Subgroups — #26003136

Order 11008.

2078 conjugacy classes of subgroups.

The subgroup lattice has 340 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, complemented
2	#19147742	C₂	1	8	yes	maximal, complemented
3	#19147734	C₂	1	8	yes	maximal, complemented
4	#19147729	C₂	1	6	yes	maximal, complemented
5	C₂×#14583335	C₂	1	6	yes	maximal, complemented
6	#19147698	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583735	—	2	1	no	—
9	#14583716	C₂²	1	4	yes	complemented
10	#14583711	—	2	4	no	—
11	#14583704	C₂²	1	4	yes	complemented
12	#14583699	—	2	3	no	—
13	C₂×#13063347	C₂²	1	6	yes	complemented
14	#14583340	C₂²	1	4	yes	complemented
15	#14583339	C₂²	1	12	yes	complemented
16	#14583335	C₂²	1	12	yes	complemented
17	#14583335	C₂²	1	24	yes	complemented
18	#14583334	C₂²	1	12	yes	complemented
19	#14583329	C₂²	1	24	yes	complemented
20	C₂×#13063269	C₂²	1	12	yes	complemented
21	C₂×#13063266	C₂²	1	12	yes	complemented
22	#14583316	C₂²	1	8	yes	complemented
23	C₂×#13063265	C₂²	1	6	yes	complemented
24	C₂×H(5,𝔽₂)×C₄₃	—	2	1	no	—
25	C₄₃×#576	—	2	4	no	—
26	C₄₃×#576	C₂²	1	8	yes	complemented
27	C₂×AGL(1,ℤ/8)×C₄₃	—	2	3	no	—
28	C₂×#135×C₄₃	C₂²	1	3	yes	complemented
29	C₂²×D₈×C₄₃	C₂²	1	3	yes	complemented
30	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
31	#13063373	—	4	1	no	—
32	#13063372	—	2	4	no	—
33	#13063371	—	2	4	no	—
34	#13063370	—	4	3	no	—
35	C₂×#10621898	—	2	6	no	—
36	#41×D₄₃	—	4	4	no	—
37	#13063358	—	2	12	no	—
38	#13063353	—	2	12	no	—
39	#13063347	—	2	6	no	—
40	#13063347	—	2	6	no	—
41	#13063347	C₂³	1	4	yes	—
42	#13063347	C₂³	1	12	yes	complemented
43	C₂×Dic₁₇₂	C₂³	1	6	yes	complemented
44	C₂×#10621873	—	2	6	no	—
45	C₂×D₃₄₄	C₂³	1	6	yes	complemented
46	#13063271	C₂³	1	8	yes	complemented
47	#13063269	C₂³	1	24	yes	complemented
48	#13063268	C₂³	1	8	yes	complemented
49	#13063266	C₂³	1	24	yes	complemented
50	#13063265	C₂³	1	4	yes	—
51	#13063265	C₂³	1	12	yes	complemented
52	C₂×C₄₃⋊C₁₆	C₂³	1	12	yes	complemented
53	H(5,𝔽₂)×C₄₃	—	2	8	no	—
54	H(5,𝔽₂)×C₄₃	—	4	4	no	—
55	C₂×#41×C₄₃	—	2	3	no	—
56	C₂×#41×C₄₃	—	4	1	no	—
57	C₂×#41×C₄₃	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₄₃	—	2	3	no	—
59	C₂²×D₄×C₄₃	—	4	3	no	—
60	AGL(1,ℤ/8)×C₄₃	—	2	24	no	—
61	#135×C₄₃	—	2	6	no	—
62	#135×C₄₃	C₂³	1	12	yes	—
63	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
64	C₂×SD₁₆×C₄₃	—	2	3	no	—
65	C₂×D₈×C₄₃	—	2	3	no	—
66	C₂×D₈×C₄₃	—	2	6	no	—
67	C₂×D₈×C₄₃	C₂³	1	12	yes	—
68	C₂×D₈×C₄₃	C₂³	1	3	yes	—
69	#131×C₄₃	D₄	1	4	yes	—
70	#131×C₄₃	C₂³	1	4	yes	—
