Subgroups — #46819783

Order 34432.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771402	C₂	1	1	yes	maximal, characteristic
2	#131×D₂₆₉	C₂	1	2	yes	maximal, complemented
3	#31770727	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770726	C₂	1	1	yes	maximal, characteristic, complemented
5	#31770724	C₂	1	2	yes	maximal, complemented
6	#31770723	C₂	1	1	yes	maximal, characteristic, complemented
7	#31770720	C₂	1	2	yes	maximal, complemented
8	#31770652	C₂	1	1	yes	maximal, characteristic, complemented
9	#31770620	C₂	1	2	yes	maximal, complemented
10	#31770559	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₂₆₉	C₂²	1	2	yes	—
13	#22575485	C₄	1	2	yes	complemented
14	#22575482	C₄	1	2	yes	complemented
15	#22575480	C₄	1	2	yes	complemented
16	#22575477	C₄	1	2	yes	complemented
17	#22575475	C₂²	1	1	yes	characteristic
18	#22575473	C₂²	1	2	yes	—
19	#22575461	—	2	2	no	—
20	#22575461	C₂²	1	2	yes	complemented
21	#22575460	C₂²	1	2	yes	complemented
22	#22575459	—	2	2	no	—
23	#22575459	C₂²	1	2	yes	complemented
24	C₂×#16864385	C₂²	1	1	yes	characteristic
25	M₁₆×D₂₆₉	—	2	2	no	—
26	M₁₆×D₂₆₉	C₂²	1	2	yes	—
27	C₂×C₈×D₂₆₉	—	2	2	no	—
28	C₂×C₈×D₂₆₉	C₂²	1	1	yes	characteristic
29	#22575410	C₂²	1	1	yes	characteristic
30	#22575389	C₂²	1	1	yes	characteristic, complemented
31	#22575388	C₂²	1	2	yes	complemented
32	#22575381	C₂²	1	2	yes	complemented
33	#22575380	C₂²	1	2	yes	complemented
34	#22575379	C₂²	1	2	yes	complemented
35	#22575364	C₂²	1	1	yes	characteristic, complemented
36	#22575363	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
38	#135×C₂₆₉	C₂²	1	1	yes	characteristic
39	#131×C₂₆₉	C₂²	1	2	yes	complemented
40	#108×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
41	#104×C₂₆₉	C₂²	1	2	yes	complemented
42	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
43	#16864403	C₂×C₄	1	2	yes	—
44	#16864402	—	2	2	no	—
45	#16864402	C₂³	1	2	yes	—
46	#16864401	—	2	2	no	—
47	#16864401	C₂×C₄	1	2	yes	—
48	Q₈×D₂₆₉	C₂×C₄	1	2	yes	—
49	D₄×D₂₆₉	—	2	2	no	—
50	D₄×D₂₆₉	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₂₆₉	—	2	2	no	—
52	C₂×C₄×D₂₆₉	C₂³	1	1	yes	characteristic
53	Dic₁₀₇₆	C₂×C₄	1	1	yes	characteristic
54	#16864392	C₂×C₄	1	2	yes	—
55	#16864391	C₂×C₄	1	2	yes	—
56	#16864390	C₂×C₄	1	2	yes	—
57	#16864389	C₂×C₄	1	2	yes	—
58	D₂₁₅₂	C₂×C₄	1	1	yes	characteristic
59	#16864387	C₂×C₄	1	2	yes	—
60	#16864386	—	2	2	no	—
61	#16864386	C₂³	1	2	yes	—
62	#16864385	—	2	2	no	—
63	#16864385	—	2	2	no	—
