Subgroups — #30647642

Order 16256.

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×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#22550641	C₂	1	2	yes	maximal, complemented
3	#22549996	C₂	1	2	yes	maximal, complemented
4	#22549995	C₂	1	2	yes	maximal, complemented
5	#22549994	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549993	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549992	C₂	1	1	yes	maximal, characteristic, complemented
8	#108×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549891	C₂	1	2	yes	maximal, complemented
10	#22549830	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₂₇	C₂²	1	2	yes	—
13	#16861232	C₄	1	1	yes	characteristic, complemented
14	#16861231	C₂²	1	1	yes	characteristic
15	#16861230	C₄	1	2	yes	complemented
16	#16861229	C₂²	1	2	yes	—
17	#16861228	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₁₂₇	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₁₂₇	C₄	1	2	yes	complemented
20	D₈×D₁₂₇	C₄	1	1	yes	characteristic, complemented
21	#16861217	—	2	2	no	—
22	#16861217	C₂²	1	2	yes	complemented
23	#16861216	C₂²	1	2	yes	complemented
24	#16861215	—	2	2	no	—
25	#16861215	C₂²	1	2	yes	complemented
26	C₂×#14523231	—	2	2	no	—
27	C₂×#14523231	C₂²	1	1	yes	characteristic
28	M₁₆×D₁₂₇	—	2	2	no	—
29	M₁₆×D₁₂₇	C₂²	1	2	yes	—
30	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
31	#16861166	C₂²	1	1	yes	characteristic
32	#16861145	C₂²	1	1	yes	characteristic, complemented
33	#16861144	C₂²	1	2	yes	complemented
34	#16861137	C₂²	1	2	yes	complemented
35	#16861136	C₂²	1	2	yes	complemented
36	#16861135	C₂²	1	2	yes	complemented
37	#16861120	C₂²	1	1	yes	characteristic, complemented
38	#16861119	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic, complemented
40	#135×C₁₂₇	C₂²	1	1	yes	characteristic
41	#131×C₁₂₇	C₂²	1	2	yes	complemented
42	#108×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
43	#104×C₁₂₇	C₂²	1	2	yes	complemented
44	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
45	#14523249	C₂×C₄	1	2	yes	—
46	#14523248	—	2	2	no	—
47	#14523248	C₂³	1	2	yes	—
48	#14523247	—	2	2	no	—
49	#14523247	C₂×C₄	1	2	yes	—
50	Q₈×D₁₂₇	C₂×C₄	1	2	yes	—
51	D₄×D₁₂₇	—	2	2	no	—
52	D₄×D₁₂₇	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₂₇	—	2	2	no	—
54	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
55	Dic₅₀₈	C₂×C₄	1	1	yes	characteristic
56	#14523238	C₂×C₄	1	2	yes	—
57	#14523237	C₂×C₄	1	2	yes	—
58	#14523236	C₂×C₄	1	2	yes	—
59	#14523235	C₂×C₄	1	2	yes	—
60	D₁₀₁₆	C₂×C₄	1	1	yes	characteristic
61	#14523233	C₂×C₄	1	2	yes	—
62	#14523232	—	2	2	no	—
63	#14523232	C₂³	1	2	yes	—
64	#14523231	—	2	2	no	—
65	#14523231	—	2	2	no	—
66	#14523231	—	2	2	no	—
67	#14523231	D₄	1	1	yes	characteristic
68	#14523231	D₄	1	1	yes	characteristic
69	#14523231	D₄	1	1	yes	characteristic
70	#14523231	D₄	1	1	yes	characteristic
71	C₂×C₁₂₇⋊C₈	—	2	2	no	—
72	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×D₁₂₇	—	2	2	no	—
75	C₈×D₁₂₇	—	2	2	no	—
76	C₈×D₁₂₇	—	2	2	no	—
77	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
78	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
79	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
80	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
81	Dic₁₂₇⋊C₄	—	2	1	no	—
82	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
83	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
84	#41×C₁₂₇	C₂³	1	2	yes	—
85	Q₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
86	SD₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
87	D₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
88	M₁₆×C₁₂₇	—	2	2	no	—
89	M₁₆×C₁₂₇	C₂³	1	2	yes	—
