Subgroups — #25014819

Order 11008.

514 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583284	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582703	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582701	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582699	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134758	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621884	—	2	1	no	—
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×#13063314	—	2	1	no	—
11	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14583284	—	2	2	no	—
13	#14583284	C₄	1	4	yes	complemented
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063256	—	2	1	no	—
16	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
17	#14582764	—	2	1	no	—
18	#14582703	—	2	2	no	—
19	#14582701	—	2	2	no	—
20	#14582699	—	2	2	no	—
21	C₁₆×Dic₄₃	—	2	2	no	—
22	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#358	—	2	2	no	—
26	C₂²×D₁₇₂	—	2	1	no	—
27	C₂×#10621884	—	2	4	no	—
28	C₂×#10621884	—	4	2	no	—
29	C₂×#10621884	—	4	2	no	—
30	C₂×D₃₄₄	—	2	1	no	—
31	C₂×D₃₄₄	—	2	1	no	—
32	C₂×D₃₄₄	—	2	1	no	—
33	C₂×D₃₄₄	—	2	1	no	—
34	C₂×D₃₄₄	—	2	1	no	—
35	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
36	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
37	#13063314	—	2	4	no	—
38	#13063314	—	4	2	no	—
39	C₂×C₄⋊Dic₄₃	—	2	1	no	—
40	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
41	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
42	C₂×C₄₃⋊C₁₆	—	2	1	no	—
43	C₂×C₄₃⋊C₁₆	—	2	1	no	—
44	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
45	#13063259	—	2	1	no	—
46	#13063259	—	2	1	no	—
47	#13063256	—	4	2	no	—
48	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₄₃	D₄	1	2	yes	—
50	C₄₃⋊#95	—	4	2	no	—
51	C₂²×Q₈×C₄₃	—	2	1	no	—
52	C₂×Q₁₆×C₄₃	—	2	1	no	—
53	C₂×Q₁₆×C₄₃	—	2	1	no	—
54	C₂×Q₁₆×C₄₃	—	2	1	no	—
55	C₂×Q₁₆×C₄₃	—	2	1	no	—
56	C₂×Q₁₆×C₄₃	—	2	1	no	—
57	C₂×Q₁₆×C₄₃	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
60	C₂×C₁₆×C₄₃	—	2	1	no	—
61	C₂×C₁₆×C₄₃	—	2	1	no	—
62	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
63	#107×C₄₃	—	2	1	no	—
64	#107×C₄₃	—	2	1	no	—
65	C₂³×D₄₃	—	4	1	no	—
66	C₂×Q₈×C₄₃	—	2	2	no	—
67	C₂×Q₈×C₄₃	—	4	1	no	—
68	C₂×Q₈×C₄₃	—	4	1	no	—
69	C₂×Q₈×C₄₃	—	4	1	no	—
70	C₂×Q₈×C₄₃	—	4	1	no	—
71	C₂×Q₈×C₄₃	—	4	1	no	—
72	C₂²×C₄×C₄₃	—	4	1	no	—
73	C₂²×C₄×C₄₃	—	4	1	no	—
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₄₃	—	4	1	no	—
76	C₂²×Dic₄₃	D₈	1	2	yes	—
77	C₂×D₁₇₂	—	2	2	no	—
78	C₂×D₁₇₂	—	4	1	no	—
79	C₂×D₁₇₂	—	4	1	no	—
80	C₂×D₁₇₂	—	4	1	no	—
81	C₂×D₁₇₂	—	4	1	no	—
82	C₂×D₁₇₂	—	4	1	no	—
83	Q₁₆×C₄₃	—	2	2	no	—
84	Q₁₆×C₄₃	—	2	2	no	—
85	Q₁₆×C₄₃	—	4	1	no	—
86	Q₁₆×C₄₃	—	4	1	no	—
87	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
93	C₄⋊C₄×C₄₃	—	4	1	no	—
94	C₄⋊C₄×C₄₃	—	4	1	no	—
95	#10621884	—	4	8	no	—
96	(C₂×D₄₃)⋊C₄	—	4	8	no	—
97	C₄⋊Dic₄₃	—	4	1	no	—
98	C₄⋊Dic₄₃	—	4	1	no	—
99	C₄×Dic₄₃	C₂²⋊C₄	1	2	yes	—
100	C₄×Dic₄₃	SD₁₆	1	2	yes	—
101	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
102	C₂×C₄₃⋊C₈	D₈	1	2	yes	—
103	C₂×C₄₃⋊C₈	SD₁₆	1	2	yes	—
104	D₃₄₄	—	2	2	no	—
105	D₃₄₄	—	2	2	no	—
106	D₃₄₄	—	4	1	no	—
107	D₃₄₄	—	4	1	no	—
108	C₁₆×C₄₃	—	2	2	no	—
109	C₄₃⋊C₁₆	—	2	2	no	—
110	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
111	C₂²×D₄₃	—	4	2	no	—
112	C₂²×D₄₃	—	8	1	no	—
