Subgroups — C₁₃×#10832 #15697171

Order 3328.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#336	C₂	1	4	yes	maximal
4	C₂³×C₈×C₁₃	C₂²	1	4	yes	—
5	C₂²×C₄²×C₁₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₈	—	2	4	no	—
10	C₂×C₁₃×C₄⋊C₈	—	2	4	no	—
11	C₂×C₁₃×C₄⋊C₈	C₄	1	8	yes	complemented
12	C₄³×C₁₃	—	2	4	no	—
13	C₁₃×#336	—	2	8	no	—
14	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	—	2	8	no	—
19	C₂²×C₈×C₁₃	—	2	4	no	—
20	C₂²×C₈×C₁₃	—	2	4	no	—
21	C₂²×C₈×C₁₃	—	2	8	no	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
23	C₂×C₄²×C₁₃	—	2	4	no	—
24	C₂×C₄²×C₁₃	—	2	4	no	—
25	C₂×C₄²×C₁₃	—	2	8	no	—
26	C₂×C₄²×C₁₃	—	2	4	no	—
27	C₂×C₄²×C₁₃	—	2	4	no	—
28	C₂×C₄²×C₁₃	—	2	4	no	—
29	C₂×C₄²×C₁₃	C₈	1	8	yes	complemented
30	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃	D₄	1	4	yes	—
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	4	yes	—
34	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
35	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
36	C₁₃×C₄⋊C₈	—	2	16	no	—
37	#10832	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	—	2	4	no	—
40	C₂²×C₄×C₁₃	—	2	4	no	—
41	C₂²×C₄×C₁₃	—	2	4	no	—
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
45	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
54	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₁₃	#41	1	4	yes	—
62	C₂×C₈×C₁₃	—	2	8	no	—
63	C₂×C₈×C₁₃	—	2	16	no	—
64	C₂×C₈×C₁₃	—	4	8	no	—
65	C₂×C₈×C₁₃	—	4	4	no	—
66	C₂×C₈×C₁₃	—	4	4	no	—
67	C₄²×C₁₃	—	2	16	no	—
68	C₄²×C₁₃	—	2	16	no	—
69	C₄²×C₁₃	—	2	8	no	—
70	C₄²×C₁₃	—	2	8	no	—
71	C₄²×C₁₃	C₂²⋊C₄	1	8	yes	—
72	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
73	C₂²×C₄⋊C₈	C₂×C₁₃	1	2	yes	—
74	C₂×C₄³	C₂×C₁₃	1	1	yes	characteristic
75	C₂×#336	C₂×C₁₃	1	4	yes	—
76	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
77	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₁₃	#117	1	2	yes	—
81	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
82	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
83	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
84	C₂×C₄×C₁₃	—	2	8	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	2	8	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	8	no	—
89	C₂×C₄×C₁₃	—	2	8	no	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	8	no	—
92	C₂×C₄×C₁₃	—	2	8	no	—
93	C₂×C₄×C₁₃	#95	1	4	yes	—
94	C₂×C₄×C₁₃	#95	1	4	yes	—
95	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
96	C₂×C₄×C₁₃	#97	1	4	yes	—
97	C₂×C₄×C₁₃	C₂²⋊C₈	1	8	yes	—
98	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
99	C₂×C₄×C₁₃	C₄⋊C₈	1	16	yes	complemented
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
103	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
104	C₂×C₄×C₁₃	#121	1	4	yes	—
105	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
106	C₂×C₄×C₁₃	#124	1	2	yes	—
107	C₂×C₄×C₁₃	#125	1	2	yes	—
108	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
111	C₈×C₁₃	—	4	8	no	—
112	C₂³×C₈	C₂²×C₁₃	1	4	yes	—
