Subgroups — C₂×M₁₆×#261 #13131390

Order 1664.

656 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#261	C₂	1	1	yes	maximal, characteristic
2	C₂×D₁₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
4	M₁₆×#261	C₂	1	8	yes	maximal, direct product factor
5	C₂×#88100	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×#261	C₂	1	2	yes	maximal, complemented
7	C₂³×#261	C₄	1	2	yes	—
8	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×#6122	C₄	1	2	yes	complemented
10	C₂×C₄×#261	C₄	1	4	yes	—
11	C₂×C₄×#261	C₄	1	2	yes	—
12	C₂×C₄×#261	C₂²	1	1	yes	characteristic
13	C₂×C₄×#261	C₂²	1	1	yes	characteristic
14	C₂×C₄×#261	C₂²	1	4	yes	—
15	D₁₃⋊M₁₆	C₄	1	4	yes	complemented
16	D₁₃⋊M₁₆	C₂²	1	4	yes	—
17	C₂×#6118	C₄	1	2	yes	complemented
18	C₂×D₁₃⋊C₈	C₂²	1	2	yes	—
19	C₂×C₁₃×M₁₆	C₄	1	1	yes	characteristic, complemented
20	C₂×#6099	C₄	1	1	yes	characteristic, complemented
21	M₁₆×D₁₃	C₄	1	4	yes	complemented
22	M₁₆×D₁₃	C₂²	1	4	yes	—
23	C₂×#6094	C₄	1	2	yes	complemented
24	C₂×C₈×D₁₃	C₂²	1	2	yes	—
25	#88100	C₂²	1	8	yes	complemented
26	C₈×#261	C₂²	1	8	yes	complemented
27	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×#261	—	2	8	no	—
29	C₂²×#261	C₂×C₄	1	2	yes	—
30	C₂²×#261	C₂×C₄	1	8	yes	—
31	C₂²×#261	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×D₁₃	C₂³	1	2	yes	—
42	#6122	C₂×C₄	1	8	yes	complemented
43	C₂×#1057	C₂×C₄	1	4	yes	complemented
44	C₄×#261	C₂×C₄	1	2	yes	—
45	C₄×#261	C₂×C₄	1	8	yes	—
46	C₄×#261	C₂×C₄	1	2	yes	—
47	C₄×#261	C₂³	1	2	yes	—
48	C₄×#261	C₂³	1	2	yes	—
49	#6118	C₂×C₄	1	8	yes	complemented
50	D₁₃⋊C₈	C₂×C₄	1	4	yes	complemented
51	D₁₃⋊C₈	C₂³	1	4	yes	—
52	C₁₃×M₁₆	C₂×C₄	1	4	yes	complemented
53	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	complemented
54	#6099	C₂×C₄	1	4	yes	complemented
55	C₂×#1055	C₂×C₄	1	2	yes	complemented
56	#6094	C₂×C₄	1	8	yes	complemented
57	C₈×D₁₃	C₂×C₄	1	4	yes	complemented
58	C₈×D₁₃	C₂³	1	4	yes	—
59	C₂×C₄×M₁₆	—	13	1	no	maximal, Sylow
60	C₂³×C₁₃	C₄²	1	1	yes	characteristic
61	C₂²×D₁₃	—	2	4	no	—
62	C₂²×D₁₃	C₄²	1	1	yes	characteristic
63	C₂²×D₁₃	C₄²	1	2	yes	—
64	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×#261	—	2	4	no	—
67	C₂×#261	—	2	16	no	—
68	C₂×#261	M₁₆	1	8	yes	direct product factor
69	C₂×#261	C₂²×C₄	1	4	yes	—
70	C₂×#261	C₂²×C₄	1	4	yes	—
71	C₂×C₄×C₁₃	C₄²	1	2	yes	—
72	C₂×C₄×C₁₃	C₄²	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₃	C₄²	1	2	yes	—
76	C₂×Dic₁₃	C₄²	1	1	yes	characteristic
77	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
79	C₄×D₁₃	C₄²	1	2	yes	—
80	C₄×D₁₃	C₄²	1	2	yes	—
81	C₄×D₁₃	C₄²	1	2	yes	—
82	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₄×D₁₃	C₂²×C₄	1	2	yes	—
84	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₁₃	C₂²×C₄	1	2	yes	—
