Subgroups — C₁₃×#13122 #15699461

Order 3328.

326 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#350	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1764	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#104	C₄	1	1	yes	characteristic
14	C₁₃×#416	C₄	1	1	yes	characteristic
15	C₁₃×#405	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#364	—	2	2	no	—
18	C₁₃×#350	—	2	2	no	—
19	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
20	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
21	C₁₃×#346	—	2	1	no	—
22	C₄×C₁₃×C₁₆	—	2	1	no	—
23	C₁₃×#341	C₄	1	1	yes	characteristic
24	C₁₃×#339	C₄	1	1	yes	characteristic
25	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
26	C₁₃×#131	—	2	1	no	—
27	C₁₃×#131	—	2	1	no	—
28	C₁₃×#131	—	2	1	no	—
29	C₁₃×#131	—	2	1	no	—
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	—	2	1	no	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₁₃×M₃₂	—	2	1	no	—
39	C₁₃×M₃₂	—	2	1	no	—
40	C₁₃×M₃₂	—	2	1	no	—
41	C₁₃×M₃₂	D₄	1	2	yes	—
42	C₂×C₁₃×C₁₆	—	2	1	no	—
43	C₂×C₁₃×C₁₆	—	2	1	no	—
44	C₂×C₁₃×C₁₆	—	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	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
50	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
51	C₁₃×#104	—	2	2	no	—
52	C₁₃×#103	C₈	1	1	yes	characteristic
53	C₁₃×#103	C₈	1	1	yes	characteristic
54	C₁₃×#102	C₈	1	1	yes	characteristic
55	C₁₃×#102	C₈	1	1	yes	characteristic
56	C₁₃×#97	—	2	1	no	—
57	C₄×C₈×C₁₃	—	2	1	no	—
58	C₄×C₈×C₁₃	—	2	1	no	—
59	C₄×C₈×C₁₃	—	2	1	no	—
60	#13122	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₁₃×#41	—	2	1	no	—
62	C₁₃×#41	—	2	1	no	—
63	C₁₃×#41	—	2	1	no	—
64	C₁₃×#41	—	2	1	no	—
65	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂×D₄×C₁₃	—	2	1	no	—
67	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₁₃×M₁₆	—	2	1	no	—
72	C₁₃×M₁₆	—	2	1	no	—
73	C₁₃×M₁₆	—	2	1	no	—
74	C₁₃×M₁₆	—	4	1	no	—
75	C₁₃×M₁₆	—	4	1	no	—
76	C₁₃×M₁₆	M₁₆	1	2	yes	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	4	1	no	—
82	C₂×C₈×C₁₃	—	4	1	no	—
83	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	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	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
93	C₁₃×C₂²⋊C₄	—	2	1	no	—
94	C₄²×C₁₃	—	2	1	no	—
95	C₄²×C₁₃	—	2	1	no	—
96	C₄²×C₁₃	—	2	1	no	—
97	C₁₃×C₁₆	—	2	2	no	—
98	C₁₃×C₁₆	—	2	1	no	—
99	C₁₃×C₁₆	—	2	1	no	—
100	C₁₃×C₁₆	—	4	1	no	—
101	C₁₃×C₁₆	—	4	1	no	—
102	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
103	C₂×#350	C₂×C₁₃	1	1	yes	characteristic
104	#2113	C₂×C₁₃	1	1	yes	characteristic
105	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#1764	C₂×C₁₃	1	1	yes	characteristic
107	#1379	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#1374	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂³×C₁₃	—	2	1	no	—
110	C₂³×C₁₃	#131	1	1	yes	characteristic
111	Q₈×C₁₃	—	2	1	no	—
112	Q₈×C₁₃	—	2	1	no	—
113	D₄×C₁₃	—	2	1	no	—
114	D₄×C₁₃	—	2	1	no	—
115	D₄×C₁₃	—	4	1	no	—
116	D₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	—	4	1	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
131	C₈×C₁₃	—	2	1	no	—
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	—	4	1	no	—
135	C₈×C₁₃	#121	1	1	yes	characteristic
136	C₈×C₁₃	#123	1	1	yes	characteristic
137	C₈×C₁₃	#124	1	1	yes	characteristic
138	C₈×C₁₃	#126	1	1	yes	characteristic
139	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
142	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
143	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×#104	C₄×C₁₃	1	1	yes	characteristic
145	#416	C₄×C₁₃	1	1	yes	characteristic
146	#405	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
148	#364	—	2	2	no	—
149	#350	—	2	2	no	—
150	C₂²⋊C₁₆	—	2	1	no	—
151	C₂²⋊C₁₆	—	2	1	no	—
152	#346	—	2	1	no	—
153	C₄×C₁₆	—	2	1	no	—
154	#341	C₄×C₁₃	1	1	yes	characteristic
155	#339	C₄×C₁₃	1	1	yes	characteristic
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	4	1	no	—
159	C₂²×C₁₃	—	4	1	no	—
160	C₂²×C₁₃	—	4	1	no	—
161	C₂²×C₁₃	#433	1	1	yes	characteristic
