Subgroups — C₂×#22550678 #30653480

Order 16256.

1512 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	C₂×#16861255	C₂	1	2	yes	maximal, complemented
2	#22550678	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16861236	C₂	1	4	yes	maximal, complemented
4	C₂×#16861234	C₂	1	2	yes	maximal, complemented
5	C₂×#16861232	C₂	1	2	yes	maximal, complemented
6	C₂²×D₁₀₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
8	C₂×#16861215	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#16861255	—	2	8	no	—
11	#16861255	C₂²	1	16	yes	complemented
12	C₂×#14523249	C₂²	1	4	yes	complemented
13	C₂×#14523248	—	2	2	no	—
14	C₂×#14523248	C₂²	1	1	yes	characteristic
15	C₂²×D₅₀₈	—	2	2	no	—
16	C₂²×D₅₀₈	C₂²	1	2	yes	complemented
17	C₂×D₄×D₁₂₇	—	2	4	no	—
18	C₂×D₄×D₁₂₇	C₂²	1	4	yes	complemented
19	#16861236	C₂²	1	32	yes	complemented
20	#16861234	C₂²	1	16	yes	complemented
21	#16861232	C₂²	1	16	yes	complemented
22	C₂×#14523235	C₂²	1	4	yes	complemented
23	C₂×D₁₀₁₆	C₂²	1	2	yes	complemented
24	C₂×D₁₀₁₆	C₂²	1	2	yes	complemented
25	C₂×D₁₀₁₆	C₂²	1	8	yes	complemented
26	C₂×#14523233	C₂²	1	4	yes	complemented
27	D₈×D₁₂₇	C₂²	1	16	yes	complemented
28	#16861215	C₂²	1	8	yes	complemented
29	C₂×#14523232	C₂²	1	1	yes	characteristic, complemented
30	C₂×#14523231	C₂²	1	2	yes	complemented
31	C₂×C₈×D₁₂₇	C₂²	1	2	yes	complemented
32	C₂×#41×C₁₂₇	C₂²	1	2	yes	complemented
33	#135×C₁₂₇	C₂²	1	8	yes	complemented
34	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×SD₁₆×C₁₂₇	C₂²	1	2	yes	complemented
36	C₂×D₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
38	C₂³×D₁₂₇	—	2	4	no	—
39	C₂³×D₁₂₇	—	2	2	no	—
40	#14523249	—	2	8	no	—
41	#14523249	C₂³	1	16	yes	complemented
42	#14523248	—	2	16	no	—
43	#14523248	D₄	1	4	yes	—
44	#14523248	C₂³	1	4	yes	—
45	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
46	C₂×D₅₀₈	—	2	4	no	—
47	C₂×D₅₀₈	—	2	16	no	—
48	C₂×D₅₀₈	—	2	2	no	—
49	C₂×D₅₀₈	—	2	4	no	—
50	C₂×D₅₀₈	D₄	1	1	yes	characteristic
51	C₂×D₅₀₈	C₂³	1	4	yes	—
52	C₂×D₅₀₈	C₂³	1	8	yes	complemented
53	C₂×C₂²⋊D₁₂₇	—	2	4	no	—
54	C₂×C₂²⋊D₁₂₇	D₄	1	2	yes	—
55	D₄×D₁₂₇	—	2	32	no	—
56	D₄×D₁₂₇	—	2	8	no	—
57	D₄×D₁₂₇	C₂³	1	16	yes	complemented
58	C₂×C₄×D₁₂₇	—	2	4	no	—
59	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
60	#14523235	C₂³	1	16	yes	complemented
61	D₁₀₁₆	C₂³	1	8	yes	complemented
62	D₁₀₁₆	C₂³	1	8	yes	complemented
63	#14523233	C₂³	1	16	yes	complemented
64	#14523232	C₂³	1	4	yes	complemented
65	#14523231	C₂³	1	8	yes	complemented
66	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
67	C₈×D₁₂₇	C₂³	1	8	yes	complemented
68	#41×C₁₂₇	—	2	4	no	—
69	#41×C₁₂₇	C₂³	1	8	yes	complemented
70	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
71	C₂×D₄×C₁₂₇	—	2	2	no	—
