Subgroups — C₂×D₁₃⋊#504 #16809616

Order 3328.

2118 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719096	C₂	1	6	yes	maximal, complemented
2	D₁₃⋊#504	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11719162	C₂	1	2	yes	maximal, complemented
4	C₂×D₁₃⋊M₃₂	C₂	1	3	yes	maximal, complemented
5	C₂²×D₁₃⋊C₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×D₁₃×#131	C₂	1	1	yes	maximal, characteristic
7	C₂×#41×D₁₃	C₄	1	1	yes	characteristic
8	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
9	C₂×#88202	C₂²	1	1	yes	characteristic
10	D₁₃×#131	C₄	1	8	yes	—
11	D₁₃×#131	C₂²	1	8	yes	—
12	C₂×#88161	C₂²	1	3	yes	—
13	C₂×M₁₆×D₁₃	C₄	1	3	yes	—
14	C₂×#88155	C₄	1	3	yes	—
15	C₂²×C₈×D₁₃	C₂²	1	3	yes	—
16	#11719162	C₂²	1	16	yes	complemented
17	C₂×#88106	C₂²	1	6	yes	complemented
18	C₂²×#6092	C₂²	1	6	yes	complemented
19	#11719096	C₂²	1	48	yes	complemented
20	D₁₃⋊M₃₂	C₂²	1	24	yes	complemented
21	C₂×#88098	C₂²	1	6	yes	complemented
22	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₂×D₁₃⋊C₁₆	C₂²	1	3	yes	—
24	C₂×D₁₃⋊C₁₆	C₂²	1	24	yes	complemented
25	C₂×D₁₃⋊C₁₆	C₂²	1	6	yes	complemented
26	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
27	#41×D₁₃	C₈	1	8	yes	—
28	#41×D₁₃	C₂×C₄	1	8	yes	—
29	C₂×#6131	C₈	1	1	yes	characteristic
30	C₂×Q₈×D₁₃	C₈	1	1	yes	characteristic
31	C₂×#6129	C₈	1	3	yes	—
32	C₂×D₄×D₁₃	C₈	1	3	yes	—
33	C₂×#6127	C₂×C₄	1	3	yes	—
34	C₂²×C₄×D₁₃	C₂×C₄	1	3	yes	—
35	C₁₃×#131	C₂×C₄	1	8	yes	—
36	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
38	#88202	C₂×C₄	1	4	yes	—
39	#88202	C₂³	1	4	yes	—
40	C₂×#6099	C₂×C₄	1	3	yes	—
41	C₂²×#1055	C₂³	1	3	yes	—
42	#88161	C₂×C₄	1	12	yes	—
43	#88161	C₂³	1	12	yes	—
44	M₁₆×D₁₃	C₂×C₄	1	24	yes	—
45	#88155	C₂×C₄	1	24	yes	—
46	C₂×#6094	C₂×C₄	1	6	yes	—
47	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
48	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
49	C₂×C₈×D₁₃	C₂×C₄	1	12	yes	—
50	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₁₃	C₂³	1	3	yes	—
52	C₂×C₈×D₁₃	C₂³	1	12	yes	—
53	#88106	C₂³	1	24	yes	complemented
54	C₂×#6092	C₂³	1	2	yes	—
55	C₂×#6092	C₂³	1	6	yes	—
56	C₂×#6092	C₂³	1	24	yes	complemented
57	#88098	C₂³	1	24	yes	complemented
58	D₁₃⋊C₁₆	C₂³	1	4	yes	—
59	D₁₃⋊C₁₆	C₂³	1	12	yes	—
60	D₁₃⋊C₁₆	C₂³	1	24	yes	complemented
61	C₂²×#504	—	13	1	no	maximal, Sylow
62	C₂³×D₁₃	C₂×C₈	1	3	yes	—
63	C₁₃×#41	C₂×C₈	1	4	yes	—
64	C₁₃×#41	C₂²×C₄	1	4	yes	—
65	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂×D₄×C₁₃	C₂×C₈	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
