Subgroups — C₂×#13116634 #16015983

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	#13116634	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718099	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₁₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂²×#6090	C₂²	1	2	yes	—
15	C₂×C₄×#1055	C₄	1	1	yes	characteristic
16	C₄×C₈×D₁₃	C₄	1	4	yes	—
17	C₄×C₈×D₁₃	C₂²	1	4	yes	—
18	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
19	#11718099	C₂²	1	8	yes	—
20	C₁₆×Dic₁₃	C₂²	1	8	yes	complemented
21	C₄⋊#6090	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₁₃	—	2	2	no	—
34	C₂×C₈×D₁₃	—	2	2	no	—
35	C₂×C₈×D₁₃	—	2	4	no	—
36	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
37	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	C₂³	1	2	yes	—
39	C₄²×D₁₃	C₈	1	4	yes	—
40	C₄²×D₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₁₃×C₁₆	—	2	4	no	—
42	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
43	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₃	C₂³	1	2	yes	—
45	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
46	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
47	C₈×Dic₁₃	C₂³	1	2	yes	—
48	C₂×#6090	—	2	4	no	—
49	C₂×#6090	C₂³	1	4	yes	—
50	C₄×#1055	C₂×C₄	1	4	yes	—
51	C₂×#2162	—	13	1	no	maximal, Sylow
52	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂×C₄×D₁₃	—	2	2	no	—
60	C₂×C₄×D₁₃	—	2	2	no	—
61	C₂×C₄×D₁₃	—	2	4	no	—
62	C₂×C₄×D₁₃	C₁₆	1	8	yes	complemented
63	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
65	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
66	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	#41	1	2	yes	—
71	C₂×C₈×C₁₃	#41	1	2	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₄²×C₁₃	C₂×C₈	1	2	yes	—
74	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
75	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
76	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
77	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
78	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	M₁₆	1	4	yes	—
81	C₂×#1055	C₂²×C₄	1	2	yes	—
82	C₂×#1055	C₂²×C₄	1	2	yes	—
83	C₈×D₁₃	—	2	4	no	—
84	C₈×D₁₃	—	2	2	no	—
85	C₈×D₁₃	—	2	2	no	—
86	C₈×D₁₃	—	2	8	no	—
87	C₁₃×C₁₆	—	2	4	no	—
88	#6090	—	2	4	no	—
89	C₂²×C₄×C₈	—	13	1	no	—
90	#2162	—	13	8	no	—
91	C₂×C₄⋊C₁₆	—	13	1	no	—
92	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic, complemented
93	C₂×C₂²⋊C₁₆	—	13	2	no	—
94	C₂×C₄×C₁₆	—	13	2	no	—
95	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	C₂²×D₁₃	—	2	2	no	—
97	C₂²×D₁₃	—	2	2	no	—
98	C₂²×D₁₃	C₂×C₁₆	1	4	yes	—
99	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
102	C₂×C₄×C₁₃	#117	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×Dic₁₃	—	2	2	no	—
109	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
110	C₂×Dic₁₃	M₃₂	1	4	yes	—
111	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
112	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
113	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
114	C₄×D₁₃	—	2	2	no	—
115	C₄×D₁₃	—	2	2	no	—
116	C₄×D₁₃	—	2	4	no	—
117	C₄×D₁₃	—	2	8	no	—
118	C₄×D₁₃	C₂×C₁₆	1	16	yes	complemented
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	#117	1	2	yes	—
121	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
122	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
123	#1055	—	2	2	no	—
124	#1055	C₂×M₁₆	1	4	yes	—
125	C₂³×C₈	—	13	1	no	—
126	C₂³×C₈	—	13	1	no	—
127	C₂²×C₄²	—	13	1	no	—
128	C₂²×C₁₆	—	13	2	no	—
129	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
130	C₂×C₄×C₈	—	13	1	no	—
131	C₂×C₄×C₈	—	13	1	no	—
132	C₂×C₄×C₈	—	13	1	no	—
133	C₂×C₄×C₈	—	13	4	no	—
134	C₂×C₄×C₈	—	13	4	no	—
135	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
136	C₄⋊C₁₆	—	13	4	no	—
137	C₄⋊C₁₆	C₂×D₁₃	1	4	yes	complemented
138	C₂²⋊C₁₆	—	13	8	no	—
139	C₄×C₁₆	—	13	8	no	—
140	C₂²×C₁₃	#431	1	2	yes	—
141	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
142	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
143	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
144	C₂×D₁₃	—	2	2	no	—
145	C₂×D₁₃	—	2	2	no	—
146	C₂×D₁₃	—	2	8	no	—
147	C₂×D₁₃	C₂²×C₁₆	1	4	yes	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	#431	1	2	yes	—
150	C₄×C₁₃	C₂²×C₁₆	1	4	yes	—
151	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
152	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
153	Dic₁₃	—	2	2	no	—
