Subgroups — #25960933

Order 11008.

2612 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19149369	C₂	1	1	yes	maximal, characteristic
2	#19149308	C₂	1	2	yes	maximal, complemented
3	#19143511	C₂	1	6	yes	maximal, complemented
4	#19143504	C₂	1	6	yes	maximal, complemented
5	#19143497	C₂	1	6	yes	maximal, complemented
6	C₂×#14583015	C₂	1	6	yes	maximal, complemented
7	#19143140	C₂	1	2	yes	maximal, complemented
8	#131×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#2916	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583735	C₂²	1	2	yes	—
11	#14583716	C₄	1	4	yes	complemented
12	#14583711	C₄	1	8	yes	complemented
13	#14583704	C₄	1	4	yes	complemented
14	#14583699	C₂²	1	6	yes	—
15	C₂×#13063347	C₂²	1	6	yes	—
16	#14583665	C₂²	1	12	yes	complemented
17	#14583664	—	2	8	no	—
18	C₂×#13063333	C₂²	1	12	yes	complemented
19	C₂×#13063332	C₂²	1	1	yes	characteristic
20	#131×D₄₃	—	2	8	no	—
21	#41×Dic₄₃	C₂²	1	1	yes	characteristic
22	#14583020	C₂²	1	24	yes	complemented
23	#14583015	C₂²	1	4	yes	complemented
24	#14583015	C₂²	1	12	yes	complemented
25	#14583015	C₂²	1	12	yes	complemented
26	#14583015	C₂²	1	12	yes	complemented
27	#14582915	C₂²	1	12	yes	complemented
28	C₂×#13063251	C₂²	1	12	yes	complemented
29	#14582857	C₂²	1	3	yes	complemented
30	#14582854	C₂²	1	3	yes	complemented
31	M₁₆×Dic₄₃	C₂²	1	3	yes	complemented
32	C₂×C₈×Dic₄₃	C₂²	1	3	yes	complemented
33	C₄₃×#568	C₂²	1	2	yes	—
34	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
35	C₄₃×#430	C₂²	1	6	yes	complemented
36	C₂×#108×C₄₃	C₂²	1	6	yes	complemented
37	#13063373	—	2	2	no	—
38	#13063372	C₂×C₄	1	8	yes	—
39	#13063371	C₂×C₄	1	8	yes	—
40	#13063370	—	2	6	no	—
41	C₂×#10621898	C₂³	1	12	yes	—
42	#41×D₄₃	—	2	8	no	—
43	#13063358	C₂×C₄	1	24	yes	complemented
44	#13063353	C₂×C₄	1	24	yes	complemented
45	#13063347	C₂×C₄	1	12	yes	complemented
46	#13063347	C₂×C₄	1	12	yes	complemented
47	#13063347	C₂³	1	4	yes	—
48	#13063347	C₂³	1	12	yes	—
49	C₂×Dic₁₇₂	C₂×C₄	1	6	yes	complemented
50	C₂×#10621873	C₂×C₄	1	12	yes	complemented
51	C₂×D₃₄₄	C₂×C₄	1	6	yes	complemented
52	#13063333	—	2	12	no	—
53	#13063333	—	2	12	no	—
54	#13063333	C₂³	1	8	yes	—
55	#13063333	C₂³	1	24	yes	—
56	#13063332	D₄	1	4	yes	—
57	#13063332	D₄	1	4	yes	—
58	#13063331	—	2	24	no	—
59	C₂×#10621877	C₂³	1	3	yes	—
60	C₂×#10621872	—	2	12	no	—
61	M₁₆×D₄₃	—	2	24	no	—
62	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
63	C₂×C₈×D₄₃	—	2	12	no	—