68	C₂×C₂²⋊D₁₃	C₂×C₈	1	6	yes	—
69	C₂²×Dic₁₃	C₂×C₈	1	3	yes	—
70	#6131	C₂×C₈	1	8	yes	—
71	Q₈×D₁₃	C₂×C₈	1	8	yes	—
72	#6129	C₂×C₈	1	24	yes	—
73	D₄×D₁₃	C₂×C₈	1	24	yes	—
74	#6127	C₂×C₈	1	12	yes	—
75	#6127	C₂²×C₄	1	12	yes	—
76	C₂×D₅₂	C₂×C₈	1	3	yes	—
77	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
78	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
79	C₂×C₄×D₁₃	C₂×C₈	1	12	yes	—
80	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×D₁₃	C₂²×C₄	1	3	yes	—
82	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
83	C₂×Dic₂₆	C₂×C₈	1	3	yes	—
84	C₁₃×M₁₆	C₂²×C₄	1	12	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	#6099	C₂²×C₄	1	12	yes	—
89	C₂×#1055	C₂²×C₄	1	3	yes	—
90	C₂×#1055	C₂²×C₄	1	6	yes	—
91	C₂×#1055	C₂⁴	1	1	yes	characteristic
92	C₂×#1055	C₂⁴	1	6	yes	—
93	#6094	C₂²×C₄	1	24	yes	—
94	C₈×D₁₃	C₂²×C₄	1	2	yes	—
95	C₈×D₁₃	C₂²×C₄	1	12	yes	—
96	C₈×D₁₃	C₂²×C₄	1	12	yes	—
97	C₈×D₁₃	C₂²×C₄	1	6	yes	—
98	C₈×D₁₃	C₂⁴	1	2	yes	—
99	C₈×D₁₃	C₂⁴	1	6	yes	—
100	#6092	C₂⁴	1	4	yes	—
101	#6092	C₂⁴	1	12	yes	—
102	C₂²×#131	—	13	1	no	—
103	C₂×#504	—	13	2	no	—
104	C₂×#504	—	13	16	no	—
105	C₂×#504	—	13	6	no	—
106	C₂²×M₃₂	—	13	3	no	—
107	C₂³×C₁₆	—	13	3	no	—
108	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
109	C₂²×D₁₃	—	2	12	no	—
110	C₂²×D₁₃	C₂²×C₈	1	3	yes	—
111	C₂²×D₁₃	C₂²×C₈	1	12	yes	—
112	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
113	Q₈×C₁₃	C₂²×C₈	1	4	yes	—
114	D₄×C₁₃	C₂²×C₈	1	12	yes	—
115	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
119	C₂²⋊D₁₃	C₂²×C₈	1	24	yes	—
120	C₂×Dic₁₃	C₂²×C₈	1	3	yes	—
121	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂²×C₈	1	12	yes	—
123	D₅₂	C₂²×C₈	1	12	yes	—
124	C₄×D₁₃	C₂²×C₈	1	2	yes	—
125	C₄×D₁₃	C₂²×C₈	1	6	yes	—
126	C₄×D₁₃	C₂²×C₈	1	12	yes	—
127	C₄×D₁₃	C₂²×C₈	1	12	yes	—
128	C₄×D₁₃	C₂³×C₄	1	2	yes	—
129	C₄×D₁₃	C₂³×C₄	1	6	yes	—
130	Dic₂₆	C₂²×C₈	1	12	yes	—
131	C₈×C₁₃	C₂³×C₄	1	2	yes	—
132	C₈×C₁₃	C₂³×C₄	1	6	yes	—
133	#1055	C₂³×C₄	1	1	yes	characteristic
134	#1055	C₂³×C₄	1	6	yes	—
135	#1055	C₂⁵	1	1	yes	characteristic
136	C₂²×#41	—	13	1	no	—
137	C₂×#131	—	13	1	no	—
138	C₂×#131	—	13	8	no	—
139	C₂×#131	—	13	8	no	—
140	C₂×#131	—	13	3	no	—
141	C₂×#131	—	13	3	no	—
142	C₂×#131	#261	1	1	yes	characteristic
143	C₂²×M₁₆	—	13	3	no	—
144	C₂³×C₈	—	13	3	no	—
145	#504	—	13	48	no	—
146	#504	—	13	16	no	—
147	C₂×M₃₂	—	13	6	no	—
148	C₂×M₃₂	—	13	24	no	—
149	C₂×M₃₂	—	13	6	no	—
150	C₂²×C₁₆	—	13	1	no	—
151	C₂²×C₁₆	—	13	3	no	—
152	C₂²×C₁₆	—	13	24	no	—
153	C₂²×C₁₆	—	13	6	no	—
