Subgroups — #30647631

Order 16256.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550674	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861215	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549996	C₂	1	4	yes	maximal, complemented
4	#22549991	C₂	1	4	yes	maximal, complemented
5	#22549924	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549893	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549892	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549831	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14523248	C₂²	1	1	yes	characteristic
11	#16861240	C₄	1	2	yes	complemented
12	#16861239	C₄	1	2	yes	complemented
13	#16861235	C₄	1	2	yes	complemented
14	#16861234	C₄	1	2	yes	complemented
15	#16861229	C₂²	1	4	yes	—
16	#16861215	—	2	1	no	—
17	#16861215	—	2	1	no	—
18	#16861215	C₂²	1	2	yes	complemented
19	#16861215	C₂²	1	2	yes	complemented
20	C₂×#14523232	C₂²	1	1	yes	characteristic
21	C₂×#14523232	C₂²	1	1	yes	characteristic
22	C₂×#14523231	—	2	1	no	—
23	C₂×C₈×D₁₂₇	—	2	1	no	—
24	#16861168	C₂²	1	1	yes	characteristic
25	#16861144	C₂²	1	4	yes	complemented
26	#16861136	C₂²	1	4	yes	complemented
27	#16861135	C₂²	1	4	yes	complemented
28	#16861119	C₂²	1	2	yes	complemented
29	C₈×Dic₁₂₇	C₂²	1	2	yes	complemented
30	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
31	#131×C₁₂₇	—	2	1	no	—
32	#131×C₁₂₇	—	2	1	no	—
33	#131×C₁₂₇	C₂²	1	2	yes	complemented
34	#131×C₁₂₇	C₂²	1	2	yes	complemented
35	C₂×M₁₆×C₁₂₇	—	2	1	no	—
36	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₂₇	—	2	1	no	—
38	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
39	#14523248	—	2	1	no	—
40	#14523248	—	2	1	no	—
41	#14523248	C₂×C₄	1	2	yes	—
42	#14523248	C₂³	1	2	yes	—
43	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
44	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
46	C₂×C₄×D₁₂₇	—	2	1	no	—
47	#14523238	C₂×C₄	1	4	yes	—
48	#14523236	C₂×C₄	1	4	yes	—
49	#14523235	C₂×C₄	1	4	yes	—
50	#14523233	C₂×C₄	1	4	yes	—
51	#14523232	—	2	1	no	—
52	#14523232	C₂×C₄	1	2	yes	—
53	#14523232	C₂×C₄	1	2	yes	—
54	#14523232	C₂³	1	2	yes	—
55	#14523231	—	2	2	no	—
56	#14523231	—	2	2	no	—
57	C₂×C₁₂₇⋊C₈	—	2	1	no	—
58	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
59	C₈×D₁₂₇	—	2	2	no	—
60	C₈×D₁₂₇	—	2	2	no	—
61	C₄⋊Dic₁₂₇	—	2	1	no	—
62	C₂²⋊Dic₁₂₇	—	2	1	no	—
63	C₄×Dic₁₂₇	C₂³	1	2	yes	—
64	#41×C₁₂₇	—	2	1	no	—
65	#41×C₁₂₇	—	2	1	no	—
66	#41×C₁₂₇	C₂×C₄	1	2	yes	—
67	#41×C₁₂₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
69	C₂×D₄×C₁₂₇	—	2	1	no	—
70	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₂₇	—	2	1	no	—
72	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
73	M₁₆×C₁₂₇	—	2	1	no	—
74	M₁₆×C₁₂₇	—	2	2	no	—
75	M₁₆×C₁₂₇	—	2	2	no	—
76	M₁₆×C₁₂₇	C₂³	1	2	yes	—
77	C₂×C₈×C₁₂₇	—	2	1	no	—
78	C₂×C₈×C₁₂₇	—	2	2	no	—
79	C₂×C₈×C₁₂₇	—	2	2	no	—
80	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
85	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
86	C₂²×D₁₂₇	—	2	1	no	—
87	Dic₂₅₄	—	2	1	no	—
88	Dic₂₅₄	C₂²×C₄	1	2	yes	—
89	D₅₀₈	—	2	1	no	—
90	D₅₀₈	C₂²×C₄	1	2	yes	—
91	C₂²⋊D₁₂₇	—	2	2	no	—
