Subgroups — C₁₃×#10812 #15697151

Order 3328.

612 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	C₁₃×#2873	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#344	C₂	1	2	yes	maximal
3	C₁₃×#1738	C₂	1	2	yes	maximal
4	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₁₃×#405	—	2	2	no	—
12	C₁₃×#405	C₄	1	4	yes	—
13	C₄×C₁₃×M₁₆	—	2	2	no	—
14	C₁₃×#344	—	2	4	no	—
15	C₁₃×#341	C₄	1	8	yes	complemented
16	C₁₃×#338	—	2	2	no	—
17	C₁₃×#338	—	2	2	no	—
18	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	—	2	1	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	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	2	no	—
30	C₂×C₁₃×M₁₆	—	2	2	no	—
31	C₂×C₁₃×M₁₆	—	2	2	no	—
32	C₂×C₁₃×M₁₆	—	2	2	no	—
33	C₂×C₁₃×M₁₆	—	2	2	no	—
34	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
35	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
37	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
38	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
39	C₂²×C₈×C₁₃	—	2	1	no	—
40	C₂²×C₈×C₁₃	—	2	2	no	—
41	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
43	C₁₃×#117	—	2	1	no	—
44	C₁₃×#117	—	2	1	no	—
45	C₁₃×#117	C₂×C₄	1	2	yes	—
46	C₁₃×#117	D₄	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₁₃×C₂²⋊C₄	Q₈	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	1	no	—
50	C₁₃×#97	—	2	4	no	—
51	C₄×C₈×C₁₃	—	2	4	no	—
52	#10812	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₁₃×M₁₆	—	2	2	no	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	4	2	no	—
72	C₁₃×M₁₆	—	4	2	no	—
73	C₁₃×M₁₆	—	4	2	no	—
74	C₁₃×M₁₆	—	4	2	no	—
75	C₁₃×M₁₆	—	4	2	no	—
76	C₁₃×M₁₆	—	4	2	no	—
77	C₁₃×M₁₆	C₄²	1	4	yes	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	4	2	no	—
87	C₂×C₈×C₁₃	—	4	2	no	—
88	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₂²⋊C₄	—	2	2	no	—
92	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
93	C₄²×C₁₃	—	2	2	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	#2873	C₂×C₁₃	1	1	yes	characteristic
96	C₂×#344	C₂×C₁₃	1	2	yes	—
97	#1738	C₂×C₁₃	1	2	yes	—
98	C₂×#338	C₂×C₁₃	1	1	yes	characteristic
99	C₂×#338	C₂×C₁₃	1	1	yes	characteristic
100	C₂³×C₁₃	—	8	1	no	—
101	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
103	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₁₃	#125	1	2	yes	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	8	2	no	—
108	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#117	1	2	yes	—
113	C₂×C₄×C₁₃	#117	1	2	yes	—
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#123	1	2	yes	—
122	C₂×C₄×C₁₃	#123	1	2	yes	—
123	C₂×C₄×C₁₃	#124	1	2	yes	—
124	C₂×C₄×C₁₃	#126	1	2	yes	—
125	C₂×C₄×C₁₃	#126	1	2	yes	—
126	C₈×C₁₃	—	2	4	no	—
127	C₈×C₁₃	—	2	4	no	—
128	C₈×C₁₃	—	4	4	no	—
129	C₈×C₁₃	—	4	4	no	—
130	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
131	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
132	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
133	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
135	#405	—	2	2	no	—
136	#405	C₄×C₁₃	1	4	yes	—
137	C₄×M₁₆	—	2	2	no	—
138	#344	—	2	4	no	—
139	#341	C₄×C₁₃	1	8	yes	complemented
140	#338	—	2	2	no	—
141	#338	—	2	2	no	—
142	C₂²×C₁₃	—	4	1	no	—
143	C₂²×C₁₃	—	4	1	no	—
144	C₂²×C₁₃	—	4	2	no	—
145	C₂²×C₁₃	—	4	1	no	—
146	C₂²×C₁₃	—	4	2	no	—
147	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₁₃	#380	1	1	yes	characteristic
149	C₂²×C₁₃	#382	1	1	yes	characteristic
150	C₂²×C₁₃	#387	1	2	yes	—
151	C₂²×C₁₃	#387	1	2	yes	—
152	C₄×C₁₃	—	4	4	no	—
153	C₄×C₁₃	#376	1	1	yes	characteristic
