Subgroups — #25014964

Order 11008.

510 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063266	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582724	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582701	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#19135462	C₂	1	1	yes	maximal, characteristic, complemented
6	#19135459	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
9	C₂×#13063266	—	2	2	no	—
10	C₂×#13063266	—	2	2	no	—
11	C₂×#13063266	C₄	1	4	yes	complemented
12	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×#13063259	C₂²	1	1	yes	characteristic
14	C₂×#13063246	—	2	1	no	—
15	C₂×#13063245	—	2	1	no	—
16	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
17	#14582802	—	2	1	no	—
18	#14582800	—	2	1	no	—
19	#14582724	—	2	2	no	—
20	#14582701	—	2	2	no	—
21	C₁₆×Dic₄₃	—	2	2	no	—
22	C₂²×D₈×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×#357	—	2	2	no	—
26	C₂²×D₁₇₂	—	2	1	no	—
27	C₂×D₃₄₄	—	2	1	no	—
28	C₂×D₃₄₄	—	2	1	no	—
29	C₂×D₃₄₄	—	2	1	no	—
30	C₂×D₃₄₄	—	2	1	no	—
31	C₂×D₃₄₄	—	2	1	no	—
32	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
33	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₄₃	—	2	1	no	—
35	C₂×C₂²⋊Dic₄₃	—	4	2	no	—
36	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
37	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
38	#13063266	—	2	8	no	—
39	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
41	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
42	#13063259	—	2	1	no	—
43	#13063259	—	2	1	no	—
44	#13063246	—	4	2	no	—
45	#13063245	—	4	2	no	—
46	C₈×Dic₄₃	—	2	1	no	—
47	C₈×Dic₄₃	—	2	1	no	—
48	C₂²×D₄×C₄₃	—	2	1	no	—
49	C₂×D₈×C₄₃	—	2	1	no	—
50	C₂×D₈×C₄₃	—	2	1	no	—
51	C₂×D₈×C₄₃	—	2	1	no	—
52	C₂×D₈×C₄₃	—	2	1	no	—
53	C₂×D₈×C₄₃	—	2	1	no	—
54	C₂×D₈×C₄₃	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
57	C₂×C₁₆×C₄₃	—	2	1	no	—
58	C₂×C₁₆×C₄₃	—	2	1	no	—
59	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
60	#107×C₄₃	—	2	1	no	—
61	#107×C₄₃	—	2	1	no	—
62	C₂⁴×C₄₃	—	4	1	no	—
63	C₂³×D₄₃	—	4	1	no	—
64	C₂×D₄×C₄₃	—	2	2	no	—
65	C₂×D₄×C₄₃	—	4	1	no	—
66	C₂×D₄×C₄₃	—	4	1	no	—
67	C₂×D₄×C₄₃	—	4	1	no	—
68	C₂×D₄×C₄₃	—	4	1	no	—
69	C₂×D₄×C₄₃	—	4	1	no	—
70	C₂²×C₄×C₄₃	—	4	1	no	—
71	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₄₃	—	4	1	no	—
73	C₂²×Dic₄₃	D₈	1	2	yes	—
74	C₂×D₁₇₂	—	2	2	no	—
75	C₂×D₁₇₂	—	4	1	no	—
76	C₂×D₁₇₂	—	4	1	no	—
77	C₂×D₁₇₂	—	4	1	no	—
78	C₂×D₁₇₂	—	4	1	no	—
79	C₂×D₁₇₂	—	4	1	no	—
80	D₈×C₄₃	—	2	2	no	—
81	D₈×C₄₃	—	2	2	no	—
82	D₈×C₄₃	—	4	1	no	—
83	D₈×C₄₃	—	4	1	no	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	—	4	1	no	—
91	C₄⋊C₄×C₄₃	—	4	1	no	—
92	C₂²⋊Dic₄₃	—	8	4	no	—
93	(C₂×D₄₃)⋊C₄	—	8	4	no	—
94	C₄⋊Dic₄₃	—	4	1	no	—
95	C₄⋊Dic₄₃	—	4	1	no	—
96	C₄×Dic₄₃	—	2	1	no	—
97	C₄×Dic₄₃	—	2	1	no	—
98	C₂×C₄₃⋊C₈	—	2	1	no	—
99	C₂×C₄₃⋊C₈	—	2	1	no	—
100	C₂×C₄₃⋊C₈	D₈	1	2	yes	—
101	D₃₄₄	—	2	2	no	—
102	D₃₄₄	—	2	2	no	—
103	D₃₄₄	—	4	1	no	—
104	D₃₄₄	—	4	1	no	—
105	C₁₆×C₄₃	—	2	2	no	—
106	C₄₃⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
107	C₂³×C₄₃	—	4	2	no	—
108	C₂³×C₄₃	—	8	1	no	—
109	C₂³×C₄₃	—	8	1	no	—
110	C₂³×C₄₃	—	8	2	no	—
111	C₂³×C₄₃	—	8	2	no	—
112	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
113	C₂²×D₄₃	—	4	2	no	—
