Subgroups — C₁₃×#12670 #15699009

Order 3328.

318 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1772	C₂	1	2	yes	maximal
5	C₁₃×#1382	C₂	1	2	yes	maximal
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#108	C₂²	1	2	yes	—
9	C₁₃×#427	C₂²	1	2	yes	—
10	C₁₃×#405	—	2	2	no	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#368	C₄	1	4	yes	complemented
13	C₁₃×#367	C₄	1	2	yes	complemented
14	C₁₃×#366	C₄	1	2	yes	complemented
15	C₄×C₁₃×C₁₆	C₄	1	4	yes	complemented
16	C₁₃×#339	—	2	2	no	—
17	C₂×C₁₃×M₁₆	—	2	2	no	—
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₁₃×#117	—	2	2	no	—
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	—	2	1	no	—
23	C₂×C₁₃×C₁₆	—	2	1	no	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
29	C₁₃×#108	—	2	2	no	—
30	C₁₃×#108	C₂×C₄	1	4	yes	—
31	C₁₃×#107	C₂×C₄	1	4	yes	—
32	C₁₃×#106	—	2	2	no	—
33	C₁₃×#97	—	4	2	no	—
34	C₄×C₈×C₁₃	—	4	2	no	—
35	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₃	Q₈	1	2	yes	—
37	#12670	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₄×C₁₃	—	2	1	no	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₁₃×M₁₆	—	2	4	no	—
41	C₁₃×M₁₆	—	4	2	no	—
42	C₂×C₈×C₁₃	—	4	2	no	—
43	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
45	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
46	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₁₃×C₂²⋊C₄	—	4	2	no	—
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	—	4	2	no	—
57	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
58	C₁₃×C₁₆	—	2	2	no	—
59	C₁₃×C₁₆	C₄²	1	2	yes	—
60	C₁₃×C₁₆	C₄²	1	2	yes	—
61	C₂×#368	C₂×C₁₃	1	1	yes	characteristic
62	#2158	C₂×C₁₃	1	1	yes	characteristic
63	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
64	#1772	C₂×C₁₃	1	2	yes	—
65	#1382	C₂×C₁₃	1	2	yes	—
66	C₂³×C₁₃	#135	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	4	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	4	4	no	—
70	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
76	C₈×C₁₃	—	4	4	no	—
77	C₈×C₁₃	#107	1	2	yes	—
78	C₈×C₁₃	#107	1	2	yes	—
79	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
80	C₈×C₁₃	#117	1	1	yes	characteristic
81	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
83	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
84	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
85	C₂×#108	C₂²×C₁₃	1	2	yes	—
86	#427	C₂²×C₁₃	1	2	yes	—
87	#405	—	2	2	no	—
88	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
89	#368	C₄×C₁₃	1	4	yes	complemented
90	#367	C₄×C₁₃	1	2	yes	complemented
91	#366	C₄×C₁₃	1	2	yes	complemented
92	C₄×C₁₆	C₄×C₁₃	1	4	yes	complemented
93	#339	—	2	2	no	—
94	C₂²×C₁₃	—	2	2	no	—
95	C₂²×C₁₃	#427	1	1	yes	characteristic
96	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
97	C₂²×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
98	C₄×C₁₃	—	2	4	no	—
99	C₄×C₁₃	—	8	4	no	—
100	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
102	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
103	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
104	C₂×M₁₆	—	2	2	no	—
105	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
107	#117	—	2	2	no	—
108	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
109	C₂×C₁₆	—	2	1	no	—
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
112	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
113	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
114	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
115	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
116	#108	—	2	2	no	—
117	#108	C₂×C₄×C₁₃	1	4	yes	—
118	#107	C₂×C₄×C₁₃	1	4	yes	—
119	#106	—	2	2	no	—
120	#97	—	4	2	no	—
121	C₄×C₈	—	4	2	no	—
122	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
123	C₄×C₈	Q₈×C₁₃	1	2	yes	—
124	C₂×C₁₃	—	2	1	no	—
125	C₂×C₁₃	—	2	1	no	—
126	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
127	C₂×C₁₃	#2178	1	2	yes	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
130	M₁₆	—	2	4	no	—
131	M₁₆	—	4	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
134	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
135	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
136	C₂×C₈	C₁₃×Q₁₆	1	2	yes	—
137	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
138	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
140	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
141	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
142	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₄²	—	2	1	no	—
146	C₄²	—	4	2	no	—
147	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₄²×C₁₃	1	2	yes	—
150	C₁₆	C₄²×C₁₃	1	2	yes	—
151	C₁₃	#12670	1	1	yes	characteristic, direct product factor, Sylow
152	C₂³	C₁₃×#135	1	1	yes	characteristic
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
158	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
159	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
160	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
161	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
162	C₈	—	4	4	no	—
163	C₈	C₁₃×#107	1	2	yes	—
164	C₈	C₁₃×#107	1	2	yes	—
165	C₈	C₂×C₄²×C₁₃	1	1	yes	characteristic
166	C₈	C₁₃×#117	1	1	yes	characteristic
167	C₈	C₄×Q₈×C₁₃	1	1	yes	characteristic
168	C₈	C₄×Q₈×C₁₃	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	C₁₃×#427	1	1	yes	characteristic
171	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
172	C₂²	C₄×C₁₃×Q₁₆	1	1	yes	characteristic
173	C₄	—	2	4	no	—
174	C₄	—	8	4	no	—
175	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
176	C₄	C₂×C₁₃×#107	1	1	yes	characteristic
177	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
178	C₄	C₄×C₁₃×Q₁₆	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₄×C₁₃×#107	1	1	yes	characteristic
182	C₂	C₁₃×#2178	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₃	D₈
commutator	C₈	C₂×C₄²×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₄×D₈
prime radical	C₂³×C₁₃	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₃×#368	C₄	semidirect
13. C₁₃×#367	C₄	semidirect
14. C₁₃×#366	C₄	semidirect
15. C₄×C₁₃×C₁₆	C₄	semidirect
37. #12670	C₁₃	direct (N × H)
89. #368	C₄×C₁₃	semidirect
90. #367	C₄×C₁₃	semidirect
91. #366	C₄×C₁₃	semidirect
92. C₄×C₁₆	C₄×C₁₃	semidirect
151. C₁₃	#12670	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×#368	2
C₁₃×#2158	2
C₂×C₄×C₁₃×C₁₆	2
C₁₃×#1772	2
C₁₃×#1382	2
#12670	13

Subgroups — C13×#12670 #15699009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12670 #15699009