Subgroups — #30652887

Order 16256.

1336 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550615	C₂	1	3	yes	maximal, complemented
2	#22550556	C₂	1	3	yes	maximal, complemented
3	#22550519	C₂	1	6	yes	maximal, complemented
4	#22550508	C₂	1	3	yes	maximal, complemented
5	#22550460	C₂	1	6	yes	maximal, complemented
6	#22550444	C₂	1	3	yes	maximal, complemented
7	#22550436	C₂	1	3	yes	maximal, complemented
8	#120×D₁₂₇	C₂	1	3	yes	maximal, complemented
9	H(3,𝔽₄)×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₂₇	C₂²	1	1	yes	characteristic, complemented
11	C₂×#14523249	C₂²	1	3	yes	complemented
12	C₂²×C₂²⋊D₁₂₇	C₂²	1	3	yes	complemented
13	C₂×D₄×D₁₂₇	C₂²	1	3	yes	complemented
14	C₂×D₄×D₁₂₇	C₂²	1	6	yes	complemented
15	#16861206	C₂²	1	1	yes	characteristic, complemented
16	#16861205	C₂²	1	3	yes	complemented
17	#16861203	C₂²	1	2	yes	complemented
18	#16861202	C₂²	1	6	yes	complemented
19	#16861198	C₂²	1	3	yes	complemented
20	#16861193	C₂²	1	3	yes	complemented
21	#16861191	C₂²	1	6	yes	complemented
22	#16861191	C₂²	1	6	yes	complemented
23	#16861190	C₂²	1	3	yes	complemented
24	#16861185	C₂²	1	6	yes	complemented
25	#16861182	C₂²	1	6	yes	complemented
26	#16861182	C₂²	1	6	yes	complemented
27	#16861182	C₂²	1	6	yes	complemented
28	#16861181	C₂²	1	6	yes	complemented
29	#16861180	C₂²	1	6	yes	complemented
30	#16861179	C₂²	1	6	yes	complemented
31	#16861179	C₂²	1	3	yes	complemented
32	#16861179	C₂²	1	6	yes	complemented
33	#16861178	C₂²	1	3	yes	complemented
34	#16861178	C₂²	1	1	yes	characteristic, complemented
35	#16861177	C₂²	1	3	yes	—
36	#16861177	C₂²	1	6	yes	complemented
37	C₄×D₅₀₈	C₂²	1	6	yes	complemented
38	(C₂²⋊D₁₂₇)⋊C₄	C₂²	1	6	yes	complemented
39	#16861166	C₂²	1	3	yes	complemented
40	C₂²⋊C₄×D₁₂₇	C₂²	1	6	yes	complemented
41	C₂²⋊C₄×D₁₂₇	C₂²	1	3	yes	complemented
42	C₂²⋊C₄×D₁₂₇	C₂²	1	3	yes	complemented
43	#124×C₁₂₇	C₂²	1	3	yes	complemented
44	#124×C₁₂₇	C₂²	1	6	yes	complemented
45	#120×C₁₂₇	C₂²	1	3	yes	complemented
46	#120×C₁₂₇	C₂²	1	3	yes	complemented
47	C₂³×D₁₂₇	—	4	6	no	—
48	C₂³×D₁₂₇	C₂³	1	3	yes	—
49	C₂³×D₁₂₇	C₂³	1	3	yes	—
50	C₂³×D₁₂₇	C₂³	1	3	yes	—
51	#14523249	—	4	6	no	—
52	C₂×D₅₀₈	C₂³	1	6	yes	—
53	C₂×D₅₀₈	C₂³	1	3	yes	—
54	C₂×D₅₀₈	C₂³	1	3	yes	complemented
55	C₂×C₂²⋊D₁₂₇	—	4	6	no	—
56	C₂×C₂²⋊D₁₂₇	C₂³	1	6	yes	—
