Subgroups — C₂×#13130059 #16792720

Order 3328.

994 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#6122	C₂	1	2	yes	maximal
2	#13130059	C₂	1	8	yes	maximal, direct product factor
3	C₂×#1057⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×#11718251	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×#1057	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#6122	C₄	1	4	yes	—
8	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₄	1	1	yes	characteristic
11	C₄×#6122	C₄	1	8	yes	—
12	C₄×#6122	C₂²	1	8	yes	—
13	C₂×#88110	C₄	1	4	yes	—
14	C₂×C₄×#1057	C₂²	1	2	yes	—
15	C₂×C₄×#1057	C₂²	1	2	yes	—
16	M₁₆×Dic₁₃	C₄	1	4	yes	—
17	M₁₆×Dic₁₃	C₂²	1	4	yes	—
18	C₂×#88055	C₄	1	1	yes	characteristic
19	C₂×#88055	C₄	1	1	yes	characteristic
20	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
22	#1057⋊C₈	C₂²	1	8	yes	complemented
23	#11718251	C₂²	1	4	yes	complemented
24	C₈×#1057	C₂²	1	4	yes	complemented
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₁₃	C₂×C₄	1	2	yes	—
27	C₂×#6122	—	2	16	no	—
28	C₂×#6122	C₂×C₄	1	16	yes	—
29	C₂²×#1057	C₂×C₄	1	4	yes	—
30	C₂²×#1057	C₂×C₄	1	4	yes	—
31	C₂×C₁₃×M₁₆	—	2	4	no	—
32	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
41	C₂×#6099	—	2	4	no	—
42	C₂×#6099	C₂×C₄	1	4	yes	—
43	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
44	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
45	#88110	C₂×C₄	1	16	yes	—
46	C₄×#1057	C₂×C₄	1	4	yes	—
47	C₄×#1057	C₂×C₄	1	4	yes	—
48	C₄×#1057	C₂³	1	4	yes	—
49	C₄×#1057	C₂³	1	4	yes	—
50	#88055	C₂×C₄	1	4	yes	—
51	#88055	C₂×C₄	1	4	yes	—
52	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₁₃	C₂³	1	2	yes	—
55	C₈×Dic₁₃	C₂³	1	2	yes	—
56	C₂×#1455	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₃	—	2	8	no	—
65	C₂²×Dic₁₃	C₄²	1	2	yes	—
66	C₂²×Dic₁₃	C₄²	1	4	yes	—
67	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
68	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
69	#6122	—	2	32	no	—
70	C₂×#1057	—	2	8	no	—
71	C₂×#1057	M₁₆	1	16	yes	complemented
72	C₂×#1057	C₂²×C₄	1	8	yes	—
73	C₂×#1057	C₂²×C₄	1	8	yes	—
74	C₁₃×M₁₆	—	2	8	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	M₁₆	1	4	yes	complemented
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₄×Dic₁₃	C₄²	1	2	yes	—
80	C₄×Dic₁₃	C₄²	1	4	yes	—
81	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
83	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
84	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
85	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
86	#6099	—	2	8	no	—
87	C₂×#1055	—	2	2	no	—
88	C₂×#1055	M₁₆	1	4	yes	complemented
89	C₂×#1055	C₂²×C₄	1	2	yes	—
90	C₂×#1055	C₂²×C₄	1	2	yes	—
91	C₂×C₄×M₁₆	—	13	2	no	—
92	C₂×C₄×M₁₆	—	13	1	no	—
93	#1455	—	13	8	no	—
94	C₂×#322	—	13	1	no	—
95	C₂×#322	—	13	2	no	—
96	C₂×C₈²	—	13	1	no	—
97	C₂³×C₁₃	—	2	4	no	—
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
100	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₁₃	—	2	4	no	—
111	C₂×Dic₁₃	—	2	8	no	—
112	C₂×Dic₁₃	—	2	8	no	—
113	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
114	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
115	C₂×Dic₁₃	C₂×M₁₆	1	8	yes	—
116	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
117	#1057	—	2	8	no	—
118	#1057	C₂×M₁₆	1	16	yes	complemented
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	C₂×M₁₆	1	4	yes	complemented
121	#1055	—	2	2	no	—
122	#1055	C₂×M₁₆	1	4	yes	complemented
123	C₂²×M₁₆	—	13	4	no	—
124	C₂²×M₁₆	—	13	1	no	—
125	C₂²×M₁₆	#261	1	1	yes	characteristic
126	C₂²×C₄²	—	13	1	no	—
127	C₄×M₁₆	—	13	8	no	—
128	C₄×M₁₆	—	13	8	no	—
129	C₄×M₁₆	—	13	4	no	—
130	C₄×M₁₆	—	13	4	no	—
131	C₂×#97	—	13	4	no	—
132	C₂×#97	—	13	1	no	—
133	C₂×#97	—	13	1	no	—
134	C₂×C₄×C₈	—	13	2	no	—
135	C₂×C₄×C₈	—	13	2	no	—
136	C₂×C₄×C₈	—	13	1	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	#322	—	13	4	no	—
139	#322	—	13	8	no	—
140	C₈²	—	13	4	no	—
141	C₂²×C₁₃	—	2	4	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	4	4	no	—
144	C₂²×C₁₃	C₄×M₁₆	1	4	yes	—
145	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
146	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
149	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
