Subgroups — C₁₃×#13129 #15699468

Order 3328.

306 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	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2117	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2115	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1764	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#104	C₄	1	1	yes	characteristic
14	C₁₃×#416	C₄	1	1	yes	characteristic
15	C₁₃×#405	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×C₄⋊C₁₆	—	2	2	no	—
18	C₁₃×#350	—	2	2	no	—
19	C₁₃×#349	—	2	2	no	—
20	C₁₃×#347	—	2	2	no	—
21	C₁₃×#341	C₄	1	1	yes	characteristic
22	C₁₃×#339	C₄	1	1	yes	characteristic
23	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
24	C₁₃×#131	—	2	1	no	—
25	C₁₃×#131	—	2	1	no	—
26	C₁₃×#131	—	2	1	no	—
27	C₁₃×#131	—	2	1	no	—
28	C₂×C₁₃×M₁₆	—	2	1	no	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₁₃×M₃₂	—	2	1	no	—
37	C₁₃×M₃₂	—	2	1	no	—
38	C₁₃×M₃₂	—	2	1	no	—
39	C₁₃×M₃₂	—	2	1	no	—
40	C₁₃×M₃₂	—	2	1	no	—
41	C₁₃×M₃₂	—	2	1	no	—
42	C₁₃×M₃₂	—	2	1	no	—
43	C₁₃×M₃₂	D₄	1	2	yes	—
44	C₂×C₁₃×C₁₆	—	2	1	no	—
45	C₂×C₁₃×C₁₆	—	2	1	no	—
46	C₂×C₁₃×C₁₆	—	2	1	no	—
47	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
48	C₁₃×#104	—	2	1	no	—
49	C₁₃×#104	—	2	1	no	—
50	C₁₃×#103	—	2	1	no	—
51	C₁₃×#102	—	2	1	no	—
52	C₁₃×#97	—	2	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₄×C₈×C₁₃	—	2	1	no	—
55	C₄×C₈×C₁₃	—	2	1	no	—
56	#13129	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₁₃×#41	—	2	1	no	—
58	C₁₃×#41	—	2	1	no	—
59	C₁₃×#41	—	2	1	no	—
60	C₁₃×#41	—	2	1	no	—
61	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
62	C₂×D₄×C₁₃	—	2	1	no	—
63	C₂×D₄×C₁₃	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₁₃×M₁₆	—	2	1	no	—
68	C₁₃×M₁₆	—	2	1	no	—
69	C₁₃×M₁₆	—	2	1	no	—
70	C₁₃×M₁₆	—	2	1	no	—
71	C₁₃×M₁₆	—	4	1	no	—
72	C₁₃×M₁₆	—	4	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	4	1	no	—
79	C₂×C₈×C₁₃	—	4	1	no	—
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	—	2	1	no	—
87	C₁₃×C₂²⋊C₄	—	2	1	no	—
88	C₄²×C₁₃	—	2	1	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
91	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
92	C₁₃×C₁₆	—	2	2	no	—
93	C₁₃×C₁₆	—	2	2	no	—
94	C₁₃×C₁₆	—	4	1	no	—
95	C₁₃×C₁₆	—	4	1	no	—
96	#2150	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#2117	C₂×C₁₃	1	1	yes	characteristic
98	#2115	C₂×C₁₃	1	1	yes	characteristic
99	#2110	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#1764	C₂×C₁₃	1	1	yes	characteristic
101	#1378	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#1376	C₂×C₁₃	1	1	yes	characteristic, complemented
103	C₂³×C₁₃	—	2	1	no	—
104	C₂³×C₁₃	#131	1	1	yes	characteristic
105	Q₈×C₁₃	—	2	1	no	—
106	Q₈×C₁₃	—	2	1	no	—
107	D₄×C₁₃	—	2	1	no	—
108	D₄×C₁₃	—	2	1	no	—
109	D₄×C₁₃	—	4	1	no	—
110	D₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
124	C₈×C₁₃	—	4	1	no	—
125	C₈×C₁₃	—	4	1	no	—
126	C₈×C₁₃	—	4	1	no	—
127	C₈×C₁₃	#121	1	1	yes	characteristic
128	C₈×C₁₃	#123	1	1	yes	characteristic
129	C₈×C₁₃	#124	1	1	yes	characteristic
130	C₈×C₁₃	#126	1	1	yes	characteristic
131	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×#104	C₄×C₁₃	1	1	yes	characteristic
137	#416	C₄×C₁₃	1	1	yes	characteristic
138	#405	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
140	C₄⋊C₁₆	—	2	2	no	—
141	#350	—	2	2	no	—
142	#349	—	2	2	no	—
143	#347	—	2	2	no	—
144	#341	C₄×C₁₃	1	1	yes	characteristic
145	#339	C₄×C₁₃	1	1	yes	characteristic
146	C₂²×C₁₃	—	4	1	no	—
147	C₂²×C₁₃	—	4	1	no	—
148	C₂²×C₁₃	—	8	1	no	—
149	C₂²×C₁₃	#433	1	1	yes	characteristic
