Subgroups — C₁₆×#6101 #15778262

Order 3328.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718093	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₈×#6101	C₂	1	1	yes	maximal, characteristic
5	#13112914	C₂	1	1	yes	maximal, characteristic
6	#13112913	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88054	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂²×#6090	C₂²	1	1	yes	characteristic
15	C₂²×#6090	C₂²	1	1	yes	characteristic
16	C₄×#6101	C₄	1	1	yes	characteristic
17	C₄×C₁₃×C₁₆	—	2	2	no	—
18	#11718142	C₄	1	1	yes	characteristic
19	#11718116	C₄	1	1	yes	characteristic
20	#11718093	C₄	1	4	yes	complemented
21	C₁₆×Dic₁₃	—	2	2	no	—
22	C₁₆×Dic₁₃	C₄	1	4	yes	complemented
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#6101	C₈	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×C₁₆	—	2	1	no	—
32	C₂×C₁₃×C₁₆	—	2	1	no	—
33	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
36	C₂×C₁₃×C₁₆	Q₈	1	2	yes	—
37	C₄×C₈×C₁₃	—	2	1	no	—
38	C₄×C₈×C₁₃	—	2	1	no	—
39	#88071	C₈	1	2	yes	—
40	#88054	C₈	1	4	yes	—
41	C₈×Dic₁₃	—	2	1	no	—
42	C₈×Dic₁₃	—	2	1	no	—
43	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
44	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
45	C₂×#6090	—	2	1	no	—
46	C₂×#6090	—	2	1	no	—
47	C₂×#6090	—	2	1	no	—
48	C₂×#6090	—	2	1	no	—
49	C₂×#6090	C₂×C₄	1	2	yes	—
50	C₂×#6090	C₂×C₄	1	2	yes	—
51	C₁₆×C₄⋊C₄	—	13	1	no	maximal, Sylow
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
56	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
66	C₄²×C₁₃	—	2	1	no	—
67	C₄²×C₁₃	—	2	1	no	—
68	#6101	C₁₆	1	8	yes	direct product factor
69	C₄×Dic₁₃	—	2	1	no	—
70	C₄×Dic₁₃	—	2	1	no	—
71	C₄×Dic₁₃	C₄²	1	2	yes	—
72	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
73	C₂×#1055	—	2	1	no	—
74	C₂×#1055	—	2	1	no	—
75	C₂×#1055	—	2	1	no	—
76	C₂×#1055	—	2	1	no	—
77	C₂×#1055	C₄²	1	2	yes	—
78	C₂×#1055	C₂×C₈	1	2	yes	—
79	C₁₃×C₁₆	—	2	2	no	—
80	C₁₃×C₁₆	C₄⋊C₄	1	4	yes	complemented
81	#6090	—	2	2	no	—
82	#6090	—	2	2	no	—
83	C₂×C₄⋊C₁₆	—	13	1	no	—
84	C₂×C₄×C₁₆	—	13	1	no	—
85	C₂×C₄×C₁₆	—	13	1	no	—
86	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
87	C₈×C₄⋊C₄	—	13	1	no	—
88	#1374	—	13	1	no	—
89	#1374	—	13	1	no	—
90	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
94	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	—	2	2	no	—
102	C₂×Dic₁₃	—	2	2	no	—
103	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
104	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
105	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
106	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₈×C₁₃	#117	1	1	yes	characteristic
110	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
112	#1055	—	2	2	no	—
113	#1055	—	2	2	no	—
114	C₂²×C₁₆	—	13	1	no	—
115	C₂²×C₁₆	—	13	1	no	—
116	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
117	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
118	C₂×C₄⋊C₈	—	13	1	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	C₂×C₄×C₈	—	13	1	no	—
121	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	13	1	no	—
123	C₄⋊C₁₆	—	13	4	no	—
124	C₄×C₁₆	—	2	2	no	—
125	C₄×C₁₆	—	13	4	no	—
126	C₄×C₁₆	—	26	2	no	—
127	#336	—	13	1	no	—
128	#336	—	13	1	no	—
129	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
130	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
132	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
133	C₂²×C₁₃	#504	1	2	yes	—
134	C₄×C₁₃	—	2	4	no	—
135	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₁₃	#405	1	1	yes	characteristic
137	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
138	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
139	Dic₁₃	—	2	2	no	—
