Subgroups — C₁₃×#3207 #13112265

Order 1664.

438 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×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#463	C₂	1	4	yes	maximal, complemented
8	C₁₃×#449	C₂	1	4	yes	maximal, complemented
9	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#41	—	2	1	no	—
11	C₂×C₁₃×#41	C₂²	1	2	yes	—
12	C₂×C₁₃×SD₁₆	C₂²	1	4	yes	—
13	C₂×C₁₃×D₈	C₂²	1	4	yes	—
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×#127	C₂²	1	1	yes	characteristic
17	C₁₃×#127	C₂²	1	1	yes	characteristic
18	C₁₃×#124	—	2	1	no	—
19	C₁₃×#124	C₂²	1	2	yes	—
20	C₁₃×#121	—	2	1	no	—
21	C₁₃×#121	—	2	1	no	—
22	C₁₃×#121	—	2	1	no	—
23	C₁₃×#121	C₂²	1	2	yes	—
24	C₄×D₄×C₁₃	—	2	1	no	—
25	C₄×D₄×C₁₃	—	2	1	no	—
26	C₄×D₄×C₁₃	—	2	1	no	—
27	C₄×D₄×C₁₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₁₃×#103	C₂²	1	4	yes	—
31	C₁₃×#102	C₂²	1	4	yes	—
32	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
35	C₁₃×#41	—	4	4	no	—
36	C₁₃×#41	—	4	4	no	—
37	C₂×Q₈×C₁₃	—	2	1	no	—
38	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
39	C₂×D₄×C₁₃	—	2	2	no	—
40	C₂×D₄×C₁₃	—	2	2	no	—
41	C₂×D₄×C₁₃	—	2	1	no	—
42	C₂×D₄×C₁₃	—	2	1	no	—
43	C₂×D₄×C₁₃	—	2	1	no	—
44	C₂×D₄×C₁₃	—	2	1	no	—
45	C₂×D₄×C₁₃	C₂³	1	2	yes	—
46	C₂×D₄×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×SD₁₆	—	4	4	no	—
53	C₁₃×D₈	—	4	4	no	—
54	C₂×C₈×C₁₃	C₂³	1	2	yes	—
55	C₂×C₈×C₁₃	C₂³	1	2	yes	—
56	C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
60	C₁₃×C₂²⋊C₄	—	2	2	no	—
61	C₁₃×C₂²⋊C₄	—	2	2	no	—
62	C₁₃×C₂²⋊C₄	—	2	2	no	—
63	C₁₃×C₂²⋊C₄	—	2	2	no	—
64	C₄²×C₁₃	D₄	1	1	yes	characteristic
65	C₄²×C₁₃	D₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂³	1	1	yes	characteristic
67	C₄²×C₁₃	C₂³	1	1	yes	characteristic
68	#3207	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₁₃	—	2	2	no	—
70	C₂³×C₁₃	—	2	2	no	—
71	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
72	Q₈×C₁₃	—	2	4	no	—
73	Q₈×C₁₃	—	4	2	no	—
74	D₄×C₁₃	—	2	4	no	—
75	D₄×C₁₃	—	2	4	no	—
76	D₄×C₁₃	—	4	4	no	—
77	D₄×C₁₃	—	4	4	no	—
78	D₄×C₁₃	—	4	2	no	—
79	D₄×C₁₃	—	4	2	no	—
80	D₄×C₁₃	—	4	2	no	—
81	D₄×C₁₃	—	4	2	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	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	—	4	4	no	—
89	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	—	2	2	no	—
97	#532	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#532	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#493	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#492	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#487	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#486	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#463	C₂×C₁₃	1	4	yes	complemented
104	#449	C₂×C₁₃	1	4	yes	complemented
105	#431	C₂×C₁₃	1	1	yes	characteristic, complemented
106	C₂²×C₁₃	—	2	1	no	—
107	C₂²×C₁₃	—	2	1	no	—
108	C₂²×C₁₃	—	2	1	no	—
109	C₂²×C₁₃	—	4	2	no	—
110	C₂²×C₁₃	—	4	2	no	—
111	C₂²×C₁₃	—	4	2	no	—
112	C₂²×C₁₃	—	4	2	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	4	2	no	—
120	C₄×C₁₃	—	4	2	no	—
121	C₄×C₁₃	#135	1	2	yes	—
122	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
124	C₂×#41	—	2	1	no	—
125	C₂×#41	C₂²×C₁₃	1	2	yes	—
126	C₂×SD₁₆	C₂²×C₁₃	1	4	yes	—
