Subgroups — C₁₃×#12900 #15699239

Order 3328.

382 conjugacy classes of subgroups.

The subgroup lattice has 148 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	4	yes	maximal
2	C₁₃×#2150	C₂	1	2	yes	maximal
3	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄³×C₁₃	C₄	1	1	yes	characteristic
10	C₁₃×#364	—	2	8	no	—
11	C₁₃×C₄⋊C₁₆	—	2	4	no	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	—	2	2	no	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₁₃×M₃₂	—	2	8	no	—
19	C₂×C₁₃×C₁₆	—	2	4	no	—
20	C₄×C₈×C₁₃	—	2	4	no	—
21	C₄×C₈×C₁₃	C₈	1	2	yes	—
22	C₄×C₈×C₁₃	C₈	1	2	yes	—
23	C₄×C₈×C₁₃	D₄	1	4	yes	—
24	C₄×C₈×C₁₃	D₄	1	2	yes	—
25	C₄×C₈×C₁₃	Q₈	1	4	yes	—
26	C₄×C₈×C₁₃	Q₈	1	2	yes	—
27	#12900	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₄×C₁₃	—	2	2	no	—
29	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃	—	2	2	no	—
32	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
33	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
36	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
37	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
40	C₄²×C₁₃	—	2	8	no	—
41	C₄²×C₁₃	—	2	2	no	—
42	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
43	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
44	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₁₃×C₁₆	—	4	4	no	—
47	C₂×#364	C₂×C₁₃	1	4	yes	—
48	#2150	C₂×C₁₃	1	2	yes	—
49	C₄²×C₈	C₂×C₁₃	1	1	yes	characteristic
50	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
51	C₂×C₄×C₁₃	—	2	8	no	—
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
60	C₈×C₁₃	—	2	2	no	—
61	C₈×C₁₃	C₄⋊C₈	1	8	yes	—
62	C₈×C₁₃	#127	1	1	yes	characteristic
63	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
64	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
66	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	—
67	C₂×C₄×C₈	C₄×C₁₃	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
69	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
70	C₄³	C₄×C₁₃	1	1	yes	characteristic
71	#364	—	2	8	no	—
72	C₄⋊C₁₆	—	2	4	no	—
73	C₂²×C₁₃	—	2	2	no	—
74	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₂²×C₁₃	#423	1	2	yes	—
76	C₄×C₁₃	—	2	8	no	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	#364	1	4	yes	complemented
79	C₄×C₁₃	#384	1	1	yes	characteristic
80	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
81	C₄×C₁₃	#423	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
83	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
84	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
85	C₂×C₄²	—	2	2	no	—
86	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
87	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
88	M₃₂	—	2	8	no	—
89	C₂×C₁₆	—	2	4	no	—
90	C₄×C₈	—	2	4	no	—
91	C₄×C₈	C₈×C₁₃	1	2	yes	—
92	C₄×C₈	C₈×C₁₃	1	2	yes	—
93	C₄×C₈	D₄×C₁₃	1	2	yes	—
94	C₄×C₈	D₄×C₁₃	1	4	yes	—
95	C₄×C₈	Q₈×C₁₃	1	4	yes	—
96	C₄×C₈	Q₈×C₁₃	1	2	yes	—
97	C₂×C₁₃	—	2	2	no	—
98	C₂×C₁₃	#1842	1	1	yes	characteristic
99	C₂×C₁₃	C₂×#364	1	2	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	C₁₃×M₁₆	1	2	yes	—
102	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
105	C₂×C₈	C₁₃×C₄⋊C₄	1	4	yes	—
106	C₂×C₈	C₂×C₈×C₁₃	1	4	yes	—
107	C₂×C₈	C₁₃×M₁₆	1	4	yes	—
108	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
110	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
111	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
112	C₄²	—	2	8	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
115	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
116	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
117	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
118	C₁₆	—	4	4	no	—
119	C₁₃	#12900	1	1	yes	characteristic, direct product factor, Sylow
120	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₁₃×C₄⋊C₈	1	2	yes	—
125	C₂×C₄	C₁₃×C₄⋊C₈	1	2	yes	—
126	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
127	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
129	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
130	C₈	—	2	2	no	—
131	C₈	C₁₃×C₄⋊C₈	1	8	yes	—
132	C₈	C₁₃×#127	1	1	yes	characteristic
133	C₈	C₁₃×C₄⋊Q₈	1	2	yes	—
134	C₈	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
137	C₂²	C₁₃×#423	1	2	yes	—
138	C₄	—	2	8	no	—
139	C₄	—	2	2	no	—
140	C₄	C₁₃×#364	1	4	yes	complemented
141	C₄	C₁₃×#384	1	1	yes	characteristic
142	C₄	C₂×C₁₃×C₄⋊C₈	1	2	yes	—
143	C₄	C₁₃×#423	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	C₁₃×#1842	1	1	yes	characteristic
146	C₂	C₂×C₁₃×#364	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
27. #12900	C₁₃	direct (N × H)
78. C₄×C₁₃	#364	semidirect
119. C₁₃	#12900	direct (N × H)
140. C₄	C₁₃×#364	semidirect

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×#2150	2
C₄²×C₈×C₁₃	2
#12900	13

Subgroups — C13×#12900 #15699239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12900 #15699239