114	C₂²×D₄₃	—	8	1	no	—
115	C₂²×D₄₃	—	8	1	no	—
116	C₂²×D₄₃	—	8	2	no	—
117	C₂²×D₄₃	—	8	2	no	—
118	D₄×C₄₃	—	4	2	no	—
119	D₄×C₄₃	—	4	2	no	—
120	D₄×C₄₃	—	8	1	no	—
121	D₄×C₄₃	—	8	1	no	—
122	C₂×C₄×C₄₃	—	8	1	no	—
123	C₂×C₄×C₄₃	—	8	2	no	—
124	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
130	C₂×Dic₄₃	—	2	2	no	—
131	C₂×Dic₄₃	—	2	2	no	—
132	C₂×Dic₄₃	—	8	1	no	—
133	C₂×Dic₄₃	—	8	2	no	—
134	C₂×Dic₄₃	D₁₆	1	4	yes	complemented
135	D₁₇₂	—	4	2	no	—
136	D₁₇₂	—	4	2	no	—
137	D₁₇₂	—	8	1	no	—
138	D₁₇₂	—	8	1	no	—
139	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
141	C₈×C₄₃	#121	1	1	yes	characteristic
142	C₈×C₄₃	#124	1	1	yes	characteristic
143	C₄₃⋊C₈	—	2	2	no	—
144	#13235	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	—	8	4	no	—
146	C₂²×C₄₃	—	8	2	no	—
147	C₂²×C₄₃	—	16	2	no	—
148	C₂²×C₄₃	—	16	1	no	—
149	C₂²×C₄₃	—	16	1	no	—
150	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
151	C₂²×C₄₃	#466	1	1	yes	characteristic
152	C₂²×C₄₃	#493	1	1	yes	characteristic
153	C₂²×C₄₃	C₂×D₁₆	1	2	yes	—
154	C₂²×C₄₃	#508	1	2	yes	—
155	C₂×D₄₃	—	8	4	no	—
156	C₂×D₄₃	—	8	2	no	—
157	C₂×D₄₃	—	16	2	no	—
158	C₂×D₄₃	—	16	1	no	—
159	C₂×D₄₃	—	16	1	no	—
160	C₄×C₄₃	—	16	1	no	—
161	C₄×C₄₃	#390	1	1	yes	characteristic
162	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
163	C₄×C₄₃	#466	1	1	yes	characteristic
164	C₄×C₄₃	#495	1	1	yes	characteristic
165	Dic₄₃	—	2	4	no	—
166	Dic₄₃	—	16	1	no	—
167	C₂²×D₁₆	—	43	1	no	—
168	C₂×#366	—	43	1	no	—
169	C₂×#357	—	43	1	no	—
170	C₂×#357	D₄₃	1	1	yes	characteristic, complemented
171	C₂×C₄×C₁₆	—	43	1	no	—
172	#1970	—	43	1	no	—
173	#1970	—	43	1	no	—
174	C₂×C₄₃	—	16	2	no	—
175	C₂×C₄₃	#1970	1	1	yes	characteristic
176	C₂×C₄₃	C₄×D₁₆	1	2	yes	—
177	C₂×C₄₃	#2215	1	2	yes	—
178	C₂×C₄₃	#2246	1	1	yes	characteristic
179	C₂×C₄₃	#2249	1	1	yes	characteristic
180	D₄₃	—	16	2	no	—
181	C₂²×D₈	—	43	1	no	—
182	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
183	C₂×D₁₆	—	43	4	no	—
184	C₂×D₁₆	—	86	2	no	—
185	C₂×D₁₆	—	86	2	no	—
186	C₂²×C₁₆	—	43	1	no	—
187	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
188	C₂×#107	—	43	1	no	—
189	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
190	C₂×#102	—	86	1	no	—
191	C₂×#102	—	86	1	no	—
192	C₂×C₄×C₈	—	43	1	no	—
193	#390	—	86	1	no	—
194	#390	—	86	1	no	—
195	#366	—	86	2	no	—
196	#357	—	2	2	no	—
197	#357	—	86	2	no	—
198	C₄×C₁₆	—	86	2	no	—
199	C₄₃	#13235	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	—	86	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	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	C₂×D₈	C₄×D₄₃	1	2	yes	—
214	C₂²×C₈	—	43	1	no	—
215	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	86	1	no	—
218	C₂×C₂²⋊C₄	—	172	2	no	—
219	C₂×C₂²⋊C₄	—	172	2	no	—
220	C₂×C₄²	—	43	1	no	—
221	D₁₆	—	86	8	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	43	2	no	—
225	C₂×C₁₆	—	43	2	no	—
226	C₂×C₁₆	—	43	2	no	—
227	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	—	86	1	no	—
231	#107	—	86	1	no	—
232	#102	—	172	2	no	—
233	#102	—	172	2	no	—
234	C₄×C₈	—	86	1	no	—
235	C₄×C₈	—	86	1	no	—
236	C₂⁴	—	4	1	no	—
237	C₂⁴	—	172	1	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	86	2	no	—
245	C₂×D₄	—	172	1	no	—
246	C₂×D₄	—	172	1	no	—