71	C₂×M₁₆×C₄₃	D₄	1	3	yes	—
72	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
73	C₂⁴×C₄₃	—	4	3	no	—
74	#41×C₄₃	—	2	12	no	—
75	#41×C₄₃	—	4	8	no	—
76	#41×C₄₃	—	4	6	no	—
77	#41×C₄₃	D₈	1	4	yes	—
78	#41×C₄₃	C₂×D₄	1	4	yes	—
79	C₂×Q₈×C₄₃	—	2	1	no	—
80	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
81	C₂×D₄×C₄₃	—	2	3	no	—
82	C₂×D₄×C₄₃	—	2	12	no	—
83	C₂×D₄×C₄₃	—	4	24	no	—
84	C₂×D₄×C₄₃	—	4	3	no	—
85	C₂×D₄×C₄₃	—	4	6	no	—
86	C₂×D₄×C₄₃	—	4	3	no	—
87	C₂×D₄×C₄₃	D₈	1	3	yes	—
88	C₂²×C₄×C₄₃	—	4	3	no	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
90	C₂×C₂²⋊D₄₃	—	4	6	no	—
91	#10621902	—	4	4	no	—
92	Q₈×D₄₃	—	4	4	no	—
93	#10621900	—	4	12	no	—
94	D₄×D₄₃	—	4	12	no	—
95	#10621898	—	2	4	no	—
96	#10621898	—	2	12	no	—
97	#10621898	—	4	6	no	—
98	#10621898	—	4	6	no	—
99	C₂×D₁₇₂	—	2	6	no	—
100	C₂×C₄×D₄₃	—	4	6	no	—
101	C₂×Dic₈₆	—	2	6	no	—
102	Q₁₆×C₄₃	—	2	1	no	—
103	Q₁₆×C₄₃	C₂⁴	1	2	yes	—
104	SD₁₆×C₄₃	—	2	12	no	—
105	D₈×C₄₃	—	2	3	no	—
106	D₈×C₄₃	—	2	12	no	—
107	D₈×C₄₃	C₂⁴	1	6	yes	—
108	M₁₆×C₄₃	C₂×D₄	1	12	yes	—
109	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
110	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
111	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
112	C₂×C₈×C₄₃	C₂⁴	1	6	yes	—
113	Dic₁₇₂	—	2	6	no	—
114	Dic₁₇₂	C₂⁴	1	4	yes	—
115	D₃₄₄	—	2	6	no	—
116	D₃₄₄	C₂⁴	1	4	yes	—
117	#10621873	—	2	4	no	—
118	#10621873	—	2	12	no	—
119	C₄₃⋊C₁₆	C₂⁴	1	8	yes	—
120	C₂³×C₄₃	—	4	3	no	—
121	C₂³×C₄₃	—	4	12	no	—
122	C₂³×C₄₃	—	16	6	no	—
123	C₂³×C₄₃	C₂×D₈	1	3	yes	—
124	C₂²×D₄₃	—	4	6	no	—
125	Q₈×C₄₃	—	2	2	no	—
126	Q₈×C₄₃	—	4	1	no	—
127	Q₈×C₄₃	C₂×D₈	1	4	yes	—
128	D₄×C₄₃	—	2	6	no	—
129	D₄×C₄₃	—	4	12	no	—
130	D₄×C₄₃	—	4	12	no	—
131	D₄×C₄₃	—	4	3	no	—
132	D₄×C₄₃	C₂×D₈	1	12	yes	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	12	no	—
135	C₂×C₄×C₄₃	C₂×D₈	1	6	yes	—
136	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
137	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
139	C₂²⋊D₄₃	—	4	4	no	—
140	C₂²⋊D₄₃	—	4	12	no	—
141	C₂×Dic₄₃	—	4	6	no	—
142	D₁₇₂	—	2	4	no	—
143	D₁₇₂	—	4	6	no	—
144	C₄×D₄₃	—	4	4	no	—
145	C₄×D₄₃	—	4	12	no	—
146	Dic₈₆	—	2	4	no	—
147	Dic₈₆	—	4	6	no	—
148	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₄₃	C₂²×D₄	1	6	yes	—
150	C₈×C₄₃	C₂⁵	1	1	yes	characteristic
151	#62636	—	43	1	no	maximal, Sylow
152	C₂²×C₄₃	—	4	3	no	—
153	C₂²×C₄₃	—	4	6	no	—
154	C₂²×C₄₃	—	8	6	no	—
155	C₂²×C₄₃	—	16	12	no	—
156	C₂²×C₄₃	C₂²×D₈	1	6	yes	—
157	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
