Subgroups — C₁₃×#12792 #15699131

Order 3328.

440 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2117	C₂	1	2	yes	maximal
5	C₁₃×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
11	C₁₃×#405	—	2	1	no	—
12	C₁₃×#405	—	2	1	no	—
13	C₁₃×#405	C₄	1	1	yes	characteristic
14	C₁₃×#405	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×M₁₆	—	2	1	no	—
16	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#364	—	2	1	no	—
20	C₁₃×#364	—	2	1	no	—
21	C₁₃×#364	—	2	1	no	—
22	C₁₃×#364	—	2	1	no	—
23	C₁₃×#350	—	2	4	no	—
24	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
25	C₁₃×#131	—	2	2	no	—
26	C₁₃×#131	—	2	1	no	—
27	C₁₃×#131	—	2	1	no	—
28	C₁₃×#131	C₈	1	4	yes	—
29	C₁₃×#131	D₄	1	2	yes	—
30	C₁₃×#131	Q₈	1	2	yes	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	—	2	1	no	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	—	2	1	no	—
36	C₂²×C₈×C₁₃	—	2	1	no	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₁₃	—	2	1	no	—
40	C₄×D₄×C₁₃	—	2	1	no	—
41	C₄×D₄×C₁₃	—	2	2	no	—
42	C₁₃×#117	—	2	1	no	—
43	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	—	2	1	no	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₁₃×M₃₂	—	2	2	no	—
47	C₁₃×M₃₂	—	2	2	no	—
48	C₁₃×M₃₂	—	2	2	no	—
49	C₁₃×M₃₂	—	2	2	no	—
50	C₂×C₁₃×C₁₆	—	2	2	no	—
51	C₂×C₁₃×C₁₆	—	2	2	no	—
52	C₁₃×#97	—	2	1	no	—
53	C₁₃×#97	—	2	1	no	—
54	C₁₃×#97	—	2	1	no	—
55	C₄×C₈×C₁₃	—	2	1	no	—
56	C₄×C₈×C₁₃	—	2	1	no	—
57	C₄×C₈×C₁₃	—	2	1	no	—
58	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	#12792	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₁₃×#41	—	2	1	no	—
64	C₁₃×#41	—	2	1	no	—
65	C₁₃×#41	C₄⋊C₄	1	2	yes	—
66	C₁₃×#41	C₂×C₈	1	2	yes	—
67	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
68	C₂×D₄×C₁₃	—	2	1	no	—
69	C₂×D₄×C₁₃	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₁₃×M₁₆	—	2	1	no	—
75	C₁₃×M₁₆	—	2	1	no	—
76	C₁₃×M₁₆	—	2	4	no	—
77	C₁₃×M₁₆	—	2	2	no	—
78	C₁₃×M₁₆	—	2	2	no	—
79	C₁₃×M₁₆	C₄⋊C₄	1	2	yes	—
80	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
89	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₂²⋊C₄	—	2	2	no	—
99	C₁₃×C₂²⋊C₄	M₁₆	1	2	yes	—
100	C₄²×C₁₃	—	2	2	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
105	C₁₃×C₁₆	—	4	2	no	—
106	C₁₃×C₁₆	—	4	2	no	—
107	C₄×#131	C₂×C₁₃	1	1	yes	characteristic
108	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#2117	C₂×C₁₃	1	2	yes	—
111	#1378	C₂×C₁₃	1	2	yes	complemented
112	C₂³×C₁₃	—	2	1	no	—
113	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
114	Q₈×C₁₃	—	2	1	no	—
115	Q₈×C₁₃	C₄⋊C₈	1	2	yes	—
116	D₄×C₁₃	—	2	2	no	—
117	D₄×C₁₃	—	2	2	no	—
118	D₄×C₁₃	—	2	1	no	—
119	D₄×C₁₃	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
137	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
138	C₈×C₁₃	#121	1	1	yes	characteristic
139	C₈×C₁₃	#121	1	1	yes	characteristic
140	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
141	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×#131	C₄×C₁₃	1	1	yes	characteristic
143	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
144	C₄×#41	C₄×C₁₃	1	1	yes	characteristic
145	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
146	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
147	#405	—	2	1	no	—
148	#405	—	2	1	no	—
149	#405	C₄×C₁₃	1	1	yes	characteristic
150	#405	C₂²×C₁₃	1	1	yes	characteristic
151	C₄×M₁₆	—	2	1	no	—
152	C₄×M₁₆	C₄×C₁₃	1	1	yes	characteristic
153	C₂×C₄×C₈	—	2	1	no	—
154	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
155	#364	—	2	1	no	—
156	#364	—	2	1	no	—
157	#364	—	2	1	no	—
158	#364	—	2	1	no	—
159	#350	—	2	4	no	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	4	1	no	—
162	C₂²×C₁₃	—	4	2	no	—
163	C₂²×C₁₃	#407	1	1	yes	characteristic
164	C₂²×C₁₃	#423	1	1	yes	characteristic
165	C₂²×C₁₃	#431	1	1	yes	characteristic
