Subgroups — C₁₃×#2579 #13111637

Order 1664.

584 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#398	C₂	1	2	yes	maximal, complemented
4	C₁₃×#396	C₂	1	2	yes	maximal, complemented
5	C₁₃×#387	C₂	1	2	yes	maximal, complemented
6	C₁₃×#386	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#386	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#385	C₂	1	2	yes	maximal, complemented
9	C₁₃×#385	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#385	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₃	—	2	4	no	—
18	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
32	C₁₃×#95	C₂²	1	2	yes	—
33	C₁₃×#95	C₂²	1	2	yes	—
34	C₁₃×#95	C₂²	1	2	yes	—
35	C₁₃×#95	C₂²	1	2	yes	—
36	C₁₃×#95	C₂²	1	2	yes	—
37	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
38	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×D₄×C₁₃	—	2	2	no	—
40	C₂×D₄×C₁₃	—	2	2	no	—
41	C₂×D₄×C₁₃	Q₈	1	4	yes	—
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	—	4	2	no	—
48	C₂²×C₄×C₁₃	—	4	2	no	—
49	C₂²×C₄×C₁₃	D₄	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
59	C₁₃×C₄⋊C₄	—	2	4	no	—
60	C₁₃×C₄⋊C₄	—	2	4	no	—
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₁₃×C₄⋊C₄	—	4	2	no	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	2	2	no	—
69	C₁₃×C₂²⋊C₄	—	2	4	no	—
70	C₁₃×C₂²⋊C₄	—	4	2	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	#2579	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	—	2	2	no	—
75	C₂³×C₁₃	—	2	1	no	—
76	C₂³×C₁₃	—	2	2	no	—
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	—	2	2	no	—
79	C₂³×C₁₃	C₂×D₄	1	2	yes	—
80	C₂³×C₁₃	C₂×Q₈	1	2	yes	—
81	C₂³×C₁₃	C₂×Q₈	1	2	yes	—
82	C₂³×C₁₃	#41	1	2	yes	—
83	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
84	D₄×C₁₃	—	2	8	no	—
85	C₂×C₄×C₁₃	—	2	8	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
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	2	no	—
105	C₂×C₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₁₃	#41	1	4	yes	—
112	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic
113	C₂²×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#398	C₂×C₁₃	1	2	yes	complemented
115	#396	C₂×C₁₃	1	2	yes	complemented
116	#387	C₂×C₁₃	1	2	yes	complemented
117	#386	C₂×C₁₃	1	1	yes	characteristic
118	#386	C₂×C₁₃	1	1	yes	characteristic
119	#385	C₂×C₁₃	1	2	yes	complemented
120	#385	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#385	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#380	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	2	4	no	—
126	C₂²×C₁₃	—	2	4	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	4	no	—
129	C₂²×C₁₃	—	4	1	no	—
130	C₂²×C₁₃	—	4	1	no	—
131	C₂²×C₁₃	—	4	2	no	—
132	C₂²×C₁₃	C₂²⋊Q₈	1	4	yes	—
133	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×#41	1	2	yes	—
137	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	—	4	1	no	—
144	C₄×C₁₃	—	4	2	no	—
145	C₄×C₁₃	—	4	2	no	—
146	C₄×C₁₃	—	4	2	no	—
147	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
148	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
149	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
150	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
151	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
152	C₄×D₄	—	2	4	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
155	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
156	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
158	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
166	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
167	#95	C₂²×C₁₃	1	2	yes	—
168	#95	C₂²×C₁₃	1	2	yes	—
169	#95	C₂²×C₁₃	1	2	yes	—
170	#95	C₂²×C₁₃	1	2	yes	—
171	#95	C₂²×C₁₃	1	2	yes	—
172	C₂×C₁₃	—	2	4	no	—
173	C₂×C₁₃	—	4	2	no	—
174	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
175	C₂×C₁₃	#525	1	1	yes	characteristic
176	C₂×C₁₃	#534	1	1	yes	characteristic
177	C₂×C₁₃	#553	1	2	yes	—
178	C₂×C₁₃	#554	1	2	yes	—
179	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
180	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	Q₈×C₁₃	1	4	yes	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	D₄×C₁₃	1	4	yes	—
192	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
193	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
194	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
197	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	4	2	no	—
213	C₄²	—	2	2	no	—
214	C₁₃	#2579	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	C₂×D₄×C₁₃	1	2	yes	—
222	C₂³	C₂×Q₈×C₁₃	1	2	yes	—
223	C₂³	C₂×Q₈×C₁₃	1	2	yes	—
224	C₂³	C₁₃×#41	1	2	yes	—
225	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
226	D₄	—	2	8	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
251	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
252	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
253	C₂×C₄	C₁₃×#41	1	4	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	C₁₃×C₂²⋊Q₈	1	4	yes	—
264	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
265	C₂²	C₂²×Q₈×C₁₃	1	1	yes	characteristic
266	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
267	C₂²	C₂×C₁₃×#41	1	2	yes	—
268	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
269	C₄	—	2	2	no	—
270	C₄	—	2	4	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	2	no	—
278	C₂	—	2	4	no	—
279	C₂	—	4	2	no	—
280	C₂	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
281	C₂	C₁₃×#525	1	1	yes	characteristic
282	C₂	C₁₃×#534	1	1	yes	characteristic
283	C₂	C₁₃×#553	1	2	yes	—
284	C₂	C₁₃×#554	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₃×C₄⋊C₄	C₂	semidirect
3. C₁₃×#398	C₂	semidirect
4. C₁₃×#396	C₂	semidirect
5. C₁₃×#387	C₂	semidirect
8. C₁₃×#385	C₂	semidirect
9. C₁₃×#385	C₂	semidirect
10. C₁₃×#385	C₂	semidirect
11. C₁₃×#380	C₂	semidirect
72. #2579	C₁₃	direct (N × H)
113. C₂²×C₄⋊C₄	C₂×C₁₃	semidirect
114. #398	C₂×C₁₃	semidirect
115. #396	C₂×C₁₃	semidirect
116. #387	C₂×C₁₃	semidirect
119. #385	C₂×C₁₃	semidirect
120. #385	C₂×C₁₃	semidirect
121. #385	C₂×C₁₃	semidirect
122. #380	C₂×C₁₃	semidirect
214. C₁₃	#2579	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×D₄×C₁₃	2
C₂²×C₁₃×C₄⋊C₄	2
C₁₃×#398	2
C₁₃×#396	2
C₁₃×#387	2
C₁₃×#386	2
C₁₃×#385	2
C₁₃×#380	2
#2579	13

Subgroups — C13×#2579 #13111637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2579 #13111637