Subgroups — #46825653

Order 34432.

1424 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	#31771440	C₂	1	2	yes	maximal, complemented
2	#31771422	C₂	1	1	yes	maximal, characteristic, complemented
3	#31771419	C₂	1	3	yes	maximal, complemented
4	#31771416	C₂	1	2	yes	maximal, complemented
5	#31771415	C₂	1	6	yes	maximal, complemented
6	#31771406	C₂	1	3	yes	maximal, complemented
7	#31771401	C₂	1	6	yes	maximal, complemented
8	C₂×#22575472	C₂	1	3	yes	maximal, complemented
9	#135×D₂₆₉	C₂	1	3	yes	maximal, complemented
10	#131×D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#576×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#22575500	—	2	2	no	—
13	#22575500	C₂²	1	2	yes	complemented
14	#22575500	C₂²	1	6	yes	complemented
15	#22575498	—	2	6	no	—
16	#22575498	C₂²	1	6	yes	complemented
17	C₂×#16864401	—	2	6	no	—
18	C₂×#16864401	C₂²	1	6	yes	complemented
19	#41×D₂₆₉	—	2	2	no	—
20	#41×D₂₆₉	C₂²	1	1	yes	characteristic
21	#41×D₂₆₉	C₂²	1	6	yes	complemented
22	#22575486	C₂²	1	6	yes	complemented
23	#22575483	C₂²	1	6	yes	complemented
24	#22575482	C₂²	1	12	yes	complemented
25	#22575479	C₂²	1	6	yes	complemented
26	#22575477	C₂²	1	12	yes	complemented
27	#22575476	C₂²	1	1	yes	characteristic, complemented
28	#22575475	C₂²	1	3	yes	complemented
29	#22575474	C₂²	1	6	yes	complemented
30	#22575473	C₂²	1	6	yes	complemented
31	#22575472	C₂²	1	6	yes	complemented
32	#22575472	C₂²	1	6	yes	complemented
33	#22575472	C₂²	1	6	yes	complemented
34	#22575472	C₂²	1	3	yes	complemented
35	C₂×Dic₁₀₇₆	C₂²	1	3	yes	complemented
36	Q₁₆×D₂₆₉	C₂²	1	3	yes	complemented
37	C₂×#16864390	C₂²	1	6	yes	complemented
38	SD₁₆×D₂₆₉	C₂²	1	6	yes	complemented
39	D₈×D₂₆₉	C₂²	1	3	yes	complemented
40	#22575461	C₂²	1	3	yes	complemented
41	#22575460	C₂²	1	1	yes	characteristic, complemented
42	#22575459	C₂²	1	3	yes	complemented
43	M₁₆×D₂₆₉	C₂²	1	3	yes	complemented
44	C₂×C₈×D₂₆₉	C₂²	1	3	yes	complemented
45	H(5,𝔽₂)×C₂₆₉	C₂²	1	2	yes	complemented
46	AGL(1,ℤ/8)×C₂₆₉	C₂²	1	6	yes	complemented
47	#135×C₂₆₉	C₂²	1	3	yes	complemented
48	C₂×D₈×C₂₆₉	C₂²	1	3	yes	complemented
49	#131×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
50	#16864403	D₄	1	1	yes	characteristic
51	#16864403	C₂³	1	2	yes	—
52	#16864402	—	2	6	no	—
53	#16864402	—	2	6	no	—
54	#16864402	C₂³	1	6	yes	—
55	#16864402	C₂³	1	3	yes	—
