Subgroups — #25013625

Order 11008.

492 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583169	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582711	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582711	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136223	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132708	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132698	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#2916	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063350	—	2	1	no	—
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×#13063265	C₂²	1	1	yes	characteristic
11	C₂×#13063265	C₂²	1	1	yes	characteristic
12	#14583169	—	2	2	no	—
13	#14583169	C₄	1	4	yes	—
14	C₂×#13063261	C₂²	1	1	yes	characteristic
15	C₂×#13063241	—	2	1	no	—
16	#14582711	—	2	2	no	—
17	#14582711	—	2	2	no	—
18	#14582683	—	2	2	no	—
19	C₄₃×#568	—	2	1	no	—
20	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#430	—	2	1	no	—
22	C₄₃×#430	—	2	2	no	—
23	C₂×#108×C₄₃	—	2	2	no	—
24	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×D₁₇₂	—	2	1	no	—
27	#13063350	—	2	4	no	—
28	#13063350	—	4	2	no	—
29	C₂×#10621884	—	4	1	no	—
30	C₂×D₃₄₄	—	2	1	no	—
31	C₂×D₃₄₄	—	2	1	no	—
32	C₂×D₃₄₄	—	2	1	no	—
33	C₂×D₃₄₄	—	2	1	no	—
34	C₂×D₃₄₄	—	2	1	no	—
35	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
36	C₂×#10621882	—	4	1	no	—
37	C₂×#10621877	—	2	1	no	—
38	#13063265	—	2	1	no	—
39	#13063265	—	2	1	no	—
40	#13063265	—	2	1	no	—
41	#13063265	—	2	1	no	—
42	C₂×C₄₃⋊C₁₆	—	2	1	no	—
43	C₂×C₄₃⋊C₁₆	—	2	1	no	—
44	#13063261	—	2	1	no	—
45	#13063261	C₂×C₄	1	2	yes	—
46	#13063241	—	2	4	no	—
47	C₂×#41×C₄₃	D₄	1	1	yes	characteristic
48	#131×C₄₃	—	2	1	no	—
49	#131×C₄₃	—	2	1	no	—
50	#131×C₄₃	—	4	2	no	—
51	#131×C₄₃	—	4	2	no	—
52	#131×C₄₃	C₂×C₄	1	2	yes	—
53	#131×C₄₃	D₄	1	2	yes	—
54	C₂×M₁₆×C₄₃	—	2	1	no	—
55	C₂×M₁₆×C₄₃	—	2	4	no	—
56	C₂×M₁₆×C₄₃	—	2	1	no	—
57	C₂×M₁₆×C₄₃	—	2	1	no	—
58	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₃	—	2	1	no	—
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	#108×C₄₃	—	2	4	no	—
62	#108×C₄₃	—	2	1	no	—
63	#108×C₄₃	—	2	1	no	—
64	#108×C₄₃	D₄	1	2	yes	—
65	#108×C₄₃	D₄	1	2	yes	—
66	C₂³×D₄₃	—	4	1	no	—
67	#41×C₄₃	—	2	1	no	—
68	#41×C₄₃	—	2	1	no	—
69	#41×C₄₃	C₂²⋊C₄	1	2	yes	—
70	#41×C₄₃	D₈	1	2	yes	—
71	C₂×Q₈×C₄₃	SD₁₆	1	1	yes	characteristic
72	C₂×D₄×C₄₃	—	2	1	no	—
73	C₂×D₄×C₄₃	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	—	2	1	no	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×D₁₇₂	—	2	2	no	—
77	C₂×D₁₇₂	—	4	1	no	—
78	C₂×D₁₇₂	—	4	1	no	—
79	C₂×D₁₇₂	—	4	1	no	—
80	C₂×D₁₇₂	—	4	1	no	—
81	C₂×D₁₇₂	—	4	1	no	—
82	M₁₆×C₄₃	—	2	1	no	—
83	M₁₆×C₄₃	—	2	2	no	—
84	M₁₆×C₄₃	—	2	2	no	—
85	M₁₆×C₄₃	—	2	2	no	—
86	M₁₆×C₄₃	—	2	2	no	—
87	M₁₆×C₄₃	—	4	4	no	—
88	M₁₆×C₄₃	—	4	1	no	—
89	M₁₆×C₄₃	—	4	1	no	—
90	M₁₆×C₄₃	C₂²⋊C₄	1	2	yes	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	4	4	no	—
95	C₂×C₈×C₄₃	—	4	1	no	—
96	C₂×C₈×C₄₃	—	4	1	no	—
97	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
103	#10621884	—	4	4	no	—
104	#10621882	—	4	4	no	—
105	#10621877	—	2	2	no	—
106	#10621877	—	4	1	no	—
107	C₂×C₄₃⋊C₈	—	4	1	no	—
108	D₃₄₄	—	2	2	no	—
109	D₃₄₄	—	2	2	no	—
110	D₃₄₄	—	4	1	no	—
111	D₃₄₄	—	4	1	no	—
112	C₄₃⋊C₁₆	—	4	1	no	—
113	C₄₃⋊C₁₆	—	4	1	no	—
114	C₂³×C₄₃	—	2	1	no	—
115	C₂³×C₄₃	#137	1	1	yes	characteristic
