Subgroups — C₁₃×#10819 #15697158

Order 3328.

694 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×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1735	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#339	C₂	1	2	yes	maximal
5	C₁₃×#1729	C₂	1	2	yes	maximal
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#97	—	2	4	no	—
12	C₂×C₁₃×#97	C₄	1	4	yes	complemented
13	C₁₃×#341	C₄	1	4	yes	complemented
14	C₁₃×#340	C₄	1	4	yes	complemented
15	C₁₃×#339	—	2	4	no	—
16	C₁₃×#336	—	2	4	no	—
17	C₂³×C₄×C₁₃	—	2	1	no	—
18	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃	D₄	1	2	yes	—
35	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
36	C₁₃×#117	—	2	2	no	—
37	C₁₃×#117	—	2	2	no	—
38	C₁₃×#117	—	2	2	no	—
39	C₁₃×#117	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
42	C₂×C₄²×C₁₃	—	2	2	no	—
43	C₂×C₄²×C₁₃	—	2	1	no	—
44	C₂×C₄²×C₁₃	—	2	2	no	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₃	D₄	1	4	yes	—
47	C₁₃×#97	—	2	8	no	—
48	#10819	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	#41	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	4	no	—
52	C₂²×C₄×C₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₃	—	4	4	no	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	#41	1	2	yes	—
61	C₂²×C₄×C₁₃	#41	1	2	yes	—
62	C₂²×C₄×C₁₃	#41	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₁₃×M₁₆	—	2	4	no	—
65	C₁₃×M₁₆	—	2	4	no	—
66	C₁₃×M₁₆	—	4	2	no	—
67	C₁₃×M₁₆	—	4	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₂×C₈×C₁₃	C₄²	1	4	yes	—
80	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₂²⋊C₄	—	4	2	no	—
84	C₁₃×C₂²⋊C₄	—	4	2	no	—
85	C₄²×C₁₃	—	2	2	no	—
86	C₄²×C₁₃	—	2	4	no	—
87	C₄²×C₁₃	—	4	2	no	—
88	C₄²×C₁₃	—	4	2	no	—
89	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
90	C₂²×#97	C₂×C₁₃	1	1	yes	characteristic
91	C₂×#341	C₂×C₁₃	1	1	yes	characteristic
92	#1735	C₂×C₁₃	1	1	yes	characteristic
93	C₂×#339	C₂×C₁₃	1	2	yes	—
94	#1729	C₂×C₁₃	1	2	yes	—
95	C₂³×C₁₃	—	2	4	no	—
96	C₂³×C₁₃	#117	1	1	yes	characteristic
97	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
99	C₂³×C₁₃	#123	1	2	yes	—
100	C₂³×C₁₃	#126	1	2	yes	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	4	8	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	4	8	no	—
106	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#117	1	2	yes	—
112	C₂×C₄×C₁₃	#117	1	2	yes	—
113	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#121	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₃	#123	1	2	yes	—
121	C₂×C₄×C₁₃	#124	1	2	yes	—
122	C₂×C₄×C₁₃	#125	1	2	yes	—
123	C₂×C₄×C₁₃	#126	1	2	yes	—
124	C₈×C₁₃	—	2	4	no	—
125	C₈×C₁₃	—	2	4	no	—
126	C₈×C₁₃	—	4	4	no	—
127	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
128	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×#117	C₂²×C₁₃	1	2	yes	—
131	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×#97	—	2	4	no	—
133	C₂×#97	C₄×C₁₃	1	4	yes	complemented
134	#341	C₄×C₁₃	1	4	yes	complemented
135	#340	C₄×C₁₃	1	4	yes	complemented
136	#339	—	2	4	no	—
137	#336	—	2	4	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₃	#381	1	1	yes	characteristic
144	C₂²×C₁₃	#383	1	1	yes	characteristic
145	C₂²×C₁₃	#387	1	2	yes	—
146	C₂²×C₁₃	#388	1	2	yes	—
147	C₂²×C₁₃	#444	1	4	yes	—
148	C₄×C₁₃	—	4	4	no	—
149	C₄×C₁₃	—	8	4	no	—
150	C₄×C₁₃	#376	1	1	yes	characteristic
