Subgroups — C₂³×D₁₃⋊M₁₆ #16809644

Order 3328.

13310 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6118	C₂	1	2	yes	maximal, complemented
2	C₂²×D₁₃⋊M₁₆	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#6122	C₂	1	2	yes	maximal, complemented
4	C₂³×D₁₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁵×D₁₃	C₄	1	1	yes	characteristic
7	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
8	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₄×D₁₃	C₄	1	28	yes	—
10	C₂³×C₄×D₁₃	C₄	1	1	yes	characteristic
11	C₂³×C₄×D₁₃	C₄	1	1	yes	characteristic
12	C₂³×C₄×D₁₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×D₁₃	C₂²	1	28	yes	—
14	C₂³×C₄×D₁₃	C₂²	1	1	yes	characteristic
15	C₂²×#6122	C₂²	1	56	yes	complemented
16	C₂³×#1057	C₂²	1	4	yes	complemented
17	C₂×D₁₃⋊M₁₆	C₂²	1	448	yes	direct product factor
18	C₂²×#6118	C₂²	1	56	yes	complemented
19	C₂²×D₁₃⋊C₈	C₂²	1	56	yes	complemented
20	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂⁴×D₁₃	—	2	16	no	—
22	C₂⁴×D₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂⁴×D₁₃	C₂×C₄	1	28	yes	—
24	C₂⁴×D₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂×C₄	1	28	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×Dic₁₃	C₂×C₄	1	14	yes	—
29	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×Dic₁₃	C₂³	1	1	yes	characteristic
31	C₂³×Dic₁₃	C₂³	1	14	yes	—
32	C₂²×C₄×D₁₃	C₂×C₄	1	336	yes	—
33	C₂²×C₄×D₁₃	C₂×C₄	1	28	yes	—
34	C₂²×C₄×D₁₃	C₂×C₄	1	14	yes	—
35	C₂²×C₄×D₁₃	C₂×C₄	1	14	yes	—
36	C₂²×C₄×D₁₃	C₂×C₄	1	28	yes	—
37	C₂²×C₄×D₁₃	C₂³	1	14	yes	—
38	C₂²×C₄×D₁₃	C₂³	1	14	yes	—
39	C₂²×C₄×D₁₃	C₂³	1	112	yes	—
40	C₂×#6122	C₂³	1	224	yes	complemented
41	C₂²×#1057	C₂³	1	56	yes	complemented
42	D₁₃⋊M₁₆	C₂³	1	512	yes	direct product factor
43	C₂×#6118	C₂³	1	224	yes	complemented
44	C₂×D₁₃⋊C₈	C₂³	1	224	yes	complemented
45	C₂⁴×M₁₆	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	—	2	8	no	—
47	C₂⁴×C₁₃	C₂²×C₄	1	14	yes	—
48	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂³×D₁₃	—	2	8	no	—
50	C₂³×D₁₃	—	2	224	no	—
51	C₂³×D₁₃	M₁₆	1	16	yes	complemented
52	C₂³×D₁₃	C₂²×C₄	1	14	yes	—
53	C₂³×D₁₃	C₂²×C₄	1	112	yes	—
