Subgroups — C₂×C₄×D₁₃⋊C₁₆ #16806710

Order 3328.

1022 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₃⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×#11718257	C₂	1	2	yes	maximal, complemented
3	C₄×D₁₃⋊C₁₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×#6092	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
9	C₂²×#6092	C₂²	1	4	yes	—
10	C₂×D₁₃⋊C₁₆	C₄	1	16	yes	direct product factor
11	C₂×D₁₃⋊C₁₆	C₂²	1	8	yes	—
12	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₁₃	C₄	1	4	yes	—
17	C₄×C₈×D₁₃	C₂²	1	4	yes	—
18	#11718257	C₂²	1	8	yes	complemented
19	C₄×#6092	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₁₃	C₈	1	2	yes	—
21	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂³	1	1	yes	characteristic
29	C₂×C₈×D₁₃	C₈	1	8	yes	—
30	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
32	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₈×D₁₃	C₂³	1	2	yes	—
34	C₄²×D₁₃	C₈	1	4	yes	—
35	C₄²×D₁₃	C₂×C₄	1	4	yes	—
36	C₂×#6092	C₂×C₄	1	16	yes	complemented
37	C₂×#6092	C₂³	1	8	yes	—
38	D₁₃⋊C₁₆	C₂×C₄	1	32	yes	direct product factor
39	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
40	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
41	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
42	C₈×Dic₁₃	C₂³	1	2	yes	—
43	C₄×#1055	C₂×C₄	1	2	yes	—
44	C₄×#1055	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₆	—	13	1	no	maximal, Sylow
46	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₁₃	C₁₆	1	16	yes	complemented
52	C₂×C₄×D₁₃	C₄²	1	4	yes	—
53	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
54	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
55	C₂×C₄×D₁₃	C₂×C₈	1	8	yes	—
56	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₄²	1	4	yes	—
58	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
61	C₄²×C₁₃	C₂×C₈	1	2	yes	—
62	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
63	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
64	C₄×Dic₁₃	C₂×C₈	1	8	yes	—
65	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
66	C₂×#1055	C₂×C₈	1	4	yes	—
67	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
68	C₂×#1055	C₂²×C₄	1	4	yes	—
69	C₂×#1055	C₂²×C₄	1	2	yes	—
70	C₂×#1055	C₂⁴	1	1	yes	characteristic
71	C₈×D₁₃	C₄²	1	8	yes	—
72	C₈×D₁₃	C₂×C₈	1	16	yes	—
73	C₈×D₁₃	C₂²×C₄	1	8	yes	—
74	#6092	C₂²×C₄	1	16	yes	complemented
75	C₂³×C₁₆	—	13	2	no	—
76	C₂²×C₄×C₈	—	13	1	no	—
77	C₂×C₄×C₁₆	—	13	2	no	—
78	C₂×C₄×C₁₆	—	13	8	no	—
79	C₂×C₄×C₁₆	—	13	2	no	—
80	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂²×D₁₃	C₄×C₈	1	4	yes	—
82	C₂²×D₁₃	C₂×C₁₆	1	8	yes	—
83	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₁₃	C₂×C₁₆	1	8	yes	complemented
85	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
