Subgroups — #30647451

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×#16861215	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550387	C₂	1	1	yes	maximal, characteristic
4	#22549951	C₂	1	2	yes	maximal, complemented
5	#22549950	C₂	1	2	yes	maximal, complemented
6	#22549944	C₂	1	2	yes	maximal, complemented
7	(C₂²⋊D₁₂₇)⋊C₈	C₂	1	2	yes	maximal, complemented
8	#22549902	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549842	C₂	1	1	yes	maximal, characteristic, complemented
10	#22549831	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	#16861215	—	2	4	no	—
14	C₂×#14523232	—	2	4	no	—
15	C₂×#14523232	C₂²	1	1	yes	characteristic
16	C₂×#14523232	C₂²	1	2	yes	—
17	C₂×#14523232	C₂²	1	1	yes	characteristic
18	C₂×#14523231	C₂²	1	2	yes	complemented
19	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
20	C₂×C₈×D₁₂₇	C₂²	1	2	yes	complemented
21	#16861192	C₄	1	2	yes	—
22	#16861188	C₄	1	1	yes	characteristic
23	#16861188	C₄	1	1	yes	characteristic
24	#16861183	C₄	1	1	yes	characteristic
25	#16861183	C₄	1	1	yes	characteristic
26	#16861178	C₄	1	2	yes	—
27	C₄×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
29	#16861168	C₂²	1	1	yes	characteristic
30	Dic₁₂₇⋊C₈	C₂²	1	2	yes	complemented
31	Dic₁₂₇⋊C₈	C₂²	1	2	yes	complemented
32	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	complemented
33	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	complemented
34	#16861119	C₂²	1	2	yes	complemented
35	C₈×Dic₁₂₇	C₂²	1	2	yes	complemented
36	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
42	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
43	#14523248	—	2	2	no	—
44	#14523248	—	2	2	no	—
45	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
46	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
51	#14523232	—	2	4	no	—
52	#14523232	—	2	2	no	—
53	#14523232	D₄	1	4	yes	complemented
54	#14523231	—	2	4	no	—
55	C₂×C₁₂₇⋊C₈	—	2	4	no	—
56	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
57	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
58	C₈×D₁₂₇	—	2	4	no	—
59	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
61	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
63	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
64	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
65	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
66	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
67	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₂₇	C₂³	1	2	yes	—
69	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₂₇	—	2	2	no	—
71	C₂²×C₄×C₁₂₇	—	2	2	no	—
72	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₂₇	—	2	2	no	—
79	C₂×C₈×C₁₂₇	—	2	2	no	—
80	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
84	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
85	Dic₂₅₄	—	2	2	no	—
86	D₅₀₈	—	2	2	no	—
87	C₂²⋊D₁₂₇	—	2	4	no	—
88	C₂²⋊D₁₂₇	—	2	4	no	—
89	C₂²⋊D₁₂₇	—	4	2	no	—
90	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
91	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
92	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
93	C₄×D₁₂₇	—	2	2	no	—
94	C₄×D₁₂₇	—	2	2	no	—
95	C₁₂₇⋊C₈	—	2	2	no	—
96	C₁₂₇⋊C₈	C₂×D₄	1	4	yes	—
97	C₂³×C₁₂₇	—	2	2	no	—
98	C₂³×C₁₂₇	—	2	2	no	—
99	C₂³×C₁₂₇	—	4	1	no	—
100	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	—	2	2	no	—
104	C₂×C₄×C₁₂₇	—	2	2	no	—
105	C₂×C₄×C₁₂₇	—	2	2	no	—
106	C₂×C₄×C₁₂₇	—	2	1	no	—
107	C₂×C₄×C₁₂₇	—	2	1	no	—
108	C₂×C₄×C₁₂₇	—	2	1	no	—
109	C₂×C₄×C₁₂₇	—	2	1	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	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₁₂₇	#41	1	2	yes	—
115	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
116	C₈×C₁₂₇	—	2	1	no	—
117	C₈×C₁₂₇	—	2	1	no	—
118	C₈×C₁₂₇	—	2	1	no	—
119	C₈×C₁₂₇	—	2	1	no	—
120	C₂²×C₁₂₇	—	2	1	no	—
121	C₂²×C₁₂₇	—	2	1	no	—
122	C₂²×C₁₂₇	—	2	1	no	—
123	C₂²×C₁₂₇	—	2	1	no	—
124	C₂²×C₁₂₇	—	2	1	no	—
125	C₂²×C₁₂₇	—	2	1	no	—
126	C₂²×C₁₂₇	—	2	1	no	—
127	C₂²×C₁₂₇	—	4	4	no	—
128	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
129	C₂²×C₁₂₇	#131	1	2	yes	complemented
130	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×D₁₂₇	—	2	2	no	—
132	C₂×D₁₂₇	—	2	2	no	—
133	C₂×D₁₂₇	—	2	2	no	—
134	C₄×C₁₂₇	—	2	1	no	—
135	C₄×C₁₂₇	—	2	1	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	2	no	—
