Subgroups — C₁₃×#3294 #13112352

Order 1664.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#480	C₂	1	2	yes	maximal, complemented
5	C₁₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#466	C₂	1	2	yes	maximal, complemented
7	C₁₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#447	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₃×#102	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#41	—	2	1	no	—
13	C₂²×D₄×C₁₃	—	2	1	no	—
14	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
15	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×D₈	C₂²	1	2	yes	complemented
17	C₂²×C₈×C₁₃	C₂²	1	2	yes	complemented
18	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₁₃×#127	C₂²	1	1	yes	characteristic
20	C₁₃×#123	—	2	2	no	—
21	C₁₃×C₂²⋊Q₈	—	2	1	no	—
22	C₁₃×#121	—	2	1	no	—
23	C₁₃×#121	C₂²	1	2	yes	—
24	C₁₃×#121	C₂²	1	2	yes	complemented
25	C₁₃×#120	—	2	2	no	—
26	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	complemented
30	C₁₃×#107	C₂²	1	1	yes	characteristic, complemented
31	C₁₃×#107	C₂²	1	2	yes	complemented
32	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₁₃×#102	C₂²	1	1	yes	characteristic
34	C₁₃×#102	C₂²	1	2	yes	—
35	C₁₃×#102	C₂²	1	2	yes	complemented
36	C₁₃×#102	C₂²	1	2	yes	complemented
37	C₁₃×#102	C₂²	1	1	yes	characteristic, complemented
38	C₁₃×#102	C₂²	1	2	yes	complemented
39	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
41	C₂⁴×C₁₃	—	2	2	no	—
42	C₁₃×#41	—	4	4	no	—
43	C₂×Q₈×C₁₃	—	2	1	no	—
44	C₂×D₄×C₁₃	—	2	4	no	—
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	—	2	2	no	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	—	2	1	no	—
49	C₂×D₄×C₁₃	—	2	1	no	—
50	C₂×D₄×C₁₃	—	4	2	no	—
51	C₂×D₄×C₁₃	—	4	4	no	—
52	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
53	C₂×D₄×C₁₃	C₂³	1	2	yes	—
54	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
60	C₁₃×D₈	—	2	2	no	—
61	C₁₃×D₈	—	4	2	no	—
62	C₂×C₈×C₁₃	—	4	2	no	—
63	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂³	1	2	yes	—
66	C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₁₃×C₄⋊C₄	—	2	1	no	—
68	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
71	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	—	2	1	no	—
74	C₁₃×C₂²⋊C₄	—	2	2	no	—
75	C₁₃×C₂²⋊C₄	—	2	2	no	—
76	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
77	C₄²×C₁₃	C₂³	1	1	yes	characteristic
78	#3294	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	—	4	4	no	—
83	C₂³×C₁₃	—	4	4	no	—
84	C₂³×C₁₃	—	8	2	no	—
85	C₂³×C₁₃	C₂×D₄	1	2	yes	—
86	Q₈×C₁₃	—	4	2	no	—
87	D₄×C₁₃	—	2	1	no	—
88	D₄×C₁₃	—	2	2	no	—
89	D₄×C₁₃	—	2	1	no	—
90	D₄×C₁₃	—	2	1	no	—
91	D₄×C₁₃	—	2	2	no	—
92	D₄×C₁₃	—	4	4	no	—
93	D₄×C₁₃	—	4	4	no	—
94	D₄×C₁₃	—	4	4	no	—
95	D₄×C₁₃	—	4	2	no	—
96	D₄×C₁₃	—	8	1	no	—
97	D₄×C₁₃	D₈	1	4	yes	complemented
98	D₄×C₁₃	#41	1	2	yes	complemented
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
111	C₈×C₁₃	—	2	1	no	—
112	C₈×C₁₃	—	2	1	no	—
113	C₈×C₁₃	—	2	2	no	—
114	#547	C₂×C₁₃	1	1	yes	characteristic, complemented
115	D₄²	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#486	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#480	C₂×C₁₃	1	2	yes	complemented
118	#474	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#466	C₂×C₁₃	1	2	yes	complemented
120	#459	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#447	C₂×C₁₃	1	2	yes	complemented
122	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₈×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂×#102	C₂×C₁₃	1	2	yes	complemented
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	4	1	no	—
129	C₂²×C₁₃	—	4	1	no	—
130	C₂²×C₁₃	—	4	2	no	—
131	C₂²×C₁₃	—	4	1	no	—
132	C₂²×C₁₃	—	4	1	no	—
133	C₂²×C₁₃	—	4	2	no	—
134	C₂²×C₁₃	—	4	2	no	—
135	C₂²×C₁₃	—	4	1	no	—
136	C₂²×C₁₃	—	8	8	no	—
137	C₂²×C₁₃	C₂×D₈	1	4	yes	—
138	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	—	4	1	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	—	4	2	no	—