64	Q₈×Dic₄₃	—	2	1	no	—
65	D₄×Dic₄₃	—	2	3	no	—
66	#13063284	C₂³	1	3	yes	—
67	C₂×C₄×Dic₄₃	C₂³	1	3	yes	—
68	#13063251	C₂³	1	8	yes	—
69	#13063251	C₂³	1	24	yes	—
70	#13063235	C₂³	1	6	yes	—
71	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
72	C₈×Dic₄₃	C₂³	1	3	yes	—
73	C₈×Dic₄₃	C₂³	1	3	yes	—
74	C₈×Dic₄₃	C₂³	1	3	yes	—
75	C₂×#41×C₄₃	C₂³	1	1	yes	characteristic
76	#131×C₄₃	—	2	8	no	—
77	#131×C₄₃	C₂×C₄	1	4	yes	—
78	#131×C₄₃	C₂×C₄	1	4	yes	—
79	C₂×M₁₆×C₄₃	C₂³	1	12	yes	—
80	C₂×M₁₆×C₄₃	C₂³	1	3	yes	—
81	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
82	#108×C₄₃	C₂³	1	4	yes	—
83	#108×C₄₃	C₂³	1	12	yes	—
84	#41×C₄₃	C₂²×C₄	1	4	yes	—
85	#41×C₄₃	C₂×D₄	1	4	yes	—
86	C₂×Q₈×C₄₃	#41	1	1	yes	characteristic
87	C₂×D₄×C₄₃	#41	1	3	yes	—
88	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
89	C₂×C₂²⋊D₄₃	—	2	12	no	—
90	C₂²×Dic₄₃	—	2	3	no	—
91	#10621902	—	2	8	no	—
92	Q₈×D₄₃	—	2	8	no	—
93	#10621900	—	2	24	no	—
94	D₄×D₄₃	—	2	24	no	—
95	#10621898	—	2	12	no	—
96	#10621898	—	2	12	no	—
97	#10621898	C₂²×C₄	1	24	yes	—
98	#10621898	C₂⁴	1	8	yes	—
99	C₂×D₁₇₂	C₂²×C₄	1	12	yes	—
100	C₂×C₄×D₄₃	—	2	12	no	—
101	C₂×Dic₈₆	C₂²×C₄	1	12	yes	—
102	M₁₆×C₄₃	—	2	12	no	—
103	M₁₆×C₄₃	C₂²×C₄	1	12	yes	—
104	M₁₆×C₄₃	C₂⁴	1	8	yes	—
105	C₂×C₈×C₄₃	—	2	12	no	—
106	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
107	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
108	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₄₃	C₂⁴	1	3	yes	—
110	C₂²⋊Dic₄₃	—	2	3	no	—
111	C₄⋊Dic₄₃	—	2	3	no	—
112	C₄×Dic₄₃	—	2	3	no	—
113	C₄×Dic₄₃	C₂⁴	1	1	yes	characteristic
114	C₄×Dic₄₃	C₂⁴	1	3	yes	—
115	#10621877	C₂×D₄	1	12	yes	—
116	C₂×C₄₃⋊C₈	C₂×D₄	1	6	yes	—
117	C₂×C₄₃⋊C₈	C₂×D₄	1	6	yes	—
118	C₂×C₄₃⋊C₈	C₂⁴	1	1	yes	characteristic
119	C₂×C₄₃⋊C₈	C₂⁴	1	3	yes	—
120	Dic₁₇₂	C₂²×C₄	1	4	yes	—
121	Dic₁₇₂	C₂²×C₄	1	12	yes	—
122	D₃₄₄	C₂²×C₄	1	4	yes	—
123	D₃₄₄	C₂²×C₄	1	12	yes	—
124	#10621873	C₂²×C₄	1	8	yes	—
125	#10621873	C₂²×C₄	1	24	yes	—
126	#10621872	—	2	24	no	—
127	#10621872	—	2	8	no	—
128	C₈×D₄₃	—	2	8	no	—
129	C₈×D₄₃	—	2	24	no	—
130	C₂³×C₄₃	C₂×#41	1	3	yes	—
131	C₂²×D₄₃	—	2	12	no	—
132	Q₈×C₄₃	C₄×D₄	1	4	yes	—
133	D₄×C₄₃	C₄×D₄	1	12	yes	—
134	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
135	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
136	C₂×C₄×C₄₃	C₂×#41	1	3	yes	—
137	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
138	C₂²⋊D₄₃	—	2	24	no	—
139	C₂²⋊D₄₃	—	2	8	no	—
