Subgroups — #30647458

Order 16256.

502 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	C₂×#16861217	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550388	C₂	1	1	yes	maximal, characteristic
4	#22549955	C₂	1	2	yes	maximal, complemented
5	#22549949	C₂	1	2	yes	maximal, complemented
6	#22549949	C₂	1	2	yes	maximal, complemented
7	C₈×D₅₀₈	C₂	1	2	yes	maximal, complemented
8	#22549903	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549842	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×(C₂×D₁₂₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
14	#16861217	—	2	4	no	—
15	C₂×#14523232	C₂²	1	1	yes	characteristic
16	C₂×#14523231	C₂²	1	2	yes	—
17	C₂×#14523231	C₂²	1	2	yes	complemented
18	M₁₆×D₁₂₇	—	2	4	no	—
19	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
21	C₂×C₈×D₁₂₇	C₂²	1	2	yes	complemented
22	#16861193	C₄	1	2	yes	—
23	#16861189	C₄	1	1	yes	characteristic
24	#16861189	C₄	1	1	yes	characteristic
25	#16861184	C₄	1	1	yes	characteristic
26	#16861184	C₄	1	1	yes	characteristic
27	#16861179	C₄	1	2	yes	—
28	C₄×D₅₀₈	C₂²	1	2	yes	—
29	C₄×D₅₀₈	C₂²	1	2	yes	—
30	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	complemented
31	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	complemented
32	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
33	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
34	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
35	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
36	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	complemented
37	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	complemented
38	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
40	#117×C₁₂₇	C₂²	1	1	yes	characteristic
41	#97×C₁₂₇	C₂²	1	2	yes	complemented
42	C₄×C₈×C₁₂₇	C₂²	1	2	yes	complemented
43	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
44	#14523248	—	2	2	no	—
45	#14523248	—	2	2	no	—
46	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
47	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
48	C₂×D₅₀₈	C₂×C₄	1	2	yes	—
49	C₂×D₅₀₈	C₂×C₄	1	2	yes	—
50	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
51	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₂₇	—	2	2	no	—
53	C₂×C₄×D₁₂₇	—	2	2	no	—
54	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
58	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
59	#14523232	—	2	2	no	—
60	#14523231	—	2	4	no	—
61	#14523231	—	2	4	no	—
62	C₂×C₁₂₇⋊C₈	—	2	2	no	—
63	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
64	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
66	C₈×D₁₂₇	—	2	4	no	—
67	C₈×D₁₂₇	—	2	4	no	—
68	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
69	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
70	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
71	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
72	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
73	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
74	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
75	M₁₆×C₁₂₇	D₄	1	4	yes	complemented
76	C₂×C₈×C₁₂₇	—	2	2	no	—
77	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
80	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
83	C₄²×C₁₂₇	C₂³	1	2	yes	—
84	C₂²×D₁₂₇	—	2	2	no	—
85	C₂²×D₁₂₇	—	2	2	no	—
86	C₂²×D₁₂₇	—	4	1	no	—
87	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
