Subgroups — #25013957

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×#14583135	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582699	C₂	1	2	yes	maximal, complemented
4	#19132686	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1808	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
7	C₂×#13063299	—	2	1	no	—
8	C₂²×C₄₃⋊C₁₆	—	2	4	no	—
9	C₂²×C₄₃⋊C₁₆	—	2	2	no	—
10	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
11	#14583135	—	2	2	no	—
12	#14583135	C₄	1	4	yes	complemented
13	C₂×#13063259	C₂²	1	1	yes	characteristic
14	C₂×#13063247	—	2	1	no	—
15	#14582699	—	2	4	no	—
16	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#107×C₄₃	C₂²	1	2	yes	—
18	C₄₃×#380	—	2	1	no	—
19	C₄₃×#340	—	2	1	no	—
20	C₂²×D₁₇₂	—	2	1	no	—
21	C₂²×C₂²⋊D₄₃	—	4	1	no	—
22	C₂×D₃₄₄	—	2	2	no	—
23	C₂×D₃₄₄	—	2	2	no	—
24	C₂×D₃₄₄	—	2	1	no	—
25	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
26	#13063299	—	2	4	no	—
27	#13063299	—	4	2	no	—
28	C₂×C₄⋊Dic₄₃	—	2	1	no	—
29	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
30	C₂×C₄₃⋊C₁₆	—	2	8	no	—
31	C₂×C₄₃⋊C₁₆	—	2	2	no	—
32	C₂×C₄₃⋊C₁₆	—	2	2	no	—
33	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	complemented
34	#13063259	—	2	1	no	—
35	#13063259	C₂×C₄	1	2	yes	—
36	#13063247	—	2	4	no	—
37	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	—	2	2	no	—
39	C₂²×C₈×C₄₃	—	2	2	no	—
40	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₄₃	D₄	1	2	yes	—
42	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₄₃	—	4	1	no	—
46	#107×C₄₃	—	2	2	no	—
47	#107×C₄₃	—	2	2	no	—
48	#95×C₄₃	—	4	1	no	—
49	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
50	C₂³×D₄₃	—	4	1	no	—
51	C₂²×C₄×C₄₃	—	2	2	no	—
52	C₂²×C₄×C₄₃	—	2	2	no	—
53	C₂²×C₄×C₄₃	—	4	2	no	—
54	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₄₃	—	4	4	no	—
59	C₂×C₂²⋊D₄₃	—	8	2	no	—
60	C₂×C₂²⋊D₄₃	—	8	2	no	—
61	C₂²×Dic₄₃	—	4	1	no	—
62	C₂×D₁₇₂	—	2	2	no	—
63	C₂×D₁₇₂	—	4	1	no	—
64	C₂×D₁₇₂	—	4	2	no	—
65	C₂×D₁₇₂	—	4	2	no	—
66	C₂×C₈×C₄₃	—	2	2	no	—
67	C₂×C₈×C₄₃	—	2	4	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₄₃	#41	1	2	yes	—
75	C₄⋊C₄×C₄₃	—	4	2	no	—
76	C₄⋊C₄×C₄₃	—	4	2	no	—
77	C₂²⋊C₄×C₄₃	—	8	2	no	—
78	(C₂×D₄₃)⋊C₄	—	4	4	no	—
79	C₄⋊Dic₄₃	—	2	2	no	—
80	C₄⋊Dic₄₃	—	4	1	no	—
81	D₃₄₄	—	2	4	no	—
82	D₃₄₄	—	4	1	no	—
83	D₃₄₄	—	4	1	no	—
84	C₄₃⋊C₁₆	—	2	4	no	—
85	C₂³×C₄₃	—	2	2	no	—
86	C₂³×C₄₃	—	2	2	no	—
87	C₂³×C₄₃	#102	1	2	yes	—
88	C₂³×C₄₃	D₁₆	1	2	yes	—
89	C₂³×C₄₃	SD₃₂	1	2	yes	—
90	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₄₃	—	4	2	no	—
92	C₂²×D₄₃	—	8	2	no	—
93	C₂²×D₄₃	—	8	1	no	—
94	C₂²×D₄₃	—	8	1	no	—
95	C₂²×D₄₃	—	8	2	no	—
96	C₂×C₄×C₄₃	—	2	2	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	8	2	no	—
101	C₂×C₄×C₄₃	—	8	2	no	—
102	C₂×C₄×C₄₃	—	8	2	no	—
103	C₂×C₄×C₄₃	#102	1	2	yes	—
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	#121	1	2	yes	—
106	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#135	1	2	yes	—
108	C₂²⋊D₄₃	—	8	8	no	—
109	C₂×Dic₄₃	—	4	2	no	—
110	C₂×Dic₄₃	—	8	1	no	—
111	C₂×Dic₄₃	—	8	1	no	—
112	D₁₇₂	—	4	2	no	—
113	D₁₇₂	—	4	2	no	—
114	D₁₇₂	—	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	#357	1	4	yes	complemented
125	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
126	C₂²×C₄₃	#466	1	2	yes	—
127	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
128	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
129	C₂²×C₄₃	#508	1	2	yes	—
130	C₂×D₄₃	—	8	2	no	—
131	C₂×D₄₃	—	8	2	no	—
132	C₂×D₄₃	—	8	2	no	—
133	C₂×D₄₃	—	16	2	no	—
134	C₂×D₄₃	—	16	2	no	—
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	—	16	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	—	8	2	no	—
141	C₂³×C₁₆	—	43	1	no	—
142	C₂×#466	—	43	1	no	—
143	C₂×#357	—	43	2	no	—
144	#1808	D₄₃	1	1	yes	characteristic, complemented
145	#1380	—	43	2	no	—
146	C₂×C₄₃	—	2	4	no	—
147	C₂×C₄₃	#1893	1	1	yes	characteristic
148	C₂×C₄₃	C₂×#357	1	1	yes	characteristic
149	C₂×C₄₃	#2138	1	1	yes	characteristic
150	C₂×C₄₃	#2215	1	2	yes	—
151	C₂×C₄₃	#2217	1	2	yes	—
152	D₄₃	—	16	2	no	—
153	C₂²×D₈	—	43	1	no	—
154	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
155	C₂×#121	—	86	1	no	—
156	C₂²×C₁₆	—	43	2	no	—