113	C₂²×D₄₃	—	8	1	no	—
114	C₂²×D₄₃	—	8	2	no	—
115	C₂²×D₄₃	—	8	2	no	—
116	Q₈×C₄₃	—	4	2	no	—
117	Q₈×C₄₃	—	4	2	no	—
118	Q₈×C₄₃	—	8	1	no	—
119	Q₈×C₄₃	—	8	1	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	8	1	no	—
122	C₂×C₄×C₄₃	—	8	1	no	—
123	C₂×C₄×C₄₃	—	8	1	no	—
124	C₂×C₄×C₄₃	—	8	2	no	—
125	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×Dic₄₃	—	2	2	no	—
132	C₂×Dic₄₃	—	2	2	no	—
133	C₂×Dic₄₃	—	8	1	no	—
134	C₂×Dic₄₃	—	8	2	no	—
135	C₂×Dic₄₃	#102	1	4	yes	—
136	D₁₇₂	—	4	2	no	—
137	D₁₇₂	—	4	2	no	—
138	D₁₇₂	—	8	1	no	—
139	D₁₇₂	—	8	1	no	—
140	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
141	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
142	C₈×C₄₃	#121	1	1	yes	characteristic
143	C₈×C₄₃	#121	1	1	yes	characteristic
144	C₄₃⋊C₈	#102	1	4	yes	—
145	#13204	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
147	C₂²×C₄₃	#486	1	1	yes	characteristic
148	C₂²×C₄₃	#493	1	1	yes	characteristic
149	C₂²×C₄₃	#508	1	2	yes	—
150	C₂²×C₄₃	#508	1	2	yes	—
151	C₂×D₄₃	—	8	4	no	—
152	C₂×D₄₃	—	8	2	no	—
153	C₂×D₄₃	—	16	2	no	—
154	C₂×D₄₃	—	16	1	no	—
155	C₂×D₄₃	—	16	1	no	—
156	C₄×C₄₃	—	8	2	no	—
157	C₄×C₄₃	—	16	1	no	—
158	C₄×C₄₃	#390	1	1	yes	characteristic
159	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
160	C₄×C₄₃	#486	1	1	yes	characteristic
161	C₄×C₄₃	#492	1	1	yes	characteristic
162	Dic₄₃	—	2	4	no	—
163	Dic₄₃	—	16	1	no	—
164	C₂×#495	—	43	1	no	—
165	C₂×#358	—	43	1	no	—
166	C₂×#358	D₄₃	1	1	yes	characteristic, complemented
167	C₂×#357	—	43	1	no	—
168	C₂×#357	—	43	1	no	—
169	C₂×C₄×C₁₆	—	43	1	no	—
170	#1846	—	43	1	no	—
171	C₂×C₄₃	#1957	1	1	yes	characteristic
172	C₂×C₄₃	#2138	1	2	yes	—
173	C₂×C₄₃	#2242	1	2	yes	—
174	C₂×C₄₃	#2249	1	1	yes	characteristic
175	C₂×C₄₃	#2249	1	1	yes	characteristic
176	D₄₃	—	16	2	no	—
177	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
178	C₂²×SD₁₆	—	86	1	no	—
179	C₂²×D₈	—	43	1	no	—
180	C₂×#124	—	86	1	no	—
181	C₂²×C₁₆	—	43	1	no	—
182	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
183	#495	—	43	4	no	—
184	#495	—	86	2	no	—
185	C₂×#107	—	43	1	no	—
186	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
187	C₂×#106	—	86	1	no	—
188	C₂×C₄×C₈	—	43	1	no	—
189	#382	—	86	1	no	—
190	#358	—	2	2	no	—
191	#358	—	86	2	no	—
192	#357	—	86	2	no	—
193	#357	—	86	2	no	—
194	C₄×C₁₆	—	86	2	no	—
195	C₄₃	#13204	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	—	2	1	no	—
197	C₂²×D₄	—	86	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₄×D₄₃	1	2	yes	—
204	C₂×SD₁₆	—	86	4	no	—
205	C₂×SD₁₆	—	172	2	no	—
206	C₂×SD₁₆	—	172	2	no	—
207	C₂×D₈	—	43	2	no	—
208	C₂×D₈	—	86	1	no	—
209	C₂×D₈	—	86	1	no	—
210	C₂×D₈	—	86	1	no	—
211	C₂×D₈	—	86	1	no	—
212	C₂×D₈	—	86	1	no	—
213	C₂²×C₈	—	43	1	no	—
214	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
215	#124	—	86	4	no	—
216	#124	—	172	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	86	1	no	—
219	C₂×C₂²⋊C₄	—	172	2	no	—
220	C₂×C₄²	—	43	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	43	2	no	—
224	C₂×C₁₆	—	86	1	no	—
225	C₂×C₁₆	—	86	1	no	—
226	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
227	#107	—	2	1	no	—
228	#107	—	2	1	no	—
229	#107	—	86	1	no	—
230	#107	—	86	1	no	—
231	#106	—	172	2	no	—
232	C₄×C₈	—	43	2	no	—
233	C₄×C₈	—	43	2	no	—
234	#95	—	172	2	no	—
235	C₂⁴	—	172	1	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×D₄	—	86	2	no	—
