Subgroups — C₁₃×#10847 #15697186

Order 3328.

898 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#423	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#343	C₂	1	4	yes	maximal
4	C₂²×C₁₃×M₁₆	C₂²	1	4	yes	—
5	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#423	—	2	4	no	—
9	C₁₃×#423	C₄	1	8	yes	complemented
10	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
11	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
12	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
13	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
14	C₂×C₁₃×#95	C₄	1	2	yes	—
15	C₁₃×#343	—	2	8	no	—
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	4	no	—
21	C₂×C₁₃×M₁₆	—	2	8	no	—
22	C₂×C₁₃×M₁₆	—	2	8	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₃	—	2	4	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
27	C₂×C₁₃×C₄⋊C₄	D₄	1	4	yes	—
28	C₂×C₁₃×C₄⋊C₄	Q₈	1	4	yes	—
29	C₂×C₄²×C₁₃	—	2	2	no	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	—	4	4	no	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	D₄	1	2	yes	—
36	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
37	C₁₃×C₄⋊C₈	—	2	8	no	—
38	C₁₃×C₄⋊C₈	—	2	8	no	—
39	C₁₃×#95	—	2	4	no	—
40	#10847	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	—	2	4	no	—
46	C₂²×C₄×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	—	4	4	no	—
48	C₂²×C₄×C₁₃	—	4	4	no	—
49	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
50	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	4	yes	—
64	C₁₃×M₁₆	—	2	16	no	—
65	C₁₃×M₁₆	—	4	4	no	—
66	C₁₃×M₁₆	—	4	4	no	—
67	C₂×C₈×C₁₃	—	2	8	no	—
68	C₂×C₈×C₁₃	—	4	8	no	—
69	C₁₃×C₄⋊C₄	—	2	8	no	—
70	C₁₃×C₄⋊C₄	—	2	8	no	—
71	C₄²×C₁₃	—	4	4	no	—
72	C₄²×C₁₃	—	4	4	no	—
73	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
74	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
75	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
76	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
77	C₂×#423	C₂×C₁₃	1	2	yes	—
78	C₂×C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
79	C₂×#343	C₂×C₁₃	1	4	yes	—
80	C₂³×C₁₃	—	2	1	no	—
81	C₂³×C₁₃	—	2	1	no	—
82	C₂³×C₁₃	—	2	2	no	—
83	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
84	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₁₃	#117	1	2	yes	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	8	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	8	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	4	8	no	—
100	C₂×C₄×C₁₃	#95	1	2	yes	—
101	C₂×C₄×C₁₃	#95	1	4	yes	—
102	C₂×C₄×C₁₃	#95	1	2	yes	—
103	C₂×C₄×C₁₃	#100	1	2	yes	—
104	C₂×C₄×C₁₃	#101	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
108	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₁₃	#121	1	4	yes	—
110	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
111	C₂×C₄×C₁₃	#124	1	2	yes	—
112	C₂×C₄×C₁₃	#125	1	2	yes	—
113	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
116	C₈×C₁₃	—	4	8	no	—
117	C₂²×M₁₆	C₂²×C₁₃	1	4	yes	—
118	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
119	C₂²×C₄²	C₄×C₁₃	1	2	yes	—
120	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
121	#423	—	2	4	no	—
122	#423	C₄×C₁₃	1	8	yes	complemented
123	C₂×C₄⋊C₈	—	2	2	no	—
124	C₂×C₄⋊C₈	—	2	2	no	—
125	C₄×C₄⋊C₄	—	2	2	no	—
126	C₄×C₄⋊C₄	—	2	2	no	—
127	C₂×#95	C₄×C₁₃	1	2	yes	—
128	#343	—	2	8	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
138	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
139	C₂²×C₁₃	#380	1	2	yes	—
140	C₂²×C₁₃	#382	1	1	yes	characteristic
141	C₂²×C₁₃	#384	1	1	yes	characteristic
142	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
143	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
144	C₂²×C₁₃	#413	1	2	yes	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	2	4	no	—
147	C₄×C₁₃	—	4	8	no	—
148	C₄×C₁₃	#343	1	4	yes	complemented
149	C₄×C₁₃	#389	1	4	yes	—
150	C₄×C₁₃	#390	1	2	yes	—
151	C₄×C₁₃	#391	1	2	yes	—
152	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
153	C₂³×C₄	C₂×C₄×C₁₃	1	4	yes	—