158	C₂×D₄₃	—	4	4	no	—
159	C₂×D₄₃	—	16	6	no	—
160	C₄×C₄₃	—	4	2	no	—
161	C₄×C₄₃	C₂²×D₈	1	6	yes	—
162	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
164	Dic₄₃	—	4	4	no	—
165	#3584	—	43	1	no	—
166	C₂×#576	D₄₃	1	1	yes	characteristic, complemented
167	#3416	—	43	8	no	—
168	#3415	—	43	8	no	—
169	#3414	—	43	6	no	—
170	C₂×#509	—	43	6	no	—
171	#3407	—	43	1	no	—
172	C₂×C₄₃	—	2	6	no	—
173	C₂×C₄₃	—	2	1	no	—
174	C₂×C₄₃	—	8	6	no	—
175	C₂×C₄₃	C₂³×D₈	1	1	yes	characteristic
176	D₄₃	—	8	4	no	—
177	#585	—	86	1	no	—
178	C₂×H(5,𝔽₂)	—	2	1	no	—
179	#578	—	43	4	no	—
180	#577	—	86	4	no	—
181	#576	—	2	4	no	—
182	#576	—	43	4	no	—
183	#576	C₂×D₄₃	1	8	yes	complemented
184	#575	—	86	3	no	—
185	C₂×AGL(1,ℤ/8)	—	2	3	no	—
186	C₂×#135	—	43	6	no	—
187	C₂×#135	C₂×D₄₃	1	3	yes	complemented
188	C₂²×D₈	C₂×D₄₃	1	3	yes	complemented
189	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
190	#510	—	43	12	no	—
191	#510	—	43	4	no	—
192	#509	—	43	12	no	—
193	#509	—	43	12	no	—
194	#509	—	43	24	no	—
195	#508	—	43	24	no	—
196	C₂×SD₃₂	—	43	12	no	—
197	C₂×D₁₆	—	43	12	no	—
198	#504	—	43	8	no	—
199	C₂×M₃₂	—	43	6	no	—
200	C₄₃	#62636	1	1	yes	characteristic, complemented, Sylow
201	2₋¹⁺⁴	—	86	4	no	—
202	2₋¹⁺⁴	—	172	1	no	—
203	H(5,𝔽₂)	—	2	8	no	—
204	H(5,𝔽₂)	—	4	4	no	—
205	H(5,𝔽₂)	—	86	4	no	—
206	H(5,𝔽₂)	—	172	3	no	—
207	C₂×#41	—	2	3	no	—
208	C₂×#41	—	4	1	no	—
209	C₂×#41	—	86	6	no	—
210	C₂×#41	—	172	4	no	—
211	C₂×#41	C₂²⋊D₄₃	1	1	yes	characteristic
212	C₂²×D₄	—	2	3	no	—
213	C₂²×D₄	—	4	3	no	—
214	#137	—	86	12	no	—
215	AGL(1,ℤ/8)	—	2	24	no	—
216	AGL(1,ℤ/8)	—	86	12	no	—
217	#135	—	2	6	no	—
218	#135	—	43	4	no	—
219	#135	—	43	12	no	—
220	#135	—	86	6	no	—
221	#135	—	86	6	no	—
222	#135	C₂²×D₄₃	1	12	yes	—
223	C₂×Q₁₆	—	43	6	no	—
224	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
225	C₂×SD₁₆	—	2	3	no	—
226	C₂×SD₁₆	—	86	6	no	—
227	C₂×D₈	—	2	3	no	—
228	C₂×D₈	—	2	6	no	—
229	C₂×D₈	—	43	6	no	—
230	C₂×D₈	C₂²×D₄₃	1	3	yes	—
231	C₂×D₈	C₂²×D₄₃	1	12	yes	—
232	#131	C₂²⋊D₄₃	1	4	yes	—
233	#131	C₂²×D₄₃	1	4	yes	—
234	C₂×M₁₆	C₂²⋊D₄₃	1	3	yes	—
235	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
236	Q₃₂	—	43	8	no	—
237	SD₃₂	—	43	8	no	—
238	SD₃₂	—	43	24	no	—
239	D₁₆	—	43	24	no	—
240	M₃₂	—	43	4	no	—
241	M₃₂	—	43	12	no	—
242	C₂×C₁₆	—	43	12	no	—
243	C₂⁴	—	4	3	no	—
244	#41	—	2	12	no	—
245	#41	—	4	8	no	—
246	#41	—	4	6	no	—
247	#41	—	86	4	no	—
248	#41	—	86	12	no	—
249	#41	—	172	4	no	—
250	#41	—	172	12	no	—
251	#41	—	172	6	no	—
252	#41	—	172	6	no	—
253	#41	#10621882	1	4	yes	—