157	C₂²×C₁₆	—	86	4	no	—
158	C₂²×C₁₆	—	86	2	no	—
159	#466	—	43	4	no	—
160	#466	—	86	2	no	—
161	C₂×#107	—	43	1	no	—
162	C₂×#107	C₂×D₄₃	1	2	yes	—
163	C₂×#102	—	86	1	no	—
164	#380	—	2	1	no	—
165	#357	—	86	4	no	—
166	#340	—	2	1	no	—
167	C₄₃	#13059	1	1	yes	characteristic, complemented, Sylow
168	C₂²×D₄	—	86	1	no	—
169	C₂²×D₄	—	172	1	no	—
170	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
171	C₂×D₈	—	43	2	no	—
172	C₂×D₈	—	86	1	no	—
173	C₂×D₈	—	86	2	no	—
174	C₂×D₈	—	86	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	C₄×D₄₃	1	2	yes	—
178	C₂²×C₈	D₁₇₂	1	2	yes	—
179	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
181	#121	—	86	4	no	—
182	#121	—	172	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	86	1	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	2	no	—
192	#107	—	2	2	no	—
193	#107	—	43	2	no	—
194	#107	—	86	1	no	—
195	#102	—	86	4	no	—
196	#95	—	4	1	no	—
197	C₂⁴	—	172	1	no	—
198	C₂⁴	#10621882	1	1	yes	characteristic
199	C₂×D₄	—	86	2	no	—
200	C₂×D₄	—	344	2	no	—
201	C₂×D₄	—	344	2	no	—
202	C₂×D₄	—	172	4	no	—
203	C₂×D₄	—	172	2	no	—
204	C₂×D₄	—	172	1	no	—
205	C₂×D₄	—	172	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	172	1	no	—
210	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
211	C₂²×C₄	#10621882	1	1	yes	characteristic
212	C₂²×C₄	#10621884	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
214	D₈	—	86	4	no	—
215	D₈	—	172	1	no	—
216	D₈	—	172	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	(C₂×D₄₃)⋊C₄	1	2	yes	—
222	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
223	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
224	C₂×C₈	D₄×D₄₃	1	2	yes	—
225	C₂×C₈	#10621900	1	2	yes	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	86	2	no	—
229	C₄⋊C₄	—	172	1	no	—
230	C₂²⋊C₄	—	8	2	no	—
231	C₂²⋊C₄	—	172	4	no	—
232	C₁₆	—	86	4	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	344	1	no	—
236	C₂³	—	344	1	no	—
237	C₂³	—	344	2	no	—
238	C₂³	—	344	2	no	—
239	C₂³	—	172	2	no	—
240	C₂³	#13063245	1	2	yes	—
241	C₂³	#13063266	1	2	yes	—
242	C₂³	#13063268	1	2	yes	—
243	C₂³	C₂×#10621882	1	1	yes	characteristic
244	D₄	—	344	8	no	—
245	D₄	—	344	2	no	—
246	D₄	—	172	2	no	—
247	D₄	—	172	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	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	—	344	1	no	—
256	C₂×C₄	—	344	1	no	—
257	C₂×C₄	—	172	2	no	—
258	C₂×C₄	#13063245	1	2	yes	—
259	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	#13063297	1	2	yes	—
261	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
262	C₂×C₄	#13063345	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
265	C₈	#13063295	1	1	yes	characteristic
266	C₈	#13063309	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	—	344	2	no	—
272	C₂²	—	344	2	no	—
273	C₂²	—	344	2	no	—
274	C₂²	—	688	2	no	—
275	C₂²	—	688	2	no	—
276	C₂²	#14582699	1	4	yes	complemented
277	C₂²	C₂×#13063245	1	1	yes	characteristic
278	C₂²	#14583133	1	2	yes	—
279	C₂²	C₂×#13063266	1	1	yes	characteristic
280	C₂²	C₂×#13063268	1	1	yes	characteristic
281	C₂²	#14583329	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	16	2	no	—
284	C₄	—	344	2	no	—
285	C₄	#14582776	1	1	yes	characteristic
286	C₄	#14582893	1	1	yes	characteristic
287	C₄	#14583130	1	2	yes	—
288	C₂	—	2	4	no	—
289	C₂	—	688	2	no	—
290	C₂	#19135021	1	1	yes	characteristic
291	C₂	C₂×#14582699	1	1	yes	characteristic
292	C₂	#19136300	1	1	yes	characteristic
293	C₂	#19136679	1	2	yes	—
294	C₂	#19136693	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1808	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂×#14583135	C₂
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1808	D₄₃
hypocenter	C₄₃	#13059
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄₃⋊C₁₆	C₂	semidirect
3. C₂×#14582699	C₂	semidirect
4. #19132686	C₂	semidirect
5. C₄₃×#1808	C₂	semidirect
12. #14583135	C₄	semidirect
33. C₂×C₄₃⋊C₁₆	D₄	semidirect
124. C₂²×C₄₃	#357	semidirect
144. #1808	D₄₃	semidirect
167. C₄₃	#13059	semidirect
276. C₂²	#14582699	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×#14583135	2
C₂×#14582699	2
#19132686	2
C₄₃×#1808	2
#13059	43