140	C₂×Dic₄₃	—	2	12	no	—
141	C₂×Dic₄₃	—	2	1	no	—
142	C₂×Dic₄₃	—	2	3	no	—
143	C₂×Dic₄₃	—	4	6	no	—
144	D₁₇₂	—	2	12	no	—
145	D₁₇₂	C₂³×C₄	1	8	yes	—
146	C₄×D₄₃	—	2	24	no	—
147	C₄×D₄₃	—	2	8	no	—
148	Dic₈₆	—	2	12	no	—
149	Dic₈₆	C₂³×C₄	1	8	yes	—
150	C₈×C₄₃	—	2	8	no	—
151	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
152	C₈×C₄₃	C₂³×C₄	1	6	yes	—
153	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
154	C₄₃⋊C₈	C₂²×D₄	1	1	yes	characteristic
155	C₄₃⋊C₈	C₂²×D₄	1	6	yes	—
156	C₄₃⋊C₈	C₂²×D₄	1	1	yes	characteristic
157	#60183	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	4	3	no	—
159	C₂²×C₄₃	C₂×C₄×D₄	1	6	yes	—
160	C₂²×C₄₃	C₂²×#41	1	1	yes	characteristic
161	C₂×D₄₃	—	2	8	no	—
162	C₂×D₄₃	—	8	12	no	—
163	C₄×C₄₃	C₂×C₄×D₄	1	6	yes	—
164	C₄×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
165	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
166	Dic₄₃	—	2	2	no	—
167	Dic₄₃	—	2	8	no	—
168	Dic₄₃	—	4	3	no	—
169	#3584	—	43	1	no	—
170	#3573	—	43	2	no	—
171	#2957	—	43	6	no	—
172	#2956	—	43	6	no	—
173	#2955	—	43	6	no	—
174	C₂×#443	—	43	6	no	—
175	#2916	D₄₃	1	1	yes	characteristic, complemented
176	#2901	—	43	2	no	—
177	C₄×#131	—	43	1	no	—
178	C₂×C₄₃	—	2	1	no	—
179	C₂×C₄₃	—	2	6	no	—
180	C₂×C₄₃	C₂²×C₄×D₄	1	1	yes	characteristic
181	D₄₃	—	4	8	no	—
182	#585	—	43	2	no	—
183	#578	—	43	4	no	—
184	#577	—	43	8	no	—
185	#576	—	43	4	no	—
186	#575	—	43	6	no	—
187	C₂×#135	—	43	6	no	—
188	#568	—	43	12	no	—
189	#568	—	86	8	no	—
190	#568	C₂×D₄₃	1	2	yes	—
191	C₂×#131	—	43	1	no	—
192	C₂×#131	—	43	12	no	—
193	C₂×#131	—	86	8	no	—
194	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
195	C₄×#41	—	43	1	no	—
196	#444	—	43	24	no	—
197	#443	—	43	4	no	—
198	#443	—	43	12	no	—
199	#443	—	43	12	no	—
200	#443	—	43	12	no	—
201	#430	C₂×D₄₃	1	6	yes	complemented
202	C₂×#108	C₂×D₄₃	1	6	yes	complemented
203	#421	—	43	12	no	—
204	C₂×#104	—	43	12	no	—
205	#405	—	43	3	no	—
206	#405	—	43	3	no	—
207	C₄×M₁₆	—	43	3	no	—
208	C₂×C₄×C₈	—	43	3	no	—
209	C₄₃	#60183	1	1	yes	characteristic, complemented, Sylow
210	2₋¹⁺⁴	—	43	8	no	—
211	2₋¹⁺⁴	—	86	2	no	—
212	H(5,𝔽₂)	—	43	8	no	—
213	H(5,𝔽₂)	—	86	6	no	—
214	C₂×#41	—	43	12	no	—
215	C₂×#41	—	86	8	no	—
216	C₂×#41	C₂²×D₄₃	1	1	yes	characteristic
217	#137	—	43	24	no	—
218	AGL(1,ℤ/8)	—	43	24	no	—
219	#135	—	43	12	no	—
220	#135	—	43	12	no	—
221	#135	—	43	4	no	—
222	#135	—	43	12	no	—
223	C₂×Q₁₆	—	43	6	no	—
224	C₂×SD₁₆	—	43	12	no	—
