Subgroups — C₁₃×#10842 #15697181

Order 3328.

950 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#382	C₂	1	2	yes	maximal
3	C₂×C₁₃×#340	C₂	1	4	yes	maximal
4	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
5	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	—	2	4	no	—
8	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	complemented
9	C₁₃×#382	—	2	4	no	—
10	C₁₃×#382	C₄	1	8	yes	complemented
11	C₂×C₁₃×#95	—	2	4	no	—
12	C₁₃×#340	—	2	8	no	—
13	C₂³×C₄×C₁₃	—	2	4	no	—
14	C₂³×C₄×C₁₃	D₄	1	2	yes	—
15	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	—	2	4	no	—
17	C₂²×C₈×C₁₃	—	2	4	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₁₃	D₄	1	4	yes	—
20	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
21	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
24	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₃	D₄	1	4	yes	—
28	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
29	C₄×C₈×C₁₃	—	2	8	no	—
30	C₁₃×#95	—	2	16	no	—
31	#10842	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	—	2	8	no	—
34	C₂²×C₄×C₁₃	—	2	8	no	—
35	C₂²×C₄×C₁₃	—	4	4	no	—
36	C₂²×C₄×C₁₃	—	4	4	no	—
37	C₂²×C₄×C₁₃	—	4	16	no	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	8	yes	—
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₁₃	#41	1	4	yes	—
47	C₂×C₈×C₁₃	—	2	4	no	—
48	C₂×C₈×C₁₃	—	2	4	no	—
49	C₂×C₈×C₁₃	—	2	8	no	—
50	C₂×C₈×C₁₃	—	2	8	no	—
51	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
52	C₁₃×C₄⋊C₄	—	2	16	no	—
53	C₁₃×C₄⋊C₄	—	4	8	no	—
54	C₄²×C₁₃	C₄²	1	4	yes	—
55	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₈	C₂×C₁₃	1	1	yes	characteristic
58	C₂×#382	C₂×C₁₃	1	2	yes	—
59	C₂×#340	C₂×C₁₃	1	4	yes	—
60	C₂³×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
61	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂³×C₁₃	#124	1	2	yes	—
63	C₂³×C₁₃	#127	1	1	yes	characteristic
64	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂³×C₁₃	#135	1	8	yes	—
66	C₂×C₄×C₁₃	—	2	16	no	—
67	C₂×C₄×C₁₃	—	2	8	no	—
68	C₂×C₄×C₁₃	—	4	8	no	—
69	C₂×C₄×C₁₃	—	4	16	no	—
70	C₂×C₄×C₁₃	—	8	8	no	—
71	C₂×C₄×C₁₃	—	8	8	no	—
72	C₂×C₄×C₁₃	#95	1	8	yes	—
73	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₁₃	#117	1	2	yes	—
78	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
79	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
80	C₂×C₄×C₁₃	#121	1	4	yes	—
81	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
82	C₂×C₄×C₁₃	#125	1	2	yes	—
83	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
84	C₈×C₁₃	—	2	8	no	—
85	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
86	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
87	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
88	C₂×C₄×C₈	—	2	4	no	—
89	C₂×C₄×C₈	C₄×C₁₃	1	4	yes	complemented
90	#382	—	2	4	no	—
91	#382	C₄×C₁₃	1	8	yes	complemented
92	C₂×#95	—	2	4	no	—
93	#340	—	2	8	no	—
94	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
95	C₂²×C₁₃	#382	1	1	yes	characteristic
96	C₂²×C₁₃	#384	1	1	yes	characteristic
97	C₂²×C₁₃	#390	1	2	yes	—
98	C₂²×C₁₃	#391	1	2	yes	—
99	C₂²×C₁₃	#416	1	8	yes	—
100	C₂²×C₁₃	#427	1	4	yes	—
101	C₂²×C₁₃	#488	1	8	yes	—
102	C₂²×C₁₃	#495	1	4	yes	—
103	C₂²×C₁₃	#499	1	4	yes	—
104	C₄×C₁₃	—	2	8	no	—
105	C₄×C₁₃	—	8	8	no	—
106	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
107	C₄×C₁₃	#380	1	2	yes	—
108	C₄×C₁₃	#389	1	4	yes	—
109	C₂³×C₄	—	2	4	no	—