151	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₁₃	#380	1	1	yes	characteristic
153	C₄×C₁₃	#382	1	1	yes	characteristic
154	C₄×C₁₃	#387	1	2	yes	—
155	C₄×C₁₃	#388	1	2	yes	—
156	C₂³×C₄	—	2	1	no	—
157	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
170	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
171	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
172	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
173	C₂²×C₈	D₄×C₁₃	1	2	yes	—
174	C₂²×C₈	Q₈×C₁₃	1	2	yes	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	—	2	1	no	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
185	C₂×C₄²	D₄×C₁₃	1	4	yes	—
186	#97	—	2	8	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	#1434	1	1	yes	characteristic
191	C₂×C₁₃	#1773	1	2	yes	—
192	C₂×C₁₃	#1938	1	4	yes	—
193	C₂⁴	C₁₃×#41	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₁₃×#41	1	2	yes	—
205	C₂²×C₄	C₁₃×#41	1	2	yes	—
206	C₂²×C₄	C₁₃×#41	1	2	yes	—
207	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₄²×C₁₃	1	4	yes	—
224	C₂×C₈	C₁₃×C₄⋊C₄	1	4	yes	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	4	2	no	—
232	C₄²	—	4	2	no	—
233	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
234	C₁₃	#10819	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³	—	2	4	no	—
236	C₂³	C₁₃×#117	1	1	yes	characteristic
237	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
238	C₂³	C₄×Q₈×C₁₃	1	1	yes	characteristic
239	C₂³	C₁₃×#123	1	2	yes	—
240	C₂³	C₁₃×#126	1	2	yes	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
250	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
251	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
252	C₂×C₄	C₁₃×#117	1	2	yes	—
253	C₂×C₄	C₁₃×#117	1	2	yes	—
254	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
255	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
256	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
257	C₂×C₄	C₄×Q₈×C₁₃	1	1	yes	characteristic
258	C₂×C₄	C₁₃×#121	1	2	yes	—
259	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
260	C₂×C₄	C₁₃×#123	1	2	yes	—
261	C₂×C₄	C₁₃×#124	1	2	yes	—
262	C₂×C₄	C₁₃×#125	1	2	yes	—
263	C₂×C₄	C₁₃×#126	1	2	yes	—
264	C₈	—	2	4	no	—
265	C₈	—	2	4	no	—
266	C₈	—	4	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
272	C₂²	C₁₃×#381	1	1	yes	characteristic
273	C₂²	C₁₃×#383	1	1	yes	characteristic
274	C₂²	C₁₃×#387	1	2	yes	—
275	C₂²	C₁₃×#388	1	2	yes	—
276	C₂²	C₁₃×#444	1	4	yes	—
277	C₄	—	4	4	no	—
278	C₄	—	8	4	no	—
279	C₄	C₁₃×#376	1	1	yes	characteristic
280	C₄	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
281	C₄	C₁₃×#380	1	1	yes	characteristic
282	C₄	C₁₃×#382	1	1	yes	characteristic
283	C₄	C₁₃×#387	1	2	yes	—
284	C₄	C₁₃×#388	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	2	no	—
288	C₂	C₁₃×#1434	1	1	yes	characteristic
289	C₂	C₁₃×#1773	1	2	yes	—
290	C₂	C₁₃×#1938	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂⁴×C₁₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×C₁₃×#97	C₄	semidirect
13. C₁₃×#341	C₄	semidirect
14. C₁₃×#340	C₄	semidirect
48. #10819	C₁₃	direct (N × H)
133. C₂×#97	C₄×C₁₃	semidirect
134. #341	C₄×C₁₃	semidirect
135. #340	C₄×C₁₃	semidirect
234. C₁₃	#10819	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×#97	2
C₂×C₁₃×#341	2
C₁₃×#1735	2
C₂×C₁₃×#339	2
C₁₃×#1729	2
#10819	13

Subgroups — C13×#10819 #15697158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10819 #15697158