Subgroups — C₁₃×#33704 #15720043

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2244	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2240	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2140	C₂	1	4	yes	maximal, complemented
6	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×D₈	—	2	1	no	—
8	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
9	C₁₃×#533	—	2	1	no	—
10	C₂×C₁₃×#127	—	2	1	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₁₃×#500	C₂²	1	2	yes	complemented
13	C₁₃×#499	—	2	1	no	—
14	C₁₃×#499	C₂²	1	2	yes	complemented
15	C₁₃×#498	—	2	1	no	—
16	C₁₃×#493	—	2	1	no	—
17	C₁₃×#493	—	2	1	no	—
18	C₁₃×#493	C₂²	1	2	yes	—
19	C₁₃×#493	C₂²	1	2	yes	—
20	C₁₃×#427	—	2	1	no	—
21	C₁₃×#427	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#357	C₂²	1	16	yes	complemented
24	C₁₃×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
26	C₂²×D₄×C₁₃	—	2	2	no	—
27	C₂²×D₄×C₁₃	—	4	2	no	—
28	C₂×C₁₃×D₈	—	2	2	no	—
29	C₂×C₁₃×D₈	—	2	2	no	—
30	C₂×C₁₃×D₈	—	2	2	no	—
31	C₂×C₁₃×D₈	—	2	8	no	—
32	C₂×C₁₃×D₈	—	4	4	no	—
33	C₂×C₁₃×D₈	C₂³	1	4	yes	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×#127	—	2	2	no	—
38	C₁₃×#127	—	2	2	no	—
39	C₁₃×#127	—	4	2	no	—
40	C₁₃×#125	—	2	2	no	—
41	C₁₃×C₂²⋊Q₈	—	4	2	no	—
42	C₄×Q₈×C₁₃	—	4	2	no	—
43	C₁₃×#117	—	2	2	no	—
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₁₃×M₃₂	—	2	4	no	—
46	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
47	C₁₃×#107	—	2	2	no	—
48	C₁₃×#107	C₂³	1	4	yes	—
49	C₁₃×#106	—	2	2	no	—
50	C₁₃×#106	—	2	2	no	—
51	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	#33704	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	—	4	2	no	—
57	C₂×Q₈×C₁₃	—	4	2	no	—
58	C₂×D₄×C₁₃	—	2	4	no	—
59	C₂×D₄×C₁₃	—	4	4	no	—
60	C₂×D₄×C₁₃	—	4	4	no	—
61	C₂×D₄×C₁₃	—	4	4	no	—
62	C₂×D₄×C₁₃	—	4	2	no	—
63	C₂×D₄×C₁₃	—	4	4	no	—
64	C₂×D₄×C₁₃	—	8	8	no	—
65	C₂²×C₄×C₁₃	D₈	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₁₃×D₈	—	2	8	no	—
68	C₁₃×D₈	—	4	4	no	—
69	C₁₃×D₈	—	4	4	no	—
70	C₁₃×D₈	—	4	4	no	—
71	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	#41	1	4	yes	—
76	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	—	4	4	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₂²⋊C₄	—	4	2	no	—
81	C₄²×C₁₃	—	4	2	no	—
82	C₄²×C₁₃	D₈	1	2	yes	—
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₁₃×C₁₆	—	2	4	no	—
86	#3160	C₂×C₁₃	1	1	yes	characteristic, complemented
87	C₂×#493	C₂×C₁₃	1	1	yes	characteristic
88	#2244	C₂×C₁₃	1	4	yes	complemented
89	#2240	C₂×C₁₃	1	4	yes	complemented
90	#2140	C₂×C₁₃	1	4	yes	complemented
91	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
92	C₂³×C₁₃	—	4	4	no	—
93	C₂³×C₁₃	—	8	8	no	—
94	C₂³×C₁₃	—	8	4	no	—
95	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
96	Q₈×C₁₃	—	8	4	no	—
97	D₄×C₁₃	—	4	4	no	—
98	D₄×C₁₃	—	4	4	no	—
99	D₄×C₁₃	—	8	8	no	—
100	D₄×C₁₃	—	8	4	no	—
101	D₄×C₁₃	—	8	8	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	#124	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₁₃	#124	1	2	yes	—
111	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
112	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
113	C₈×C₁₃	C₂×#41	1	2	yes	—
114	C₂²×D₈	—	2	1	no	—
115	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
116	#533	—	2	1	no	—
117	C₂×#127	—	2	1	no	—
118	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
119	#500	C₂²×C₁₃	1	2	yes	complemented
120	#499	—	2	1	no	—
121	#499	C₂²×C₁₃	1	2	yes	complemented
122	#498	—	2	1	no	—
123	#493	—	2	1	no	—
124	#493	—	2	1	no	—
125	#493	C₂²×C₁₃	1	2	yes	—
126	#493	C₂²×C₁₃	1	2	yes	—
127	#427	—	2	1	no	—
128	#427	C₂²×C₁₃	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
130	#357	C₂²×C₁₃	1	16	yes	complemented
131	#346	C₂²×C₁₃	1	2	yes	complemented
132	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	8	4	no	—
135	C₂²×C₁₃	—	8	8	no	—
136	C₂²×C₁₃	—	16	8	no	—
137	C₂²×C₁₃	#486	1	2	yes	—
138	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	8	4	no	—
141	C₄×C₁₃	#486	1	2	yes	—
142	C₄×C₁₃	#509	1	4	yes	—