162	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
163	C₂²×C₁₃	#436	1	1	yes	characteristic
164	C₄×C₁₃	—	4	2	no	—
165	C₄×C₁₃	—	4	1	no	—
166	C₄×C₁₃	—	4	1	no	—
167	C₄×C₁₃	—	4	1	no	—
168	C₄×C₁₃	#388	1	1	yes	characteristic
169	C₄×C₁₃	#431	1	1	yes	characteristic
170	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
171	C₄×C₁₃	#435	1	1	yes	characteristic
172	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
173	#131	—	2	1	no	—
174	#131	—	2	1	no	—
175	#131	—	2	1	no	—
176	#131	—	2	1	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
179	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
183	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	D₄×C₁₃	1	2	yes	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
197	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
198	#104	—	2	2	no	—
199	#103	C₈×C₁₃	1	1	yes	characteristic
200	#103	C₈×C₁₃	1	1	yes	characteristic
201	#102	C₈×C₁₃	1	1	yes	characteristic
202	#102	C₈×C₁₃	1	1	yes	characteristic
203	#97	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₂×C₁₃	—	2	1	no	—
208	C₂×C₁₃	—	2	1	no	—
209	C₂×C₁₃	—	8	1	no	—
210	C₂×C₁₃	#1923	1	1	yes	characteristic
211	C₂×C₁₃	#2170	1	1	yes	characteristic
212	C₂×C₁₃	#2171	1	1	yes	characteristic
213	#41	—	2	1	no	—
214	#41	—	2	1	no	—
215	#41	—	2	1	no	—
216	#41	—	2	1	no	—
217	C₂×Q₈	C₂×C₈×C₁₃	1	1	yes	characteristic
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	C₂×C₈×C₁₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
223	M₁₆	—	2	1	no	—
224	M₁₆	—	2	1	no	—
225	M₁₆	—	2	1	no	—
226	M₁₆	—	4	1	no	—
227	M₁₆	—	4	1	no	—
228	M₁₆	C₁₃×M₁₆	1	2	yes	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
239	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
240	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
241	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
242	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
243	C₄⋊C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
244	C₄⋊C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
245	C₂²⋊C₄	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₁₆	—	2	2	no	—
250	C₁₆	—	2	1	no	—
251	C₁₆	—	2	1	no	—
252	C₁₆	—	4	1	no	—
253	C₁₆	—	4	1	no	—
254	C₁₃	#13122	1	1	yes	characteristic, direct product factor, Sylow
255	C₂³	—	2	1	no	—
256	C₂³	C₁₃×#131	1	1	yes	characteristic
257	Q₈	—	2	1	no	—
258	Q₈	—	2	1	no	—
259	D₄	—	2	1	no	—
260	D₄	—	2	1	no	—
261	D₄	—	4	1	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
272	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
276	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	2	no	—
280	C₈	—	4	1	no	—
281	C₈	C₁₃×#121	1	1	yes	characteristic
282	C₈	C₁₃×#123	1	1	yes	characteristic
283	C₈	C₁₃×#124	1	1	yes	characteristic
284	C₈	C₁₃×#126	1	1	yes	characteristic
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	C₁₃×#433	1	1	yes	characteristic
291	C₂²	C₈×D₄×C₁₃	1	1	yes	characteristic
292	C₂²	C₁₃×#436	1	1	yes	characteristic
293	C₄	—	4	2	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	C₁₃×#388	1	1	yes	characteristic
298	C₄	C₁₃×#431	1	1	yes	characteristic
299	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
300	C₄	C₁₃×#435	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	8	1	no	—
304	C₂	C₁₃×#1923	1	1	yes	characteristic
305	C₂	C₁₃×#2170	1	1	yes	characteristic
306	C₂	C₁₃×#2171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×C₁₃×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ C₂²×C₁₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#364	C₂	semidirect
4. C₁₃×#2108	C₂	semidirect
6. C₁₃×#1379	C₂	semidirect
7. C₁₃×#1374	C₂	semidirect
60. #13122	C₁₃	direct (N × H)
102. C₂×#364	C₂×C₁₃	semidirect
105. #2108	C₂×C₁₃	semidirect
107. #1379	C₂×C₁₃	semidirect
108. #1374	C₂×C₁₃	semidirect
254. C₁₃	#13122	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#364	2
C₂×C₁₃×#350	2
C₁₃×#2113	2
C₁₃×#2108	2
C₁₃×#1764	2
C₁₃×#1379	2
C₁₃×#1374	2
#13122	13

Subgroups — C13×#13122 #15699461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13122 #15699461