Subgroups — #30647562

Order 16256.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861235	C₂	1	1	yes	maximal, characteristic
2	#22550356	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549986	C₂	1	2	yes	maximal, complemented
4	#22549976	C₂	1	2	yes	maximal, complemented
5	C₄×#14523236	C₂	1	2	yes	maximal, complemented
6	C₄×#14523233	C₂	1	2	yes	maximal, complemented
7	#22549919	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549870	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16861130	C₂	1	1	yes	maximal, characteristic, complemented
10	#22549832	C₂	1	1	yes	maximal, characteristic, complemented
11	#518×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	#16861235	C₄	1	8	yes	complemented
14	C₂×#14523236	C₂²	1	2	yes	—
15	C₂×#14523233	C₂²	1	2	yes	—
16	C₂×#14523232	C₂²	1	1	yes	characteristic
17	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₅₄	C₂²	1	2	yes	complemented
19	C₄×D₅₀₈	C₂²	1	2	yes	complemented
20	(C₂²⋊D₁₂₇)⋊C₄	—	2	2	no	—
21	#16861167	—	2	1	no	—
22	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₁₂₇	—	2	1	no	—
24	#16861142	C₂²	1	2	yes	complemented
25	#16861140	C₂²	1	2	yes	complemented
26	#16861132	C₂²	1	1	yes	characteristic, complemented
27	#16861132	C₂²	1	1	yes	characteristic, complemented
28	#16861130	C₂²	1	2	yes	—
29	#16861130	C₂²	1	2	yes	—
30	#16861129	C₂²	1	1	yes	characteristic, complemented
31	#16861129	C₂²	1	1	yes	characteristic, complemented
32	#16861120	C₂²	1	2	yes	complemented
33	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₁₂₇	—	2	1	no	—
36	C₄×D₄×C₁₂₇	—	2	1	no	—
37	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
38	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
39	#117×C₁₂₇	C₂²	1	1	yes	characteristic
40	#117×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
42	#14523248	—	2	2	no	—
43	#14523248	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
45	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
47	C₂²×Dic₁₂₇	—	2	1	no	—
48	C₂×C₄×D₁₂₇	—	2	1	no	—
49	C₂×C₄×D₁₂₇	—	2	2	no	—
50	#14523236	C₂×C₄	1	8	yes	complemented
51	#14523233	C₂×C₄	1	8	yes	complemented
52	#14523232	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₂₇⋊C₈	—	2	2	no	—
54	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
56	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
57	C₄⋊Dic₁₂₇	—	2	1	no	—
58	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
59	Dic₁₂₇⋊C₄	—	2	2	no	—
60	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
61	C₄×Dic₁₂₇	—	2	2	no	—
62	C₂⁴×C₁₂₇	—	2	1	no	—
63	C₂×D₄×C₁₂₇	—	2	1	no	—
64	C₂×D₄×C₁₂₇	—	2	2	no	—
65	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
66	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₂₇	—	2	2	no	—
68	C₂²×C₄×C₁₂₇	—	2	2	no	—
69	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
75	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
76	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
77	C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
78	C₄²×C₁₂₇	—	2	1	no	—
79	C₄²×C₁₂₇	C₂³	1	2	yes	—
80	C₂²×D₁₂₇	—	2	1	no	—
81	Dic₂₅₄	—	2	1	no	—
82	Dic₂₅₄	C₂²×C₄	1	2	yes	—
83	D₅₀₈	—	2	1	no	—
84	D₅₀₈	C₂²×C₄	1	2	yes	—