54	C₂³×D₁₃	C₂²×C₄	1	14	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	112	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	14	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	14	yes	—
58	C₂²×Dic₁₃	C₂²×C₄	1	84	yes	—
59	C₂²×Dic₁₃	C₂²×C₄	1	7	yes	—
60	C₂²×Dic₁₃	C₂²×C₄	1	14	yes	—
61	C₂²×Dic₁₃	C₂⁴	1	7	yes	—
62	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
63	C₂×C₄×D₁₃	C₂²×C₄	1	84	yes	—
64	C₂×C₄×D₁₃	C₂²×C₄	1	448	yes	—
65	C₂×C₄×D₁₃	C₂²×C₄	1	112	yes	—
66	C₂×C₄×D₁₃	C₂²×C₄	1	84	yes	—
67	C₂×C₄×D₁₃	C₂²×C₄	1	112	yes	—
68	C₂×C₄×D₁₃	C₂⁴	1	28	yes	—
69	C₂×C₄×D₁₃	C₂⁴	1	28	yes	—
70	C₂×C₄×D₁₃	C₂⁴	1	64	yes	—
71	#6122	C₂⁴	1	128	yes	complemented
72	C₂×#1057	C₂⁴	1	112	yes	complemented
73	#6118	C₂⁴	1	128	yes	complemented
74	D₁₃⋊C₈	C₂⁴	1	128	yes	complemented
75	C₂⁵×C₄	—	13	1	no	—
76	C₂³×M₁₆	—	13	2	no	—
77	C₂³×M₁₆	—	13	56	no	—
78	C₂³×M₁₆	—	13	2	no	—
79	C₂⁴×C₈	—	13	2	no	—
80	C₂³×C₁₃	—	2	56	no	—
81	C₂³×C₁₃	C₂×M₁₆	1	8	yes	complemented
82	C₂³×C₁₃	C₂³×C₄	1	28	yes	—
83	C₂³×C₁₃	C₂³×C₄	1	7	yes	—
84	C₂²×D₁₃	—	2	56	no	—
85	C₂²×D₁₃	—	2	448	no	—
86	C₂²×D₁₃	C₂×M₁₆	1	112	yes	complemented
87	C₂²×D₁₃	C₂³×C₄	1	28	yes	—
88	C₂²×D₁₃	C₂³×C₄	1	64	yes	—
89	C₂²×D₁₃	C₂³×C₄	1	28	yes	—
90	C₂×C₄×C₁₃	C₂³×C₄	1	64	yes	—
91	C₂×C₄×C₁₃	C₂³×C₄	1	28	yes	—
92	C₂×C₄×C₁₃	C₂³×C₄	1	28	yes	—
93	C₂×Dic₁₃	C₂³×C₄	1	56	yes	—
94	C₂×Dic₁₃	C₂³×C₄	1	21	yes	—
95	C₂×Dic₁₃	C₂³×C₄	1	28	yes	—
96	C₂×Dic₁₃	C₂⁵	1	7	yes	—
97	C₂×Dic₁₃	C₂⁵	1	8	yes	—
98	C₄×D₁₃	C₂³×C₄	1	56	yes	—
99	C₄×D₁₃	C₂³×C₄	1	64	yes	—
100	C₄×D₁₃	C₂³×C₄	1	56	yes	—
101	C₄×D₁₃	C₂³×C₄	1	64	yes	—
102	C₄×D₁₃	C₂⁵	1	8	yes	—
103	C₄×D₁₃	C₂⁵	1	8	yes	—
104	#1057	C₂⁵	1	32	yes	complemented
105	C₂⁶	—	13	1	no	—
106	C₂⁴×C₄	—	13	1	no	—
107	C₂⁴×C₄	—	13	1	no	—
108	C₂⁴×C₄	—	13	28	no	—
109	C₂⁴×C₄	—	13	1	no	—
110	C₂⁴×C₄	—	13	28	no	—
111	C₂⁴×C₄	—	13	1	no	—
112	C₂⁴×C₄	—	13	1	no	—
113	C₂⁴×C₄	#261	1	1	yes	characteristic
114	C₂²×M₁₆	—	13	56	no	—
115	C₂²×M₁₆	—	13	448	no	—
116	C₂²×M₁₆	—	13	56	no	—
117	C₂³×C₈	—	13	56	no	—
118	C₂³×C₈	—	13	4	no	—
119	C₂²×C₁₃	—	2	56	no	—
120	C₂²×C₁₃	C₂²×M₁₆	1	28	yes	complemented
121	C₂²×C₁₃	C₂⁴×C₄	1	8	yes	—
122	C₂²×C₁₃	C₂⁴×C₄	1	7	yes	—
123	C₂×D₁₃	—	2	56	no	—
124	C₂×D₁₃	—	2	128	no	—
125	C₂×D₁₃	C₂²×M₁₆	1	112	yes	complemented