56	#16864401	—	2	6	no	—
57	#16864401	—	2	12	no	—
58	#16864401	—	2	6	no	—
59	#16864401	—	2	6	no	—
60	#16864401	—	2	6	no	—
61	#16864401	—	2	6	no	—
62	#16864401	D₄	1	3	yes	—
63	#16864401	C₂³	1	6	yes	—
64	#16864401	C₂³	1	12	yes	—
65	C₂×Dic₅₃₈	—	2	6	no	—
66	C₂×Dic₅₃₈	C₂³	1	6	yes	—
67	Q₈×D₂₆₉	—	2	2	no	—
68	Q₈×D₂₆₉	D₄	1	1	yes	characteristic
69	Q₈×D₂₆₉	C₂³	1	6	yes	—
70	C₂×C₂²⋊D₂₆₉	—	2	6	no	—
71	C₂×C₂²⋊D₂₆₉	—	2	12	no	—
72	D₄×D₂₆₉	—	2	6	no	—
73	D₄×D₂₆₉	—	2	6	no	—
74	D₄×D₂₆₉	D₄	1	3	yes	—
75	D₄×D₂₆₉	C₂³	1	6	yes	—
76	C₂²×Dic₂₆₉	—	2	6	no	—
77	C₂×C₄×D₂₆₉	—	2	6	no	—
78	C₂×C₄×D₂₆₉	C₂³	1	3	yes	—
79	Dic₁₀₇₆	C₂³	1	3	yes	—
80	Dic₁₀₇₆	C₂³	1	6	yes	—
81	#16864392	C₂³	1	6	yes	—
82	#16864391	C₂³	1	6	yes	—
83	#16864390	C₂³	1	6	yes	—
84	#16864390	C₂³	1	12	yes	—
85	#16864389	C₂³	1	2	yes	—
86	D₂₁₅₂	C₂³	1	1	yes	characteristic
87	#16864387	C₂³	1	6	yes	—
88	#16864386	C₂³	1	3	yes	—
89	#16864385	C₂³	1	6	yes	—
90	C₂×#14524130	C₂³	1	3	yes	—
91	C₈×D₂₆₉	C₂³	1	1	yes	characteristic
92	C₈×D₂₆₉	C₂³	1	3	yes	—
93	C₈×D₂₆₉	C₂³	1	3	yes	—
94	C₈×D₂₆₉	C₂³	1	3	yes	—
95	#41×C₂₆₉	—	2	2	no	—
96	#41×C₂₆₉	C₂³	1	6	yes	—
97	#41×C₂₆₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₂₆₉	—	2	6	no	—
99	C₂×D₄×C₂₆₉	C₂³	1	6	yes	—
100	Q₁₆×C₂₆₉	C₂³	1	1	yes	characteristic
101	SD₁₆×C₂₆₉	C₂³	1	6	yes	—
102	D₈×C₂₆₉	C₂³	1	3	yes	—
103	D₈×C₂₆₉	C₂³	1	6	yes	—
104	M₁₆×C₂₆₉	C₂³	1	3	yes	—
105	C₂×C₈×C₂₆₉	C₂³	1	3	yes	—
106	C₂²×D₂₆₉	—	2	6	no	—
107	C₂²×D₂₆₉	C₂×D₄	1	3	yes	—
108	Dic₅₃₈	—	2	6	no	—
109	Dic₅₃₈	—	2	6	no	—
110	Dic₅₃₈	C₂×D₄	1	3	yes	—
111	Dic₅₃₈	C₂⁴	1	6	yes	—
112	D₁₀₇₆	—	2	2	no	—
113	D₁₀₇₆	C₂×D₄	1	3	yes	—
114	D₁₀₇₆	C₂⁴	1	2	yes	—
115	C₂²⋊D₂₆₉	—	2	6	no	—
116	C₂²⋊D₂₆₉	—	2	12	no	—
117	C₂²⋊D₂₆₉	—	2	6	no	—
118	C₂²⋊D₂₆₉	—	2	6	no	—
119	C₂²⋊D₂₆₉	C₂×D₄	1	6	yes	—
120	C₂×Dic₂₆₉	—	2	6	no	—
121	C₂×Dic₂₆₉	—	2	6	no	—
122	C₂×Dic₂₆₉	—	2	12	no	—
123	C₂×Dic₂₆₉	C₂×D₄	1	3	yes	—
124	C₄×D₂₆₉	—	2	6	no	—
125	C₄×D₂₆₉	—	2	6	no	—
126	C₄×D₂₆₉	—	2	2	no	—
127	C₄×D₂₆₉	C₂×D₄	1	3	yes	—
128	C₄×D₂₆₉	C₂×D₄	1	3	yes	—
129	C₄×D₂₆₉	C₂⁴	1	1	yes	characteristic
130	C₄×D₂₆₉	C₂⁴	1	3	yes	—
131	#14524130	C₂⁴	1	1	yes	characteristic
132	#14524130	C₂⁴	1	3	yes	—
133	C₂³×C₂₆₉	—	2	6	no	—
134	Q₈×C₂₆₉	C₂×D₄	1	1	yes	characteristic
135	Q₈×C₂₆₉	C₂⁴	1	2	yes	—
136	D₄×C₂₆₉	—	2	6	no	—
137	D₄×C₂₆₉	—	2	6	no	—
138	D₄×C₂₆₉	C₂×D₄	1	3	yes	—
