Subgroups — C₂×#22550677 #30653479

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×#16861254	C₂	1	2	yes	maximal, complemented
2	#22550677	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16861235	C₂	1	2	yes	maximal, complemented
4	C₂×#16861233	C₂	1	4	yes	maximal, complemented
5	C₂×#16861231	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16861228	C₂	1	2	yes	maximal, complemented
7	C₂×D₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
8	C₂×#16861215	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#16861254	—	2	8	no	—
11	#16861254	C₂²	1	16	yes	complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	C₂×#14523248	C₂²	1	2	yes	complemented
14	C₂×#14523247	—	2	2	no	—
15	C₂×#14523247	C₂²	1	4	yes	complemented
16	C₂²×C₂²⋊D₁₂₇	—	2	4	no	—
17	C₂×D₄×D₁₂₇	—	2	2	no	—
18	C₂×D₄×D₁₂₇	C₂²	1	4	yes	complemented
19	#16861235	C₂²	1	16	yes	complemented
20	#16861233	C₂²	1	32	yes	complemented
21	#16861231	C₂²	1	8	yes	complemented
22	#16861228	C₂²	1	16	yes	complemented
23	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×#14523237	C₂²	1	2	yes	complemented
25	C₂×#14523236	C₂²	1	4	yes	complemented
26	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic, complemented
27	C₂×#14523233	C₂²	1	4	yes	complemented
28	D₈×D₁₂₇	C₂²	1	16	yes	complemented
29	#16861215	C₂²	1	8	yes	complemented
30	C₂×#14523232	C₂²	1	1	yes	characteristic, complemented
31	C₂×#14523231	C₂²	1	2	yes	complemented
32	C₂×C₈×D₁₂₇	C₂²	1	2	yes	complemented
33	C₂²×D₄×C₁₂₇	C₂²	1	2	yes	complemented
34	C₂×D₈×C₁₂₇	C₂²	1	2	yes	complemented
35	C₂×D₈×C₁₂₇	C₂²	1	2	yes	complemented
36	C₂×D₈×C₁₂₇	C₂²	1	8	yes	complemented
37	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
38	C₂³×D₁₂₇	—	2	4	no	—
39	#14523248	—	2	4	no	—
40	#14523248	D₄	1	4	yes	—
41	#14523248	C₂³	1	4	yes	—
42	#14523248	C₂³	1	8	yes	complemented
43	#14523247	—	2	16	no	—
44	#14523247	—	2	8	no	—
45	#14523247	C₂³	1	16	yes	complemented
46	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
47	C₂×Dic₂₅₄	C₂³	1	2	yes	—
48	C₂×D₅₀₈	D₄	1	1	yes	characteristic
49	C₂×D₅₀₈	C₂³	1	2	yes	—
50	C₂×C₂²⋊D₁₂₇	—	2	4	no	—
51	C₂×C₂²⋊D₁₂₇	—	2	32	no	—
52	C₂×C₂²⋊D₁₂₇	—	2	4	no	—
53	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
54	C₂×C₂²⋊D₁₂₇	D₄	1	2	yes	—
55	D₄×D₁₂₇	—	2	16	no	—
56	D₄×D₁₂₇	—	2	8	no	—
57	D₄×D₁₂₇	C₂³	1	16	yes	complemented
58	C₂²×Dic₁₂₇	—	2	4	no	—
59	C₂×C₄×D₁₂₇	—	2	2	no	—
60	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
61	Dic₅₀₈	C₂³	1	4	yes	complemented
62	#14523237	C₂³	1	8	yes	complemented
63	#14523236	C₂³	1	16	yes	complemented
64	D₁₀₁₆	C₂³	1	4	yes	complemented
65	#14523233	C₂³	1	16	yes	complemented
66	#14523232	C₂³	1	4	yes	complemented
67	#14523231	C₂³	1	8	yes	complemented
68	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
69	C₈×D₁₂₇	C₂³	1	8	yes	complemented
70	C₂⁴×C₁₂₇	—	2	2	no	—
71	C₂×D₄×C₁₂₇	—	2	2	no	—
72	C₂×D₄×C₁₂₇	—	2	4	no	—
73	C₂×D₄×C₁₂₇	C₂³	1	4	yes	—
74	C₂×D₄×C₁₂₇	C₂³	1	8	yes	complemented
75	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
76	D₈×C₁₂₇	C₂³	1	8	yes	complemented
77	D₈×C₁₂₇	C₂³	1	8	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	2	no	—
82	C₂²×D₁₂₇	—	2	16	no	—
83	C₂²×D₁₂₇	—	8	8	no	—
84	C₂²×D₁₂₇	C₂×D₄	1	2	yes	—
