Subgroups — C₁₃×#3500 #13112558

Order 1664.

1540 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#548	C₂	1	4	yes	maximal, complemented
4	C₁₃×#546	C₂	1	12	yes	maximal, complemented
5	C₁₃×#536	C₂	1	3	yes	maximal, complemented
6	C₁₃×#535	C₂	1	3	yes	maximal, complemented
7	C₁₃×#532	C₂	1	3	yes	maximal, complemented
8	C₂×C₁₃×#124	C₂	1	3	yes	maximal, complemented
9	C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×2₋¹⁺⁴	—	2	8	no	—
11	C₁₃×H(5,𝔽₂)	—	2	8	no	—
12	C₂×C₁₃×#41	C₂²	1	3	yes	—
13	C₂×C₁₃×#41	C₂²	1	2	yes	—
14	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#41	C₂²	1	3	yes	complemented
16	C₂×C₁₃×#41	C₂²	1	6	yes	complemented
17	C₂²×Q₈×C₁₃	C₂²	1	3	yes	complemented
18	C₂²×Q₈×C₁₃	C₂²	1	2	yes	complemented
19	C₂²×D₄×C₁₃	C₂²	1	3	yes	—
20	C₂²×D₄×C₁₃	C₂²	1	6	yes	—
21	C₁₃×C₄⋊Q₈	C₂²	1	3	yes	complemented
22	C₁₃×#127	C₂²	1	3	yes	complemented
23	C₁₃×#124	C₂²	1	3	yes	complemented
24	C₁₃×#124	C₂²	1	6	yes	complemented
25	C₁₃×#124	C₂²	1	12	yes	complemented
26	C₁₃×#124	C₂²	1	1	yes	characteristic, complemented
27	C₁₃×#123	C₂²	1	12	yes	complemented
28	C₁₃×C₂²⋊Q₈	C₂²	1	12	yes	complemented
29	C₁₃×#121	C₂²	1	12	yes	complemented
30	C₁₃×#121	C₂²	1	12	yes	complemented
31	C₁₃×#120	C₂²	1	12	yes	complemented
32	C₄×Q₈×C₁₃	C₂²	1	2	yes	complemented
33	C₄×D₄×C₁₃	C₂²	1	6	yes	complemented
34	C₄×D₄×C₁₃	C₂²	1	12	yes	complemented
35	C₁₃×#117	C₂²	1	3	yes	complemented
36	C₂×C₁₃×C₂²⋊C₄	C₂²	1	12	yes	complemented
37	C₂×C₄²×C₁₃	C₂²	1	3	yes	complemented
38	C₂⁴×C₁₃	C₂³	1	6	yes	—
39	C₁₃×#41	—	2	8	no	—
40	C₁₃×#41	—	2	24	no	—
41	C₁₃×#41	—	2	12	no	—
42	C₁₃×#41	—	2	12	no	—
43	C₁₃×#41	D₄	1	8	yes	complemented
44	C₂×Q₈×C₁₃	—	2	8	no	—
45	C₂×Q₈×C₁₃	—	2	12	no	—
46	C₂×Q₈×C₁₃	C₂³	1	6	yes	—
47	C₂×Q₈×C₁₃	C₂³	1	3	yes	—
48	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₃	—	2	24	no	—
51	C₂×D₄×C₁₃	—	2	12	no	—
52	C₂×D₄×C₁₃	C₂³	1	6	yes	—
53	C₂×D₄×C₁₃	C₂³	1	6	yes	—
54	C₂×D₄×C₁₃	C₂³	1	6	yes	—
55	C₂×D₄×C₁₃	C₂³	1	3	yes	—
56	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₁₃	C₂³	1	12	yes	—
58	C₂×D₄×C₁₃	C₂³	1	3	yes	—
59	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
60	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
61	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
62	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
63	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
65	C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
66	C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
67	C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
68	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
