Subgroups — C₁₃×#32808 #15719147

Order 3328.

732 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2862	C₂	1	2	yes	maximal
2	C₁₃×#2056	C₂	1	4	yes	maximal
3	C₁₃×#2024	C₂	1	4	yes	maximal
4	C₁₃×#2016	C₂	1	4	yes	maximal
5	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	—
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₁₃×#103	C₂²	1	8	yes	—
11	C₁₃×#391	—	2	2	no	—
12	C₁₃×#391	—	2	2	no	—
13	C₁₃×#391	C₂²	1	4	yes	—
14	C₁₃×#391	C₂²	1	4	yes	—
15	C₁₃×#382	—	2	2	no	—
16	C₁₃×#382	—	2	2	no	—
17	C₄³×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#340	C₂²	1	4	yes	—
19	C₁₃×#336	C₂²	1	4	yes	—
20	C₂²×Q₈×C₁₃	—	2	2	no	—
21	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
22	C₂×C₁₃×Q₁₆	—	4	8	no	—
23	C₂×C₁₃×Q₁₆	—	4	4	no	—
24	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
25	C₁₃×C₄⋊Q₈	—	4	4	no	—
26	C₁₃×C₄⋊Q₈	—	4	4	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
31	C₂×C₄²×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
36	C₁₃×C₄⋊C₈	—	4	2	no	—
37	C₁₃×#103	—	4	8	no	—
38	C₁₃×#95	—	2	4	no	—
39	C₁₃×#95	—	2	4	no	—
40	C₁₃×#95	—	2	4	no	—
41	C₁₃×#95	—	2	4	no	—
42	#32808	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂×Q₈×C₁₃	—	2	8	no	—
44	C₂×Q₈×C₁₃	—	2	8	no	—
45	C₂×Q₈×C₁₃	—	2	4	no	—
46	C₂×Q₈×C₁₃	—	2	4	no	—
47	C₂×Q₈×C₁₃	—	4	4	no	—
48	C₂×Q₈×C₁₃	—	4	4	no	—
49	C₂×Q₈×C₁₃	—	4	4	no	—
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	—	2	4	no	—
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₁₃×Q₁₆	—	4	16	no	—
58	C₂×C₈×C₁₃	—	2	8	no	—
59	C₂×C₈×C₁₃	—	2	4	no	—
60	C₁₃×C₄⋊C₄	—	2	8	no	—
61	C₁₃×C₄⋊C₄	—	4	4	no	—
62	C₁₃×C₄⋊C₄	—	4	4	no	—
63	C₁₃×C₄⋊C₄	—	4	4	no	—
64	C₄²×C₁₃	—	2	4	no	—
65	C₄²×C₁₃	—	2	1	no	—
66	C₄²×C₁₃	—	2	4	no	—
67	C₄²×C₁₃	—	2	1	no	—
68	C₄²×C₁₃	—	4	1	no	—
69	C₄²×C₁₃	Q₁₆	1	4	yes	complemented
70	#2862	C₂×C₁₃	1	2	yes	—
71	#2056	C₂×C₁₃	1	4	yes	—
72	#2024	C₂×C₁₃	1	4	yes	—
73	#2016	C₂×C₁₃	1	4	yes	—
74	#1831	C₂×C₁₃	1	1	yes	characteristic
75	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
76	Q₈×C₁₃	—	4	8	no	—
77	Q₈×C₁₃	—	4	8	no	—
78	Q₈×C₁₃	—	8	8	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	1	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	—	4	4	no	—
90	C₂×C₄×C₁₃	—	4	4	no	—
91	C₂×C₄×C₁₃	—	4	4	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
95	C₂×C₄×C₁₃	#137	1	4	yes	—
96	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
97	C₂×C₄×C₁₃	H(5,𝔽₂)	1	4	yes	—
98	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
99	C₈×C₁₃	—	4	4	no	—
100	C₂²×Q₁₆	C₂²×C₁₃	1	4	yes	—
101	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
102	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
103	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
104	C₂×#103	C₂²×C₁₃	1	8	yes	—
105	#391	—	2	2	no	—
106	#391	—	2	2	no	—
107	#391	C₂²×C₁₃	1	4	yes	—
108	#391	C₂²×C₁₃	1	4	yes	—
109	#382	—	2	2	no	—
110	#382	—	2	2	no	—
111	C₄³	C₂²×C₁₃	1	1	yes	characteristic
112	#340	C₂²×C₁₃	1	4	yes	—
113	#336	C₂²×C₁₃	1	4	yes	—
114	C₂²×C₁₃	#535	1	2	yes	—
115	C₂²×C₁₃	#537	1	1	yes	characteristic