225	C₂×D₈	—	43	6	no	—
226	#131	—	2	8	no	—
227	#131	—	43	4	no	—
228	#131	—	43	4	no	—
229	#131	—	43	8	no	—
230	#131	—	43	24	no	—
231	#131	—	86	24	no	—
232	#131	—	86	12	no	—
233	#131	—	86	12	no	—
234	#131	C₄×D₄₃	1	4	yes	—
235	#131	C₄×D₄₃	1	4	yes	—
236	C₂×M₁₆	—	43	3	no	—
237	C₂×M₁₆	—	86	12	no	—
238	C₂×M₁₆	—	86	24	no	—
239	C₂×M₁₆	C₂²×D₄₃	1	12	yes	—
240	C₂×M₁₆	C₂²×D₄₃	1	3	yes	—
241	C₂²×C₈	—	43	3	no	—
242	C₂²×C₈	—	86	12	no	—
243	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
244	C₄×Q₈	—	86	1	no	—
245	C₄×D₄	—	86	3	no	—
246	#117	—	43	3	no	—
247	C₂×C₄²	—	43	3	no	—
248	#108	C₂²×D₄₃	1	4	yes	—
249	#108	C₂²×D₄₃	1	12	yes	—
250	#104	—	43	8	no	—
251	#104	—	43	24	no	—
252	#97	—	43	6	no	—
253	C₄×C₈	—	43	1	no	—
254	C₄×C₈	—	43	3	no	—
255	C₄×C₈	—	43	3	no	—
256	C₄×C₈	—	43	3	no	—
257	#41	—	43	24	no	—
258	#41	—	43	8	no	—
259	#41	—	86	8	no	—
260	#41	—	86	24	no	—
261	#41	—	86	12	no	—
262	#41	—	86	12	no	—
263	#41	C₂×C₄×D₄₃	1	4	yes	—
264	#41	D₄×D₄₃	1	4	yes	—
265	C₂×Q₈	—	43	12	no	—
266	C₂×Q₈	—	86	8	no	—
267	C₂×Q₈	#10621902	1	1	yes	characteristic
268	C₂×D₄	—	43	12	no	—
269	C₂×D₄	—	86	12	no	—
270	C₂×D₄	—	86	24	no	—
271	C₂×D₄	#10621902	1	3	yes	—
272	C₂²×C₄	—	86	12	no	—
273	C₂²×C₄	—	86	3	no	—
274	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
275	Q₁₆	—	43	4	no	—
276	Q₁₆	—	43	12	no	—
277	SD₁₆	—	43	8	no	—
278	SD₁₆	—	43	24	no	—
279	D₈	—	43	4	no	—
280	D₈	—	43	12	no	—
281	M₁₆	—	2	12	no	—
282	M₁₆	—	43	12	no	—
283	M₁₆	—	86	8	no	—
284	M₁₆	—	86	24	no	—
285	M₁₆	C₂×C₄×D₄₃	1	12	yes	—
286	M₁₆	C₂³×D₄₃	1	8	yes	—
287	C₂×C₈	—	2	12	no	—
288	C₂×C₈	—	43	6	no	—
289	C₂×C₈	—	43	6	no	—
290	C₂×C₈	—	43	1	no	—
291	C₂×C₈	—	43	3	no	—
292	C₂×C₈	—	86	8	no	—
293	C₂×C₈	—	86	24	no	—
294	C₂×C₈	C₂×C₄×D₄₃	1	6	yes	—
295	C₂×C₈	C₂×C₄×D₄₃	1	6	yes	—
296	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₄₃	1	3	yes	—
298	C₄⋊C₄	—	86	3	no	—
299	C₂²⋊C₄	—	86	3	no	—
300	C₄²	—	43	1	no	—
301	C₄²	—	43	3	no	—
302	C₄²	—	86	3	no	—
303	C₂³	—	86	12	no	—
304	C₂³	C₂×#10621902	1	3	yes	—
305	Q₈	—	43	8	no	—
306	Q₈	—	86	12	no	—
307	Q₈	#13063287	1	4	yes	—
308	D₄	—	43	8	no	—
309	D₄	—	86	24	no	—
310	D₄	—	86	8	no	—
311	D₄	—	86	12	no	—
312	D₄	#13063287	1	12	yes	—
313	C₂×C₄	—	86	12	no	—
314	C₂×C₄	—	86	1	no	—
315	C₂×C₄	—	86	24	no	—
316	C₂×C₄	—	86	8	no	—
317	C₂×C₄	—	86	3	no	—
318	C₂×C₄	—	172	6	no	—
319	C₂×C₄	C₂²×C₄×D₄₃	1	6	yes	—
320	C₂×C₄	C₂×D₄×D₄₃	1	6	yes	—
321	C₂×C₄	C₂×#10621902	1	3	yes	—
