Subgroups — C₁₃×#8390 #15694729

Order 3328.

706 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#557	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1801	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1546	C₂	1	2	yes	maximal
4	C₁₃×#1487	C₂	1	2	yes	maximal
5	C₁₃×#1461	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#557	—	2	4	no	—
7	C₁₃×#557	—	2	4	no	—
8	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#125	—	2	2	no	—
11	C₂×C₁₃×#125	C₄	1	2	yes	—
12	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
13	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#431	C₂²	1	2	yes	—
17	C₁₃×#423	C₂²	1	2	yes	—
18	C₂×C₁₃×#101	—	2	1	no	—
19	C₁₃×#343	—	2	1	no	—
20	C₁₃×#332	—	2	4	no	—
21	C₁₃×Q₈⋊C₈	—	2	4	no	—
22	C₁₃×#324	—	2	2	no	—
23	C₂²×Q₈×C₁₃	—	2	1	no	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₁₃×C₄⋊Q₈	—	2	8	no	—
28	C₁₃×C₄⋊Q₈	—	2	2	no	—
29	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
30	C₁₃×#125	—	2	16	no	—
31	C₁₃×#125	—	2	4	no	—
32	C₁₃×#125	—	2	4	no	—
33	C₄×Q₈×C₁₃	—	2	8	no	—
34	C₄×Q₈×C₁₃	—	2	4	no	—
35	C₄×Q₈×C₁₃	—	2	4	no	—
36	C₄×Q₈×C₁₃	—	2	2	no	—
37	C₄×Q₈×C₁₃	—	2	2	no	—
38	C₄×Q₈×C₁₃	—	2	2	no	—
39	C₄×Q₈×C₁₃	D₄	1	4	yes	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	—	2	1	no	—
51	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×C₄⋊C₈	—	2	2	no	—
54	C₁₃×C₄⋊C₈	—	2	2	no	—
55	C₁₃×C₄⋊C₈	—	2	2	no	—
56	C₁₃×#101	—	4	2	no	—
57	C₁₃×C₂²⋊C₈	—	2	2	no	—
58	C₄×C₈×C₁₃	—	2	2	no	—
59	#8390	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×Q₈×C₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₃	—	2	1	no	—
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	2	4	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	—	2	1	no	—
66	C₂×Q₈×C₁₃	—	2	1	no	—
67	C₂×Q₈×C₁₃	—	4	2	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×Q₈×C₁₃	—	4	1	no	—
70	C₂×Q₈×C₁₃	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₁₃×M₁₆	—	4	2	no	—
80	C₁₃×M₁₆	—	4	2	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₁₃×C₄⋊C₄	—	2	8	no	—
84	C₁₃×C₄⋊C₄	—	2	8	no	—
85	C₁₃×C₄⋊C₄	—	2	8	no	—
86	C₁₃×C₄⋊C₄	—	2	1	no	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
93	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₁₃	—	2	8	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	2	4	no	—
98	C₄²×C₁₃	C₂×D₄	1	2	yes	—
99	C₄²×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×#557	C₂×C₁₃	1	1	yes	characteristic
101	#1801	C₂×C₁₃	1	1	yes	characteristic
102	#1546	C₂×C₁₃	1	2	yes	—
103	#1487	C₂×C₁₃	1	2	yes	—
104	#1461	C₂×C₁₃	1	1	yes	characteristic
105	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	Q₈×C₁₃	—	2	4	no	—
107	Q₈×C₁₃	—	2	2	no	—
108	Q₈×C₁₃	—	4	2	no	—
109	Q₈×C₁₃	—	4	4	no	—
110	Q₈×C₁₃	—	4	2	no	—
111	Q₈×C₁₃	—	4	4	no	—
112	Q₈×C₁₃	#101	1	4	yes	complemented
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	4	1	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#120	1	2	yes	—
130	C₂×C₄×C₁₃	#120	1	2	yes	—
131	C₂×C₄×C₁₃	#135	1	2	yes	—
132	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
134	C₈×C₁₃	—	8	2	no	—
135	C₈×C₁₃	—	8	2	no	—
136	#557	—	2	4	no	—
137	#557	—	2	4	no	—
138	C₂×C₄⋊Q₈	—	2	1	no	—
139	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×#125	—	2	2	no	—
