Subgroups — C₂×#22550676 #30653477

Order 16256.

1484 conjugacy classes of subgroups.

The subgroup lattice has 222 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	2	yes	maximal, complemented
2	#22550676	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16861242	C₂	1	4	yes	maximal, complemented
4	C₂×#16861237	C₂	1	4	yes	maximal, complemented
5	C₂×#16861231	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14523248	—	2	2	no	—
8	C₂×#14523248	—	2	2	no	—
9	C₂×#14523248	—	2	8	no	—
10	C₂×#14523248	C₂²	1	4	yes	—
11	C₂×#14523248	C₂²	1	4	yes	complemented
12	C₂×#14523248	C₂²	1	16	yes	complemented
13	C₂²×Dic₂₅₄	C₂²	1	2	yes	complemented
14	C₂²×D₅₀₈	C₂²	1	2	yes	complemented
15	C₂²×C₂²⋊D₁₂₇	—	2	2	no	—
16	C₂²×C₄×D₁₂₇	—	2	2	no	—
17	#16861242	C₂²	1	32	yes	complemented
18	#16861237	C₂²	1	32	yes	complemented
19	#16861231	C₂²	1	32	yes	complemented
20	C₂×Dic₅₀₈	C₂²	1	4	yes	complemented
21	C₂×#14523237	C₂²	1	8	yes	complemented
22	C₂×D₁₀₁₆	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	complemented
25	C₂×M₁₆×C₁₂₇	C₂²	1	8	yes	complemented
26	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	complemented
27	C₂³×D₁₂₇	—	2	2	no	—
28	#14523248	—	2	8	no	—
29	#14523248	—	2	16	no	—
30	#14523248	—	2	16	no	—
31	#14523248	—	2	8	no	—
32	#14523248	C₂³	1	16	yes	—
33	#14523248	C₂³	1	16	yes	complemented
34	C₂×Dic₂₅₄	—	2	2	no	—
35	C₂×Dic₂₅₄	—	2	4	no	—
36	C₂×Dic₂₅₄	C₂³	1	4	yes	—
37	C₂×Dic₂₅₄	C₂³	1	8	yes	complemented
38	C₂×D₅₀₈	—	2	2	no	—
39	C₂×D₅₀₈	—	2	4	no	—
40	C₂×D₅₀₈	C₂³	1	4	yes	—
41	C₂×D₅₀₈	C₂³	1	8	yes	complemented
42	C₂×C₂²⋊D₁₂₇	—	2	4	no	—
43	C₂×C₂²⋊D₁₂₇	—	2	4	no	—
44	C₂×C₂²⋊D₁₂₇	—	2	16	no	—
45	C₂²×Dic₁₂₇	—	2	2	no	—
46	C₂×C₄×D₁₂₇	—	2	4	no	—
47	C₂×C₄×D₁₂₇	—	2	16	no	—
48	C₂×C₄×D₁₂₇	—	2	4	no	—
49	Dic₅₀₈	C₂³	1	16	yes	complemented
50	#14523237	C₂³	1	32	yes	complemented
51	D₁₀₁₆	C₂³	1	16	yes	complemented
52	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
54	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
56	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
59	M₁₆×C₁₂₇	C₂³	1	16	yes	complemented
60	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
61	C₂×C₈×C₁₂₇	C₂³	1	8	yes	complemented
62	C₂²×D₁₂₇	—	2	4	no	—
63	C₂²×D₁₂₇	—	2	8	no	—
64	C₂²×D₁₂₇	—	4	8	no	—
65	Dic₂₅₄	—	2	4	no	—
66	Dic₂₅₄	—	2	8	no	—
67	Dic₂₅₄	C₂⁴	1	8	yes	—
68	D₅₀₈	—	2	4	no	—
69	D₅₀₈	—	2	8	no	—
70	D₅₀₈	C₂⁴	1	8	yes	—
71	C₂²⋊D₁₂₇	—	2	16	no	—
72	C₂²⋊D₁₂₇	—	2	16	no	—
73	C₂×Dic₁₂₇	—	2	4	no	—
74	C₂×Dic₁₂₇	—	2	8	no	—
75	C₄×D₁₂₇	—	2	16	no	—
76	C₄×D₁₂₇	—	2	16	no	—
77	C₂³×C₁₂₇	—	4	2	no	—
78	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
79	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
80	C₂³×C₁₂₇	C₂×D₄	1	4	yes	—
81	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂×D₄	1	8	yes	—
83	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
