Subgroups — C₁₃×#12664 #15699003

Order 3328.

334 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2108	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1379	C₂	1	2	yes	maximal
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
10	C₁₃×#427	C₄	1	1	yes	characteristic
11	C₁₃×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×#364	C₄	1	4	yes	complemented
14	C₁₃×#346	—	2	2	no	—
15	C₄×C₁₃×C₁₆	—	2	2	no	—
16	C₄×C₁₃×C₁₆	C₄	1	4	yes	complemented
17	C₁₃×#339	C₄	1	2	yes	—
18	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₁₃×#117	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₁₃×M₃₂	—	2	2	no	—
24	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
25	C₂×C₁₃×C₁₆	—	2	2	no	—
26	C₂×C₁₃×C₁₆	—	2	1	no	—
27	C₂×C₁₃×C₁₆	—	2	1	no	—
28	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
31	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
32	C₂×C₁₃×C₁₆	Q₈	1	2	yes	—
33	C₁₃×#108	C₈	1	4	yes	—
34	C₁₃×#107	C₈	1	2	yes	—
35	C₁₃×#106	C₈	1	2	yes	—
36	C₁₃×#97	—	2	2	no	—
37	C₄×C₈×C₁₃	—	2	2	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
40	#12664	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×C₄×C₁₃	—	2	1	no	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₁₃×M₁₆	—	2	2	no	—
44	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	C₄²	1	2	yes	—
47	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
56	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
57	C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	1	no	—
60	C₄²×C₁₃	C₄²	1	2	yes	—
61	C₁₃×C₁₆	—	2	2	no	—
62	C₁₃×C₁₆	—	2	4	no	—
63	C₁₃×C₁₆	C₄⋊C₄	1	2	yes	—
64	C₁₃×C₁₆	C₄⋊C₄	1	2	yes	—
65	C₂×#364	C₂×C₁₃	1	1	yes	characteristic
66	#2108	C₂×C₁₃	1	2	yes	—
67	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
68	#1772	C₂×C₁₃	1	1	yes	characteristic
69	#1379	C₂×C₁₃	1	2	yes	—
70	C₂³×C₁₃	#131	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
80	C₈×C₁₃	—	2	4	no	—
81	C₈×C₁₃	C₄×C₈	1	2	yes	—
82	C₈×C₁₃	C₄×C₈	1	2	yes	—
83	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₈×C₁₃	#117	1	1	yes	characteristic
85	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
86	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
87	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
88	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
89	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
90	C₂×#108	C₄×C₁₃	1	1	yes	characteristic
91	#427	C₄×C₁₃	1	1	yes	characteristic
92	#405	C₂²×C₁₃	1	2	yes	—
93	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
94	#364	C₄×C₁₃	1	4	yes	complemented
95	#346	—	2	2	no	—
96	C₄×C₁₆	—	2	2	no	—
97	C₄×C₁₆	C₄×C₁₃	1	4	yes	complemented
98	#339	C₄×C₁₃	1	2	yes	—
99	C₂²×C₁₃	—	2	2	no	—
100	C₂²×C₁₃	#405	1	1	yes	characteristic
101	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
102	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
103	C₄×C₁₃	—	2	4	no	—
104	C₄×C₁₃	—	4	4	no	—
105	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
107	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
108	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
109	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
110	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
111	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
112	#117	C₂×C₄×C₁₃	1	2	yes	—
113	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
114	M₃₂	—	2	2	no	—
115	M₃₂	C₂×C₄×C₁₃	1	4	yes	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
120	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
121	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
122	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
123	C₂×C₁₆	Q₈×C₁₃	1	2	yes	—
124	#108	C₈×C₁₃	1	4	yes	—
125	#107	C₈×C₁₃	1	2	yes	—
126	#106	C₈×C₁₃	1	2	yes	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
130	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
131	C₂×C₁₃	—	2	1	no	—
132	C₂×C₁₃	—	2	1	no	—
133	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₁₃	#2170	1	2	yes	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
137	M₁₆	—	2	2	no	—
138	M₁₆	C₂×C₈×C₁₃	1	4	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₄²×C₁₃	1	2	yes	—
141	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
142	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
143	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
144	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
145	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
146	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
147	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
148	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
149	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
150	C₄⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	C₄²×C₁₃	1	2	yes	—
155	C₁₆	—	2	2	no	—
156	C₁₆	—	2	4	no	—
157	C₁₆	C₁₃×C₄⋊C₄	1	2	yes	—
158	C₁₆	C₁₃×C₄⋊C₄	1	2	yes	—
159	C₁₃	#12664	1	1	yes	characteristic, direct product factor, Sylow
160	C₂³	C₁₃×#131	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
166	C₂×C₄	C₄×Q₈×C₁₃	1	1	yes	characteristic
167	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
169	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	C₄×C₈×C₁₃	1	2	yes	—
172	C₈	C₄×C₈×C₁₃	1	2	yes	—
173	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
174	C₈	C₁₃×#117	1	1	yes	characteristic
175	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
176	C₈	C₄×Q₈×C₁₃	1	1	yes	characteristic
177	C₂²	—	2	2	no	—
178	C₂²	C₁₃×#405	1	1	yes	characteristic
179	C₂²	C₈×D₄×C₁₃	1	1	yes	characteristic
180	C₂²	C₈×Q₈×C₁₃	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	—	4	4	no	—
183	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
184	C₄	C₂×C₄×C₈×C₁₃	1	1	yes	characteristic
185	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
186	C₄	C₈×Q₈×C₁₃	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	C₈×C₁₃×C₄⋊C₄	1	1	yes	characteristic
190	C₂	C₁₃×#2170	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₃×C₁₆	D₄
commutator	C₄	C₂×C₄×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂³×C₁₃	#131
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃×C₁₆ ◁ C₂²×C₁₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₁₃×#364	C₄	semidirect
16. C₄×C₁₃×C₁₆	C₄	semidirect
40. #12664	C₁₃	direct (N × H)
94. #364	C₄×C₁₃	semidirect
97. C₄×C₁₆	C₄×C₁₃	semidirect
159. C₁₃	#12664	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#364	2
C₁₃×#2108	2
C₂×C₄×C₁₃×C₁₆	2
C₁₃×#1772	2
C₁₃×#1379	2
#12664	13

Subgroups — C13×#12664 #15699003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12664 #15699003