127	C₂×D₈	C₂²×C₁₃	1	4	yes	—
128	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
130	#127	C₂²×C₁₃	1	1	yes	characteristic
131	#127	C₂²×C₁₃	1	1	yes	characteristic
132	#124	—	2	1	no	—
133	#124	C₂²×C₁₃	1	2	yes	—
134	#121	—	2	1	no	—
135	#121	—	2	1	no	—
136	#121	—	2	1	no	—
137	#121	C₂²×C₁₃	1	2	yes	—
138	C₄×D₄	—	2	1	no	—
139	C₄×D₄	—	2	1	no	—
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	C₂²×C₁₃	1	2	yes	—
142	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
143	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
144	#103	C₂²×C₁₃	1	4	yes	—
145	#102	C₂²×C₁₃	1	4	yes	—
146	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
147	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
148	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×C₁₃	—	4	1	no	—
150	C₂×C₁₃	—	8	2	no	—
151	C₂×C₁₃	—	8	2	no	—
152	C₂×C₁₃	#535	1	1	yes	characteristic
153	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
154	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	#41	—	4	4	no	—
156	#41	—	4	4	no	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	—	2	1	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	1	no	—
165	C₂×D₄	C₂³×C₁₃	1	2	yes	—
166	C₂×D₄	C₂³×C₁₃	1	2	yes	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
170	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
172	SD₁₆	—	4	4	no	—
173	D₈	—	4	4	no	—
174	C₂×C₈	C₂³×C₁₃	1	2	yes	—
175	C₂×C₈	C₂³×C₁₃	1	2	yes	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₄²	D₄×C₁₃	1	1	yes	characteristic
185	C₄²	D₄×C₁₃	1	1	yes	characteristic
186	C₄²	C₂³×C₁₃	1	1	yes	characteristic
187	C₄²	C₂³×C₁₃	1	1	yes	characteristic
188	C₁₃	#3207	1	1	yes	characteristic, direct product factor, Sylow
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
192	Q₈	—	2	4	no	—
193	Q₈	—	4	2	no	—
194	D₄	—	2	4	no	—
195	D₄	—	2	4	no	—
196	D₄	—	4	4	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
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	C₁₃×#135	1	2	yes	—
233	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
234	C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
235	C₂	—	4	1	no	—
236	C₂	—	8	2	no	—
237	C₂	—	8	2	no	—
238	C₂	C₁₃×#535	1	1	yes	characteristic
239	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
240	C₂	C₂×C₁₃×AGL(1,ℤ/8)	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 ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#532	C₂	semidirect
2. C₁₃×#532	C₂	semidirect
3. C₁₃×#493	C₂	semidirect
4. C₁₃×#492	C₂	semidirect
5. C₁₃×#487	C₂	semidirect
6. C₁₃×#486	C₂	semidirect
7. C₁₃×#463	C₂	semidirect
8. C₁₃×#449	C₂	semidirect
9. C₁₃×#431	C₂	semidirect
68. #3207	C₁₃	direct (N × H)
97. #532	C₂×C₁₃	semidirect
98. #532	C₂×C₁₃	semidirect
99. #493	C₂×C₁₃	semidirect
100. #492	C₂×C₁₃	semidirect
101. #487	C₂×C₁₃	semidirect
102. #486	C₂×C₁₃	semidirect
103. #463	C₂×C₁₃	semidirect
104. #449	C₂×C₁₃	semidirect
105. #431	C₂×C₁₃	semidirect
188. C₁₃	#3207	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×#532	2
C₁₃×#493	2
C₁₃×#492	2
C₁₃×#487	2
C₁₃×#486	2
C₁₃×#463	2
C₁₃×#449	2
C₁₃×#431	2
#3207	13

Subgroups — C13×#3207 #13112265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3207 #13112265