72	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₂₇	—	2	2	no	—
74	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
75	Q₁₆×C₁₂₇	C₂³	1	4	yes	complemented
76	SD₁₆×C₁₂₇	C₂³	1	8	yes	complemented
77	D₈×C₁₂₇	C₂³	1	4	yes	complemented
78	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₂₇	C₂³	1	4	yes	complemented
81	C₂²×D₁₂₇	—	2	4	no	—
82	C₂²×D₁₂₇	—	2	8	no	—
83	C₂²×D₁₂₇	—	2	16	no	—
84	C₂²×D₁₂₇	—	8	8	no	—
85	C₂²×D₁₂₇	—	8	4	no	—
86	C₂²×D₁₂₇	C₂×D₄	1	2	yes	—
87	Dic₂₅₄	C₂×D₄	1	4	yes	—
88	D₅₀₈	—	2	16	no	—
89	D₅₀₈	—	2	4	no	—
90	D₅₀₈	—	2	8	no	—
91	D₅₀₈	C₂×D₄	1	4	yes	—
92	D₅₀₈	C₂⁴	1	8	yes	—
93	C₂²⋊D₁₂₇	—	2	16	no	—
94	C₂²⋊D₁₂₇	C₂×D₄	1	8	yes	—
95	C₂×Dic₁₂₇	C₂×D₄	1	2	yes	—
96	C₄×D₁₂₇	—	2	16	no	—
97	C₄×D₁₂₇	C₂×D₄	1	4	yes	—
98	C₄×D₁₂₇	C₂⁴	1	4	yes	—
99	C₁₂₇⋊C₈	C₂⁴	1	4	yes	—
100	C₂³×C₁₂₇	—	2	2	no	—
101	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₁₂₇	—	2	2	no	—
103	Q₈×C₁₂₇	C₂⁴	1	4	yes	—
104	D₄×C₁₂₇	—	2	8	no	—
105	D₄×C₁₂₇	—	2	2	no	—
106	D₄×C₁₂₇	C₂⁴	1	4	yes	—
107	C₂×C₄×C₁₂₇	—	2	2	no	—
108	C₂×C₄×C₁₂₇	—	2	8	no	—
109	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
113	C₈×C₁₂₇	C₂⁴	1	2	yes	—
114	C₈×C₁₂₇	C₂⁴	1	2	yes	—
115	C₂²×C₁₂₇	—	2	2	no	—
116	C₂²×C₁₂₇	—	2	4	no	—
117	C₂²×C₁₂₇	—	4	4	no	—
118	C₂²×C₁₂₇	C₂²×D₄	1	2	yes	—
119	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₁₂₇	—	2	4	no	—
121	C₂×D₁₂₇	—	2	8	no	—
122	C₂×D₁₂₇	—	4	8	no	—
123	C₂×D₁₂₇	—	8	16	no	—
124	C₂×D₁₂₇	—	8	8	no	—
125	C₂×D₁₂₇	C₂²×D₄	1	4	yes	—
126	C₄×C₁₂₇	—	2	4	no	—
127	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
128	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
130	Dic₁₂₇	C₂²×D₄	1	4	yes	—
131	C₂×C₁₂₇	—	2	2	no	—
132	C₂×C₁₂₇	—	4	4	no	—
133	C₂×C₁₂₇	#576	1	2	yes	complemented
134	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
135	D₁₂₇	—	2	4	no	—
136	D₁₂₇	—	4	8	no	—
137	C₂×#576	—	127	1	no	maximal, Sylow
138	C₁₂₇	C₂×#576	1	1	yes	characteristic, complemented, Sylow
139	C₂×H(5,𝔽₂)	—	127	2	no	—
140	#576	—	127	16	no	—
141	C₂×AGL(1,ℤ/8)	—	127	2	no	—
142	C₂×AGL(1,ℤ/8)	—	127	4	no	—
143	C₂×#135	—	127	2	no	—
144	C₂×#135	D₁₂₇	1	1	yes	characteristic, complemented
145	C₂²×D₈	—	127	2	no	—
146	C₂²×D₈	—	127	1	no	—
147	C₂×#131	—	127	1	no	—
148	H(5,𝔽₂)	—	127	16	no	—
149	H(5,𝔽₂)	—	254	8	no	—
150	C₂×#41	—	127	1	no	—
151	C₂×#41	—	127	4	no	—
152	C₂×#41	—	254	2	no	—
153	C₂×#41	C₂×D₁₂₇	1	2	yes	complemented
154	C₂²×D₄	—	127	4	no	—
155	C₂²×D₄	—	127	2	no	—
156	C₂²×D₄	—	254	2	no	—
157	C₂²×D₄	—	254	4	no	—
158	AGL(1,ℤ/8)	—	127	16	no	—
159	AGL(1,ℤ/8)	—	127	32	no	—
160	#135	—	127	16	no	—
161	#135	C₂×D₁₂₇	1	8	yes	complemented
162	C₂×Q₁₆	C₂×D₁₂₇	1	1	yes	characteristic, complemented
163	C₂×SD₁₆	—	127	4	no	—
164	C₂×SD₁₆	C₂×D₁₂₇	1	2	yes	complemented
