Subgroups — C₁₃×#13061 #15699400

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#466	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#357	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1808	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#121	—	2	1	no	—
9	C₂×C₁₃×M₃₂	—	2	2	no	—
10	C₂×C₁₃×M₃₂	—	2	4	no	—
11	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₁₃×#466	—	2	2	no	—
13	C₁₃×#466	C₄	1	4	yes	complemented
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#102	—	2	1	no	—
17	C₁₃×#380	—	2	1	no	—
18	C₁₃×#357	—	2	4	no	—
19	C₁₃×#340	—	2	1	no	—
20	C₂²×D₄×C₁₃	—	2	1	no	—
21	C₂²×D₄×C₁₃	—	4	1	no	—
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₁₃×D₈	—	2	2	no	—
24	C₂×C₁₃×D₈	—	2	1	no	—
25	C₂×C₁₃×D₈	—	2	2	no	—
26	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×#121	—	2	4	no	—
34	C₁₃×#121	—	4	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
39	C₁₃×M₃₂	—	2	8	no	—
40	C₂×C₁₃×C₁₆	—	2	2	no	—
41	C₂×C₁₃×C₁₆	—	2	2	no	—
42	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
43	C₁₃×#107	—	2	1	no	—
44	C₁₃×#107	—	2	2	no	—
45	C₁₃×#107	—	2	2	no	—
46	C₁₃×#107	C₂×C₄	1	2	yes	—
47	C₁₃×#102	—	2	4	no	—
48	C₁₃×#95	—	4	1	no	—
49	#13061	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁴×C₁₃	—	4	1	no	—
51	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	4	4	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	4	1	no	—
56	C₂×D₄×C₁₃	—	4	2	no	—
57	C₂×D₄×C₁₃	—	8	4	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₁₃×D₈	—	2	4	no	—
67	C₁₃×D₈	—	4	1	no	—
68	C₁₃×D₈	—	4	1	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	2	yes	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	4	1	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₁₃×C₂²⋊C₄	—	4	4	no	—
83	C₁₃×C₂²⋊C₄	—	8	2	no	—
84	C₁₃×C₁₆	—	2	4	no	—
85	C₂²×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
86	C₂×#466	C₂×C₁₃	1	1	yes	characteristic
87	C₂×#357	C₂×C₁₃	1	2	yes	complemented
88	#1808	C₂×C₁₃	1	1	yes	characteristic, complemented
89	#1380	C₂×C₁₃	1	2	yes	complemented
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	—	2	1	no	—
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	—	2	1	no	—
94	C₂³×C₁₃	—	4	2	no	—
95	C₂³×C₁₃	—	8	2	no	—
96	C₂³×C₁₃	—	8	2	no	—
97	C₂³×C₁₃	—	8	1	no	—
98	C₂³×C₁₃	—	8	1	no	—
99	C₂³×C₁₃	#102	1	2	yes	—
100	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
101	D₄×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	8	8	no	—
104	D₄×C₁₃	—	8	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	8	2	no	—
111	C₂×C₄×C₁₃	—	8	2	no	—
112	C₂×C₄×C₁₃	—	8	2	no	—
113	C₂×C₄×C₁₃	—	8	2	no	—
114	C₂×C₄×C₁₃	#102	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#121	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#135	1	2	yes	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	C₄×D₄	1	2	yes	—
121	C₈×C₁₃	#120	1	1	yes	characteristic
122	C₈×C₁₃	#123	1	1	yes	characteristic
123	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
124	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×#121	—	2	1	no	—
126	C₂×M₃₂	—	2	2	no	—
127	C₂×M₃₂	—	2	4	no	—
128	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
129	#466	—	2	2	no	—
130	#466	C₄×C₁₃	1	4	yes	complemented
131	C₂×#107	C₂²×C₁₃	1	2	yes	—
132	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×#102	—	2	1	no	—
134	#380	—	2	1	no	—
135	#357	—	2	4	no	—
136	#340	—	2	1	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	2	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	4	2	no	—
141	C₂²×C₁₃	—	4	2	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	8	2	no	—
145	C₂²×C₁₃	—	16	2	no	—
146	C₂²×C₁₃	—	16	2	no	—
147	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
148	C₂²×C₁₃	#466	1	2	yes	—
149	C₂²×C₁₃	#509	1	1	yes	characteristic
