Subgroups — #46819797

Order 34432.

452 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₂₆₉	C₂	1	1	yes	maximal, characteristic
2	#31771370	C₂	1	2	yes	maximal, complemented
3	#31770725	C₂	1	2	yes	maximal, complemented
4	#31770724	C₂	1	2	yes	maximal, complemented
5	#31770723	C₂	1	1	yes	maximal, characteristic, complemented
6	#31770722	C₂	1	1	yes	maximal, characteristic, complemented
7	#31770721	C₂	1	1	yes	maximal, characteristic, complemented
8	#108×D₂₆₉	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	#22575476	C₄	1	1	yes	characteristic, complemented
14	#22575475	C₂²	1	1	yes	characteristic
15	#22575474	C₄	1	2	yes	complemented
16	#22575473	C₂²	1	2	yes	—
17	#22575472	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₂₆₉	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₂₆₉	C₄	1	2	yes	complemented
20	D₈×D₂₆₉	C₄	1	1	yes	characteristic, complemented
21	#22575461	—	2	2	no	—
22	#22575461	C₂²	1	2	yes	complemented
23	#22575460	C₂²	1	2	yes	complemented
24	#22575459	—	2	2	no	—
25	#22575459	C₂²	1	2	yes	complemented
26	C₂×#16864385	—	2	2	no	—
27	C₂×#16864385	C₂²	1	1	yes	characteristic
28	M₁₆×D₂₆₉	—	2	2	no	—
29	M₁₆×D₂₆₉	C₂²	1	2	yes	—
30	C₂×C₈×D₂₆₉	C₂²	1	1	yes	characteristic
31	#22575410	C₂²	1	1	yes	characteristic
32	#22575389	C₂²	1	1	yes	characteristic, complemented
33	#22575388	C₂²	1	2	yes	complemented
34	#22575381	C₂²	1	2	yes	complemented
35	#22575380	C₂²	1	2	yes	complemented
36	#22575379	C₂²	1	2	yes	complemented
37	#22575364	C₂²	1	1	yes	characteristic, complemented
38	#22575363	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
40	#135×C₂₆₉	C₂²	1	1	yes	characteristic
41	#131×C₂₆₉	C₂²	1	2	yes	complemented
42	#108×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
43	#104×C₂₆₉	C₂²	1	2	yes	complemented
44	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
45	#16864403	C₂×C₄	1	2	yes	—
46	#16864402	—	2	2	no	—
47	#16864402	C₂³	1	2	yes	—
48	#16864401	—	2	2	no	—
49	#16864401	C₂×C₄	1	2	yes	—
50	Q₈×D₂₆₉	C₂×C₄	1	2	yes	—
51	D₄×D₂₆₉	—	2	2	no	—
52	D₄×D₂₆₉	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₂₆₉	—	2	2	no	—
54	C₂×C₄×D₂₆₉	C₂³	1	1	yes	characteristic
55	Dic₁₀₇₆	C₂×C₄	1	1	yes	characteristic
56	#16864392	C₂×C₄	1	2	yes	—
57	#16864391	C₂×C₄	1	2	yes	—
58	#16864390	C₂×C₄	1	2	yes	—
59	#16864389	C₂×C₄	1	2	yes	—
60	D₂₁₅₂	C₂×C₄	1	1	yes	characteristic
61	#16864387	C₂×C₄	1	2	yes	—
62	#16864386	—	2	2	no	—
63	#16864386	C₂³	1	2	yes	—
64	#16864385	—	2	2	no	—
65	#16864385	—	2	2	no	—
66	#16864385	—	2	2	no	—
67	#16864385	D₄	1	1	yes	characteristic
68	#16864385	D₄	1	1	yes	characteristic
69	#16864385	D₄	1	1	yes	characteristic
70	#16864385	D₄	1	1	yes	characteristic
71	C₂×#14524130	—	2	2	no	—
