Subgroups — #24984614

Order 11008.

568 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134663	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582654	C₂	1	1	yes	maximal, characteristic
3	#19132418	C₂	1	2	yes	maximal, complemented
4	#19132418	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2296	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₃⋊C₈	—	2	1	no	—
7	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
8	#14582654	—	2	1	no	—
9	#14582654	—	2	1	no	—
10	#14582654	C₂²	1	2	yes	—
11	#14582654	C₂²	1	2	yes	—
12	C₂×#117×C₄₃	C₄	1	1	yes	characteristic
13	C₂×#117×C₄₃	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#389	C₄	1	1	yes	characteristic
16	C₄₃×#389	C₄	1	1	yes	characteristic
17	C₄₃×#389	C₂²	1	1	yes	characteristic
18	C₄₃×#389	C₂²	1	1	yes	characteristic
19	C₄₃×#385	—	2	1	no	—
20	C₄₃×#385	—	2	1	no	—
21	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
22	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
23	C₂²×C₄₃⋊C₈	—	2	4	no	—
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	C₂²×C₄₃⋊C₈	—	4	2	no	—
27	C₂²×C₄₃⋊C₈	—	4	2	no	—
28	C₂²⋊(C₄₃⋊C₈)	—	4	2	no	—
29	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
32	#117×C₄₃	—	2	2	no	—
33	#117×C₄₃	—	4	2	no	—
34	#117×C₄₃	C₈	1	4	yes	complemented
35	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
45	C₂×C₄²×C₄₃	—	2	1	no	—
46	C₂×C₄²×C₄₃	—	2	1	no	—
47	C₂×C₄²×C₄₃	—	4	2	no	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
52	#95×C₄₃	—	2	1	no	—
53	#95×C₄₃	—	2	1	no	—
54	#95×C₄₃	—	2	1	no	—
55	#95×C₄₃	—	2	1	no	—
56	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	—	2	1	no	—
59	C₂²×C₄×C₄₃	—	2	1	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	—	2	2	no	—
64	C₂²×C₄×C₄₃	—	2	2	no	—
65	C₂²×C₄×C₄₃	—	2	1	no	—
66	C₂²×C₄×C₄₃	—	2	1	no	—
67	C₂²×C₄×C₄₃	—	2	1	no	—
68	C₂²×C₄×C₄₃	—	2	1	no	—
69	C₂²×C₄×C₄₃	—	4	1	no	—
70	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₄₃	M₁₆	1	2	yes	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	—	2	1	no	—
79	C₄⋊C₄×C₄₃	—	2	1	no	—
80	C₄⋊C₄×C₄₃	—	4	1	no	—
81	C₄⋊C₄×C₄₃	—	4	1	no	—
82	C₄⋊C₄×C₄₃	—	4	2	no	—
83	C₄⋊C₄×C₄₃	—	4	2	no	—
84	C₄⋊C₄×C₄₃	—	4	2	no	—
85	C₄⋊C₄×C₄₃	—	4	1	no	—
86	C₄⋊C₄×C₄₃	—	4	1	no	—
87	C₄⋊C₄×C₄₃	C₂×C₈	1	2	yes	complemented
88	C₄⋊C₄×C₄₃	C₂×C₈	1	2	yes	complemented
89	C₂²⋊C₄×C₄₃	—	2	4	no	—
90	C₂²⋊C₄×C₄₃	—	4	2	no	—
91	C₄²×C₄₃	—	2	4	no	—
92	C₄²×C₄₃	—	4	1	no	—
93	C₄²×C₄₃	—	4	1	no	—
94	C₄²×C₄₃	—	4	1	no	—
95	C₄²×C₄₃	—	4	1	no	—
96	C₄²×C₄₃	—	4	2	no	—
97	C₂×C₄₃⋊C₈	—	2	4	no	—
98	C₂×C₄₃⋊C₈	—	4	8	no	—
99	C₂×C₄₃⋊C₈	—	4	2	no	—
100	C₂×C₄₃⋊C₈	—	4	2	no	—
101	C₂×C₄₃⋊C₈	—	4	2	no	—
102	C₂×C₄₃⋊C₈	—	4	2	no	—
103	C₂×C₄₃⋊C₈	—	4	2	no	—
104	C₂³×C₄₃	—	2	1	no	—
105	C₂³×C₄₃	—	2	1	no	—
106	C₂³×C₄₃	—	2	1	no	—
107	C₂³×C₄₃	—	2	1	no	—
108	C₂³×C₄₃	C₂²⋊C₈	1	2	yes	—
109	C₂³×C₄₃	#99	1	2	yes	—
110	C₂³×C₄₃	#101	1	2	yes	—
111	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	2	4	no	—
115	C₂×C₄×C₄₃	—	2	4	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	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	4	4	no	—