57	C₂×C₂²⋊D₁₂₇	C₂³	1	6	yes	—
58	C₂×C₂²⋊D₁₂₇	C₂³	1	3	yes	—
59	C₂×C₂²⋊D₁₂₇	C₂³	1	3	yes	—
60	C₂×C₂²⋊D₁₂₇	C₂³	1	6	yes	complemented
61	D₄×D₁₂₇	—	4	6	no	—
62	D₄×D₁₂₇	—	4	12	no	—
63	C₂²×Dic₁₂₇	C₂³	1	3	yes	complemented
64	C₂×C₄×D₁₂₇	C₂³	1	3	yes	—
65	C₂×C₄×D₁₂₇	C₂³	1	6	yes	—
66	C₂×C₄×D₁₂₇	C₂³	1	6	yes	complemented
67	C₄⋊Dic₁₂₇	C₂³	1	6	yes	complemented
68	Dic₁₂₇⋊C₄	C₂³	1	6	yes	complemented
69	Dic₁₂₇⋊C₄	C₂³	1	6	yes	complemented
70	C₂²⋊Dic₁₂₇	C₂³	1	6	yes	—
71	C₂²⋊Dic₁₂₇	C₂³	1	3	yes	—
72	C₂²⋊Dic₁₂₇	C₂³	1	3	yes	complemented
73	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	—
74	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	—
75	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	—
76	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	complemented
77	C₄×Dic₁₂₇	C₂³	1	3	yes	complemented
78	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
79	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic, complemented
80	C₂×Q₈×C₁₂₇	C₂³	1	3	yes	complemented
81	C₂×D₄×C₁₂₇	C₂³	1	3	yes	—
82	C₂×D₄×C₁₂₇	C₂³	1	6	yes	—
83	C₂²⋊C₄×C₁₂₇	C₂³	1	3	yes	—
84	C₂²⋊C₄×C₁₂₇	C₂³	1	6	yes	—
85	C₂²⋊C₄×C₁₂₇	C₂³	1	6	yes	complemented
86	C₂²⋊C₄×C₁₂₇	C₂³	1	3	yes	complemented
87	C₄²×C₁₂₇	C₂³	1	2	yes	complemented
88	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic, complemented
89	C₂²×D₁₂₇	—	2	12	no	—
90	C₂²×D₁₂₇	—	2	6	no	—
91	C₂²×D₁₂₇	—	2	6	no	—
92	C₂²×D₁₂₇	—	4	3	no	—
93	C₂²×D₁₂₇	—	4	6	no	—
94	C₂²×D₁₂₇	—	4	6	no	—
95	C₂²×D₁₂₇	—	4	16	no	—
96	C₂²×D₁₂₇	C₂⁴	1	1	yes	characteristic
97	C₂²×D₁₂₇	C₂⁴	1	3	yes	—
98	C₂²×D₁₂₇	C₂⁴	1	3	yes	—
99	D₅₀₈	—	2	12	no	—
100	D₅₀₈	—	4	3	no	—
101	D₅₀₈	—	4	3	no	—
102	C₂²⋊D₁₂₇	—	2	12	no	—
103	C₂²⋊D₁₂₇	—	4	3	no	—
104	C₂²⋊D₁₂₇	—	4	3	no	—
105	C₂²⋊D₁₂₇	—	4	6	no	—
106	C₂²⋊D₁₂₇	—	4	6	no	—
107	C₂×Dic₁₂₇	—	4	3	no	—
108	C₂×Dic₁₂₇	C₂⁴	1	6	yes	—
109	C₂×Dic₁₂₇	C₂⁴	1	3	yes	—
110	C₄×D₁₂₇	—	2	6	no	—
111	C₄×D₁₂₇	—	4	6	no	—
112	C₄×D₁₂₇	—	4	6	no	—
113	C₂³×C₁₂₇	—	4	3	no	—
114	C₂³×C₁₂₇	—	4	3	no	—
115	C₂³×C₁₂₇	C₂⁴	1	3	yes	—
116	C₂³×C₁₂₇	C₂⁴	1	3	yes	—
117	Q₈×C₁₂₇	—	4	3	no	—
118	D₄×C₁₂₇	—	4	3	no	—
119	D₄×C₁₂₇	—	4	6	no	—
120	C₂×C₄×C₁₂₇	C₂⁴	1	3	yes	—
121	C₂×C₄×C₁₂₇	C₂⁴	1	6	yes	—
122	C₂²×C₁₂₇	—	2	3	no	—
123	C₂²×C₁₂₇	—	2	6	no	—
124	C₂²×C₁₂₇	—	2	3	no	—
125	C₂²×C₁₂₇	—	2	6	no	—
126	C₂²×C₁₂₇	—	4	4	no	—
127	C₂²×C₁₂₇	—	8	2	no	—