86	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄×D₁₃	C₂²×C₄	1	2	yes	—
88	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
89	#1057	C₂²×C₄	1	8	yes	complemented
90	C₈×C₁₃	C₂²×C₄	1	4	yes	complemented
91	#1055	C₂²×C₄	1	4	yes	complemented
92	C₂²×M₁₆	—	13	1	no	—
93	C₂²×M₁₆	—	13	1	no	—
94	C₂²×C₄²	—	13	1	no	—
95	C₄×M₁₆	—	13	8	no	—
96	C₂×#97	—	13	2	no	—
97	C₂×C₄×C₈	—	13	2	no	—
98	C₂²×C₁₃	—	2	2	no	—
99	C₂²×C₁₃	C₂×C₄²	1	2	yes	—
100	C₂²×C₁₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×D₁₃	—	2	2	no	—
102	C₂×D₁₃	—	2	4	no	—
103	C₂×D₁₃	—	2	4	no	—
104	C₂×D₁₃	C₂×C₄²	1	2	yes	—
105	C₂×D₁₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×D₁₃	C₂×M₁₆	1	4	yes	—
107	C₂×D₁₃	C₂³×C₄	1	1	yes	characteristic
108	#261	—	2	4	no	—
109	#261	C₂×M₁₆	1	8	yes	direct product factor
110	C₄×C₁₃	C₂×C₄²	1	2	yes	—
111	C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
112	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	Dic₁₃	C₂×C₄²	1	1	yes	characteristic
114	Dic₁₃	C₂×C₄²	1	2	yes	—
115	Dic₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂³×C₄	—	13	1	no	—
117	C₂³×C₄	—	13	2	no	—
118	C₂×M₁₆	—	13	4	no	—
119	C₂×M₁₆	—	13	2	no	—
120	C₂×M₁₆	—	13	4	no	—
121	C₂×M₁₆	—	13	2	no	—
122	C₂×M₁₆	—	13	1	no	—
123	C₂×M₁₆	—	13	4	no	—
124	C₂×M₁₆	—	13	4	no	—
125	C₂×M₁₆	—	13	2	no	—
126	C₂×M₁₆	#261	1	1	yes	characteristic, direct product factor
127	C₂²×C₈	—	13	2	no	—
128	C₂²×C₈	—	13	2	no	—
129	C₂×C₄²	—	13	1	no	—
130	C₂×C₄²	—	13	1	no	—
131	C₂×C₄²	—	13	4	no	—
132	C₂×C₄²	—	13	2	no	—
133	C₂×C₄²	—	13	4	no	—
134	#97	—	13	8	no	—
135	C₄×C₈	—	13	8	no	—
136	C₂×C₁₃	—	2	2	no	—
137	C₂×C₁₃	C₄×M₁₆	1	2	yes	complemented
138	C₂×C₁₃	C₂²×C₄²	1	1	yes	characteristic
139	D₁₃	—	2	2	no	—
140	D₁₃	C₄×M₁₆	1	2	yes	complemented
141	D₁₃	C₂²×M₁₆	1	2	yes	—
142	C₂⁴	—	13	1	no	—
143	C₂²×C₄	—	13	2	no	—
144	C₂²×C₄	—	13	1	no	—
145	C₂²×C₄	—	13	1	no	—
146	C₂²×C₄	—	13	2	no	—
147	C₂²×C₄	—	13	1	no	—
148	C₂²×C₄	—	13	2	no	—
149	C₂²×C₄	—	13	2	no	—
150	C₂²×C₄	—	13	2	no	—
151	C₂²×C₄	—	13	1	no	—
152	C₂²×C₄	—	13	1	no	—
153	C₂²×C₄	—	13	8	no	—
154	C₂²×C₄	—	13	2	no	—
155	C₂²×C₄	—	26	8	no	—
156	C₂²×C₄	C₂×#261	1	1	yes	characteristic
157	M₁₆	—	13	8	no	—
158	M₁₆	—	13	8	no	—
159	M₁₆	—	13	4	no	—
160	M₁₆	—	13	8	no	—
161	M₁₆	C₂×#261	1	4	yes	direct product factor
162	C₂×C₈	—	13	4	no	—
163	C₂×C₈	—	13	4	no	—
164	C₂×C₈	—	13	4	no	—
165	C₂×C₈	—	13	2	no	—
166	C₂×C₈	—	13	4	no	—
167	C₂×C₈	—	13	4	no	—
168	C₂×C₈	C₂×#261	1	2	yes	complemented
169	C₄²	—	13	2	no	—
170	C₄²	—	13	2	no	—
171	C₄²	—	13	2	no	—
172	C₄²	—	13	8	no	—
173	C₄²	—	13	2	no	—
174	C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
175	C₂³	—	13	1	no	—
176	C₂³	—	13	2	no	—
177	C₂³	—	13	2	no	—
178	C₂³	—	13	1	no	—
179	C₂³	—	26	4	no	—
180	C₂³	C₄×#261	1	1	yes	characteristic
181	C₂×C₄	—	13	4	no	—