254	#41	C₂×C₂²⋊D₄₃	1	4	yes	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	86	6	no	—
257	C₂×Q₈	—	172	4	no	—
258	C₂×Q₈	#10621882	1	1	yes	characteristic
259	C₂×D₄	—	2	3	no	—
260	C₂×D₄	—	2	12	no	—
261	C₂×D₄	—	4	3	no	—
262	C₂×D₄	—	4	24	no	—
263	C₂×D₄	—	4	3	no	—
264	C₂×D₄	—	4	6	no	—
265	C₂×D₄	—	86	6	no	—
266	C₂×D₄	—	172	6	no	—
267	C₂×D₄	—	172	12	no	—
268	C₂×D₄	#10621882	1	3	yes	—
269	C₂²×C₄	—	4	3	no	—
270	C₂²×C₄	—	172	6	no	—
271	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
272	Q₁₆	—	2	1	no	—
273	Q₁₆	—	43	4	no	—
274	Q₁₆	—	86	6	no	—
275	Q₁₆	C₂³×D₄₃	1	2	yes	—
276	SD₁₆	—	2	12	no	—
277	SD₁₆	—	86	4	no	—
278	SD₁₆	—	86	12	no	—
279	D₈	—	2	3	no	—
280	D₈	—	2	12	no	—
281	D₈	—	43	4	no	—
282	D₈	—	86	6	no	—
283	D₈	C₂³×D₄₃	1	6	yes	—
284	M₁₆	C₂×C₂²⋊D₄₃	1	12	yes	—
285	C₂×C₈	C₂×C₂²⋊D₄₃	1	3	yes	—
286	C₂×C₈	C₂×C₂²⋊D₄₃	1	6	yes	—
287	C₂×C₈	C₂³×D₄₃	1	6	yes	—
288	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
289	C₁₆	—	43	8	no	—
290	C₂³	—	4	3	no	—
291	C₂³	—	4	12	no	—
292	C₂³	—	16	6	no	—
293	C₂³	—	172	6	no	—
294	C₂³	C₂×#10621882	1	3	yes	—
295	Q₈	—	2	2	no	—
296	Q₈	—	4	1	no	—
297	Q₈	—	86	4	no	—
298	Q₈	—	172	6	no	—
299	Q₈	C₂×#10621882	1	4	yes	—
300	D₄	—	2	6	no	—
301	D₄	—	4	12	no	—
302	D₄	—	4	12	no	—
303	D₄	—	4	3	no	—
304	D₄	—	86	4	no	—
305	D₄	—	172	4	no	—
306	D₄	—	172	12	no	—
307	D₄	—	172	6	no	—
308	D₄	C₂×#10621882	1	12	yes	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	12	no	—
311	C₂×C₄	—	172	6	no	—
312	C₂×C₄	—	172	4	no	—
313	C₂×C₄	—	172	12	no	—
314	C₂×C₄	C₂×#10621882	1	3	yes	—
315	C₂×C₄	C₂×#10621882	1	6	yes	—
316	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
318	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
319	C₈	C₂²×C₂²⋊D₄₃	1	6	yes	—
320	C₈	C₂⁴×D₄₃	1	1	yes	characteristic
321	C₂²	—	4	3	no	—
322	C₂²	—	4	6	no	—
323	C₂²	—	8	6	no	—
324	C₂²	—	16	12	no	—
325	C₂²	—	172	4	no	—
326	C₂²	—	688	6	no	—
327	C₂²	C₂²×#10621882	1	1	yes	characteristic
328	C₂²	C₂²×#10621882	1	6	yes	—
329	C₄	—	4	2	no	—
330	C₄	—	172	4	no	—
331	C₄	C₂²×#10621882	1	6	yes	—
332	C₄	C₂²×#10621882	1	1	yes	characteristic
333	C₄	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	6	no	—
336	C₂	—	8	6	no	—
337	C₂	—	344	4	no	—
338	C₂	C₂³×#10621882	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂³×#10621882
commutator, cosocle	C₈×C₄₃	C₂⁵
Frattini	C₈	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#576	C₂
socle	C₂×C₄₃	C₂³×D₈
hypercenter	C₂×#576	D₄₃
hypocenter	C₄₃	#62636
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#576

