Subgroups — C₁₃×#2748 #13111806

Order 1664.

464 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#399	C₂	1	2	yes	maximal, complemented
2	C₁₃×#398	C₂	1	2	yes	maximal, complemented
3	C₁₃×#396	C₂	1	2	yes	maximal, complemented
4	C₁₃×#390	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#387	C₂	1	2	yes	maximal, complemented
6	C₁₃×#386	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#386	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#385	C₂	1	2	yes	maximal, complemented
9	C₁₃×#380	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
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×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	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	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×D₄×C₁₃	—	4	1	no	—
34	C₂×D₄×C₁₃	—	4	1	no	—
35	C₂×D₄×C₁₃	Q₈	1	4	yes	—
36	C₂²×C₄×C₁₃	—	2	4	no	—
37	C₂²×C₄×C₁₃	—	4	1	no	—
38	C₂²×C₄×C₁₃	—	4	2	no	—
39	C₂²×C₄×C₁₃	—	4	1	no	—
40	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₁₃×C₄⋊C₄	—	4	2	no	—
49	C₁₃×C₄⋊C₄	—	4	1	no	—
50	C₁₃×C₄⋊C₄	—	4	2	no	—
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₁₃×C₄⋊C₄	—	4	2	no	—
53	C₁₃×C₂²⋊C₄	—	2	4	no	—
54	C₁₃×C₂²⋊C₄	—	4	2	no	—
55	C₁₃×C₂²⋊C₄	—	4	2	no	—
56	C₁₃×C₂²⋊C₄	—	4	1	no	—
57	C₁₃×C₂²⋊C₄	—	4	1	no	—
58	C₁₃×C₂²⋊C₄	—	4	2	no	—
59	C₄²×C₁₃	—	4	1	no	—
60	#2748	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₁₃	—	2	2	no	—
62	C₂³×C₁₃	—	2	2	no	—
63	C₂³×C₁₃	—	2	2	no	—
64	C₂³×C₁₃	—	2	2	no	—
65	C₂³×C₁₃	—	2	1	no	—
66	C₂³×C₁₃	—	2	1	no	—
67	C₂³×C₁₃	—	2	1	no	—
68	C₂³×C₁₃	—	2	1	no	—
69	C₂³×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂³×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
72	D₄×C₁₃	—	4	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	1	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	—	2	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	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₁₃	#41	1	4	yes	—
96	#399	C₂×C₁₃	1	2	yes	complemented
97	#398	C₂×C₁₃	1	2	yes	complemented
98	#396	C₂×C₁₃	1	2	yes	complemented
99	#390	C₂×C₁₃	1	1	yes	characteristic
100	#387	C₂×C₁₃	1	2	yes	complemented
101	#386	C₂×C₁₃	1	1	yes	characteristic
102	#386	C₂×C₁₃	1	1	yes	characteristic
103	#385	C₂×C₁₃	1	2	yes	complemented
104	#380	C₂×C₁₃	1	2	yes	complemented
105	C₂²×C₁₃	—	2	2	no	—
106	C₂²×C₁₃	—	2	2	no	—
107	C₂²×C₁₃	—	2	4	no	—
108	C₂²×C₁₃	—	2	4	no	—
109	C₂²×C₁₃	—	4	2	no	—
110	C₂²×C₁₃	—	4	2	no	—
111	C₂²×C₁₃	—	4	1	no	—
112	C₂²×C₁₃	—	4	1	no	—
113	C₂²×C₁₃	—	4	1	no	—
114	C₂²×C₁₃	—	4	1	no	—
115	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂²×C₁₃	C₂×#41	1	2	yes	—
117	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	4	no	—
122	C₄×C₁₃	—	4	2	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	—	4	2	no	—
126	C₄×C₁₃	—	4	2	no	—
127	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
128	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
129	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
131	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
132	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
137	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
140	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
143	#95	C₂²×C₁₃	1	2	yes	—
144	#95	C₂²×C₁₃	1	2	yes	—
145	#95	C₂²×C₁₃	1	2	yes	—
146	#95	C₂²×C₁₃	1	2	yes	—
147	#95	C₂²×C₁₃	1	2	yes	—
148	C₂×C₁₃	—	4	2	no	—
149	C₂×C₁₃	—	4	2	no	—
150	C₂×C₁₃	#540	1	2	yes	—
151	C₂×C₁₃	#543	1	1	yes	characteristic
152	C₂×C₁₃	#551	1	1	yes	characteristic
153	C₂×C₁₃	#554	1	2	yes	—
154	C₂×C₁₃	#560	1	1	yes	characteristic
155	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
156	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
157	C₂×D₄	—	4	1	no	—
158	C₂×D₄	—	4	1	no	—
159	C₂×D₄	Q₈×C₁₃	1	4	yes	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
165	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
167	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	1	no	—
181	C₂²⋊C₄	—	4	1	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₄²	—	4	1	no	—
184	C₁₃	#2748	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	1	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	1	no	—
193	C₂³	C₂×Q₈×C₁₃	1	2	yes	—
194	C₂³	C₂×Q₈×C₁₃	1	2	yes	—
195	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
196	D₄	—	4	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	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	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
219	C₂×C₄	C₁₃×#41	1	4	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	C₂²×Q₈×C₁₃	1	1	yes	characteristic
231	C₂²	C₂×C₁₃×#41	1	2	yes	—
232	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
233	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
234	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₂	—	4	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₁₃×#540	1	2	yes	—
245	C₂	C₁₃×#543	1	1	yes	characteristic
246	C₂	C₁₃×#551	1	1	yes	characteristic
247	C₂	C₁₃×#554	1	2	yes	—
248	C₂	C₁₃×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
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×#399	C₂	semidirect
2. C₁₃×#398	C₂	semidirect
3. C₁₃×#396	C₂	semidirect
5. C₁₃×#387	C₂	semidirect
8. C₁₃×#385	C₂	semidirect
9. C₁₃×#380	C₂	semidirect
60. #2748	C₁₃	direct (N × H)
96. #399	C₂×C₁₃	semidirect
97. #398	C₂×C₁₃	semidirect
98. #396	C₂×C₁₃	semidirect
100. #387	C₂×C₁₃	semidirect
103. #385	C₂×C₁₃	semidirect
104. #380	C₂×C₁₃	semidirect
184. C₁₃	#2748	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×#399	2
C₁₃×#398	2
C₁₃×#396	2
C₁₃×#390	2
C₁₃×#387	2
C₁₃×#386	2
C₁₃×#385	2
C₁₃×#380	2
#2748	13

Subgroups — C13×#2748 #13111806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2748 #13111806