Subgroups — #24989446

Order 11008.

478 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582950	C₂	1	1	yes	maximal, characteristic
2	#19134572	C₂	1	2	yes	maximal
3	C₂×#14582652	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132429	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
7	#14582950	—	2	2	no	—
8	#14582950	—	2	2	no	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×Dic₄₃⋊C₈	C₄	1	1	yes	characteristic
11	C₂×Dic₄₃⋊C₈	C₄	1	1	yes	characteristic
12	C₂×(C₂×D₄₃)⋊C₈	—	2	2	no	—
13	C₂×(C₂×D₄₃)⋊C₈	C₄	1	1	yes	characteristic
14	C₂×(C₂×D₄₃)⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
16	#14582653	—	2	2	no	—
17	#14582652	—	2	2	no	—
18	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#335	—	2	2	no	—
21	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×D₄₃	Q₈	1	1	yes	characteristic
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
29	C₂×C₄×Dic₄₃	Q₈	1	1	yes	characteristic
30	C₄²×D₄₃	—	2	1	no	—
31	C₄²×D₄₃	—	2	1	no	—
32	C₄²×D₄₃	—	2	1	no	—
33	C₄²×D₄₃	—	2	1	no	—
34	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
35	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
36	Dic₄₃⋊C₈	—	2	2	no	—
37	Dic₄₃⋊C₈	—	2	2	no	—
38	(C₂×D₄₃)⋊C₈	—	2	2	no	—
39	(C₂×D₄₃)⋊C₈	—	2	2	no	—
40	(C₂×D₄₃)⋊C₈	—	4	4	no	—
41	C₄×C₄₃⋊C₈	—	2	1	no	—
42	C₄×C₄₃⋊C₈	—	2	1	no	—
43	C₂²×C₈×C₄₃	—	2	2	no	—
44	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₄₃	—	2	2	no	—
48	C₄⋊C₈×C₄₃	—	2	2	no	—
49	C₄×C₈×C₄₃	D₄	1	2	yes	—
50	C₄×C₈×C₄₃	D₄	1	2	yes	—
51	C₂³×D₄₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×D₄₃	—	2	1	no	—
59	C₂×C₄×D₄₃	—	2	1	no	—
60	C₂×C₄×D₄₃	—	2	1	no	—
61	C₂×C₄×D₄₃	—	2	1	no	—
62	C₂×C₄×D₄₃	—	2	1	no	—
63	C₂×C₄×D₄₃	—	2	1	no	—
64	C₂×C₄×D₄₃	—	2	1	no	—
65	C₂×C₄×D₄₃	—	2	1	no	—
66	C₂×C₄×D₄₃	—	2	2	no	—
67	C₂×C₄×D₄₃	—	2	2	no	—
68	C₂×C₄×D₄₃	—	4	2	no	—
69	C₂×C₄×D₄₃	D₈	1	2	yes	—
70	C₂×C₄×D₄₃	Q₁₆	1	2	yes	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	—	4	2	no	—
74	C₂×C₈×C₄₃	—	4	2	no	—
75	C₂×C₈×C₄₃	—	4	2	no	—
76	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₃	#41	1	1	yes	characteristic
81	C₄²×C₄₃	#41	1	1	yes	characteristic
82	C₄×Dic₄₃	—	2	1	no	—
83	C₄×Dic₄₃	—	2	1	no	—
84	C₄×Dic₄₃	—	2	1	no	—
85	C₄×Dic₄₃	—	2	1	no	—
86	C₄×Dic₄₃	—	2	1	no	—
87	C₄×Dic₄₃	—	2	1	no	—
88	C₂×C₄₃⋊C₈	—	2	1	no	—
89	C₂×C₄₃⋊C₈	—	2	1	no	—
90	C₂×C₄₃⋊C₈	—	2	2	no	—
91	C₂×C₄₃⋊C₈	—	2	2	no	—
92	C₂×C₄₃⋊C₈	—	4	2	no	—
93	C₂×C₄₃⋊C₈	—	4	2	no	—
94	C₂×C₄₃⋊C₈	—	4	2	no	—
95	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×D₄₃	—	2	1	no	—
97	C₂²×D₄₃	—	2	2	no	—
98	C₂²×D₄₃	—	2	1	no	—
99	C₂²×D₄₃	—	2	2	no	—
100	C₂²×D₄₃	#107	1	2	yes	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#131	1	2	yes	—
109	C₂×C₄×C₄₃	#131	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
114	C₂×Dic₄₃	—	2	1	no	—
115	C₂×Dic₄₃	—	2	1	no	—
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	—	2	2	no	—
119	C₂×Dic₄₃	—	2	2	no	—
120	C₂×Dic₄₃	#107	1	2	yes	—
121	C₄×D₄₃	—	2	2	no	—
122	C₄×D₄₃	—	2	2	no	—
123	C₄×D₄₃	—	2	2	no	—
124	C₄×D₄₃	—	2	2	no	—
125	C₄×D₄₃	—	4	2	no	—
126	C₄×D₄₃	—	4	2	no	—
127	C₄×D₄₃	—	4	1	no	—
128	C₄×D₄₃	—	4	1	no	—
