Subgroups — #24984634

Order 11008.

522 conjugacy classes of subgroups.

The subgroup lattice has 364 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	C₄×C₂²⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#19132421	C₂	1	2	yes	maximal, complemented
4	#19132418	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
9	#14582654	—	2	1	no	—
10	#14582654	C₂²	1	2	yes	—
11	#14582654	C₂²	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₄₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₄₃	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#391	C₂²	1	1	yes	characteristic
16	C₄₃×#389	C₂²	1	1	yes	characteristic
17	C₄₃×#387	—	2	1	no	—
18	C₄₃×#385	—	2	1	no	—
19	C₄₃×#380	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
21	C₄×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊C₄×C₄₃	—	2	1	no	—
23	C₂²×C₄₃⋊C₈	—	2	2	no	—
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
26	C₂²⋊(C₄₃⋊C₈)	—	4	2	no	—
27	C₄×C₄₃⋊C₈	—	4	2	no	—
28	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
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	C₂²⋊Q₈×C₄₃	—	2	2	no	—
33	C₂²⋊Q₈×C₄₃	C₈	1	4	yes	complemented
34	C₄×Q₈×C₄₃	—	4	2	no	—
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	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
43	C₂×C₄²×C₄₃	—	2	1	no	—
44	C₂×C₄²×C₄₃	—	2	1	no	—
45	C₂×C₄²×C₄₃	—	4	2	no	—
46	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
50	#95×C₄₃	—	2	1	no	—
51	#95×C₄₃	—	2	1	no	—
52	#95×C₄₃	—	2	1	no	—
53	#95×C₄₃	—	2	1	no	—
54	#95×C₄₃	—	2	1	no	—
55	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Q₈×C₄₃	—	2	1	no	—
57	C₂×Q₈×C₄₃	—	4	1	no	—
58	C₂×Q₈×C₄₃	—	4	1	no	—
59	C₂×Q₈×C₄₃	C₂×C₈	1	2	yes	—
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	1	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	—	4	1	no	—
69	C₂²×C₄×C₄₃	—	4	1	no	—
70	C₂²×C₄×C₄₃	—	4	1	no	—
71	C₂²×C₄×C₄₃	—	4	1	no	—
72	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₄₃	M₁₆	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₄₃	—	2	4	no	—
81	C₄⋊C₄×C₄₃	—	2	1	no	—
82	C₄⋊C₄×C₄₃	—	4	1	no	—
83	C₄⋊C₄×C₄₃	—	4	1	no	—
84	C₄⋊C₄×C₄₃	—	4	2	no	—
85	C₄⋊C₄×C₄₃	—	4	2	no	—
86	C₄⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
87	C₂²⋊C₄×C₄₃	—	2	4	no	—
88	C₂²⋊C₄×C₄₃	—	4	2	no	—
89	C₄²×C₄₃	—	2	2	no	—
90	C₄²×C₄₃	—	2	2	no	—
91	C₄²×C₄₃	—	4	1	no	—
92	C₄²×C₄₃	—	4	1	no	—
93	C₄²×C₄₃	—	4	2	no	—
94	C₄²×C₄₃	—	4	2	no	—
95	C₂×C₄₃⋊C₈	—	2	4	no	—
96	C₂×C₄₃⋊C₈	—	4	2	no	—
97	C₂×C₄₃⋊C₈	—	4	2	no	—
98	C₂×C₄₃⋊C₈	—	4	2	no	—
99	C₂×C₄₃⋊C₈	—	4	2	no	—
100	C₂×C₄₃⋊C₈	—	4	2	no	—
101	C₂³×C₄₃	—	2	1	no	—
102	C₂³×C₄₃	—	2	1	no	—
103	C₂³×C₄₃	—	2	1	no	—
104	C₂³×C₄₃	—	2	1	no	—
105	C₂³×C₄₃	—	2	1	no	—
106	C₂³×C₄₃	—	2	1	no	—
107	C₂³×C₄₃	C₂²⋊C₈	1	2	yes	—
108	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	Q₈×C₄₃	—	4	2	no	—
110	Q₈×C₄₃	—	4	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	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	—	4	2	no	—
127	C₂×C₄×C₄₃	—	4	4	no	—
128	C₂×C₄×C₄₃	—	4	4	no	—
129	C₂×C₄×C₄₃	—	4	1	no	—
130	C₂×C₄×C₄₃	—	4	1	no	—
131	C₂×C₄×C₄₃	—	4	1	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	C₂²⋊C₈	1	2	yes	—
140	C₂×C₄×C₄₃	#99	1	2	yes	—
141	C₂×C₄×C₄₃	#104	1	2	yes	—
142	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#135	1	2	yes	—
145	C₂×C₄×C₄₃	#137	1	2	yes	—
146	C₄₃⋊C₈	—	4	4	no	—
147	C₄₃⋊C₈	—	8	2	no	—
148	#7583	—	43	1	no	maximal, Sylow
149	C₂²×C₄₃	—	2	2	no	—
150	C₂²×C₄₃	—	2	2	no	—
151	C₂²×C₄₃	—	2	2	no	—
152	C₂²×C₄₃	—	4	1	no	—
153	C₂²×C₄₃	—	4	1	no	—
154	C₂²×C₄₃	—	4	1	no	—
155	C₂²×C₄₃	—	4	1	no	—
156	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂²×C₄₃	C₂×#99	1	1	yes	characteristic
158	C₂²×C₄₃	#413	1	1	yes	characteristic
159	C₂²×C₄₃	#416	1	1	yes	characteristic
160	C₂²×C₄₃	#419	1	1	yes	characteristic
161	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
162	C₂²×C₄₃	#421	1	1	yes	characteristic
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	2	2	no	—
167	C₄×C₄₃	—	2	2	no	—
168	C₄×C₄₃	—	2	2	no	—
169	C₄×C₄₃	—	2	2	no	—
170	C₄×C₄₃	—	4	2	no	—
171	C₄×C₄₃	—	4	2	no	—
