Subgroups — #30650517

Order 16256.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550693	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550683	C₂	1	2	yes	maximal, complemented
3	#22550681	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550260	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550151	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550146	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550140	C₂	1	2	yes	maximal, complemented
8	#22550055	C₂	1	2	yes	maximal, complemented
9	#22550050	C₂	1	2	yes	maximal, complemented
10	#22549994	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16861256	C₂²	1	2	yes	complemented
13	#16861255	C₂²	1	2	yes	—
14	#16861242	—	2	2	no	—
15	#16861242	C₂²	1	2	yes	complemented
16	#16861241	—	2	2	no	—
17	#16861239	C₂²	1	2	yes	—
18	#16861237	—	2	2	no	—
19	#16861237	C₂²	1	2	yes	—
20	#16861234	C₂²	1	2	yes	complemented
21	#16861233	—	2	2	no	—
22	#16861231	—	2	1	no	—
23	#16861231	—	2	1	no	—
24	#16861231	C₂²	1	1	yes	characteristic
25	#16861230	—	2	2	no	—
26	C₂×Dic₅₀₈	—	2	1	no	—
27	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic, complemented
28	C₂×#14523237	C₂²	1	2	yes	complemented
29	SD₁₆×D₁₂₇	—	2	2	no	—
30	C₂×D₁₀₁₆	—	2	1	no	—
31	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic
32	#16861215	C₂²	1	2	yes	complemented
33	#16861197	C₂²	1	2	yes	complemented
34	#16861145	C₂²	1	1	yes	characteristic, complemented
35	#16861137	C₂²	1	2	yes	complemented
36	#16861127	C₂²	1	2	yes	complemented
37	#16861125	C₂²	1	2	yes	complemented
38	#135×C₁₂₇	C₂²	1	1	yes	characteristic
39	#131×C₁₂₇	C₂²	1	1	yes	characteristic
40	#131×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
41	#108×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
42	#104×C₁₂₇	C₂²	1	1	yes	characteristic
43	#104×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
45	#14523249	—	2	2	no	—
46	#14523248	—	2	2	no	—
47	#14523248	—	2	2	no	—
48	#14523248	C₂³	1	2	yes	—
49	#14523247	—	2	2	no	—
50	#14523247	—	4	2	no	—
51	C₂×Dic₂₅₄	—	2	2	no	—
52	C₂×Dic₂₅₄	C₂³	1	2	yes	—
53	Q₈×D₁₂₇	—	2	2	no	—
54	C₂×D₅₀₈	—	2	2	no	—
55	C₂×D₅₀₈	C₂³	1	2	yes	—
56	D₄×D₁₂₇	—	2	2	no	—
57	D₄×D₁₂₇	—	4	2	no	—
58	Dic₅₀₈	—	2	2	no	—
59	Dic₅₀₈	—	2	1	no	—
60	Dic₅₀₈	—	2	1	no	—
61	Dic₅₀₈	D₄	1	1	yes	characteristic
62	#14523238	—	2	2	no	—
63	#14523237	—	2	2	no	—
64	#14523237	—	2	2	no	—
65	#14523237	—	2	2	no	—
66	#14523237	—	2	2	no	—
67	#14523237	D₄	1	2	yes	—
68	#14523236	—	2	2	no	—
69	#14523235	—	2	2	no	—
70	D₁₀₁₆	—	2	2	no	—
71	D₁₀₁₆	—	2	1	no	—
72	D₁₀₁₆	—	2	1	no	—
73	D₁₀₁₆	D₄	1	1	yes	characteristic
74	#14523233	—	2	2	no	—
75	#14523232	C₂³	1	2	yes	—
76	#14523231	—	2	2	no	—
77	C₈×D₁₂₇	—	2	2	no	—
78	(C₂×D₁₂₇)⋊C₄	—	4	2	no	—
79	#41×C₁₂₇	C₂³	1	1	yes	characteristic
80	#41×C₁₂₇	C₂³	1	1	yes	characteristic
81	Q₁₆×C₁₂₇	D₄	1	1	yes	characteristic
82	SD₁₆×C₁₂₇	D₄	1	1	yes	characteristic
83	SD₁₆×C₁₂₇	D₄	1	1	yes	characteristic
84	D₈×C₁₂₇	D₄	1	1	yes	characteristic
85	M₁₆×C₁₂₇	—	2	1	no	—
86	M₁₆×C₁₂₇	—	2	1	no	—
87	M₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
88	M₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₁₂₇	—	2	1	no	—
90	C₂×C₈×C₁₂₇	—	2	1	no	—
91	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
93	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
94	C₂²×D₁₂₇	—	2	2	no	—
95	C₂²×D₁₂₇	—	4	2	no	—
