Subgroups — C₁₃×#9615 #15695954

Order 3328.

382 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#1948	C₂	1	2	yes	maximal
2	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1296	C₂	1	2	yes	maximal
6	C₂×C₁₃×#107	—	2	1	no	—
7	C₂×C₁₃×#107	—	2	1	no	—
8	C₂×C₁₃×#106	—	2	2	no	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×#384	C₂²	1	2	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₁₃×#340	—	2	2	no	—
16	C₁₃×#336	—	2	1	no	—
17	C₁₃×#335	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₃	—	2	1	no	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
26	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	—	2	1	no	—
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×#107	—	4	2	no	—
30	C₁₃×#107	—	4	2	no	—
31	C₁₃×#106	—	4	4	no	—
32	C₁₃×C₄⋊C₈	—	2	2	no	—
33	C₁₃×C₄⋊C₈	—	2	1	no	—
34	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
35	C₁₃×C₄⋊C₈	Q₈	1	2	yes	—
36	C₄×C₈×C₁₃	—	4	2	no	—
37	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃	Q₈	1	2	yes	—
39	C₁₃×#95	—	2	1	no	—
40	#9615	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×C₄×C₁₃	—	2	1	no	—
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	—	4	2	no	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	2	4	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	—	4	2	no	—
52	C₂×C₈×C₁₃	—	4	1	no	—
53	C₂×C₈×C₁₃	—	4	2	no	—
54	C₂×C₈×C₁₃	—	4	1	no	—
55	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
58	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	1	no	—
61	C₁₃×C₄⋊C₄	—	4	2	no	—
62	C₁₃×C₄⋊C₄	—	4	2	no	—
63	C₁₃×C₄⋊C₄	—	4	2	no	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₁₃×C₄⋊C₄	—	8	2	no	—
66	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
67	C₁₃×C₄⋊C₄	Q₁₆	1	2	yes	—
68	C₄²×C₁₃	—	4	1	no	—
69	C₄²×C₁₃	—	4	1	no	—
70	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₁₃	#41	1	1	yes	characteristic
74	#1948	C₂×C₁₃	1	2	yes	—
75	#1845	C₂×C₁₃	1	1	yes	characteristic
76	C₈×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
77	C₂×#335	C₂×C₁₃	1	1	yes	characteristic
78	#1296	C₂×C₁₃	1	2	yes	—
79	C₂³×C₁₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	4	1	no	—
85	C₂×C₄×C₁₃	—	4	1	no	—
86	C₂×C₄×C₁₃	—	4	1	no	—
87	C₂×C₄×C₁₃	—	4	1	no	—
88	C₂×C₄×C₁₃	—	8	2	no	—
89	C₂×C₄×C₁₃	—	8	2	no	—
90	C₂×C₄×C₁₃	—	8	2	no	—
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
103	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	—	4	4	no	—
107	C₈×C₁₃	—	4	2	no	—
108	C₈×C₁₃	#106	1	4	yes	complemented
109	C₂×#107	—	2	1	no	—
110	C₂×#107	—	2	1	no	—
111	C₂×#106	—	2	2	no	—
112	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
114	C₂×C₄×C₈	—	2	1	no	—
115	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
116	#384	C₂²×C₁₃	1	2	yes	—
117	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
118	#340	—	2	2	no	—
119	#336	—	2	1	no	—
120	#335	C₄×C₁₃	1	4	yes	complemented
121	C₂²×C₁₃	#389	1	1	yes	characteristic
122	C₂²×C₁₃	#460	1	1	yes	characteristic
123	C₂²×C₁₃	#461	1	1	yes	characteristic
124	C₂²×C₁₃	#466	1	1	yes	characteristic
125	C₂²×C₁₃	#467	1	1	yes	characteristic
126	C₂²×C₁₃	#472	1	1	yes	characteristic
127	C₂²×C₁₃	#473	1	1	yes	characteristic
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	4	2	no	—
131	C₄×C₁₃	—	4	2	no	—
132	C₄×C₁₃	—	16	2	no	—
133	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
134	C₄×C₁₃	#428	1	1	yes	characteristic
135	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
136	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
137	C₄×C₁₃	#475	1	1	yes	characteristic
138	C₄×C₁₃	#476	1	1	yes	characteristic
139	C₄×C₁₃	#499	1	1	yes	characteristic