182	C₂×C₄	—	13	8	no	—
183	C₂×C₄	—	13	1	no	—
184	C₂×C₄	—	13	2	no	—
185	C₂×C₄	—	13	2	no	—
186	C₂×C₄	—	13	1	no	—
187	C₂×C₄	—	13	2	no	—
188	C₂×C₄	—	13	1	no	—
189	C₂×C₄	—	13	1	no	—
190	C₂×C₄	—	13	2	no	—
191	C₂×C₄	—	13	2	no	—
192	C₂×C₄	—	13	2	no	—
193	C₂×C₄	—	13	2	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	4	no	—
198	C₂×C₄	—	26	16	no	—
199	C₂×C₄	—	26	4	no	—
200	C₂×C₄	C₄×#261	1	1	yes	characteristic
201	C₂×C₄	C₄×#261	1	2	yes	—
202	C₂×C₄	C₂²×#261	1	2	yes	—
203	C₂×C₄	C₂²×#261	1	1	yes	characteristic
204	C₈	—	13	8	no	—
205	C₈	—	13	4	no	—
206	C₈	C₂²×#261	1	4	yes	complemented
207	C₂²	—	2	2	no	—
208	C₂²	—	13	4	no	—
209	C₂²	—	13	1	no	—
210	C₂²	—	13	1	no	—
211	C₂²	—	13	2	no	—
212	C₂²	—	26	4	no	—
213	C₂²	—	26	4	no	—
214	C₂²	—	26	2	no	—
215	C₂²	C₂×C₄×#261	1	2	yes	—
216	C₂²	C₂×C₄×#261	1	1	yes	characteristic
217	C₄	—	13	8	no	—
218	C₄	—	13	1	no	—
219	C₄	—	13	2	no	—
220	C₄	—	13	1	no	—
221	C₄	—	26	4	no	—
222	C₄	C₂×C₄×#261	1	2	yes	—
223	C₄	C₂×C₄×#261	1	1	yes	characteristic
224	C₄	C₂³×#261	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	13	2	no	—
227	C₂	—	13	2	no	—
228	C₂	—	26	2	no	—
229	C₂	M₁₆×#261	1	2	yes	direct product factor
230	C₂	C₂²×C₄×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂²×C₄²
Frattini	C₄	C₂³×#261
Fitting	C₂×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄²
prime radical	C₂³×D₁₃	C₂×C₄
cosocle	C₄×D₁₃	C₂⁴
hypercenter	C₂×M₁₆	#261
hypocenter	C₁₃	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. M₁₆×#261	C₂	direct (N × H)
5. C₂×#88100	C₂	semidirect
6. C₂×C₈×#261	C₂	semidirect
9. C₂×#6122	C₄	semidirect
15. D₁₃⋊M₁₆	C₄	semidirect
17. C₂×#6118	C₄	semidirect
19. C₂×C₁₃×M₁₆	C₄	semidirect
20. C₂×#6099	C₄	semidirect
21. M₁₆×D₁₃	C₄	semidirect
23. C₂×#6094	C₄	semidirect
25. #88100	C₂²	semidirect
26. C₈×#261	C₂²	semidirect
42. #6122	C₂×C₄	semidirect
43. C₂×#1057	C₂×C₄	semidirect
49. #6118	C₂×C₄	semidirect
50. D₁₃⋊C₈	C₂×C₄	semidirect
52. C₁₃×M₁₆	C₂×C₄	semidirect
53. C₂×C₈×C₁₃	C₂×C₄	semidirect
54. #6099	C₂×C₄	semidirect
55. C₂×#1055	C₂×C₄	semidirect
56. #6094	C₂×C₄	semidirect
57. C₈×D₁₃	C₂×C₄	semidirect
68. C₂×#261	M₁₆	direct (N × H)
89. #1057	C₂²×C₄	semidirect
90. C₈×C₁₃	C₂²×C₄	semidirect
91. #1055	C₂²×C₄	semidirect
109. #261	C₂×M₁₆	direct (N × H)
126. C₂×M₁₆	#261	direct (N × H)
137. C₂×C₁₃	C₄×M₁₆	semidirect
140. D₁₃	C₄×M₁₆	semidirect
161. M₁₆	C₂×#261	direct (N × H)
168. C₂×C₈	C₂×#261	semidirect
174. C₁₃	C₂×C₄×M₁₆	semidirect
206. C₈	C₂²×#261	semidirect
229. C₂	M₁₆×#261	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×C₄×#261	2
C₂×D₁₃⋊M₁₆	2
C₂×M₁₆×D₁₃	2
M₁₆×#261	2
C₂×#88100	2
C₂×C₈×#261	2
C₂×C₄×M₁₆	13

Subgroups — C2×M16×#261 #13131390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₆×#261 #13131390