140	Dic₁₃	—	2	4	no	—
141	Dic₁₃	C₄×C₁₆	1	4	yes	complemented
142	C₂²×C₈	—	13	1	no	—
143	C₂²×C₈	—	13	1	no	—
144	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	13	2	no	—
147	C₂×C₄²	—	13	1	no	—
148	C₂×C₄²	—	13	1	no	—
149	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	13	2	no	—
153	C₂×C₁₆	—	13	2	no	—
154	C₂×C₁₆	—	26	1	no	—
155	C₂×C₁₆	—	26	1	no	—
156	C₂×C₁₆	—	26	1	no	—
157	C₂×C₁₆	—	26	1	no	—
158	C₂×C₁₆	Dic₂₆	1	2	yes	—
159	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
160	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
161	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
162	C₄⋊C₈	—	13	4	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	13	2	no	—
166	C₄×C₈	—	13	2	no	—
167	C₄×C₈	—	26	1	no	—
168	C₄×C₈	—	26	1	no	—
169	#95	—	13	2	no	—
170	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
172	C₂×C₁₃	#2108	1	1	yes	characteristic
173	C₂×C₁₃	D₄×C₁₆	1	2	yes	—
174	C₂×C₁₃	Q₈×C₁₆	1	2	yes	—
175	C₂²×C₄	—	13	1	no	—
176	C₂²×C₄	—	13	1	no	—
177	C₂²×C₄	—	13	2	no	—
178	C₂²×C₄	C₈×D₁₃	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	13	2	no	—
183	C₂×C₈	—	13	2	no	—
184	C₂×C₈	—	26	1	no	—
185	C₂×C₈	—	26	1	no	—
186	C₂×C₈	—	26	1	no	—
187	C₂×C₈	—	26	1	no	—
188	C₂×C₈	C₈×D₁₃	1	2	yes	—
189	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
192	C₂×C₈	#6127	1	1	yes	characteristic
193	C₂×C₈	#6127	1	1	yes	characteristic
194	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
195	C₄⋊C₄	—	13	8	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	13	2	no	—
199	C₄²	—	13	2	no	—
200	C₄²	—	26	1	no	—
201	C₄²	—	26	1	no	—
202	C₁₆	—	2	2	no	—
203	C₁₆	—	26	2	no	—
204	C₁₆	—	26	2	no	—
205	C₁₆	#6101	1	4	yes	direct product factor
206	C₁₃	C₁₆×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
207	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	13	4	no	—
211	C₂×C₄	—	13	2	no	—
212	C₂×C₄	—	13	4	no	—
213	C₂×C₄	—	13	2	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	—	26	2	no	—
216	C₂×C₄	—	26	2	no	—
217	C₂×C₄	C₁₆×D₁₃	1	4	yes	—
218	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
219	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
220	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
221	C₂×C₄	#88155	1	1	yes	characteristic
222	C₂×C₄	#88155	1	1	yes	characteristic
223	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	26	2	no	—
226	C₈	—	26	2	no	—
227	C₈	C₄×Dic₂₆	1	1	yes	characteristic
228	C₈	#88126	1	1	yes	characteristic
229	C₈	C₂×#6101	1	1	yes	characteristic
230	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
231	C₂²	C₈×Dic₂₆	1	1	yes	characteristic
232	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
233	C₂²	C₂×C₁₆×D₁₃	1	2	yes	—
234	C₂²	#11718562	1	2	yes	—
235	C₂²	C₈×C₂²⋊D₁₃	1	1	yes	characteristic
236	C₄	—	2	4	no	—
237	C₄	—	13	4	no	—
238	C₄	—	26	2	no	—
239	C₄	—	26	4	no	—
240	C₄	C₈×Dic₂₆	1	1	yes	characteristic
241	C₄	#11718350	1	1	yes	characteristic
242	C₄	C₄×#6101	1	1	yes	characteristic
243	C₄	C₈×C₂²⋊D₁₃	1	1	yes	characteristic
244	C₂	C₈×#6101	1	1	yes	characteristic
245	C₂	C₄×C₁₆×D₁₃	1	1	yes	characteristic
246	C₂	#13116431	1	1	yes	characteristic
247	C₂	C₁₆×C₂²⋊D₁₃	1	2	yes	—
248	C₂	C₁₆×Dic₂₆	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
20. #11718093	C₄	semidirect
22. C₁₆×Dic₁₃	C₄	semidirect
68. #6101	C₁₆	direct (N × H)
80. C₁₃×C₁₆	C₄⋊C₄	semidirect
141. Dic₁₃	C₄×C₁₆	semidirect
205. C₁₆	#6101	direct (N × H)
206. C₁₃	C₁₆×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718093	2
C₂×C₁₆×Dic₁₃	2
C₈×#6101	2
#13112914	2
#13112913	2
C₂×C₄×C₁₃×C₁₆	2
C₁₆×C₄⋊C₄	13

Subgroups — C16×#6101 #15778262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#6101 #15778262