Subgroups — C₂×#22550642 #30653439

Order 16256.

1460 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523248	C₂	1	1	yes	maximal, characteristic
2	#22550642	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16861217	C₂	1	4	yes	maximal, complemented
4	C₂×#16861215	C₂	1	4	yes	maximal, complemented
5	C₂²×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆×D₁₂₇	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14523248	C₄	1	2	yes	—
9	C₂×#14523248	C₄	1	2	yes	—
10	C₂×#14523248	C₄	1	8	yes	—
11	C₂×#14523248	C₂²	1	2	yes	—
12	C₂×#14523248	C₂²	1	2	yes	—
13	C₂×#14523248	C₂²	1	8	yes	—
14	C₂²×Dic₂₅₄	C₄	1	1	yes	characteristic
15	C₂²×D₅₀₈	C₄	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₂₇	C₄	1	2	yes	—
17	C₂²×C₄×D₁₂₇	C₂²	1	2	yes	—
18	#16861217	C₂²	1	32	yes	complemented
19	#16861215	C₂²	1	32	yes	complemented
20	C₂×#14523232	C₂²	1	2	yes	—
21	C₂×#14523232	C₂²	1	2	yes	complemented
22	C₂×#14523232	C₂²	1	8	yes	complemented
23	C₂×#14523231	C₂²	1	8	yes	complemented
24	M₁₆×D₁₂₇	C₂²	1	32	yes	complemented
25	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
26	C₂×C₈×D₁₂₇	C₂²	1	8	yes	complemented
27	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
29	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	complemented
30	C₂×M₁₆×C₁₂₇	C₂²	1	8	yes	complemented
31	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	complemented
32	C₂³×D₁₂₇	C₂×C₄	1	2	yes	—
33	#14523248	C₂×C₄	1	8	yes	—
34	#14523248	C₂×C₄	1	8	yes	—
35	#14523248	C₂×C₄	1	16	yes	—
36	#14523248	C₂×C₄	1	16	yes	—
37	#14523248	C₂³	1	8	yes	—
38	#14523248	C₂³	1	8	yes	—
39	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
40	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
41	C₂×Dic₂₅₄	C₂×C₄	1	8	yes	—
42	C₂×D₅₀₈	C₂×C₄	1	2	yes	—
43	C₂×D₅₀₈	C₂×C₄	1	2	yes	—
44	C₂×D₅₀₈	C₂×C₄	1	8	yes	—
45	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	4	yes	—
46	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	4	yes	—
47	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	16	yes	—
48	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
50	C₂×C₄×D₁₂₇	C₂×C₄	1	8	yes	—
51	C₂×C₄×D₁₂₇	C₂³	1	4	yes	—
52	C₂×C₄×D₁₂₇	C₂³	1	8	yes	—
53	#14523232	C₂³	1	8	yes	—
54	#14523232	C₂³	1	8	yes	complemented
55	#14523231	C₂³	1	32	yes	complemented
56	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
57	C₂×C₁₂₇⋊C₈	C₂³	1	8	yes	complemented
58	C₈×D₁₂₇	C₂³	1	32	yes	complemented
59	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
64	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
68	M₁₆×C₁₂₇	C₂³	1	8	yes	—
69	M₁₆×C₁₂₇	C₂³	1	8	yes	complemented
70	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
71	C₂×C₈×C₁₂₇	C₂³	1	8	yes	complemented
72	C₂²×D₁₂₇	—	4	4	no	—
73	C₂²×D₁₂₇	C₂²×C₄	1	4	yes	—
74	C₂²×D₁₂₇	C₂²×C₄	1	8	yes	—
75	Dic₂₅₄	C₂²×C₄	1	8	yes	—
76	Dic₂₅₄	C₂²×C₄	1	8	yes	—
77	D₅₀₈	C₂²×C₄	1	8	yes	—
78	D₅₀₈	C₂²×C₄	1	8	yes	—
79	C₂²⋊D₁₂₇	C₂²×C₄	1	16	yes	—
80	C₂²⋊D₁₂₇	C₂²×C₄	1	16	yes	—
81	C₂×Dic₁₂₇	C₂²×C₄	1	8	yes	—
82	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
83	C₄×D₁₂₇	C₂²×C₄	1	8	yes	—
84	C₄×D₁₂₇	C₂²×C₄	1	16	yes	—
85	C₄×D₁₂₇	C₂⁴	1	8	yes	—
86	C₁₂₇⋊C₈	C₂⁴	1	8	yes	—
87	C₂³×C₁₂₇	—	4	2	no	—
88	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
90	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
91	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