139	D₄×C₂₆₉	C₂⁴	1	6	yes	—
140	C₂×C₄×C₂₆₉	—	2	6	no	—
141	C₂×C₄×C₂₆₉	C₂⁴	1	3	yes	—
142	C₈×C₂₆₉	C₂⁴	1	1	yes	characteristic
143	C₈×C₂₆₉	C₂⁴	1	3	yes	—
144	C₂²×C₂₆₉	—	2	6	no	—
145	C₂²×C₂₆₉	—	8	6	no	—
146	C₂²×C₂₆₉	C₂²×D₄	1	3	yes	—
147	C₂×D₂₆₉	—	2	2	no	—
148	C₂×D₂₆₉	—	4	3	no	—
149	C₂×D₂₆₉	—	8	6	no	—
150	C₂×D₂₆₉	C₂²×D₄	1	3	yes	—
151	C₂×D₂₆₉	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₂₆₉	—	2	2	no	—
153	C₄×C₂₆₉	C₂²×D₄	1	3	yes	—
154	C₄×C₂₆₉	C₂⁵	1	1	yes	characteristic
155	Dic₂₆₉	—	2	6	no	—
156	Dic₂₆₉	C₂²×D₄	1	1	yes	characteristic
157	Dic₂₆₉	C₂²×D₄	1	3	yes	—
158	C₂×C₂₆₉	—	2	3	no	—
159	C₂×C₂₆₉	—	4	6	no	—
160	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
161	D₂₆₉	—	2	1	no	—
162	D₂₆₉	—	2	3	no	—
163	D₂₆₉	—	4	2	no	—
164	C₂₆₉	#3584	1	1	yes	characteristic, complemented, Sylow
165	#3584	—	269	1	no	maximal, Sylow
166	#585	—	269	2	no	—
167	#578	—	269	3	no	—
168	#578	—	269	1	no	—
169	#577	—	269	2	no	—
170	#577	—	269	6	no	—
171	#576	—	269	3	no	—
172	#576	D₂₆₉	1	1	yes	characteristic, complemented
173	#575	—	269	6	no	—
174	C₂×#135	—	269	3	no	—
175	C₂×#135	—	269	3	no	—
176	C₂×#131	—	269	1	no	—
177	2₋¹⁺⁴	—	269	6	no	—
178	2₋¹⁺⁴	—	269	2	no	—
179	2₋¹⁺⁴	—	538	2	no	—
180	H(5,𝔽₂)	—	269	6	no	—
181	H(5,𝔽₂)	—	538	6	no	—
182	H(5,𝔽₂)	C₂×D₂₆₉	1	2	yes	complemented
183	C₂×#41	—	269	1	no	—
184	C₂×#41	—	269	6	no	—
185	C₂×#41	—	269	6	no	—
186	C₂×#41	—	538	6	no	—
187	C₂×#41	—	538	2	no	—
188	#137	—	269	6	no	—
189	#137	—	269	12	no	—
190	#137	—	269	6	no	—
191	AGL(1,ℤ/8)	—	269	6	no	—
192	AGL(1,ℤ/8)	—	269	12	no	—
193	AGL(1,ℤ/8)	C₂×D₂₆₉	1	6	yes	complemented
194	#135	—	269	6	no	—
195	#135	—	269	1	no	—
196	#135	—	269	6	no	—
197	#135	—	269	3	no	—
198	#135	—	269	6	no	—
199	#135	—	269	6	no	—
200	#135	—	269	6	no	—
201	#135	—	269	3	no	—
202	#135	C₂×D₂₆₉	1	3	yes	complemented
203	C₂×Q₁₆	—	269	3	no	—
204	C₂×Q₁₆	—	269	3	no	—
205	C₂×SD₁₆	—	269	6	no	—
206	C₂×SD₁₆	—	269	6	no	—
207	C₂×D₈	—	269	3	no	—
208	C₂×D₈	C₂×D₂₆₉	1	3	yes	complemented
209	#131	—	269	1	no	—
210	#131	—	269	3	no	—
211	#131	—	269	3	no	—
212	#131	C₂×D₂₆₉	1	1	yes	characteristic, complemented
213	C₂×M₁₆	—	269	3	no	—
214	C₂²×C₈	—	269	3	no	—
215	#41	—	2	2	no	—
216	#41	—	269	1	no	—
217	#41	—	269	3	no	—
218	#41	—	269	3	no	—
219	#41	—	269	2	no	—
220	#41	—	269	6	no	—
221	#41	—	269	12	no	—
222	#41	—	269	6	no	—
