Subgroups — C₁₃×#13126 #15699465

Order 3328.

300 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2151	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×#2110	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×C₄⋊C₁₆	—	2	2	no	—
18	C₁₃×#350	—	2	2	no	—
19	C₁₃×#350	—	2	2	no	—
20	C₁₃×#347	—	2	2	no	—
21	C₁₃×#344	C₄	1	1	yes	characteristic
22	C₁₃×#344	C₄	1	1	yes	characteristic
23	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
24	C₁₃×#131	—	2	1	no	—
25	C₁₃×#131	—	2	1	no	—
26	C₁₃×#131	—	2	1	no	—
27	C₁₃×#131	—	2	1	no	—
28	C₂×C₁₃×M₁₆	—	2	1	no	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	—	2	1	no	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
35	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
36	C₁₃×M₃₂	—	2	1	no	—
37	C₁₃×M₃₂	—	2	1	no	—
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	D₄	1	2	yes	—
48	C₁₃×#101	—	2	1	no	—
49	C₁₃×#100	—	2	1	no	—
50	C₁₃×#99	—	2	2	no	—
51	C₁₃×#97	—	2	1	no	—
52	C₁₃×#97	—	2	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₄×C₈×C₁₃	—	2	1	no	—
55	#13126	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₁₃×#41	—	2	1	no	—
57	C₁₃×#41	—	2	1	no	—
58	C₁₃×#41	—	2	1	no	—
59	C₁₃×#41	—	2	1	no	—
60	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
61	C₂×D₄×C₁₃	—	2	1	no	—
62	C₂×D₄×C₁₃	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₁₃×M₁₆	—	2	1	no	—
66	C₁₃×M₁₆	—	2	1	no	—
67	C₁₃×M₁₆	—	2	1	no	—
68	C₁₃×M₁₆	—	2	1	no	—
69	C₁₃×M₁₆	—	2	1	no	—
70	C₁₃×M₁₆	—	2	1	no	—
71	C₁₃×M₁₆	—	4	1	no	—
72	C₁₃×M₁₆	—	4	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	2	1	no	—
85	C₁₃×C₄⋊C₄	—	2	1	no	—
86	C₁₃×C₂²⋊C₄	—	2	1	no	—
87	C₁₃×C₂²⋊C₄	M₁₆	1	2	yes	—
88	C₄²×C₁₃	—	2	1	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	C₁₃×C₁₆	—	2	2	no	—
91	C₁₃×C₁₆	—	2	2	no	—
92	C₁₃×C₁₆	—	4	1	no	—
93	C₁₃×C₁₆	—	4	1	no	—
94	#2151	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#2117	C₂×C₁₃	1	1	yes	characteristic
96	C₂×#350	C₂×C₁₃	1	1	yes	characteristic
97	#2110	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#1757	C₂×C₁₃	1	1	yes	characteristic
99	#1379	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#1378	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₂³×C₁₃	—	2	1	no	—
102	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
103	Q₈×C₁₃	—	2	1	no	—
104	Q₈×C₁₃	—	2	1	no	—
105	D₄×C₁₃	—	2	1	no	—
106	D₄×C₁₃	—	2	1	no	—
107	D₄×C₁₃	—	4	1	no	—
108	D₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
120	C₈×C₁₃	—	4	1	no	—
121	C₈×C₁₃	—	4	1	no	—
122	C₈×C₁₃	—	4	1	no	—
123	C₈×C₁₃	#121	1	1	yes	characteristic
124	C₈×C₁₃	#123	1	1	yes	characteristic
125	C₈×C₁₃	#124	1	1	yes	characteristic
126	C₈×C₁₃	#126	1	1	yes	characteristic
127	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
128	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
131	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
132	#413	C₄×C₁₃	1	1	yes	characteristic
133	#410	C₄×C₁₃	1	1	yes	characteristic
134	#405	C₂²×C₁₃	1	1	yes	characteristic
135	#405	C₂²×C₁₃	1	1	yes	characteristic
136	C₄⋊C₁₆	—	2	2	no	—
137	#350	—	2	2	no	—
138	#350	—	2	2	no	—
139	#347	—	2	2	no	—
140	#344	C₄×C₁₃	1	1	yes	characteristic
141	#344	C₄×C₁₃	1	1	yes	characteristic
142	C₂²×C₁₃	—	4	1	no	—
143	C₂²×C₁₃	—	4	1	no	—
144	C₂²×C₁₃	—	8	1	no	—
145	C₂²×C₁₃	#432	1	1	yes	characteristic
146	C₂²×C₁₃	#435	1	1	yes	characteristic
147	C₂²×C₁₃	#436	1	1	yes	characteristic
148	C₄×C₁₃	—	4	2	no	—
149	C₄×C₁₃	—	4	1	no	—
150	C₄×C₁₃	—	8	1	no	—
151	C₄×C₁₃	#388	1	1	yes	characteristic
152	C₄×C₁₃	#433	1	1	yes	characteristic
153	C₄×C₁₃	#435	1	1	yes	characteristic
