Subgroups — C₁₃×#3339 #13112397

Order 1664.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#553	C₂	1	2	yes	maximal, complemented
2	C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#482	C₂	1	2	yes	maximal, complemented
4	C₁₃×#481	C₂	1	2	yes	maximal, complemented
5	C₁₃×#478	C₂	1	2	yes	maximal, complemented
6	C₁₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#126	—	2	2	no	—
16	C₁₃×#125	C₂²	1	2	yes	—
17	C₁₃×#123	—	2	2	no	—
18	C₁₃×#121	—	2	2	no	—
19	C₁₃×#121	C₂²	1	2	yes	—
20	C₁₃×#121	C₂²	1	2	yes	—
21	C₄×D₄×C₁₃	—	2	2	no	—
22	C₄×D₄×C₁₃	C₂²	1	2	yes	—
23	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#117	C₂²	1	2	yes	—
25	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₁₃×#107	C₂²	1	1	yes	characteristic
27	C₁₃×#107	C₂²	1	1	yes	characteristic
28	C₁₃×#107	C₂²	1	2	yes	—
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×#106	C₂²	1	2	yes	—
31	C₁₃×#106	C₂²	1	2	yes	—
32	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₁₃×#102	C₂²	1	2	yes	—
34	C₁₃×#102	C₂²	1	2	yes	—
35	C₁₃×#102	C₂²	1	2	yes	—
36	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
39	C₂×D₄×C₁₃	—	2	2	no	—
40	C₂×D₄×C₁₃	—	2	2	no	—
41	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
42	C₂×D₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×D₈	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
58	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
59	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
60	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
61	C₁₃×C₂²⋊C₄	—	2	2	no	—
62	C₁₃×C₂²⋊C₄	—	2	2	no	—
63	C₁₃×C₂²⋊C₄	—	2	2	no	—
64	C₁₃×C₂²⋊C₄	—	2	2	no	—
65	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
66	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
67	C₄²×C₁₃	—	2	2	no	—
68	C₄²×C₁₃	C₂³	1	1	yes	characteristic
69	#3339	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₁₃	—	2	2	no	—
71	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
73	D₄×C₁₃	—	2	2	no	—
74	D₄×C₁₃	—	2	2	no	—
75	D₄×C₁₃	—	2	2	no	—
76	D₄×C₁₃	—	4	4	no	—
77	D₄×C₁₃	—	8	2	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	8	2	no	—
86	C₂×C₄×C₁₃	—	8	2	no	—
87	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₃	—	2	1	no	—
93	C₈×C₁₃	—	2	1	no	—
94	C₈×C₁₃	—	2	1	no	—
95	C₈×C₁₃	#41	1	2	yes	—
96	#553	C₂×C₁₃	1	2	yes	complemented
97	#499	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#482	C₂×C₁₃	1	2	yes	complemented
99	#481	C₂×C₁₃	1	2	yes	complemented
100	#478	C₂×C₁₃	1	2	yes	complemented
101	#466	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#466	C₂×C₁₃	1	1	yes	characteristic, complemented
103	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
104	C₈×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#427	C₂×C₁₃	1	1	yes	characteristic, complemented
106	C₂×#107	C₂×C₁₃	1	1	yes	characteristic, complemented
107	C₂²×C₁₃	—	2	1	no	—
108	C₂²×C₁₃	—	2	1	no	—
109	C₂²×C₁₃	—	2	1	no	—
110	C₂²×C₁₃	—	2	1	no	—
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	—	4	2	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	#135	1	2	yes	—
116	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	4	2	no	—
122	C₄×C₁₃	—	4	2	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
128	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
129	#126	—	2	2	no	—
130	#125	C₂²×C₁₃	1	2	yes	—
131	#123	—	2	2	no	—
132	#121	—	2	2	no	—
133	#121	C₂²×C₁₃	1	2	yes	—
134	#121	C₂²×C₁₃	1	2	yes	—
135	C₄×D₄	—	2	2	no	—
136	C₄×D₄	C₂²×C₁₃	1	2	yes	—
137	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
138	#117	C₂²×C₁₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
140	#107	C₂²×C₁₃	1	1	yes	characteristic
141	#107	C₂²×C₁₃	1	1	yes	characteristic
142	#107	C₂²×C₁₃	1	2	yes	—
143	#107	C₂²×C₁₃	1	1	yes	characteristic
144	#106	C₂²×C₁₃	1	2	yes	—
145	#106	C₂²×C₁₃	1	2	yes	—
146	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
147	#102	C₂²×C₁₃	1	2	yes	—
148	#102	C₂²×C₁₃	1	2	yes	—
149	#102	C₂²×C₁₃	1	2	yes	—
150	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
152	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×C₁₃	—	2	2	no	—
154	C₂×C₁₃	—	4	1	no	—
155	C₂×C₁₃	—	8	2	no	—
156	C₂×C₁₃	#547	1	1	yes	characteristic
157	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
158	C₂×C₁₃	#576	1	1	yes	characteristic
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
162	C₂×D₄	C₂³×C₁₃	1	2	yes	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
167	D₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
171	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
179	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
180	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
186	C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
187	C₄²	—	2	2	no	—
188	C₄²	C₂³×C₁₃	1	1	yes	characteristic
189	C₁₃	#3339	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³	—	2	2	no	—
191	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
192	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
193	D₄	—	2	2	no	—
194	D₄	—	2	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	4	4	no	—
197	D₄	—	8	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	C₁₃×#41	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₁₃×#135	1	2	yes	—
225	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
234	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	8	2	no	—
238	C₂	C₁₃×#547	1	1	yes	characteristic
239	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
240	C₂	C₁₃×#576	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₁₃×#553	C₂	semidirect
2. C₁₃×#499	C₂	semidirect
3. C₁₃×#482	C₂	semidirect
4. C₁₃×#481	C₂	semidirect
5. C₁₃×#478	C₂	semidirect
6. C₁₃×#466	C₂	semidirect
7. C₁₃×#466	C₂	semidirect
8. C₄×C₁₃×D₈	C₂	semidirect
9. C₈×D₄×C₁₃	C₂	semidirect
10. C₁₃×#427	C₂	semidirect
11. C₂×C₁₃×#107	C₂	semidirect
69. #3339	C₁₃	direct (N × H)
96. #553	C₂×C₁₃	semidirect
97. #499	C₂×C₁₃	semidirect
98. #482	C₂×C₁₃	semidirect
99. #481	C₂×C₁₃	semidirect
100. #478	C₂×C₁₃	semidirect
101. #466	C₂×C₁₃	semidirect
102. #466	C₂×C₁₃	semidirect
103. C₄×D₈	C₂×C₁₃	semidirect
104. C₈×D₄	C₂×C₁₃	semidirect
105. #427	C₂×C₁₃	semidirect
106. C₂×#107	C₂×C₁₃	semidirect
189. C₁₃	#3339	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×#553	2
C₁₃×#499	2
C₁₃×#482	2
C₁₃×#481	2
C₁₃×#478	2
C₁₃×#466	2
C₄×C₁₃×D₈	2
C₈×D₄×C₁₃	2
C₁₃×#427	2
C₂×C₁₃×#107	2
#3339	13

Subgroups — C13×#3339 #13112397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3339 #13112397