96	Dic₂₅₄	—	2	2	no	—
97	Dic₂₅₄	—	2	2	no	—
98	Dic₂₅₄	—	4	2	no	—
99	Dic₂₅₄	C₂×D₄	1	2	yes	—
100	D₅₀₈	—	2	2	no	—
101	D₅₀₈	—	2	2	no	—
102	D₅₀₈	—	4	2	no	—
103	D₅₀₈	C₂×D₄	1	2	yes	—
104	C₂²⋊D₁₂₇	—	2	2	no	—
105	C₂²⋊D₁₂₇	—	4	2	no	—
106	C₂²⋊D₁₂₇	—	4	2	no	—
107	C₂×Dic₁₂₇	—	2	2	no	—
108	C₂×Dic₁₂₇	—	4	2	no	—
109	C₄×D₁₂₇	—	2	2	no	—
110	C₄×D₁₂₇	—	4	2	no	—
111	C₄×D₁₂₇	—	4	2	no	—
112	C₁₂₇⋊C₈	—	2	2	no	—
113	Q₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
115	D₄×C₁₂₇	—	2	1	no	—
116	D₄×C₁₂₇	—	2	1	no	—
117	D₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
118	D₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₂₇	—	2	1	no	—
120	C₂×C₄×C₁₂₇	—	2	1	no	—
121	C₂×C₄×C₁₂₇	—	2	1	no	—
122	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
123	C₈×C₁₂₇	—	2	1	no	—
124	C₈×C₁₂₇	—	2	1	no	—
125	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
126	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
127	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
128	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₁₂₇	—	2	1	no	—
130	C₂²×C₁₂₇	—	2	1	no	—
131	C₂²×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×D₁₂₇	—	2	2	no	—
133	C₂×D₁₂₇	—	4	2	no	—
134	C₂×D₁₂₇	—	8	2	no	—
135	C₂×D₁₂₇	—	16	2	no	—
136	C₄×C₁₂₇	—	2	1	no	—
137	C₄×C₁₂₇	—	2	1	no	—
138	C₄×C₁₂₇	—	4	1	no	—
139	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
141	Dic₁₂₇	—	2	2	no	—
142	Dic₁₂₇	—	4	2	no	—
143	C₂×C₁₂₇	—	2	1	no	—
144	C₂×C₁₂₇	—	4	1	no	—
145	C₂×C₁₂₇	—	4	1	no	—
146	C₂×C₁₂₇	D₄²	1	1	yes	characteristic
147	D₁₂₇	—	4	2	no	—
148	D₁₂₇	—	8	2	no	—
149	#3290	—	127	1	no	maximal, Sylow
150	C₁₂₇	#3290	1	1	yes	characteristic, complemented, Sylow
151	#578	—	127	1	no	—
152	#577	—	127	2	no	—
153	#576	—	127	1	no	—
154	#495	—	127	1	no	—
155	#473	—	127	1	no	—
156	#472	—	127	1	no	—
157	#471	—	127	2	no	—
158	#455	—	127	2	no	—
159	#454	—	127	2	no	—
160	#443	—	127	1	no	—
161	#443	D₁₂₇	1	1	yes	characteristic, complemented
162	2₋¹⁺⁴	—	127	2	no	—
163	H(5,𝔽₂)	—	127	2	no	—
164	#137	—	127	2	no	—
165	#137	—	127	2	no	—
166	#137	—	254	2	no	—
167	#137	—	254	2	no	—
168	AGL(1,ℤ/8)	—	127	2	no	—
169	AGL(1,ℤ/8)	—	127	2	no	—
170	AGL(1,ℤ/8)	—	254	2	no	—
171	AGL(1,ℤ/8)	—	254	2	no	—
172	#135	—	127	1	no	—
173	#135	—	254	1	no	—
174	#135	—	254	1	no	—
175	#135	—	254	2	no	—
176	#135	C₂×D₁₂₇	1	1	yes	characteristic
177	C₂×Q₁₆	—	127	1	no	—
178	C₂×Q₁₆	—	254	1	no	—
179	C₂×SD₁₆	—	127	2	no	—
180	C₂×SD₁₆	—	254	2	no	—
181	C₂×D₈	—	127	1	no	—
182	C₂×D₈	—	254	1	no	—
183	#131	—	127	2	no	—
184	#131	C₂×D₁₂₇	1	1	yes	characteristic
185	#131	C₂×D₁₂₇	1	1	yes	characteristic, complemented
186	#124	—	127	2	no	—
187	#108	—	127	1	no	—
188	#108	C₂×D₁₂₇	1	1	yes	characteristic, complemented
189	#104	—	127	2	no	—
190	#104	C₂×D₁₂₇	1	1	yes	characteristic
191	#104	C₂×D₁₂₇	1	1	yes	characteristic, complemented
192	#101	—	127	2	no	—
193	#100	—	127	2	no	—
194	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
195	#41	—	127	2	no	—
196	#41	—	508	2	no	—
197	#41	—	254	2	no	—
198	#41	—	254	2	no	—
199	#41	—	254	2	no	—
200	#41	—	254	2	no	—
201	#41	C₂²×D₁₂₇	1	1	yes	characteristic
202	#41	C₂²×D₁₂₇	1	1	yes	characteristic
203	C₂×Q₈	—	127	2	no	—
204	C₂×Q₈	—	254	2	no	—
205	C₂×Q₈	—	254	2	no	—
206	C₂×D₄	—	127	2	no	—
207	C₂×D₄	—	508	2	no	—
208	C₂×D₄	—	254	2	no	—
209	C₂×D₄	—	254	2	no	—
210	Q₁₆	—	127	1	no	—