85	C₂²⋊D₁₂₇	—	2	4	no	—
86	C₂×Dic₁₂₇	—	2	1	no	—
87	C₂×Dic₁₂₇	—	2	2	no	—
88	C₂×Dic₁₂₇	—	8	1	no	—
89	C₄×D₁₂₇	—	2	4	no	—
90	C₄×D₁₂₇	—	8	2	no	—
91	C₁₂₇⋊C₈	—	2	1	no	—
92	C₁₂₇⋊C₈	—	2	1	no	—
93	C₁₂₇⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂³×C₁₂₇	—	2	2	no	—
95	C₂³×C₁₂₇	—	2	4	no	—
96	C₂³×C₁₂₇	—	8	1	no	—
97	C₂³×C₁₂₇	—	8	1	no	—
98	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
99	D₄×C₁₂₇	—	2	4	no	—
100	D₄×C₁₂₇	—	2	2	no	—
101	D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
102	C₂×C₄×C₁₂₇	—	2	1	no	—
103	C₂×C₄×C₁₂₇	—	2	1	no	—
104	C₂×C₄×C₁₂₇	—	8	2	no	—
105	C₂×C₄×C₁₂₇	—	8	2	no	—
106	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	#41	1	2	yes	—
111	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
112	C₂²×C₁₂₇	—	2	1	no	—
113	C₂²×C₁₂₇	—	2	1	no	—
114	C₂²×C₁₂₇	—	2	4	no	—
115	C₂²×C₁₂₇	—	4	2	no	—
116	C₂²×C₁₂₇	—	4	2	no	—
117	C₂²×C₁₂₇	—	8	4	no	—
118	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
119	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₁₂₇	—	2	2	no	—
121	C₂×D₁₂₇	—	4	1	no	—
122	C₂×D₁₂₇	—	4	1	no	—
123	C₄×C₁₂₇	—	2	2	no	—
124	C₄×C₁₂₇	—	2	2	no	—
125	C₄×C₁₂₇	—	4	1	no	—
126	C₄×C₁₂₇	—	4	1	no	—
127	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
128	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
130	Dic₁₂₇	—	2	2	no	—
131	Dic₁₂₇	—	4	2	no	—
132	C₂×C₁₂₇	—	2	2	no	—
133	C₂×C₁₂₇	—	4	4	no	—
134	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₁₂₇	#576	1	1	yes	characteristic
136	C₂×C₁₂₇	#577	1	1	yes	characteristic
137	D₁₂₇	—	4	2	no	—
138	#2946	—	127	1	no	maximal, Sylow
139	C₁₂₇	#2946	1	1	yes	characteristic, complemented, Sylow
140	C₂×AGL(1,ℤ/8)	—	127	1	no	—
141	#520	—	127	1	no	—
142	#518	D₁₂₇	1	1	yes	characteristic, complemented
143	#442	—	127	2	no	—
144	#440	—	127	2	no	—
145	C₄×SD₁₆	—	127	2	no	—
146	C₄×D₈	—	127	2	no	—
147	#428	—	127	1	no	—
148	#416	—	127	1	no	—
149	C₂×#102	—	127	1	no	—
150	#405	—	127	1	no	—
151	C₂×#41	—	127	1	no	—
152	C₂²×D₄	C₂×D₁₂₇	1	1	yes	characteristic
153	AGL(1,ℤ/8)	—	127	8	no	—
154	C₂×SD₁₆	—	127	2	no	—
155	C₂×D₈	—	127	2	no	—
156	C₂×M₁₆	—	127	1	no	—
157	C₂²×C₈	—	127	1	no	—
158	C₄×Q₈	—	127	2	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	127	2	no	—
162	C₄×D₄	—	254	2	no	—
163	C₄×D₄	C₂×D₁₂₇	1	2	yes	—
164	C₄×D₄	C₂×D₁₂₇	1	2	yes	—
165	#117	—	254	1	no	—
166	#117	C₂×D₁₂₇	1	1	yes	characteristic
167	#117	C₂×D₁₂₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	127	1	no	—
169	C₂×C₄⋊C₄	—	254	1	no	—
170	C₂×C₂²⋊C₄	—	2	2	no	—
171	#107	—	127	2	no	—
172	#106	—	127	2	no	—
173	#103	—	127	1	no	—
174	#103	—	127	1	no	—
175	#102	—	127	2	no	—
176	#102	—	127	2	no	—
177	#102	—	127	1	no	—
178	#102	—	127	1	no	—
179	#97	—	127	2	no	—
180	C₄×C₈	—	127	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	127	4	no	—
183	#41	—	254	2	no	—
184	C₂×Q₈	—	127	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	127	1	no	—
188	C₂×D₄	—	254	1	no	—
189	C₂×D₄	C₄×D₁₂₇	1	4	yes	complemented
190	C₂×D₄	C₂²×D₁₂₇	1	2	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	254	1	no	—
194	C₂²×C₄	—	254	2	no	—
195	C₂²×C₄	—	254	1	no	—
196	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