165	C₂×D₈	—	127	4	no	—
166	C₂×D₈	—	127	16	no	—
167	C₂×D₈	—	127	2	no	—
168	C₂×D₈	—	127	2	no	—
169	C₂×D₈	—	127	8	no	—
170	C₂×D₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
171	#131	—	127	8	no	—
172	C₂×M₁₆	—	127	1	no	—
173	C₂×M₁₆	—	127	2	no	—
174	C₂²×C₈	—	127	2	no	—
175	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
176	C₂⁴	—	254	4	no	—
177	C₂⁴	—	254	2	no	—
178	#41	—	2	4	no	—
179	#41	—	127	4	no	—
180	#41	—	127	4	no	—
181	#41	—	127	16	no	—
182	#41	—	254	16	no	—
183	#41	—	254	8	no	—
184	#41	C₂²×D₁₂₇	1	8	yes	complemented
185	C₂×Q₈	—	127	1	no	—
186	C₂×Q₈	C₂²×D₁₂₇	1	2	yes	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	127	2	no	—
189	C₂×D₄	—	127	16	no	—
190	C₂×D₄	—	127	1	no	—
191	C₂×D₄	—	127	4	no	—
192	C₂×D₄	—	127	8	no	—
193	C₂×D₄	—	254	16	no	—
194	C₂×D₄	—	254	4	no	—
195	C₂×D₄	—	254	8	no	—
196	C₂×D₄	—	254	4	no	—
197	C₂×D₄	—	254	32	no	—
198	C₂×D₄	—	254	2	no	—
199	C₂×D₄	—	254	4	no	—
200	C₂×D₄	C₂²×D₁₂₇	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	127	2	no	—
203	C₂²×C₄	—	254	4	no	—
204	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
205	Q₁₆	C₂²×D₁₂₇	1	4	yes	complemented
206	SD₁₆	—	127	16	no	—
207	SD₁₆	C₂²×D₁₂₇	1	8	yes	complemented
208	D₈	—	127	16	no	—
209	D₈	—	127	8	no	—
210	D₈	—	127	8	no	—
211	D₈	C₂²×D₁₂₇	1	4	yes	complemented
212	M₁₆	—	127	4	no	—
213	M₁₆	—	127	8	no	—
214	C₂×C₈	—	127	2	no	—
215	C₂×C₈	—	127	8	no	—
216	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
217	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
218	C₂×C₈	C₂²×D₁₂₇	1	4	yes	complemented
219	C₂³	—	2	2	no	—
220	C₂³	—	127	2	no	—
221	C₂³	—	1016	4	no	—
222	C₂³	—	1016	8	no	—
223	C₂³	—	254	16	no	—
224	C₂³	—	254	8	no	—
225	C₂³	—	254	4	no	—
226	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	127	4	no	—
229	Q₈	C₂³×D₁₂₇	1	4	yes	—
230	D₄	—	2	8	no	—
231	D₄	—	2	2	no	—
232	D₄	—	127	8	no	—
233	D₄	—	127	4	no	—
234	D₄	—	127	8	no	—
235	D₄	—	254	16	no	—
236	D₄	—	254	16	no	—
237	D₄	—	254	8	no	—
238	D₄	—	254	4	no	—
239	D₄	C₂³×D₁₂₇	1	4	yes	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	127	2	no	—
243	C₂×C₄	—	127	4	no	—
244	C₂×C₄	—	127	4	no	—
245	C₂×C₄	—	254	16	no	—
246	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
247	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
248	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
249	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
250	C₈	—	127	4	no	—
251	C₈	C₂³×D₁₂₇	1	2	yes	—
252	C₈	C₂³×D₁₂₇	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	127	4	no	—
257	C₂²	—	1016	8	no	—
258	C₂²	—	1016	16	no	—
259	C₂²	—	508	8	no	—
260	C₂²	—	254	8	no	—
261	C₂²	—	254	4	no	—
262	C₂²	C₂×D₄×D₁₂₇	1	2	yes	—
263	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	127	4	no	—
