Subgroups — C₁₃×#3496 #13112554

Order 1664.

1660 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₁₃×#547	C₂	1	4	yes	maximal, complemented
3	C₁₃×#546	C₂	1	8	yes	maximal, complemented
4	D₄²×C₁₃	C₂	1	4	yes	maximal, complemented
5	C₁₃×#537	C₂	1	2	yes	maximal, complemented
6	C₁₃×#534	C₂	1	4	yes	maximal, complemented
7	C₁₃×#532	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#121	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×H(5,𝔽₂)	—	2	16	no	—
11	C₂×C₁₃×#41	C₂²	1	4	yes	complemented
12	C₂×C₁₃×#41	C₂²	1	8	yes	complemented
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₃	C₂²	1	8	yes	complemented
15	C₂²×D₄×C₁₃	C₂²	1	8	yes	complemented
16	C₂²×D₄×C₁₃	C₂²	1	4	yes	complemented
17	C₂³×C₄×C₁₃	C₂²	1	2	yes	complemented
18	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
19	C₁₃×#127	C₂²	1	2	yes	complemented
20	C₁₃×#124	C₂²	1	4	yes	complemented
21	C₁₃×#123	C₂²	1	16	yes	complemented
22	C₁₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
23	C₁₃×#121	C₂²	1	4	yes	complemented
24	C₁₃×#121	C₂²	1	8	yes	complemented
25	C₁₃×#121	C₂²	1	8	yes	complemented
26	C₁₃×#121	C₂²	1	8	yes	complemented
27	C₁₃×#121	C₂²	1	4	yes	complemented
28	C₁₃×#121	C₂²	1	8	yes	complemented
29	C₁₃×#120	C₂²	1	16	yes	complemented
30	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
31	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
32	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
33	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
34	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	complemented
36	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
37	C₂×C₁₃×C₂²⋊C₄	C₂²	1	8	yes	complemented
38	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×C₁₃	C₂³	1	8	yes	—
40	C₂⁴×C₁₃	C₂³	1	2	yes	—
41	C₁₃×#41	—	2	32	no	—
42	C₁₃×#41	—	2	16	no	—
43	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
44	C₂×D₄×C₁₃	—	2	32	no	—
45	C₂×D₄×C₁₃	—	2	32	no	—
46	C₂×D₄×C₁₃	—	4	8	no	—
47	C₂×D₄×C₁₃	D₄	1	8	yes	complemented
48	C₂×D₄×C₁₃	C₂³	1	4	yes	—
49	C₂×D₄×C₁₃	C₂³	1	16	yes	—
50	C₂×D₄×C₁₃	C₂³	1	8	yes	—
51	C₂×D₄×C₁₃	C₂³	1	4	yes	—
52	C₂×D₄×C₁₃	C₂³	1	4	yes	—
53	C₂×D₄×C₁₃	C₂³	1	8	yes	—
54	C₂×D₄×C₁₃	C₂³	1	4	yes	—
55	C₂²×C₄×C₁₃	—	4	4	no	—
56	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂³	1	8	yes	—
58	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂³	1	8	yes	—
60	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
65	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
66	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
68	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
69	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
70	C₁₃×C₂²⋊C₄	C₂³	1	16	yes	—
71	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
72	C₄²×C₁₃	C₂³	1	2	yes	—
73	C₄²×C₁₃	C₂³	1	2	yes	—
74	#3496	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₃	—	2	16	no	—
76	C₂³×C₁₃	—	4	2	no	—
77	C₂³×C₁₃	—	4	2	no	—
78	C₂³×C₁₃	—	4	8	no	—