150	C₂²×C₁₃	#435	1	1	yes	characteristic
151	C₂²×C₁₃	#435	1	1	yes	characteristic
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	4	1	no	—
154	C₄×C₁₃	—	4	1	no	—
155	C₄×C₁₃	—	8	1	no	—
156	C₄×C₁₃	#388	1	1	yes	characteristic
157	C₄×C₁₃	#432	1	1	yes	characteristic
158	C₄×C₁₃	#435	1	1	yes	characteristic
159	C₄×C₁₃	#436	1	1	yes	characteristic
160	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
161	#131	—	2	1	no	—
162	#131	—	2	1	no	—
163	#131	—	2	1	no	—
164	#131	—	2	1	no	—
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
167	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
171	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
172	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
173	M₃₂	—	2	1	no	—
174	M₃₂	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	D₄×C₁₃	1	2	yes	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
185	#104	—	2	1	no	—
186	#104	—	2	1	no	—
187	#103	—	2	1	no	—
188	#102	—	2	1	no	—
189	#97	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₂×C₁₃	—	2	1	no	—
194	C₂×C₁₃	—	2	1	no	—
195	C₂×C₁₃	—	2	1	no	—
196	C₂×C₁₃	—	8	1	no	—
197	C₂×C₁₃	#1924	1	1	yes	characteristic
198	#41	—	2	1	no	—
199	#41	—	2	1	no	—
200	#41	—	2	1	no	—
201	#41	—	2	1	no	—
202	C₂×Q₈	C₁₃×M₁₆	1	1	yes	characteristic
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	C₁₃×M₁₆	1	1	yes	characteristic
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
208	M₁₆	—	2	1	no	—
209	M₁₆	—	2	1	no	—
210	M₁₆	—	2	1	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
224	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
225	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
226	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	2	1	no	—
231	C₄²	C₁₃×M₁₆	1	1	yes	characteristic
232	C₄²	C₁₃×M₁₆	1	1	yes	characteristic
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₁₆	—	4	1	no	—
236	C₁₆	—	4	1	no	—
237	C₁₃	#13129	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³	—	2	1	no	—
239	C₂³	C₁₃×#131	1	1	yes	characteristic
240	Q₈	—	2	1	no	—
241	Q₈	—	2	1	no	—
242	D₄	—	2	1	no	—
243	D₄	—	2	1	no	—
244	D₄	—	4	1	no	—
245	D₄	—	4	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
257	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
258	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	C₁₃×#121	1	1	yes	characteristic
263	C₈	C₁₃×#123	1	1	yes	characteristic
264	C₈	C₁₃×#124	1	1	yes	characteristic
265	C₈	C₁₃×#126	1	1	yes	characteristic
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	C₁₃×#433	1	1	yes	characteristic
270	C₂²	C₁₃×#435	1	1	yes	characteristic
271	C₂²	C₁₃×#435	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	8	1	no	—
276	C₄	C₁₃×#388	1	1	yes	characteristic
277	C₄	C₁₃×#432	1	1	yes	characteristic
278	C₄	C₁₃×#435	1	1	yes	characteristic
279	C₄	C₁₃×#436	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₁₃×#1924	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃	#124
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1924
prime radical	C₂×C₁₃×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2150	C₂	semidirect
4. C₁₃×#2110	C₂	semidirect
6. C₁₃×#1378	C₂	semidirect
7. C₁₃×#1376	C₂	semidirect
56. #13129	C₁₃	direct (N × H)
96. #2150	C₂×C₁₃	semidirect
99. #2110	C₂×C₁₃	semidirect
101. #1378	C₂×C₁₃	semidirect
102. #1376	C₂×C₁₃	semidirect
237. C₁₃	#13129	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#2150	2
C₁₃×#2117	2
C₁₃×#2115	2
C₁₃×#2110	2
C₁₃×#1764	2
C₁₃×#1378	2
C₁₃×#1376	2
#13129	13

Subgroups — C13×#13129 #15699468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13129 #15699468