Subgroups — C₁₃×#3396 #13112454

Order 1664.

384 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#559	C₂	1	2	yes	maximal, complemented
2	C₁₃×#495	C₂	1	2	yes	maximal, complemented
3	C₁₃×#493	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#463	C₂	1	4	yes	maximal, complemented
5	C₁₃×#460	C₂	1	2	yes	maximal, complemented
6	C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×D₈	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×SD₁₆	C₂²	1	4	yes	—
11	C₂×C₁₃×D₈	C₂²	1	2	yes	—
12	C₂×C₁₃×D₈	C₂²	1	2	yes	—
13	C₁₃×#127	C₂²	1	2	yes	—
14	C₁₃×#124	—	2	2	no	—
15	C₁₃×#124	C₂²	1	4	yes	—
16	C₁₃×#123	—	2	4	no	—
17	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
18	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₃	—	2	2	no	—
20	C₄×D₄×C₁₃	C₂²	1	4	yes	—
21	C₁₃×#107	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₁₃×#103	C₂²	1	2	yes	—
25	C₁₃×#102	C₂²	1	4	yes	—
26	C₄×C₈×C₁₃	C₂²	1	2	yes	—
27	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
30	C₂×D₄×C₁₃	—	2	2	no	—
31	C₂×D₄×C₁₃	—	2	2	no	—
32	C₂×D₄×C₁₃	C₂³	1	4	yes	—
33	C₂²×C₄×C₁₃	—	2	4	no	—
34	C₁₃×Q₁₆	—	2	2	no	—
35	C₁₃×SD₁₆	—	4	4	no	—
36	C₁₃×D₈	—	2	4	no	—
37	C₁₃×D₈	—	4	2	no	—
38	C₂×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
44	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₁₃×C₂²⋊C₄	—	2	4	no	—
46	C₁₃×C₂²⋊C₄	—	2	4	no	—
47	C₁₃×C₂²⋊C₄	—	2	4	no	—
48	C₄²×C₁₃	—	2	2	no	—
49	C₄²×C₁₃	C₂³	1	2	yes	—
50	C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	#3396	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₁₃	—	2	4	no	—
53	Q₈×C₁₃	—	2	2	no	—
54	Q₈×C₁₃	—	2	2	no	—
55	Q₈×C₁₃	—	2	2	no	—
56	D₄×C₁₃	—	2	4	no	—
57	D₄×C₁₃	—	2	4	no	—
58	D₄×C₁₃	—	4	4	no	—
59	D₄×C₁₃	—	8	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	8	4	no	—
63	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₈×C₁₃	—	2	1	no	—
69	C₈×C₁₃	—	2	2	no	—
70	C₈×C₁₃	#41	1	2	yes	—
71	#559	C₂×C₁₃	1	2	yes	complemented
72	#495	C₂×C₁₃	1	2	yes	complemented
73	#493	C₂×C₁₃	1	1	yes	characteristic
74	#463	C₂×C₁₃	1	4	yes	complemented
75	#460	C₂×C₁₃	1	2	yes	complemented
76	C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
77	C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
78	C₄×D₈	C₂×C₁₃	1	2	yes	complemented
79	C₂²×C₁₃	—	4	4	no	—
80	C₂²×C₁₃	—	4	4	no	—
81	C₂²×C₁₃	—	4	4	no	—
82	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₁₃	—	2	2	no	—
84	C₄×C₁₃	—	2	2	no	—
85	C₄×C₁₃	—	2	1	no	—
86	C₄×C₁₃	—	4	2	no	—
87	C₄×C₁₃	—	4	2	no	—
88	C₄×C₁₃	#135	1	2	yes	—
89	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
90	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
91	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
92	C₂×SD₁₆	C₂²×C₁₃	1	4	yes	—
93	C₂×D₈	C₂²×C₁₃	1	2	yes	—
94	C₂×D₈	C₂²×C₁₃	1	2	yes	—
95	#127	C₂²×C₁₃	1	2	yes	—
96	#124	—	2	2	no	—
97	#124	C₂²×C₁₃	1	4	yes	—
98	#123	—	2	4	no	—
99	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
100	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
101	C₄×D₄	—	2	2	no	—
102	C₄×D₄	C₂²×C₁₃	1	4	yes	—
103	#107	C₂²×C₁₃	1	1	yes	characteristic
104	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
105	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
106	#103	C₂²×C₁₃	1	2	yes	—
107	#102	C₂²×C₁₃	1	4	yes	—
108	C₄×C₈	C₂²×C₁₃	1	2	yes	—
109	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
110	C₂×C₁₃	—	8	4	no	—
111	C₂×C₁₃	#550	1	1	yes	characteristic
112	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
113	C₂×C₁₃	#576	1	1	yes	characteristic
114	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
115	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	C₂³×C₁₃	1	4	yes	—
119	C₂²×C₄	—	2	4	no	—
120	Q₁₆	—	2	2	no	—
121	SD₁₆	—	4	4	no	—
122	D₈	—	2	4	no	—
123	D₈	—	4	2	no	—
124	C₂×C₈	C₂³×C₁₃	1	2	yes	—
125	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
126	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
129	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
130	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	2	4	no	—
134	C₄²	—	2	2	no	—
135	C₄²	C₂³×C₁₃	1	2	yes	—
136	C₄²	C₂³×C₁₃	1	1	yes	characteristic
137	C₁₃	#3396	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³	—	2	4	no	—
139	Q₈	—	2	2	no	—
140	Q₈	—	2	2	no	—
141	Q₈	—	2	2	no	—
142	D₄	—	2	4	no	—
143	D₄	—	2	4	no	—
144	D₄	—	4	4	no	—
145	D₄	—	8	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	8	4	no	—
149	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
151	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
152	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
153	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	2	no	—
156	C₈	C₁₃×#41	1	2	yes	—
157	C₂²	—	4	4	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	4	2	no	—
166	C₄	C₁₃×#135	1	2	yes	—
167	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
168	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
169	C₂	—	8	4	no	—
170	C₂	C₁₃×#550	1	1	yes	characteristic
171	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
172	C₂	C₁₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#559	C₂	semidirect
2. C₁₃×#495	C₂	semidirect
4. C₁₃×#463	C₂	semidirect
5. C₁₃×#460	C₂	semidirect
6. C₈×Q₈×C₁₃	C₂	semidirect
7. C₄×C₁₃×Q₁₆	C₂	semidirect
8. C₄×C₁₃×D₈	C₂	semidirect
51. #3396	C₁₃	direct (N × H)
71. #559	C₂×C₁₃	semidirect
72. #495	C₂×C₁₃	semidirect
74. #463	C₂×C₁₃	semidirect
75. #460	C₂×C₁₃	semidirect
76. C₈×Q₈	C₂×C₁₃	semidirect
77. C₄×Q₁₆	C₂×C₁₃	semidirect
78. C₄×D₈	C₂×C₁₃	semidirect
137. C₁₃	#3396	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#559	2
C₁₃×#495	2
C₁₃×#493	2
C₁₃×#463	2
C₁₃×#460	2
C₈×Q₈×C₁₃	2
C₄×C₁₃×Q₁₆	2
C₄×C₁₃×D₈	2
#3396	13

Subgroups — C13×#3396 #13112454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3396 #13112454