84	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
89	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
90	C₈×C₁₂₇	C₂⁴	1	8	yes	—
91	C₂²×C₁₂₇	—	2	2	no	—
92	C₂²×C₁₂₇	—	2	4	no	—
93	C₂²×C₁₂₇	—	4	4	no	—
94	C₂²×C₁₂₇	C₂²×D₄	1	4	yes	—
95	C₂²×C₁₂₇	C₂²×D₄	1	2	yes	—
96	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×D₁₂₇	—	2	8	no	—
98	C₂×D₁₂₇	—	4	16	no	—
99	C₂×D₁₂₇	—	4	8	no	—
100	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
101	C₄×C₁₂₇	C₂²×D₄	1	4	yes	—
102	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
104	Dic₁₂₇	—	2	8	no	—
105	C₂×C₁₂₇	—	2	4	no	—
106	C₂×C₁₂₇	—	2	2	no	—
107	C₂×C₁₂₇	#575	1	2	yes	complemented
108	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
109	D₁₂₇	—	4	8	no	—
110	C₂×#575	—	127	1	no	maximal, Sylow
111	C₁₂₇	C₂×#575	1	1	yes	characteristic, complemented, Sylow
112	C₂²×#41	—	127	2	no	—
113	#575	—	127	16	no	—
114	C₂×#137	—	127	4	no	—
115	C₂×AGL(1,ℤ/8)	—	127	4	no	—
116	C₂×#135	—	127	4	no	—
117	C₂²×M₁₆	D₁₂₇	1	1	yes	characteristic, complemented
118	C₂×#41	—	127	4	no	—
119	C₂×#41	—	127	4	no	—
120	C₂×#41	—	127	16	no	—
121	C₂×#41	—	254	2	no	—
122	C₂×#41	—	254	8	no	—
123	C₂×#41	—	254	2	no	—
124	C₂²×Q₈	—	127	2	no	—
125	C₂²×D₄	—	127	2	no	—
126	C₂²×D₄	—	254	2	no	—
127	C₂³×C₄	—	254	2	no	—
128	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
129	#137	—	127	32	no	—
130	AGL(1,ℤ/8)	—	127	32	no	—
131	#135	—	127	32	no	—
132	C₂×Q₁₆	—	127	4	no	—
133	C₂×SD₁₆	—	127	8	no	—
134	C₂×D₈	—	127	4	no	—
135	C₂×M₁₆	C₂×D₁₂₇	1	4	yes	complemented
136	C₂×M₁₆	C₂×D₁₂₇	1	8	yes	complemented
137	C₂²×C₈	C₂×D₁₂₇	1	2	yes	complemented
138	C₂⁴	—	254	2	no	—
139	C₂⁴	D₅₀₈	1	1	yes	characteristic
140	#41	—	127	16	no	—
141	#41	—	127	16	no	—
142	#41	—	254	16	no	—
143	#41	—	254	8	no	—
144	#41	—	254	8	no	—
145	#41	—	254	16	no	—
146	C₂×Q₈	—	127	4	no	—
147	C₂×Q₈	—	127	8	no	—
148	C₂×Q₈	—	254	2	no	—
149	C₂×Q₈	—	254	4	no	—
150	C₂×D₄	—	127	4	no	—
151	C₂×D₄	—	127	8	no	—
152	C₂×D₄	—	254	4	no	—
153	C₂×D₄	—	254	4	no	—
154	C₂×D₄	—	254	16	no	—
155	C₂×D₄	—	254	2	no	—
156	C₂×D₄	—	254	4	no	—
157	C₂²×C₄	—	254	2	no	—
158	C₂²×C₄	—	254	4	no	—
159	C₂²×C₄	—	254	16	no	—
160	C₂²×C₄	—	254	4	no	—
161	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
162	C₂²×C₄	D₅₀₈	1	4	yes	—
163	C₂²×C₄	D₅₀₈	1	2	yes	—
164	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
166	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
167	Q₁₆	—	127	16	no	—
168	SD₁₆	—	127	32	no	—
169	D₈	—	127	16	no	—
170	M₁₆	C₂²×D₁₂₇	1	16	yes	complemented
171	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
172	C₂×C₈	C₂²×D₁₂₇	1	8	yes	complemented
173	C₂³	—	4	2	no	—
174	C₂³	—	508	8	no	—
175	C₂³	—	254	8	no	—
176	C₂³	—	254	4	no	—
177	C₂³	C₂×D₅₀₈	1	2	yes	—
178	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
179	C₂³	C₂×D₅₀₈	1	4	yes	—
180	Q₈	—	127	8	no	—
181	Q₈	—	254	8	no	—
182	Q₈	—	254	4	no	—
183	D₄	—	127	8	no	—