113	C₂²×C₄²	C₄×C₁₃	1	2	yes	—
114	C₂²×C₄²	C₄×C₁₃	1	2	yes	—
115	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
116	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
117	C₂×C₄⋊C₈	—	2	4	no	—
118	C₂×C₄⋊C₈	—	2	4	no	—
119	C₂×C₄⋊C₈	C₄×C₁₃	1	8	yes	complemented
120	C₄³	—	2	4	no	—
121	#336	—	2	8	no	—
122	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
123	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
124	C₂²×C₁₃	#380	1	2	yes	—
125	C₂²×C₁₃	#382	1	1	yes	characteristic
126	C₂²×C₁₃	#384	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
129	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
130	C₂²×C₁₃	C₂×C₂²⋊C₈	1	2	yes	—
131	C₂²×C₁₃	#407	1	2	yes	—
132	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
133	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
134	C₂²×C₁₃	#423	1	4	yes	—
135	C₂²×C₁₃	#423	1	2	yes	—
136	C₂²×C₁₃	#431	1	4	yes	—
137	C₂²×C₁₃	#432	1	4	yes	—
138	C₄×C₁₃	—	2	16	no	—
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	#336	1	8	yes	complemented
141	C₄×C₁₃	#389	1	4	yes	—
142	C₄×C₁₃	#390	1	2	yes	—
143	C₄×C₁₃	#391	1	2	yes	—
144	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
145	C₂³×C₄	C₂×C₄×C₁₃	1	4	yes	—
146	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
147	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
148	C₂²×C₈	—	2	8	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	—	2	4	no	—
151	C₂²×C₈	—	2	8	no	—
152	C₂²×C₈	C₂×C₄×C₁₃	1	8	yes	—
153	C₂×C₄²	—	2	4	no	—
154	C₂×C₄²	—	2	4	no	—
155	C₂×C₄²	—	2	8	no	—
156	C₂×C₄²	—	2	4	no	—
157	C₂×C₄²	—	2	4	no	—
158	C₂×C₄²	—	2	4	no	—
159	C₂×C₄²	C₈×C₁₃	1	8	yes	complemented
160	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
161	C₂×C₄²	D₄×C₁₃	1	4	yes	—
162	C₂×C₄²	D₄×C₁₃	1	2	yes	—
163	C₂×C₄²	D₄×C₁₃	1	4	yes	—
164	C₂×C₄²	Q₈×C₁₃	1	4	yes	—
165	C₂×C₄²	Q₈×C₁₃	1	2	yes	—
166	C₄⋊C₈	—	2	16	no	—
167	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
168	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
169	C₂×C₁₃	#1728	1	1	yes	characteristic
170	C₂×C₁₃	C₄×C₄⋊C₈	1	2	yes	—
171	C₂×C₁₃	#1767	1	2	yes	—
172	C₂×C₁₃	#1797	1	4	yes	—
173	C₂×C₁₃	#1831	1	2	yes	—
174	C₂×C₁₃	#1842	1	2	yes	—
175	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
182	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
183	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
184	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
185	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
186	C₂²×C₄	C₁₃×C₄⋊C₄	1	8	yes	—
187	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
188	C₂²×C₄	C₂×C₈×C₁₃	1	8	yes	—
189	C₂²×C₄	C₂×C₈×C₁₃	1	4	yes	—
190	C₂²×C₄	C₁₃×M₁₆	1	8	yes	—
191	C₂²×C₄	C₁₃×M₁₆	1	4	yes	—
192	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
196	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
198	C₂²×C₄	C₁₃×#41	1	4	yes	—
199	C₂×C₈	—	2	8	no	—
200	C₂×C₈	—	2	16	no	—
201	C₂×C₈	—	4	8	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	4	4	no	—