128	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
129	C₂×D₁₂₇	—	2	6	no	—
130	C₂×D₁₂₇	—	2	6	no	—
131	C₂×D₁₂₇	—	2	3	no	—
132	C₂×D₁₂₇	—	2	3	no	—
133	C₂×D₁₂₇	—	4	12	no	—
134	C₂×D₁₂₇	—	4	12	no	—
135	C₂×D₁₂₇	—	8	6	no	—
136	C₂×D₁₂₇	H(5,𝔽₂)	1	6	yes	—
137	C₄×C₁₂₇	—	2	3	no	—
138	C₄×C₁₂₇	—	2	6	no	—
139	Dic₁₂₇	—	2	3	no	—
140	Dic₁₂₇	—	2	6	no	—
141	C₂×C₁₂₇	—	4	3	no	—
142	C₂×C₁₂₇	—	4	3	no	—
143	C₂×C₁₂₇	C₂×H(5,𝔽₂)	1	3	yes	—
144	D₁₂₇	—	2	6	no	—
145	D₁₂₇	—	4	1	no	—
146	D₁₂₇	—	4	3	no	—
147	#3532	—	127	1	no	maximal, Sylow
148	C₁₂₇	#3532	1	1	yes	characteristic, complemented, Sylow
149	H(3,𝔽₄)	D₁₂₇	1	1	yes	characteristic, complemented
150	#560	—	127	3	no	—
151	#551	—	127	3	no	—
152	#546	—	127	6	no	—
153	D₄²	—	127	3	no	—
154	#538	—	127	6	no	—
155	#535	—	127	3	no	—
156	#534	—	127	3	no	—
157	C₂×#120	—	127	3	no	—
158	C₂⁵	—	127	1	no	—
159	C₂×#41	—	127	3	no	—
160	C₂²×D₄	—	127	3	no	—
161	C₂²×D₄	—	127	3	no	—
162	C₂²×D₄	—	127	6	no	—
163	#127	—	127	3	no	—
164	#127	—	127	1	no	—
165	#126	—	127	6	no	—
166	#126	—	127	2	no	—
167	#124	—	127	3	no	—
168	#124	C₂×D₁₂₇	1	3	yes	complemented
169	#124	C₂×D₁₂₇	1	6	yes	complemented
170	#123	—	127	3	no	—
171	#123	—	127	6	no	—
172	#123	—	127	6	no	—
173	#123	—	127	3	no	—
174	C₂²⋊Q₈	—	127	6	no	—
175	#121	—	127	6	no	—
176	#121	—	127	6	no	—
177	#121	—	127	6	no	—
178	#121	—	127	6	no	—
179	#121	—	127	6	no	—
180	#120	—	127	3	no	—
181	#120	—	127	1	no	—
182	#120	—	127	6	no	—
183	#120	—	127	3	no	—
184	#120	—	127	6	no	—
185	#120	—	127	3	no	—
186	#120	—	127	6	no	—
187	#120	C₂×D₁₂₇	1	3	yes	complemented
188	#120	C₂×D₁₂₇	1	3	yes	complemented
189	C₄×D₄	—	127	6	no	—
190	C₄×D₄	—	127	6	no	—
191	#117	—	127	3	no	—
192	C₂×C₂²⋊C₄	—	127	3	no	—
193	C₂×C₂²⋊C₄	—	127	6	no	—
194	C₂×C₂²⋊C₄	—	127	3	no	—
195	C₂⁴	—	127	3	no	—
196	C₂⁴	—	127	3	no	—
197	C₂⁴	—	127	3	no	—
198	C₂⁴	—	508	6	no	—
199	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
200	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic, complemented
201	#41	—	508	6	no	—
202	C₂×Q₈	C₂²×D₁₂₇	1	3	yes	complemented
203	C₂×D₄	—	127	3	no	—
204	C₂×D₄	—	127	3	no	—
205	C₂×D₄	—	127	6	no	—
206	C₂×D₄	—	127	6	no	—
207	C₂×D₄	—	127	6	no	—
208	C₂×D₄	—	127	3	no	—
209	C₂×D₄	—	127	3	no	—
210	C₂×D₄	—	127	6	no	—
211	C₂×D₄	—	508	12	no	—
212	C₂×D₄	—	508	6	no	—
