Subgroups — C₁₃×#13369 #15699708

Order 3328.

378 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2140	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1978	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₁₃×#544	—	2	1	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#501	—	2	2	no	—
13	C₁₃×#500	—	2	1	no	—
14	C₁₃×#499	—	2	1	no	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₁₃×#428	—	2	1	no	—
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
19	C₁₃×#418	—	2	1	no	—
20	C₂×C₁₃×#100	—	2	1	no	—
21	C₁₃×#405	C₂²	1	1	yes	characteristic
22	C₁₃×#357	—	2	2	no	—
23	C₁₃×#357	—	2	1	no	—
24	C₁₃×#357	—	2	1	no	—
25	C₂²×D₄×C₁₃	—	2	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	1	no	—
29	C₂×C₁₃×D₈	—	2	1	no	—
30	C₂×C₁₃×D₈	—	2	1	no	—
31	C₂×C₁₃×D₈	Q₈	1	2	yes	complemented
32	C₂×C₁₃×M₁₆	—	2	1	no	—
33	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×C₄⋊Q₈	—	2	2	no	—
36	C₁₃×C₄⋊Q₈	—	4	1	no	—
37	C₁₃×#125	—	2	2	no	—
38	C₁₃×#125	—	4	1	no	—
39	C₁₃×C₂²⋊Q₈	—	4	2	no	—
40	C₁₃×#117	—	2	1	no	—
41	C₁₃×#117	D₄	1	1	yes	characteristic
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₁₃×M₃₂	—	2	1	no	—
44	C₁₃×M₃₂	—	2	1	no	—
45	C₂×C₁₃×C₁₆	—	2	1	no	—
46	C₂×C₁₃×C₁₆	—	2	1	no	—
47	C₂×C₁₃×C₁₆	—	2	1	no	—
48	C₂×C₁₃×C₁₆	—	2	1	no	—
49	C₁₃×#108	—	2	1	no	—
50	C₁₃×#108	—	2	1	no	—
51	C₁₃×#107	—	2	2	no	—
52	C₁₃×#107	—	2	1	no	—
53	C₁₃×#107	D₄	1	2	yes	—
54	C₁₃×#107	D₄	1	1	yes	characteristic
55	C₁₃×#107	D₄	1	1	yes	characteristic
56	C₁₃×#106	—	2	2	no	—
57	C₁₃×#106	—	2	1	no	—
58	C₁₃×#102	—	4	1	no	—
59	C₁₃×#102	—	4	1	no	—
60	C₁₃×#100	—	4	2	no	—
61	C₁₃×#97	—	2	1	no	—
62	C₄×C₈×C₁₃	—	2	1	no	—
63	#13369	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂⁴×C₁₃	—	4	1	no	—
65	C₂×Q₈×C₁₃	—	4	2	no	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	4	1	no	—
68	C₂×D₄×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	4	1	no	—
70	C₂×D₄×C₁₃	—	4	1	no	—
71	C₂×D₄×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₁₃×D₈	—	2	2	no	—
75	C₁₃×D₈	—	2	2	no	—
76	C₁₃×D₈	—	4	1	no	—
77	C₁₃×D₈	—	4	1	no	—
78	C₁₃×M₁₆	—	2	1	no	—
79	C₁₃×M₁₆	—	4	1	no	—
80	C₁₃×M₁₆	—	4	1	no	—
81	C₁₃×M₁₆	SD₁₆	1	2	yes	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	1	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	1	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
99	C₁₃×C₂²⋊C₄	—	2	1	no	—
100	C₁₃×C₂²⋊C₄	—	4	1	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	4	1	no	—
103	C₁₃×C₁₆	—	4	1	no	—
104	C₁₃×C₁₆	—	4	1	no	—
105	#3164	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#2140	C₂×C₁₃	1	1	yes	characteristic
107	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
108	#1978	C₂×C₁₃	1	1	yes	characteristic
109	#1387	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#1382	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#1379	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂³×C₁₃	—	4	2	no	—
113	C₂³×C₁₃	—	8	1	no	—
114	C₂³×C₁₃	—	8	1	no	—
115	C₂³×C₁₃	—	16	1	no	—
116	C₂³×C₁₃	—	16	1	no	—
117	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
118	Q₈×C₁₃	—	16	2	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	8	1	no	—
122	D₄×C₁₃	—	8	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	16	1	no	—
129	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	8	1	no	—
137	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
139	C₈×C₁₃	#124	1	1	yes	characteristic
140	C₈×C₁₃	#124	1	1	yes	characteristic
141	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
142	#544	—	2	1	no	—
143	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
144	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
145	#501	—	2	2	no	—
146	#500	—	2	1	no	—
147	#499	—	2	1	no	—
148	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
149	#428	—	2	1	no	—
150	#427	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
152	#418	—	2	1	no	—
153	C₂×#100	—	2	1	no	—
154	#405	C₂²×C₁₃	1	1	yes	characteristic
155	#357	—	2	2	no	—
156	#357	—	2	1	no	—
157	#357	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	8	2	no	—
161	C₂²×C₁₃	—	16	2	no	—
162	C₂²×C₁₃	—	16	2	no	—
163	C₂²×C₁₃	—	16	1	no	—
164	C₂²×C₁₃	—	16	1	no	—
165	C₂²×C₁₃	#447	1	1	yes	characteristic
166	C₂²×C₁₃	#476	1	1	yes	characteristic
167	C₂²×C₁₃	#489	1	1	yes	characteristic
