Subgroups — C₂×#13116716 #16016318

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718565	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718556	C₂	1	2	yes	maximal, complemented
3	#13116716	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718089	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88161	—	2	2	no	—
8	C₂×#88158	C₂²	1	2	yes	—
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#108	C₂²	1	2	yes	—
11	C₂×#88065	—	2	2	no	—
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂²×#6090	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₀₄	C₄	1	1	yes	characteristic, complemented
15	#11718565	C₄	1	4	yes	complemented
16	#11718565	C₂²	1	4	yes	—
17	C₂×#88040	C₄	1	2	yes	complemented
18	C₂×D₂₀₈	C₄	1	1	yes	characteristic, complemented
19	#11718556	—	2	4	no	—
20	#11718556	C₂²	1	8	yes	complemented
21	C₁₃×#368	C₂²	1	4	yes	complemented
22	C₁₆×Dic₁₃	C₂²	1	4	yes	complemented
23	#11718089	C₂²	1	8	yes	complemented
24	C₂×#6127	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
28	C₂²×#1055	D₄	1	1	yes	characteristic
29	#88161	—	2	8	no	—
30	#88161	—	4	4	no	—
31	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
32	#88158	—	2	2	no	—
33	#88158	—	2	2	no	—
34	#88158	C₂×C₄	1	4	yes	—
35	#88158	C₂³	1	4	yes	—
36	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
37	C₂×#6095	—	2	2	no	—
38	C₂×#6094	—	4	2	no	—
39	C₂×C₈×D₁₃	—	4	2	no	—
40	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
44	C₁₃×#108	—	2	2	no	—
45	C₁₃×#108	C₂³	1	4	yes	—
46	#88065	—	2	8	no	—
47	C₈×Dic₁₃	D₄	1	2	yes	—
48	C₈×Dic₁₃	C₂³	1	2	yes	—
49	C₂×#6090	—	2	1	no	—
50	C₂×#6090	—	2	1	no	—
51	C₂×#6090	C₂³	1	2	yes	—
52	Dic₁₀₄	C₂×C₄	1	4	yes	complemented
53	#88040	C₂×C₄	1	8	yes	complemented
54	D₂₀₈	C₂×C₄	1	4	yes	complemented
55	C₂×#2178	—	13	1	no	maximal, Sylow
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₃	—	4	2	no	—
58	C₂²×Dic₁₃	—	2	1	no	—
59	#6127	—	2	4	no	—
60	#6127	—	2	4	no	—
61	#6127	—	4	2	no	—
62	#6127	—	4	2	no	—
63	C₂×D₅₂	—	2	2	no	—
64	C₂×C₄×D₁₃	—	4	2	no	—
65	C₂×Dic₂₆	—	2	2	no	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	—	4	2	no	—
68	C₂×C₈×C₁₃	—	4	2	no	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₄×Dic₁₃	—	2	1	no	—
76	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
77	C₂×#1055	D₈	1	1	yes	characteristic
78	C₂×#1055	D₈	1	1	yes	characteristic
79	C₂×#1055	D₈	1	2	yes	—
80	C₂×#1055	C₂×D₄	1	2	yes	—
81	Dic₅₂	—	2	2	no	—
82	Dic₅₂	C₂²×C₄	1	4	yes	—
83	D₁₀₄	—	2	2	no	—
84	D₁₀₄	C₂²×C₄	1	4	yes	—
85	#6095	—	2	4	no	—
86	#6095	—	2	4	no	—
87	#6094	—	4	8	no	—
88	C₈×D₁₃	—	4	8	no	—
89	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
90	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
91	#6090	—	2	2	no	—
92	C₂×#508	—	13	1	no	—
93	C₂×#443	—	13	2	no	—
94	#2178	—	13	8	no	—
95	C₂×#368	D₁₃	1	1	yes	characteristic, complemented
96	C₂×#359	—	13	2	no	—
97	C₂×C₄×C₁₆	—	13	1	no	—
98	C₂³×C₁₃	#135	1	1	yes	characteristic
