Subgroups — #24983149

Order 11008.

622 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582654	C₂	1	1	yes	maximal, characteristic
2	#19132825	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₃×#336	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₃⋊C₈	—	2	1	no	—
5	C₂²×C₄⋊Dic₄₃	C₄	1	2	yes	—
6	C₂×C₄₃⋊#95	—	2	1	no	—
7	C₂×C₄₃⋊#95	—	2	1	no	—
8	#14582654	—	2	1	no	—
9	#14582654	C₄	1	2	yes	complemented
10	C₂³×C₈×C₄₃	—	2	1	no	—
11	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#336	—	2	1	no	—
13	C₄₃×#336	C₄	1	2	yes	complemented
14	C₂³×Dic₄₃	—	2	2	no	—
15	C₂²×C₄₃⋊C₈	—	2	4	no	—
16	C₂²×C₄₃⋊C₈	—	2	2	no	—
17	C₂²×C₄₃⋊C₈	—	4	2	no	—
18	C₂²×C₄₃⋊C₈	—	4	1	no	—
19	C₂²×C₄₃⋊C₈	—	4	1	no	—
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	C₂×C₄⋊Dic₄₃	—	4	2	no	—
22	C₂×C₄⋊Dic₄₃	—	4	2	no	—
23	C₂×C₄⋊Dic₄₃	C₈	1	4	yes	complemented
24	C₄₃⋊#95	—	4	1	no	—
25	C₄₃⋊#95	—	4	1	no	—
26	C₄₃⋊#95	—	4	1	no	—
27	C₄₃⋊#95	—	4	1	no	—
28	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	—	2	4	no	—
32	C₂²×C₈×C₄₃	—	2	2	no	—
33	C₂²×C₈×C₄₃	—	4	2	no	—
34	C₂²×C₈×C₄₃	—	4	1	no	—
35	C₂²×C₈×C₄₃	—	4	1	no	—
36	C₂×C₄²×C₄₃	—	4	2	no	—
37	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	Q₈	1	1	yes	characteristic
41	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	—	2	2	no	—
43	C₂²×C₄×C₄₃	—	2	2	no	—
44	C₂²×C₄×C₄₃	—	2	2	no	—
45	C₂²×C₄×C₄₃	—	2	1	no	—
46	C₂²×C₄×C₄₃	—	2	1	no	—
47	C₂²×C₄×C₄₃	—	2	2	no	—
48	C₂²×C₄×C₄₃	—	2	2	no	—
49	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₄₃	M₁₆	1	2	yes	—
57	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
65	C₂²×Dic₄₃	—	2	4	no	—
66	C₂²×Dic₄₃	—	4	2	no	—
67	C₂²×Dic₄₃	—	4	2	no	—
68	C₂²×Dic₄₃	—	4	2	no	—
69	C₂²×Dic₄₃	—	4	2	no	—
70	C₂²×Dic₄₃	—	4	2	no	—
71	C₂²×Dic₄₃	—	4	2	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	—	4	4	no	—
74	C₂×C₈×C₄₃	—	4	4	no	—
75	C₂×C₈×C₄₃	—	4	2	no	—
76	C₂×C₈×C₄₃	—	4	2	no	—
77	C₄²×C₄₃	—	4	1	no	—
78	C₄²×C₄₃	—	4	1	no	—
79	C₄²×C₄₃	—	4	1	no	—
80	C₄²×C₄₃	—	4	1	no	—
81	C₄⋊Dic₄₃	—	4	4	no	—
82	C₄⋊Dic₄₃	—	4	4	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	4	4	no	—
85	C₂×C₄₃⋊C₈	—	4	4	no	—
86	C₂×C₄₃⋊C₈	—	4	2	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂³×C₄₃	#95	1	1	yes	characteristic
89	C₂³×C₄₃	C₂²⋊C₈	1	2	yes	—
90	C₂³×C₄₃	#99	1	2	yes	—
91	C₂³×C₄₃	#100	1	1	yes	characteristic
92	C₂³×C₄₃	#101	1	1	yes	characteristic
93	C₂³×C₄₃	#102	1	1	yes	characteristic
94	C₂³×C₄₃	#102	1	1	yes	characteristic
95	C₂³×C₄₃	#103	1	1	yes	characteristic
96	C₂³×C₄₃	#103	1	1	yes	characteristic
97	C₂³×C₄₃	#104	1	2	yes	—
98	C₂³×C₄₃	#104	1	2	yes	—
99	C₂×C₄×C₄₃	—	2	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	2	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#97	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
130	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
134	C₂×Dic₄₃	—	4	4	no	—
135	C₂×Dic₄₃	—	4	4	no	—
136	C₂×Dic₄₃	—	4	4	no	—
137	C₂×Dic₄₃	—	8	2	no	—
138	C₂×Dic₄₃	—	8	2	no	—
139	C₂×Dic₄₃	—	8	2	no	—
140	C₂×Dic₄₃	—	8	2	no	—
141	C₈×C₄₃	—	4	4	no	—
142	C₄₃⋊C₈	—	4	4	no	—
143	#7025	—	43	1	no	maximal, Sylow
144	C₂²×C₄₃	#323	1	2	yes	—
145	C₂²×C₄₃	#324	1	2	yes	—
146	C₂²×C₄₃	D₄⋊C₈	1	2	yes	—
147	C₂²×C₄₃	D₄⋊C₈	1	2	yes	—
148	C₂²×C₄₃	Q₈⋊C₈	1	2	yes	—
149	C₂²×C₄₃	Q₈⋊C₈	1	2	yes	—