168	C₄×C₁₃	—	4	1	no	—
169	C₄×C₁₃	—	4	1	no	—
170	C₄×C₁₃	—	8	2	no	—
171	C₄×C₁₃	—	8	1	no	—
172	C₄×C₁₃	—	8	1	no	—
173	C₄×C₁₃	#393	1	1	yes	characteristic
174	C₄×C₁₃	#448	1	1	yes	characteristic
175	C₄×C₁₃	#474	1	1	yes	characteristic
176	C₄×C₁₃	#486	1	1	yes	characteristic
177	C₂²×D₄	—	2	1	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	—	2	1	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	—	2	1	no	—
183	C₂×D₈	Q₈×C₁₃	1	2	yes	complemented
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	—	4	1	no	—
189	#125	—	2	2	no	—
190	#125	—	4	1	no	—
191	C₂²⋊Q₈	—	4	2	no	—
192	#117	—	2	1	no	—
193	#117	D₄×C₁₃	1	1	yes	characteristic
194	C₂×C₄⋊C₄	—	2	1	no	—
195	M₃₂	—	2	1	no	—
196	M₃₂	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	#108	—	2	1	no	—
202	#108	—	2	1	no	—
203	#107	—	2	2	no	—
204	#107	—	2	1	no	—
205	#107	D₄×C₁₃	1	2	yes	—
206	#107	D₄×C₁₃	1	1	yes	characteristic
207	#107	D₄×C₁₃	1	1	yes	characteristic
208	#106	—	2	2	no	—
209	#106	—	2	1	no	—
210	#102	—	4	1	no	—
211	#102	—	4	1	no	—
212	#100	—	4	2	no	—
213	#97	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₂×C₁₃	—	2	1	no	—
216	C₂×C₁₃	—	2	1	no	—
217	C₂×C₁₃	—	16	2	no	—
218	C₂×C₁₃	#2023	1	1	yes	characteristic
219	C₂×C₁₃	#2192	1	1	yes	characteristic
220	C₂×C₁₃	#2193	1	1	yes	characteristic
221	C₂⁴	—	4	1	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	1	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	D₈	—	2	2	no	—
232	D₈	—	2	2	no	—
233	D₈	—	4	1	no	—
234	D₈	—	4	1	no	—
235	M₁₆	—	2	1	no	—
236	M₁₆	—	4	1	no	—
237	M₁₆	—	4	1	no	—
238	M₁₆	C₁₃×SD₁₆	1	2	yes	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
244	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
245	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
246	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
255	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	4	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	4	1	no	—
260	C₁₆	—	4	1	no	—
261	C₁₆	—	4	1	no	—
262	C₁₃	#13369	1	1	yes	characteristic, direct product factor, Sylow
263	C₂³	—	4	2	no	—
264	C₂³	—	8	1	no	—
265	C₂³	—	8	1	no	—
266	C₂³	—	16	1	no	—
267	C₂³	—	16	1	no	—
268	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
269	Q₈	—	16	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	8	1	no	—
273	D₄	—	8	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	16	1	no	—
280	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
281	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
282	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
283	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
284	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
285	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	8	1	no	—
288	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
289	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
290	C₈	C₁₃×#124	1	1	yes	characteristic
291	C₈	C₁₃×#124	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	16	2	no	—
296	C₂²	—	16	2	no	—
297	C₂²	—	16	1	no	—
298	C₂²	—	16	1	no	—
299	C₂²	C₁₃×#447	1	1	yes	characteristic
300	C₂²	C₁₃×#476	1	1	yes	characteristic
301	C₂²	C₁₃×#489	1	1	yes	characteristic
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	1	no	—
306	C₄	—	8	1	no	—
307	C₄	C₁₃×#393	1	1	yes	characteristic
308	C₄	C₁₃×#448	1	1	yes	characteristic
309	C₄	C₁₃×#474	1	1	yes	characteristic
310	C₄	C₁₃×#486	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	16	2	no	—
314	C₂	C₁₃×#2023	1	1	yes	characteristic
315	C₂	C₁₃×#2192	1	1	yes	characteristic
316	C₂	C₁₃×#2193	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂²×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
1. C₁₃×#3164	C₂	semidirect
5. C₁₃×#1387	C₂	semidirect
6. C₁₃×#1382	C₂	semidirect
7. C₁₃×#1379	C₂	semidirect
31. C₂×C₁₃×D₈	Q₈	semidirect
63. #13369	C₁₃	direct (N × H)
105. #3164	C₂×C₁₃	semidirect
109. #1387	C₂×C₁₃	semidirect
110. #1382	C₂×C₁₃	semidirect
111. #1379	C₂×C₁₃	semidirect
183. C₂×D₈	Q₈×C₁₃	semidirect
262. C₁₃	#13369	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×#3164	2
C₁₃×#2140	2
C₂×C₁₃×#357	2
C₁₃×#1978	2
C₁₃×#1387	2
C₁₃×#1382	2
C₁₃×#1379	2
#13369	13

Subgroups — C13×#13369 #15699708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13369 #15699708