90	C₂×C₈×C₁₂₇	—	2	2	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	#41	1	1	yes	characteristic
96	Dic₂₅₄	—	2	2	no	—
97	Dic₂₅₄	C₂²×C₄	1	2	yes	—
98	D₅₀₈	—	2	2	no	—
99	D₅₀₈	C₂²×C₄	1	2	yes	—
100	C₂²⋊D₁₂₇	—	2	2	no	—
101	C₂²⋊D₁₂₇	—	2	2	no	—
102	C₂²⋊D₁₂₇	—	2	2	no	—
103	C₂×Dic₁₂₇	—	2	2	no	—
104	C₂×Dic₁₂₇	—	2	1	no	—
105	C₂×Dic₁₂₇	#41	1	1	yes	characteristic
106	C₄×D₁₂₇	—	2	2	no	—
107	C₄×D₁₂₇	—	2	2	no	—
108	C₄×D₁₂₇	—	2	2	no	—
109	C₄×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
110	C₄×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
111	C₄×D₁₂₇	C₂×D₄	1	1	yes	characteristic
112	C₄×D₁₂₇	C₂×D₄	1	1	yes	characteristic
113	C₁₂₇⋊C₈	—	2	2	no	—
114	C₁₂₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
115	C₁₂₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
116	C₁₂₇⋊C₈	C₂×D₄	1	1	yes	characteristic
117	C₁₂₇⋊C₈	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
119	D₄×C₁₂₇	—	2	2	no	—
120	D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
121	C₂×C₄×C₁₂₇	—	2	2	no	—
122	C₂×C₄×C₁₂₇	—	2	1	no	—
123	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
124	C₈×C₁₂₇	—	2	2	no	—
125	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
126	C₈×C₁₂₇	C₂²×C₄	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	2	no	—
130	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
131	C₂×D₁₂₇	—	2	2	no	—
132	C₂×D₁₂₇	—	4	1	no	—
133	C₂×D₁₂₇	—	8	2	no	—
134	C₂×D₁₂₇	C₄×D₄	1	1	yes	characteristic
135	C₂×D₁₂₇	C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₂₇	—	2	2	no	—
137	C₄×C₁₂₇	—	4	1	no	—
138	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
140	Dic₁₂₇	—	2	2	no	—
141	Dic₁₂₇	—	4	1	no	—
142	Dic₁₂₇	C₄×D₄	1	1	yes	characteristic
143	Dic₁₂₇	C₄×D₄	1	1	yes	characteristic
144	C₂×C₁₂₇	—	2	1	no	—
145	C₂×C₁₂₇	—	4	2	no	—
146	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
147	D₁₂₇	—	2	1	no	—
148	D₁₂₇	—	2	1	no	—
149	D₁₂₇	—	4	2	no	—
150	#2957	—	127	1	no	maximal, Sylow
151	C₁₂₇	#2957	1	1	yes	characteristic, complemented, Sylow
152	C₂×#135	—	127	1	no	—
153	#568	—	127	2	no	—
154	#444	—	127	2	no	—
155	#444	—	127	2	no	—
156	#443	—	127	1	no	—
157	#443	—	127	1	no	—
158	#443	—	127	1	no	—
159	#443	D₁₂₇	1	1	yes	characteristic, complemented
160	C₂×#108	—	127	1	no	—
161	#421	—	127	2	no	—
162	#405	—	127	1	no	—
163	C₂×#41	—	127	2	no	—
164	#135	—	127	2	no	—
165	#135	—	127	1	no	—
166	#135	—	127	2	no	—
167	#135	—	127	1	no	—
168	#135	—	127	1	no	—
169	#135	C₂×D₁₂₇	1	1	yes	characteristic
170	C₂×Q₁₆	—	127	1	no	—
171	C₂×SD₁₆	—	127	2	no	—
172	C₂×D₈	—	127	1	no	—
173	#131	—	127	2	no	—
174	#131	—	127	2	no	—
175	#131	—	127	2	no	—
176	#131	—	254	2	no	—
177	#131	—	254	2	no	—
178	#131	C₂×D₁₂₇	1	2	yes	complemented
179	C₂×M₁₆	—	127	1	no	—
180	C₂×M₁₆	—	127	2	no	—
181	C₂×M₁₆	—	254	2	no	—
182	C₂×M₁₆	—	254	2	no	—
183	C₂²×C₈	—	127	1	no	—
184	#117	—	127	1	no	—
185	#108	—	127	2	no	—
186	#108	—	127	1	no	—
187	#108	C₂×D₁₂₇	1	1	yes	characteristic, complemented
188	#104	—	127	2	no	—
189	#104	—	127	2	no	—
190	#104	—	127	2	no	—
191	#104	C₂×D₁₂₇	1	2	yes	complemented
192	#97	—	127	1	no	—
193	#97	—	127	1	no	—
194	C₄×C₈	—	127	1	no	—
195	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
196	#41	—	127	2	no	—
197	#41	—	127	2	no	—
198	#41	—	127	2	no	—
199	#41	—	254	2	no	—
200	#41	—	254	2	no	—
201	#41	C₂²×D₁₂₇	1	2	yes	—
202	C₂×Q₈	—	127	2	no	—
203	C₂×D₄	—	127	2	no	—
204	C₂×D₄	—	254	2	no	—
205	C₂²×C₄	—	127	1	no	—
206	C₂²×C₄	—	254	2	no	—
207	Q₁₆	—	127	2	no	—
208	Q₁₆	—	127	1	no	—
209	Q₁₆	C₄×D₁₂₇	1	1	yes	characteristic
210	SD₁₆	—	127	2	no	—
211	SD₁₆	—	127	2	no	—
212	SD₁₆	—	127	2	no	—
213	SD₁₆	C₄×D₁₂₇	1	2	yes	—
214	D₈	—	127	2	no	—
215	D₈	—	127	1	no	—