Complements

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

N	H ≅ G/N	type
1. #19149369	C₂	semidirect
2. #19147742	C₂	semidirect
3. #19147734	C₂	semidirect
4. #19147729	C₂	semidirect
5. C₂×#14583335	C₂	semidirect
6. #19147698	C₂	semidirect
7. C₂×C₄₃×#576	C₂	semidirect
9. #14583716	C₂²	semidirect
11. #14583704	C₂²	semidirect
13. C₂×#13063347	C₂²	semidirect
14. #14583340	C₂²	semidirect
15. #14583339	C₂²	semidirect
16. #14583335	C₂²	semidirect
17. #14583335	C₂²	semidirect
18. #14583334	C₂²	semidirect
19. #14583329	C₂²	semidirect
20. C₂×#13063269	C₂²	semidirect
21. C₂×#13063266	C₂²	semidirect
22. #14583316	C₂²	semidirect
23. C₂×#13063265	C₂²	semidirect
26. C₄₃×#576	C₂²	semidirect
28. C₂×#135×C₄₃	C₂²	semidirect
29. C₂²×D₈×C₄₃	C₂²	semidirect
42. #13063347	C₂³	semidirect
43. C₂×Dic₁₇₂	C₂³	semidirect
45. C₂×D₃₄₄	C₂³	semidirect
46. #13063271	C₂³	semidirect
47. #13063269	C₂³	semidirect
48. #13063268	C₂³	semidirect
49. #13063266	C₂³	semidirect
51. #13063265	C₂³	semidirect
52. C₂×C₄₃⋊C₁₆	C₂³	semidirect
166. C₂×#576	D₄₃	semidirect
183. #576	C₂×D₄₃	semidirect
187. C₂×#135	C₂×D₄₃	semidirect
188. C₂²×D₈	C₂×D₄₃	semidirect
200. C₄₃	#62636	semidirect

Maximal subgroups

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

subgroup	index
#19149369	2
#19147742	2
#19147734	2
#19147729	2
C₂×#14583335	2
#19147698	2
C₂×C₄₃×#576	2
#62636	43