197	SD₁₆	—	127	8	no	—
198	D₈	—	127	8	no	—
199	M₁₆	—	127	4	no	—
200	C₂×C₈	—	127	2	no	—
201	C₂×C₈	—	127	1	no	—
202	C₂×C₈	—	127	1	no	—
203	C₂×C₈	—	254	2	no	—
204	C₄⋊C₄	—	127	2	no	—
205	C₄⋊C₄	—	254	2	no	—
206	C₄⋊C₄	—	254	1	no	—
207	C₄⋊C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
208	C₄⋊C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
209	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	254	2	no	—
215	C₂²⋊C₄	C₂²⋊D₁₂₇	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	254	2	no	—
218	C₄²	C₂²×D₁₂₇	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	254	1	no	—
224	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
225	Q₈	—	127	2	no	—
226	Q₈	—	254	1	no	—
227	D₄	—	2	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	127	2	no	—
230	D₄	—	254	4	no	—
231	D₄	—	254	1	no	—
232	D₄	C₂×C₄×D₁₂₇	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	1016	2	no	—
238	C₂×C₄	—	1016	1	no	—
239	C₂×C₄	—	254	2	no	—
240	C₂×C₄	—	254	1	no	—
241	C₂×C₄	—	254	4	no	—
242	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
246	C₂×C₄	#14523248	1	2	yes	—
247	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
248	C₈	—	127	4	no	—
249	C₈	—	254	1	no	—
250	C₈	—	254	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	508	1	no	—
258	C₂²	—	508	1	no	—
259	C₂²	—	254	2	no	—
260	C₂²	C₄×C₂²⋊D₁₂₇	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	508	2	no	—
267	C₄	—	254	2	no	—
268	C₄	C₄×C₂²⋊D₁₂₇	1	2	yes	—
269	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
270	C₄	C₂×#14523248	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	508	2	no	—
274	C₂	C₂×C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
275	C₂	#22550679	1	1	yes	characteristic
276	C₂	#22550687	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#518×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#16861235	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#518	D₁₂₇
hypocenter	C₁₂₇	#2946
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #518

Complements

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

N	H ≅ G/N	type
2. #22550356	C₂	semidirect
3. #22549986	C₂	semidirect
4. #22549976	C₂	semidirect
5. C₄×#14523236	C₂	semidirect
6. C₄×#14523233	C₂	semidirect
7. #22549919	C₂	semidirect
8. #22549870	C₂	semidirect
9. C₂×#16861130	C₂	semidirect
10. #22549832	C₂	semidirect
11. #518×C₁₂₇	C₂	semidirect
13. #16861235	C₄	semidirect
18. C₄×Dic₂₅₄	C₂²	semidirect
19. C₄×D₅₀₈	C₂²	semidirect
24. #16861142	C₂²	semidirect
25. #16861140	C₂²	semidirect
26. #16861132	C₂²	semidirect
27. #16861132	C₂²	semidirect
30. #16861129	C₂²	semidirect
31. #16861129	C₂²	semidirect
32. #16861120	C₂²	semidirect
33. C₄×C₁₂₇⋊C₈	C₂²	semidirect
43. #14523248	C₂×C₄	semidirect
50. #14523236	C₂×C₄	semidirect
51. #14523233	C₂×C₄	semidirect
52. #14523232	C₂×C₄	semidirect
65. C₂×D₄×C₁₂₇	C₂×C₄	semidirect
139. C₁₂₇	#2946	semidirect
142. #518	D₁₂₇	semidirect
189. C₂×D₄	C₄×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861235	2
#22550356	2
#22549986	2
#22549976	2
C₄×#14523236	2
C₄×#14523233	2
#22549919	2
#22549870	2
C₂×#16861130	2
#22549832	2
#518×C₁₂₇	2
#2946	127