141	C₂×#125	C₄×C₁₃	1	2	yes	—
142	C₂×C₄×Q₈	—	2	1	no	—
143	C₂×C₄×Q₈	C₄×C₁₃	1	2	yes	—
144	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
146	#431	C₂²×C₁₃	1	2	yes	—
147	#423	C₂²×C₁₃	1	2	yes	—
148	C₂×#101	—	2	1	no	—
149	#343	—	2	1	no	—
150	#332	—	2	4	no	—
151	Q₈⋊C₈	—	2	4	no	—
152	#324	—	2	2	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	2	1	no	—
155	C₂²×C₁₃	#379	1	1	yes	characteristic
156	C₂²×C₁₃	#416	1	1	yes	characteristic
157	C₂²×C₁₃	#417	1	1	yes	characteristic
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	4	2	no	—
162	C₄×C₁₃	—	4	2	no	—
163	C₄×C₁₃	—	4	2	no	—
164	C₄×C₁₃	—	4	4	no	—
165	C₄×C₁₃	C₂×#101	1	2	yes	—
166	C₄×C₁₃	#449	1	2	yes	—
167	C₄×C₁₃	#452	1	2	yes	—
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
170	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₄⋊Q₈	—	2	8	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	D₄×C₁₃	1	4	yes	—
175	#125	—	2	16	no	—
176	#125	—	2	4	no	—
177	#125	—	2	4	no	—
178	C₄×Q₈	—	2	8	no	—
179	C₄×Q₈	—	2	4	no	—
180	C₄×Q₈	—	2	4	no	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×Q₈	—	2	2	no	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	D₄×C₁₃	1	4	yes	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
192	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
193	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	2	2	no	—
201	#101	—	4	2	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₂×C₁₃	—	2	1	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	#1784	1	1	yes	characteristic
207	C₂×C₁₃	#1788	1	1	yes	characteristic
208	C₂×C₁₃	#1796	1	1	yes	characteristic
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	C₁₃×C₂²⋊C₄	1	2	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
224	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
242	C₄⋊C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
243	C₄²	—	2	8	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	4	no	—
247	C₄²	C₂×D₄×C₁₃	1	2	yes	—
248	C₄²	C₂×D₄×C₁₃	1	2	yes	—
249	C₁₃	#8390	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	2	2	no	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	4	no	—
255	Q₈	—	4	2	no	—
256	Q₈	—	4	4	no	—
257	Q₈	C₁₃×#101	1	4	yes	complemented
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₄	C₁₃×#120	1	2	yes	—
275	C₂×C₄	C₁₃×#120	1	2	yes	—
276	C₂×C₄	C₁₃×#135	1	2	yes	—
277	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
279	C₈	—	8	2	no	—
280	C₈	—	8	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₁₃×#379	1	1	yes	characteristic
284	C₂²	C₁₃×#416	1	1	yes	characteristic
285	C₂²	C₁₃×#417	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	C₂×C₁₃×#101	1	2	yes	—
294	C₄	C₁₃×#449	1	2	yes	—
295	C₄	C₁₃×#452	1	2	yes	—
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	C₁₃×#1784	1	1	yes	characteristic
299	C₂	C₁₃×#1788	1	1	yes	characteristic
300	C₂	C₁₃×#1796	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
59. #8390	C₁₃	direct (N × H)
112. Q₈×C₁₃	#101	semidirect
249. C₁₃	#8390	direct (N × H)
257. Q₈	C₁₃×#101	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#557	2
C₁₃×#1801	2
C₁₃×#1546	2
C₁₃×#1487	2
C₁₃×#1461	2
#8390	13

Subgroups — C13×#8390 #15694729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#8390 #15694729