Subgroups — #30647628

Order 16256.

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	#22550673	C₂	1	1	yes	maximal, characteristic
2	#131×D₁₂₇	C₂	1	2	yes	maximal, complemented
3	#22549998	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549997	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549995	C₂	1	2	yes	maximal, complemented
6	#22549994	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549991	C₂	1	2	yes	maximal, complemented
8	#22549923	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	#16861241	C₄	1	2	yes	complemented
14	#16861238	C₄	1	2	yes	complemented
15	#16861236	C₄	1	2	yes	complemented
16	#16861233	C₄	1	2	yes	complemented
17	#16861231	C₂²	1	1	yes	characteristic
18	#16861229	C₂²	1	2	yes	—
19	#16861217	—	2	2	no	—
20	#16861217	C₂²	1	2	yes	complemented
21	#16861216	C₂²	1	2	yes	complemented
22	#16861215	—	2	2	no	—
23	#16861215	C₂²	1	2	yes	complemented
24	C₂×#14523231	C₂²	1	1	yes	characteristic
25	M₁₆×D₁₂₇	—	2	2	no	—
26	M₁₆×D₁₂₇	C₂²	1	2	yes	—
27	C₂×C₈×D₁₂₇	—	2	2	no	—
28	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
29	#16861166	C₂²	1	1	yes	characteristic
30	#16861145	C₂²	1	1	yes	characteristic, complemented
31	#16861144	C₂²	1	2	yes	complemented
32	#16861137	C₂²	1	2	yes	complemented
33	#16861136	C₂²	1	2	yes	complemented
34	#16861135	C₂²	1	2	yes	complemented
35	#16861120	C₂²	1	1	yes	characteristic, complemented
36	#16861119	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic, complemented
38	#135×C₁₂₇	C₂²	1	1	yes	characteristic
39	#131×C₁₂₇	C₂²	1	2	yes	complemented
40	#108×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
41	#104×C₁₂₇	C₂²	1	2	yes	complemented
42	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
43	#14523249	C₂×C₄	1	2	yes	—
44	#14523248	—	2	2	no	—
45	#14523248	C₂³	1	2	yes	—
46	#14523247	—	2	2	no	—
47	#14523247	C₂×C₄	1	2	yes	—
48	Q₈×D₁₂₇	C₂×C₄	1	2	yes	—
49	D₄×D₁₂₇	—	2	2	no	—
50	D₄×D₁₂₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₂₇	—	2	2	no	—
52	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
53	Dic₅₀₈	C₂×C₄	1	1	yes	characteristic
54	#14523238	C₂×C₄	1	2	yes	—
55	#14523237	C₂×C₄	1	2	yes	—
56	#14523236	C₂×C₄	1	2	yes	—
57	#14523235	C₂×C₄	1	2	yes	—
58	D₁₀₁₆	C₂×C₄	1	1	yes	characteristic
59	#14523233	C₂×C₄	1	2	yes	—
60	#14523232	—	2	2	no	—
61	#14523232	C₂³	1	2	yes	—
62	#14523231	—	2	2	no	—
63	#14523231	—	2	2	no	—
64	#14523231	—	2	2	no	—
65	#14523231	C₂×C₄	1	2	yes	—
66	#14523231	C₂×C₄	1	2	yes	—
67	C₂×C₁₂₇⋊C₈	—	2	2	no	—
68	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₈×D₁₂₇	—	2	2	no	—
71	C₈×D₁₂₇	—	2	2	no	—
72	C₈×D₁₂₇	—	2	2	no	—
73	C₈×D₁₂₇	D₄	1	2	yes	—
74	C₈×D₁₂₇	D₄	1	2	yes	—
75	Dic₁₂₇⋊C₄	—	2	1	no	—
76	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
77	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
78	#41×C₁₂₇	C₂³	1	2	yes	—
79	Q₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
80	SD₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
81	D₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
82	M₁₆×C₁₂₇	—	2	2	no	—
83	M₁₆×C₁₂₇	C₂³	1	2	yes	—
84	C₂×C₈×C₁₂₇	—	2	2	no	—
85	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
87	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
88	C₂²×D₁₂₇	—	2	2	no	—
89	C₂²×D₁₂₇	#41	1	1	yes	characteristic