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	4	2	no	—
169	C₄×C₁₃	—	4	1	no	—
170	C₄×C₁₃	#380	1	1	yes	characteristic
171	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
172	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
173	C₄×C₁₃	#432	1	1	yes	characteristic
174	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
175	#131	—	2	2	no	—
176	#131	—	2	1	no	—
177	#131	—	2	1	no	—
178	#131	C₈×C₁₃	1	4	yes	—
179	#131	D₄×C₁₃	1	2	yes	—
180	#131	Q₈×C₁₃	1	2	yes	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
184	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₄×Q₈	—	2	1	no	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	2	no	—
192	#117	—	2	1	no	—
193	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
196	M₃₂	—	2	2	no	—
197	M₃₂	—	2	2	no	—
198	M₃₂	—	2	2	no	—
199	M₃₂	—	2	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	2	no	—
202	#97	—	2	1	no	—
203	#97	—	2	1	no	—
204	#97	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
209	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
210	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
211	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
212	C₂×C₁₃	—	2	1	no	—
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	4	2	no	—
216	C₂×C₁₃	#1797	1	1	yes	characteristic
217	#41	—	2	1	no	—
218	#41	—	2	1	no	—
219	#41	C₁₃×C₄⋊C₄	1	2	yes	—
220	#41	C₂×C₈×C₁₃	1	2	yes	—
221	C₂×Q₈	C₁₃×M₁₆	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₁₃×M₁₆	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
228	M₁₆	—	2	1	no	—
229	M₁₆	—	2	1	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	C₁₃×C₄⋊C₄	1	2	yes	—
234	M₁₆	C₂×C₈×C₁₃	1	2	yes	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
243	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
248	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
249	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	C₁₃×M₁₆	1	2	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
258	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
259	C₁₆	—	4	2	no	—
260	C₁₆	—	4	2	no	—
261	C₁₃	#12792	1	1	yes	characteristic, direct product factor, Sylow
262	C₂³	—	2	1	no	—
263	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	C₁₃×C₄⋊C₈	1	2	yes	—
266	D₄	—	2	2	no	—
267	D₄	—	2	2	no	—
268	D₄	—	2	1	no	—
269	D₄	C₁₃×C₄⋊C₈	1	2	yes	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
282	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
283	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
287	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
288	C₈	C₁₃×#121	1	1	yes	characteristic
289	C₈	C₁₃×#121	1	1	yes	characteristic
290	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
291	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₁₃×#407	1	1	yes	characteristic
296	C₂²	C₁₃×#423	1	1	yes	characteristic
297	C₂²	C₁₃×#431	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	1	no	—
302	C₄	C₁₃×#380	1	1	yes	characteristic
303	C₄	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
304	C₄	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
305	C₄	C₁₃×#432	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	C₁₃×#1797	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃	#121
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1797
prime radical	C₂×C₁₃×#41	C₂×C₄
LCS	G ▷ C₂×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
2. C₂×C₁₃×#364	C₂	semidirect
3. C₂×C₁₃×#364	C₂	semidirect
5. C₁₃×#1378	C₂	semidirect
62. #12792	C₁₃	direct (N × H)
108. C₂×#364	C₂×C₁₃	semidirect
109. C₂×#364	C₂×C₁₃	semidirect
111. #1378	C₂×C₁₃	semidirect
261. C₁₃	#12792	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#131	2
C₂×C₁₃×#364	2
C₁₃×#2117	2
C₁₃×#1378	2
#12792	13

Subgroups — C13×#12792 #15699131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12792 #15699131