64	#16864385	—	2	2	no	—
65	#16864385	C₂×C₄	1	2	yes	—
66	#16864385	C₂×C₄	1	2	yes	—
67	C₂×#14524130	—	2	2	no	—
68	C₂×#14524130	C₂³	1	1	yes	characteristic
69	C₂×#14524130	C₂³	1	1	yes	characteristic
70	C₈×D₂₆₉	—	2	2	no	—
71	C₈×D₂₆₉	—	2	2	no	—
72	C₈×D₂₆₉	—	2	2	no	—
73	C₈×D₂₆₉	D₄	1	2	yes	—
74	C₈×D₂₆₉	D₄	1	2	yes	—
75	#16864381	—	2	1	no	—
76	#16864379	—	2	1	no	—
77	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
78	#41×C₂₆₉	C₂³	1	2	yes	—
79	Q₁₆×C₂₆₉	C₂×C₄	1	1	yes	characteristic
80	SD₁₆×C₂₆₉	C₂×C₄	1	2	yes	—
81	D₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
82	M₁₆×C₂₆₉	—	2	2	no	—
83	M₁₆×C₂₆₉	C₂³	1	2	yes	—
84	C₂×C₈×C₂₆₉	—	2	2	no	—
85	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
87	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
88	C₂²×D₂₆₉	—	2	2	no	—
89	C₂²×D₂₆₉	#41	1	1	yes	characteristic
90	Dic₅₃₈	—	2	2	no	—
91	Dic₅₃₈	C₂²×C₄	1	2	yes	—
92	D₁₀₇₆	—	2	2	no	—
93	D₁₀₇₆	C₂²×C₄	1	2	yes	—
94	C₂²⋊D₂₆₉	—	2	2	no	—
95	C₂²⋊D₂₆₉	—	2	2	no	—
96	C₂²⋊D₂₆₉	—	2	2	no	—
97	C₂×Dic₂₆₉	—	2	2	no	—
98	C₂×Dic₂₆₉	—	2	1	no	—
99	C₂×Dic₂₆₉	#41	1	1	yes	characteristic
100	C₄×D₂₆₉	—	2	2	no	—
101	C₄×D₂₆₉	—	2	2	no	—
102	C₄×D₂₆₉	—	2	2	no	—
103	C₄×D₂₆₉	C₂²×C₄	1	2	yes	—
104	C₄×D₂₆₉	C₂×D₄	1	2	yes	—
105	#14524130	—	2	2	no	—
106	#14524130	C₂²×C₄	1	2	yes	—
107	#14524130	C₂×D₄	1	2	yes	—
108	Q₈×C₂₆₉	C₂²×C₄	1	2	yes	—
109	D₄×C₂₆₉	—	2	2	no	—
110	D₄×C₂₆₉	C₂²×C₄	1	2	yes	—
111	C₂×C₄×C₂₆₉	—	2	2	no	—
112	C₂×C₄×C₂₆₉	—	2	1	no	—
113	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
114	C₈×C₂₆₉	—	2	2	no	—
115	C₈×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
116	C₈×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
117	C₈×C₂₆₉	C₂×D₄	1	1	yes	characteristic
118	C₈×C₂₆₉	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₂₆₉	—	2	2	no	—
120	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
121	C₂×D₂₆₉	—	2	2	no	—
122	C₂×D₂₆₉	—	4	1	no	—
123	C₂×D₂₆₉	—	4	4	no	—
124	C₂×D₂₆₉	C₄×D₄	1	2	yes	—
125	C₄×C₂₆₉	—	2	2	no	—
126	C₄×C₂₆₉	—	4	1	no	—
127	C₄×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
129	Dic₂₆₉	—	2	2	no	—
130	Dic₂₆₉	—	4	1	no	—
131	Dic₂₆₉	C₄×D₄	1	2	yes	—
132	C₂×C₂₆₉	—	2	1	no	—
133	C₂×C₂₆₉	—	4	2	no	—
134	C₂×C₂₆₉	C₂×C₄×D₄	1	1	yes	characteristic
135	D₂₆₉	—	2	2	no	—
136	D₂₆₉	—	4	2	no	—
137	C₂₆₉	#2955	1	1	yes	characteristic, complemented, Sylow
138	#2955	—	269	1	no	maximal, Sylow
139	#575	—	269	1	no	—
140	C₂×#131	—	269	2	no	—
141	#444	—	269	1	no	—
142	#444	—	269	1	no	—
143	#444	—	269	2	no	—