92	C₂²⋊D₁₂₇	—	2	2	no	—
93	C₂×Dic₁₂₇	—	2	1	no	—
94	C₂×Dic₁₂₇	—	2	2	no	—
95	C₄×D₁₂₇	—	2	2	no	—
96	C₄×D₁₂₇	—	2	2	no	—
97	C₁₂₇⋊C₈	—	2	2	no	—
98	C₁₂₇⋊C₈	C₂²×C₄	1	4	yes	—
99	C₂³×C₁₂₇	—	2	1	no	—
100	C₂³×C₁₂₇	#41	1	1	yes	characteristic
101	Q₈×C₁₂₇	—	2	1	no	—
102	Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
103	D₄×C₁₂₇	—	2	2	no	—
104	D₄×C₁₂₇	—	2	2	no	—
105	D₄×C₁₂₇	—	2	1	no	—
106	D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₂₇	—	2	1	no	—
108	C₂×C₄×C₁₂₇	—	2	2	no	—
109	C₂×C₄×C₁₂₇	—	2	2	no	—
110	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
115	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
116	C₈×C₁₂₇	—	2	2	no	—
117	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
118	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
119	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
120	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₁₂₇	—	2	2	no	—
122	C₂²×C₁₂₇	—	4	1	no	—
123	C₂²×C₁₂₇	—	4	2	no	—
124	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
126	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
127	C₂×D₁₂₇	—	2	2	no	—
128	C₂×D₁₂₇	—	4	2	no	—
129	C₄×C₁₂₇	—	2	2	no	—
130	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
131	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
134	Dic₁₂₇	—	2	2	no	—
135	Dic₁₂₇	—	4	2	no	—
136	C₂×C₁₂₇	—	2	1	no	—
137	C₂×C₁₂₇	—	2	1	no	—
138	C₂×C₁₂₇	—	2	1	no	—
139	C₂×C₁₂₇	—	4	2	no	—
140	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
141	D₁₂₇	—	4	2	no	—
142	#2955	—	127	1	no	maximal, Sylow
143	C₁₂₇	#2955	1	1	yes	characteristic, complemented, Sylow
144	#575	—	127	1	no	—
145	C₂×#131	—	127	1	no	—
146	C₂×#131	D₁₂₇	1	1	yes	characteristic, complemented
147	#444	—	127	4	no	—
148	#443	—	127	4	no	—
149	#430	—	127	1	no	—
150	#421	—	127	1	no	—
151	#421	—	127	1	no	—
152	#405	—	127	1	no	—
153	C₂×#41	—	127	1	no	—
154	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
155	#137	—	127	2	no	—
156	#137	—	127	2	no	—
157	AGL(1,ℤ/8)	—	127	2	no	—
158	AGL(1,ℤ/8)	—	127	2	no	—
159	#135	—	127	4	no	—
160	#131	—	2	1	no	—
161	#131	—	2	1	no	—
162	#131	—	127	2	no	—
163	#131	—	127	2	no	—
164	#131	—	254	1	no	—
165	#131	—	254	1	no	—
166	#131	C₂×D₁₂₇	1	2	yes	complemented
167	#131	C₂×D₁₂₇	1	2	yes	complemented
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	127	1	no	—
170	C₂×M₁₆	—	127	1	no	—
171	C₂×M₁₆	—	254	1	no	—
172	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	254	1	no	—
175	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
176	#117	—	127	1	no	—
177	#108	—	127	4	no	—
178	#104	—	127	4	no	—
179	#104	—	127	4	no	—
180	#97	—	127	2	no	—
181	C₄×C₈	—	127	2	no	—
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	127	2	no	—
185	#41	—	127	2	no	—
186	#41	—	254	1	no	—
187	#41	—	254	1	no	—
188	#41	C₄×D₁₂₇	1	2	yes	—
189	#41	C₂²×D₁₂₇	1	2	yes	—
190	C₂×Q₈	—	127	1	no	—
191	C₂×Q₈	C₄×D₁₂₇	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	127	1	no	—
194	C₂×D₄	—	254	1	no	—
195	C₂×D₄	C₄×D₁₂₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	254	1	no	—
198	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