116	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
117	C₂²×C₁₃	C₂×#137	1	2	yes	—
118	C₂²×C₁₃	#575	1	1	yes	characteristic
119	C₄×C₁₃	—	2	4	no	—
120	C₄×C₁₃	—	2	4	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	4	1	no	—
123	C₄×C₁₃	—	8	4	no	—
124	C₄×C₁₃	—	8	4	no	—
125	C₄×C₁₃	H(3,𝔽₄)	1	2	yes	—
126	C₄×C₁₃	#562	1	2	yes	—
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
129	C₂×Q₁₆	—	4	8	no	—
130	C₂×Q₁₆	—	4	4	no	—
131	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
132	C₄⋊Q₈	—	4	4	no	—
133	C₄⋊Q₈	—	4	4	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
138	C₂×C₄²	D₄×C₁₃	1	2	yes	—
139	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
140	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
141	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
142	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
143	C₄⋊C₈	—	4	2	no	—
144	#103	—	4	8	no	—
145	#95	—	2	4	no	—
146	#95	—	2	4	no	—
147	#95	—	2	4	no	—
148	#95	—	2	4	no	—
149	C₂×C₁₃	#2819	1	1	yes	characteristic
150	C₂×C₁₃	#3209	1	2	yes	—
151	C₂×C₁₃	#3215	1	2	yes	—
152	C₂×C₁₃	#3249	1	1	yes	characteristic
153	C₂×C₁₃	#3255	1	1	yes	characteristic
154	C₂×Q₈	—	2	8	no	—
155	C₂×Q₈	—	2	8	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	4	no	—
159	C₂×Q₈	—	4	4	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
166	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
167	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
168	Q₁₆	—	4	16	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	2	4	no	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	4	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₁₃×Q₁₆	1	4	yes	complemented
181	C₁₃	#32808	1	1	yes	characteristic, direct product factor, Sylow
182	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
183	Q₈	—	4	8	no	—
184	Q₈	—	4	8	no	—
185	Q₈	—	8	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₂×C₁₃×Q₁₆	1	4	yes	—
201	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
202	C₂×C₄	C₁₃×#137	1	4	yes	—
203	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₄	C₁₃×H(5,𝔽₂)	1	4	yes	—
205	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
206	C₈	—	4	4	no	—
207	C₂²	C₁₃×#535	1	2	yes	—
208	C₂²	C₁₃×#537	1	1	yes	characteristic
209	C₂²	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
210	C₂²	C₂×C₁₃×#137	1	2	yes	—
211	C₂²	C₁₃×#575	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	1	no	—
216	C₄	—	8	4	no	—
217	C₄	—	8	4	no	—
218	C₄	C₁₃×H(3,𝔽₄)	1	2	yes	—
219	C₄	C₁₃×#562	1	2	yes	—
220	C₂	C₁₃×#2819	1	1	yes	characteristic
221	C₂	C₁₃×#3209	1	2	yes	—
222	C₂	C₁₃×#3215	1	2	yes	—
223	C₂	C₁₃×#3249	1	1	yes	characteristic
224	C₂	C₁₃×#3255	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
42. #32808	C₁₃	direct (N × H)
69. C₄²×C₁₃	Q₁₆	semidirect
180. C₄²	C₁₃×Q₁₆	semidirect
181. C₁₃	#32808	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×#2862	2
C₁₃×#2056	2
C₁₃×#2024	2
C₁₃×#2016	2
C₁₃×#1831	2
#32808	13

Subgroups — C13×#32808 #15719147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32808 #15719147