90	Dic₂₅₄	—	2	2	no	—
91	Dic₂₅₄	C₂²×C₄	1	2	yes	—
92	D₅₀₈	—	2	2	no	—
93	D₅₀₈	C₂²×C₄	1	2	yes	—
94	C₂²⋊D₁₂₇	—	2	2	no	—
95	C₂²⋊D₁₂₇	—	2	2	no	—
96	C₂²⋊D₁₂₇	—	2	2	no	—
97	C₂×Dic₁₂₇	—	2	2	no	—
98	C₂×Dic₁₂₇	—	2	1	no	—
99	C₂×Dic₁₂₇	#41	1	1	yes	characteristic
100	C₄×D₁₂₇	—	2	2	no	—
101	C₄×D₁₂₇	—	2	2	no	—
102	C₄×D₁₂₇	—	2	2	no	—
103	C₄×D₁₂₇	C₂²×C₄	1	2	yes	—
104	C₄×D₁₂₇	C₂×D₄	1	2	yes	—
105	C₁₂₇⋊C₈	—	2	2	no	—
106	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
107	C₁₂₇⋊C₈	C₂×D₄	1	2	yes	—
108	Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
109	D₄×C₁₂₇	—	2	2	no	—
110	D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
111	C₂×C₄×C₁₂₇	—	2	2	no	—
112	C₂×C₄×C₁₂₇	—	2	1	no	—
113	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
114	C₈×C₁₂₇	—	2	2	no	—
115	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
116	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
117	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
118	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₁₂₇	—	2	2	no	—
120	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
121	C₂×D₁₂₇	—	2	2	no	—
122	C₂×D₁₂₇	—	4	1	no	—
123	C₂×D₁₂₇	—	4	4	no	—
124	C₂×D₁₂₇	C₄×D₄	1	2	yes	—
125	C₄×C₁₂₇	—	2	2	no	—
126	C₄×C₁₂₇	—	4	1	no	—
127	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
129	Dic₁₂₇	—	2	2	no	—
130	Dic₁₂₇	—	4	1	no	—
131	Dic₁₂₇	C₄×D₄	1	2	yes	—
132	C₂×C₁₂₇	—	2	1	no	—
133	C₂×C₁₂₇	—	4	2	no	—
134	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
135	D₁₂₇	—	2	2	no	—
136	D₁₂₇	—	4	2	no	—
137	#2955	—	127	1	no	maximal, Sylow
138	C₁₂₇	#2955	1	1	yes	characteristic, complemented, Sylow
139	#575	—	127	1	no	—
140	C₂×#131	—	127	2	no	—
141	#444	—	127	1	no	—
142	#444	—	127	1	no	—
143	#444	—	127	2	no	—
144	#443	—	127	2	no	—
145	#443	—	127	1	no	—
146	#443	D₁₂₇	1	1	yes	characteristic, complemented
147	#430	—	127	1	no	—
148	#421	—	127	2	no	—
149	#405	—	127	1	no	—
150	C₂×#41	—	127	2	no	—
151	#137	—	127	2	no	—
152	#137	—	127	2	no	—
153	AGL(1,ℤ/8)	—	127	2	no	—
154	AGL(1,ℤ/8)	—	127	2	no	—
155	#135	—	127	2	no	—
156	#135	—	127	1	no	—
157	#135	C₂×D₁₂₇	1	1	yes	characteristic
158	#131	—	127	2	no	—
159	#131	—	127	2	no	—
160	#131	—	127	2	no	—
161	#131	—	254	2	no	—
162	#131	—	254	2	no	—
163	#131	C₂×D₁₂₇	1	2	yes	complemented
164	C₂×M₁₆	—	127	1	no	—
165	C₂×M₁₆	—	127	2	no	—
166	C₂×M₁₆	—	254	2	no	—
167	C₂²×C₈	—	127	1	no	—
168	C₂²×C₈	—	254	2	no	—
169	#117	—	127	1	no	—
170	#108	—	127	2	no	—
171	#108	—	127	1	no	—
172	#108	C₂×D₁₂₇	1	1	yes	characteristic, complemented
173	#104	—	127	2	no	—
174	#104	—	127	2	no	—
175	#104	—	127	2	no	—
176	#104	C₂×D₁₂₇	1	2	yes	complemented
177	#97	—	127	1	no	—
178	#97	—	127	1	no	—
179	C₄×C₈	—	127	1	no	—
180	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
181	#41	—	127	2	no	—
182	#41	—	127	2	no	—
183	#41	—	127	2	no	—
184	#41	—	254	2	no	—
185	#41	—	254	2	no	—
186	#41	C₂²×D₁₂₇	1	2	yes	—
187	C₂×Q₈	—	127	2	no	—
188	C₂×D₄	—	127	2	no	—
189	C₂×D₄	—	254	2	no	—
190	C₂²×C₄	—	127	1	no	—
191	C₂²×C₄	—	254	2	no	—
192	Q₁₆	—	127	2	no	—
193	Q₁₆	—	127	1	no	—
194	Q₁₆	C₄×D₁₂₇	1	1	yes	characteristic
195	SD₁₆	—	127	2	no	—
196	SD₁₆	—	127	2	no	—
197	SD₁₆	—	127	2	no	—
198	SD₁₆	C₄×D₁₂₇	1	2	yes	—
199	D₈	—	127	2	no	—
200	D₈	—	127	1	no	—
201	D₈	C₄×D₁₂₇	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	127	2	no	—
