Subgroups — C₁₃×#13210 #15699549

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#360	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#135	—	2	1	no	—
6	C₂×C₁₃×#135	C₂²	1	2	yes	—
7	C₁₃×#532	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₁₃×#495	—	2	1	no	—
10	C₁₃×#495	C₄	1	2	yes	complemented
11	C₁₃×#494	C₄	1	1	yes	characteristic, complemented
12	C₁₃×#493	C₄	1	1	yes	characteristic, complemented
13	C₁₃×#492	—	2	1	no	—
14	C₂×C₁₃×#106	—	2	1	no	—
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#384	—	2	1	no	—
18	C₁₃×#360	—	2	8	no	—
19	C₁₃×#346	—	2	1	no	—
20	C₄×C₁₃×C₁₆	—	2	1	no	—
21	C₂×C₁₃×#41	—	2	2	no	—
22	C₁₃×#135	—	2	4	no	—
23	C₁₃×#135	—	2	4	no	—
24	C₁₃×#135	—	4	2	no	—
25	C₁₃×#135	—	4	2	no	—
26	C₂×C₁₃×Q₁₆	—	2	1	no	—
27	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×SD₁₆	—	2	2	no	—
29	C₂×C₁₃×SD₁₆	—	2	2	no	—
30	C₂×C₁₃×D₈	—	2	1	no	—
31	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×#127	—	2	1	no	—
36	C₁₃×#124	—	2	2	no	—
37	C₁₃×#121	—	4	2	no	—
38	C₄×D₄×C₁₃	—	4	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₁₃×M₃₂	—	2	2	no	—
43	C₁₃×M₃₂	D₄	1	4	yes	complemented
44	C₂×C₁₃×C₁₆	—	2	2	no	—
45	C₁₃×#106	—	2	4	no	—
46	C₁₃×#106	—	4	2	no	—
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
51	#13210	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₁₃×#41	—	4	2	no	—
53	C₁₃×#41	—	4	2	no	—
54	C₁₃×#41	—	4	2	no	—
55	C₁₃×#41	—	4	2	no	—
56	C₂×Q₈×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	D₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×Q₁₆	—	2	4	no	—
65	C₁₃×Q₁₆	—	4	2	no	—
66	C₁₃×SD₁₆	—	4	4	no	—
67	C₁₃×SD₁₆	—	4	4	no	—
68	C₁₃×D₈	—	2	4	no	—
69	C₁₃×D₈	—	4	2	no	—
70	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	D₈	1	2	yes	—
73	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	2	yes	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	8	4	no	—
82	C₁₃×C₂²⋊C₄	—	4	4	no	—
83	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₁₃	SD₁₆	1	2	yes	—
86	C₁₃×C₁₆	—	2	4	no	—
87	#3147	C₂×C₁₃	1	1	yes	characteristic
88	C₂×#360	C₂×C₁₃	1	4	yes	complemented
89	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
90	#1844	C₂×C₁₃	1	1	yes	characteristic, complemented
91	C₂³×C₁₃	—	4	2	no	—
92	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₁₃	—	4	2	no	—
94	Q₈×C₁₃	—	4	2	no	—
95	D₄×C₁₃	—	4	2	no	—
96	D₄×C₁₃	—	4	2	no	—
97	D₄×C₁₃	—	8	2	no	—
98	D₄×C₁₃	—	8	2	no	—
99	D₄×C₁₃	—	8	4	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	8	2	no	—
103	C₂×C₄×C₁₃	—	8	2	no	—
104	C₂×C₄×C₁₃	—	8	4	no	—
105	C₂×C₄×C₁₃	—	8	2	no	—
106	C₂×C₄×C₁₃	#102	1	2	yes	—
107	C₂×C₄×C₁₃	#102	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₈×C₁₃	#102	1	2	yes	—
115	C₈×C₁₃	#102	1	2	yes	—
116	C₈×C₁₃	C₄×D₄	1	2	yes	—
117	C₈×C₁₃	#121	1	2	yes	—
118	C₂×#135	—	2	1	no	—
119	C₂×#135	C₂²×C₁₃	1	2	yes	—
120	#532	—	2	1	no	—
121	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
122	#495	—	2	1	no	—
123	#495	C₄×C₁₃	1	2	yes	complemented
124	#494	C₄×C₁₃	1	1	yes	characteristic, complemented
125	#493	C₄×C₁₃	1	1	yes	characteristic, complemented
126	#492	—	2	1	no	—
127	C₂×#106	—	2	1	no	—
128	C₂×#106	C₂²×C₁₃	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
130	#384	—	2	1	no	—
131	#360	—	2	8	no	—
132	#346	—	2	1	no	—
133	C₄×C₁₆	—	2	1	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	8	4	no	—
136	C₂²×C₁₃	—	8	2	no	—
137	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₁₃	#486	1	1	yes	characteristic
139	C₂²×C₁₃	#493	1	1	yes	characteristic
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	8	2	no	—
142	C₄×C₁₃	—	16	2	no	—
143	C₄×C₁₃	#360	1	4	yes	complemented
144	C₄×C₁₃	#390	1	1	yes	characteristic
145	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
146	C₄×C₁₃	#486	1	1	yes	characteristic