85	Dic₂₅₄	—	2	2	no	—
86	Dic₂₅₄	C₂×D₄	1	4	yes	—
87	Dic₂₅₄	C₂⁴	1	4	yes	—
88	D₅₀₈	—	2	2	no	—
89	D₅₀₈	C₂×D₄	1	4	yes	—
90	D₅₀₈	C₂⁴	1	4	yes	—
91	C₂²⋊D₁₂₇	—	2	8	no	—
92	C₂²⋊D₁₂₇	—	2	16	no	—
93	C₂²⋊D₁₂₇	—	2	16	no	—
94	C₂²⋊D₁₂₇	C₂×D₄	1	8	yes	—
95	C₂×Dic₁₂₇	—	2	2	no	—
96	C₂×Dic₁₂₇	—	2	16	no	—
97	C₂×Dic₁₂₇	C₂×D₄	1	2	yes	—
98	C₄×D₁₂₇	—	2	8	no	—
99	C₄×D₁₂₇	C₂×D₄	1	4	yes	—
100	C₄×D₁₂₇	C₂⁴	1	4	yes	—
101	C₁₂₇⋊C₈	C₂⁴	1	4	yes	—
102	C₂³×C₁₂₇	—	2	4	no	—
103	C₂³×C₁₂₇	—	2	8	no	—
104	C₂³×C₁₂₇	—	8	4	no	—
105	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
106	D₄×C₁₂₇	—	2	4	no	—
107	D₄×C₁₂₇	—	2	8	no	—
108	D₄×C₁₂₇	C₂⁴	1	8	yes	—
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	8	no	—
117	C₂²×C₁₂₇	—	4	8	no	—
118	C₂²×C₁₂₇	—	8	8	no	—
119	C₂²×C₁₂₇	C₂²×D₄	1	2	yes	—
120	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₂₇	—	2	4	no	—
122	C₂×D₁₂₇	—	2	4	no	—
123	C₂×D₁₂₇	—	4	4	no	—
124	C₂×D₁₂₇	—	8	16	no	—
125	C₂×D₁₂₇	C₂²×D₄	1	4	yes	—
126	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
127	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
129	Dic₁₂₇	—	2	4	no	—
130	Dic₁₂₇	C₂²×D₄	1	4	yes	—
131	C₂×C₁₂₇	—	2	2	no	—
132	C₂×C₁₂₇	—	4	8	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	4	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	—	127	1	no	—
145	C₂²×D₈	—	127	2	no	—
146	C₂²×D₈	D₁₂₇	1	1	yes	characteristic, complemented
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	—	127	2	no	—
153	C₂×#41	—	254	2	no	—
154	C₂²×D₄	—	127	4	no	—
155	C₂²×D₄	—	254	2	no	—
156	C₂²×D₄	—	254	4	no	—
157	C₂²×D₄	C₂×D₁₂₇	1	2	yes	complemented
158	AGL(1,ℤ/8)	—	127	16	no	—
159	AGL(1,ℤ/8)	—	127	32	no	—
160	#135	—	127	16	no	—
161	#135	—	127	8	no	—
162	C₂×Q₁₆	—	127	1	no	—
163	C₂×SD₁₆	—	127	4	no	—
164	C₂×SD₁₆	—	127	2	no	—
165	C₂×D₈	—	127	4	no	—
166	C₂×D₈	—	127	16	no	—
167	C₂×D₈	—	127	1	no	—
168	C₂×D₈	C₂×D₁₂₇	1	2	yes	complemented
169	C₂×D₈	C₂×D₁₂₇	1	2	yes	complemented
170	C₂×D₈	C₂×D₁₂₇	1	8	yes	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	—	2	2	no	—
177	C₂⁴	—	254	4	no	—
178	#41	—	127	4	no	—
179	#41	—	127	4	no	—
180	#41	—	127	16	no	—
181	#41	—	127	8	no	—
182	#41	—	254	8	no	—
183	#41	—	254	16	no	—
184	#41	—	254	4	no	—
185	C₂×Q₈	—	127	1	no	—
186	C₂×Q₈	—	127	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	127	2	no	—
190	C₂×D₄	—	127	16	no	—
191	C₂×D₄	—	127	1	no	—
192	C₂×D₄	—	127	2	no	—
193	C₂×D₄	—	254	8	no	—
194	C₂×D₄	—	254	4	no	—
195	C₂×D₄	—	254	16	no	—
196	C₂×D₄	—	254	32	no	—
197	C₂×D₄	—	254	4	no	—
198	C₂×D₄	—	254	2	no	—
199	C₂×D₄	C₂²×D₁₂₇	1	4	yes	—
200	C₂×D₄	C₂²×D₁₂₇	1	8	yes	complemented
201	C₂²×C₄	—	127	2	no	—
202	C₂²×C₄	—	254	4	no	—
203	C₂²×C₄	—	254	2	no	—
204	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
205	Q₁₆	—	127	4	no	—
206	SD₁₆	—	127	16	no	—
207	SD₁₆	—	127	8	no	—
208	D₈	—	127	16	no	—
209	D₈	—	127	4	no	—
210	D₈	C₂²×D₁₂₇	1	8	yes	complemented
211	D₈	C₂²×D₁₂₇	1	8	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	4	no	—
220	C₂³	—	2	8	no	—
221	C₂³	—	8	4	no	—
222	C₂³	—	127	2	no	—
223	C₂³	—	1016	8	no	—
224	C₂³	—	254	16	no	—
225	C₂³	—	254	2	no	—
226	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
227	Q₈	—	127	4	no	—
228	Q₈	—	127	4	no	—
229	Q₈	—	254	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	2	8	no	—