116	C₂²×D₄₃	—	4	2	no	—
117	C₂²×D₄₃	—	8	1	no	—
118	C₂²×D₄₃	—	8	1	no	—
119	C₂²×D₄₃	—	16	2	no	—
120	Q₈×C₄₃	—	2	1	no	—
121	Q₈×C₄₃	#102	1	2	yes	—
122	D₄×C₄₃	—	2	2	no	—
123	D₄×C₄₃	—	2	2	no	—
124	D₄×C₄₃	—	2	1	no	—
125	D₄×C₄₃	#102	1	2	yes	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	D₁₇₂	—	4	2	no	—
136	D₁₇₂	—	4	2	no	—
137	D₁₇₂	—	8	1	no	—
138	D₁₇₂	—	8	1	no	—
139	C₈×C₄₃	—	2	2	no	—
140	C₈×C₄₃	—	4	2	no	—
141	C₈×C₄₃	—	4	2	no	—
142	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
143	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
144	C₈×C₄₃	#121	1	1	yes	characteristic
145	C₈×C₄₃	#121	1	1	yes	characteristic
146	C₄₃⋊C₈	—	4	2	no	—
147	#13007	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	2	2	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	—	8	1	no	—
151	C₂²×C₄₃	#417	1	1	yes	characteristic
152	C₂²×C₄₃	#448	1	1	yes	characteristic
153	C₂²×C₄₃	#480	1	1	yes	characteristic
154	C₂×D₄₃	—	8	2	no	—
155	C₂×D₄₃	—	16	2	no	—
156	C₂×D₄₃	—	16	2	no	—
157	C₂×D₄₃	—	16	2	no	—
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	#386	1	1	yes	characteristic
160	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
161	C₄×C₄₃	#447	1	1	yes	characteristic
162	C₄×C₄₃	#481	1	1	yes	characteristic
163	C₂×#472	—	43	1	no	—
164	#2916	D₄₃	1	1	yes	characteristic, complemented
165	C₂×#361	—	43	1	no	—
166	C₂×#361	—	43	1	no	—
167	#2117	—	43	1	no	—
168	#1388	—	43	1	no	—
169	#1385	—	43	1	no	—
170	C₂×C₄₃	—	2	1	no	—
171	C₂×C₄₃	—	2	1	no	—
172	C₂×C₄₃	—	2	1	no	—
173	C₂×C₄₃	—	4	2	no	—
174	C₂×C₄₃	#1874	1	1	yes	characteristic
175	D₄₃	—	16	2	no	—
176	C₂×AGL(1,ℤ/8)	—	86	1	no	—
177	C₂²×D₈	—	43	1	no	—
178	#568	—	2	1	no	—
179	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
180	C₂×M₃₂	—	43	1	no	—
181	C₂×M₃₂	—	43	1	no	—
182	#472	—	43	4	no	—
183	#472	—	86	2	no	—
184	#430	—	2	1	no	—
185	#430	—	2	2	no	—
186	C₂×#108	—	2	2	no	—
187	C₂×#108	—	43	1	no	—
188	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
189	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
190	C₂×#100	—	86	1	no	—
191	#361	—	86	2	no	—
192	#361	—	86	2	no	—
193	#350	—	86	2	no	—
194	C₄₃	#13007	1	1	yes	characteristic, complemented, Sylow
195	C₂×#41	C₂²⋊D₄₃	1	1	yes	characteristic
196	C₂²×D₄	—	86	1	no	—
197	AGL(1,ℤ/8)	—	86	4	no	—
198	AGL(1,ℤ/8)	—	172	2	no	—
199	C₂×SD₁₆	—	172	1	no	—
200	C₂×D₈	—	43	2	no	—
201	C₂×D₈	—	86	1	no	—
202	C₂×D₈	—	86	1	no	—
203	C₂×D₈	—	86	1	no	—
204	C₂×D₈	—	86	1	no	—
205	C₂×D₈	—	86	1	no	—
206	C₂×D₈	—	172	1	no	—
207	#131	—	2	1	no	—
208	#131	—	2	1	no	—
209	#131	—	4	2	no	—
210	#131	—	4	2	no	—
211	#131	C₄×D₄₃	1	2	yes	—
212	#131	D₁₇₂	1	2	yes	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	2	4	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₂×M₁₆	—	86	1	no	—
218	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
219	C₂²×C₈	—	2	1	no	—
220	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
221	M₃₂	—	86	1	no	—
222	M₃₂	—	86	1	no	—
223	M₃₂	—	86	1	no	—
224	M₃₂	—	86	1	no	—
225	C₂×C₁₆	—	86	1	no	—
226	C₂×C₁₆	—	86	1	no	—
227	#108	—	2	4	no	—
228	#108	—	2	1	no	—
229	#108	—	2	1	no	—
230	#108	—	43	2	no	—
231	#108	—	86	1	no	—
232	#108	C₂²⋊D₄₃	1	2	yes	—
233	#108	C₂²⋊D₄₃	1	2	yes	—
234	#100	—	86	4	no	—
235	C₂⁴	—	172	1	no	—
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	#41	(C₂×D₄₃)⋊C₄	1	2	yes	—