144	#443	—	269	2	no	—
145	#443	—	269	1	no	—
146	#443	D₂₆₉	1	1	yes	characteristic, complemented
147	#430	—	269	1	no	—
148	#421	—	269	2	no	—
149	#405	—	269	1	no	—
150	C₂×#41	—	269	2	no	—
151	#137	—	269	2	no	—
152	#137	—	269	2	no	—
153	AGL(1,ℤ/8)	—	269	2	no	—
154	AGL(1,ℤ/8)	—	269	2	no	—
155	#135	—	269	2	no	—
156	#135	—	269	1	no	—
157	#135	C₂×D₂₆₉	1	1	yes	characteristic
158	#131	—	269	2	no	—
159	#131	—	269	2	no	—
160	#131	—	269	2	no	—
161	#131	—	538	2	no	—
162	#131	—	538	2	no	—
163	#131	C₂×D₂₆₉	1	2	yes	complemented
164	C₂×M₁₆	—	269	1	no	—
165	C₂×M₁₆	—	269	2	no	—
166	C₂×M₁₆	—	538	2	no	—
167	C₂²×C₈	—	269	1	no	—
168	C₂²×C₈	—	538	2	no	—
169	#117	—	269	1	no	—
170	#108	—	269	2	no	—
171	#108	—	269	1	no	—
172	#108	C₂×D₂₆₉	1	1	yes	characteristic, complemented
173	#104	—	269	2	no	—
174	#104	—	269	2	no	—
175	#104	—	269	2	no	—
176	#104	C₂×D₂₆₉	1	2	yes	complemented
177	#97	—	269	1	no	—
178	#97	—	269	1	no	—
179	C₄×C₈	—	269	1	no	—
180	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic, complemented
181	#41	—	269	2	no	—
182	#41	—	269	2	no	—
183	#41	—	269	2	no	—
184	#41	—	538	2	no	—
185	#41	—	538	2	no	—
186	#41	C₂²×D₂₆₉	1	2	yes	—
187	C₂×Q₈	—	269	2	no	—
188	C₂×D₄	—	269	2	no	—
189	C₂×D₄	—	538	2	no	—
190	C₂²×C₄	—	269	1	no	—
191	C₂²×C₄	—	538	2	no	—
192	Q₁₆	—	269	2	no	—
193	Q₁₆	—	269	1	no	—
194	Q₁₆	C₄×D₂₆₉	1	1	yes	characteristic
195	SD₁₆	—	269	2	no	—
196	SD₁₆	—	269	2	no	—
197	SD₁₆	—	269	2	no	—
198	SD₁₆	C₄×D₂₆₉	1	2	yes	—
199	D₈	—	269	2	no	—
200	D₈	—	269	1	no	—
201	D₈	C₄×D₂₆₉	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	269	2	no	—
204	M₁₆	—	269	2	no	—
205	M₁₆	—	269	2	no	—
206	M₁₆	—	538	2	no	—
207	M₁₆	—	538	2	no	—
208	M₁₆	—	538	2	no	—
209	M₁₆	—	538	2	no	—
210	M₁₆	C₂²×D₂₆₉	1	2	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	269	1	no	—
213	C₂×C₈	—	269	1	no	—
214	C₂×C₈	—	269	2	no	—
215	C₂×C₈	—	269	2	no	—
216	C₂×C₈	—	538	2	no	—
217	C₂×C₈	—	538	2	no	—
218	C₂×C₈	—	538	2	no	—
219	C₂×C₈	—	538	2	no	—
220	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
222	C₄⋊C₄	—	538	1	no	—
223	C₂²⋊C₄	—	538	1	no	—
224	C₄²	—	269	1	no	—
225	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
226	C₂³	—	269	1	no	—
227	C₂³	—	538	2	no	—
228	Q₈	—	269	2	no	—
229	Q₈	—	538	2	no	—
230	Q₈	C₂×C₄×D₂₆₉	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	269	2	no	—
233	D₄	—	538	2	no	—
234	D₄	—	538	2	no	—
235	D₄	—	538	2	no	—
236	D₄	—	538	2	no	—
237	D₄	C₂×C₄×D₂₆₉	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	269	1	no	—
