Subgroups — C₁₃×#12674 #15699013

Order 3328.

358 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×#367	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₃×#107	C₂	1	2	yes	maximal
5	C₁₃×#1380	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×#107	C₂²	1	2	yes	—
9	C₂×C₁₃×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
12	C₁₃×#367	C₄	1	4	yes	complemented
13	C₁₃×#366	C₄	1	4	yes	complemented
14	C₁₃×#346	C₄	1	4	yes	complemented
15	C₁₃×#340	—	2	2	no	—
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄²×C₁₃	—	4	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	2	yes	—
27	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
28	C₁₃×#107	—	2	2	no	—
29	C₁₃×#107	—	2	2	no	—
30	C₁₃×#107	C₂×C₄	1	4	yes	—
31	C₁₃×#107	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₃	Q₈	1	2	yes	—
34	C₁₃×#95	—	4	2	no	—
35	#12674	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₄×C₁₃	—	4	2	no	—
37	C₂²×C₄×C₁₃	—	4	2	no	—
38	C₂²×C₄×C₁₃	D₈	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	—	2	1	no	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
48	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
49	C₁₃×C₄⋊C₄	—	2	4	no	—
50	C₁₃×C₄⋊C₄	—	2	4	no	—
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₁₃×C₄⋊C₄	—	4	2	no	—
53	C₄²×C₁₃	—	8	4	no	—
54	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
55	C₄²×C₁₃	Q₁₆	1	2	yes	—
56	C₁₃×C₁₆	—	2	2	no	—
57	C₁₃×C₁₆	C₄²	1	4	yes	complemented
58	C₂×#367	C₂×C₁₃	1	1	yes	characteristic
59	C₂×#366	C₂×C₁₃	1	1	yes	characteristic
60	C₂×#346	C₂×C₁₃	1	1	yes	characteristic
61	C₄×#107	C₂×C₁₃	1	2	yes	—
62	#1380	C₂×C₁₃	1	2	yes	—
63	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	4	4	no	—
67	C₂×C₄×C₁₃	—	4	4	no	—
68	C₂×C₄×C₁₃	—	8	2	no	—
69	C₂×C₄×C₁₃	—	8	2	no	—
70	C₂×C₄×C₁₃	—	8	2	no	—
71	C₂×C₄×C₁₃	—	8	2	no	—
72	C₂×C₄×C₁₃	#107	1	4	yes	—
73	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
79	C₈×C₁₃	—	2	2	no	—
80	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
81	C₈×C₁₃	#117	1	1	yes	characteristic
82	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
83	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
84	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
85	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
86	C₂×#107	C₂²×C₁₃	1	2	yes	—
87	C₂×#107	C₂²×C₁₃	1	2	yes	—
88	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
89	C₄×C₄⋊C₄	—	2	2	no	—
90	#367	C₄×C₁₃	1	4	yes	complemented
91	#366	C₄×C₁₃	1	4	yes	complemented
92	#346	C₄×C₁₃	1	4	yes	complemented
93	#340	—	2	2	no	—
94	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
95	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
96	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
97	C₂²×C₁₃	#509	1	2	yes	—
98	C₂²×C₁₃	#510	1	2	yes	—
99	C₄×C₁₃	—	2	4	no	—
100	C₄×C₁₃	—	8	4	no	—
101	C₄×C₁₃	—	8	4	no	—
102	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₄×C₁₃	#427	1	1	yes	characteristic
104	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
105	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
106	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄²	—	4	2	no	—
111	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
115	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
116	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
117	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
118	#107	—	2	2	no	—
119	#107	—	2	2	no	—
120	#107	C₂×C₄×C₁₃	1	4	yes	—
121	#107	C₂×C₄×C₁₃	1	4	yes	—
122	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
123	C₄×C₈	Q₈×C₁₃	1	2	yes	—
124	#95	—	4	2	no	—
125	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
126	C₂×C₁₃	#2159	1	2	yes	—
127	C₂×C₁₃	#2175	1	2	yes	—
128	C₂×C₁₃	#2176	1	1	yes	characteristic
129	C₂×C₁₃	#2177	1	1	yes	characteristic
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	C₁₃×D₈	1	2	yes	—
133	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	C₁₃×C₄⋊C₄	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	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄²	—	8	4	no	—
148	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
149	C₄²	C₁₃×Q₁₆	1	2	yes	—
150	C₁₆	—	2	2	no	—
151	C₁₆	C₄²×C₁₃	1	4	yes	complemented
152	C₁₃	#12674	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	C₁₃×#107	1	4	yes	—
163	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
166	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
167	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
168	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂×C₄²×C₁₃	1	1	yes	characteristic
171	C₈	C₁₃×#117	1	1	yes	characteristic
172	C₈	C₄×Q₈×C₁₃	1	1	yes	characteristic
173	C₈	C₄×Q₈×C₁₃	1	1	yes	characteristic
174	C₂²	C₂×C₁₃×#107	1	1	yes	characteristic
175	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
176	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
177	C₂²	C₁₃×#509	1	2	yes	—
178	C₂²	C₁₃×#510	1	2	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	8	4	no	—
181	C₄	—	8	4	no	—
182	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
183	C₄	C₁₃×#427	1	1	yes	characteristic
184	C₄	C₄×C₁₃×Q₁₆	1	1	yes	characteristic
185	C₄	C₄×C₁₃×Q₁₆	1	1	yes	characteristic
186	C₂	C₄×C₁₃×#107	1	1	yes	characteristic
187	C₂	C₁₃×#2159	1	2	yes	—
188	C₂	C₁₃×#2175	1	2	yes	—
189	C₂	C₁₃×#2176	1	1	yes	characteristic
190	C₂	C₁₃×#2177	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
12. C₁₃×#367	C₄	semidirect
13. C₁₃×#366	C₄	semidirect
14. C₁₃×#346	C₄	semidirect
35. #12674	C₁₃	direct (N × H)
57. C₁₃×C₁₆	C₄²	semidirect
90. #367	C₄×C₁₃	semidirect
91. #366	C₄×C₁₃	semidirect
92. #346	C₄×C₁₃	semidirect
151. C₁₆	C₄²×C₁₃	semidirect
152. C₁₃	#12674	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×#367	2
C₂×C₁₃×#366	2
C₂×C₁₃×#346	2
C₄×C₁₃×#107	2
C₁₃×#1380	2
#12674	13

Subgroups — C13×#12674 #15699013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12674 #15699013