132	C₂×C₄×C₄₃	—	4	1	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	—	4	1	no	—
139	C₂×C₄×C₄₃	—	4	1	no	—
140	C₂×C₄×C₄₃	—	4	1	no	—
141	C₂×C₄×C₄₃	—	4	1	no	—
142	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
143	C₂×C₄×C₄₃	#104	1	2	yes	—
144	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
146	C₂×C₄×C₄₃	#135	1	2	yes	—
147	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
149	C₄₃⋊C₈	—	4	4	no	—
150	C₄₃⋊C₈	—	8	2	no	—
151	#7579	—	43	1	no	maximal, Sylow
152	C₂²×C₄₃	—	2	2	no	—
153	C₂²×C₄₃	—	2	2	no	—
154	C₂²×C₄₃	—	2	2	no	—
155	C₂²×C₄₃	—	2	2	no	—
156	C₂²×C₄₃	—	2	2	no	—
157	C₂²×C₄₃	—	2	2	no	—
158	C₂²×C₄₃	#324	1	4	yes	—
159	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₂²×C₄₃	C₂×#99	1	1	yes	characteristic
161	C₂²×C₄₃	C₂×#101	1	1	yes	characteristic
162	C₂²×C₄₃	#416	1	1	yes	characteristic
163	C₂²×C₄₃	#417	1	1	yes	characteristic
164	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
165	C₂²×C₄₃	#421	1	1	yes	characteristic
166	C₄×C₄₃	—	2	2	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	2	2	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	—	2	2	no	—
173	C₄×C₄₃	—	4	2	no	—
174	C₄×C₄₃	—	4	2	no	—
175	C₄×C₄₃	—	4	1	no	—
176	C₄×C₄₃	—	8	1	no	—
177	C₄×C₄₃	—	8	1	no	—
178	#2296	D₄₃	1	1	yes	characteristic
179	#1817	—	43	1	no	—
180	C₂×#336	—	43	1	no	—
181	#1271	—	43	2	no	—
182	#1271	—	43	2	no	—
183	C₂×C₄₃	—	2	4	no	—
184	C₂×C₄₃	C₂×#324	1	1	yes	characteristic
185	C₂×C₄₃	#1476	1	1	yes	characteristic
186	C₂×C₄₃	#1477	1	1	yes	characteristic
187	C₂×C₄₃	#1504	1	1	yes	characteristic
188	C₂×C₄₃	#1516	1	1	yes	characteristic
189	C₂×C₄₃	#1531	1	1	yes	characteristic
190	C₂×C₄₃	#1552	1	1	yes	characteristic
191	C₂³×C₈	—	86	1	no	—
192	C₂×#117	Dic₄₃	1	1	yes	characteristic
193	C₂×#117	Dic₄₃	1	1	yes	characteristic
194	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
195	C₂×C₂²⋊C₈	—	86	1	no	—
196	#389	Dic₄₃	1	1	yes	characteristic
197	#389	Dic₄₃	1	1	yes	characteristic
198	#389	C₂×D₄₃	1	1	yes	characteristic
199	#389	C₂×D₄₃	1	1	yes	characteristic
200	#385	—	2	1	no	—
201	#385	—	2	1	no	—
202	C₄×C₄⋊C₄	—	2	1	no	—
203	C₄×C₄⋊C₄	—	2	1	no	—
204	#336	—	43	2	no	—
205	#336	—	43	2	no	—
206	#336	—	86	1	no	—
207	#336	—	86	1	no	—
208	C₄₃	#7579	1	1	yes	characteristic, complemented, Sylow
209	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
210	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
211	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
212	C₂²×C₈	—	86	4	no	—
213	C₂²×C₈	—	86	2	no	—
214	C₂²×C₈	—	86	2	no	—
215	C₂²×C₈	—	172	2	no	—
216	C₂²×C₈	—	172	2	no	—
217	#117	—	2	2	no	—
218	#117	—	4	2	no	—
219	#117	C₄₃⋊C₈	1	4	yes	complemented
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
227	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	—	2	1	no	—
229	C₂×C₂²⋊C₄	—	2	1	no	—
230	C₂×C₄²	—	2	1	no	—
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	—	4	2	no	—