266	C₄	C₂×D₄×D₁₂₇	1	2	yes	—
267	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
268	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	4	4	no	—
271	C₂	—	508	8	no	—
272	C₂	—	254	4	no	—
273	C₂	#22550678	1	2	yes	direct product factor
274	C₂	C₂²×D₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator, cosocle	C₄×C₁₂₇	C₂⁵
Frattini	C₄	C₂⁴×D₁₂₇
Fitting	C₂×#135×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
hypercenter	C₂×#135	D₁₂₇
hypocenter	C₁₂₇	C₂×#576
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#16861255	C₂	semidirect
2. #22550678	C₂	direct (N × H)
3. C₂×#16861236	C₂	semidirect
4. C₂×#16861234	C₂	semidirect
5. C₂×#16861232	C₂	semidirect
6. C₂²×D₁₀₁₆	C₂	semidirect
7. C₂×D₈×D₁₂₇	C₂	semidirect
8. C₂×#16861215	C₂	semidirect
9. C₂×#135×C₁₂₇	C₂	semidirect
11. #16861255	C₂²	semidirect
12. C₂×#14523249	C₂²	semidirect
16. C₂²×D₅₀₈	C₂²	semidirect
18. C₂×D₄×D₁₂₇	C₂²	semidirect
19. #16861236	C₂²	semidirect
20. #16861234	C₂²	semidirect
21. #16861232	C₂²	semidirect
22. C₂×#14523235	C₂²	semidirect
23. C₂×D₁₀₁₆	C₂²	semidirect
24. C₂×D₁₀₁₆	C₂²	semidirect
25. C₂×D₁₀₁₆	C₂²	semidirect
26. C₂×#14523233	C₂²	semidirect
27. D₈×D₁₂₇	C₂²	semidirect
28. #16861215	C₂²	semidirect
29. C₂×#14523232	C₂²	semidirect
30. C₂×#14523231	C₂²	semidirect
31. C₂×C₈×D₁₂₇	C₂²	semidirect
32. C₂×#41×C₁₂₇	C₂²	semidirect
33. #135×C₁₂₇	C₂²	semidirect
34. C₂×Q₁₆×C₁₂₇	C₂²	semidirect
35. C₂×SD₁₆×C₁₂₇	C₂²	semidirect
36. C₂×D₈×C₁₂₇	C₂²	semidirect
37. C₂²×C₈×C₁₂₇	C₂²	semidirect
41. #14523249	C₂³	semidirect
52. C₂×D₅₀₈	C₂³	semidirect
57. D₄×D₁₂₇	C₂³	semidirect
60. #14523235	C₂³	semidirect
61. D₁₀₁₆	C₂³	semidirect
62. D₁₀₁₆	C₂³	semidirect
63. #14523233	C₂³	semidirect
64. #14523232	C₂³	semidirect
65. #14523231	C₂³	semidirect
67. C₈×D₁₂₇	C₂³	semidirect
69. #41×C₁₂₇	C₂³	semidirect
75. Q₁₆×C₁₂₇	C₂³	semidirect
76. SD₁₆×C₁₂₇	C₂³	semidirect
77. D₈×C₁₂₇	C₂³	semidirect
80. C₂×C₈×C₁₂₇	C₂³	semidirect
133. C₂×C₁₂₇	#576	semidirect
138. C₁₂₇	C₂×#576	semidirect
144. C₂×#135	D₁₂₇	semidirect
153. C₂×#41	C₂×D₁₂₇	semidirect
161. #135	C₂×D₁₂₇	semidirect
162. C₂×Q₁₆	C₂×D₁₂₇	semidirect
164. C₂×SD₁₆	C₂×D₁₂₇	semidirect
170. C₂×D₈	C₂×D₁₂₇	semidirect
175. C₂²×C₈	C₂×D₁₂₇	semidirect
184. #41	C₂²×D₁₂₇	semidirect
205. Q₁₆	C₂²×D₁₂₇	semidirect
207. SD₁₆	C₂²×D₁₂₇	semidirect
211. D₈	C₂²×D₁₂₇	semidirect
218. C₂×C₈	C₂²×D₁₂₇	semidirect
273. C₂	#22550678	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16861255	2
#22550678	2
C₂×#16861236	2
C₂×#16861234	2
C₂×#16861232	2
C₂²×D₁₀₁₆	2
C₂×D₈×D₁₂₇	2
C₂×#16861215	2
C₂×#135×C₁₂₇	2
C₂×#576	127

Subgroups — C2×#22550678 #30653480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550678 #30653480