72	C₂×#14524130	C₂³	1	1	yes	characteristic
73	C₂×#14524130	C₂³	1	1	yes	characteristic
74	C₈×D₂₆₉	—	2	2	no	—
75	C₈×D₂₆₉	—	2	2	no	—
76	C₈×D₂₆₉	—	2	2	no	—
77	C₈×D₂₆₉	C₂×C₄	1	1	yes	characteristic
78	C₈×D₂₆₉	C₂×C₄	1	1	yes	characteristic
79	C₈×D₂₆₉	C₂×C₄	1	1	yes	characteristic
80	C₈×D₂₆₉	C₂×C₄	1	1	yes	characteristic
81	#16864381	—	2	1	no	—
82	#16864379	—	2	1	no	—
83	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
84	#41×C₂₆₉	C₂³	1	2	yes	—
85	Q₁₆×C₂₆₉	C₂×C₄	1	1	yes	characteristic
86	SD₁₆×C₂₆₉	C₂×C₄	1	2	yes	—
87	D₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
88	M₁₆×C₂₆₉	—	2	2	no	—
89	M₁₆×C₂₆₉	C₂³	1	2	yes	—
90	C₂×C₈×C₂₆₉	—	2	2	no	—
91	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
93	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
94	C₂²×D₂₆₉	—	2	2	no	—
95	C₂²×D₂₆₉	#41	1	1	yes	characteristic
96	Dic₅₃₈	—	2	2	no	—
97	Dic₅₃₈	C₂²×C₄	1	2	yes	—
98	D₁₀₇₆	—	2	2	no	—
99	D₁₀₇₆	C₂²×C₄	1	2	yes	—
100	C₂²⋊D₂₆₉	—	2	2	no	—
101	C₂²⋊D₂₆₉	—	2	2	no	—
102	C₂²⋊D₂₆₉	—	2	2	no	—
103	C₂×Dic₂₆₉	—	2	2	no	—
104	C₂×Dic₂₆₉	—	2	1	no	—
105	C₂×Dic₂₆₉	#41	1	1	yes	characteristic
106	C₄×D₂₆₉	—	2	2	no	—
107	C₄×D₂₆₉	—	2	2	no	—
108	C₄×D₂₆₉	—	2	2	no	—
109	C₄×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
110	C₄×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
111	C₄×D₂₆₉	C₂×D₄	1	1	yes	characteristic
112	C₄×D₂₆₉	C₂×D₄	1	1	yes	characteristic
113	#14524130	—	2	2	no	—
114	#14524130	C₂²×C₄	1	1	yes	characteristic
115	#14524130	C₂²×C₄	1	1	yes	characteristic
116	#14524130	C₂×D₄	1	1	yes	characteristic
117	#14524130	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₂₆₉	C₂²×C₄	1	2	yes	—
119	D₄×C₂₆₉	—	2	2	no	—
120	D₄×C₂₆₉	C₂²×C₄	1	2	yes	—
121	C₂×C₄×C₂₆₉	—	2	2	no	—
122	C₂×C₄×C₂₆₉	—	2	1	no	—
123	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
124	C₈×C₂₆₉	—	2	2	no	—
125	C₈×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
126	C₈×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
127	C₈×C₂₆₉	C₂×D₄	1	1	yes	characteristic
128	C₈×C₂₆₉	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₂₆₉	—	2	2	no	—
130	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
131	C₂×D₂₆₉	—	2	2	no	—
132	C₂×D₂₆₉	—	4	1	no	—
133	C₂×D₂₆₉	—	8	2	no	—
134	C₂×D₂₆₉	C₄×D₄	1	1	yes	characteristic
135	C₂×D₂₆₉	C₄×D₄	1	1	yes	characteristic
136	C₄×C₂₆₉	—	2	2	no	—
137	C₄×C₂₆₉	—	4	1	no	—
138	C₄×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
140	Dic₂₆₉	—	2	2	no	—
141	Dic₂₆₉	—	4	1	no	—
142	Dic₂₆₉	C₄×D₄	1	1	yes	characteristic
143	Dic₂₆₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₂₆₉	—	2	1	no	—
145	C₂×C₂₆₉	—	4	2	no	—
146	C₂×C₂₆₉	C₂×C₄×D₄	1	1	yes	characteristic
147	D₂₆₉	—	2	1	no	—
148	D₂₆₉	—	2	1	no	—
149	D₂₆₉	—	4	2	no	—