69	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
70	C₄²×C₁₃	C₂³	1	1	yes	characteristic
71	C₄²×C₁₃	C₂³	1	3	yes	—
72	C₄²×C₁₃	C₂³	1	6	yes	—
73	#3500	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₃	—	2	12	no	—
75	C₂³×C₁₃	—	4	12	no	—
76	C₂³×C₁₃	C₂⁴	1	6	yes	—
77	C₂³×C₁₃	C₂⁴	1	2	yes	—
78	C₂³×C₁₃	C₂⁴	1	3	yes	—
79	Q₈×C₁₃	—	2	12	no	—
80	Q₈×C₁₃	—	2	6	no	—
81	Q₈×C₁₃	—	2	2	no	—
82	Q₈×C₁₃	C₂×D₄	1	4	yes	—
83	D₄×C₁₃	—	2	12	no	—
84	D₄×C₁₃	—	2	12	no	—
85	D₄×C₁₃	—	2	12	no	—
86	D₄×C₁₃	—	2	6	no	—
87	D₄×C₁₃	—	2	2	no	—
88	D₄×C₁₃	—	4	12	no	—
89	D₄×C₁₃	C₂×D₄	1	12	yes	—
90	C₂×C₄×C₁₃	—	2	12	no	—
91	C₂×C₄×C₁₃	—	2	12	no	—
92	C₂×C₄×C₁₃	—	2	12	no	—
93	C₂×C₄×C₁₃	—	4	6	no	—
94	C₂×C₄×C₁₃	C₂×D₄	1	12	yes	—
95	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
96	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
97	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
98	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
99	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
101	C₂×2₋¹⁺⁴	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂×H(5,𝔽₂)	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#548	C₂×C₁₃	1	4	yes	complemented
104	#546	C₂×C₁₃	1	12	yes	complemented
105	#536	C₂×C₁₃	1	3	yes	complemented
106	#535	C₂×C₁₃	1	3	yes	complemented
107	#532	C₂×C₁₃	1	3	yes	complemented
108	C₂×#124	C₂×C₁₃	1	3	yes	complemented
109	C₄×#41	C₂×C₁₃	1	1	yes	characteristic, complemented
110	C₂²×C₁₃	—	2	6	no	—
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	—	2	6	no	—
113	C₂²×C₁₃	—	2	4	no	—
114	C₂²×C₁₃	—	2	12	no	—
115	C₂²×C₁₃	—	8	12	no	—
116	C₂²×C₁₃	C₂²×D₄	1	6	yes	—
117	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	—	2	6	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	6	no	—
122	C₄×C₁₃	C₂²×D₄	1	6	yes	—
123	C₄×C₁₃	C₂²×D₄	1	2	yes	—
124	2₋¹⁺⁴	—	2	8	no	—
125	H(5,𝔽₂)	—	2	8	no	—
126	C₂×#41	C₂²×C₁₃	1	3	yes	—
127	C₂×#41	C₂²×C₁₃	1	2	yes	—
128	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×#41	C₂²×C₁₃	1	3	yes	complemented
130	C₂×#41	C₂²×C₁₃	1	6	yes	complemented
131	C₂²×Q₈	C₂²×C₁₃	1	3	yes	complemented
132	C₂²×Q₈	C₂²×C₁₃	1	2	yes	complemented
133	C₂²×D₄	C₂²×C₁₃	1	3	yes	—
134	C₂²×D₄	C₂²×C₁₃	1	6	yes	—
135	C₄⋊Q₈	C₂²×C₁₃	1	3	yes	complemented
136	#127	C₂²×C₁₃	1	3	yes	complemented
137	#124	C₂²×C₁₃	1	3	yes	complemented
138	#124	C₂²×C₁₃	1	6	yes	complemented
139	#124	C₂²×C₁₃	1	12	yes	complemented
140	#124	C₂²×C₁₃	1	1	yes	characteristic, complemented
141	#123	C₂²×C₁₃	1	12	yes	complemented
142	C₂²⋊Q₈	C₂²×C₁₃	1	12	yes	complemented
143	#121	C₂²×C₁₃	1	12	yes	complemented
