Subgroups — C₁₃×#32806 #15719145

Order 3328.

980 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2858	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2054	C₂	1	4	yes	maximal, complemented
3	C₁₃×#2023	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2011	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	C₂²	1	4	yes	—
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₁₃×#127	C₂²	1	2	yes	—
9	C₂×C₁₃×#124	—	2	2	no	—
10	C₂×C₁₃×#124	—	2	2	no	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×#102	C₂²	1	8	yes	—
13	C₁₃×#390	—	2	2	no	—
14	C₁₃×#390	—	2	2	no	—
15	C₁₃×#390	C₂²	1	4	yes	—
16	C₁₃×#390	C₂²	1	4	yes	—
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×D₄×C₁₃	—	2	2	no	—
22	C₂²×D₄×C₁₃	—	2	2	no	—
23	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
24	C₂×C₁₃×D₈	—	4	8	no	—
25	C₂×C₁₃×D₈	—	4	4	no	—
26	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
27	C₁₃×C₄⋊Q₈	—	4	4	no	—
28	C₁₃×#127	—	4	4	no	—
29	C₁₃×#124	—	4	8	no	—
30	C₁₃×#124	—	4	8	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
33	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
34	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
37	C₂×C₄²×C₁₃	D₄	1	2	yes	—
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
42	C₁₃×C₄⋊C₈	—	4	2	no	—
43	C₁₃×#102	—	4	8	no	—
44	#32806	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂⁴×C₁₃	—	2	4	no	—
46	C₂×Q₈×C₁₃	—	4	4	no	—
47	C₂×Q₈×C₁₃	—	4	4	no	—
48	C₂×Q₈×C₁₃	—	4	4	no	—
49	C₂×D₄×C₁₃	—	2	8	no	—
50	C₂×D₄×C₁₃	—	2	8	no	—
51	C₂×D₄×C₁₃	—	2	4	no	—
52	C₂×D₄×C₁₃	—	2	4	no	—
53	C₂×D₄×C₁₃	—	4	4	no	—
54	C₂×D₄×C₁₃	—	4	4	no	—
55	C₂×D₄×C₁₃	—	4	4	no	—
56	C₂×D₄×C₁₃	—	4	4	no	—
57	C₂×D₄×C₁₃	—	4	4	no	—
58	C₂×D₄×C₁₃	—	4	4	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₁₃×D₈	—	4	16	no	—
66	C₂×C₈×C₁₃	—	2	8	no	—
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₁₃×C₄⋊C₄	—	2	8	no	—
69	C₁₃×C₄⋊C₄	—	4	4	no	—
70	C₁₃×C₂²⋊C₄	—	4	8	no	—
71	C₁₃×C₂²⋊C₄	—	4	8	no	—
72	C₁₃×C₂²⋊C₄	—	4	8	no	—
73	C₁₃×C₂²⋊C₄	—	4	8	no	—
74	C₄²×C₁₃	—	2	4	no	—
75	C₄²×C₁₃	—	2	1	no	—
76	C₄²×C₁₃	—	2	4	no	—
77	C₄²×C₁₃	—	2	1	no	—
78	C₄²×C₁₃	—	4	1	no	—
79	C₄²×C₁₃	D₈	1	4	yes	complemented
80	#2858	C₂×C₁₃	1	2	yes	complemented
81	#2054	C₂×C₁₃	1	4	yes	complemented
82	#2023	C₂×C₁₃	1	4	yes	complemented
83	#2011	C₂×C₁₃	1	4	yes	complemented
84	#1831	C₂×C₁₃	1	1	yes	characteristic, complemented
85	C₂³×C₁₃	—	4	4	no	—
86	C₂³×C₁₃	—	4	4	no	—
87	C₂³×C₁₃	—	4	4	no	—
88	C₂³×C₁₃	—	4	4	no	—
89	C₂³×C₁₃	—	4	4	no	—
90	C₂³×C₁₃	—	4	4	no	—
91	C₂³×C₁₃	—	4	4	no	—
92	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₁₃	—	8	8	no	—
94	D₄×C₁₃	—	4	8	no	—
95	D₄×C₁₃	—	4	8	no	—
96	D₄×C₁₃	—	8	8	no	—
97	D₄×C₁₃	—	8	8	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
111	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
112	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₁₃	H(5,𝔽₂)	1	4	yes	—
114	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
115	C₈×C₁₃	—	4	4	no	—
116	C₂²×D₈	C₂²×C₁₃	1	4	yes	—
117	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
118	C₂×#127	C₂²×C₁₃	1	2	yes	—
119	C₂×#124	—	2	2	no	—
120	C₂×#124	—	2	2	no	—
121	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
122	C₂×#102	C₂²×C₁₃	1	8	yes	—
123	#390	—	2	2	no	—
124	#390	—	2	2	no	—
125	#390	C₂²×C₁₃	1	4	yes	—
126	#390	C₂²×C₁₃	1	4	yes	—
127	C₄³	C₂²×C₁₃	1	1	yes	characteristic
128	#340	C₂²×C₁₃	1	4	yes	—
129	#336	C₂²×C₁₃	1	4	yes	—
130	C₂²×C₁₃	—	8	4	no	—
131	C₂²×C₁₃	—	8	4	no	—
132	C₂²×C₁₃	—	8	4	no	—
133	C₂²×C₁₃	—	8	4	no	—
134	C₂²×C₁₃	—	8	4	no	—
135	C₂²×C₁₃	—	8	4	no	—
136	C₂²×C₁₃	—	8	4	no	—
137	C₂²×C₁₃	#535	1	2	yes	—
138	C₂²×C₁₃	#537	1	1	yes	characteristic
139	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
141	C₂²×C₁₃	#575	1	1	yes	characteristic
142	C₄×C₁₃	—	2	4	no	—
143	C₄×C₁₃	—	2	4	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	4	1	no	—
