Subgroups — C₁₃×#13123 #15699462

Order 3328.

312 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2117	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#350	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1757	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1378	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	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×#413	C₄	1	1	yes	characteristic
14	C₁₃×#410	C₄	1	1	yes	characteristic
15	C₁₃×#405	C₂²	1	1	yes	characteristic
16	C₁₃×#405	C₂²	1	1	yes	characteristic
17	C₁₃×#364	—	2	2	no	—
18	C₁₃×#350	—	2	2	no	—
19	C₁₃×#350	—	2	1	no	—
20	C₁₃×#350	—	2	1	no	—
21	C₁₃×#346	—	2	1	no	—
22	C₄×C₁₃×C₁₆	—	2	1	no	—
23	C₁₃×#344	C₄	1	1	yes	characteristic
24	C₁₃×#344	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	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	—	2	1	no	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
37	C₁₃×#117	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	—	2	1	no	—
42	C₁₃×M₃₂	—	2	1	no	—
43	C₁₃×M₃₂	D₄	1	2	yes	—
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×#101	C₈	1	2	yes	—
52	C₁₃×#100	C₈	1	2	yes	—
53	C₁₃×#99	—	2	2	no	—
54	C₁₃×#97	—	2	1	no	—
55	C₁₃×#97	—	2	1	no	—
56	C₄×C₈×C₁₃	—	2	1	no	—
57	C₄×C₈×C₁₃	—	2	1	no	—
58	#13123	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₁₃×#41	—	2	1	no	—
60	C₁₃×#41	—	2	1	no	—
61	C₁₃×#41	—	2	1	no	—
62	C₁₃×#41	—	2	1	no	—
63	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂×D₄×C₁₃	—	2	1	no	—
65	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₁₃×M₁₆	—	2	1	no	—
69	C₁₃×M₁₆	—	2	1	no	—
70	C₁₃×M₁₆	—	2	1	no	—
71	C₁₃×M₁₆	—	2	1	no	—
72	C₁₃×M₁₆	—	4	1	no	—
73	C₁₃×M₁₆	—	4	1	no	—
74	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
75	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	4	1	no	—
80	C₂×C₈×C₁₃	—	4	1	no	—
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	—	2	1	no	—
88	C₁₃×C₄⋊C₄	—	2	1	no	—
89	C₁₃×C₂²⋊C₄	—	2	1	no	—
90	C₁₃×C₂²⋊C₄	M₁₆	1	2	yes	—
91	C₄²×C₁₃	—	2	1	no	—
92	C₄²×C₁₃	—	2	1	no	—
93	C₁₃×C₁₆	—	2	2	no	—
94	C₁₃×C₁₆	—	2	1	no	—
95	C₁₃×C₁₆	—	2	1	no	—
96	C₁₃×C₁₆	—	4	1	no	—
97	C₁₃×C₁₆	—	4	1	no	—
98	#2153	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#2117	C₂×C₁₃	1	1	yes	characteristic
100	C₂×#350	C₂×C₁₃	1	1	yes	characteristic
101	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#1757	C₂×C₁₃	1	1	yes	characteristic
103	#1379	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#1378	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₂³×C₁₃	—	2	1	no	—
106	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	Q₈×C₁₃	—	2	1	no	—
108	Q₈×C₁₃	—	2	1	no	—
109	D₄×C₁₃	—	2	1	no	—
110	D₄×C₁₃	—	2	1	no	—
111	D₄×C₁₃	—	4	1	no	—
112	D₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
124	C₈×C₁₃	—	2	2	no	—
125	C₈×C₁₃	—	2	2	no	—
126	C₈×C₁₃	—	4	1	no	—
127	C₈×C₁₃	#121	1	1	yes	characteristic
128	C₈×C₁₃	#123	1	1	yes	characteristic
129	C₈×C₁₃	#124	1	1	yes	characteristic
130	C₈×C₁₃	#126	1	1	yes	characteristic
131	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
135	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
136	#413	C₄×C₁₃	1	1	yes	characteristic
137	#410	C₄×C₁₃	1	1	yes	characteristic
138	#405	C₂²×C₁₃	1	1	yes	characteristic
139	#405	C₂²×C₁₃	1	1	yes	characteristic
140	#364	—	2	2	no	—
141	#350	—	2	2	no	—
142	#350	—	2	1	no	—
143	#350	—	2	1	no	—
144	#346	—	2	1	no	—
145	C₄×C₁₆	—	2	1	no	—
146	#344	C₄×C₁₃	1	1	yes	characteristic
147	#344	C₄×C₁₃	1	1	yes	characteristic
148	C₂²×C₁₃	—	4	1	no	—
149	C₂²×C₁₃	—	4	1	no	—
150	C₂²×C₁₃	—	8	1	no	—
151	C₂²×C₁₃	#431	1	1	yes	characteristic
152	C₂²×C₁₃	#435	1	1	yes	characteristic
153	C₂²×C₁₃	#436	1	1	yes	characteristic