99	C₂²×D₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₂²⋊D₁₃	—	4	4	no	—
107	C₂²⋊D₁₃	—	4	4	no	—
108	C₂×Dic₁₃	—	2	4	no	—
109	C₂×Dic₁₃	—	2	2	no	—
110	C₂×Dic₁₃	—	4	2	no	—
111	D₅₂	—	2	4	no	—
112	D₅₂	—	4	2	no	—
113	C₄×D₁₃	—	4	4	no	—
114	C₄×D₁₃	—	4	4	no	—
115	Dic₂₆	—	2	4	no	—
116	Dic₂₆	—	4	2	no	—
117	C₈×C₁₃	—	4	4	no	—
118	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
122	#1055	C₂×D₈	1	2	yes	—
123	#1055	C₂×D₈	1	2	yes	—
124	C₂×#135	—	13	2	no	—
125	C₂×#131	—	26	2	no	—
126	#508	—	13	4	no	—
127	#508	—	13	4	no	—
128	C₂×Q₃₂	—	13	1	no	—
129	C₂×SD₃₂	—	13	2	no	—
130	C₂×D₁₆	—	13	1	no	—
131	C₂²×C₁₆	—	13	1	no	—
132	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
133	#443	—	13	8	no	—
134	#443	—	26	4	no	—
135	C₂×#108	C₂×D₁₃	1	2	yes	—
136	C₂×#104	—	26	2	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	#368	C₂×D₁₃	1	4	yes	complemented
139	#359	—	13	8	no	—
140	C₄×C₁₆	—	13	4	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
143	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
145	C₂×D₁₃	—	4	4	no	—
146	C₂×D₁₃	—	8	4	no	—
147	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
148	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
149	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
151	Dic₁₃	—	2	4	no	—
152	Dic₁₃	—	4	4	no	—
153	C₂×#41	—	26	2	no	—
154	#135	—	13	4	no	—
155	#135	—	13	4	no	—
156	#135	—	26	2	no	—
157	#135	—	26	2	no	—
158	C₂×Q₁₆	—	13	2	no	—
159	C₂×SD₁₆	—	26	2	no	—
160	C₂×D₈	—	13	2	no	—
161	#131	—	26	8	no	—
162	#131	—	52	4	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	52	2	no	—
165	C₂²×C₈	—	13	1	no	—
166	C₂²×C₈	—	52	2	no	—
167	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₂×C₄²	—	13	1	no	—
169	Q₃₂	—	13	4	no	—
170	SD₃₂	—	13	8	no	—
171	D₁₆	—	13	4	no	—
172	C₂×C₁₆	—	13	2	no	—
173	C₂×C₁₆	—	26	1	no	—
174	C₂×C₁₆	—	26	1	no	—
175	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
176	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
177	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
178	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
179	#108	—	2	2	no	—
180	#108	C₂²×D₁₃	1	4	yes	—
181	#104	—	26	8	no	—
182	C₄×C₈	—	13	2	no	—
183	C₄×C₈	—	13	2	no	—
184	C₂×C₁₃	—	2	1	no	—
185	C₂×C₁₃	—	2	1	no	—
186	C₂×C₁₃	#2178	1	2	yes	complemented
187	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
188	D₁₃	—	8	4	no	—
189	#41	—	26	4	no	—
190	#41	—	26	4	no	—
191	#41	—	52	2	no	—
192	#41	—	52	2	no	—
193	C₂×Q₈	—	26	2	no	—
194	C₂×D₄	—	26	2	no	—
195	C₂×D₄	—	52	2	no	—
196	C₂²×C₄	—	26	1	no	—
197	C₂²×C₄	—	52	2	no	—
198	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
199	Q₁₆	—	13	4	no	—
200	Q₁₆	—	26	2	no	—
201	SD₁₆	—	26	4	no	—
202	SD₁₆	—	26	4	no	—
203	D₈	—	13	4	no	—
204	D₈	—	26	2	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	52	8	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	13	2	no	—
210	C₂×C₈	—	13	1	no	—