243	C₂×D₄	—	172	1	no	—
244	C₂×D₄	—	172	1	no	—
245	C₂×D₄	—	172	1	no	—
246	C₂×D₄	—	172	1	no	—
247	C₂×D₄	—	172	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	43	2	no	—
251	C₂²×C₄	—	172	1	no	—
252	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	4	1	no	—
256	Q₁₆	—	4	1	no	—
257	SD₁₆	—	172	8	no	—
258	D₈	—	86	2	no	—
259	D₈	—	86	2	no	—
260	D₈	—	172	1	no	—
261	D₈	—	172	1	no	—
262	C₂×C₈	—	43	2	no	—
263	C₂×C₈	—	43	2	no	—
264	C₂×C₈	—	43	2	no	—
265	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
266	C₂×C₈	#10621898	1	1	yes	characteristic
267	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
268	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
269	C₂×C₈	#10621900	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	172	1	no	—
274	C₄⋊C₄	—	172	1	no	—
275	C₂²⋊C₄	—	172	8	no	—
276	C₄²	—	43	2	no	—
277	C₄²	—	43	2	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	86	2	no	—
280	C₂³	—	344	2	no	—
281	C₂³	—	344	2	no	—
282	C₂³	—	344	1	no	—
283	C₂³	—	344	1	no	—
284	C₂³	—	172	2	no	—
285	C₂³	D₈×D₄₃	1	1	yes	characteristic
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	1	no	—
289	Q₈	—	8	1	no	—
290	D₄	—	344	1	no	—
291	D₄	—	344	1	no	—
292	D₄	—	172	2	no	—
293	D₄	—	172	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	43	4	no	—
300	C₂×C₄	—	86	2	no	—
301	C₂×C₄	—	86	2	no	—
302	C₂×C₄	—	344	2	no	—
303	C₂×C₄	—	344	1	no	—
304	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
305	C₂×C₄	#13063311	1	1	yes	characteristic
306	C₂×C₄	#13063321	1	1	yes	characteristic
307	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
308	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
309	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
310	C₈	—	43	4	no	—
311	C₈	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
312	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
313	C₈	#13063297	1	1	yes	characteristic
314	C₈	#13063298	1	1	yes	characteristic
315	C₂²	—	344	4	no	—
316	C₂²	—	344	2	no	—
317	C₂²	—	688	2	no	—
318	C₂²	—	688	1	no	—
319	C₂²	—	688	1	no	—
320	C₂²	#102×D₄₃	1	1	yes	characteristic
321	C₂²	#14583244	1	1	yes	characteristic
322	C₂²	#14583274	1	1	yes	characteristic
323	C₂²	#14583330	1	2	yes	—
324	C₂²	#14583332	1	2	yes	—
325	C₄	—	8	2	no	—
326	C₄	—	16	1	no	—
327	C₄	—	86	4	no	—
328	C₄	—	688	1	no	—
329	C₄	#14582799	1	1	yes	characteristic
330	C₄	#102×D₄₃	1	1	yes	characteristic
331	C₄	#14583244	1	1	yes	characteristic
332	C₄	#14583270	1	1	yes	characteristic
333	C₂	—	688	2	no	—
334	C₂	#19135399	1	1	yes	characteristic
335	C₂	#19136301	1	2	yes	—
336	C₂	#19136828	1	2	yes	—
337	C₂	#19136855	1	1	yes	characteristic
338	C₂	#19136857	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#358	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#358	D₄₃
hypocenter	C₄₃	#13204
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

Complements

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

N	H ≅ G/N	type
2. C₂×#14582703	C₂	semidirect
5. C₂×C₁₆×Dic₄₃	C₂	semidirect
6. #19134758	C₂	semidirect
7. C₂×C₄₃×#358	C₂	semidirect
13. #14583284	C₄	semidirect
166. C₂×#358	D₄₃	semidirect
195. C₄₃	#13204	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583284	2
C₂×#14582703	2
C₂×#14582701	2
C₂×#14582699	2
C₂×C₁₆×Dic₄₃	2
#19134758	2
C₂×C₄₃×#358	2
#13204	43