79	C₂³×C₁₃	—	4	8	no	—
80	C₂³×C₁₃	C₂×D₄	1	8	yes	complemented
81	C₂³×C₁₃	C₂⁴	1	8	yes	—
82	C₂³×C₁₃	C₂⁴	1	4	yes	—
83	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
84	Q₈×C₁₃	—	2	8	no	—
85	D₄×C₁₃	—	2	32	no	—
86	D₄×C₁₃	—	2	16	no	—
87	D₄×C₁₃	—	2	8	no	—
88	D₄×C₁₃	—	2	8	no	—
89	D₄×C₁₃	—	4	8	no	—
90	D₄×C₁₃	—	4	4	no	—
91	D₄×C₁₃	C₂×D₄	1	16	yes	complemented
92	C₂×C₄×C₁₃	—	2	16	no	—
93	C₂×C₄×C₁₃	—	2	16	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	8	no	—
96	C₂×C₄×C₁₃	—	4	4	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	4	1	no	—
99	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	complemented
100	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
105	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
106	C₂×H(5,𝔽₂)	C₂×C₁₃	1	2	yes	complemented
107	#547	C₂×C₁₃	1	4	yes	complemented
108	#546	C₂×C₁₃	1	8	yes	complemented
109	D₄²	C₂×C₁₃	1	4	yes	complemented
110	#537	C₂×C₁₃	1	2	yes	complemented
111	#534	C₂×C₁₃	1	4	yes	complemented
112	#532	C₂×C₁₃	1	2	yes	complemented
113	C₂×#121	C₂×C₁₃	1	4	yes	complemented
114	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂²×C₁₃	—	2	1	no	—
116	C₂²×C₁₃	—	2	4	no	—
117	C₂²×C₁₃	—	2	1	no	—
118	C₂²×C₁₃	—	2	4	no	—
119	C₂²×C₁₃	—	2	8	no	—
120	C₂²×C₁₃	—	2	8	no	—
121	C₂²×C₁₃	—	2	8	no	—
122	C₂²×C₁₃	—	4	8	no	—
123	C₂²×C₁₃	—	8	16	no	—
124	C₂²×C₁₃	C₂²×D₄	1	8	yes	—
125	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
126	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
127	C₄×C₁₃	—	2	8	no	—
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	—	2	4	no	—
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	C₂²×D₄	1	4	yes	—
132	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	complemented
133	H(5,𝔽₂)	—	2	16	no	—
134	C₂×#41	C₂²×C₁₃	1	4	yes	complemented
135	C₂×#41	C₂²×C₁₃	1	8	yes	complemented
136	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
137	C₂²×D₄	C₂²×C₁₃	1	8	yes	complemented
138	C₂²×D₄	C₂²×C₁₃	1	8	yes	complemented
139	C₂²×D₄	C₂²×C₁₃	1	4	yes	complemented
140	C₂³×C₄	C₂²×C₁₃	1	2	yes	complemented
141	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	complemented
142	#127	C₂²×C₁₃	1	2	yes	complemented
143	#124	C₂²×C₁₃	1	4	yes	complemented
144	#123	C₂²×C₁₃	1	16	yes	complemented
145	C₂²⋊Q₈	C₂²×C₁₃	1	8	yes	complemented
146	#121	C₂²×C₁₃	1	4	yes	complemented
147	#121	C₂²×C₁₃	1	8	yes	complemented
148	#121	C₂²×C₁₃	1	8	yes	complemented
149	#121	C₂²×C₁₃	1	8	yes	complemented
150	#121	C₂²×C₁₃	1	4	yes	complemented
151	#121	C₂²×C₁₃	1	8	yes	complemented
152	#120	C₂²×C₁₃	1	16	yes	complemented
153	C₄×D₄	C₂²×C₁₃	1	4	yes	complemented
154	C₄×D₄	C₂²×C₁₃	1	4	yes	complemented
155	C₄×D₄	C₂²×C₁₃	1	4	yes	complemented
156	C₄×D₄	C₂²×C₁₃	1	4	yes	complemented
157	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic, complemented
158	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	complemented
159	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	complemented