154	Dic₁₃	C₂²×C₁₆	1	4	yes	—
155	Dic₁₃	C₂×M₃₂	1	4	yes	—
156	C₂³×C₄	—	13	1	no	—
157	C₂³×C₄	—	13	1	no	—
158	C₂³×C₄	—	13	1	no	—
159	C₂²×C₈	—	13	1	no	—
160	C₂²×C₈	—	13	1	no	—
161	C₂²×C₈	—	13	4	no	—
162	C₂²×C₈	—	13	2	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	26	4	no	—
165	C₂²×C₈	—	26	2	no	—
166	C₂²×C₈	—	26	2	no	—
167	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×C₄²	—	13	1	no	—
170	C₂×C₄²	—	13	1	no	—
171	C₂×C₄²	—	13	1	no	—
172	C₂×C₄²	—	13	4	no	—
173	C₂×C₄²	—	13	4	no	—
174	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
175	C₂×C₁₆	—	2	4	no	—
176	C₂×C₁₆	—	13	4	no	—
177	C₂×C₁₆	—	26	4	no	—
178	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
179	C₄×C₈	—	13	2	no	—
180	C₄×C₈	—	13	4	no	—
181	C₄×C₈	—	13	2	no	—
182	C₄×C₈	—	13	4	no	—
183	C₄×C₈	C₄×D₁₃	1	2	yes	—
184	C₄×C₈	C₂²×D₁₃	1	2	yes	—
185	C₂×C₁₃	#2162	1	4	yes	complemented
186	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
187	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
188	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
189	D₁₃	—	2	4	no	—
190	C₂⁴	—	13	1	no	—
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	—	13	1	no	—
193	C₂²×C₄	—	13	4	no	—
194	C₂²×C₄	—	13	8	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	4	no	—
197	C₂²×C₄	—	13	2	no	—
198	C₂²×C₄	—	13	2	no	—
199	C₂²×C₄	—	26	4	no	—
200	C₂²×C₄	—	26	2	no	—
201	C₂²×C₄	—	26	2	no	—
202	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	13	4	no	—
207	C₂×C₈	—	13	2	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	26	8	no	—
210	C₂×C₈	—	26	2	no	—
211	C₂×C₈	—	26	4	no	—
212	C₂×C₈	—	26	2	no	—
213	C₂×C₈	—	26	2	no	—
214	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
215	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂×C₈	#6129	1	2	yes	—
217	C₂×C₈	#6131	1	2	yes	—
218	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
219	C₄²	—	13	2	no	—
220	C₄²	—	13	4	no	—
221	C₄²	—	13	4	no	—
222	C₄²	—	13	2	no	—
223	C₄²	C₈×D₁₃	1	2	yes	—
224	C₄²	C₂×C₄×D₁₃	1	2	yes	—
225	C₁₆	—	2	4	no	—
226	C₁₆	—	26	4	no	—
227	C₁₃	C₂×#2162	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	13	2	no	—
229	C₂³	—	13	4	no	—
230	C₂³	—	26	2	no	—
231	C₂³	—	26	2	no	—
232	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	13	2	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	13	4	no	—
237	C₂×C₄	—	13	4	no	—
238	C₂×C₄	—	13	16	no	—
239	C₂×C₄	—	13	2	no	—
240	C₂×C₄	—	26	8	no	—
241	C₂×C₄	—	26	2	no	—
242	C₂×C₄	—	26	2	no	—
243	C₂×C₄	—	26	2	no	—
244	C₂×C₄	—	26	4	no	—
245	C₂×C₄	C₁₆×D₁₃	1	4	yes	—
246	C₂×C₄	#88146	1	2	yes	—
247	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
248	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
250	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
251	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	13	4	no	—
254	C₈	—	26	2	no	—
255	C₈	#88146	1	2	yes	—
256	C₈	C₂×#6129	1	1	yes	characteristic
257	C₈	C₂×#6131	1	1	yes	characteristic
258	C₂²	—	13	4	no	—
259	C₂²	—	26	2	no	—
260	C₂²	—	26	8	no	—
261	C₂²	—	26	2	no	—
262	C₂²	#11718494	1	2	yes	—
263	C₂²	C₂×C₁₆×D₁₃	1	2	yes	—
264	C₂²	D₁₃×M₃₂	1	2	yes	—
265	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	13	4	no	—
268	C₄	—	13	4	no	—
269	C₄	—	26	2	no	—
270	C₄	#11718494	1	2	yes	—
271	C₄	C₂×C₁₆×D₁₃	1	4	yes	—
272	C₄	C₂×#88146	1	1	yes	characteristic
273	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
274	C₂	—	26	4	no	—
275	C₂	#13116634	1	4	yes	direct product factor
276	C₂	C₂×#11718494	1	1	yes	characteristic
277	C₂	C₂²×C₁₆×D₁₃	1	1	yes	characteristic
278	C₂	C₂×D₁₃×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	C₂×#2162
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #13116634	C₂	direct (N × H)
3. C₂×C₄⋊#6090	C₂	semidirect
5. C₂×C₁₆×Dic₁₃	C₂	semidirect
6. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
18. C₁₃×C₄⋊C₁₆	C₂²	semidirect
20. C₁₆×Dic₁₃	C₂²	semidirect
21. C₄⋊#6090	C₂²	semidirect
62. C₂×C₄×D₁₃	C₁₆	semidirect
92. C₂×C₄⋊C₁₆	D₁₃	semidirect
118. C₄×D₁₃	C₂×C₁₆	semidirect
137. C₄⋊C₁₆	C₂×D₁₃	semidirect
185. C₂×C₁₃	#2162	semidirect
227. C₁₃	C₂×#2162	semidirect
275. C₂	#13116634	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×C₈×D₁₃	2
#13116634	2
C₂×C₄⋊#6090	2
C₂×#11718099	2
C₂×C₁₆×Dic₁₃	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×#2162	13

Subgroups — C2×#13116634 #16015983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116634 #16015983