223	#41	—	538	6	no	—
224	#41	—	538	6	no	—
225	#41	—	538	6	no	—
226	#41	—	538	6	no	—
227	#41	—	538	6	no	—
228	#41	—	538	12	no	—
229	#41	—	538	6	no	—
230	#41	—	538	6	no	—
231	#41	C₂²×D₂₆₉	1	6	yes	—
232	#41	C₂²×D₂₆₉	1	1	yes	characteristic
233	C₂×Q₈	—	269	1	no	—
234	C₂×Q₈	—	269	6	no	—
235	C₂×Q₈	—	269	6	no	—
236	C₂×Q₈	—	538	6	no	—
237	C₂×Q₈	—	538	2	no	—
238	C₂×D₄	—	2	6	no	—
239	C₂×D₄	—	269	3	no	—
240	C₂×D₄	—	269	6	no	—
241	C₂×D₄	—	538	6	no	—
242	C₂×D₄	—	538	6	no	—
243	C₂×D₄	—	538	12	no	—
244	C₂×D₄	—	538	6	no	—
245	C₂×D₄	C₂²×D₂₆₉	1	6	yes	—
246	C₂²×C₄	—	269	3	no	—
247	C₂²×C₄	—	538	6	no	—
248	C₂²×C₄	—	538	6	no	—
249	Q₁₆	—	269	6	no	—
250	Q₁₆	—	269	3	no	—
251	Q₁₆	—	269	6	no	—
252	Q₁₆	C₂²×D₂₆₉	1	1	yes	characteristic
253	SD₁₆	—	269	2	no	—
254	SD₁₆	—	269	6	no	—
255	SD₁₆	—	269	12	no	—
256	SD₁₆	—	269	6	no	—
257	SD₁₆	C₂²×D₂₆₉	1	6	yes	—
258	D₈	—	269	6	no	—
259	D₈	—	269	1	no	—
260	D₈	C₂²×D₂₆₉	1	3	yes	—
261	D₈	C₂²×D₂₆₉	1	6	yes	—
262	M₁₆	—	269	3	no	—
263	M₁₆	—	269	6	no	—
264	M₁₆	C₂²×D₂₆₉	1	3	yes	—
265	C₂×C₈	—	269	3	no	—
266	C₂×C₈	—	269	1	no	—
267	C₂×C₈	—	269	3	no	—
268	C₂×C₈	—	269	3	no	—
269	C₂×C₈	—	269	3	no	—
270	C₂×C₈	C₂²×D₂₆₉	1	3	yes	—
271	C₂³	—	2	6	no	—
272	C₂³	—	269	3	no	—
273	C₂³	—	538	6	no	—
274	Q₈	—	269	3	no	—
275	Q₈	—	269	6	no	—
276	Q₈	—	538	6	no	—
277	Q₈	—	538	6	no	—
278	Q₈	D₄×D₂₆₉	1	1	yes	characteristic
279	Q₈	C₂³×D₂₆₉	1	2	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	2	6	no	—
282	D₄	—	269	6	no	—
283	D₄	—	269	3	no	—
284	D₄	—	269	2	no	—
285	D₄	—	538	2	no	—
286	D₄	—	538	6	no	—
287	D₄	—	538	6	no	—
288	D₄	—	538	12	no	—
289	D₄	—	538	6	no	—
290	D₄	D₄×D₂₆₉	1	3	yes	—
291	D₄	C₂³×D₂₆₉	1	6	yes	—
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	269	3	no	—
294	C₂×C₄	—	269	3	no	—
295	C₂×C₄	—	269	3	no	—
296	C₂×C₄	—	269	1	no	—
297	C₂×C₄	—	269	3	no	—
298	C₂×C₄	—	538	2	no	—
299	C₂×C₄	—	538	6	no	—
300	C₂×C₄	—	538	6	no	—
301	C₂×C₄	—	538	12	no	—
302	C₂×C₄	—	538	6	no	—
303	C₂×C₄	—	538	6	no	—
304	C₂×C₄	C₂³×D₂₆₉	1	3	yes	—
305	C₈	—	269	1	no	—
306	C₈	—	269	3	no	—
307	C₈	C₂³×D₂₆₉	1	1	yes	characteristic
308	C₈	C₂³×D₂₆₉	1	3	yes	—
309	C₂²	—	2	6	no	—
310	C₂²	—	8	6	no	—
311	C₂²	—	269	1	no	—
312	C₂²	—	269	3	no	—
313	C₂²	—	538	2	no	—
314	C₂²	—	1076	3	no	—
315	C₂²	—	2152	6	no	—
316	C₂²	C₂×D₄×D₂₆₉	1	3	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	269	1	no	—