172	C₄×C₄₃	—	8	1	no	—
173	C₄×C₄₃	—	8	1	no	—
174	C₄×C₂²⋊Q₈	D₄₃	1	1	yes	characteristic
175	#1817	—	43	1	no	—
176	C₄×C₂²⋊C₈	—	43	1	no	—
177	#1272	—	43	2	no	—
178	#1271	—	43	2	no	—
179	C₂×C₄₃	—	4	2	no	—
180	C₂×C₄₃	#1459	1	1	yes	characteristic
181	C₂×C₄₃	#1476	1	1	yes	characteristic
182	C₂×C₄₃	#1479	1	1	yes	characteristic
183	C₂×C₄₃	#1508	1	1	yes	characteristic
184	C₂×C₄₃	#1516	1	1	yes	characteristic
185	C₂×C₄₃	#1529	1	1	yes	characteristic
186	C₂×C₄₃	#1554	1	1	yes	characteristic
187	C₂×C₂²⋊Q₈	Dic₄₃	1	1	yes	characteristic
188	C₂×C₄×Q₈	Dic₄₃	1	1	yes	characteristic
189	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
190	C₂×C₂²⋊C₈	—	86	1	no	—
191	C₂×C₂²⋊C₈	—	86	1	no	—
192	C₂×C₄×C₈	—	86	1	no	—
193	#391	C₂×D₄₃	1	1	yes	characteristic
194	#389	C₂×D₄₃	1	1	yes	characteristic
195	#387	—	2	1	no	—
196	#385	—	2	1	no	—
197	#380	Dic₄₃	1	1	yes	characteristic
198	C₄×C₄⋊C₄	—	2	1	no	—
199	C₄×C₄⋊C₄	Dic₄₃	1	1	yes	characteristic
200	C₄×C₂²⋊C₄	—	2	1	no	—
201	#336	—	43	2	no	—
202	#336	—	43	2	no	—
203	#336	—	86	1	no	—
204	C₄₃	#7583	1	1	yes	characteristic, complemented, Sylow
205	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
206	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
207	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
208	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
209	C₂²×C₈	—	86	2	no	—
210	C₂²×C₈	—	86	2	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	C₂²⋊Q₈	C₄₃⋊C₈	1	4	yes	complemented
213	C₄×Q₈	—	4	2	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	—	4	2	no	—
225	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
227	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
229	C₂²⋊C₈	—	86	4	no	—
230	C₂²⋊C₈	—	172	2	no	—
231	C₄×C₈	—	172	2	no	—
232	#95	—	2	1	no	—
233	#95	—	2	1	no	—
234	#95	—	2	1	no	—
235	#95	—	2	1	no	—
236	#95	—	2	1	no	—
237	C₂⁴	C₂²⋊Dic₄₃	1	1	yes	characteristic
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	C₂×C₄₃⋊C₈	1	2	yes	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
255	C₂²×C₄	#10621877	1	2	yes	—
256	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
258	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
262	C₂×C₈	—	86	4	no	—
263	C₂×C₈	—	172	2	no	—
264	C₂×C₈	—	172	2	no	—
265	C₂×C₈	—	172	2	no	—
266	C₂×C₈	—	172	2	no	—
267	C₂×C₈	—	172	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	C₂×C₄₃⋊C₈	1	2	yes	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	4	2	no	—
282	C₄²	—	4	2	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
290	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	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	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
322	C₂×C₄	#13063240	1	2	yes	—
323	C₂×C₄	#13063252	1	2	yes	—
324	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
325	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
326	C₂×C₄	#13063345	1	2	yes	—
327	C₂×C₄	#13063356	1	2	yes	—
328	C₈	—	344	2	no	—
329	C₈	—	172	4	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
338	C₂²	C₂×#13063240	1	1	yes	characteristic
339	C₂²	#14582883	1	1	yes	characteristic
340	C₂²	#14582895	1	1	yes	characteristic
341	C₂²	#14582908	1	1	yes	characteristic
342	C₂²	C₂×#13063252	1	1	yes	characteristic
343	C₂²	#14582916	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	1	no	—
354	C₄	—	8	1	no	—
355	C₂	—	4	2	no	—
356	C₂	#19132913	1	1	yes	characteristic
357	C₂	#19132977	1	1	yes	characteristic
358	C₂	#19132990	1	1	yes	characteristic
359	C₂	#19133115	1	1	yes	characteristic
360	C₂	#19133151	1	1	yes	characteristic
361	C₂	#19133202	1	1	yes	characteristic
362	C₂	#19133305	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×C₂²⋊Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×C₂²⋊Q₈	D₄₃
hypocenter	C₄₃	#7583
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #19132421	C₂	semidirect
4. #19132418	C₂	semidirect
33. C₂²⋊Q₈×C₄₃	C₈	semidirect
204. C₄₃	#7583	semidirect
212. C₂²⋊Q₈	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
C₄×C₂²⋊(C₄₃⋊C₈)	2
#19132421	2
#19132418	2
C₄×C₂²⋊Q₈×C₄₃	2
#7583	43