89	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
90	Dic₂₅₄	—	2	2	no	—
91	D₅₀₈	—	2	4	no	—
92	D₅₀₈	—	2	2	no	—
93	D₅₀₈	—	4	2	no	—
94	C₂²⋊D₁₂₇	—	2	4	no	—
95	C₂×Dic₁₂₇	—	2	2	no	—
96	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
99	C₄×D₁₂₇	—	2	2	no	—
100	C₄×D₁₂₇	—	2	2	no	—
101	C₄×D₁₂₇	—	2	1	no	—
102	C₄×D₁₂₇	—	2	2	no	—
103	C₄×D₁₂₇	—	2	1	no	—
104	C₄×D₁₂₇	—	2	1	no	—
105	C₄×D₁₂₇	—	2	1	no	—
106	C₄×D₁₂₇	—	2	2	no	—
107	C₁₂₇⋊C₈	—	2	1	no	—
108	C₁₂₇⋊C₈	—	2	2	no	—
109	C₁₂₇⋊C₈	—	2	1	no	—
110	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
113	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₁₂₇	#41	1	2	yes	—
116	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
117	C₈×C₁₂₇	—	2	1	no	—
118	C₈×C₁₂₇	—	2	1	no	—
119	C₈×C₁₂₇	C₂×D₄	1	4	yes	—
120	C₂²×C₁₂₇	—	2	1	no	—
121	C₂²×C₁₂₇	—	2	1	no	—
122	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
123	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₁₂₇	—	2	1	no	—
125	C₂×D₁₂₇	—	2	1	no	—
126	C₂×D₁₂₇	—	2	1	no	—
127	C₂×D₁₂₇	—	2	2	no	—
128	C₂×D₁₂₇	—	2	2	no	—
129	C₂×D₁₂₇	—	2	1	no	—
130	C₂×D₁₂₇	—	2	1	no	—
131	C₂×D₁₂₇	—	2	2	no	—
132	C₂×D₁₂₇	—	4	4	no	—
133	C₂×D₁₂₇	#131	1	2	yes	complemented
134	C₄×C₁₂₇	—	2	2	no	—
135	C₄×C₁₂₇	—	2	2	no	—
136	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
137	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
139	Dic₁₂₇	—	2	1	no	—
140	Dic₁₂₇	—	2	1	no	—
141	Dic₁₂₇	—	2	2	no	—
142	C₂×C₁₂₇	—	2	2	no	—
143	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
145	C₂×C₁₂₇	#568	1	1	yes	characteristic
146	D₁₂₇	—	2	2	no	—
147	D₁₂₇	—	4	2	no	—
148	D₁₂₇	—	4	1	no	—
149	#2933	—	127	1	no	maximal, Sylow
150	C₁₂₇	#2933	1	1	yes	characteristic, complemented, Sylow
151	C₂×#131	—	127	1	no	—
152	C₂²×M₁₆	—	127	1	no	—
153	#525	—	127	1	no	—
154	#436	—	127	2	no	—
155	#435	—	127	2	no	—
156	#435	—	127	2	no	—
157	C₈×D₄	—	127	2	no	—
158	#424	—	127	1	no	—
159	#408	—	127	1	no	—
160	C₂×C₂²⋊C₈	—	127	1	no	—
161	#405	D₁₂₇	1	1	yes	characteristic, complemented
162	C₂×#41	—	127	1	no	—
163	C₂³×C₄	—	127	1	no	—
164	#131	—	254	4	no	—
165	C₂×M₁₆	—	127	1	no	—
166	C₂×M₁₆	—	127	2	no	—
167	C₂×M₁₆	—	127	2	no	—
168	C₂×M₁₆	—	254	4	no	—
169	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
170	C₂²×C₈	—	127	1	no	—
171	C₂²×C₈	—	127	2	no	—
172	C₂²×C₈	—	127	2	no	—
173	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
174	#123	—	127	2	no	—
175	C₂²⋊Q₈	—	127	1	no	—
176	C₂²⋊Q₈	—	127	1	no	—
177	#121	—	127	1	no	—
178	#121	—	127	1	no	—
179	#120	—	127	2	no	—
180	C₄×D₄	—	127	2	no	—
181	C₄×D₄	—	127	2	no	—
182	#117	C₂×D₁₂₇	1	1	yes	characteristic
183	C₄⋊C₈	—	127	2	no	—
184	C₄⋊C₈	—	127	2	no	—
185	C₂²⋊C₈	—	127	1	no	—
186	C₂²⋊C₈	—	127	1	no	—
187	C₂²⋊C₈	—	127	1	no	—
188	C₂²⋊C₈	—	127	1	no	—
189	C₂²⋊C₈	—	127	2	no	—
190	C₂²⋊C₈	—	127	2	no	—
191	#97	C₂×D₁₂₇	1	2	yes	complemented
192	C₄×C₈	C₂×D₁₂₇	1	2	yes	complemented
193	C₂⁴	—	127	1	no	—
194	#41	—	254	2	no	—
195	#41	—	254	2	no	—
196	C₂×Q₈	—	127	1	no	—
197	C₂×D₄	—	127	2	no	—
198	C₂×D₄	—	127	2	no	—
199	C₂×D₄	—	127	2	no	—
200	C₂×D₄	—	127	1	no	—
201	C₂²×C₄	—	127	1	no	—
202	C₂²×C₄	—	127	1	no	—
203	C₂²×C₄	—	127	1	no	—
204	C₂²×C₄	—	127	1	no	—
205	C₂²×C₄	—	127	2	no	—
206	C₂²×C₄	—	127	1	no	—
207	C₂²×C₄	—	254	2	no	—
208	C₂²×C₄	—	254	2	no	—
209	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
210	M₁₆	—	254	2	no	—
211	M₁₆	—	254	4	no	—
212	M₁₆	—	254	4	no	—
213	M₁₆	D₅₀₈	1	4	yes	complemented