247	C₂×D₄	—	172	1	no	—
248	C₂×D₄	—	172	1	no	—
249	C₂×D₄	—	172	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	43	2	no	—
252	C₂²×C₄	—	172	1	no	—
253	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
254	D₈	—	2	2	no	—
255	D₈	—	2	2	no	—
256	D₈	—	4	1	no	—
257	D₈	—	4	1	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	—	86	1	no	—
264	C₂×C₈	—	86	1	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	—	344	4	no	—
276	C₂²⋊C₄	—	344	4	no	—
277	C₄²	—	86	1	no	—
278	C₄²	—	86	1	no	—
279	C₁₆	—	2	2	no	—
280	C₁₆	—	43	4	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	8	2	no	—
283	C₂³	—	8	2	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	344	2	no	—
287	C₂³	—	344	2	no	—
288	C₂³	—	344	1	no	—
289	C₂³	—	344	1	no	—
290	C₂³	—	172	2	no	—
291	C₂³	D₈×D₄₃	1	1	yes	characteristic
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	8	1	no	—
295	D₄	—	8	1	no	—
296	D₄	—	344	1	no	—
297	D₄	—	344	1	no	—
298	D₄	—	172	2	no	—
299	D₄	—	172	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	43	4	no	—
303	C₂×C₄	—	86	2	no	—
304	C₂×C₄	—	86	2	no	—
305	C₂×C₄	—	344	2	no	—
306	C₂×C₄	—	344	1	no	—
307	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
308	C₂×C₄	#13063298	1	1	yes	characteristic
309	C₂×C₄	#13063321	1	1	yes	characteristic
310	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
311	C₂×C₄	#13063346	1	1	yes	characteristic
312	C₂×C₄	#13063346	1	1	yes	characteristic
313	C₈	—	86	2	no	—
314	C₈	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
315	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
316	C₈	#13063297	1	1	yes	characteristic
317	C₈	#13063311	1	1	yes	characteristic
318	C₂²	—	8	4	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	16	1	no	—
321	C₂²	—	16	1	no	—
322	C₂²	—	16	2	no	—
323	C₂²	—	344	4	no	—
324	C₂²	—	344	2	no	—
325	C₂²	—	688	1	no	—
326	C₂²	—	688	1	no	—
327	C₂²	—	688	2	no	—
328	C₂²	#14582979	1	1	yes	characteristic
329	C₂²	#14583134	1	1	yes	characteristic
330	C₂²	#14583274	1	1	yes	characteristic
331	C₂²	D₁₆×D₄₃	1	2	yes	—
332	C₂²	#14583330	1	2	yes	—
333	C₄	—	16	1	no	—
334	C₄	—	86	4	no	—
335	C₄	—	688	1	no	—
336	C₄	#14582799	1	1	yes	characteristic
337	C₄	#14582979	1	1	yes	characteristic
338	C₄	#14583134	1	1	yes	characteristic
339	C₄	#14583280	1	1	yes	characteristic
340	C₂	—	16	2	no	—
341	C₂	—	688	2	no	—
342	C₂	#19135458	1	1	yes	characteristic
343	C₂	#19136439	1	2	yes	—
344	C₂	#19136680	1	2	yes	—
345	C₂	#19136843	1	1	yes	characteristic
346	C₂	#19136855	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×#357	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×#13063266	C₂
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#357	D₄₃
hypocenter	C₄₃	#13235
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
2. C₂×#14582724	C₂	semidirect
3. C₂×#14582701	C₂	semidirect
4. C₂×C₁₆×Dic₄₃	C₂	semidirect
5. #19135462	C₂	semidirect
6. #19135459	C₂	semidirect
7. C₂×C₄₃×#357	C₂	semidirect
11. C₂×#13063266	C₄	semidirect
106. C₄₃⋊C₁₆	C₂²⋊C₄	semidirect
134. C₂×Dic₄₃	D₁₆	semidirect
170. C₂×#357	D₄₃	semidirect
199. C₄₃	#13235	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063266	2
C₂×#14582724	2
C₂×#14582701	2
C₂×C₁₆×Dic₄₃	2
#19135462	2
#19135459	2
C₂×C₄₃×#357	2
#13235	43