150	C₂₆₉	#2957	1	1	yes	characteristic, complemented, Sylow
151	#2957	—	269	1	no	maximal, Sylow
152	C₂×#135	—	269	1	no	—
153	#568	—	269	2	no	—
154	#444	—	269	2	no	—
155	#444	—	269	2	no	—
156	#443	—	269	1	no	—
157	#443	—	269	1	no	—
158	#443	—	269	1	no	—
159	#443	D₂₆₉	1	1	yes	characteristic, complemented
160	C₂×#108	—	269	1	no	—
161	#421	—	269	2	no	—
162	#405	—	269	1	no	—
163	C₂×#41	—	269	2	no	—
164	#135	—	269	2	no	—
165	#135	—	269	1	no	—
166	#135	—	269	2	no	—
167	#135	—	269	1	no	—
168	#135	—	269	1	no	—
169	#135	C₂×D₂₆₉	1	1	yes	characteristic
170	C₂×Q₁₆	—	269	1	no	—
171	C₂×SD₁₆	—	269	2	no	—
172	C₂×D₈	—	269	1	no	—
173	#131	—	269	2	no	—
174	#131	—	269	2	no	—
175	#131	—	269	2	no	—
176	#131	—	538	2	no	—
177	#131	—	538	2	no	—
178	#131	C₂×D₂₆₉	1	2	yes	complemented
179	C₂×M₁₆	—	269	1	no	—
180	C₂×M₁₆	—	269	2	no	—
181	C₂×M₁₆	—	538	2	no	—
182	C₂×M₁₆	—	538	2	no	—
183	C₂²×C₈	—	269	1	no	—
184	#117	—	269	1	no	—
185	#108	—	269	2	no	—
186	#108	—	269	1	no	—
187	#108	C₂×D₂₆₉	1	1	yes	characteristic, complemented
188	#104	—	269	2	no	—
189	#104	—	269	2	no	—
190	#104	—	269	2	no	—
191	#104	C₂×D₂₆₉	1	2	yes	complemented
192	#97	—	269	1	no	—
193	#97	—	269	1	no	—
194	C₄×C₈	—	269	1	no	—
195	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic, complemented
196	#41	—	269	2	no	—
197	#41	—	269	2	no	—
198	#41	—	269	2	no	—
199	#41	—	538	2	no	—
200	#41	—	538	2	no	—
201	#41	C₂²×D₂₆₉	1	2	yes	—
202	C₂×Q₈	—	269	2	no	—
203	C₂×D₄	—	269	2	no	—
204	C₂×D₄	—	538	2	no	—
205	C₂²×C₄	—	269	1	no	—
206	C₂²×C₄	—	538	2	no	—
207	Q₁₆	—	269	2	no	—
208	Q₁₆	—	269	1	no	—
209	Q₁₆	C₄×D₂₆₉	1	1	yes	characteristic
210	SD₁₆	—	269	2	no	—
211	SD₁₆	—	269	2	no	—
212	SD₁₆	—	269	2	no	—
213	SD₁₆	C₄×D₂₆₉	1	2	yes	—
214	D₈	—	269	2	no	—
215	D₈	—	269	1	no	—
216	D₈	C₄×D₂₆₉	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	269	1	no	—
219	M₁₆	—	269	1	no	—
220	M₁₆	—	269	1	no	—
221	M₁₆	—	269	1	no	—
222	M₁₆	—	269	2	no	—
223	M₁₆	—	538	2	no	—
224	M₁₆	—	538	2	no	—
225	M₁₆	—	538	2	no	—
226	M₁₆	—	538	2	no	—
227	M₁₆	C₂²×D₂₆₉	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	269	1	no	—
230	C₂×C₈	—	269	1	no	—
231	C₂×C₈	—	269	1	no	—
232	C₂×C₈	—	269	1	no	—
233	C₂×C₈	—	269	1	no	—
234	C₂×C₈	—	269	1	no	—
235	C₂×C₈	—	538	2	no	—
236	C₂×C₈	—	538	2	no	—
237	C₂×C₈	—	538	2	no	—
238	C₂×C₈	—	538	2	no	—
239	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
241	C₄⋊C₄	—	538	1	no	—
242	C₂²⋊C₄	—	538	1	no	—
243	C₄²	—	269	1	no	—
244	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
245	C₂³	—	269	1	no	—
246	C₂³	—	538	2	no	—
247	Q₈	—	269	2	no	—
248	Q₈	—	538	2	no	—
249	Q₈	C₂×C₄×D₂₆₉	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	269	2	no	—
252	D₄	—	538	2	no	—