213	C₂×D₄	—	508	6	no	—
214	C₂×D₄	C₂²×D₁₂₇	1	6	yes	—
215	C₂×D₄	C₂²×D₁₂₇	1	3	yes	—
216	C₂²×C₄	—	127	3	no	—
217	C₂²×C₄	—	127	6	no	—
218	C₂²×C₄	—	127	6	no	—
219	C₂²×C₄	—	127	3	no	—
220	C₄⋊C₄	—	127	6	no	—
221	C₄⋊C₄	—	127	6	no	—
222	C₄⋊C₄	—	127	6	no	—
223	C₂²⋊C₄	—	127	3	no	—
224	C₂²⋊C₄	—	127	6	no	—
225	C₂²⋊C₄	—	127	3	no	—
226	C₂²⋊C₄	—	127	6	no	—
227	C₂²⋊C₄	—	127	6	no	—
228	C₂²⋊C₄	—	127	6	no	—
229	C₂²⋊C₄	—	127	6	no	—
230	C₂²⋊C₄	C₂²×D₁₂₇	1	6	yes	—
231	C₂²⋊C₄	C₂²×D₁₂₇	1	3	yes	—
232	C₂²⋊C₄	C₂²×D₁₂₇	1	3	yes	complemented
233	C₂²⋊C₄	C₂²×D₁₂₇	1	6	yes	complemented
234	C₄²	—	127	3	no	—
235	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic, complemented
236	C₄²	C₂²×D₁₂₇	1	2	yes	complemented
237	C₂³	—	4	3	no	—
238	C₂³	—	4	3	no	—
239	C₂³	—	127	3	no	—
240	C₂³	—	127	1	no	—
241	C₂³	—	127	3	no	—
242	C₂³	—	508	6	no	—
243	C₂³	—	508	6	no	—
244	C₂³	—	508	3	no	—
245	C₂³	—	508	16	no	—
246	C₂³	—	254	6	no	—
247	C₂³	—	254	6	no	—
248	C₂³	—	254	12	no	—
249	C₂³	C₂³×D₁₂₇	1	3	yes	—
250	C₂³	C₂³×D₁₂₇	1	3	yes	—
251	Q₈	—	4	3	no	—
252	D₄	—	4	6	no	—
253	D₄	—	4	3	no	—
254	D₄	—	508	3	no	—
255	D₄	—	508	3	no	—
256	D₄	—	508	3	no	—
257	D₄	—	508	3	no	—
258	D₄	—	508	6	no	—
259	D₄	—	508	6	no	—
260	D₄	—	254	12	no	—
261	D₄	—	254	12	no	—
262	C₂×C₄	—	127	6	no	—
263	C₂×C₄	—	127	3	no	—
264	C₂×C₄	—	508	6	no	—
265	C₂×C₄	—	508	6	no	—
266	C₂×C₄	—	508	3	no	—
267	C₂×C₄	—	254	6	no	—
268	C₂×C₄	C₂³×D₁₂₇	1	3	yes	—
269	C₂×C₄	C₂³×D₁₂₇	1	6	yes	—
270	C₂²	—	2	3	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	6	no	—
273	C₂²	—	2	3	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	127	6	no	—
277	C₂²	—	1016	6	no	—
278	C₂²	—	508	12	no	—
279	C₂²	—	508	12	no	—
280	C₂²	—	254	6	no	—
281	C₂²	—	254	3	no	—
282	C₂²	—	254	3	no	—
283	C₂²	—	254	6	no	—
284	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
285	C₄	—	2	6	no	—
286	C₄	—	2	3	no	—
287	C₄	—	254	3	no	—
288	C₄	—	254	6	no	—
289	C₂	—	4	3	no	—
290	C₂	—	4	3	no	—
291	C₂	—	508	3	no	—
292	C₂	—	508	1	no	—
293	C₂	—	254	6	no	—
294	C₂	H(5,𝔽₂)×D₁₂₇	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₂₇
commutator, socle, cosocle	C₂²×C₁₂₇	C₂⁵
Fitting	H(3,𝔽₄)×C₁₂₇	C₂
hypercenter	H(3,𝔽₄)	D₁₂₇
hypocenter	C₁₂₇	#3532
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #22550615	C₂	semidirect
2. #22550556	C₂	semidirect