150	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
151	Dic₁₃	—	2	4	no	—
152	Dic₁₃	C₄×M₁₆	1	4	yes	—
153	Dic₁₃	C₂²×M₁₆	1	4	yes	—
154	C₂³×C₄	—	13	2	no	—
155	C₂³×C₄	C₂×#261	1	1	yes	characteristic
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	13	16	no	—
158	C₂×M₁₆	—	13	4	no	—
159	C₂×M₁₆	—	26	4	no	—
160	C₂×M₁₆	—	26	16	no	—
161	C₂×M₁₆	C₂×#261	1	4	yes	—
162	C₂²×C₈	—	13	4	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	13	4	no	—
165	C₂²×C₈	—	13	1	no	—
166	C₂²×C₈	C₂×#261	1	1	yes	characteristic
167	C₂²×C₈	C₂×#261	1	1	yes	characteristic
168	C₂×C₄²	—	13	2	no	—
169	C₂×C₄²	—	13	4	no	—
170	C₂×C₄²	—	13	2	no	—
171	C₂×C₄²	—	13	2	no	—
172	C₂×C₄²	—	13	1	no	—
173	C₂×C₄²	—	13	1	no	—
174	#97	—	13	16	no	—
175	#97	—	13	4	no	—
176	#97	—	13	4	no	—
177	C₄×C₈	—	13	4	no	—
178	C₄×C₈	—	13	4	no	—
179	C₄×C₈	—	13	4	no	—
180	C₄×C₈	—	13	4	no	—
181	C₄×C₈	—	13	2	no	—
182	C₄×C₈	—	13	2	no	—
183	C₄×C₈	—	13	2	no	—
184	C₄×C₈	—	13	2	no	—
185	C₂×C₁₃	—	4	2	no	—
186	C₂×C₁₃	#1455	1	4	yes	complemented
187	C₂×C₁₃	C₂×C₄×M₁₆	1	2	yes	—
188	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
189	C₂⁴	C₄×#261	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	13	2	no	—
193	C₂²×C₄	—	13	4	no	—
194	C₂²×C₄	—	13	4	no	—
195	C₂²×C₄	—	13	2	no	—
196	C₂²×C₄	—	26	8	no	—
197	C₂²×C₄	C₄×#261	1	1	yes	characteristic
198	C₂²×C₄	C₄×#261	1	2	yes	—
199	C₂²×C₄	C₂²×#261	1	2	yes	—
200	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
201	M₁₆	—	2	8	no	—
202	M₁₆	—	26	8	no	—
203	M₁₆	—	26	32	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	13	8	no	—
206	C₂×C₈	—	13	16	no	—
207	C₂×C₈	—	13	4	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	13	8	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	26	8	no	—
212	C₂×C₈	—	26	2	no	—
213	C₂×C₈	#6122	1	4	yes	complemented
214	C₂×C₈	C₂²×#261	1	2	yes	—
215	C₂×C₈	C₂²×#261	1	2	yes	—
216	C₄²	—	13	2	no	—
217	C₄²	—	13	4	no	—
218	C₄²	—	13	4	no	—
219	C₄²	—	13	2	no	—
220	C₄²	—	13	1	no	—
221	C₄²	—	13	1	no	—
222	C₄²	—	13	2	no	—
223	C₁₃	C₂×#1455	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	2	4	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₄×#261	1	2	yes	—
227	C₂³	C₂×C₄×#261	1	1	yes	characteristic
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	13	8	no	—
233	C₂×C₄	—	13	2	no	—
234	C₂×C₄	—	13	2	no	—
235	C₂×C₄	—	13	4	no	—
236	C₂×C₄	—	26	8	no	—
237	C₂×C₄	—	26	8	no	—
238	C₂×C₄	—	26	4	no	—
239	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
240	C₂×C₄	C₂×C₄×#261	1	2	yes	—
241	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
242	C₂×C₄	C₂×#6122	1	2	yes	—
243	C₂×C₄	C₂³×#261	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	13	16	no	—
246	C₈	—	13	4	no	—
247	C₈	—	26	8	no	—
248	C₈	—	26	2	no	—
249	C₈	C₂×#6122	1	4	yes	complemented
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	4	no	—
253	C₂²	M₁₆×#261	1	4	yes	—
254	C₂²	C₄×#6122	1	2	yes	—
255	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	13	4	no	—
258	C₄	—	13	4	no	—
259	C₄	—	26	4	no	—
260	C₄	C₄×#6122	1	2	yes	—
261	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
262	C₄	C₂²×#6122	1	1	yes	characteristic
263	C₂	—	4	2	no	—
264	C₂	#13130059	1	4	yes	direct product factor
265	C₂	C₂×M₁₆×#261	1	2	yes	—
266	C₂	C₂×C₄×#6122	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂⁴×C₁₃	C₄²
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	C₂×#1455
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13130059	C₂	direct (N × H)
3. C₂×#1057⋊C₈	C₂	semidirect
4. C₂×#11718251	C₂	semidirect
5. C₂×C₈×#1057	C₂	semidirect
22. #1057⋊C₈	C₂²	semidirect
23. #11718251	C₂²	semidirect
24. C₈×#1057	C₂²	semidirect
71. C₂×#1057	M₁₆	semidirect
76. C₂×C₈×C₁₃	M₁₆	semidirect
88. C₂×#1055	M₁₆	semidirect
118. #1057	C₂×M₁₆	semidirect
120. C₈×C₁₃	C₂×M₁₆	semidirect
122. #1055	C₂×M₁₆	semidirect
186. C₂×C₁₃	#1455	semidirect
213. C₂×C₈	#6122	semidirect
223. C₁₃	C₂×#1455	semidirect
249. C₈	C₂×#6122	semidirect
264. C₂	#13130059	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×C₄×#6122	2
#13130059	2
C₂×#1057⋊C₈	2
C₂×#11718251	2
C₂×C₈×#1057	2
C₂×M₁₆×Dic₁₃	2
C₂×#1455	13

Subgroups — C2×#13130059 #16792720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130059 #16792720