211	Q₁₆	—	254	2	no	—
212	Q₁₆	—	254	2	no	—
213	Q₁₆	—	254	1	no	—
214	Q₁₆	—	254	1	no	—
215	Q₁₆	D₅₀₈	1	1	yes	characteristic
216	SD₁₆	—	127	2	no	—
217	SD₁₆	—	254	2	no	—
218	SD₁₆	—	254	2	no	—
219	SD₁₆	—	254	2	no	—
220	SD₁₆	—	254	2	no	—
221	SD₁₆	—	254	2	no	—
222	SD₁₆	—	254	2	no	—
223	SD₁₆	D₅₀₈	1	1	yes	characteristic
224	SD₁₆	D₅₀₈	1	1	yes	characteristic
225	D₈	—	127	1	no	—
226	D₈	—	254	2	no	—
227	D₈	—	254	2	no	—
228	D₈	—	254	1	no	—
229	D₈	—	254	1	no	—
230	D₈	D₅₀₈	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	127	2	no	—
234	M₁₆	—	254	2	no	—
235	M₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
236	M₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	254	2	no	—
240	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
242	C₂²⋊C₄	—	508	2	no	—
243	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
244	C₂³	—	508	2	no	—
245	C₂³	—	254	2	no	—
246	Q₈	—	127	2	no	—
247	Q₈	—	508	2	no	—
248	Q₈	—	254	2	no	—
249	Q₈	—	254	2	no	—
250	Q₈	C₂×D₅₀₈	1	1	yes	characteristic
251	Q₈	C₂×D₅₀₈	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	127	2	no	—
255	D₄	—	508	2	no	—
256	D₄	—	508	2	no	—
257	D₄	—	508	2	no	—
258	D₄	—	254	2	no	—
259	D₄	—	254	2	no	—
260	D₄	—	254	2	no	—
261	D₄	C₂×D₅₀₈	1	1	yes	characteristic
262	D₄	C₂×D₅₀₈	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	508	2	no	—
267	C₂×C₄	—	508	2	no	—
268	C₂×C₄	—	508	2	no	—
269	C₂×C₄	—	254	2	no	—
270	C₂×C₄	—	254	2	no	—
271	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	254	2	no	—
275	C₈	D₄×D₁₂₇	1	1	yes	characteristic
276	C₈	D₄×D₁₂₇	1	1	yes	characteristic
277	C₈	C₂×D₅₀₈	1	1	yes	characteristic
278	C₈	C₂×D₅₀₈	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2032	2	no	—
282	C₂²	—	1016	2	no	—
283	C₂²	—	508	2	no	—
284	C₂²	—	254	2	no	—
285	C₂²	#16861255	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	508	2	no	—
290	C₄	—	254	2	no	—
291	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
292	C₄	C₂²×D₅₀₈	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	1016	2	no	—
297	C₂	—	508	2	no	—
298	C₂	D₄×D₅₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#443×C₁₂₇	C₂
socle	C₂×C₁₂₇	D₄²
hypercenter	#443	D₁₂₇
hypocenter	C₁₂₇	#3290
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #22550693	C₂	semidirect
2. #22550683	C₂	semidirect
3. #22550681	C₂	semidirect
4. #22550260	C₂	semidirect
5. #22550151	C₂	semidirect
6. #22550146	C₂	semidirect
7. #22550140	C₂	semidirect
8. #22550055	C₂	semidirect
9. #22550050	C₂	semidirect
10. #22549994	C₂	semidirect
11. #443×C₁₂₇	C₂	semidirect
12. #16861256	C₂²	semidirect
15. #16861242	C₂²	semidirect
20. #16861234	C₂²	semidirect
27. C₂×Dic₅₀₈	C₂²	semidirect
28. C₂×#14523237	C₂²	semidirect
32. #16861215	C₂²	semidirect
33. #16861197	C₂²	semidirect
34. #16861145	C₂²	semidirect
35. #16861137	C₂²	semidirect
36. #16861127	C₂²	semidirect
37. #16861125	C₂²	semidirect
40. #131×C₁₂₇	C₂²	semidirect
41. #108×C₁₂₇	C₂²	semidirect
43. #104×C₁₂₇	C₂²	semidirect
44. C₄×C₈×C₁₂₇	C₂²	semidirect
150. C₁₂₇	#3290	semidirect
161. #443	D₁₂₇	semidirect
185. #131	C₂×D₁₂₇	semidirect
188. #108	C₂×D₁₂₇	semidirect
191. #104	C₂×D₁₂₇	semidirect
194. C₄×C₈	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550693	2
#22550683	2
#22550681	2
#22550260	2
#22550151	2
#22550146	2
#22550140	2
#22550055	2
#22550050	2
#22549994	2
#443×C₁₂₇	2
#3290	127