160	C₂×C₂²⋊C₄	C₂²×C₁₃	1	8	yes	complemented
161	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic, complemented
162	C₂×C₁₃	—	2	2	no	—
163	C₂×C₁₃	—	2	8	no	—
164	C₂×C₁₃	—	4	8	no	—
165	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
166	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×C₁₃	#585	1	1	yes	characteristic
168	C₂⁴	C₂³×C₁₃	1	8	yes	—
169	C₂⁴	C₂³×C₁₃	1	2	yes	—
170	#41	—	2	32	no	—
171	#41	—	2	16	no	—
172	C₂×Q₈	C₂³×C₁₃	1	4	yes	—
173	C₂×D₄	—	2	32	no	—
174	C₂×D₄	—	2	32	no	—
175	C₂×D₄	—	4	8	no	—
176	C₂×D₄	D₄×C₁₃	1	8	yes	complemented
177	C₂×D₄	C₂³×C₁₃	1	4	yes	—
178	C₂×D₄	C₂³×C₁₃	1	16	yes	—
179	C₂×D₄	C₂³×C₁₃	1	8	yes	—
180	C₂×D₄	C₂³×C₁₃	1	4	yes	—
181	C₂×D₄	C₂³×C₁₃	1	4	yes	—
182	C₂×D₄	C₂³×C₁₃	1	8	yes	—
183	C₂×D₄	C₂³×C₁₃	1	4	yes	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₁₃	1	8	yes	—
187	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
188	C₂²×C₄	C₂³×C₁₃	1	8	yes	—
189	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
191	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
192	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
194	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
195	C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
196	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
197	C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
198	C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
199	C₂²⋊C₄	C₂³×C₁₃	1	16	yes	—
200	C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
201	C₄²	C₂³×C₁₃	1	2	yes	—
202	C₄²	C₂³×C₁₃	1	2	yes	—
203	C₁₃	#3496	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³	—	2	16	no	—
205	C₂³	—	4	2	no	—
206	C₂³	—	4	2	no	—
207	C₂³	—	4	8	no	—
208	C₂³	—	4	8	no	—
209	C₂³	C₂×D₄×C₁₃	1	8	yes	complemented
210	C₂³	C₂⁴×C₁₃	1	8	yes	—
211	C₂³	C₂⁴×C₁₃	1	4	yes	—
212	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
213	Q₈	—	2	8	no	—
214	D₄	—	2	32	no	—
215	D₄	—	2	16	no	—
216	D₄	—	2	8	no	—
217	D₄	—	2	8	no	—
218	D₄	—	4	8	no	—
219	D₄	—	4	4	no	—
220	D₄	C₂×D₄×C₁₃	1	16	yes	complemented
221	C₂×C₄	—	2	16	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	complemented
229	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
230	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
231	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
232	C₂×C₄	C₂⁴×C₁₃	1	8	yes	—
233	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
234	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
235	C₂²	—	2	1	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	4	8	no	—
243	C₂²	—	8	16	no	—
244	C₂²	C₂²×D₄×C₁₃	1	8	yes	—
245	C₂²	C₂²×D₄×C₁₃	1	2	yes	—
246	C₂²	C₂⁵×C₁₃	1	1	yes	characteristic
247	C₄	—	2	8	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	4	no	—
250	C₄	—	2	2	no	—
251	C₄	C₂²×D₄×C₁₃	1	4	yes	—
252	C₄	C₁₃×H(5,𝔽₂)	1	2	yes	complemented
253	C₂	—	2	2	no	—
254	C₂	—	2	8	no	—
255	C₂	—	4	8	no	—
256	C₂	C₂³×D₄×C₁₃	1	1	yes	characteristic