154	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
155	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	8	no	—
158	C₂×M₁₆	—	2	8	no	—
159	C₂×M₁₆	C₂×C₄×C₁₃	1	8	yes	—
160	C₂²×C₈	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	D₄×C₁₃	1	4	yes	—
164	C₂×C₄⋊C₄	Q₈×C₁₃	1	4	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	4	4	no	—
168	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
169	C₂×C₄²	D₄×C₁₃	1	2	yes	—
170	C₂×C₄²	D₄×C₁₃	1	2	yes	—
171	C₂×C₄²	D₄×C₁₃	1	2	yes	—
172	C₂×C₄²	Q₈×C₁₃	1	2	yes	—
173	C₄⋊C₈	—	2	8	no	—
174	C₄⋊C₈	—	2	8	no	—
175	#95	—	2	4	no	—
176	C₂×C₁₃	—	2	2	no	—
177	C₂×C₁₃	—	2	1	no	—
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#343	1	1	yes	characteristic
181	C₂×C₁₃	#1742	1	1	yes	characteristic
182	C₂×C₁₃	#1800	1	2	yes	—
183	C₂×C₁₃	#1801	1	2	yes	—
184	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
193	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
194	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
195	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
198	C₂²×C₄	C₁₃×C₄⋊C₄	1	8	yes	—
199	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
204	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
206	C₂²×C₄	C₁₃×#41	1	4	yes	—
207	M₁₆	—	2	16	no	—
208	M₁₆	—	4	4	no	—
209	M₁₆	—	4	4	no	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	4	8	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄²	—	4	4	no	—
215	C₄²	—	4	4	no	—
216	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
217	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
218	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
219	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
220	C₁₃	#10847	1	1	yes	characteristic, direct product factor, Sylow
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	2	no	—
224	C₂³	C₂×C₄²×C₁₃	1	1	yes	characteristic
225	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
226	C₂³	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
227	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
228	C₂³	C₁₃×#117	1	2	yes	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	C₁₃×#95	1	2	yes	—
242	C₂×C₄	C₁₃×#95	1	4	yes	—
243	C₂×C₄	C₁₃×#95	1	2	yes	—
244	C₂×C₄	C₁₃×#100	1	2	yes	—
245	C₂×C₄	C₁₃×#101	1	2	yes	—
246	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
247	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
248	C₂×C₄	C₄×D₄×C₁₃	1	4	yes	—
249	C₂×C₄	C₄×Q₈×C₁₃	1	4	yes	—
250	C₂×C₄	C₁₃×#121	1	4	yes	—
251	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
252	C₂×C₄	C₁₃×#124	1	2	yes	—
253	C₂×C₄	C₁₃×#125	1	2	yes	—
254	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
255	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
256	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
257	C₈	—	4	8	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	C₂×C₁₃×#95	1	1	yes	characteristic
267	C₂²	C₄×C₁₃×C₄⋊C₄	1	2	yes	—
268	C₂²	C₁₃×#380	1	2	yes	—
269	C₂²	C₁₃×#382	1	1	yes	characteristic
270	C₂²	C₁₃×#384	1	1	yes	characteristic
271	C₂²	C₂×C₁₃×#100	1	1	yes	characteristic
272	C₂²	C₂×C₁₃×#101	1	1	yes	characteristic
273	C₂²	C₁₃×#413	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	4	8	no	—
277	C₄	C₁₃×#343	1	4	yes	complemented
278	C₄	C₁₃×#389	1	4	yes	—
279	C₄	C₁₃×#390	1	2	yes	—
280	C₄	C₁₃×#391	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	C₁₃×C₄⋊#95	1	1	yes	characteristic
285	C₂	C₂×C₁₃×#343	1	1	yes	characteristic
286	C₂	C₁₃×#1742	1	1	yes	characteristic
287	C₂	C₁₃×#1800	1	2	yes	—
288	C₂	C₁₃×#1801	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
9. C₁₃×#423	C₄	semidirect
40. #10847	C₁₃	direct (N × H)
122. #423	C₄×C₁₃	semidirect
148. C₄×C₁₃	#343	semidirect
220. C₁₃	#10847	direct (N × H)
277. C₄	C₁₃×#343	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#423	2
C₂×C₄×C₁₃×C₄⋊C₄	2
C₂×C₁₃×#343	2
#10847	13

Subgroups — C13×#10847 #15697186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10847 #15697186