Subgroups — #24992621

Order 11008.

524 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	#19143159	C₂	1	1	yes	maximal, characteristic
2	#19133267	C₂	1	2	yes	maximal, complemented
3	#19132909	C₂	1	1	yes	maximal, characteristic
4	#19132453	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2621	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	—	2	1	no	—
7	#14582952	C₂²	1	2	yes	—
8	#14582926	—	2	4	no	—
9	#14582923	—	2	1	no	—
10	#14582923	C₂²	1	2	yes	—
11	#14582645	—	2	4	no	—
12	#14582625	—	2	1	no	—
13	#14582623	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₄₃	—	2	1	no	—
16	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#398	—	2	1	no	—
18	C₄₃×#398	C₄	1	2	yes	—
19	C₄₃×#397	—	2	1	no	—
20	C₄₃×#397	C₄	1	2	yes	—
21	C₄₃×#391	—	2	1	no	—
22	C₄₃×#382	C₂²	1	1	yes	characteristic
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	—	4	2	no	—
26	C₂×#10621877	—	4	2	no	—
27	C₂²×C₄₃⋊C₈	—	2	2	no	—
28	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
29	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
32	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
33	C₄×C₄₃⋊C₈	—	2	2	no	—
34	C₂²×Q₈×C₄₃	—	2	1	no	—
35	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₄₃	—	2	2	no	—
37	C₄⋊Q₈×C₄₃	—	2	2	no	—
38	C₂²⋊Q₈×C₄₃	—	4	2	no	—
39	C₂²⋊Q₈×C₄₃	—	4	2	no	—
40	#117×C₄₃	—	2	2	no	—
41	#117×C₄₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	#95×C₄₃	—	2	2	no	—
54	#95×C₄₃	—	2	2	no	—
55	#95×C₄₃	—	2	1	no	—
56	#95×C₄₃	C₂×C₄	1	2	yes	—
57	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Q₈×C₄₃	—	4	1	no	—
59	C₂×Q₈×C₄₃	—	4	1	no	—
60	C₂×Q₈×C₄₃	—	4	1	no	—
61	C₂×Q₈×C₄₃	—	4	1	no	—
62	C₂×Q₈×C₄₃	—	4	1	no	—
63	C₂×Q₈×C₄₃	—	4	1	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	—	2	2	no	—
69	C₂²×C₄×C₄₃	—	2	2	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	D₈	1	2	yes	—
74	C₂²×C₄×C₄₃	SD₁₆	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	2	no	—
79	C₄⋊C₄×C₄₃	—	2	2	no	—
80	C₄⋊C₄×C₄₃	—	2	4	no	—
81	C₄⋊C₄×C₄₃	—	4	4	no	—
82	C₄⋊C₄×C₄₃	—	4	2	no	—
83	C₄⋊C₄×C₄₃	—	4	1	no	—
84	C₄⋊C₄×C₄₃	—	4	1	no	—
85	C₂²⋊C₄×C₄₃	—	2	2	no	—
86	C₂²⋊C₄×C₄₃	—	2	2	no	—
87	C₂²⋊C₄×C₄₃	—	4	4	no	—
88	C₄²×C₄₃	—	2	1	no	—
89	C₄²×C₄₃	—	2	1	no	—
90	C₄²×C₄₃	C₂×D₄	1	2	yes	—
91	C₄²×C₄₃	#41	1	2	yes	—
92	#10621877	—	4	4	no	—
93	#10621877	—	4	2	no	—
94	#10621877	—	4	2	no	—
95	C₂×C₄₃⋊C₈	—	2	4	no	—
96	C₂×C₄₃⋊C₈	—	4	4	no	—
97	C₂×C₄₃⋊C₈	—	4	2	no	—
98	C₂³×C₄₃	—	2	1	no	—
99	C₂³×C₄₃	—	2	1	no	—
100	C₂³×C₄₃	—	4	1	no	—
101	C₂³×C₄₃	—	4	1	no	—
102	C₂³×C₄₃	#102	1	2	yes	—
103	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₄₃	—	8	1	no	—
105	Q₈×C₄₃	—	8	1	no	—
106	Q₈×C₄₃	—	8	1	no	—
107	Q₈×C₄₃	—	8	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	4	1	no	—
117	C₂×C₄×C₄₃	—	4	1	no	—
118	C₂×C₄×C₄₃	—	4	1	no	—
119	C₂×C₄×C₄₃	—	4	1	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	#102	1	2	yes	—
127	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#121	1	2	yes	—
130	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
135	C₄₃⋊C₈	—	4	4	no	—
136	C₄₃⋊C₈	—	8	2	no	—
137	#9195	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	—	2	1	no	—
139	C₂²×C₄₃	—	2	1	no	—
140	C₂²×C₄₃	—	2	2	no	—
141	C₂²×C₄₃	—	4	2	no	—
142	C₂²×C₄₃	—	4	1	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	—	8	1	no	—
145	C₂²×C₄₃	#386	1	1	yes	characteristic
146	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
147	C₂²×C₄₃	#417	1	1	yes	characteristic
148	C₄×C₄₃	—	2	2	no	—
149	C₄×C₄₃	—	2	2	no	—
150	C₄×C₄₃	—	2	2	no	—
151	C₄×C₄₃	—	4	1	no	—
152	C₄×C₄₃	—	4	1	no	—
153	C₄×C₄₃	—	8	2	no	—
154	C₄×C₄₃	—	8	2	no	—
155	C₄×C₄₃	#447	1	1	yes	characteristic
156	C₄×C₄₃	#448	1	1	yes	characteristic
157	C₄×C₄₃	#480	1	1	yes	characteristic
158	C₄×C₄₃	#481	1	1	yes	characteristic
159	#2905	—	43	1	no	—