199	Q₁₆	—	127	4	no	—
200	SD₁₆	—	127	4	no	—
201	SD₁₆	—	127	4	no	—
202	D₈	—	127	4	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	127	2	no	—
207	M₁₆	—	127	2	no	—
208	M₁₆	—	127	2	no	—
209	M₁₆	—	254	1	no	—
210	M₁₆	—	254	2	no	—
211	M₁₆	—	254	2	no	—
212	M₁₆	C₂²×D₁₂₇	1	2	yes	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	127	2	no	—
217	C₂×C₈	—	254	1	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	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
226	C₄⋊C₄	—	254	1	no	—
227	C₂²⋊C₄	—	254	1	no	—
228	C₄²	—	127	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	254	1	no	—
231	C₂³	#14523248	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	127	2	no	—
234	Q₈	—	254	1	no	—
235	Q₈	C₂×C₄×D₁₂₇	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	127	2	no	—
240	D₄	—	254	2	no	—
241	D₄	—	254	2	no	—
242	D₄	—	254	1	no	—
243	D₄	C₂×C₄×D₁₂₇	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	254	2	no	—
248	C₂×C₄	—	254	1	no	—
249	C₂×C₄	—	254	2	no	—
250	C₂×C₄	—	254	2	no	—
251	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
255	C₂×C₄	#14523248	1	1	yes	characteristic
256	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	127	4	no	—
259	C₈	—	254	2	no	—
260	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
261	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
262	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
263	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	508	2	no	—
268	C₂²	—	254	2	no	—
269	C₂²	C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
270	C₂²	C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
271	C₂²	C₂×#14523248	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	508	2	no	—
274	C₄	—	254	2	no	—
275	C₄	C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
276	C₄	C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
278	C₄	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	508	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#131×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₄×D₄
prime radical	#22550674	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#131	D₁₂₇
hypocenter	C₁₂₇	#2955
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#16861215	C₂	semidirect
3. #22549996	C₂	semidirect
4. #22549991	C₂	semidirect
5. #22549924	C₂	semidirect
6. #22549893	C₂	semidirect
7. #22549892	C₂	semidirect
8. #22549831	C₂	semidirect
9. C₂×#131×C₁₂₇	C₂	semidirect
11. #16861240	C₄	semidirect
12. #16861239	C₄	semidirect
13. #16861235	C₄	semidirect
14. #16861234	C₄	semidirect
18. #16861215	C₂²	semidirect
19. #16861215	C₂²	semidirect
25. #16861144	C₂²	semidirect
26. #16861136	C₂²	semidirect
27. #16861135	C₂²	semidirect
28. #16861119	C₂²	semidirect
29. C₈×Dic₁₂₇	C₂²	semidirect
33. #131×C₁₂₇	C₂²	semidirect
34. #131×C₁₂₇	C₂²	semidirect
143. C₁₂₇	#2955	semidirect
146. C₂×#131	D₁₂₇	semidirect
166. #131	C₂×D₁₂₇	semidirect
167. #131	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550674	2
C₂×#16861215	2
#22549996	2
#22549991	2
#22549924	2
#22549893	2
#22549892	2
#22549831	2
C₂×#131×C₁₂₇	2
#2955	127