232	D₄	—	127	8	no	—
233	D₄	—	127	4	no	—
234	D₄	—	127	4	no	—
235	D₄	—	254	16	no	—
236	D₄	—	254	16	no	—
237	D₄	—	254	8	no	—
238	D₄	—	254	2	no	—
239	D₄	C₂³×D₁₂₇	1	8	yes	—
240	C₂×C₄	—	127	2	no	—
241	C₂×C₄	—	127	4	no	—
242	C₂×C₄	—	127	4	no	—
243	C₂×C₄	—	254	16	no	—
244	C₂×C₄	—	254	2	no	—
245	C₂×C₄	—	254	8	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	8	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	8	8	no	—
257	C₂²	—	127	4	no	—
258	C₂²	—	1016	16	no	—
259	C₂²	—	508	4	no	—
260	C₂²	—	254	4	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	—	127	4	no	—
265	C₄	—	254	4	no	—
266	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
267	C₄	C₂×D₄×D₁₂₇	1	2	yes	—
268	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	4	8	no	—
271	C₂	—	508	4	no	—
272	C₂	—	254	4	no	—
273	C₂	#22550677	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	C₂²×D₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
hypercenter	C₂²×D₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#576
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16861254	C₂	semidirect
2. #22550677	C₂	direct (N × H)
3. C₂×#16861235	C₂	semidirect
4. C₂×#16861233	C₂	semidirect
5. C₂×#16861231	C₂	semidirect
6. C₂×#16861228	C₂	semidirect
7. C₂×D₈×D₁₂₇	C₂	semidirect
8. C₂×#16861215	C₂	semidirect
9. C₂²×D₈×C₁₂₇	C₂	semidirect
11. #16861254	C₂²	semidirect
13. C₂×#14523248	C₂²	semidirect
15. C₂×#14523247	C₂²	semidirect
18. C₂×D₄×D₁₂₇	C₂²	semidirect
19. #16861235	C₂²	semidirect
20. #16861233	C₂²	semidirect
21. #16861231	C₂²	semidirect
22. #16861228	C₂²	semidirect
23. C₂×Dic₅₀₈	C₂²	semidirect
24. C₂×#14523237	C₂²	semidirect
25. C₂×#14523236	C₂²	semidirect
26. C₂×D₁₀₁₆	C₂²	semidirect
27. C₂×#14523233	C₂²	semidirect
28. D₈×D₁₂₇	C₂²	semidirect
29. #16861215	C₂²	semidirect
30. C₂×#14523232	C₂²	semidirect
31. C₂×#14523231	C₂²	semidirect
32. C₂×C₈×D₁₂₇	C₂²	semidirect
33. C₂²×D₄×C₁₂₇	C₂²	semidirect
34. C₂×D₈×C₁₂₇	C₂²	semidirect
35. C₂×D₈×C₁₂₇	C₂²	semidirect
36. C₂×D₈×C₁₂₇	C₂²	semidirect
37. C₂²×C₈×C₁₂₇	C₂²	semidirect
42. #14523248	C₂³	semidirect
45. #14523247	C₂³	semidirect
57. D₄×D₁₂₇	C₂³	semidirect
61. Dic₅₀₈	C₂³	semidirect
62. #14523237	C₂³	semidirect
63. #14523236	C₂³	semidirect
64. D₁₀₁₆	C₂³	semidirect
65. #14523233	C₂³	semidirect
66. #14523232	C₂³	semidirect
67. #14523231	C₂³	semidirect
69. C₈×D₁₂₇	C₂³	semidirect
74. C₂×D₄×C₁₂₇	C₂³	semidirect
76. D₈×C₁₂₇	C₂³	semidirect
77. D₈×C₁₂₇	C₂³	semidirect
80. C₂×C₈×C₁₂₇	C₂³	semidirect
133. C₂×C₁₂₇	#576	semidirect
138. C₁₂₇	C₂×#576	semidirect
146. C₂²×D₈	D₁₂₇	semidirect
157. C₂²×D₄	C₂×D₁₂₇	semidirect
168. C₂×D₈	C₂×D₁₂₇	semidirect
169. C₂×D₈	C₂×D₁₂₇	semidirect
170. C₂×D₈	C₂×D₁₂₇	semidirect
175. C₂²×C₈	C₂×D₁₂₇	semidirect
200. C₂×D₄	C₂²×D₁₂₇	semidirect
210. D₈	C₂²×D₁₂₇	semidirect
211. D₈	C₂²×D₁₂₇	semidirect
218. C₂×C₈	C₂²×D₁₂₇	semidirect
273. C₂	#22550677	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×#16861254	2
#22550677	2
C₂×#16861235	2
C₂×#16861233	2
C₂×#16861231	2
C₂×#16861228	2
C₂×D₈×D₁₂₇	2
C₂×#16861215	2
C₂²×D₈×C₁₂₇	2
C₂×#576	127

Subgroups — C2×#22550677 #30653479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550677 #30653479