Subgroups — C₁₃×#13246 #15699585

Order 3328.

422 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1978	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×SD₃₂	—	2	1	no	—
11	C₂×C₁₃×SD₃₂	—	2	1	no	—
12	C₂×C₁₃×SD₃₂	—	2	2	no	—
13	C₂×C₁₃×SD₃₂	C₄	1	4	yes	complemented
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
18	C₁₃×#419	—	2	1	no	—
19	C₁₃×#418	—	2	1	no	—
20	C₂×C₁₃×#101	—	2	1	no	—
21	C₂×C₁₃×#100	—	2	1	no	—
22	C₁₃×#405	C₂²	1	1	yes	characteristic
23	C₁₃×#368	—	2	2	no	—
24	C₁₃×#362	—	2	2	no	—
25	C₁₃×#361	—	2	2	no	—
26	C₁₃×#346	—	2	1	no	—
27	C₄×C₁₃×C₁₆	—	2	1	no	—
28	C₂²×Q₈×C₁₃	—	2	1	no	—
29	C₂²×D₄×C₁₃	—	2	1	no	—
30	C₂×C₁₃×Q₁₆	—	2	1	no	—
31	C₂×C₁₃×Q₁₆	—	2	1	no	—
32	C₂×C₁₃×Q₁₆	—	2	1	no	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	—	2	1	no	—
39	C₂×C₁₃×D₈	—	2	1	no	—
40	C₂×C₁₃×D₈	—	2	1	no	—
41	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
42	C₂×C₁₃×M₁₆	—	2	1	no	—
43	C₂×C₁₃×M₁₆	—	2	1	no	—
44	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×#117	D₄	1	1	yes	characteristic
47	C₁₃×SD₃₂	—	2	4	no	—
48	C₁₃×SD₃₂	—	2	4	no	—
49	C₁₃×M₃₂	—	2	1	no	—
50	C₁₃×M₃₂	D₄	1	2	yes	—
51	C₂×C₁₃×C₁₆	—	2	1	no	—
52	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
53	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
54	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
55	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
56	C₁₃×#108	—	2	1	no	—
57	C₁₃×#108	—	2	1	no	—
58	C₁₃×#108	—	2	1	no	—
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#103	—	4	2	no	—
61	C₁₃×#102	—	4	2	no	—
62	C₁₃×#101	—	4	2	no	—
63	C₁₃×#100	—	4	2	no	—
64	C₁₃×#97	—	2	1	no	—
65	C₄×C₈×C₁₃	—	2	1	no	—
66	#13246	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂⁴×C₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	2	2	no	—
69	C₂×Q₈×C₁₃	—	4	1	no	—
70	C₂×Q₈×C₁₃	—	4	1	no	—
71	C₂×Q₈×C₁₃	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	4	1	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂×D₄×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	4	1	no	—
78	C₂×D₄×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×Q₁₆	—	2	2	no	—
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₁₃×Q₁₆	—	4	1	no	—
85	C₁₃×Q₁₆	—	4	1	no	—
86	C₁₃×D₈	—	2	2	no	—
87	C₁₃×D₈	—	2	2	no	—
88	C₁₃×D₈	—	4	1	no	—
89	C₁₃×D₈	—	4	1	no	—
90	C₁₃×M₁₆	—	2	1	no	—
91	C₁₃×M₁₆	—	4	1	no	—
92	C₁₃×M₁₆	—	4	1	no	—
93	C₁₃×M₁₆	—	4	1	no	—
94	C₁₃×M₁₆	—	4	1	no	—
95	C₁₃×M₁₆	D₈	1	2	yes	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₂×C₈×C₁₃	—	4	1	no	—
99	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
106	C₁₃×C₂²⋊C₄	—	2	1	no	—
107	C₄²×C₁₃	—	2	1	no	—
108	C₁₃×C₁₆	—	2	2	no	—
109	C₁₃×C₁₆	C₂²⋊C₄	1	2	yes	—
110	C₁₃×C₁₆	C₂²⋊C₄	1	2	yes	—
111	C₂²×SD₃₂	C₂×C₁₃	1	1	yes	characteristic
112	C₂×#368	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂×#361	C₂×C₁₃	1	1	yes	characteristic
115	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#1979	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#1978	C₂×C₁₃	1	1	yes	characteristic
118	C₂³×C₁₃	—	4	2	no	—
119	C₂³×C₁₃	—	8	1	no	—
120	C₂³×C₁₃	—	8	1	no	—
121	C₂³×C₁₃	—	16	2	no	—
122	C₂³×C₁₃	#135	1	1	yes	characteristic
123	Q₈×C₁₃	—	4	2	no	—
124	Q₈×C₁₃	—	4	2	no	—
125	Q₈×C₁₃	—	8	1	no	—
126	Q₈×C₁₃	—	8	1	no	—
127	D₄×C₁₃	—	4	2	no	—
128	D₄×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	8	1	no	—
130	D₄×C₁₃	—	8	1	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	8	1	no	—
134	C₂×C₄×C₁₃	—	8	1	no	—
135	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	—	8	1	no	—
143	C₈×C₁₃	—	8	1	no	—
144	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
146	C₈×C₁₃	#121	1	1	yes	characteristic
147	C₈×C₁₃	#124	1	1	yes	characteristic
148	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
149	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×SD₃₂	—	2	1	no	—
151	C₂×SD₃₂	—	2	1	no	—
152	C₂×SD₃₂	—	2	2	no	—
153	C₂×SD₃₂	C₄×C₁₃	1	4	yes	complemented
154	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
155	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
158	#419	—	2	1	no	—
159	#418	—	2	1	no	—
160	C₂×#101	—	2	1	no	—