154	C₂²×C₁₆	—	13	6	no	—
155	C₂²×C₁₃	—	2	6	no	—
156	C₂²×C₁₃	C₂³×C₈	1	6	yes	—
157	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
158	C₂×D₁₃	—	2	6	no	—
159	C₂×D₁₃	—	2	24	no	—
160	C₂×D₁₃	#504	1	4	yes	complemented
161	C₂×D₁₃	C₂³×C₈	1	6	yes	—
162	C₂×D₁₃	C₂³×C₈	1	2	yes	—
163	C₄×C₁₃	C₂³×C₈	1	6	yes	—
164	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
165	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
166	Dic₁₃	C₂³×C₈	1	6	yes	—
167	Dic₁₃	C₂³×C₈	1	2	yes	—
168	C₂×#41	—	13	8	no	—
169	C₂×#41	—	13	8	no	—
170	C₂×#41	—	13	1	no	—
171	C₂×#41	—	13	3	no	—
172	C₂×#41	—	13	3	no	—
173	C₂×#41	C₂×#261	1	1	yes	characteristic
174	C₂²×Q₈	—	13	1	no	—
175	C₂²×D₄	—	13	3	no	—
176	C₂³×C₄	—	13	3	no	—
177	#131	—	13	4	no	—
178	#131	—	13	4	no	—
179	#131	—	13	12	no	—
180	#131	—	13	12	no	—
181	#131	—	13	24	no	—
182	#131	C₂×#261	1	8	yes	—
183	C₂×M₁₆	—	13	3	no	—
184	C₂×M₁₆	—	13	24	no	—
185	C₂×M₁₆	—	13	6	no	—
186	C₂×M₁₆	C₂×#261	1	3	yes	—
187	C₂²×C₈	—	13	3	no	—
188	C₂²×C₈	—	13	3	no	—
189	C₂²×C₈	—	13	12	no	—
190	C₂²×C₈	—	13	3	no	—
191	C₂²×C₈	—	13	12	no	—
192	C₂²×C₈	—	13	1	no	—
193	C₂²×C₈	—	13	3	no	—
194	C₂²×C₈	C₂×#261	1	3	yes	—
195	M₃₂	—	13	24	no	—
196	M₃₂	—	13	24	no	—
197	C₂×C₁₆	—	13	4	no	—
198	C₂×C₁₆	—	13	12	no	—
199	C₂×C₁₆	—	13	24	no	—
200	C₂×C₁₆	—	13	2	no	—
201	C₂×C₁₆	—	13	6	no	—
202	C₂×C₁₆	—	13	24	no	—
203	C₂×C₁₃	—	2	6	no	—
204	C₂×C₁₃	C₂×#504	1	2	yes	complemented
205	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
206	D₁₃	—	2	6	no	—
207	D₁₃	C₂×#504	1	4	yes	complemented
208	C₂⁴	—	13	3	no	—
209	#41	—	13	8	no	—
210	#41	—	13	24	no	—
211	#41	—	13	12	no	—
212	#41	—	13	12	no	—
213	#41	D₁₃⋊C₈	1	4	yes	—
214	#41	C₂²×#261	1	4	yes	—
215	C₂×Q₈	—	13	8	no	—
216	C₂×Q₈	—	13	3	no	—
217	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
218	C₂×D₄	—	13	6	no	—
219	C₂×D₄	—	13	24	no	—
220	C₂×D₄	—	13	3	no	—
221	C₂×D₄	D₁₃⋊C₈	1	3	yes	—
222	C₂²×C₄	—	13	3	no	—
223	C₂²×C₄	—	13	3	no	—
224	C₂²×C₄	—	13	3	no	—
225	C₂²×C₄	—	13	12	no	—
226	C₂²×C₄	—	13	1	no	—
227	C₂²×C₄	—	13	3	no	—
228	C₂²×C₄	—	13	12	no	—
229	C₂²×C₄	C₂²×#261	1	3	yes	—
230	M₁₆	—	13	12	no	—
231	M₁₆	—	13	24	no	—
232	M₁₆	C₂²×#261	1	12	yes	—
233	C₂×C₈	—	13	1	no	—
234	C₂×C₈	—	13	6	no	—
235	C₂×C₈	—	13	3	no	—
236	C₂×C₈	—	13	6	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	13	6	no	—
239	C₂×C₈	—	13	2	no	—
240	C₂×C₈	—	13	12	no	—
241	C₂×C₈	—	13	6	no	—
242	C₂×C₈	—	13	12	no	—
243	C₂×C₈	C₂²×#261	1	3	yes	—
244	C₂×C₈	C₂²×#261	1	12	yes	—
245	C₂×C₈	C₂²×#261	1	1	yes	characteristic
246	C₁₆	—	13	4	no	—