160	#2621	D₄₃	1	1	yes	characteristic
161	#1546	—	43	2	no	—
162	#1458	—	43	1	no	—
163	#1286	—	43	2	no	—
164	C₂×C₄₃	—	2	1	no	—
165	C₂×C₄₃	—	2	1	no	—
166	C₂×C₄₃	—	4	2	no	—
167	C₂×C₄₃	#1874	1	1	yes	characteristic
168	C₂×C₄₃	#1909	1	1	yes	characteristic
169	C₂×C₄₃	#1915	1	1	yes	characteristic
170	C₂²×M₁₆	—	86	1	no	—
171	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
172	C₂×C₂²⋊Q₈	—	2	1	no	—
173	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
174	#431	—	43	2	no	—
175	#424	—	86	4	no	—
176	#423	—	43	2	no	—
177	#423	—	86	1	no	—
178	#398	—	2	1	no	—
179	#398	Dic₄₃	1	2	yes	—
180	#397	—	2	1	no	—
181	#397	Dic₄₃	1	2	yes	—
182	#391	—	2	1	no	—
183	#382	C₂×D₄₃	1	1	yes	characteristic
184	#332	—	86	4	no	—
185	#324	—	86	1	no	—
186	#323	—	86	1	no	—
187	C₄₃	#9195	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	2	1	no	—
189	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
190	C₂×M₁₆	—	86	2	no	—
191	C₂×M₁₆	—	86	2	no	—
192	C₂×M₁₆	—	172	2	no	—
193	C₂×M₁₆	—	172	2	no	—
194	C₂²×C₈	—	86	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₂²⋊Q₈	—	4	2	no	—
198	C₂²⋊Q₈	—	4	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
205	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
212	C₄⋊C₈	—	43	4	no	—
213	C₄⋊C₈	—	86	4	no	—
214	C₄⋊C₈	—	86	2	no	—
215	C₄⋊C₈	—	86	2	no	—
216	C₂²⋊C₈	—	86	2	no	—
217	C₄×C₈	—	86	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	1	no	—
221	#95	C₂×Dic₄₃	1	2	yes	—
222	C₂⁴	C₂²⋊Dic₄₃	1	1	yes	characteristic
223	C₂×Q₈	—	4	1	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	#10621882	1	2	yes	—
236	C₂²×C₄	#10621883	1	2	yes	—
237	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
243	M₁₆	—	172	4	no	—
244	M₁₆	—	172	2	no	—
245	M₁₆	—	172	2	no	—
246	C₂×C₈	—	86	4	no	—
247	C₂×C₈	—	172	2	no	—
248	C₂×C₈	—	172	4	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	D₄×D₄₃	1	2	yes	—
262	C₄²	#10621900	1	2	yes	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	4	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	#13063246	1	2	yes	—
268	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
269	Q₈	—	8	1	no	—
270	Q₈	—	8	1	no	—
271	Q₈	—	8	1	no	—
272	Q₈	—	8	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	#13063246	1	2	yes	—
292	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
293	C₂×C₄	#13063293	1	1	yes	characteristic
294	C₂×C₄	#13063297	1	2	yes	—
295	C₂×C₄	#13063306	1	1	yes	characteristic
296	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
297	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
298	C₂×C₄	#13063350	1	1	yes	characteristic
299	C₂×C₄	#13063355	1	1	yes	characteristic
300	C₈	—	344	2	no	—
301	C₈	—	172	4	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	8	1	no	—
309	C₂²	#14582779	1	1	yes	characteristic
310	C₂²	C₂×#13063246	1	1	yes	characteristic
311	C₂²	#14582900	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	#14583032	1	1	yes	characteristic
320	C₄	#14583037	1	1	yes	characteristic
321	C₄	#14583210	1	1	yes	characteristic
322	C₄	#14583215	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	4	2	no	—
326	C₂	#19134913	1	1	yes	characteristic
327	C₂	#19135118	1	1	yes	characteristic
328	C₂	#19135148	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582779
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2621	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2621	D₄₃
hypocenter	C₄₃	#9195
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2621

Complements

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

N	H ≅ G/N	type
2. #19133267	C₂	semidirect
4. #19132453	C₂	semidirect
187. C₄₃	#9195	semidirect

Maximal subgroups

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

subgroup	index
#19143159	2
#19133267	2
#19132909	2
#19132453	2
C₄₃×#2621	2
#9195	43