161	C₂×#100	—	2	1	no	—
162	#405	C₂²×C₁₃	1	1	yes	characteristic
163	#368	—	2	2	no	—
164	#362	—	2	2	no	—
165	#361	—	2	2	no	—
166	#346	—	2	1	no	—
167	C₄×C₁₆	—	2	1	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	8	2	no	—
170	C₂²×C₁₃	—	16	2	no	—
171	C₂²×C₁₃	—	16	2	no	—
172	C₂²×C₁₃	—	16	2	no	—
173	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
174	C₂²×C₁₃	#466	1	1	yes	characteristic
175	C₂²×C₁₃	#495	1	1	yes	characteristic
176	C₄×C₁₃	—	4	2	no	—
177	C₄×C₁₃	—	8	2	no	—
178	C₄×C₁₃	#390	1	1	yes	characteristic
179	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
180	C₄×C₁₃	#466	1	1	yes	characteristic
181	C₄×C₁₃	#493	1	1	yes	characteristic
182	C₂²×Q₈	—	2	1	no	—
183	C₂²×D₄	—	2	1	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
200	#117	D₄×C₁₃	1	1	yes	characteristic
201	SD₃₂	—	2	4	no	—
202	SD₃₂	—	2	4	no	—
203	M₃₂	—	2	1	no	—
204	M₃₂	D₄×C₁₃	1	2	yes	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
207	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
210	#108	—	2	1	no	—
211	#108	—	2	1	no	—
212	#108	—	2	1	no	—
213	#108	—	2	1	no	—
214	#103	—	4	2	no	—
215	#102	—	4	2	no	—
216	#101	—	4	2	no	—
217	#100	—	4	2	no	—
218	#97	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₂×C₁₃	—	2	1	no	—
221	C₂×C₁₃	—	2	1	no	—
222	C₂×C₁₃	—	16	2	no	—
223	C₂×C₁₃	#1970	1	1	yes	characteristic
224	C₂×C₁₃	#2223	1	2	yes	—
225	C₂⁴	—	4	1	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	4	1	no	—
243	Q₁₆	—	4	1	no	—
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	D₈	—	4	1	no	—
247	D₈	—	4	1	no	—
248	M₁₆	—	2	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	C₁₃×D₈	1	2	yes	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
262	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
263	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₁₆	—	2	2	no	—
267	C₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
268	C₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
269	C₁₃	#13246	1	1	yes	characteristic, direct product factor, Sylow
270	C₂³	—	4	2	no	—
271	C₂³	—	8	1	no	—
272	C₂³	—	8	1	no	—
273	C₂³	—	16	2	no	—
274	C₂³	C₁₃×#135	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	8	1	no	—
278	Q₈	—	8	1	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	8	1	no	—
282	D₄	—	8	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
288	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
289	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
290	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
291	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
292	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	8	1	no	—
295	C₈	—	8	1	no	—
296	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
297	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
298	C₈	C₁₃×#121	1	1	yes	characteristic
299	C₈	C₁₃×#124	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	16	2	no	—
303	C₂²	—	16	2	no	—
304	C₂²	—	16	2	no	—
305	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
306	C₂²	C₁₃×#466	1	1	yes	characteristic
307	C₂²	C₁₃×#495	1	1	yes	characteristic
308	C₄	—	4	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₁₃×#390	1	1	yes	characteristic
311	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
312	C₄	C₁₃×#466	1	1	yes	characteristic
313	C₄	C₁₃×#493	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	16	2	no	—
317	C₂	C₁₃×#1970	1	1	yes	characteristic
318	C₂	C₁₃×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#368	C₂	semidirect
3. C₂×C₁₃×#362	C₂	semidirect
5. C₁₃×#2108	C₂	semidirect
6. C₁₃×#1979	C₂	semidirect
13. C₂×C₁₃×SD₃₂	C₄	semidirect
66. #13246	C₁₃	direct (N × H)
112. C₂×#368	C₂×C₁₃	semidirect
113. C₂×#362	C₂×C₁₃	semidirect
115. #2108	C₂×C₁₃	semidirect
116. #1979	C₂×C₁₃	semidirect
153. C₂×SD₃₂	C₄×C₁₃	semidirect
269. C₁₃	#13246	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×SD₃₂	2
C₂×C₁₃×#368	2
C₂×C₁₃×#362	2
C₂×C₁₃×#361	2
C₁₃×#2108	2
C₁₃×#1979	2
C₁₃×#1978	2
#13246	13

Subgroups — C13×#13246 #15699585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13246 #15699585