253	D₄	—	538	2	no	—
254	D₄	—	538	2	no	—
255	D₄	—	538	2	no	—
256	D₄	C₂×C₄×D₂₆₉	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	269	1	no	—
260	C₂×C₄	—	269	1	no	—
261	C₂×C₄	—	269	1	no	—
262	C₂×C₄	—	269	1	no	—
263	C₂×C₄	—	269	1	no	—
264	C₂×C₄	—	538	2	no	—
265	C₂×C₄	—	538	2	no	—
266	C₂×C₄	—	538	2	no	—
267	C₂×C₄	—	538	2	no	—
268	C₂×C₄	—	538	1	no	—
269	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	269	1	no	—
272	C₈	—	269	1	no	—
273	C₈	—	269	1	no	—
274	C₈	—	269	1	no	—
275	C₈	—	538	2	no	—
276	C₈	C₂×C₄×D₂₆₉	1	1	yes	characteristic
277	C₈	C₂×C₄×D₂₆₉	1	1	yes	characteristic
278	C₈	D₄×D₂₆₉	1	1	yes	characteristic
279	C₈	D₄×D₂₆₉	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	269	1	no	—
282	C₂²	—	269	1	no	—
283	C₂²	—	538	2	no	—
284	C₂²	—	1076	1	no	—
285	C₂²	—	2152	2	no	—
286	C₂²	#41×D₂₆₉	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	269	1	no	—
290	C₄	—	269	1	no	—
291	C₄	—	538	2	no	—
292	C₄	—	1076	1	no	—
293	C₄	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
294	C₄	C₂×D₄×D₂₆₉	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	538	1	no	—
298	C₂	—	538	1	no	—
299	C₂	—	1076	2	no	—
300	C₂	C₄×D₄×D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₆₉
commutator	C₄×C₂₆₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#443×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₂×C₄×D₄
prime radical	#135×D₂₆₉	C₂
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	#443	D₂₆₉
hypocenter	C₂₆₉	#2957
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #31771370	C₂	semidirect
3. #31770725	C₂	semidirect
4. #31770724	C₂	semidirect
5. #31770723	C₂	semidirect
6. #31770722	C₂	semidirect
7. #31770721	C₂	semidirect
8. #108×D₂₆₉	C₂	semidirect
9. #31770620	C₂	semidirect
10. #31770559	C₂	semidirect
11. #443×C₂₆₉	C₂	semidirect
13. #22575476	C₄	semidirect
15. #22575474	C₄	semidirect
17. #22575472	C₄	semidirect
18. Q₁₆×D₂₆₉	C₄	semidirect
19. SD₁₆×D₂₆₉	C₄	semidirect
20. D₈×D₂₆₉	C₄	semidirect
22. #22575461	C₂²	semidirect
23. #22575460	C₂²	semidirect
25. #22575459	C₂²	semidirect
32. #22575389	C₂²	semidirect
33. #22575388	C₂²	semidirect
34. #22575381	C₂²	semidirect
35. #22575380	C₂²	semidirect
36. #22575379	C₂²	semidirect
37. #22575364	C₂²	semidirect
38. #22575363	C₂²	semidirect
39. C₈×Dic₂₆₉	C₂²	semidirect
41. #131×C₂₆₉	C₂²	semidirect
42. #108×C₂₆₉	C₂²	semidirect
43. #104×C₂₆₉	C₂²	semidirect
44. C₄×C₈×C₂₆₉	C₂²	semidirect
150. C₂₆₉	#2957	semidirect
159. #443	D₂₆₉	semidirect
178. #131	C₂×D₂₆₉	semidirect
187. #108	C₂×D₂₆₉	semidirect
191. #104	C₂×D₂₆₉	semidirect
195. 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
#135×D₂₆₉	2
#31771370	2
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