129	C₄×D₄₃	—	4	2	no	—
130	C₈×C₄₃	—	2	2	no	—
131	C₈×C₄₃	—	2	2	no	—
132	C₈×C₄₃	—	8	2	no	—
133	C₄₃⋊C₈	—	4	1	no	—
134	C₄₃⋊C₈	—	4	1	no	—
135	C₄₃⋊C₈	—	8	2	no	—
136	#8599	—	43	1	no	maximal, Sylow
137	C₂²×C₄₃	#380	1	1	yes	characteristic
138	C₂²×C₄₃	#424	1	2	yes	—
139	C₂²×C₄₃	C₂×#107	1	1	yes	characteristic
140	C₂²×C₄₃	#427	1	1	yes	characteristic
141	C₂²×C₄₃	#430	1	2	yes	—
142	C₂×D₄₃	—	2	2	no	—
143	C₂×D₄₃	—	2	4	no	—
144	C₂×D₄₃	—	4	1	no	—
145	C₂×D₄₃	—	4	1	no	—
146	C₂×D₄₃	—	4	2	no	—
147	C₄×C₄₃	—	2	1	no	—
148	C₄×C₄₃	—	2	1	no	—
149	C₄×C₄₃	#408	1	1	yes	characteristic
150	C₄×C₄₃	#408	1	1	yes	characteristic
151	C₄×C₄₃	#433	1	2	yes	—
152	C₄×C₄₃	#466	1	1	yes	characteristic
153	C₄×C₄₃	#466	1	1	yes	characteristic
154	C₄×C₄₃	#467	1	1	yes	characteristic
155	C₄×C₄₃	#467	1	1	yes	characteristic
156	Dic₄₃	—	2	2	no	—
157	Dic₄₃	—	4	1	no	—
158	Dic₄₃	—	4	1	no	—
159	C₂×#431	—	43	1	no	—
160	#1797	—	43	2	no	—
161	C₂×#335	—	43	1	no	—
162	C₂×#335	D₄₃	1	1	yes	characteristic, complemented
163	#1277	—	43	2	no	—
164	C₂×C₄₃	#1571	1	2	yes	—
165	C₂×C₄₃	#1572	1	2	yes	—
166	C₂×C₄₃	#1798	1	1	yes	characteristic
167	C₂×C₄₃	#1808	1	1	yes	characteristic
168	C₂×C₄₃	#1811	1	1	yes	characteristic
169	D₄₃	—	4	2	no	—
170	C₂²×C₄²	—	43	1	no	—
171	#431	—	86	2	no	—
172	#431	—	86	2	no	—
173	C₂×C₄⋊C₈	—	43	1	no	—
174	C₂×C₄⋊C₈	—	43	1	no	—
175	C₂×C₄⋊C₈	—	43	2	no	—
176	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
177	C₂×C₂²⋊C₈	—	43	1	no	—
178	C₂×C₂²⋊C₈	—	43	1	no	—
179	C₂×C₂²⋊C₈	—	86	2	no	—
180	C₂×C₄×C₈	—	43	1	no	—
181	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
182	#336	—	86	2	no	—
183	#335	—	2	2	no	—
184	#335	—	86	2	no	—
185	C₄₃	#8599	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₄	—	43	1	no	—
187	C₂³×C₄	—	43	1	no	—
188	C₂³×C₄	—	43	1	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	43	1	no	—
191	C₂²×C₈	—	43	1	no	—
192	C₂²×C₈	—	86	2	no	—
193	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
194	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
195	C₂×C₄²	—	43	1	no	—
196	C₂×C₄²	—	43	1	no	—
197	C₂×C₄²	—	43	1	no	—
198	C₂×C₄²	—	86	1	no	—
199	C₂×C₄²	—	86	1	no	—
200	C₂×C₄²	—	86	1	no	—
201	C₂×C₄²	—	86	1	no	—
202	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	86	2	no	—
206	C₄⋊C₈	—	86	2	no	—
207	C₄⋊C₈	—	86	2	no	—
208	C₄⋊C₈	—	86	2	no	—
209	C₂²⋊C₈	—	86	2	no	—
210	C₂²⋊C₈	—	86	2	no	—
211	C₂²⋊C₈	—	172	4	no	—
212	C₄×C₈	—	86	1	no	—
213	C₄×C₈	—	86	1	no	—
214	C₄×C₈	D₁₇₂	1	2	yes	—
215	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
216	C₂⁴	—	43	1	no	—
217	C₂²×C₄	—	43	1	no	—
218	C₂²×C₄	—	43	2	no	—
219	C₂²×C₄	—	43	2	no	—
220	C₂²×C₄	—	43	1	no	—
221	C₂²×C₄	—	43	1	no	—
222	C₂²×C₄	—	86	1	no	—
223	C₂²×C₄	—	86	1	no	—
224	C₂²×C₄	—	86	1	no	—
225	C₂²×C₄	—	86	1	no	—
226	C₂²×C₄	—	86	1	no	—
227	C₂²×C₄	—	86	1	no	—
228	C₂²×C₄	—	86	1	no	—
229	C₂²×C₄	—	86	1	no	—
230	C₂²×C₄	—	86	2	no	—
231	C₂²×C₄	—	86	2	no	—
232	C₂²×C₄	—	172	2	no	—
233	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
234	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
235	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	86	1	no	—