126	C₂×D₁₃	C₂⁴×C₄	1	8	yes	—
127	C₂×D₁₃	C₂⁴×C₄	1	8	yes	—
128	C₄×C₁₃	C₂⁴×C₄	1	8	yes	—
129	C₄×C₁₃	C₂⁴×C₄	1	8	yes	—
130	Dic₁₃	C₂⁴×C₄	1	7	yes	—
131	Dic₁₃	C₂⁴×C₄	1	8	yes	—
132	Dic₁₃	C₂⁶	1	1	yes	characteristic
133	C₂⁵	—	13	1	no	—
134	C₂⁵	—	13	28	no	—
135	C₂⁵	—	13	1	no	—
136	C₂⁵	—	26	16	no	—
137	C₂⁵	C₂×#261	1	1	yes	characteristic
138	C₂³×C₄	—	13	1	no	—
139	C₂³×C₄	—	13	14	no	—
140	C₂³×C₄	—	13	14	no	—
141	C₂³×C₄	—	13	1	no	—
142	C₂³×C₄	—	13	14	no	—
143	C₂³×C₄	—	13	14	no	—
144	C₂³×C₄	—	13	112	no	—
145	C₂³×C₄	—	13	28	no	—
146	C₂³×C₄	—	13	336	no	—
147	C₂³×C₄	—	13	14	no	—
148	C₂³×C₄	—	13	14	no	—
149	C₂³×C₄	—	13	28	no	—
150	C₂³×C₄	C₂×#261	1	1	yes	characteristic
151	C₂³×C₄	C₂×#261	1	28	yes	—
152	C₂³×C₄	C₂×#261	1	1	yes	characteristic
153	C₂×M₁₆	—	13	224	no	—
154	C₂×M₁₆	—	13	512	no	—
155	C₂×M₁₆	—	13	224	no	—
156	C₂²×C₈	—	13	224	no	—
157	C₂²×C₈	—	13	56	no	—
158	C₂×C₁₃	—	2	8	no	—
159	C₂×C₁₃	C₂³×M₁₆	1	14	yes	complemented
160	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
161	D₁₃	—	2	8	no	—
162	D₁₃	C₂³×M₁₆	1	16	yes	complemented
163	C₂⁴	—	2	8	no	—
164	C₂⁴	—	13	16	no	—
165	C₂⁴	—	13	14	no	—
166	C₂⁴	—	13	112	no	—
167	C₂⁴	—	13	14	no	—
168	C₂⁴	—	26	224	no	—
169	C₂⁴	—	26	8	no	—
170	C₂⁴	C₂²×#261	1	14	yes	—
171	C₂⁴	C₂²×#261	1	1	yes	characteristic
172	C₂²×C₄	—	13	7	no	—
173	C₂²×C₄	—	13	28	no	—
174	C₂²×C₄	—	13	84	no	—
175	C₂²×C₄	—	13	7	no	—
176	C₂²×C₄	—	13	14	no	—
177	C₂²×C₄	—	13	28	no	—
178	C₂²×C₄	—	13	28	no	—
179	C₂²×C₄	—	13	64	no	—
180	C₂²×C₄	—	13	84	no	—
181	C₂²×C₄	—	13	112	no	—
182	C₂²×C₄	—	13	448	no	—
183	C₂²×C₄	—	13	84	no	—
184	C₂²×C₄	—	13	112	no	—
185	C₂²×C₄	C₂²×#261	1	112	yes	—
186	C₂²×C₄	C₂²×#261	1	14	yes	—
187	C₂²×C₄	C₂²×#261	1	14	yes	—
188	M₁₆	—	13	128	no	—
189	M₁₆	—	13	128	no	—
190	C₂×C₈	—	13	128	no	—
191	C₂×C₈	—	13	112	no	—
192	C₁₃	C₂⁴×M₁₆	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	2	56	no	—
194	C₂³	—	13	112	no	—
195	C₂³	—	13	28	no	—
196	C₂³	—	13	64	no	—
197	C₂³	—	13	28	no	—
198	C₂³	—	26	448	no	—
199	C₂³	—	26	56	no	—
200	C₂³	D₁₃⋊M₁₆	1	8	yes	direct product factor
201	C₂³	C₂³×#261	1	28	yes	—
202	C₂³	C₂³×#261	1	7	yes	—
203	C₂×C₄	—	13	7	no	—
204	C₂×C₄	—	13	8	no	—
205	C₂×C₄	—	13	56	no	—