211	C₂×C₈	—	13	1	no	—
212	C₂×C₈	—	13	2	no	—
213	C₂×C₈	—	52	8	no	—
214	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
217	C₂×C₈	#6131	1	1	yes	characteristic
218	C₂×C₈	#6131	1	1	yes	characteristic
219	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
220	C₄²	—	13	2	no	—
221	C₄²	—	26	1	no	—
222	C₁₆	—	26	2	no	—
223	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
224	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
225	C₁₃	C₂×#2178	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	52	2	no	—
227	C₂³	#88173	1	1	yes	characteristic
228	Q₈	—	26	4	no	—
229	Q₈	—	52	2	no	—
230	D₄	—	26	4	no	—
231	D₄	—	52	4	no	—
232	D₄	—	52	4	no	—
233	D₄	—	52	2	no	—
234	C₂×C₄	—	26	4	no	—
235	C₂×C₄	—	26	2	no	—
236	C₂×C₄	—	52	2	no	—
237	C₂×C₄	—	52	4	no	—
238	C₂×C₄	—	52	4	no	—
239	C₂×C₄	#88147	1	1	yes	characteristic
240	C₂×C₄	#88147	1	1	yes	characteristic
241	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
242	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
243	C₂×C₄	#88173	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
245	C₈	—	4	4	no	—
246	C₈	—	13	2	no	—
247	C₈	—	13	2	no	—
248	C₈	#88147	1	1	yes	characteristic
249	C₈	#88147	1	1	yes	characteristic
250	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
251	C₈	C₂×#6131	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	52	4	no	—
254	C₂²	—	104	4	no	—
255	C₂²	#11718533	1	1	yes	characteristic
256	C₂²	#11718533	1	1	yes	characteristic
257	C₂²	C₂×#88173	1	1	yes	characteristic
258	C₄	—	26	4	no	—
259	C₄	—	52	4	no	—
260	C₄	#11718533	1	1	yes	characteristic
261	C₄	#11718533	1	1	yes	characteristic
262	C₄	C₂×#88147	1	1	yes	characteristic
263	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	104	4	no	—
267	C₂	#13116716	1	2	yes	direct product factor
268	C₂	C₂×#11718533	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#368	C₂
socle	C₂²×C₁₃	C₄×D₈
prime radical	C₂×#11718565	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#368	D₁₃
hypocenter	C₁₃	C₂×#2178
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
2. C₂×#11718556	C₂	semidirect
3. #13116716	C₂	direct (N × H)
4. C₂×#11718089	C₂	semidirect
5. C₂×C₁₆×Dic₁₃	C₂	semidirect
6. C₂×C₁₃×#368	C₂	semidirect
14. C₂×Dic₁₀₄	C₄	semidirect
15. #11718565	C₄	semidirect
17. C₂×#88040	C₄	semidirect
18. C₂×D₂₀₈	C₄	semidirect
20. #11718556	C₂²	semidirect
21. C₁₃×#368	C₂²	semidirect
22. C₁₆×Dic₁₃	C₂²	semidirect
23. #11718089	C₂²	semidirect
52. Dic₁₀₄	C₂×C₄	semidirect
53. #88040	C₂×C₄	semidirect
54. D₂₀₈	C₂×C₄	semidirect
95. C₂×#368	D₁₃	semidirect
138. #368	C₂×D₁₃	semidirect
186. C₂×C₁₃	#2178	semidirect
225. C₁₃	C₂×#2178	semidirect
267. C₂	#13116716	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11718565	2
C₂×#11718556	2
#13116716	2
C₂×#11718089	2
C₂×C₁₆×Dic₁₃	2
C₂×C₁₃×#368	2
C₂×#2178	13

Subgroups — C2×#13116716 #16016318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116716 #16016318