154	C₄×C₁₃	—	4	2	no	—
155	C₄×C₁₃	—	4	1	no	—
156	C₄×C₁₃	—	8	1	no	—
157	C₄×C₁₃	#388	1	1	yes	characteristic
158	C₄×C₁₃	#433	1	1	yes	characteristic
159	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
160	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
161	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
162	#131	—	2	1	no	—
163	#131	—	2	1	no	—
164	#131	—	2	1	no	—
165	#131	—	2	1	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
168	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
169	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
172	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
173	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
174	M₃₂	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	D₄×C₁₃	1	2	yes	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
186	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
187	#101	C₈×C₁₃	1	2	yes	—
188	#100	C₈×C₁₃	1	2	yes	—
189	#99	—	2	2	no	—
190	#97	—	2	1	no	—
191	#97	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₂×C₁₃	—	2	1	no	—
195	C₂×C₁₃	—	2	1	no	—
196	C₂×C₁₃	—	2	1	no	—
197	C₂×C₁₃	—	8	1	no	—
198	C₂×C₁₃	#1923	1	1	yes	characteristic
199	#41	—	2	1	no	—
200	#41	—	2	1	no	—
201	#41	—	2	1	no	—
202	#41	—	2	1	no	—
203	C₂×Q₈	C₂×C₈×C₁₃	1	1	yes	characteristic
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	C₂×C₈×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
208	M₁₆	—	2	1	no	—
209	M₁₆	—	2	1	no	—
210	M₁₆	—	2	1	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	C₂×C₈×C₁₃	1	2	yes	—
215	M₁₆	C₂×C₈×C₁₃	1	2	yes	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
224	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
225	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
226	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	C₁₃×M₁₆	1	2	yes	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	1	no	—
235	C₁₆	—	2	1	no	—
236	C₁₆	—	4	1	no	—
237	C₁₆	—	4	1	no	—
238	C₁₃	#13123	1	1	yes	characteristic, direct product factor, Sylow
239	C₂³	—	2	1	no	—
240	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
241	Q₈	—	2	1	no	—
242	Q₈	—	2	1	no	—
243	D₄	—	2	1	no	—
244	D₄	—	2	1	no	—
245	D₄	—	4	1	no	—
246	D₄	—	4	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
256	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
257	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	4	1	no	—
261	C₈	C₁₃×#121	1	1	yes	characteristic
262	C₈	C₁₃×#123	1	1	yes	characteristic
263	C₈	C₁₃×#124	1	1	yes	characteristic
264	C₈	C₁₃×#126	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	1	no	—
268	C₂²	C₁₃×#431	1	1	yes	characteristic
269	C₂²	C₁₃×#435	1	1	yes	characteristic
270	C₂²	C₁₃×#436	1	1	yes	characteristic
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	8	1	no	—
274	C₄	C₁₃×#388	1	1	yes	characteristic
275	C₄	C₁₃×#433	1	1	yes	characteristic
276	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
277	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	8	1	no	—
282	C₂	C₁₃×#1923	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	#1923
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
1. C₁₃×#2153	C₂	semidirect
4. C₁₃×#2108	C₂	semidirect
6. C₁₃×#1379	C₂	semidirect
7. C₁₃×#1378	C₂	semidirect
58. #13123	C₁₃	direct (N × H)
98. #2153	C₂×C₁₃	semidirect
101. #2108	C₂×C₁₃	semidirect
103. #1379	C₂×C₁₃	semidirect
104. #1378	C₂×C₁₃	semidirect
238. C₁₃	#13123	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×#2153	2
C₁₃×#2117	2
C₂×C₁₃×#350	2
C₁₃×#2108	2
C₁₃×#1757	2
C₁₃×#1379	2
C₁₃×#1378	2
#13123	13

Subgroups — C13×#13123 #15699462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13123 #15699462