Subgroups — C₁₃×#12839 #15699178

Order 3328.

362 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×#504	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2158	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1812	C₂	1	2	yes	maximal
4	C₁₃×#1384	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
6	C₁₃×#504	—	2	2	no	—
7	C₁₃×#504	—	2	2	no	—
8	C₂×C₁₃×M₃₂	C₄	1	2	yes	complemented
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₄	1	2	yes	complemented
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#427	C₂²	1	4	yes	—
13	C₁₃×#410	—	2	2	no	—
14	C₁₃×#367	—	2	2	no	—
15	C₁₃×#366	—	2	2	no	—
16	C₁₃×H(3,ℤ/4)	—	2	2	no	—
17	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
18	C₁₃×#131	—	2	2	no	—
19	C₁₃×#131	—	2	2	no	—
20	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	Q₈	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₁₃×#117	—	2	4	no	—
25	C₁₃×M₃₂	—	2	2	no	—
26	C₁₃×M₃₂	—	2	4	no	—
27	C₂×C₁₃×C₁₆	—	2	2	no	—
28	C₂×C₁₃×C₁₆	—	2	2	no	—
29	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
33	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
34	C₁₃×#107	—	2	4	no	—
35	C₁₃×#106	—	2	4	no	—
36	C₁₃×#99	—	4	4	no	—
37	#12839	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₁₃×#41	—	2	2	no	—
39	C₁₃×#41	—	2	2	no	—
40	C₂×Q₈×C₁₃	Q₁₆	1	1	yes	characteristic
41	C₂×D₄×C₁₃	D₈	1	2	yes	—
42	C₂×D₄×C₁₃	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₁₃×M₁₆	—	2	2	no	—
46	C₁₃×M₁₆	—	2	4	no	—
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₁₃	#41	1	2	yes	—
54	C₁₃×C₄⋊C₄	—	4	4	no	—
55	C₁₃×C₂²⋊C₄	—	4	4	no	—
56	C₄²×C₁₃	—	4	4	no	—
57	C₁₃×C₁₆	—	2	2	no	—
58	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
59	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
60	C₁₃×C₁₆	C₂²⋊C₄	1	2	yes	—
61	C₂×#504	C₂×C₁₃	1	1	yes	characteristic
62	#2158	C₂×C₁₃	1	2	yes	complemented
63	#1812	C₂×C₁₃	1	2	yes	—
64	#1384	C₂×C₁₃	1	2	yes	complemented
65	C₂³×C₁₃	#107	1	2	yes	—
66	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₁₃	—	2	2	no	—
68	D₄×C₁₃	—	2	4	no	—
69	D₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	8	4	no	—
73	C₂×C₄×C₁₃	#107	1	2	yes	—
74	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#121	1	2	yes	—
76	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#135	1	2	yes	—
78	C₈×C₁₃	—	2	2	no	—
79	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₈×C₁₃	#117	1	1	yes	characteristic
81	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
82	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
83	#504	—	2	2	no	—
84	#504	—	2	2	no	—
85	C₂×M₃₂	C₄×C₁₃	1	2	yes	complemented
86	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
87	C₂²×C₁₆	C₄×C₁₃	1	2	yes	complemented
88	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
89	#427	C₂²×C₁₃	1	4	yes	—
90	#410	—	2	2	no	—
91	#367	—	2	2	no	—
92	#366	—	2	2	no	—
93	H(3,ℤ/4)	—	2	2	no	—
94	C₂²×C₁₃	—	2	2	no	—
95	C₂²×C₁₃	—	4	1	no	—
96	C₂²×C₁₃	—	4	2	no	—
97	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
98	C₂²×C₁₃	#466	1	2	yes	—
99	C₄×C₁₃	—	2	2	no	—
100	C₄×C₁₃	—	16	4	no	—
101	C₄×C₁₃	#380	1	1	yes	characteristic
102	C₄×C₁₃	#427	1	1	yes	characteristic
103	C₄×C₁₃	#467	1	2	yes	—
104	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
105	#131	—	2	2	no	—
106	#131	—	2	2	no	—
107	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
108	C₂×M₁₆	Q₈×C₁₃	1	2	yes	—
109	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
110	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
111	#117	—	2	4	no	—
112	M₃₂	—	2	2	no	—
113	M₃₂	—	2	4	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
117	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
118	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
119	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
120	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
121	#107	—	2	4	no	—
122	#106	—	2	4	no	—
123	#99	—	4	4	no	—
124	C₂×C₁₃	—	2	2	no	—
125	C₂×C₁₃	—	2	1	no	—
126	C₂×C₁₃	—	4	2	no	—
127	C₂×C₁₃	#1808	1	1	yes	characteristic
128	#41	—	2	2	no	—
129	#41	—	2	2	no	—
130	C₂×Q₈	C₁₃×Q₁₆	1	1	yes	characteristic
131	C₂×D₄	C₁₃×D₈	1	2	yes	—
132	C₂×D₄	C₁₃×Q₁₆	1	1	yes	characteristic
133	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
135	M₁₆	—	2	2	no	—
136	M₁₆	—	2	4	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
140	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
141	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
142	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
143	C₂×C₈	C₁₃×#41	1	2	yes	—
144	C₄⋊C₄	—	4	4	no	—
145	C₂²⋊C₄	—	4	4	no	—
146	C₄²	—	4	4	no	—
147	C₁₆	—	2	2	no	—
148	C₁₆	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
149	C₁₆	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
150	C₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
151	C₁₃	#12839	1	1	yes	characteristic, direct product factor, Sylow
152	C₂³	C₁₃×#107	1	2	yes	—
153	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
154	Q₈	—	2	2	no	—
155	D₄	—	2	4	no	—
156	D₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	8	4	no	—
160	C₂×C₄	C₁₃×#107	1	2	yes	—
161	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄	C₁₃×#121	1	2	yes	—
163	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
164	C₂×C₄	C₁₃×#135	1	2	yes	—
165	C₈	—	2	2	no	—
166	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
167	C₈	C₁₃×#117	1	1	yes	characteristic
168	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	2	no	—
172	C₂²	C₂×C₁₃×#107	1	1	yes	characteristic
173	C₂²	C₁₃×#466	1	2	yes	—
174	C₄	—	2	2	no	—
175	C₄	—	16	4	no	—
176	C₄	C₁₃×#380	1	1	yes	characteristic
177	C₄	C₁₃×#427	1	1	yes	characteristic
178	C₄	C₁₃×#467	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	2	1	no	—
181	C₂	—	4	2	no	—
182	C₂	C₁₃×#1808	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	#427
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1808
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2158	C₂	semidirect
4. C₁₃×#1384	C₂	semidirect
8. C₂×C₁₃×M₃₂	C₄	semidirect
10. C₂²×C₁₃×C₁₆	C₄	semidirect
37. #12839	C₁₃	direct (N × H)
62. #2158	C₂×C₁₃	semidirect
64. #1384	C₂×C₁₃	semidirect
85. C₂×M₃₂	C₄×C₁₃	semidirect
87. C₂²×C₁₆	C₄×C₁₃	semidirect
151. C₁₃	#12839	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×#504	2
C₁₃×#2158	2
C₁₃×#1812	2
C₁₃×#1384	2
#12839	13

Subgroups — C13×#12839 #15699178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12839 #15699178