146	C₄×C₁₃	—	8	4	no	—
147	C₄×C₁₃	H(3,𝔽₄)	1	2	yes	—
148	C₄×C₁₃	#562	1	2	yes	—
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×D₄	—	2	2	no	—
151	C₂²×D₄	—	2	2	no	—
152	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
153	C₂×D₈	—	4	8	no	—
154	C₂×D₈	—	4	4	no	—
155	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
156	C₄⋊Q₈	—	4	4	no	—
157	#127	—	4	4	no	—
158	#124	—	4	8	no	—
159	#124	—	4	8	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
162	C₂×C₂²⋊C₄	—	2	4	no	—
163	C₂×C₂²⋊C₄	—	2	4	no	—
164	C₂×C₂²⋊C₄	—	2	4	no	—
165	C₂×C₂²⋊C₄	—	2	4	no	—
166	C₂×C₄²	D₄×C₁₃	1	2	yes	—
167	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
168	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
170	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
171	C₄⋊C₈	—	4	2	no	—
172	#102	—	4	8	no	—
173	C₂×C₁₃	—	16	4	no	—
174	C₂×C₁₃	#2819	1	1	yes	characteristic
175	C₂×C₁₃	#3205	1	2	yes	—
176	C₂×C₁₃	#3213	1	2	yes	—
177	C₂×C₁₃	#3245	1	1	yes	characteristic
178	C₂×C₁₃	#3253	1	1	yes	characteristic
179	C₂⁴	—	2	4	no	—
180	C₂×Q₈	—	4	4	no	—
181	C₂×Q₈	—	4	4	no	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×D₄	—	2	8	no	—
184	C₂×D₄	—	2	8	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	4	4	no	—
188	C₂×D₄	—	4	4	no	—
189	C₂×D₄	—	4	4	no	—
190	C₂×D₄	—	4	4	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
198	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
199	D₈	—	4	16	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	2	4	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₂²⋊C₄	—	4	8	no	—
205	C₂²⋊C₄	—	4	8	no	—
206	C₂²⋊C₄	—	4	8	no	—
207	C₂²⋊C₄	—	4	8	no	—
208	C₄²	—	2	4	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	C₁₃×D₈	1	4	yes	complemented
214	C₁₃	#32806	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	4	4	no	—
216	C₂³	—	4	4	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	4	no	—
220	C₂³	—	4	4	no	—
221	C₂³	—	4	4	no	—
222	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
223	Q₈	—	8	8	no	—
224	D₄	—	4	8	no	—
225	D₄	—	4	8	no	—
226	D₄	—	8	8	no	—
227	D₄	—	8	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
240	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
241	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
242	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂×C₄	C₁₃×H(5,𝔽₂)	1	4	yes	—
244	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
245	C₈	—	4	4	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	4	no	—
249	C₂²	—	8	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	—	8	4	no	—
252	C₂²	—	8	4	no	—
253	C₂²	C₁₃×#535	1	2	yes	—
254	C₂²	C₁₃×#537	1	1	yes	characteristic
255	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
256	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
257	C₂²	C₁₃×#575	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	1	no	—
262	C₄	—	8	4	no	—
263	C₄	C₁₃×H(3,𝔽₄)	1	2	yes	—
264	C₄	C₁₃×#562	1	2	yes	—
265	C₂	—	16	4	no	—
266	C₂	C₁₃×#2819	1	1	yes	characteristic
267	C₂	C₁₃×#3205	1	2	yes	—
268	C₂	C₁₃×#3213	1	2	yes	—
269	C₂	C₁₃×#3245	1	1	yes	characteristic
270	C₂	C₁₃×#3253	1	1	yes	characteristic

	subgroup	G/H
center, socle	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
1. C₁₃×#2858	C₂	semidirect
2. C₁₃×#2054	C₂	semidirect
3. C₁₃×#2023	C₂	semidirect
4. C₁₃×#2011	C₂	semidirect
5. C₁₃×#1831	C₂	semidirect
44. #32806	C₁₃	direct (N × H)
79. C₄²×C₁₃	D₈	semidirect
80. #2858	C₂×C₁₃	semidirect
81. #2054	C₂×C₁₃	semidirect
82. #2023	C₂×C₁₃	semidirect
83. #2011	C₂×C₁₃	semidirect
84. #1831	C₂×C₁₃	semidirect
213. C₄²	C₁₃×D₈	semidirect
214. C₁₃	#32806	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×#2858	2
C₁₃×#2054	2
C₁₃×#2023	2
C₁₃×#2011	2
C₁₃×#1831	2
#32806	13

Subgroups — C13×#32806 #15719145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32806 #15719145