144	#121	C₂²×C₁₃	1	12	yes	complemented
145	#120	C₂²×C₁₃	1	12	yes	complemented
146	C₄×Q₈	C₂²×C₁₃	1	2	yes	complemented
147	C₄×D₄	C₂²×C₁₃	1	6	yes	complemented
148	C₄×D₄	C₂²×C₁₃	1	12	yes	complemented
149	#117	C₂²×C₁₃	1	3	yes	complemented
150	C₂×C₂²⋊C₄	C₂²×C₁₃	1	12	yes	complemented
151	C₂×C₄²	C₂²×C₁₃	1	3	yes	complemented
152	C₂×C₁₃	—	2	6	no	—
153	C₂×C₁₃	—	4	2	no	—
154	C₂×C₁₃	—	4	6	no	—
155	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
156	C₂×C₁₃	#585	1	2	yes	—
157	C₂⁴	C₂³×C₁₃	1	6	yes	—
158	#41	—	2	8	no	—
159	#41	—	2	24	no	—
160	#41	—	2	12	no	—
161	#41	—	2	12	no	—
162	#41	D₄×C₁₃	1	8	yes	complemented
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	—	2	12	no	—
165	C₂×Q₈	C₂³×C₁₃	1	6	yes	—
166	C₂×Q₈	C₂³×C₁₃	1	3	yes	—
167	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
168	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
169	C₂×D₄	—	2	24	no	—
170	C₂×D₄	—	2	12	no	—
171	C₂×D₄	C₂³×C₁₃	1	6	yes	—
172	C₂×D₄	C₂³×C₁₃	1	6	yes	—
173	C₂×D₄	C₂³×C₁₃	1	6	yes	—
174	C₂×D₄	C₂³×C₁₃	1	3	yes	—
175	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
176	C₂×D₄	C₂³×C₁₃	1	12	yes	—
177	C₂×D₄	C₂³×C₁₃	1	3	yes	—
178	C₂²×C₄	C₂³×C₁₃	1	6	yes	—
179	C₂²×C₄	C₂³×C₁₃	1	6	yes	—
180	C₂²×C₄	C₂³×C₁₃	1	6	yes	—
181	C₂²×C₄	C₂³×C₁₃	1	6	yes	—
182	C₂²×C₄	C₂³×C₁₃	1	3	yes	—
183	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
184	C₄⋊C₄	C₂³×C₁₃	1	12	yes	—
185	C₂²⋊C₄	C₂³×C₁₃	1	12	yes	—
186	C₂²⋊C₄	C₂³×C₁₃	1	24	yes	—
187	C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
188	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
189	C₄²	C₂³×C₁₃	1	1	yes	characteristic
190	C₄²	C₂³×C₁₃	1	3	yes	—
191	C₄²	C₂³×C₁₃	1	6	yes	—
192	C₁₃	#3500	1	1	yes	characteristic, direct product factor, Sylow
193	C₂³	—	2	12	no	—
194	C₂³	—	4	12	no	—
195	C₂³	C₂⁴×C₁₃	1	6	yes	—
196	C₂³	C₂⁴×C₁₃	1	2	yes	—
197	C₂³	C₂⁴×C₁₃	1	3	yes	—
198	Q₈	—	2	12	no	—
199	Q₈	—	2	6	no	—
200	Q₈	—	2	2	no	—
201	Q₈	C₂×D₄×C₁₃	1	4	yes	—
202	D₄	—	2	12	no	—
203	D₄	—	2	12	no	—
204	D₄	—	2	12	no	—
205	D₄	—	2	6	no	—
206	D₄	—	2	2	no	—
207	D₄	—	4	12	no	—
208	D₄	C₂×D₄×C₁₃	1	12	yes	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	—	2	12	no	—
212	C₂×C₄	—	4	6	no	—
213	C₂×C₄	C₂×D₄×C₁₃	1	12	yes	—
214	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
215	C₂×C₄	C₂⁴×C₁₃	1	6	yes	—
216	C₂×C₄	C₂⁴×C₁₃	1	6	yes	—
217	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
218	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₁₃	1	3	yes	—
220	C₂²	—	2	6	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	6	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	12	no	—