242	C₂×C₈	—	86	1	no	—
243	C₂×C₈	—	86	2	no	—
244	C₂×C₈	—	86	2	no	—
245	C₂×C₈	—	172	2	no	—
246	C₂×C₈	—	172	2	no	—
247	C₂×C₈	—	172	2	no	—
248	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
249	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
250	C₄²	—	86	1	no	—
251	C₄²	—	86	1	no	—
252	C₄²	—	86	1	no	—
253	C₄²	—	86	1	no	—
254	C₄²	—	86	1	no	—
255	C₄²	—	86	1	no	—
256	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
257	C₄²	#10621900	1	1	yes	characteristic
258	C₄²	#10621902	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
260	C₂³	—	43	2	no	—
261	C₂³	—	86	2	no	—
262	C₂³	—	86	1	no	—
263	C₂³	—	86	2	no	—
264	C₂³	—	86	1	no	—
265	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	43	2	no	—
268	C₂×C₄	—	86	1	no	—
269	C₂×C₄	—	86	1	no	—
270	C₂×C₄	—	86	2	no	—
271	C₂×C₄	—	86	2	no	—
272	C₂×C₄	—	86	2	no	—
273	C₂×C₄	—	86	2	no	—
274	C₂×C₄	—	86	2	no	—
275	C₂×C₄	—	86	2	no	—
276	C₂×C₄	—	86	1	no	—
277	C₂×C₄	—	86	1	no	—
278	C₂×C₄	—	172	2	no	—
279	C₂×C₄	—	172	2	no	—
280	C₂×C₄	—	172	1	no	—
281	C₂×C₄	—	172	1	no	—
282	C₂×C₄	—	172	2	no	—
283	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#13063283	1	1	yes	characteristic
285	C₂×C₄	#13063296	1	1	yes	characteristic
286	C₂×C₄	#13063300	1	1	yes	characteristic
287	C₂×C₄	#13063301	1	1	yes	characteristic
288	C₂×C₄	#13063305	1	1	yes	characteristic
289	C₂×C₄	#13063331	1	2	yes	—
290	C₂×C₄	#13063333	1	2	yes	—
291	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
292	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
293	C₂×C₄	#13063344	1	1	yes	characteristic
294	C₂×C₄	#13063348	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	8	2	no	—
298	C₈	—	344	2	no	—
299	C₈	—	172	1	no	—
300	C₈	—	172	1	no	—
301	C₂²	—	86	2	no	—
302	C₂²	—	86	4	no	—
303	C₂²	—	172	1	no	—
304	C₂²	—	172	1	no	—
305	C₂²	—	172	2	no	—
306	C₂²	#14582756	1	1	yes	characteristic
307	C₂²	#14582924	1	2	yes	—
308	C₂²	#107×D₄₃	1	1	yes	characteristic
309	C₂²	#14582936	1	1	yes	characteristic
310	C₂²	#14582946	1	2	yes	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	86	2	no	—
314	C₄	—	172	1	no	—
315	C₄	—	172	1	no	—
316	C₄	#14582865	1	1	yes	characteristic
317	C₄	#14582867	1	1	yes	characteristic
318	C₄	#14582959	1	2	yes	—
319	C₄	#14583132	1	1	yes	characteristic
320	C₄	#14583136	1	1	yes	characteristic
321	C₄	#14583137	1	1	yes	characteristic
322	C₄	#14583141	1	1	yes	characteristic
323	C₂	—	172	2	no	—
324	C₂	#19133368	1	2	yes	—
325	C₂	#19133374	1	2	yes	—
326	C₂	#19134576	1	1	yes	characteristic
327	C₂	#19134621	1	1	yes	characteristic
328	C₂	#19134634	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#335	C₂
socle	C₂³×C₄₃	C₂×C₄⋊C₄
prime radical	C₂³×D₄₃	C₄⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#335	D₄₃
hypocenter	C₄₃	#8599
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. C₂×#14582652	C₂	semidirect
4. #19132429	C₂	semidirect
5. C₂×C₄₃×#335	C₂	semidirect
162. C₂×#335	D₄₃	semidirect
185. C₄₃	#8599	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582950	2
#19134572	2
C₂×#14582652	2
#19132429	2
C₂×C₄₃×#335	2
#8599	43