	no	—
289	C₂	—	16	2	no	—
290	C₂	#19135024	1	1	yes	characteristic
291	C₂	C₂×#14582700	1	1	yes	characteristic
292	C₂	#19136302	1	1	yes	characteristic
293	C₂	#19136679	1	2	yes	—
294	C₂	#19136689	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting, prime radical	C₂×C₄₃×#466	C₂
socle	C₂³×C₄₃	C₂×D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#466	D₄₃
hypocenter	C₄₃	#13059
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄₃⋊C₁₆	C₂	semidirect
2. C₂×#14582700	C₂	semidirect
3. #19134624	C₂	semidirect
4. #19132686	C₂	semidirect
17. C₄₃×#466	C₄	semidirect
25. C₂×C₄₃⋊C₁₆	D₄	semidirect
129. C₂²×C₄₃	#357	semidirect
160. #466	Dic₄₃	semidirect
167. C₄₃	#13059	semidirect
276. C₂²	#14582700	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄₃⋊C₁₆	2
C₂×#14582700	2
#19134624	2
#19132686	2
C₂×C₄₃×#466	2
#13059	43