239	#41	#10621882	1	2	yes	—
240	C₂×Q₈	#10621884	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	86	2	no	—
243	C₂×D₄	—	172	1	no	—
244	C₂×D₄	—	172	1	no	—
245	C₂×D₄	—	172	1	no	—
246	C₂×D₄	—	172	1	no	—
247	C₂×D₄	—	172	1	no	—
248	C₂×D₄	#10621884	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
251	SD₁₆	—	172	4	no	—
252	D₈	—	86	2	no	—
253	D₈	—	86	2	no	—
254	D₈	—	172	4	no	—
255	D₈	—	172	1	no	—
256	D₈	—	172	1	no	—
257	M₁₆	—	2	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	4	4	no	—
263	M₁₆	—	4	1	no	—
264	M₁₆	—	4	1	no	—
265	M₁₆	—	86	2	no	—
266	M₁₆	—	172	1	no	—
267	M₁₆	(C₂×D₄₃)⋊C₄	1	2	yes	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	4	4	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	172	1	no	—
275	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
276	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
277	C₂×C₈	#10621898	1	1	yes	characteristic
278	C₂×C₈	#10621898	1	1	yes	characteristic
279	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
280	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
281	C₁₆	—	172	1	no	—
282	C₁₆	—	172	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	344	1	no	—
285	C₂³	—	344	1	no	—
286	C₂³	—	172	2	no	—
287	C₂³	—	688	2	no	—
288	C₂³	#13063356	1	1	yes	characteristic
289	Q₈	—	2	1	no	—
290	Q₈	#13063245	1	2	yes	—
291	D₄	—	2	2	no	—
292	D₄	—	2	2	no	—
293	D₄	—	2	1	no	—
294	D₄	—	344	1	no	—
295	D₄	—	344	1	no	—
296	D₄	—	172	2	no	—
297	D₄	—	172	2	no	—
298	D₄	#13063245	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
303	C₂×C₄	#13063294	1	1	yes	characteristic
304	C₂×C₄	#13063308	1	1	yes	characteristic
305	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
306	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
307	C₂×C₄	#13063351	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	172	2	no	—
312	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
313	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
314	C₈	#13063299	1	1	yes	characteristic
315	C₈	#13063299	1	1	yes	characteristic
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	344	2	no	—
320	C₂²	—	688	2	no	—
321	C₂²	—	688	2	no	—
322	C₂²	—	688	2	no	—
323	C₂²	#14582898	1	1	yes	characteristic
324	C₂²	#14583038	1	1	yes	characteristic
325	C₂²	#14583212	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	#14582778	1	1	yes	characteristic
328	C₄	C₂×#13063245	1	1	yes	characteristic
329	C₄	#14583033	1	1	yes	characteristic
330	C₄	#14583217	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	4	2	no	—
335	C₂	—	688	2	no	—
336	C₂	#19134912	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19134912
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#2916	C₂
socle	C₂×C₄₃	#1874
prime radical	C₂×#14583169	C₂
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#2916	D₄₃
hypocenter	C₄₃	#13007
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

Complements

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

N	H ≅ G/N	type
2. C₂×#14582711	C₂	semidirect
3. C₂×#14582711	C₂	semidirect
4. #19136223	C₂	semidirect
5. #19132708	C₂	semidirect
6. #19132698	C₂	semidirect
7. C₄₃×#2916	C₂	semidirect
164. #2916	D₄₃	semidirect
194. C₄₃	#13007	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583169	2
C₂×#14582711	2
#19136223	2
#19132708	2
#19132698	2
C₄₃×#2916	2
#13007	43