3. #22550519	C₂	semidirect
4. #22550508	C₂	semidirect
5. #22550460	C₂	semidirect
6. #22550444	C₂	semidirect
7. #22550436	C₂	semidirect
8. #120×D₁₂₇	C₂	semidirect
9. H(3,𝔽₄)×C₁₂₇	C₂	semidirect
10. C₂⁴×D₁₂₇	C₂²	semidirect
11. C₂×#14523249	C₂²	semidirect
12. C₂²×C₂²⋊D₁₂₇	C₂²	semidirect
13. C₂×D₄×D₁₂₇	C₂²	semidirect
14. C₂×D₄×D₁₂₇	C₂²	semidirect
15. #16861206	C₂²	semidirect
16. #16861205	C₂²	semidirect
17. #16861203	C₂²	semidirect
18. #16861202	C₂²	semidirect
19. #16861198	C₂²	semidirect
20. #16861193	C₂²	semidirect
21. #16861191	C₂²	semidirect
22. #16861191	C₂²	semidirect
23. #16861190	C₂²	semidirect
24. #16861185	C₂²	semidirect
25. #16861182	C₂²	semidirect
26. #16861182	C₂²	semidirect
27. #16861182	C₂²	semidirect
28. #16861181	C₂²	semidirect
29. #16861180	C₂²	semidirect
30. #16861179	C₂²	semidirect
31. #16861179	C₂²	semidirect
32. #16861179	C₂²	semidirect
33. #16861178	C₂²	semidirect
34. #16861178	C₂²	semidirect
36. #16861177	C₂²	semidirect
37. C₄×D₅₀₈	C₂²	semidirect
38. (C₂²⋊D₁₂₇)⋊C₄	C₂²	semidirect
39. #16861166	C₂²	semidirect
40. C₂²⋊C₄×D₁₂₇	C₂²	semidirect
41. C₂²⋊C₄×D₁₂₇	C₂²	semidirect
42. C₂²⋊C₄×D₁₂₇	C₂²	semidirect
43. #124×C₁₂₇	C₂²	semidirect
44. #124×C₁₂₇	C₂²	semidirect
45. #120×C₁₂₇	C₂²	semidirect
46. #120×C₁₂₇	C₂²	semidirect
54. C₂×D₅₀₈	C₂³	semidirect
60. C₂×C₂²⋊D₁₂₇	C₂³	semidirect
63. C₂²×Dic₁₂₇	C₂³	semidirect
66. C₂×C₄×D₁₂₇	C₂³	semidirect
67. C₄⋊Dic₁₂₇	C₂³	semidirect
68. Dic₁₂₇⋊C₄	C₂³	semidirect
69. Dic₁₂₇⋊C₄	C₂³	semidirect
72. C₂²⋊Dic₁₂₇	C₂³	semidirect
76. (C₂×D₁₂₇)⋊C₄	C₂³	semidirect
77. C₄×Dic₁₂₇	C₂³	semidirect
79. C₂⁴×C₁₂₇	C₂³	semidirect
80. C₂×Q₈×C₁₂₇	C₂³	semidirect
85. C₂²⋊C₄×C₁₂₇	C₂³	semidirect
86. C₂²⋊C₄×C₁₂₇	C₂³	semidirect
87. C₄²×C₁₂₇	C₂³	semidirect
88. C₄²×C₁₂₇	C₂³	semidirect
148. C₁₂₇	#3532	semidirect
149. H(3,𝔽₄)	D₁₂₇	semidirect
168. #124	C₂×D₁₂₇	semidirect
169. #124	C₂×D₁₂₇	semidirect
187. #120	C₂×D₁₂₇	semidirect
188. #120	C₂×D₁₂₇	semidirect
200. C₂⁴	C₂²×D₁₂₇	semidirect
202. C₂×Q₈	C₂²×D₁₂₇	semidirect
232. C₂²⋊C₄	C₂²×D₁₂₇	semidirect
233. C₂²⋊C₄	C₂²×D₁₂₇	semidirect
235. C₄²	C₂²×D₁₂₇	semidirect
236. C₄²	C₂²×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550615	2
#22550556	2
#22550519	2
#22550508	2
#22550460	2
#22550444	2
#22550436	2
#120×D₁₂₇	2
H(3,𝔽₄)×C₁₂₇	2
#3532	127