257	C₂	C₂×C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
258	C₂	C₁₃×#585	1	1	yes	characteristic

	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×H(5,𝔽₂)	C₂	semidirect
2. C₁₃×#547	C₂	semidirect
3. C₁₃×#546	C₂	semidirect
4. D₄²×C₁₃	C₂	semidirect
5. C₁₃×#537	C₂	semidirect
6. C₁₃×#534	C₂	semidirect
7. C₁₃×#532	C₂	semidirect
8. C₂×C₁₃×#121	C₂	semidirect
9. C₂×C₄×D₄×C₁₃	C₂	semidirect
11. C₂×C₁₃×#41	C₂²	semidirect
12. C₂×C₁₃×#41	C₂²	semidirect
14. C₂²×D₄×C₁₃	C₂²	semidirect
15. C₂²×D₄×C₁₃	C₂²	semidirect
16. C₂²×D₄×C₁₃	C₂²	semidirect
17. C₂³×C₄×C₁₃	C₂²	semidirect
18. C₁₃×C₄⋊Q₈	C₂²	semidirect
19. C₁₃×#127	C₂²	semidirect
20. C₁₃×#124	C₂²	semidirect
21. C₁₃×#123	C₂²	semidirect
22. C₁₃×C₂²⋊Q₈	C₂²	semidirect
23. C₁₃×#121	C₂²	semidirect
24. C₁₃×#121	C₂²	semidirect
25. C₁₃×#121	C₂²	semidirect
26. C₁₃×#121	C₂²	semidirect
27. C₁₃×#121	C₂²	semidirect
28. C₁₃×#121	C₂²	semidirect
29. C₁₃×#120	C₂²	semidirect
30. C₄×D₄×C₁₃	C₂²	semidirect
31. C₄×D₄×C₁₃	C₂²	semidirect
32. C₄×D₄×C₁₃	C₂²	semidirect
33. C₄×D₄×C₁₃	C₂²	semidirect
34. C₂×C₁₃×C₄⋊C₄	C₂²	semidirect
35. C₂×C₁₃×C₄⋊C₄	C₂²	semidirect
36. C₂×C₁₃×C₂²⋊C₄	C₂²	semidirect
37. C₂×C₁₃×C₂²⋊C₄	C₂²	semidirect
38. C₂×C₄²×C₁₃	C₂²	semidirect
47. C₂×D₄×C₁₃	D₄	semidirect
74. #3496	C₁₃	direct (N × H)
80. C₂³×C₁₃	C₂×D₄	semidirect
91. D₄×C₁₃	C₂×D₄	semidirect
99. C₂×C₄×C₁₃	C₂×D₄	semidirect
106. C₂×H(5,𝔽₂)	C₂×C₁₃	semidirect
107. #547	C₂×C₁₃	semidirect
108. #546	C₂×C₁₃	semidirect
109. D₄²	C₂×C₁₃	semidirect
110. #537	C₂×C₁₃	semidirect
111. #534	C₂×C₁₃	semidirect
112. #532	C₂×C₁₃	semidirect
113. C₂×#121	C₂×C₁₃	semidirect
114. C₂×C₄×D₄	C₂×C₁₃	semidirect
132. C₄×C₁₃	H(5,𝔽₂)	semidirect
134. C₂×#41	C₂²×C₁₃	semidirect
135. C₂×#41	C₂²×C₁₃	semidirect
137. C₂²×D₄	C₂²×C₁₃	semidirect
138. C₂²×D₄	C₂²×C₁₃	semidirect
139. C₂²×D₄	C₂²×C₁₃	semidirect
140. C₂³×C₄	C₂²×C₁₃	semidirect
141. C₄⋊Q₈	C₂²×C₁₃	semidirect
142. #127	C₂²×C₁₃	semidirect
143. #124	C₂²×C₁₃	semidirect
144. #123	C₂²×C₁₃	semidirect
145. C₂²⋊Q₈	C₂²×C₁₃	semidirect
146. #121	C₂²×C₁₃	semidirect
147. #121	C₂²×C₁₃	semidirect
148. #121	C₂²×C₁₃	semidirect
149. #121	C₂²×C₁₃	semidirect
150. #121	C₂²×C₁₃	semidirect
151. #121	C₂²×C₁₃	semidirect
152. #120	C₂²×C₁₃	semidirect
153. C₄×D₄	C₂²×C₁₃	semidirect
154. C₄×D₄	C₂²×C₁₃	semidirect
155. C₄×D₄	C₂²×C₁₃	semidirect
156. C₄×D₄	C₂²×C₁₃	semidirect
157. C₂×C₄⋊C₄	C₂²×C₁₃	semidirect
158. C₂×C₄⋊C₄	C₂²×C₁₃	semidirect
159. C₂×C₂²⋊C₄	C₂²×C₁₃	semidirect
160. C₂×C₂²⋊C₄	C₂²×C₁₃	semidirect
161. C₂×C₄²	C₂²×C₁₃	semidirect
176. C₂×D₄	D₄×C₁₃	semidirect
203. C₁₃	#3496	direct (N × H)
209. C₂³	C₂×D₄×C₁₃	semidirect
220. D₄	C₂×D₄×C₁₃	semidirect
228. C₂×C₄	C₂×D₄×C₁₃	semidirect
252. C₄	C₁₃×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×H(5,𝔽₂)	2
C₁₃×#547	2
C₁₃×#546	2
D₄²×C₁₃	2
C₁₃×#537	2
C₁₃×#534	2
C₁₃×#532	2
C₂×C₁₃×#121	2
C₂×C₄×D₄×C₁₃	2
#3496	13

Subgroups — C13×#3496 #13112554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3496 #13112554