154	C₄×C₁₃	#435	1	1	yes	characteristic
155	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
156	#131	—	2	1	no	—
157	#131	—	2	1	no	—
158	#131	—	2	1	no	—
159	#131	—	2	1	no	—
160	C₂×M₁₆	—	2	1	no	—
161	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
163	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
166	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
167	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
168	M₃₂	—	2	1	no	—
169	M₃₂	—	2	1	no	—
170	M₃₂	—	2	1	no	—
171	M₃₂	—	2	1	no	—
172	M₃₂	—	2	1	no	—
173	M₃₂	D₄×C₁₃	1	2	yes	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
180	#101	—	2	1	no	—
181	#100	—	2	1	no	—
182	#99	—	2	2	no	—
183	#97	—	2	1	no	—
184	#97	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	2	1	no	—
190	C₂×C₁₃	—	8	1	no	—
191	C₂×C₁₃	#1924	1	1	yes	characteristic
192	#41	—	2	1	no	—
193	#41	—	2	1	no	—
194	#41	—	2	1	no	—
195	#41	—	2	1	no	—
196	C₂×Q₈	C₁₃×M₁₆	1	1	yes	characteristic
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	C₁₃×M₁₆	1	1	yes	characteristic
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
201	M₁₆	—	2	1	no	—
202	M₁₆	—	2	1	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	2	1	no	—
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
216	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
217	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
218	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
219	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	C₁₃×M₁₆	1	2	yes	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₁₆	—	2	2	no	—
227	C₁₆	—	2	2	no	—
228	C₁₆	—	4	1	no	—
229	C₁₆	—	4	1	no	—
230	C₁₃	#13126	1	1	yes	characteristic, direct product factor, Sylow
231	C₂³	—	2	1	no	—
232	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
233	Q₈	—	2	1	no	—
234	Q₈	—	2	1	no	—
235	D₄	—	2	1	no	—
236	D₄	—	2	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
245	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
246	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
248	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
249	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
250	C₈	—	4	1	no	—
251	C₈	—	4	1	no	—
252	C₈	—	4	1	no	—
253	C₈	C₁₃×#121	1	1	yes	characteristic
254	C₈	C₁₃×#123	1	1	yes	characteristic
255	C₈	C₁₃×#124	1	1	yes	characteristic
256	C₈	C₁₃×#126	1	1	yes	characteristic
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	1	no	—
260	C₂²	C₁₃×#432	1	1	yes	characteristic
261	C₂²	C₁₃×#435	1	1	yes	characteristic
262	C₂²	C₁₃×#436	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	8	1	no	—
266	C₄	C₁₃×#388	1	1	yes	characteristic
267	C₄	C₁₃×#433	1	1	yes	characteristic
268	C₄	C₁₃×#435	1	1	yes	characteristic
269	C₄	C₁₃×#435	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	8	1	no	—
274	C₂	C₁₃×#1924	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃	#124
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1924
prime radical	C₂×C₁₃×#41	C₂×C₄
LCS	G ▷ C₂²×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×#2151	C₂	semidirect
4. C₁₃×#2110	C₂	semidirect
6. C₁₃×#1379	C₂	semidirect
7. C₁₃×#1378	C₂	semidirect
55. #13126	C₁₃	direct (N × H)
94. #2151	C₂×C₁₃	semidirect
97. #2110	C₂×C₁₃	semidirect
99. #1379	C₂×C₁₃	semidirect
100. #1378	C₂×C₁₃	semidirect
230. C₁₃	#13126	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×#2151	2
C₁₃×#2117	2
C₂×C₁₃×#350	2
C₁₃×#2110	2
C₁₃×#1757	2
C₁₃×#1379	2
C₁₃×#1378	2
#13126	13

Subgroups — C13×#13126 #15699465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13126 #15699465