241	C₂×C₄	—	269	2	no	—
242	C₂×C₄	—	269	2	no	—
243	C₂×C₄	—	538	2	no	—
244	C₂×C₄	—	538	2	no	—
245	C₂×C₄	—	538	2	no	—
246	C₂×C₄	—	538	2	no	—
247	C₂×C₄	—	538	1	no	—
248	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	269	2	no	—
251	C₈	—	269	2	no	—
252	C₈	—	538	2	no	—
253	C₈	C₂×C₄×D₂₆₉	1	1	yes	characteristic
254	C₈	C₂×C₄×D₂₆₉	1	1	yes	characteristic
255	C₈	D₄×D₂₆₉	1	1	yes	characteristic
256	C₈	D₄×D₂₆₉	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	269	2	no	—
259	C₂²	—	538	2	no	—
260	C₂²	—	1076	1	no	—
261	C₂²	—	1076	4	no	—
262	C₂²	#41×D₂₆₉	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	269	2	no	—
266	C₄	—	538	2	no	—
267	C₄	—	1076	1	no	—
268	C₄	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
269	C₄	C₂×D₄×D₂₆₉	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	538	2	no	—
273	C₂	—	1076	2	no	—
274	C₂	C₄×D₄×D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×D₂₆₉
commutator	C₄×C₂₆₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#443×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₂×C₄×D₄
prime radical	#31771402	C₂
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	#443	D₂₆₉
hypocenter	C₂₆₉	#2955
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #131×D₂₆₉	C₂	semidirect
3. #31770727	C₂	semidirect
4. #31770726	C₂	semidirect
5. #31770724	C₂	semidirect
6. #31770723	C₂	semidirect
7. #31770720	C₂	semidirect
8. #31770652	C₂	semidirect
9. #31770620	C₂	semidirect
10. #31770559	C₂	semidirect
11. #443×C₂₆₉	C₂	semidirect
13. #22575485	C₄	semidirect
14. #22575482	C₄	semidirect
15. #22575480	C₄	semidirect
16. #22575477	C₄	semidirect
20. #22575461	C₂²	semidirect
21. #22575460	C₂²	semidirect
23. #22575459	C₂²	semidirect
30. #22575389	C₂²	semidirect
31. #22575388	C₂²	semidirect
32. #22575381	C₂²	semidirect
33. #22575380	C₂²	semidirect
34. #22575379	C₂²	semidirect
35. #22575364	C₂²	semidirect
36. #22575363	C₂²	semidirect
37. C₈×Dic₂₆₉	C₂²	semidirect
39. #131×C₂₆₉	C₂²	semidirect
40. #108×C₂₆₉	C₂²	semidirect
41. #104×C₂₆₉	C₂²	semidirect
42. C₄×C₈×C₂₆₉	C₂²	semidirect
137. C₂₆₉	#2955	semidirect
146. #443	D₂₆₉	semidirect
163. #131	C₂×D₂₆₉	semidirect
172. #108	C₂×D₂₆₉	semidirect
176. #104	C₂×D₂₆₉	semidirect
180. C₄×C₈	C₂×D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771402	2
#131×D₂₆₉	2
#31770727	2
#31770726	2
#31770724	2
#31770723	2
#31770720	2
#31770652	2
#31770620	2
#31770559	2
#443×C₂₆₉	2
#2955	269