110	C₂³×C₄	D₄×C₁₃	1	2	yes	—
111	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
112	C₂²×C₈	—	2	4	no	—
113	C₂²×C₈	—	2	4	no	—
114	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
115	C₂²×C₈	D₄×C₁₃	1	4	yes	—
116	C₂²×C₈	Q₈×C₁₃	1	4	yes	—
117	C₂×C₄⋊C₄	—	2	8	no	—
118	C₂×C₄⋊C₄	—	2	8	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	8	yes	—
121	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
122	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
123	C₂×C₄²	D₄×C₁₃	1	4	yes	—
124	C₂×C₄²	Q₈×C₁₃	1	2	yes	—
125	C₄×C₈	—	2	8	no	—
126	#95	—	2	16	no	—
127	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
128	C₂×C₁₃	#1735	1	2	yes	—
129	C₂×C₁₃	#1835	1	2	yes	—
130	C₂×C₁₃	#1846	1	2	yes	—
131	C₂×C₁₃	#1958	1	4	yes	—
132	C₂×C₁₃	#1991	1	4	yes	—
133	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	4	4	no	—
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	—	4	16	no	—
139	C₂²×C₄	C₁₃×C₂²⋊C₄	1	8	yes	—
140	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
141	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
143	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
145	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
147	C₂²×C₄	C₁₃×#41	1	4	yes	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	C₁₃×C₄⋊C₄	1	8	yes	—
153	C₄⋊C₄	—	2	16	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄²	C₄²×C₁₃	1	4	yes	—
156	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
157	C₄²	C₁₃×C₄⋊C₄	1	8	yes	—
158	C₁₃	#10842	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
160	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
161	C₂³	C₁₃×#124	1	2	yes	—
162	C₂³	C₁₃×#127	1	1	yes	characteristic
163	C₂³	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
164	C₂³	C₁₃×#135	1	8	yes	—
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	4	8	no	—
168	C₂×C₄	—	4	16	no	—
169	C₂×C₄	—	8	8	no	—
170	C₂×C₄	—	8	8	no	—
171	C₂×C₄	C₁₃×#95	1	8	yes	—
172	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
175	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
176	C₂×C₄	C₁₃×#117	1	2	yes	—
177	C₂×C₄	C₄×D₄×C₁₃	1	4	yes	—
178	C₂×C₄	C₄×Q₈×C₁₃	1	4	yes	—
179	C₂×C₄	C₁₃×#121	1	4	yes	—
180	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
181	C₂×C₄	C₁₃×#125	1	2	yes	—
182	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
183	C₈	—	2	8	no	—
184	C₂²	C₂×C₁₃×#95	1	1	yes	characteristic
185	C₂²	C₁₃×#382	1	1	yes	characteristic
186	C₂²	C₁₃×#384	1	1	yes	characteristic
187	C₂²	C₁₃×#390	1	2	yes	—
188	C₂²	C₁₃×#391	1	2	yes	—
189	C₂²	C₁₃×#416	1	8	yes	—
190	C₂²	C₁₃×#427	1	4	yes	—
191	C₂²	C₁₃×#488	1	8	yes	—
192	C₂²	C₁₃×#495	1	4	yes	—
193	C₂²	C₁₃×#499	1	4	yes	—
194	C₄	—	2	8	no	—
195	C₄	—	8	8	no	—
196	C₄	C₄×C₁₃×C₄⋊C₄	1	2	yes	—
197	C₄	C₁₃×#380	1	2	yes	—
198	C₄	C₁₃×#389	1	4	yes	—
199	C₂	C₁₃×C₄⋊#95	1	1	yes	characteristic
200	C₂	C₁₃×#1735	1	2	yes	—
201	C₂	C₁₃×#1835	1	2	yes	—
202	C₂	C₁₃×#1846	1	2	yes	—
203	C₂	C₁₃×#1958	1	4	yes	—
204	C₂	C₁₃×#1991	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂×C₄×C₈×C₁₃	C₄	semidirect
10. C₁₃×#382	C₄	semidirect
31. #10842	C₁₃	direct (N × H)
89. C₂×C₄×C₈	C₄×C₁₃	semidirect
91. #382	C₄×C₁₃	semidirect
158. C₁₃	#10842	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₁₃	2
C₂×C₁₃×#382	2
C₂×C₁₃×#340	2
#10842	13

Subgroups — C13×#10842 #15697181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10842 #15697181