95	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
100	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
101	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
102	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
103	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
104	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
105	C₈×C₁₂₇	C₂⁴	1	8	yes	—
106	C₂²×C₁₂₇	—	2	2	no	—
107	C₂²×C₁₂₇	—	2	2	no	—
108	C₂²×C₁₂₇	—	2	2	no	—
109	C₂²×C₁₂₇	—	4	4	no	—
110	C₂²×C₁₂₇	C₂³×C₄	1	2	yes	—
111	C₂²×C₁₂₇	C₂³×C₄	1	2	yes	—
112	C₂²×C₁₂₇	C₂³×C₄	1	2	yes	—
113	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×D₁₂₇	—	2	8	no	—
115	C₂×D₁₂₇	—	4	8	no	—
116	C₂×D₁₂₇	C₂³×C₄	1	8	yes	—
117	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
118	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
120	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
121	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
122	Dic₁₂₇	C₂³×C₄	1	8	yes	—
123	C₂×C₁₂₇	—	2	2	no	—
124	C₂×C₁₂₇	—	2	2	no	—
125	C₂×C₁₂₇	—	2	2	no	—
126	C₂×C₁₂₇	#568	1	2	yes	complemented
127	C₂×C₁₂₇	C₂⁴×C₄	1	1	yes	characteristic
128	D₁₂₇	—	2	8	no	—
129	C₂×#568	—	127	1	no	maximal, Sylow
130	C₁₂₇	C₂×#568	1	1	yes	characteristic, complemented, Sylow
131	C₂²×#41	—	127	1	no	—
132	#568	—	127	16	no	—
133	C₂×#131	—	127	4	no	—
134	C₂×#131	—	127	4	no	—
135	C₂²×M₁₆	—	127	1	no	—
136	C₂²×M₁₆	—	127	4	no	—
137	C₂²×M₁₆	D₁₂₇	1	1	yes	characteristic, complemented
138	C₂×#41	—	127	8	no	—
139	C₂×#41	—	127	2	no	—
140	C₂×#41	—	127	2	no	—
141	C₂×#41	—	127	2	no	—
142	C₂×#41	—	127	2	no	—
143	C₂×#41	—	127	8	no	—
144	C₂²×Q₈	—	127	1	no	—
145	C₂²×D₄	—	127	2	no	—
146	C₂²×D₄	—	127	1	no	—
147	C₂³×C₄	—	127	2	no	—
148	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
149	#131	—	127	32	no	—
150	#131	—	127	32	no	—
151	C₂×M₁₆	—	127	2	no	—
152	C₂×M₁₆	—	127	2	no	—
153	C₂×M₁₆	—	127	8	no	—
154	C₂×M₁₆	—	127	32	no	—
155	C₂×M₁₆	—	127	8	no	—
156	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
157	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	complemented
158	C₂×M₁₆	C₂×D₁₂₇	1	8	yes	complemented
159	C₂²×C₈	—	127	2	no	—
160	C₂²×C₈	—	127	8	no	—
161	C₂²×C₈	C₂×D₁₂₇	1	2	yes	complemented
162	C₂⁴	—	127	2	no	—
163	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
164	#41	—	127	8	no	—
165	#41	—	127	8	no	—
166	#41	—	127	16	no	—
167	#41	—	127	16	no	—
168	#41	—	127	8	no	—
169	#41	—	127	8	no	—
170	C₂×Q₈	—	127	2	no	—
171	C₂×Q₈	—	127	2	no	—
172	C₂×Q₈	—	127	8	no	—
173	C₂×D₄	—	127	16	no	—
174	C₂×D₄	—	127	4	no	—
175	C₂×D₄	—	127	4	no	—
176	C₂×D₄	—	127	8	no	—
177	C₂×D₄	—	127	2	no	—
178	C₂×D₄	—	127	2	no	—
179	C₂²×C₄	—	127	2	no	—
180	C₂²×C₄	—	127	8	no	—
181	C₂²×C₄	—	127	4	no	—
182	C₂²×C₄	—	127	8	no	—
183	C₂²×C₄	—	127	4	no	—
184	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
185	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
186	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
187	C₂²×C₄	C₄×D₁₂₇	1	4	yes	—
188	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
189	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
192	M₁₆	—	127	8	no	—
193	M₁₆	—	127	8	no	—
194	M₁₆	—	127	32	no	—
195	M₁₆	C₂²×D₁₂₇	1	8	yes	—
196	M₁₆	C₂²×D₁₂₇	1	8	yes	complemented
197	C₂×C₈	—	127	8	no	—
198	C₂×C₈	—	127	4	no	—
199	C₂×C₈	—	127	32	no	—
200	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