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	127	1	no	—
216	C₂×C₈	—	127	2	no	—
217	C₂×C₈	—	127	1	no	—
218	C₂×C₈	—	254	2	no	—
219	C₂×C₈	—	254	4	no	—
220	C₂×C₈	—	254	4	no	—
221	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
224	C₄⋊C₄	—	127	2	no	—
225	C₄⋊C₄	—	127	2	no	—
226	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
227	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
228	C₂²⋊C₄	—	127	2	no	—
229	C₂²⋊C₄	—	127	2	no	—
230	C₂²⋊C₄	—	127	2	no	—
231	C₂²⋊C₄	—	127	2	no	—
232	C₂²⋊C₄	C₄×D₁₂₇	1	2	yes	—
233	C₄²	C₂²×D₁₂₇	1	2	yes	—
234	C₂³	—	127	1	no	—
235	C₂³	—	127	1	no	—
236	C₂³	—	127	2	no	—
237	C₂³	—	508	1	no	—
238	C₂³	—	254	2	no	—
239	C₂³	—	254	2	no	—
240	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
241	Q₈	—	254	2	no	—
242	D₄	—	508	2	no	—
243	D₄	—	254	4	no	—
244	D₄	—	254	4	no	—
245	D₄	—	254	2	no	—
246	C₂×C₄	—	127	1	no	—
247	C₂×C₄	—	127	1	no	—
248	C₂×C₄	—	127	2	no	—
249	C₂×C₄	—	254	2	no	—
250	C₂×C₄	—	254	1	no	—
251	C₂×C₄	—	254	2	no	—
252	C₂×C₄	—	254	1	no	—
253	C₂×C₄	—	254	1	no	—
254	C₂×C₄	—	254	1	no	—
255	C₂×C₄	—	254	2	no	—
256	C₂×C₄	—	254	2	no	—
257	C₂×C₄	—	254	2	no	—
258	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
259	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
261	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
262	C₂×C₄	#14523248	1	2	yes	—
263	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	254	1	no	—
267	C₈	—	254	1	no	—
268	C₈	—	254	2	no	—
269	C₈	C₂×D₅₀₈	1	4	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	127	2	no	—
273	C₂²	—	508	4	no	—
274	C₂²	—	254	1	no	—
275	C₂²	—	254	2	no	—
276	C₂²	—	254	2	no	—
277	C₂²	—	254	1	no	—
278	C₂²	—	254	1	no	—
279	C₂²	—	254	2	no	—
280	C₂²	—	254	1	no	—
281	C₂²	—	254	1	no	—
282	C₂²	C₄×D₅₀₈	1	2	yes	—
283	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	254	1	no	—
287	C₄	—	254	1	no	—
288	C₄	—	254	2	no	—
289	C₄	C₄×D₅₀₈	1	2	yes	—
290	C₄	C₂²×D₅₀₈	1	1	yes	characteristic
291	C₄	C₂×#14523248	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	508	1	no	—
294	C₂	—	508	2	no	—
295	C₂	—	254	2	no	—
296	C₂	C₂×C₄×D₅₀₈	1	1	yes	characteristic
297	C₂	#131×D₁₂₇	1	1	yes	characteristic
298	C₂	#22550641	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#405×C₁₂₇	C₂
prime radical	#22550388	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#405	D₁₂₇
hypocenter	C₁₂₇	#2933
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#16861217	C₂	semidirect
2. C₂×M₁₆×D₁₂₇	C₂	semidirect
4. #22549955	C₂	semidirect
5. #22549949	C₂	semidirect
6. #22549949	C₂	semidirect
7. C₈×D₅₀₈	C₂	semidirect
8. #22549903	C₂	semidirect
9. #22549842	C₂	semidirect
10. C₂×(C₂×D₁₂₇)⋊C₈	C₂	semidirect
11. #405×C₁₂₇	C₂	semidirect
17. C₂×#14523231	C₂²	semidirect
21. C₂×C₈×D₁₂₇	C₂²	semidirect
30. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
31. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
36. (C₂×D₁₂₇)⋊C₈	C₂²	semidirect
37. (C₂×D₁₂₇)⋊C₈	C₂²	semidirect
41. #97×C₁₂₇	C₂²	semidirect
42. C₄×C₈×C₁₂₇	C₂²	semidirect
75. M₁₆×C₁₂₇	D₄	semidirect
133. C₂×D₁₂₇	#131	semidirect
150. C₁₂₇	#2933	semidirect
161. #405	D₁₂₇	semidirect
191. #97	C₂×D₁₂₇	semidirect
192. C₄×C₈	C₂×D₁₂₇	semidirect
213. M₁₆	D₅₀₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861217	2
C₂×M₁₆×D₁₂₇	2
#22550388	2
#22549955	2
#22549949	2
C₈×D₅₀₈	2
#22549903	2
#22549842	2
C₂×(C₂×D₁₂₇)⋊C₈	2
#405×C₁₂₇	2
#2933	127