Subgroups — C₁₃×#9876 #15696215

Order 3328.

486 conjugacy classes of subgroups.

The subgroup lattice has 212 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×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×#97	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	—	2	2	no	—
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₁₃×#97	—	2	2	no	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
10	C₁₃×#389	C₄	1	2	yes	—
11	C₁₃×#387	—	2	2	no	—
12	C₁₃×#382	C₄	1	1	yes	characteristic
13	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄³×C₁₃	C₂²	1	1	yes	characteristic
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	C₁₃×C₄⋊C₈	—	4	4	no	—
31	C₁₃×#97	—	2	8	no	—
32	C₄×C₈×C₁₃	—	2	2	no	—
33	C₄×C₈×C₁₃	Q₈	1	4	yes	—
34	C₁₃×#95	—	2	2	no	—
35	C₁₃×#95	—	2	2	no	—
36	C₁₃×#95	—	2	2	no	—
37	#9876	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	—	2	2	no	—
40	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	—	2	8	no	—
46	C₂×C₈×C₁₃	—	2	4	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₂×C₈×C₁₃	—	4	4	no	—
49	C₁₃×C₄⋊C₄	—	2	8	no	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	2	4	no	—
52	C₁₃×C₄⋊C₄	—	4	4	no	—
53	C₁₃×C₄⋊C₄	M₁₆	1	4	yes	—
54	C₄²×C₁₃	—	2	4	no	—
55	C₄²×C₁₃	—	2	2	no	—
56	C₄²×C₁₃	—	2	2	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	4	2	no	—
60	C₄²×C₁₃	—	4	2	no	—
61	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
62	C₄²×C₁₃	#41	1	2	yes	—
63	C₄×#125	C₂×C₁₃	1	1	yes	characteristic
64	#1842	C₂×C₁₃	1	1	yes	characteristic
65	C₄×#97	C₂×C₁₃	1	1	yes	characteristic
66	#1279	C₂×C₁₃	1	4	yes	—
67	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	4	2	no	—
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	—	4	2	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	#104	1	4	yes	—
81	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₁₃	#131	1	2	yes	—
87	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×C₄×C₁₃	#137	1	2	yes	—
89	C₈×C₁₃	—	2	8	no	—
90	C₈×C₁₃	—	8	2	no	—
91	C₂×#125	C₄×C₁₃	1	1	yes	characteristic
92	C₂×C₄⋊C₈	—	2	2	no	—
93	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
94	C₂×#97	—	2	2	no	—
95	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
96	#389	C₄×C₁₃	1	2	yes	—
97	#387	—	2	2	no	—
98	#382	C₄×C₁₃	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	2	2	no	—
100	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
101	C₄³	C₂²×C₁₃	1	1	yes	characteristic
102	C₂²×C₁₃	#391	1	1	yes	characteristic
103	C₂²×C₁₃	#407	1	1	yes	characteristic
104	C₂²×C₁₃	#408	1	1	yes	characteristic
105	C₂²×C₁₃	#417	1	2	yes	—
106	C₂²×C₁₃	C₂×#104	1	2	yes	—
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	—	2	4	no	—
109	C₄×C₁₃	—	4	2	no	—
110	C₄×C₁₃	—	4	4	no	—
111	C₄×C₁₃	—	8	2	no	—
112	C₄×C₁₃	#446	1	4	yes	—
113	C₄×C₁₃	#490	1	1	yes	characteristic
114	C₄×C₁₃	#491	1	1	yes	characteristic
115	C₄×C₁₃	#501	1	2	yes	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	2	2	no	—
118	#125	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
123	C₂×C₄²	—	2	2	no	—
124	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
126	C₂×C₄²	D₄×C₁₃	1	2	yes	—
127	C₂×C₄²	D₄×C₁₃	1	2	yes	—
128	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
129	C₄⋊C₈	—	2	4	no	—
130	C₄⋊C₈	—	4	4	no	—
131	#97	—	2	8	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	Q₈×C₁₃	1	4	yes	—
134	#95	—	2	2	no	—
135	#95	—	2	2	no	—
136	#95	—	2	2	no	—
137	C₂×C₁₃	#1483	1	2	yes	—
138	C₂×C₁₃	#1489	1	2	yes	—
139	C₂×C₁₃	#1985	1	1	yes	characteristic
140	C₂×C₁₃	#1992	1	1	yes	characteristic
141	C₂×C₁₃	#1994	1	1	yes	characteristic
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
145	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
146	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	4	no	—
153	C₄⋊C₄	—	2	8	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	C₁₃×M₁₆	1	4	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	4	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
166	C₄²	C₁₃×#41	1	2	yes	—
167	C₁₃	#9876	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₁₃×#104	1	4	yes	—
182	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
183	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
184	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
185	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
187	C₂×C₄	C₁₃×#131	1	2	yes	—
188	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
189	C₂×C₄	C₁₃×#137	1	2	yes	—
190	C₈	—	2	8	no	—
191	C₈	—	8	2	no	—
192	C₂²	C₁₃×#391	1	1	yes	characteristic
193	C₂²	C₁₃×#407	1	1	yes	characteristic
194	C₂²	C₁₃×#408	1	1	yes	characteristic
195	C₂²	C₁₃×#417	1	2	yes	—
196	C₂²	C₂×C₁₃×#104	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	8	2	no	—
202	C₄	C₁₃×#446	1	4	yes	—
203	C₄	C₁₃×#490	1	1	yes	characteristic
204	C₄	C₁₃×#491	1	1	yes	characteristic
205	C₄	C₁₃×#501	1	2	yes	—
206	C₂	C₁₃×#1483	1	2	yes	—
207	C₂	C₁₃×#1489	1	2	yes	—
208	C₂	C₁₃×#1985	1	1	yes	characteristic
209	C₂	C₁₃×#1992	1	1	yes	characteristic
210	C₂	C₁₃×#1994	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
37. #9876	C₁₃	direct (N × H)
167. C₁₃	#9876	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×#1842	2
C₄×C₁₃×#97	2
C₁₃×#1279	2
#9876	13

Subgroups — C13×#9876 #15696215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9876 #15696215