150	C₂²×C₄₃	#327	1	2	yes	—
151	C₂²×C₄₃	#327	1	2	yes	—
152	C₂²×C₄₃	#328	1	2	yes	—
153	C₂²×C₄₃	#328	1	2	yes	—
154	C₂²×C₄₃	#329	1	2	yes	—
155	C₂²×C₄₃	#330	1	2	yes	—
156	C₂²×C₄₃	#331	1	1	yes	characteristic
157	C₂²×C₄₃	#332	1	1	yes	characteristic
158	C₂²×C₄₃	#332	1	1	yes	characteristic
159	C₂²×C₄₃	#333	1	1	yes	characteristic
160	C₂²×C₄₃	#336	1	1	yes	characteristic
161	C₂²×C₄₃	#339	1	1	yes	characteristic
162	C₂²×C₄₃	#339	1	1	yes	characteristic
163	C₂²×C₄₃	#340	1	1	yes	characteristic
164	C₂²×C₄₃	#340	1	1	yes	characteristic
165	C₂²×C₄₃	#342	1	1	yes	characteristic
166	C₂²×C₄₃	#343	1	1	yes	characteristic
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	2	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	—	2	1	no	—
173	C₄×C₄₃	—	2	1	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	2	1	no	—
177	C₄×C₄₃	—	2	1	no	—
178	Dic₄₃	—	8	4	no	—
179	C₂×#336	—	43	1	no	—
180	C₂×#336	D₄₃	1	1	yes	characteristic
181	C₄⋊#95	—	43	1	no	—
182	C₂×C₄₃	#1270	1	2	yes	—
183	C₂×C₄₃	#1271	1	2	yes	—
184	C₂×C₄₃	#1271	1	2	yes	—
185	C₂×C₄₃	#1272	1	2	yes	—
186	C₂×C₄₃	#1279	1	1	yes	characteristic
187	C₂×C₄₃	#1279	1	1	yes	characteristic
188	C₂×C₄₃	#1280	1	1	yes	characteristic
189	C₂×C₄₃	#1294	1	1	yes	characteristic
190	C₂×C₄₃	#1294	1	1	yes	characteristic
191	C₂×C₄₃	#1295	1	1	yes	characteristic
192	C₂×C₄₃	#1296	1	1	yes	characteristic
193	C₂³×C₈	—	2	1	no	—
194	C₂³×C₈	—	86	1	no	—
195	C₂²×C₄⋊C₄	—	43	2	no	—
196	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
197	C₂×#95	—	86	1	no	—
198	C₂×#95	—	86	1	no	—
199	#336	—	2	1	no	—
200	#336	—	43	2	no	—
201	#336	—	86	1	no	—
202	#336	Dic₄₃	1	2	yes	complemented
203	C₄₃	#7025	1	1	yes	characteristic, complemented, Sylow
204	C₂³×C₄	—	86	2	no	—
205	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
206	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
207	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
208	C₂²×C₈	—	2	4	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	4	2	no	—
211	C₂²×C₈	—	4	1	no	—
212	C₂²×C₈	—	4	1	no	—
213	C₂²×C₈	—	86	4	no	—
214	C₂²×C₈	—	86	2	no	—
215	C₂²×C₈	—	172	2	no	—
216	C₂²×C₈	—	172	1	no	—
217	C₂²×C₈	—	172	1	no	—
218	C₂×C₄⋊C₄	—	43	4	no	—
219	C₂×C₄⋊C₄	—	86	2	no	—
220	C₂×C₄⋊C₄	—	172	2	no	—
221	C₂×C₄⋊C₄	—	172	2	no	—
222	C₂×C₄²	—	4	2	no	—
223	C₂×C₄²	Dic₈₆	1	1	yes	characteristic
224	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
227	#95	—	172	1	no	—
228	#95	—	172	1	no	—
229	#95	—	172	1	no	—
230	#95	—	172	1	no	—
231	C₂⁴	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	86	4	no	—
240	C₂²×C₄	—	172	2	no	—
241	C₂²×C₄	—	172	2	no	—
242	C₂²×C₄	—	172	2	no	—
243	C₂²×C₄	—	172	2	no	—
244	C₂²×C₄	—	172	2	no	—
245	C₂²×C₄	—	172	2	no	—
246	C₂²×C₄	C₈×D₄₃	1	2	yes	—
247	C₂²×C₄	#10621872	1	2	yes	—
248	C₂²×C₄	#10621873	1	1	yes	characteristic
249	C₂²×C₄	#10621873	1	1	yes	characteristic
250	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
251	C₂²×C₄	Dic₁₇₂	1	1	yes	characteristic
252	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
253	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
256	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
257	C₂²×C₄	#10621882	1	1	yes	characteristic
258	C₂²×C₄	#10621883	1	1	yes	characteristic
259	C₂²×C₄	#10621884	1	1	yes	characteristic