319	C₄	—	269	3	no	—
320	C₄	—	538	6	no	—
321	C₄	C₂×D₄×D₂₆₉	1	3	yes	—
322	C₄	C₂⁴×D₂₆₉	1	1	yes	characteristic
323	C₂	—	2	3	no	—
324	C₂	—	4	6	no	—
325	C₂	—	538	1	no	—
326	C₂	—	538	3	no	—
327	C₂	—	1076	2	no	—
328	C₂	C₂²×D₄×D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₂₆₉
commutator, cosocle	C₄×C₂₆₉	C₂⁵
Frattini	C₄	C₂⁴×D₂₆₉
Fitting	#576×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₂³×D₄
hypercenter	#576	D₂₆₉
hypocenter	C₂₆₉	#3584
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

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

N	H ≅ G/N	type
1. #31771440	C₂	semidirect
2. #31771422	C₂	semidirect
3. #31771419	C₂	semidirect
4. #31771416	C₂	semidirect
5. #31771415	C₂	semidirect
6. #31771406	C₂	semidirect
7. #31771401	C₂	semidirect
8. C₂×#22575472	C₂	semidirect
9. #135×D₂₆₉	C₂	semidirect
10. #131×D₂₆₉	C₂	semidirect
11. #576×C₂₆₉	C₂	semidirect
13. #22575500	C₂²	semidirect
14. #22575500	C₂²	semidirect
16. #22575498	C₂²	semidirect
18. C₂×#16864401	C₂²	semidirect
21. #41×D₂₆₉	C₂²	semidirect
22. #22575486	C₂²	semidirect
23. #22575483	C₂²	semidirect
24. #22575482	C₂²	semidirect
25. #22575479	C₂²	semidirect
26. #22575477	C₂²	semidirect
27. #22575476	C₂²	semidirect
28. #22575475	C₂²	semidirect
29. #22575474	C₂²	semidirect
30. #22575473	C₂²	semidirect
31. #22575472	C₂²	semidirect
32. #22575472	C₂²	semidirect
33. #22575472	C₂²	semidirect
34. #22575472	C₂²	semidirect
35. C₂×Dic₁₀₇₆	C₂²	semidirect
36. Q₁₆×D₂₆₉	C₂²	semidirect
37. C₂×#16864390	C₂²	semidirect
38. SD₁₆×D₂₆₉	C₂²	semidirect
39. D₈×D₂₆₉	C₂²	semidirect
40. #22575461	C₂²	semidirect
41. #22575460	C₂²	semidirect
42. #22575459	C₂²	semidirect
43. M₁₆×D₂₆₉	C₂²	semidirect
44. C₂×C₈×D₂₆₉	C₂²	semidirect
45. H(5,𝔽₂)×C₂₆₉	C₂²	semidirect
46. AGL(1,ℤ/8)×C₂₆₉	C₂²	semidirect
47. #135×C₂₆₉	C₂²	semidirect
48. C₂×D₈×C₂₆₉	C₂²	semidirect
49. #131×C₂₆₉	C₂²	semidirect
164. C₂₆₉	#3584	semidirect
172. #576	D₂₆₉	semidirect
182. H(5,𝔽₂)	C₂×D₂₆₉	semidirect
193. AGL(1,ℤ/8)	C₂×D₂₆₉	semidirect
202. #135	C₂×D₂₆₉	semidirect
208. C₂×D₈	C₂×D₂₆₉	semidirect
212. #131	C₂×D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771440	2
#31771422	2
#31771419	2
#31771416	2
#31771415	2
#31771406	2
#31771401	2
C₂×#22575472	2
#135×D₂₆₉	2
#131×D₂₆₉	2
#576×C₂₆₉	2
#3584	269