204	C₄²	—	2	16	no	—
205	C₄²	—	2	16	no	—
206	C₄²	—	2	8	no	—
207	C₄²	—	2	8	no	—
208	C₄²	C₁₃×C₂²⋊C₄	1	8	yes	—
209	C₄²	C₁₃×C₄⋊C₄	1	8	yes	—
210	C₁₃	#10832	1	1	yes	characteristic, direct product factor, Sylow
211	C₂³	C₂×C₄²×C₁₃	1	1	yes	characteristic
212	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
213	C₂³	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
214	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
215	C₂³	C₁₃×#117	1	2	yes	—
216	C₂³	C₂²×C₈×C₁₃	1	2	yes	—
217	C₂³	C₂×C₁₃×M₁₆	1	2	yes	—
218	C₂³	C₂×C₁₃×M₁₆	1	4	yes	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	C₁₃×#95	1	4	yes	—
229	C₂×C₄	C₁₃×#95	1	4	yes	—
230	C₂×C₄	C₄×C₈×C₁₃	1	4	yes	—
231	C₂×C₄	C₁₃×#97	1	4	yes	—
232	C₂×C₄	C₁₃×C₂²⋊C₈	1	8	yes	—
233	C₂×C₄	C₁₃×C₄⋊C₈	1	8	yes	—
234	C₂×C₄	C₁₃×C₄⋊C₈	1	16	yes	complemented
235	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
236	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
237	C₂×C₄	C₄×D₄×C₁₃	1	4	yes	—
238	C₂×C₄	C₄×Q₈×C₁₃	1	4	yes	—
239	C₂×C₄	C₁₃×#121	1	4	yes	—
240	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
241	C₂×C₄	C₁₃×#124	1	2	yes	—
242	C₂×C₄	C₁₃×#125	1	2	yes	—
243	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
244	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
245	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
246	C₈	—	4	8	no	—
247	C₂²	C₂×C₁₃×#95	1	1	yes	characteristic
248	C₂²	C₄×C₁₃×C₄⋊C₄	1	2	yes	—
249	C₂²	C₁₃×#380	1	2	yes	—
250	C₂²	C₁₃×#382	1	1	yes	characteristic
251	C₂²	C₁₃×#384	1	1	yes	characteristic
252	C₂²	C₂×C₄×C₈×C₁₃	1	1	yes	characteristic
253	C₂²	C₂×C₁₃×#97	1	1	yes	characteristic
254	C₂²	C₄×C₁₃×M₁₆	1	2	yes	—
255	C₂²	C₂×C₁₃×C₂²⋊C₈	1	2	yes	—
256	C₂²	C₁₃×#407	1	2	yes	—
257	C₂²	C₂×C₁₃×C₄⋊C₈	1	2	yes	—
258	C₂²	C₂×C₁₃×C₄⋊C₈	1	4	yes	—
259	C₂²	C₁₃×#423	1	2	yes	—
260	C₂²	C₁₃×#423	1	4	yes	—
261	C₂²	C₁₃×#431	1	4	yes	—
262	C₂²	C₁₃×#432	1	4	yes	—
263	C₄	—	2	4	no	—
264	C₄	—	2	16	no	—
265	C₄	C₁₃×#336	1	8	yes	complemented
266	C₄	C₁₃×#389	1	4	yes	—
267	C₄	C₁₃×#390	1	2	yes	—
268	C₄	C₁₃×#391	1	2	yes	—
269	C₂	C₁₃×C₄⋊#95	1	1	yes	characteristic
270	C₂	C₂×C₁₃×#336	1	1	yes	characteristic
271	C₂	C₁₃×#1728	1	1	yes	characteristic
272	C₂	C₄×C₁₃×C₄⋊C₈	1	2	yes	—
273	C₂	C₁₃×#1767	1	2	yes	—
274	C₂	C₁₃×#1797	1	4	yes	—
275	C₂	C₁₃×#1831	1	2	yes	—
276	C₂	C₁₃×#1842	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₃	C₂³
commutator	C₂²	C₂×C₄×C₈×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂×C₁₃×C₄⋊C₈	C₄	semidirect
29. C₂×C₄²×C₁₃	C₈	semidirect
37. #10832	C₁₃	direct (N × H)
99. C₂×C₄×C₁₃	C₄⋊C₈	semidirect
119. C₂×C₄⋊C₈	C₄×C₁₃	semidirect
140. C₄×C₁₃	#336	semidirect
159. C₂×C₄²	C₈×C₁₃	semidirect
210. C₁₃	#10832	direct (N × H)
234. C₂×C₄	C₁₃×C₄⋊C₈	semidirect
265. C₄	C₁₃×#336	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×C₄⋊C₈	2
C₂×C₄³×C₁₃	2
C₂×C₁₃×#336	2
#10832	13

Subgroups — C13×#10832 #15697171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10832 #15697171