140	C₄×C₁₃	#500	1	1	yes	characteristic
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
144	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
152	#107	—	4	2	no	—
153	#107	—	4	2	no	—
154	#106	—	4	4	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
158	C₄⋊C₈	Q₈×C₁₃	1	2	yes	—
159	C₄×C₈	—	4	2	no	—
160	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
161	C₄×C₈	Q₈×C₁₃	1	2	yes	—
162	#95	—	2	1	no	—
163	C₂×C₁₃	#1668	1	1	yes	characteristic
164	C₂×C₁₃	#1670	1	1	yes	characteristic
165	C₂×C₁₃	#1673	1	1	yes	characteristic
166	C₂×C₁₃	#1675	1	1	yes	characteristic
167	C₂×C₁₃	#1942	1	1	yes	characteristic
168	C₂×C₁₃	#1948	1	1	yes	characteristic
169	C₂×C₁₃	#1952	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
175	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
185	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
186	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
187	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	8	2	no	—
195	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
196	C₄⋊C₄	C₁₃×Q₁₆	1	2	yes	—
197	C₄²	—	4	1	no	—
198	C₄²	—	4	1	no	—
199	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
200	C₄²	C₂×Q₈×C₁₃	1	1	yes	characteristic
201	C₄²	C₂×Q₈×C₁₃	1	1	yes	characteristic
202	C₄²	C₁₃×#41	1	1	yes	characteristic
203	C₁₃	#9615	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³	C₁₃×#121	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₄	C₄×Q₈×C₁₃	1	1	yes	characteristic
219	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
220	C₂×C₄	C₁₃×#125	1	1	yes	characteristic
221	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
222	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
223	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
224	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
225	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
226	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
227	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
228	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
229	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₈	—	4	2	no	—
233	C₈	C₁₃×#106	1	4	yes	complemented
234	C₂²	C₁₃×#389	1	1	yes	characteristic
235	C₂²	C₁₃×#460	1	1	yes	characteristic
236	C₂²	C₁₃×#461	1	1	yes	characteristic
237	C₂²	C₁₃×#466	1	1	yes	characteristic
238	C₂²	C₁₃×#467	1	1	yes	characteristic
239	C₂²	C₁₃×#472	1	1	yes	characteristic
240	C₂²	C₁₃×#473	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	16	2	no	—
246	C₄	C₂×C₁₃×#106	1	1	yes	characteristic
247	C₄	C₁₃×#428	1	1	yes	characteristic
248	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
249	C₄	C₄×C₁₃×Q₁₆	1	1	yes	characteristic
250	C₄	C₁₃×#475	1	1	yes	characteristic
251	C₄	C₁₃×#476	1	1	yes	characteristic
252	C₄	C₁₃×#499	1	1	yes	characteristic
253	C₄	C₁₃×#500	1	1	yes	characteristic
254	C₂	C₁₃×#1668	1	1	yes	characteristic
255	C₂	C₁₃×#1670	1	1	yes	characteristic
256	C₂	C₁₃×#1673	1	1	yes	characteristic
257	C₂	C₁₃×#1675	1	1	yes	characteristic
258	C₂	C₁₃×#1942	1	1	yes	characteristic
259	C₂	C₁₃×#1948	1	1	yes	characteristic
260	C₂	C₁₃×#1952	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	#121
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₁₃×#335	C₄	semidirect
40. #9615	C₁₃	direct (N × H)
108. C₈×C₁₃	#106	semidirect
120. #335	C₄×C₁₃	semidirect
203. C₁₃	#9615	direct (N × H)
233. C₈	C₁₃×#106	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#1948	2
C₁₃×#1845	2
C₈×C₁₃×C₄⋊C₄	2
C₂×C₁₃×#335	2
C₁₃×#1296	2
#9615	13

Subgroups — C13×#9615 #15695954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9615 #15695954