233	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
237	C₂²⋊C₈	—	172	2	no	—
238	#95	—	2	1	no	—
239	#95	—	2	1	no	—
240	#95	—	2	1	no	—
241	#95	—	2	1	no	—
242	C₂⁴	C₂²⋊Dic₄₃	1	1	yes	characteristic
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
257	C₂²×C₄	#10621877	1	2	yes	—
258	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
260	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
261	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₂×C₈	—	86	4	no	—
265	C₂×C₈	—	172	2	no	—
266	C₂×C₈	—	172	2	no	—
267	C₂×C₈	—	172	2	no	—
268	C₂×C₈	—	172	8	no	—
269	C₂×C₈	—	172	2	no	—
270	C₂×C₈	—	172	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	complemented
281	C₄⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	complemented
282	C₂²⋊C₄	—	2	4	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	4	1	no	—
287	C₄²	—	4	1	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	4	2	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
295	C₂³	#13063240	1	2	yes	—
296	C₂³	#13063244	1	2	yes	—
297	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	1	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	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
329	C₂×C₄	#13063252	1	2	yes	—
330	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
331	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
332	C₂×C₄	#13063345	1	2	yes	—
333	C₂×C₄	#13063350	1	1	yes	characteristic
334	C₂×C₄	#13063355	1	1	yes	characteristic
335	C₈	—	344	2	no	—
336	C₈	—	172	4	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	#14582625	1	4	yes	—
344	C₂²	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
345	C₂²	C₂×#13063240	1	1	yes	characteristic
346	C₂²	C₂×#13063244	1	1	yes	characteristic
347	C₂²	#14582895	1	1	yes	characteristic
348	C₂²	#14582900	1	1	yes	characteristic
349	C₂²	C₂×#13063252	1	1	yes	characteristic
350	C₂²	#14582916	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	4	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	4	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₂	—	2	4	no	—
364	C₂	C₂×#14582625	1	1	yes	characteristic
365	C₂	#19132977	1	1	yes	characteristic
366	C₂	#19132982	1	1	yes	characteristic
367	C₂	#19133099	1	1	yes	characteristic
368	C₂	#19133151	1	1	yes	characteristic
369	C₂	#19133210	1	1	yes	characteristic
370	C₂	#19133295	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2296	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2296	D₄₃
hypocenter	C₄₃	#7579
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2296

Complements

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

N	H ≅ G/N	type
3. #19132418	C₂	semidirect
4. #19132418	C₂	semidirect
34. #117×C₄₃	C₈	semidirect
87. C₄⋊C₄×C₄₃	C₂×C₈	semidirect
88. C₄⋊C₄×C₄₃	C₂×C₈	semidirect
208. C₄₃	#7579	semidirect
219. #117	C₄₃⋊C₈	semidirect
280. C₄⋊C₄	C₂×C₄₃⋊C₈	semidirect
281. C₄⋊C₄	C₂×C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#19134663	2
C₂×#14582654	2
#19132418	2
C₄₃×#2296	2
#7579	43