322	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
323	C₈	—	2	8	no	—
324	C₈	—	43	1	no	—
325	C₈	—	43	1	no	—
326	C₈	—	43	6	no	—
327	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
328	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
329	C₈	C₂²×C₄×D₄₃	1	6	yes	—
330	C₂²	—	4	3	no	—
331	C₂²	—	86	8	no	—
332	C₂²	—	344	12	no	—
333	C₂²	C₂×#13063287	1	6	yes	—
334	C₂²	C₂²×#10621902	1	1	yes	characteristic
335	C₄	—	86	2	no	—
336	C₄	—	86	8	no	—
337	C₄	—	172	3	no	—
338	C₄	C₂×#13063287	1	6	yes	—
339	C₄	C₂³×C₄×D₄₃	1	1	yes	characteristic
340	C₄	C₂²×D₄×D₄₃	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	6	no	—
343	C₂	—	172	8	no	—
344	C₂	C₂²×#13063287	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₄×D₄₃
commutator	C₄×C₄₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₄₃×#2916	C₂
socle	C₂×C₄₃	C₂²×C₄×D₄
prime radical	#19149369	C₂
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	#2916	D₄₃
hypocenter	C₄₃	#60183
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #2916

Complements

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

N	H ≅ G/N	type
2. #19149308	C₂	semidirect
3. #19143511	C₂	semidirect
4. #19143504	C₂	semidirect
5. #19143497	C₂	semidirect
6. C₂×#14583015	C₂	semidirect
7. #19143140	C₂	semidirect
8. #131×Dic₄₃	C₂	semidirect
9. C₄₃×#2916	C₂	semidirect
11. #14583716	C₄	semidirect
12. #14583711	C₄	semidirect
13. #14583704	C₄	semidirect
16. #14583665	C₂²	semidirect
18. C₂×#13063333	C₂²	semidirect
22. #14583020	C₂²	semidirect
23. #14583015	C₂²	semidirect
24. #14583015	C₂²	semidirect
25. #14583015	C₂²	semidirect
26. #14583015	C₂²	semidirect
27. #14582915	C₂²	semidirect
28. C₂×#13063251	C₂²	semidirect
29. #14582857	C₂²	semidirect
30. #14582854	C₂²	semidirect
31. M₁₆×Dic₄₃	C₂²	semidirect
32. C₂×C₈×Dic₄₃	C₂²	semidirect
35. C₄₃×#430	C₂²	semidirect
36. C₂×#108×C₄₃	C₂²	semidirect
43. #13063358	C₂×C₄	semidirect
44. #13063353	C₂×C₄	semidirect
45. #13063347	C₂×C₄	semidirect
46. #13063347	C₂×C₄	semidirect
49. C₂×Dic₁₇₂	C₂×C₄	semidirect
50. C₂×#10621873	C₂×C₄	semidirect
51. C₂×D₃₄₄	C₂×C₄	semidirect
175. #2916	D₄₃	semidirect
201. #430	C₂×D₄₃	semidirect
202. C₂×#108	C₂×D₄₃	semidirect
209. C₄₃	#60183	semidirect

Maximal subgroups

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

subgroup	index
#19149369	2
#19149308	2
#19143511	2
#19143504	2
#19143497	2
C₂×#14583015	2
#19143140	2
#131×Dic₄₃	2
C₄₃×#2916	2
#60183	43