Subgroups — C₁₃×#12867 #15699206

Order 3328.

470 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1808	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1380	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
7	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
8	C₂×C₁₃×#107	—	2	1	no	—
9	C₂×C₁₃×#107	—	2	2	no	—
10	C₂×C₁₃×#107	C₄	1	4	yes	complemented
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₁₃×#380	—	2	1	no	—
14	C₁₃×C₂²⋊C₁₆	—	2	2	no	—
15	C₁₃×#340	—	2	1	no	—
16	C₂³×C₄×C₁₃	—	4	1	no	—
17	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	—	2	4	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	D₄	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
27	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
28	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
29	C₂×C₁₃×C₁₆	—	2	2	no	—
30	C₂×C₁₃×C₁₆	—	2	4	no	—
31	C₁₃×#107	—	2	4	no	—
32	C₁₃×#107	—	2	2	no	—
33	C₁₃×#107	—	2	4	no	—
34	C₁₃×#107	C₂×C₄	1	4	yes	—
35	C₁₃×#95	—	4	1	no	—
36	#12867	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	—	2	2	no	—
40	C₂²×C₄×C₁₃	—	4	4	no	—
41	C₂²×C₄×C₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	—	4	2	no	—
43	C₂²×C₄×C₁₃	—	8	4	no	—
44	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	—	2	4	no	—
50	C₂×C₈×C₁₃	—	2	4	no	—
51	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	#41	1	4	yes	—
55	C₁₃×C₄⋊C₄	—	2	4	no	—
56	C₁₃×C₄⋊C₄	—	4	4	no	—
57	C₁₃×C₄⋊C₄	—	4	2	no	—
58	C₁₃×C₄⋊C₄	—	4	2	no	—
59	C₁₃×C₄⋊C₄	—	4	2	no	—
60	C₁₃×C₂²⋊C₄	—	8	2	no	—
61	C₁₃×C₁₆	—	4	2	no	—
62	C₂²×#107	C₂×C₁₃	1	1	yes	characteristic
63	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
64	#1808	C₂×C₁₃	1	1	yes	characteristic
65	#1380	C₂×C₁₃	1	4	yes	complemented
66	C₂³×C₁₃	—	2	2	no	—
67	C₂³×C₁₃	—	2	2	no	—
68	C₂³×C₁₃	#102	1	2	yes	—
69	C₂³×C₁₃	SD₃₂	1	2	yes	—
70	C₂³×C₁₃	Q₃₂	1	2	yes	—
71	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	4	4	no	—
77	C₂×C₄×C₁₃	—	8	8	no	—
78	C₂×C₄×C₁₃	—	8	4	no	—
79	C₂×C₄×C₁₃	—	8	2	no	—
80	C₂×C₄×C₁₃	—	8	2	no	—
81	C₂×C₄×C₁₃	—	8	2	no	—
82	C₂×C₄×C₁₃	#102	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#123	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#137	1	2	yes	—
87	C₈×C₁₃	—	2	2	no	—
88	C₈×C₁₃	#117	1	2	yes	—
89	C₈×C₁₃	#123	1	2	yes	—
90	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
91	C₂²×C₄⋊C₄	—	2	1	no	—
92	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
93	C₂×#107	—	2	1	no	—
94	C₂×#107	—	2	2	no	—
95	C₂×#107	C₄×C₁₃	1	4	yes	complemented
96	C₂×#107	C₂²×C₁₃	1	2	yes	—
97	C₂×#107	C₂²×C₁₃	1	2	yes	—
98	#380	—	2	1	no	—
99	C₂²⋊C₁₆	—	2	2	no	—
100	#340	—	2	1	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	—	2	4	no	—
104	C₂²×C₁₃	—	2	4	no	—
105	C₂²×C₁₃	#358	1	4	yes	—
106	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
107	C₂²×C₁₃	#480	1	2	yes	—
108	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
109	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
110	C₂²×C₁₃	#509	1	2	yes	—
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	—	8	4	no	—
113	C₄×C₁₃	—	16	2	no	—
114	C₄×C₁₃	#381	1	1	yes	characteristic
115	C₄×C₁₃	#419	1	1	yes	characteristic
116	C₄×C₁₃	#483	1	2	yes	—
117	C₂³×C₄	—	4	1	no	—
118	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
121	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
122	C₂²×C₈	D₄×C₁₃	1	2	yes	—
123	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	4	1	no	—
128	C₂×C₄⋊C₄	—	4	2	no	—
129	C₂×C₂²⋊C₄	—	4	1	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	4	no	—
132	#107	—	2	4	no	—
133	#107	—	2	2	no	—
134	#107	—	2	4	no	—
135	#107	C₂×C₄×C₁₃	1	4	yes	—
136	#95	—	4	1	no	—
137	C₂×C₁₃	—	2	4	no	—
138	C₂×C₁₃	#1823	1	1	yes	characteristic
139	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
140	C₂×C₁₃	#2140	1	1	yes	characteristic
141	C₂×C₁₃	#2233	1	2	yes	—
142	C₂×C₁₃	#2236	1	2	yes	—
143	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	4	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	8	4	no	—
150	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
151	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
152	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
153	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
158	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
159	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
160	C₂×C₈	C₁₃×#41	1	4	yes	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₂²⋊C₄	—	8	2	no	—
167	C₁₆	—	4	2	no	—
168	C₁₃	#12867	1	1	yes	characteristic, direct product factor, Sylow
169	C₂³	—	2	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	C₁₃×#102	1	2	yes	—
172	C₂³	C₁₃×SD₃₂	1	2	yes	—
173	C₂³	C₁₃×Q₃₂	1	2	yes	—
174	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	8	no	—
181	C₂×C₄	—	8	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	C₁₃×#102	1	2	yes	—
186	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₄	C₁₃×#123	1	2	yes	—
188	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
189	C₂×C₄	C₁₃×#137	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	C₁₃×#117	1	2	yes	—
192	C₈	C₁₃×#123	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	4	no	—
197	C₂²	C₁₃×#358	1	4	yes	—
198	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
199	C₂²	C₁₃×#480	1	2	yes	—
200	C₂²	C₂×C₁₃×SD₃₂	1	1	yes	characteristic
201	C₂²	C₂×C₁₃×Q₃₂	1	1	yes	characteristic
202	C₂²	C₁₃×#509	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	8	4	no	—
205	C₄	—	16	2	no	—
206	C₄	C₁₃×#381	1	1	yes	characteristic
207	C₄	C₁₃×#419	1	1	yes	characteristic
208	C₄	C₁₃×#483	1	2	yes	—
209	C₂	—	2	4	no	—
210	C₂	C₁₃×#1823	1	1	yes	characteristic
211	C₂	C₂×C₁₃×#358	1	1	yes	characteristic
212	C₂	C₁₃×#2140	1	1	yes	characteristic
213	C₂	C₁₃×#2233	1	2	yes	—
214	C₂	C₁₃×#2236	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂⁴×C₁₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1380	C₂	semidirect
10. C₂×C₁₃×#107	C₄	semidirect
36. #12867	C₁₃	direct (N × H)
65. #1380	C₂×C₁₃	semidirect
95. C₂×#107	C₄×C₁₃	semidirect
168. C₁₃	#12867	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×C₁₃×#107	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₁₃×#1808	2
C₁₃×#1380	2
#12867	13

Subgroups — C13×#12867 #15699206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12867 #15699206