150	C₂²×C₁₃	#509	1	2	yes	—
151	C₂²×C₁₃	#510	1	1	yes	characteristic
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	8	2	no	—
154	C₄×C₁₃	—	16	2	no	—
155	C₄×C₁₃	#386	1	1	yes	characteristic
156	C₄×C₁₃	#416	1	1	yes	characteristic
157	C₄×C₁₃	#465	1	2	yes	—
158	C₂²×D₄	—	2	1	no	—
159	C₂²×D₄	—	4	1	no	—
160	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
161	C₂×D₈	—	2	2	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	—	2	2	no	—
164	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
168	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
169	C₂²×C₈	D₄×C₁₃	1	2	yes	—
170	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
171	#121	—	2	4	no	—
172	#121	—	4	2	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₂²⋊C₄	—	4	1	no	—
176	C₂×C₂²⋊C₄	—	4	1	no	—
177	M₃₂	—	2	8	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	D₄×C₁₃	1	4	yes	complemented
181	#107	—	2	1	no	—
182	#107	—	2	2	no	—
183	#107	—	2	2	no	—
184	#107	C₂×C₄×C₁₃	1	2	yes	—
185	#102	—	2	4	no	—
186	#95	—	4	1	no	—
187	C₂×C₁₃	—	4	2	no	—
188	C₂×C₁₃	—	16	2	no	—
189	C₂×C₁₃	#1893	1	1	yes	characteristic
190	C₂×C₁₃	#2139	1	1	yes	characteristic
191	C₂×C₁₃	#2140	1	1	yes	characteristic
192	C₂×C₁₃	#2218	1	2	yes	—
193	C₂×C₁₃	#2220	1	2	yes	—
194	C₂⁴	—	4	1	no	—
195	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	8	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
207	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
208	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	D₈	—	2	4	no	—
211	D₈	—	4	1	no	—
212	D₈	—	4	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₂×C₈	C₁₃×#41	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	4	no	—
227	C₂²⋊C₄	—	8	2	no	—
228	C₁₆	—	2	4	no	—
229	C₁₃	#13061	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	8	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	1	no	—
239	C₂³	C₁₃×#102	1	2	yes	—
240	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
241	D₄	—	4	2	no	—
242	D₄	—	4	2	no	—
243	D₄	—	8	8	no	—
244	D₄	—	8	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	C₁₃×#102	1	2	yes	—
255	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₁₃×#121	1	2	yes	—
257	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
258	C₂×C₄	C₁₃×#135	1	2	yes	—
259	C₈	—	2	2	no	—
260	C₈	C₄×D₄×C₁₃	1	2	yes	—
261	C₈	C₁₃×#120	1	1	yes	characteristic
262	C₈	C₁₃×#123	1	1	yes	characteristic
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	8	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	16	2	no	—
272	C₂²	—	16	2	no	—
273	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
274	C₂²	C₁₃×#466	1	2	yes	—
275	C₂²	C₁₃×#509	1	1	yes	characteristic
276	C₂²	C₁₃×#509	1	2	yes	—
277	C₂²	C₁₃×#510	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	16	2	no	—
281	C₄	C₁₃×#386	1	1	yes	characteristic
282	C₄	C₁₃×#416	1	1	yes	characteristic
283	C₄	C₁₃×#465	1	2	yes	—
284	C₂	—	4	2	no	—
285	C₂	—	16	2	no	—
286	C₂	C₁₃×#1893	1	1	yes	characteristic
287	C₂	C₁₃×#2139	1	1	yes	characteristic
288	C₂	C₁₃×#2140	1	1	yes	characteristic
289	C₂	C₁₃×#2218	1	2	yes	—
290	C₂	C₁₃×#2220	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×M₃₂	C₂	semidirect
3. C₂×C₁₃×#357	C₂	semidirect
4. C₁₃×#1808	C₂	semidirect
5. C₁₃×#1380	C₂	semidirect
13. C₁₃×#466	C₄	semidirect
42. C₂×C₁₃×C₁₆	D₄	semidirect
49. #13061	C₁₃	direct (N × H)
85. C₂²×M₃₂	C₂×C₁₃	semidirect
87. C₂×#357	C₂×C₁₃	semidirect
88. #1808	C₂×C₁₃	semidirect
89. #1380	C₂×C₁₃	semidirect
130. #466	C₄×C₁₃	semidirect
180. C₂×C₁₆	D₄×C₁₃	semidirect
229. C₁₃	#13061	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×M₃₂	2
C₂×C₁₃×#466	2
C₂×C₁₃×#357	2
C₁₃×#1808	2
C₁₃×#1380	2
#13061	13

Subgroups — C13×#13061 #15699400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13061 #15699400