Subgroups — C₄×C₁₃×#125 #13111363

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#389	C₂	1	2	yes	maximal
3	C₁₃×#387	C₂	1	4	yes	maximal
4	C₁₃×#382	C₂	1	1	yes	maximal, characteristic
5	C₄×C₁₃×C₄⋊C₄	C₂	1	4	yes	maximal
6	C₄×C₁₃×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#125	C₄	1	8	yes	direct product factor
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	C₂²	1	4	yes	—
16	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#95	C₂²	1	4	yes	—
20	C₁₃×#95	C₂²	1	4	yes	—
21	C₁₃×#95	C₂²	1	4	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×C₄⋊C₄	—	2	8	no	—
30	C₁₃×C₄⋊C₄	—	2	4	no	—
31	C₁₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
32	C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
33	C₄²×C₁₃	—	2	4	no	—
34	C₄²×C₁₃	—	2	2	no	—
35	C₄²×C₁₃	—	2	4	no	—
36	C₄²×C₁₃	—	2	8	no	—
37	C₄²×C₁₃	C₂×C₄	1	4	yes	—
38	C₄²×C₁₃	Q₈	1	4	yes	—
39	C₄×#125	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	2	4	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	2	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	2	4	no	—
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	—	2	4	no	—
50	C₂×C₄×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₁₃	C₂²×C₄	1	8	yes	—
53	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₁₃	#41	1	4	yes	—
58	C₂×C₄×C₁₃	#41	1	2	yes	—
59	C₂×C₄×C₁₃	#41	1	4	yes	—
60	C₂×#125	C₂×C₁₃	1	1	yes	characteristic
61	#389	C₂×C₁₃	1	2	yes	—
62	#387	C₂×C₁₃	1	4	yes	—
63	#382	C₂×C₁₃	1	1	yes	characteristic
64	C₄×C₄⋊C₄	C₂×C₁₃	1	4	yes	—
65	C₄×C₄⋊C₄	C₂×C₁₃	1	2	yes	—
66	C₄³	C₂×C₁₃	1	1	yes	characteristic
67	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
68	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
70	C₂²×C₁₃	C₂×#41	1	2	yes	—
71	C₂²×C₁₃	C₂×#41	1	2	yes	—
72	C₄×C₁₃	—	2	4	no	—
73	C₄×C₁₃	—	2	8	no	—
74	C₄×C₁₃	—	2	4	no	—
75	C₄×C₁₃	—	2	2	no	—
76	C₄×C₁₃	—	4	4	no	—
77	C₄×C₁₃	C₄×Q₈	1	4	yes	—
78	C₄×C₁₃	#125	1	4	yes	direct product factor
79	#125	C₄×C₁₃	1	8	yes	direct product factor
80	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
82	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
83	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
84	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
85	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
86	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
87	C₂×C₄²	C₂²×C₁₃	1	4	yes	—
88	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
89	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
90	#95	C₂²×C₁₃	1	4	yes	—
91	#95	C₂²×C₁₃	1	4	yes	—
92	#95	C₂²×C₁₃	1	4	yes	—
93	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
94	C₂×C₁₃	C₄×#41	1	2	yes	—
95	C₂×C₁₃	C₂×#125	1	1	yes	characteristic
96	C₂×C₁₃	#529	1	2	yes	—
97	C₂×C₁₃	#533	1	1	yes	characteristic
98	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
99	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
100	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
101	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
102	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
103	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
104	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
105	C₄⋊C₄	—	2	8	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	C₂×C₄×C₁₃	1	16	yes	—
108	C₄⋊C₄	C₂×C₄×C₁₃	1	8	yes	—
109	C₄²	—	2	4	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	2	4	no	—
112	C₄²	—	2	8	no	—
113	C₄²	C₂×C₄×C₁₃	1	4	yes	—
114	C₄²	Q₈×C₁₃	1	4	yes	—
115	C₁₃	C₄×#125	1	1	yes	characteristic, direct product factor, Sylow
116	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	C₂²×C₄×C₁₃	1	4	yes	—
128	C₂×C₄	C₂²×C₄×C₁₃	1	8	yes	—
129	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
130	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
131	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
132	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
133	C₂×C₄	C₁₃×#41	1	4	yes	—
134	C₂×C₄	C₁₃×#41	1	2	yes	—
135	C₂×C₄	C₁₃×#41	1	4	yes	—
136	C₂²	C₂³×C₄×C₁₃	1	1	yes	characteristic
137	C₂²	C₂²×Q₈×C₁₃	1	1	yes	characteristic
138	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
139	C₂²	C₂×C₁₃×#41	1	2	yes	—
140	C₂²	C₂×C₁₃×#41	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	2	8	no	—
143	C₄	—	2	4	no	—
144	C₄	—	2	2	no	—
145	C₄	—	4	4	no	—
146	C₄	C₄×Q₈×C₁₃	1	4	yes	—
147	C₄	C₁₃×#125	1	4	yes	direct product factor
148	C₂	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
149	C₂	C₄×C₁₃×#41	1	2	yes	—
150	C₂	C₂×C₁₃×#125	1	1	yes	characteristic
151	C₂	C₁₃×#529	1	2	yes	—
152	C₂	C₁₃×#533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂³
commutator	C₂²	C₂³×C₄×C₁₃
Frattini, cosocle	C₂³	C₂⁴×C₁₃
socle, prime radical	C₂³×C₁₃	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₁₃×#125	C₄	direct (N × H)
39. C₄×#125	C₁₃	direct (N × H)
78. C₄×C₁₃	#125	direct (N × H)
79. #125	C₄×C₁₃	direct (N × H)
115. C₁₃	C₄×#125	direct (N × H)
147. C₄	C₁₃×#125	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×#125	2
C₁₃×#389	2
C₁₃×#387	2
C₁₃×#382	2
C₄×C₁₃×C₄⋊C₄	2
C₄³×C₁₃	2
C₄×#125	13

Subgroups — C4×C13×#125 #13111363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₃×#125 #13111363