86	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
89	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×Dic₁₃	C₄×C₈	1	4	yes	—
91	C₂×Dic₁₃	C₂×C₁₆	1	8	yes	complemented
92	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
93	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
94	C₄×D₁₃	C₄×C₈	1	8	yes	—
95	C₄×D₁₃	C₂×C₁₆	1	32	yes	complemented
96	C₄×D₁₃	C₂×C₄²	1	4	yes	—
97	C₄×D₁₃	C₂²×C₈	1	4	yes	—
98	C₈×C₁₃	C₂×C₄²	1	4	yes	—
99	C₈×C₁₃	C₂²×C₈	1	4	yes	—
100	#1055	C₂×C₄²	1	2	yes	—
101	#1055	C₂²×C₈	1	4	yes	—
102	#1055	C₂³×C₄	1	2	yes	—
103	C₂³×C₈	—	13	1	no	—
104	C₂³×C₈	—	13	1	no	—
105	C₂²×C₄²	—	13	1	no	—
106	C₂²×C₁₆	—	13	8	no	—
107	C₂²×C₁₆	—	13	4	no	—
108	C₂²×C₁₆	—	13	16	no	—
109	C₂×C₄×C₈	—	13	1	no	—
110	C₂×C₄×C₈	—	13	1	no	—
111	C₂×C₄×C₈	—	13	1	no	—
112	C₂×C₄×C₈	—	13	4	no	—
113	C₂×C₄×C₈	—	13	4	no	—
114	C₂×C₄×C₈	#261	1	1	yes	characteristic
115	C₄×C₁₆	—	13	8	no	—
116	C₄×C₁₆	—	13	8	no	—
117	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
118	C₂²×C₁₃	C₂²×C₁₆	1	4	yes	—
119	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
120	C₂×D₁₃	C₄×C₁₆	1	16	yes	complemented
121	C₂×D₁₃	C₂×C₄×C₈	1	4	yes	—
122	C₂×D₁₃	C₂²×C₁₆	1	8	yes	—
123	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
124	C₄×C₁₃	C₂²×C₁₆	1	8	yes	complemented
125	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
126	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
127	Dic₁₃	C₂×C₄×C₈	1	4	yes	—
128	Dic₁₃	C₂²×C₁₆	1	8	yes	complemented
129	C₂³×C₄	—	13	2	no	—
130	C₂³×C₄	—	13	1	no	—
131	C₂²×C₈	—	13	1	no	—
132	C₂²×C₈	—	13	1	no	—
133	C₂²×C₈	—	13	8	no	—
134	C₂²×C₈	—	13	8	no	—
135	C₂²×C₈	—	13	4	no	—
136	C₂²×C₈	—	13	2	no	—
137	C₂²×C₈	—	13	2	no	—
138	C₂²×C₈	C₂×#261	1	1	yes	characteristic
139	C₂²×C₈	C₂×#261	1	1	yes	characteristic
140	C₂×C₄²	—	13	2	no	—
141	C₂×C₄²	—	13	1	no	—
142	C₂×C₄²	—	13	4	no	—
143	C₂×C₄²	—	13	4	no	—
144	C₂×C₄²	C₂×#261	1	1	yes	characteristic
145	C₂×C₁₆	—	13	32	no	—
146	C₂×C₁₆	—	13	16	no	—
147	C₂×C₁₆	—	13	8	no	—
148	C₄×C₈	—	13	2	no	—
149	C₄×C₈	—	13	2	no	—
150	C₄×C₈	—	13	4	no	—
151	C₄×C₈	—	13	2	no	—
152	C₄×C₈	—	13	2	no	—
153	C₄×C₈	C₂×#261	1	4	yes	—
154	C₂×C₁₃	C₂×C₄×C₁₆	1	4	yes	complemented
155	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
156	C₂×C₁₃	C₂³×C₁₆	1	2	yes	—
157	D₁₃	C₂×C₄×C₁₆	1	8	yes	complemented
158	C₂⁴	—	13	1	no	—
159	C₂²×C₄	—	13	1	no	—
160	C₂²×C₄	—	13	2	no	—
161	C₂²×C₄	—	13	16	no	—
162	C₂²×C₄	—	13	2	no	—
163	C₂²×C₄	—	13	8	no	—
164	C₂²×C₄	—	13	4	no	—
165	C₂²×C₄	—	13	4	no	—
166	C₂²×C₄	—	13	2	no	—
167	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
168	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
169	C₂×C₈	—	13	4	no	—
170	C₂×C₈	—	13	4	no	—
171	C₂×C₈	—	13	1	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	—	13	1	no	—