201	C₂×C₈	C₂²×D₁₂₇	1	8	yes	complemented
202	C₂³	—	4	2	no	—
203	C₂³	—	127	8	no	—
204	C₂³	—	127	4	no	—
205	C₂³	—	508	4	no	—
206	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
207	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
208	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
209	C₂³	C₂×C₄×D₁₂₇	1	4	yes	—
210	Q₈	—	127	8	no	—
211	Q₈	—	127	8	no	—
212	D₄	—	127	16	no	—
213	D₄	—	127	16	no	—
214	D₄	—	127	8	no	—
215	D₄	—	127	8	no	—
216	C₂×C₄	—	127	8	no	—
217	C₂×C₄	—	127	4	no	—
218	C₂×C₄	—	127	16	no	—
219	C₂×C₄	—	127	8	no	—
220	C₂×C₄	—	127	8	no	—
221	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
223	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
224	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
227	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
228	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
229	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
230	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
231	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
232	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
233	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
234	C₈	—	127	8	no	—
235	C₈	C₂³×D₁₂₇	1	8	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	127	8	no	—
241	C₂²	—	508	8	no	—
242	C₂²	—	254	8	no	—
243	C₂²	C₂²×C₄×D₁₂₇	1	2	yes	—
244	C₂²	C₂²×C₄×D₁₂₇	1	2	yes	—
245	C₂²	C₂²×C₄×D₁₂₇	1	2	yes	—
246	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
247	C₄	—	127	8	no	—
248	C₄	C₂²×C₄×D₁₂₇	1	2	yes	—
249	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
250	C₄	C₂²×C₄×D₁₂₇	1	2	yes	—
251	C₄	C₂²×C₄×D₁₂₇	1	2	yes	—
252	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	—	254	8	no	—
257	C₂	#22550642	1	2	yes	direct product factor
258	C₂	C₂³×C₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₁₂₇
Fitting	C₂²×M₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	C₂²×#14523248	C₂
cosocle	C₄×C₁₂₇	C₂⁵
hypercenter	C₂²×M₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×#568
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22550642	C₂	direct (N × H)
3. C₂×#16861217	C₂	semidirect
4. C₂×#16861215	C₂	semidirect
5. C₂²×#14523232	C₂	semidirect
6. C₂×M₁₆×D₁₂₇	C₂	semidirect
7. C₂²×M₁₆×C₁₂₇	C₂	semidirect
18. #16861217	C₂²	semidirect
19. #16861215	C₂²	semidirect
21. C₂×#14523232	C₂²	semidirect
22. C₂×#14523232	C₂²	semidirect
23. C₂×#14523231	C₂²	semidirect
24. M₁₆×D₁₂₇	C₂²	semidirect
25. C₂²×C₁₂₇⋊C₈	C₂²	semidirect
26. C₂×C₈×D₁₂₇	C₂²	semidirect
29. C₂×M₁₆×C₁₂₇	C₂²	semidirect
30. C₂×M₁₆×C₁₂₇	C₂²	semidirect
31. C₂²×C₈×C₁₂₇	C₂²	semidirect
54. #14523232	C₂³	semidirect
55. #14523231	C₂³	semidirect
57. C₂×C₁₂₇⋊C₈	C₂³	semidirect
58. C₈×D₁₂₇	C₂³	semidirect
69. M₁₆×C₁₂₇	C₂³	semidirect
71. C₂×C₈×C₁₂₇	C₂³	semidirect
126. C₂×C₁₂₇	#568	semidirect
130. C₁₂₇	C₂×#568	semidirect
137. C₂²×M₁₆	D₁₂₇	semidirect
157. C₂×M₁₆	C₂×D₁₂₇	semidirect
158. C₂×M₁₆	C₂×D₁₂₇	semidirect
161. C₂²×C₈	C₂×D₁₂₇	semidirect
196. M₁₆	C₂²×D₁₂₇	semidirect
201. C₂×C₈	C₂²×D₁₂₇	semidirect
257. C₂	#22550642	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523248	2
#22550642	2
C₂×#16861217	2
C₂×#16861215	2
C₂²×#14523232	2
C₂×M₁₆×D₁₂₇	2
C₂²×M₁₆×C₁₂₇	2
C₂×#568	127

Subgroups — C2×#22550642 #30653439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550642 #30653439