247	C₁₆	—	13	12	no	—
248	C₁₃	C₂²×#504	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	13	1	no	—
250	C₂³	—	13	12	no	—
251	C₂³	—	13	3	no	—
252	C₂³	—	26	12	no	—
253	C₂³	C₂×D₁₃⋊C₈	1	3	yes	—
254	Q₈	—	13	12	no	—
255	Q₈	C₂×D₁₃⋊C₈	1	4	yes	—
256	D₄	—	13	24	no	—
257	D₄	—	13	12	no	—
258	D₄	C₂×D₁₃⋊C₈	1	12	yes	—
259	C₂×C₄	—	13	1	no	—
260	C₂×C₄	—	13	3	no	—
261	C₂×C₄	—	13	12	no	—
262	C₂×C₄	—	13	2	no	—
263	C₂×C₄	—	13	12	no	—
264	C₂×C₄	—	13	12	no	—
265	C₂×C₄	—	13	6	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	13	6	no	—
268	C₂×C₄	C₂×D₁₃⋊C₈	1	3	yes	—
269	C₂×C₄	C₂×D₁₃⋊C₈	1	6	yes	—
270	C₂×C₄	C₂³×#261	1	1	yes	characteristic
271	C₂×C₄	C₂³×#261	1	6	yes	—
272	C₈	—	13	1	no	—
273	C₈	—	13	1	no	—
274	C₈	—	13	6	no	—
275	C₈	C₂³×#261	1	2	yes	—
276	C₈	C₂³×#261	1	6	yes	—
277	C₂²	—	2	6	no	—
278	C₂²	—	13	4	no	—
279	C₂²	—	13	2	no	—
280	C₂²	—	13	6	no	—
281	C₂²	—	26	24	no	—
282	C₂²	—	26	6	no	—
283	C₂²	C₂²×D₁₃⋊C₈	1	6	yes	—
284	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
285	C₄	—	13	2	no	—
286	C₄	—	13	6	no	—
287	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
288	C₄	C₂²×D₁₃⋊C₈	1	6	yes	—
289	C₄	C₂⁴×#261	1	1	yes	characteristic
290	C₂	—	2	6	no	—
291	C₂	—	13	4	no	—
292	C₂	—	26	6	no	—
293	C₂	D₁₃⋊#504	1	2	yes	direct product factor
294	C₂	C₂³×D₁₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#261
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#261
Fitting	C₂×C₁₃×#131	C₄
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂×#41×D₁₃	C₄
cosocle	#1055	C₂⁵
hypercenter	C₂×#131	#261
hypocenter	C₁₃	C₂²×#504
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#11719096	C₂	semidirect
2. D₁₃⋊#504	C₂	direct (N × H)
3. C₂×#11719162	C₂	semidirect
4. C₂×D₁₃⋊M₃₂	C₂	semidirect
5. C₂²×D₁₃⋊C₁₆	C₂	semidirect
16. #11719162	C₂²	semidirect
17. C₂×#88106	C₂²	semidirect
18. C₂²×#6092	C₂²	semidirect
19. #11719096	C₂²	semidirect
20. D₁₃⋊M₃₂	C₂²	semidirect
21. C₂×#88098	C₂²	semidirect
24. C₂×D₁₃⋊C₁₆	C₂²	semidirect
25. C₂×D₁₃⋊C₁₆	C₂²	semidirect
53. #88106	C₂³	semidirect
56. C₂×#6092	C₂³	semidirect
57. #88098	C₂³	semidirect
60. D₁₃⋊C₁₆	C₂³	semidirect
160. C₂×D₁₃	#504	semidirect
204. C₂×C₁₃	C₂×#504	semidirect
207. D₁₃	C₂×#504	semidirect
248. C₁₃	C₂²×#504	semidirect
293. C₂	D₁₃⋊#504	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×#11719096	2
D₁₃⋊#504	2
C₂×#11719162	2
C₂×D₁₃⋊M₃₂	2
C₂²×D₁₃⋊C₁₆	2
C₂×D₁₃×#131	2
C₂²×#504	13

Subgroups — C2×D13⋊#504 #16809616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₃⋊#504 #16809616