184	D₄	—	254	16	no	—
185	D₄	—	254	16	no	—
186	D₄	—	254	8	no	—
187	D₄	—	254	4	no	—
188	C₂×C₄	—	254	8	no	—
189	C₂×C₄	—	254	4	no	—
190	C₂×C₄	—	254	16	no	—
191	C₂×C₄	—	254	16	no	—
192	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
193	C₂×C₄	C₂×D₅₀₈	1	8	yes	—
194	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
195	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
196	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
197	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
198	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
199	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
200	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
201	C₈	C₂³×D₁₂₇	1	8	yes	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	4	4	no	—
205	C₂²	—	508	16	no	—
206	C₂²	—	508	8	no	—
207	C₂²	—	254	8	no	—
208	C₂²	C₂²×D₅₀₈	1	4	yes	—
209	C₂²	C₂²×D₅₀₈	1	2	yes	—
210	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
211	C₄	—	254	8	no	—
212	C₄	C₂²×D₅₀₈	1	2	yes	—
213	C₄	C₂²×D₅₀₈	1	4	yes	—
214	C₄	C₂²×D₅₀₈	1	1	yes	characteristic
215	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	—	2	2	no	—
218	C₂	—	508	8	no	—
219	C₂	#22550676	1	2	yes	direct product factor
220	C₂	C₂³×D₅₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₀₈
commutator, cosocle	C₄×C₁₂₇	C₂⁵
Frattini	C₄	C₂⁴×D₁₂₇
Fitting	C₂²×M₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
hypercenter	C₂²×M₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×#575
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523248	C₂	semidirect
2. #22550676	C₂	direct (N × H)
3. C₂×#16861242	C₂	semidirect
4. C₂×#16861237	C₂	semidirect
5. C₂×#16861231	C₂	semidirect
6. C₂²×M₁₆×C₁₂₇	C₂	semidirect
11. C₂×#14523248	C₂²	semidirect
12. C₂×#14523248	C₂²	semidirect
13. C₂²×Dic₂₅₄	C₂²	semidirect
14. C₂²×D₅₀₈	C₂²	semidirect
17. #16861242	C₂²	semidirect
18. #16861237	C₂²	semidirect
19. #16861231	C₂²	semidirect
20. C₂×Dic₅₀₈	C₂²	semidirect
21. C₂×#14523237	C₂²	semidirect
22. C₂×D₁₀₁₆	C₂²	semidirect
24. C₂×M₁₆×C₁₂₇	C₂²	semidirect
25. C₂×M₁₆×C₁₂₇	C₂²	semidirect
26. C₂²×C₈×C₁₂₇	C₂²	semidirect
33. #14523248	C₂³	semidirect
37. C₂×Dic₂₅₄	C₂³	semidirect
41. C₂×D₅₀₈	C₂³	semidirect
49. Dic₅₀₈	C₂³	semidirect
50. #14523237	C₂³	semidirect
51. D₁₀₁₆	C₂³	semidirect
59. M₁₆×C₁₂₇	C₂³	semidirect
61. C₂×C₈×C₁₂₇	C₂³	semidirect
107. C₂×C₁₂₇	#575	semidirect
111. C₁₂₇	C₂×#575	semidirect
117. C₂²×M₁₆	D₁₂₇	semidirect
135. C₂×M₁₆	C₂×D₁₂₇	semidirect
136. C₂×M₁₆	C₂×D₁₂₇	semidirect
137. C₂²×C₈	C₂×D₁₂₇	semidirect
170. M₁₆	C₂²×D₁₂₇	semidirect
172. C₂×C₈	C₂²×D₁₂₇	semidirect
219. C₂	#22550676	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
#22550676	2
C₂×#16861242	2
C₂×#16861237	2
C₂×#16861231	2
C₂²×M₁₆×C₁₂₇	2
C₂×#575	127

Subgroups — C2×#22550676 #30653477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550676 #30653477