206	C₂×C₄	—	13	21	no	—
207	C₂×C₄	—	13	28	no	—
208	C₂×C₄	—	13	8	no	—
209	C₂×C₄	—	13	8	no	—
210	C₂×C₄	—	13	56	no	—
211	C₂×C₄	—	13	64	no	—
212	C₂×C₄	—	13	56	no	—
213	C₂×C₄	—	13	64	no	—
214	C₂×C₄	C₂³×#261	1	64	yes	—
215	C₂×C₄	C₂³×#261	1	28	yes	—
216	C₂×C₄	C₂³×#261	1	28	yes	—
217	C₈	—	13	32	no	—
218	C₂²	—	2	56	no	—
219	C₂²	—	13	112	no	—
220	C₂²	—	13	8	no	—
221	C₂²	—	13	8	no	—
222	C₂²	—	26	128	no	—
223	C₂²	—	26	56	no	—
224	C₂²	C₂×D₁₃⋊M₁₆	1	28	yes	direct product factor
225	C₂²	C₂⁴×#261	1	8	yes	—
226	C₂²	C₂⁴×#261	1	7	yes	—
227	C₄	—	13	7	no	—
228	C₄	—	13	8	no	—
229	C₄	—	13	1	no	—
230	C₄	C₂⁴×#261	1	8	yes	—
231	C₄	C₂⁴×#261	1	8	yes	—
232	C₂	—	2	8	no	—
233	C₂	—	13	16	no	—
234	C₂	—	26	8	no	—
235	C₂	C₂²×D₁₃⋊M₁₆	1	14	yes	direct product factor
236	C₂	C₂⁵×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#261
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#261
Fitting	C₂⁴×C₄×C₁₃	C₄
socle	C₂⁴×C₁₃	C₂²×C₄
prime radical	C₂⁵×D₁₃	C₄
cosocle	Dic₁₃	C₂⁶
hypercenter	C₂⁴×C₄	#261
hypocenter	C₁₃	C₂⁴×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#6118	C₂	semidirect
2. C₂²×D₁₃⋊M₁₆	C₂	direct (N × H)
3. C₂³×#6122	C₂	semidirect
4. C₂³×D₁₃⋊C₈	C₂	semidirect
15. C₂²×#6122	C₂²	semidirect
16. C₂³×#1057	C₂²	semidirect
17. C₂×D₁₃⋊M₁₆	C₂²	direct (N × H)
18. C₂²×#6118	C₂²	semidirect
19. C₂²×D₁₃⋊C₈	C₂²	semidirect
40. C₂×#6122	C₂³	semidirect
41. C₂²×#1057	C₂³	semidirect
42. D₁₃⋊M₁₆	C₂³	direct (N × H)
43. C₂×#6118	C₂³	semidirect
44. C₂×D₁₃⋊C₈	C₂³	semidirect
51. C₂³×D₁₃	M₁₆	semidirect
71. #6122	C₂⁴	semidirect
72. C₂×#1057	C₂⁴	semidirect
73. #6118	C₂⁴	semidirect
74. D₁₃⋊C₈	C₂⁴	semidirect
81. C₂³×C₁₃	C₂×M₁₆	semidirect
86. C₂²×D₁₃	C₂×M₁₆	semidirect
104. #1057	C₂⁵	semidirect
120. C₂²×C₁₃	C₂²×M₁₆	semidirect
125. C₂×D₁₃	C₂²×M₁₆	semidirect
159. C₂×C₁₃	C₂³×M₁₆	semidirect
162. D₁₃	C₂³×M₁₆	semidirect
192. C₁₃	C₂⁴×M₁₆	semidirect
200. C₂³	D₁₃⋊M₁₆	direct (N × H)
224. C₂²	C₂×D₁₃⋊M₁₆	direct (N × H)
235. C₂	C₂²×D₁₃⋊M₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6118	2
C₂²×D₁₃⋊M₁₆	2
C₂³×#6122	2
C₂³×D₁₃⋊C₈	2
C₂⁴×C₄×D₁₃	2
C₂⁴×M₁₆	13

Subgroups — C23×D13⋊M16 #16809644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₁₃⋊M₁₆ #16809644