140	Dic₁₂₇	—	2	2	no	—
141	Dic₁₂₇	—	2	2	no	—
142	C₂×C₁₂₇	—	2	2	no	—
143	C₂×C₁₂₇	—	2	2	no	—
144	C₂×C₁₂₇	—	4	1	no	—
145	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
147	C₂×C₁₂₇	#568	1	1	yes	characteristic
148	D₁₂₇	—	4	2	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	D₁₂₇	1	1	yes	characteristic, complemented
161	#405	—	127	1	no	—
162	C₂×#41	—	127	1	no	—
163	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
164	#131	—	254	4	no	—
165	C₂×M₁₆	—	127	1	no	—
166	C₂×M₁₆	—	127	1	no	—
167	C₂×M₁₆	—	127	2	no	—
168	C₂×M₁₆	—	127	2	no	—
169	C₂×M₁₆	—	254	4	no	—
170	C₂²×C₈	—	127	2	no	—
171	C₂²×C₈	—	127	2	no	—
172	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
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	—	127	1	no	—
183	C₄⋊C₈	—	127	2	no	—
184	C₄⋊C₈	—	127	2	no	—
185	C₂²⋊C₈	—	127	2	no	—
186	C₂²⋊C₈	—	127	2	no	—
187	C₂²⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
188	C₂²⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
189	C₂²⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
190	C₂²⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
191	#97	—	127	2	no	—
192	C₄×C₈	—	127	2	no	—
193	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
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	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	127	2	no	—
204	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
205	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
206	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
210	M₁₆	—	127	4	no	—
211	M₁₆	—	254	4	no	—
212	M₁₆	—	254	4	no	—
213	M₁₆	—	254	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	127	2	no	—
217	C₂×C₈	—	127	2	no	—
218	C₂×C₈	—	254	4	no	—
219	C₂×C₈	—	254	4	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	C₄⋊C₄	—	127	2	no	—
225	C₄⋊C₄	—	127	2	no	—
226	C₄⋊C₄	—	127	1	no	—
227	C₄⋊C₄	—	127	1	no	—
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	—	127	2	no	—
233	C₄²	—	127	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	127	2	no	—
238	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
239	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
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	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	127	2	no	—
254	C₂×C₄	—	127	2	no	—
255	C₂×C₄	—	127	2	no	—
256	C₂×C₄	—	254	2	no	—
257	C₂×C₄	—	254	2	no	—
258	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
261	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
262	C₂×C₄	#14523247	1	2	yes	—
263	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	127	4	no	—
269	C₈	—	254	2	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	254	2	no	—
279	C₂²	—	254	2	no	—
280	C₂²	—	254	2	no	—
281	C₂²	(C₂²⋊D₁₂₇)⋊C₄	1	2	yes	—
282	C₂²	#16861215	1	2	yes	complemented
283	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	254	2	no	—
287	C₄	—	254	2	no	—
288	C₄	—	254	2	no	—
289	C₄	(C₂²⋊D₁₂₇)⋊C₄	1	2	yes	—
290	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
291	C₄	C₂×#14523247	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	508	2	no	—
296	C₂	C₂×(C₂²⋊D₁₂₇)⋊C₄	1	1	yes	characteristic
297	C₂	C₂×#16861215	1	1	yes	characteristic
298	C₂	#22550642	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	C₂×C₂²⋊C₈×C₁₂₇	C₂
prime radical	#22550387	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#2933
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16861215	C₂	semidirect
2. C₂²×#14523232	C₂	semidirect
4. #22549951	C₂	semidirect
5. #22549950	C₂	semidirect
6. #22549944	C₂	semidirect
7. (C₂²⋊D₁₂₇)⋊C₈	C₂	semidirect
8. #22549902	C₂	semidirect
9. #22549842	C₂	semidirect
10. #22549831	C₂	semidirect
11. C₂×C₂²⋊C₈×C₁₂₇	C₂	semidirect
18. C₂×#14523231	C₂²	semidirect
20. C₂×C₈×D₁₂₇	C₂²	semidirect
30. Dic₁₂₇⋊C₈	C₂²	semidirect
31. Dic₁₂₇⋊C₈	C₂²	semidirect
32. (C₂×D₁₂₇)⋊C₈	C₂²	semidirect
33. (C₂×D₁₂₇)⋊C₈	C₂²	semidirect
34. #16861119	C₂²	semidirect
35. C₈×Dic₁₂₇	C₂²	semidirect
53. #14523232	D₄	semidirect
129. C₂²×C₁₂₇	#131	semidirect
150. C₁₂₇	#2933	semidirect
160. C₂×C₂²⋊C₈	D₁₂₇	semidirect
282. C₂²	#16861215	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861215	2
C₂²×#14523232	2
#22550387	2
#22549951	2
#22549950	2
#22549944	2
(C₂²⋊D₁₂₇)⋊C₈	2
#22549902	2
#22549842	2
#22549831	2
C₂×C₂²⋊C₈×C₁₂₇	2
#2933	127