204	M₁₆	—	127	2	no	—
205	M₁₆	—	127	2	no	—
206	M₁₆	—	254	2	no	—
207	M₁₆	—	254	2	no	—
208	M₁₆	—	254	2	no	—
209	M₁₆	—	254	2	no	—
210	M₁₆	C₂²×D₁₂₇	1	2	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	127	1	no	—
213	C₂×C₈	—	127	1	no	—
214	C₂×C₈	—	127	2	no	—
215	C₂×C₈	—	127	2	no	—
216	C₂×C₈	—	254	2	no	—
217	C₂×C₈	—	254	2	no	—
218	C₂×C₈	—	254	2	no	—
219	C₂×C₈	—	254	2	no	—
220	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
222	C₄⋊C₄	—	254	1	no	—
223	C₂²⋊C₄	—	254	1	no	—
224	C₄²	—	127	1	no	—
225	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
226	C₂³	—	127	1	no	—
227	C₂³	—	254	2	no	—
228	Q₈	—	127	2	no	—
229	Q₈	—	254	2	no	—
230	Q₈	C₂×C₄×D₁₂₇	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	127	2	no	—
233	D₄	—	254	2	no	—
234	D₄	—	254	2	no	—
235	D₄	—	254	2	no	—
236	D₄	—	254	2	no	—
237	D₄	C₂×C₄×D₁₂₇	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	127	1	no	—
241	C₂×C₄	—	127	2	no	—
242	C₂×C₄	—	127	2	no	—
243	C₂×C₄	—	254	2	no	—
244	C₂×C₄	—	254	2	no	—
245	C₂×C₄	—	254	2	no	—
246	C₂×C₄	—	254	1	no	—
247	C₂×C₄	—	254	2	no	—
248	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	127	2	no	—
251	C₈	—	127	2	no	—
252	C₈	—	254	2	no	—
253	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
254	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
255	C₈	D₄×D₁₂₇	1	1	yes	characteristic
256	C₈	D₄×D₁₂₇	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	127	2	no	—
259	C₂²	—	508	1	no	—
260	C₂²	—	508	4	no	—
261	C₂²	—	254	2	no	—
262	C₂²	#41×D₁₂₇	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	127	2	no	—
266	C₄	—	508	1	no	—
267	C₄	—	254	2	no	—
268	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
269	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	508	2	no	—
273	C₂	—	254	2	no	—
274	C₂	C₄×D₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#443×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₄×D₄
prime radical	#22550673	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#443	D₁₂₇
hypocenter	C₁₂₇	#2955
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #131×D₁₂₇	C₂	semidirect
3. #22549998	C₂	semidirect
4. #22549997	C₂	semidirect
5. #22549995	C₂	semidirect
6. #22549994	C₂	semidirect
7. #22549991	C₂	semidirect
8. #22549923	C₂	semidirect
9. #22549891	C₂	semidirect
10. #22549830	C₂	semidirect
11. #443×C₁₂₇	C₂	semidirect
13. #16861241	C₄	semidirect
14. #16861238	C₄	semidirect
15. #16861236	C₄	semidirect
16. #16861233	C₄	semidirect
20. #16861217	C₂²	semidirect
21. #16861216	C₂²	semidirect
23. #16861215	C₂²	semidirect
30. #16861145	C₂²	semidirect
31. #16861144	C₂²	semidirect
32. #16861137	C₂²	semidirect
33. #16861136	C₂²	semidirect
34. #16861135	C₂²	semidirect
35. #16861120	C₂²	semidirect
36. #16861119	C₂²	semidirect
37. C₈×Dic₁₂₇	C₂²	semidirect
39. #131×C₁₂₇	C₂²	semidirect
40. #108×C₁₂₇	C₂²	semidirect
41. #104×C₁₂₇	C₂²	semidirect
42. C₄×C₈×C₁₂₇	C₂²	semidirect
138. C₁₂₇	#2955	semidirect
146. #443	D₁₂₇	semidirect
163. #131	C₂×D₁₂₇	semidirect
172. #108	C₂×D₁₂₇	semidirect
176. #104	C₂×D₁₂₇	semidirect
180. 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
#22550673	2
#131×D₁₂₇	2
#22549998	2
#22549997	2
#22549995	2
#22549994	2
#22549991	2
#22549923	2
#22549891	2
#22549830	2
#443×C₁₂₇	2
#2955	127