260	C₂²×C₄	#10621885	1	1	yes	characteristic
261	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	4	no	—
264	C₂×C₈	—	4	4	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	86	4	no	—
268	C₂×C₈	—	172	4	no	—
269	C₂×C₈	—	172	4	no	—
270	C₂×C₈	—	172	2	no	—
271	C₂×C₈	—	172	2	no	—
272	C₄⋊C₄	—	172	4	no	—
273	C₄⋊C₄	—	172	4	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₂³	C₄₃⋊#95	1	1	yes	characteristic
279	C₂³	(C₂×D₄₃)⋊C₈	1	2	yes	—
280	C₂³	#13063239	1	2	yes	—
281	C₂³	#13063241	1	1	yes	characteristic
282	C₂³	#13063243	1	1	yes	characteristic
283	C₂³	#13063245	1	1	yes	characteristic
284	C₂³	#13063247	1	1	yes	characteristic
285	C₂³	#13063248	1	1	yes	characteristic
286	C₂³	#13063250	1	1	yes	characteristic
287	C₂³	#13063251	1	2	yes	—
288	C₂³	#13063253	1	2	yes	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	344	2	no	—
314	C₂×C₄	—	344	2	no	—
315	C₂×C₄	—	344	2	no	—
316	C₂×C₄	—	344	2	no	—
317	C₂×C₄	—	172	4	no	—
318	C₂×C₄	—	172	4	no	—
319	C₂×C₄	—	172	4	no	—
320	C₂×C₄	C₈×Dic₄₃	1	1	yes	characteristic
321	C₂×C₄	#13063235	1	1	yes	characteristic
322	C₂×C₄	#13063246	1	1	yes	characteristic
323	C₂×C₄	#13063247	1	1	yes	characteristic
324	C₂×C₄	#13063249	1	1	yes	characteristic
325	C₂×C₄	#13063250	1	1	yes	characteristic
326	C₂×C₄	Dic₄₃⋊C₈	1	2	yes	—
327	C₂×C₄	#13063256	1	1	yes	characteristic
328	C₂×C₄	#13063257	1	1	yes	characteristic
329	C₂×C₄	#13063258	1	1	yes	characteristic
330	C₂×C₄	#13063259	1	1	yes	characteristic
331	C₈	—	4	4	no	—
332	C₈	—	172	4	no	—
333	C₂²	#14582622	1	2	yes	—
334	C₂²	#14582624	1	2	yes	—
335	C₂²	#14582626	1	2	yes	—
336	C₂²	#14582628	1	2	yes	—
337	C₂²	#14582629	1	2	yes	—
338	C₂²	#14582631	1	2	yes	—
339	C₂²	#14582632	1	2	yes	—
340	C₂²	#14582634	1	2	yes	—
341	C₂²	#14582635	1	2	yes	—
342	C₂²	#14582637	1	2	yes	—
343	C₂²	#14582638	1	2	yes	—
344	C₂²	#14582640	1	2	yes	—
345	C₂²	#14582642	1	1	yes	characteristic
346	C₂²	#14582644	1	1	yes	characteristic
347	C₂²	#14582646	1	1	yes	characteristic
348	C₂²	#14582647	1	1	yes	characteristic
349	C₂²	#14582653	1	1	yes	characteristic
350	C₂²	#14582659	1	1	yes	characteristic
351	C₂²	#14582660	1	1	yes	characteristic
352	C₂²	#14582661	1	1	yes	characteristic
353	C₂²	#14582662	1	1	yes	characteristic
354	C₂²	#14582665	1	1	yes	characteristic
355	C₂²	#14582667	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	344	4	no	—
368	C₂	#19132415	1	2	yes	—
369	C₂	#19132417	1	2	yes	—
370	C₂	#19132419	1	2	yes	—
371	C₂	#19132420	1	2	yes	—
372	C₂	#19132434	1	1	yes	characteristic
373	C₂	#19132435	1	1	yes	characteristic
374	C₂	#19132437	1	1	yes	characteristic
375	C₂	#19132474	1	1	yes	characteristic
376	C₂	#19132475	1	1	yes	characteristic
377	C₂	#19132477	1	1	yes	characteristic
378	C₂	#19132479	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₄₃
Fitting	C₂×C₄₃×#336	C₂
socle, prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂²×C₄²×C₄₃	C₂²
hypercenter	C₂×#336	D₄₃
hypocenter	C₄₃	#7025
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
9. #14582654	C₄	semidirect
13. C₄₃×#336	C₄	semidirect
23. C₂×C₄⋊Dic₄₃	C₈	semidirect
202. #336	Dic₄₃	semidirect
203. C₄₃	#7025	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582654	2
#19132825	2
C₂×C₄₃×#336	2
#7025	43