174	C₂×C₈	—	13	16	no	—
175	C₂×C₈	—	13	8	no	—
176	C₂×C₈	—	13	8	no	—
177	C₂×C₈	D₁₃⋊C₈	1	4	yes	—
178	C₂×C₈	C₄×#261	1	4	yes	—
179	C₂×C₈	C₂²×#261	1	2	yes	—
180	C₂×C₈	C₂²×#261	1	2	yes	—
181	C₄²	—	13	2	no	—
182	C₄²	—	13	8	no	—
183	C₄²	—	13	2	no	—
184	C₄²	D₁₃⋊C₈	1	2	yes	—
185	C₄²	C₂²×#261	1	2	yes	—
186	C₁₆	—	13	16	no	—
187	C₁₃	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
188	C₂³	—	13	4	no	—
189	C₂³	—	13	8	no	—
190	C₂³	—	13	2	no	—
191	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
192	C₂×C₄	—	13	4	no	—
193	C₂×C₄	—	13	8	no	—
194	C₂×C₄	—	13	4	no	—
195	C₂×C₄	—	13	2	no	—
196	C₂×C₄	—	13	8	no	—
197	C₂×C₄	—	13	32	no	—
198	C₂×C₄	—	13	4	no	—
199	C₂×C₄	—	13	4	no	—
200	C₂×C₄	D₁₃⋊C₁₆	1	8	yes	direct product factor
201	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
202	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
203	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
204	C₂×C₄	C₂×C₄×#261	1	2	yes	—
205	C₂×C₄	C₂³×#261	1	1	yes	characteristic
206	C₈	—	13	4	no	—
207	C₈	—	13	2	no	—
208	C₈	—	13	2	no	—
209	C₈	C₂×D₁₃⋊C₈	1	4	yes	—
210	C₈	C₂×C₄×#261	1	4	yes	—
211	C₂²	—	13	8	no	—
212	C₂²	—	13	16	no	—
213	C₂²	—	13	4	no	—
214	C₂²	C₄×D₁₃⋊C₈	1	2	yes	—
215	C₂²	C₂×D₁₃⋊C₁₆	1	4	yes	—
216	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
217	C₄	—	13	8	no	—
218	C₄	—	13	4	no	—
219	C₄	C₄×D₁₃⋊C₈	1	2	yes	—
220	C₄	C₂×D₁₃⋊C₁₆	1	8	yes	direct product factor
221	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
222	C₄	C₂²×C₄×#261	1	1	yes	characteristic
223	C₂	—	13	8	no	—
224	C₂	C₄×D₁₃⋊C₁₆	1	4	yes	direct product factor
225	C₂	C₂×C₄×D₁₃⋊C₈	1	1	yes	characteristic
226	C₂	C₂²×D₁₃⋊C₁₆	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	#261
commutator, hypocenter	C₁₃	C₂²×C₄×C₁₆
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718257	C₂	semidirect
3. C₄×D₁₃⋊C₁₆	C₂	direct (N × H)
4. C₂×C₄×#6092	C₂	semidirect
10. C₂×D₁₃⋊C₁₆	C₄	direct (N × H)
18. #11718257	C₂²	semidirect
19. C₄×#6092	C₂²	semidirect
36. C₂×#6092	C₂×C₄	semidirect
38. D₁₃⋊C₁₆	C₂×C₄	direct (N × H)
51. C₂×C₄×D₁₃	C₁₆	semidirect
74. #6092	C₂²×C₄	semidirect
84. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
91. C₂×Dic₁₃	C₂×C₁₆	semidirect
95. C₄×D₁₃	C₂×C₁₆	semidirect
120. C₂×D₁₃	C₄×C₁₆	semidirect
124. C₄×C₁₃	C₂²×C₁₆	semidirect
128. Dic₁₃	C₂²×C₁₆	semidirect
154. C₂×C₁₃	C₂×C₄×C₁₆	semidirect
157. D₁₃	C₂×C₄×C₁₆	semidirect
187. C₁₃	C₂²×C₄×C₁₆	semidirect
200. C₂×C₄	D₁₃⋊C₁₆	direct (N × H)
220. C₄	C₂×D₁₃⋊C₁₆	direct (N × H)
224. C₂	C₄×D₁₃⋊C₁₆	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×D₁₃⋊C₁₆	2
C₂×#11718257	2
C₄×D₁₃⋊C₁₆	2
C₂×C₄×#6092	2
C₂×C₄×C₈×D₁₃	2
C₂²×C₄×C₁₆	13

Subgroups — C2×C4×D13⋊C16 #16806710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₁₃⋊C₁₆ #16806710