154	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₁₃	#381	1	1	yes	characteristic
156	C₄×C₁₃	#383	1	1	yes	characteristic
157	C₄×C₁₃	#388	1	2	yes	—
158	C₄×C₁₃	#388	1	2	yes	—
159	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
160	C₂×M₁₆	—	2	1	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
176	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
177	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
178	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
179	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
183	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
184	#117	—	2	1	no	—
185	#117	—	2	1	no	—
186	#117	C₂×C₄×C₁₃	1	2	yes	—
187	#117	D₄×C₁₃	1	2	yes	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₂²⋊C₄	Q₈×C₁₃	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	—	2	4	no	—
193	C₂×C₁₃	—	2	1	no	—
194	C₂×C₁₃	—	2	1	no	—
195	C₂×C₁₃	—	2	1	no	—
196	C₂×C₁₃	—	2	2	no	—
197	C₂×C₁₃	—	2	2	no	—
198	C₂×C₁₃	#1434	1	1	yes	characteristic
199	C₂⁴	C₂×Q₈×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
207	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
208	C₂²×C₄	C₁₃×#41	1	2	yes	—
209	C₂²×C₄	C₁₃×#41	1	2	yes	—
210	C₂²×C₄	C₁₃×#41	1	2	yes	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	C₄²×C₁₃	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	C₁₃×C₄⋊C₄	1	4	yes	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₁₃	#10812	1	1	yes	characteristic, direct product factor, Sylow
242	C₂³	—	8	1	no	—
243	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
244	C₂³	C₄×Q₈×C₁₃	1	2	yes	—
245	C₂³	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂³	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂³	C₁₃×#125	1	2	yes	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
253	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
254	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
255	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
256	C₂×C₄	C₁₃×#117	1	2	yes	—
257	C₂×C₄	C₁₃×#117	1	2	yes	—
258	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
259	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
261	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
262	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
263	C₂×C₄	C₁₃×#123	1	2	yes	—
264	C₂×C₄	C₁₃×#123	1	2	yes	—
265	C₂×C₄	C₁₃×#124	1	2	yes	—
266	C₂×C₄	C₁₃×#126	1	2	yes	—
267	C₂×C₄	C₁₃×#126	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	2	4	no	—
270	C₈	—	4	4	no	—
271	C₈	—	4	4	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
278	C₂²	C₁₃×#380	1	1	yes	characteristic
279	C₂²	C₁₃×#382	1	1	yes	characteristic
280	C₂²	C₁₃×#387	1	2	yes	—
281	C₂²	C₁₃×#387	1	2	yes	—
282	C₄	—	4	4	no	—
283	C₄	C₁₃×#376	1	1	yes	characteristic
284	C₄	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
285	C₄	C₁₃×#381	1	1	yes	characteristic
286	C₄	C₁₃×#383	1	1	yes	characteristic
287	C₄	C₁₃×#388	1	2	yes	—
288	C₄	C₁₃×#388	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	C₁₃×#1434	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂×C₁₃	#1434
prime radical	C₂⁴×C₁₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂²×C₁₃×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₁₃×#341	C₄	semidirect
52. #10812	C₁₃	direct (N × H)
139. #341	C₄×C₁₃	semidirect
241. C₁₃	#10812	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×#2873	2
C₂×C₁₃×#344	2
C₁₃×#1738	2
C₂×C₁₃×#338	2
#10812	13

Subgroups — C13×#10812 #15697151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10812 #15697151