143	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
144	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
145	C₂²×D₄	—	2	2	no	—
146	C₂²×D₄	—	4	2	no	—
147	C₂×D₈	—	2	2	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	—	2	8	no	—
151	C₂×D₈	—	4	4	no	—
152	C₂×D₈	C₂³×C₁₃	1	4	yes	—
153	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
155	C₄⋊Q₈	—	2	2	no	—
156	#127	—	2	2	no	—
157	#127	—	2	2	no	—
158	#127	—	4	2	no	—
159	#125	—	2	2	no	—
160	C₂²⋊Q₈	—	4	2	no	—
161	C₄×Q₈	—	4	2	no	—
162	#117	—	2	2	no	—
163	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
164	M₃₂	—	2	4	no	—
165	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
166	#107	—	2	2	no	—
167	#107	C₂³×C₁₃	1	4	yes	—
168	#106	—	2	2	no	—
169	#106	—	2	2	no	—
170	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
171	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
172	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×C₁₃	—	2	2	no	—
175	C₂×C₁₃	—	16	4	no	—
176	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
177	C₂×C₁₃	C₂×#509	1	2	yes	—
178	C₂⁴	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂×D₄	—	4	4	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂×D₄	—	8	8	no	—
187	C₂²×C₄	C₁₃×D₈	1	2	yes	—
188	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
189	D₈	—	2	8	no	—
190	D₈	—	4	4	no	—
191	D₈	—	4	4	no	—
192	D₈	—	4	4	no	—
193	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
194	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
195	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
196	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
197	C₂×C₈	C₁₃×#41	1	4	yes	—
198	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₄²	—	4	2	no	—
204	C₄²	C₁₃×D₈	1	2	yes	—
205	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
207	C₁₆	—	2	4	no	—
208	C₁₃	#33704	1	1	yes	characteristic, direct product factor, Sylow
209	C₂³	—	4	4	no	—
210	C₂³	—	8	8	no	—
211	C₂³	—	8	4	no	—
212	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
213	Q₈	—	8	4	no	—
214	D₄	—	4	4	no	—
215	D₄	—	4	4	no	—
216	D₄	—	8	8	no	—
217	D₄	—	8	4	no	—
218	D₄	—	8	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	C₁₃×#124	1	2	yes	—
222	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
223	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
224	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
225	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
226	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
227	C₈	C₁₃×#124	1	2	yes	—
228	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
229	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
230	C₈	C₂×C₁₃×#41	1	2	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	8	4	no	—
233	C₂²	—	8	8	no	—
234	C₂²	—	16	8	no	—
235	C₂²	C₁₃×#486	1	2	yes	—
236	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	8	4	no	—
239	C₄	C₁₃×#486	1	2	yes	—
240	C₄	C₁₃×#509	1	4	yes	—
241	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
242	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	16	4	no	—
245	C₂	C₂×C₁₃×#486	1	1	yes	characteristic
246	C₂	C₂×C₁₃×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3160	C₂	semidirect
3. C₁₃×#2244	C₂	semidirect
4. C₁₃×#2240	C₂	semidirect
5. C₁₃×#2140	C₂	semidirect
6. C₄×C₁₃×M₃₂	C₂	semidirect
12. C₁₃×#500	C₂²	semidirect
14. C₁₃×#499	C₂²	semidirect
23. C₁₃×#357	C₂²	semidirect
24. C₁₃×#346	C₂²	semidirect
25. C₄×C₁₃×C₁₆	C₂²	semidirect
55. #33704	C₁₃	direct (N × H)
86. #3160	C₂×C₁₃	semidirect
88. #2244	C₂×C₁₃	semidirect
89. #2240	C₂×C₁₃	semidirect
90. #2140	C₂×C₁₃	semidirect
91. C₄×M₃₂	C₂×C₁₃	semidirect
119. #500	C₂²×C₁₃	semidirect
121. #499	C₂²×C₁₃	semidirect
130. #357	C₂²×C₁₃	semidirect
131. #346	C₂²×C₁₃	semidirect
132. C₄×C₁₆	C₂²×C₁₃	semidirect
208. C₁₃	#33704	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×#3160	2
C₂×C₁₃×#493	2
C₁₃×#2244	2
C₁₃×#2240	2
C₁₃×#2140	2
C₄×C₁₃×M₃₂	2
#33704	13

Subgroups — C13×#33704 #15720043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33704 #15720043