216	D₈	C₄×D₁₂₇	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	127	1	no	—
219	M₁₆	—	127	1	no	—
220	M₁₆	—	127	1	no	—
221	M₁₆	—	127	1	no	—
222	M₁₆	—	127	2	no	—
223	M₁₆	—	254	2	no	—
224	M₁₆	—	254	2	no	—
225	M₁₆	—	254	2	no	—
226	M₁₆	—	254	2	no	—
227	M₁₆	C₂²×D₁₂₇	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	127	1	no	—
230	C₂×C₈	—	127	1	no	—
231	C₂×C₈	—	127	1	no	—
232	C₂×C₈	—	127	1	no	—
233	C₂×C₈	—	127	1	no	—
234	C₂×C₈	—	127	1	no	—
235	C₂×C₈	—	254	2	no	—
236	C₂×C₈	—	254	2	no	—
237	C₂×C₈	—	254	2	no	—
238	C₂×C₈	—	254	2	no	—
239	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
241	C₄⋊C₄	—	254	1	no	—
242	C₂²⋊C₄	—	254	1	no	—
243	C₄²	—	127	1	no	—
244	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
245	C₂³	—	127	1	no	—
246	C₂³	—	254	2	no	—
247	Q₈	—	127	2	no	—
248	Q₈	—	254	2	no	—
249	Q₈	C₂×C₄×D₁₂₇	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	127	2	no	—
252	D₄	—	254	2	no	—
253	D₄	—	254	2	no	—
254	D₄	—	254	2	no	—
255	D₄	—	254	2	no	—
256	D₄	C₂×C₄×D₁₂₇	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	127	1	no	—
260	C₂×C₄	—	127	1	no	—
261	C₂×C₄	—	127	1	no	—
262	C₂×C₄	—	127	1	no	—
263	C₂×C₄	—	127	1	no	—
264	C₂×C₄	—	254	2	no	—
265	C₂×C₄	—	254	2	no	—
266	C₂×C₄	—	254	2	no	—
267	C₂×C₄	—	254	1	no	—
268	C₂×C₄	—	254	2	no	—
269	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	127	1	no	—
272	C₈	—	127	1	no	—
273	C₈	—	127	1	no	—
274	C₈	—	127	1	no	—
275	C₈	—	254	2	no	—
276	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
277	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
278	C₈	D₄×D₁₂₇	1	1	yes	characteristic
279	C₈	D₄×D₁₂₇	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	127	1	no	—
282	C₂²	—	127	1	no	—
283	C₂²	—	1016	2	no	—
284	C₂²	—	508	1	no	—
285	C₂²	—	254	2	no	—
286	C₂²	#41×D₁₂₇	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	127	1	no	—
290	C₄	—	127	1	no	—
291	C₄	—	508	1	no	—
292	C₄	—	254	2	no	—
293	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
294	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	508	2	no	—
298	C₂	—	254	1	no	—
299	C₂	—	254	1	no	—
300	C₂	C₄×D₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#443×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₄×D₄
prime radical	#135×D₁₂₇	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#443	D₁₂₇
hypocenter	C₁₂₇	#2957
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #22550641	C₂	semidirect
3. #22549996	C₂	semidirect
4. #22549995	C₂	semidirect
5. #22549994	C₂	semidirect
6. #22549993	C₂	semidirect
7. #22549992	C₂	semidirect
8. #108×D₁₂₇	C₂	semidirect
9. #22549891	C₂	semidirect
10. #22549830	C₂	semidirect
11. #443×C₁₂₇	C₂	semidirect
13. #16861232	C₄	semidirect
15. #16861230	C₄	semidirect
17. #16861228	C₄	semidirect
18. Q₁₆×D₁₂₇	C₄	semidirect
19. SD₁₆×D₁₂₇	C₄	semidirect
20. D₈×D₁₂₇	C₄	semidirect
22. #16861217	C₂²	semidirect
23. #16861216	C₂²	semidirect
25. #16861215	C₂²	semidirect
32. #16861145	C₂²	semidirect
33. #16861144	C₂²	semidirect
34. #16861137	C₂²	semidirect
35. #16861136	C₂²	semidirect
36. #16861135	C₂²	semidirect
37. #16861120	C₂²	semidirect
38. #16861119	C₂²	semidirect
39. C₈×Dic₁₂₇	C₂²	semidirect
41. #131×C₁₂₇	C₂²	semidirect
42. #108×C₁₂₇	C₂²	semidirect
43. #104×C₁₂₇	C₂²	semidirect
44. C₄×C₈×C₁₂₇	C₂²	semidirect
151. C₁₂₇	#2957	semidirect
159. #443	D₁₂₇	semidirect
178. #131	C₂×D₁₂₇	semidirect
187. #108	C₂×D₁₂₇	semidirect
191. #104	C₂×D₁₂₇	semidirect
195. 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
#135×D₁₂₇	2
#22550641	2
#22549996	2
#22549995	2
#22549994	2
#22549993	2
#22549992	2
#108×D₁₂₇	2
#22549891	2
#22549830	2
#443×C₁₂₇	2
#2957	127