147	C₄×C₁₃	#492	1	1	yes	characteristic
148	C₂×#41	—	2	2	no	—
149	#135	—	2	4	no	—
150	#135	—	2	4	no	—
151	#135	—	4	2	no	—
152	#135	—	4	2	no	—
153	C₂×Q₁₆	—	2	1	no	—
154	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
155	C₂×SD₁₆	—	2	2	no	—
156	C₂×SD₁₆	—	2	2	no	—
157	C₂×D₈	—	2	1	no	—
158	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
159	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
161	C₄⋊Q₈	—	2	1	no	—
162	#127	—	2	1	no	—
163	#124	—	2	2	no	—
164	#121	—	4	2	no	—
165	C₄×D₄	—	4	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	4	2	no	—
168	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
169	M₃₂	—	2	2	no	—
170	M₃₂	D₄×C₁₃	1	4	yes	complemented
171	C₂×C₁₆	—	2	2	no	—
172	#106	—	2	4	no	—
173	#106	—	4	2	no	—
174	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
175	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
176	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
177	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	—	2	1	no	—
180	C₂×C₁₃	—	16	2	no	—
181	C₂×C₁₃	#1957	1	1	yes	characteristic
182	C₂×C₁₃	C₂×#360	1	2	yes	—
183	#41	—	4	2	no	—
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₁₃×D₈	1	2	yes	—
194	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
195	Q₁₆	—	2	4	no	—
196	Q₁₆	—	4	2	no	—
197	SD₁₆	—	4	4	no	—
198	SD₁₆	—	4	4	no	—
199	D₈	—	2	4	no	—
200	D₈	—	4	2	no	—
201	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₈	C₁₃×D₈	1	2	yes	—
204	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
205	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₈	C₁₃×#41	1	2	yes	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	8	4	no	—
213	C₂²⋊C₄	—	4	4	no	—
214	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
215	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
216	C₄²	C₁₃×SD₁₆	1	2	yes	—
217	C₁₆	—	2	4	no	—
218	C₁₃	#13210	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³	—	4	2	no	—
220	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	Q₈	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	C₁₃×#102	1	2	yes	—
235	C₂×C₄	C₁₃×#102	1	2	yes	—
236	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
238	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
239	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
240	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
241	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
242	C₈	C₁₃×#102	1	2	yes	—
243	C₈	C₁₃×#102	1	2	yes	—
244	C₈	C₄×D₄×C₁₃	1	2	yes	—
245	C₈	C₁₃×#121	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
250	C₂²	C₁₃×#486	1	1	yes	characteristic
251	C₂²	C₁₃×#493	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	16	2	no	—
255	C₄	C₁₃×#360	1	4	yes	complemented
256	C₄	C₁₃×#390	1	1	yes	characteristic
257	C₄	C₂×C₁₃×#102	1	1	yes	characteristic
258	C₄	C₁₃×#486	1	1	yes	characteristic
259	C₄	C₁₃×#492	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	C₁₃×#1957	1	1	yes	characteristic
264	C₂	C₂×C₁₃×#360	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₁₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#360	C₂	semidirect
3. C₄×C₁₃×M₃₂	C₂	semidirect
4. C₁₃×#1844	C₂	semidirect
10. C₁₃×#495	C₄	semidirect
11. C₁₃×#494	C₄	semidirect
12. C₁₃×#493	C₄	semidirect
43. C₁₃×M₃₂	D₄	semidirect
51. #13210	C₁₃	direct (N × H)
88. C₂×#360	C₂×C₁₃	semidirect
89. C₄×M₃₂	C₂×C₁₃	semidirect
90. #1844	C₂×C₁₃	semidirect
123. #495	C₄×C₁₃	semidirect
124. #494	C₄×C₁₃	semidirect
125. #493	C₄×C₁₃	semidirect
143. C₄×C₁₃	#360	semidirect
170. M₃₂	D₄×C₁₃	semidirect
218. C₁₃	#13210	direct (N × H)
255. C₄	C₁₃×#360	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#3147	2
C₂×C₁₃×#360	2
C₄×C₁₃×M₃₂	2
C₁₃×#1844	2
#13210	13

Subgroups — C13×#13210 #15699549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13210 #15699549