Subgroups — C₁₃×#9875 #15696214

Order 3328.

434 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1279	C₂	1	4	yes	maximal
5	C₂×C₁₃×#125	C₄	1	1	yes	characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
8	C₁₃×#389	C₄	1	2	yes	—
9	C₁₃×#387	—	2	2	no	—
10	C₁₃×#382	C₄	1	1	yes	characteristic
11	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
12	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₄³×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#336	—	2	2	no	—
15	C₁₃×#336	—	2	2	no	—
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₁₃×#125	—	2	4	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
22	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₃	—	2	2	no	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	2	yes	—
27	C₂×C₄²×C₁₃	D₄	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×C₄⋊C₈	—	2	4	no	—
30	C₄×C₈×C₁₃	—	2	2	no	—
31	C₄×C₈×C₁₃	Q₈	1	4	yes	—
32	C₁₃×#95	—	2	2	no	—
33	C₁₃×#95	—	2	2	no	—
34	C₁₃×#95	—	2	2	no	—
35	#9875	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₄×C₁₃	—	2	2	no	—
37	C₂²×C₄×C₁₃	—	2	2	no	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	—	2	4	no	—
44	C₂×C₈×C₁₃	—	4	4	no	—
45	C₂×C₈×C₁₃	—	4	2	no	—
46	C₂×C₈×C₁₃	—	4	4	no	—
47	C₁₃×C₄⋊C₄	—	2	8	no	—
48	C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₁₃×C₄⋊C₄	—	2	4	no	—
50	C₁₃×C₄⋊C₄	—	4	4	no	—
51	C₁₃×C₄⋊C₄	M₁₆	1	4	yes	—
52	C₄²×C₁₃	—	2	2	no	—
53	C₄²×C₁₃	—	2	2	no	—
54	C₄²×C₁₃	—	2	4	no	—
55	C₄²×C₁₃	—	4	2	no	—
56	C₄²×C₁₃	—	4	4	no	—
57	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
58	C₄²×C₁₃	#41	1	2	yes	—
59	C₄×#125	C₂×C₁₃	1	1	yes	characteristic
60	#1831	C₂×C₁₃	1	1	yes	characteristic
61	#1744	C₂×C₁₃	1	1	yes	characteristic
62	#1279	C₂×C₁₃	1	4	yes	—
63	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	4	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	4	2	no	—
73	C₂×C₄×C₁₃	—	4	2	no	—
74	C₂×C₄×C₁₃	—	4	2	no	—
75	C₂×C₄×C₁₃	—	4	2	no	—
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₁₃	#131	1	2	yes	—
83	C₂×C₄×C₁₃	#135	1	4	yes	—
84	C₈×C₁₃	—	4	4	no	—
85	C₈×C₁₃	—	8	2	no	—
86	C₂×#125	C₄×C₁₃	1	1	yes	characteristic
87	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
88	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
89	#389	C₄×C₁₃	1	2	yes	—
90	#387	—	2	2	no	—
91	#382	C₄×C₁₃	1	1	yes	characteristic
92	C₄×C₄⋊C₄	—	2	2	no	—
93	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
94	C₄³	C₂²×C₁₃	1	1	yes	characteristic
95	#336	—	2	2	no	—
96	#336	—	2	2	no	—
97	C₂²×C₁₃	#391	1	1	yes	characteristic
98	C₂²×C₁₃	#407	1	1	yes	characteristic
99	C₂²×C₁₃	#408	1	1	yes	characteristic
100	C₂²×C₁₃	#416	1	2	yes	—
101	C₂²×C₁₃	#421	1	2	yes	—
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	4	2	no	—
104	C₄×C₁₃	—	4	2	no	—
105	C₄×C₁₃	—	4	4	no	—
106	C₄×C₁₃	—	8	2	no	—
107	C₄×C₁₃	#446	1	4	yes	—
108	C₄×C₁₃	#488	1	2	yes	—
109	C₄×C₁₃	#499	1	2	yes	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	2	2	no	—
112	#125	—	2	4	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
116	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
117	C₂×C₄²	—	2	2	no	—
118	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
119	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
120	C₂×C₄²	D₄×C₁₃	1	2	yes	—
121	C₂×C₄²	D₄×C₁₃	1	2	yes	—
122	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
123	C₄⋊C₈	—	2	4	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	Q₈×C₁₃	1	4	yes	—
126	#95	—	2	2	no	—
127	#95	—	2	2	no	—
128	#95	—	2	2	no	—
129	C₂×C₁₃	#1482	1	2	yes	—
130	C₂×C₁₃	#1488	1	2	yes	—
131	C₂×C₁₃	#1985	1	1	yes	characteristic
132	C₂×C₁₃	#1991	1	1	yes	characteristic
133	C₂×C₁₃	#1995	1	1	yes	characteristic
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
137	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
138	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	4	4	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	4	no	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	C₁₃×M₁₆	1	4	yes	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	4	no	—
153	C₄²	—	4	2	no	—
154	C₄²	—	4	4	no	—
155	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
156	C₄²	C₁₃×#41	1	2	yes	—
157	C₁₃	#9875	1	1	yes	characteristic, direct product factor, Sylow
158	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
173	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
174	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
175	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
176	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
177	C₂×C₄	C₁₃×#131	1	2	yes	—
178	C₂×C₄	C₁₃×#135	1	4	yes	—
179	C₈	—	4	4	no	—
180	C₈	—	8	2	no	—
181	C₂²	C₁₃×#391	1	1	yes	characteristic
182	C₂²	C₁₃×#407	1	1	yes	characteristic
183	C₂²	C₁₃×#408	1	1	yes	characteristic
184	C₂²	C₁₃×#416	1	2	yes	—
185	C₂²	C₁₃×#421	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	4	no	—
190	C₄	—	8	2	no	—
191	C₄	C₁₃×#446	1	4	yes	—
192	C₄	C₁₃×#488	1	2	yes	—
193	C₄	C₁₃×#499	1	2	yes	—
194	C₂	C₁₃×#1482	1	2	yes	—
195	C₂	C₁₃×#1488	1	2	yes	—
196	C₂	C₁₃×#1985	1	1	yes	characteristic
197	C₂	C₁₃×#1991	1	1	yes	characteristic
198	C₂	C₁₃×#1995	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
35. #9875	C₁₃	direct (N × H)
157. C₁₃	#9875	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#125	2
C₁₃×#1831	2
C₁₃×#1744	2
C₁₃×#1279	2
#9875	13

Subgroups — C13×#9875 #15696214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9875 #15696214