Subgroups — C₁₃×#2665 #13111723

Order 1664.

424 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#400	C₂	1	2	yes	maximal, complemented
2	C₁₃×#396	C₂	1	2	yes	maximal
3	C₁₃×#395	C₂	1	2	yes	maximal, complemented
4	C₁₃×#393	C₂	1	2	yes	maximal
5	C₁₃×#391	C₂	1	2	yes	maximal, complemented
6	C₁₃×#387	C₂	1	2	yes	maximal, complemented
7	C₁₃×#381	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#380	C₂	1	1	yes	maximal, characteristic
9	C₄×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
21	C₁₃×#95	C₂²	1	1	yes	characteristic
22	C₁₃×#95	C₂²	1	1	yes	characteristic
23	C₁₃×#95	C₂²	1	2	yes	—
24	C₁₃×#95	C₂²	1	1	yes	characteristic
25	C₁₃×#95	C₂²	1	1	yes	characteristic
26	C₁₃×#95	C₂²	1	2	yes	—
27	C₁₃×#95	C₂²	1	2	yes	—
28	C₁₃×#95	C₂²	1	2	yes	—
29	C₁₃×#95	C₂²	1	2	yes	—
30	C₁₃×#95	C₂²	1	2	yes	—
31	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₁₃	—	2	4	no	—
33	C₂×Q₈×C₁₃	—	4	2	no	—
34	C₂×Q₈×C₁₃	—	4	2	no	—
35	C₂²×C₄×C₁₃	—	4	2	no	—
36	C₂²×C₄×C₁₃	Q₈	1	4	yes	complemented
37	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₁₃×C₄⋊C₄	—	2	4	no	—
46	C₁₃×C₄⋊C₄	—	4	2	no	—
47	C₁₃×C₄⋊C₄	—	4	2	no	—
48	C₁₃×C₄⋊C₄	—	4	2	no	—
49	C₁₃×C₂²⋊C₄	—	2	4	no	—
50	C₁₃×C₂²⋊C₄	—	4	1	no	—
51	C₁₃×C₂²⋊C₄	—	4	2	no	—
52	C₁₃×C₂²⋊C₄	—	4	1	no	—
53	C₄²×C₁₃	—	4	2	no	—
54	#2665	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₁₃	—	2	2	no	—
56	C₂³×C₁₃	—	2	1	no	—
57	C₂³×C₁₃	—	2	2	no	—
58	C₂³×C₁₃	—	2	1	no	—
59	C₂³×C₁₃	C₂×Q₈	1	2	yes	—
60	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
61	Q₈×C₁₃	—	4	8	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₁₃	#41	1	4	yes	—
85	#400	C₂×C₁₃	1	2	yes	complemented
86	#396	C₂×C₁₃	1	2	yes	—
87	#395	C₂×C₁₃	1	2	yes	complemented
88	#393	C₂×C₁₃	1	2	yes	—
89	#391	C₂×C₁₃	1	2	yes	complemented
90	#387	C₂×C₁₃	1	2	yes	complemented
91	#381	C₂×C₁₃	1	1	yes	characteristic
92	#380	C₂×C₁₃	1	1	yes	characteristic
93	C₄×C₂²⋊C₄	C₂×C₁₃	1	1	yes	characteristic
94	C₂²×C₁₃	—	2	2	no	—
95	C₂²×C₁₃	—	2	2	no	—
96	C₂²×C₁₃	—	2	2	no	—
97	C₂²×C₁₃	—	4	2	no	—
98	C₂²×C₁₃	—	4	2	no	—
99	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×C₁₃	C₂×#41	1	2	yes	—
101	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
103	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₁₃	—	2	4	no	—
105	C₄×C₁₃	—	2	2	no	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	—	4	2	no	—
108	C₄×C₁₃	—	4	2	no	—
109	C₄×C₁₃	—	4	2	no	—
110	C₄×C₁₃	—	4	2	no	—
111	C₄×C₁₃	—	4	2	no	—
112	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
113	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
115	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
118	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
119	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
120	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
122	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
123	#95	C₂²×C₁₃	1	1	yes	characteristic
124	#95	C₂²×C₁₃	1	1	yes	characteristic
125	#95	C₂²×C₁₃	1	2	yes	—
126	#95	C₂²×C₁₃	1	1	yes	characteristic
127	#95	C₂²×C₁₃	1	1	yes	characteristic
128	#95	C₂²×C₁₃	1	2	yes	—
129	#95	C₂²×C₁₃	1	2	yes	—
130	#95	C₂²×C₁₃	1	2	yes	—
131	#95	C₂²×C₁₃	1	2	yes	—
132	#95	C₂²×C₁₃	1	2	yes	—
133	C₂×C₁₃	—	4	2	no	—
134	C₂×C₁₃	#533	1	1	yes	characteristic
135	C₂×C₁₃	#538	1	2	yes	—
136	C₂×C₁₃	#542	1	2	yes	—
137	C₂×C₁₃	#543	1	1	yes	characteristic
138	C₂×C₁₃	#544	1	1	yes	characteristic
139	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
140	C₂×Q₈	—	2	4	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	Q₈×C₁₃	1	4	yes	complemented
145	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
149	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
152	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₂²⋊C₄	—	4	1	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₂²⋊C₄	—	4	1	no	—
161	C₄²	—	4	2	no	—
162	C₁₃	#2665	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³	—	2	2	no	—
164	C₂³	—	2	1	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	1	no	—
167	C₂³	C₂×Q₈×C₁₃	1	2	yes	—
168	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
191	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
192	C₂×C₄	C₁₃×#41	1	4	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₂²×Q₈×C₁₃	1	1	yes	characteristic
199	C₂²	C₂×C₁₃×#41	1	2	yes	—
200	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
201	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
202	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₂	—	4	2	no	—
212	C₂	C₁₃×#533	1	1	yes	characteristic
213	C₂	C₁₃×#538	1	2	yes	—
214	C₂	C₁₃×#542	1	2	yes	—
215	C₂	C₁₃×#543	1	1	yes	characteristic
216	C₂	C₁₃×#544	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₁₃×#400	C₂	semidirect
3. C₁₃×#395	C₂	semidirect
5. C₁₃×#391	C₂	semidirect
6. C₁₃×#387	C₂	semidirect
36. C₂²×C₄×C₁₃	Q₈	semidirect
54. #2665	C₁₃	direct (N × H)
85. #400	C₂×C₁₃	semidirect
87. #395	C₂×C₁₃	semidirect
89. #391	C₂×C₁₃	semidirect
90. #387	C₂×C₁₃	semidirect
144. C₂²×C₄	Q₈×C₁₃	semidirect
162. C₁₃	#2665	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×#400	2
C₁₃×#396	2
C₁₃×#395	2
C₁₃×#393	2
C₁₃×#391	2
C₁₃×#387	2
C₁₃×#381	2
C₁₃×#380	2
C₄×C₁₃×C₂²⋊C₄	2
#2665	13

Subgroups — C13×#2665 #13111723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2665 #13111723