225	C₂²	—	8	12	no	—
226	C₂²	C₂²×D₄×C₁₃	1	6	yes	—
227	C₂²	C₂⁵×C₁₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	6	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	6	no	—
232	C₄	C₂²×D₄×C₁₃	1	6	yes	—
233	C₄	C₂²×D₄×C₁₃	1	2	yes	—
234	C₂	—	2	6	no	—
235	C₂	—	4	6	no	—
236	C₂	—	4	2	no	—
237	C₂	C₂³×D₄×C₁₃	1	1	yes	characteristic
238	C₂	C₁₃×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×2₋¹⁺⁴	C₂	semidirect
2. C₂×C₁₃×H(5,𝔽₂)	C₂	semidirect
3. C₁₃×#548	C₂	semidirect
4. C₁₃×#546	C₂	semidirect
5. C₁₃×#536	C₂	semidirect
6. C₁₃×#535	C₂	semidirect
7. C₁₃×#532	C₂	semidirect
8. C₂×C₁₃×#124	C₂	semidirect
9. C₄×C₁₃×#41	C₂	semidirect
15. C₂×C₁₃×#41	C₂²	semidirect
16. C₂×C₁₃×#41	C₂²	semidirect
17. C₂²×Q₈×C₁₃	C₂²	semidirect
18. C₂²×Q₈×C₁₃	C₂²	semidirect
21. C₁₃×C₄⋊Q₈	C₂²	semidirect
22. C₁₃×#127	C₂²	semidirect
23. C₁₃×#124	C₂²	semidirect
24. C₁₃×#124	C₂²	semidirect
25. C₁₃×#124	C₂²	semidirect
26. C₁₃×#124	C₂²	semidirect
27. C₁₃×#123	C₂²	semidirect
28. C₁₃×C₂²⋊Q₈	C₂²	semidirect
29. C₁₃×#121	C₂²	semidirect
30. C₁₃×#121	C₂²	semidirect
31. C₁₃×#120	C₂²	semidirect
32. C₄×Q₈×C₁₃	C₂²	semidirect
33. C₄×D₄×C₁₃	C₂²	semidirect
34. C₄×D₄×C₁₃	C₂²	semidirect
35. C₁₃×#117	C₂²	semidirect
36. C₂×C₁₃×C₂²⋊C₄	C₂²	semidirect
37. C₂×C₄²×C₁₃	C₂²	semidirect
43. C₁₃×#41	D₄	semidirect
73. #3500	C₁₃	direct (N × H)
101. C₂×2₋¹⁺⁴	C₂×C₁₃	semidirect
102. C₂×H(5,𝔽₂)	C₂×C₁₃	semidirect
103. #548	C₂×C₁₃	semidirect
104. #546	C₂×C₁₃	semidirect
105. #536	C₂×C₁₃	semidirect
106. #535	C₂×C₁₃	semidirect
107. #532	C₂×C₁₃	semidirect
108. C₂×#124	C₂×C₁₃	semidirect
109. C₄×#41	C₂×C₁₃	semidirect
129. C₂×#41	C₂²×C₁₃	semidirect
130. C₂×#41	C₂²×C₁₃	semidirect
131. C₂²×Q₈	C₂²×C₁₃	semidirect
132. C₂²×Q₈	C₂²×C₁₃	semidirect
135. C₄⋊Q₈	C₂²×C₁₃	semidirect
136. #127	C₂²×C₁₃	semidirect
137. #124	C₂²×C₁₃	semidirect
138. #124	C₂²×C₁₃	semidirect
139. #124	C₂²×C₁₃	semidirect
140. #124	C₂²×C₁₃	semidirect
141. #123	C₂²×C₁₃	semidirect
142. C₂²⋊Q₈	C₂²×C₁₃	semidirect
143. #121	C₂²×C₁₃	semidirect
144. #121	C₂²×C₁₃	semidirect
145. #120	C₂²×C₁₃	semidirect
146. C₄×Q₈	C₂²×C₁₃	semidirect
147. C₄×D₄	C₂²×C₁₃	semidirect
148. C₄×D₄	C₂²×C₁₃	semidirect
149. #117	C₂²×C₁₃	semidirect
150. C₂×C₂²⋊C₄	C₂²×C₁₃	semidirect
151. C₂×C₄²	C₂²×C₁₃	semidirect
162. #41	D₄×C₁₃	semidirect
192. C₁₃	#3500	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×2₋¹⁺⁴	2
C₂×C₁₃×H(5,𝔽₂)	2
C₁₃×#548	2
C₁₃×#546	2
C₁₃×#536	2
C₁₃×#535	2
C₁₃×#532	2
C₂×C₁₃×#124	2
C₄×C₁₃×#41	2
#3500	13

Subgroups — C13×#3500 #13112558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3500 #13112558