144	C₄×C₁₃	C₂×D₈	1	2	yes	—
145	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
146	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×#41	—	2	1	no	—
148	C₂²×D₄	—	2	1	no	—
149	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
150	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×D₈	C₂²×C₁₃	1	2	yes	complemented
152	C₂²×C₈	C₂²×C₁₃	1	2	yes	complemented
153	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
154	#127	C₂²×C₁₃	1	1	yes	characteristic
155	#123	—	2	2	no	—
156	C₂²⋊Q₈	—	2	1	no	—
157	#121	—	2	1	no	—
158	#121	C₂²×C₁₃	1	2	yes	—
159	#121	C₂²×C₁₃	1	2	yes	complemented
160	#120	—	2	2	no	—
161	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
162	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
163	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic, complemented
164	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	complemented
165	#107	C₂²×C₁₃	1	1	yes	characteristic, complemented
166	#107	C₂²×C₁₃	1	2	yes	complemented
167	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
168	#102	C₂²×C₁₃	1	1	yes	characteristic
169	#102	C₂²×C₁₃	1	2	yes	—
170	#102	C₂²×C₁₃	1	2	yes	complemented
171	#102	C₂²×C₁₃	1	2	yes	complemented
172	#102	C₂²×C₁₃	1	1	yes	characteristic, complemented
173	#102	C₂²×C₁₃	1	2	yes	complemented
174	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	complemented
175	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
176	C₂×C₁₃	—	2	4	no	—
177	C₂×C₁₃	—	4	2	no	—
178	C₂×C₁₃	—	8	1	no	—
179	C₂×C₁₃	—	8	2	no	—
180	C₂×C₁₃	#546	1	1	yes	characteristic
181	C₂×C₁₃	C₂²×D₈	1	1	yes	characteristic
182	C₂×C₁₃	#577	1	1	yes	characteristic
183	C₂⁴	—	2	2	no	—
184	#41	—	4	4	no	—
185	C₂×Q₈	—	2	1	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	4	no	—
194	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
195	C₂×D₄	C₂³×C₁₃	1	2	yes	—
196	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
202	D₈	—	2	2	no	—
203	D₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
207	C₂×C₈	C₂³×C₁₃	1	2	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
211	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
212	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
213	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
219	C₄²	C₂³×C₁₃	1	1	yes	characteristic
220	C₁₃	#3294	1	1	yes	characteristic, direct product factor, Sylow
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	—	8	2	no	—
227	C₂³	C₂×D₄×C₁₃	1	2	yes	—
228	Q₈	—	4	2	no	—
229	D₄	—	2	1	no	—
230	D₄	—	2	2	no	—
231	D₄	—	2	1	no	—
232	D₄	—	2	1	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	2	no	—
238	D₄	—	8	1	no	—
239	D₄	C₁₃×D₈	1	4	yes	complemented
240	D₄	C₁₃×#41	1	2	yes	complemented
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
251	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
252	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	8	no	—
268	C₂²	C₂×C₁₃×D₈	1	4	yes	—
269	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	2	no	—
275	C₄	C₂×C₁₃×D₈	1	2	yes	—
276	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
277	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
278	C₂	—	2	4	no	—
279	C₂	—	4	2	no	—
280	C₂	—	8	2	no	—
281	C₂	—	8	1	no	—
282	C₂	C₁₃×#546	1	1	yes	characteristic
283	C₂	C₂²×C₁₃×D₈	1	1	yes	characteristic
284	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#547	C₂	semidirect
2. D₄²×C₁₃	C₂	semidirect
3. C₁₃×#486	C₂	semidirect
4. C₁₃×#480	C₂	semidirect
5. C₁₃×#474	C₂	semidirect
6. C₁₃×#466	C₂	semidirect
7. C₁₃×#459	C₂	semidirect
8. C₁₃×#447	C₂	semidirect
9. C₄×C₁₃×D₈	C₂	semidirect
10. C₈×D₄×C₁₃	C₂	semidirect
11. C₂×C₁₃×#102	C₂	semidirect
16. C₂×C₁₃×D₈	C₂²	semidirect
17. C₂²×C₈×C₁₃	C₂²	semidirect
18. C₁₃×C₄⋊Q₈	C₂²	semidirect
24. C₁₃×#121	C₂²	semidirect
28. C₄×D₄×C₁₃	C₂²	semidirect
29. C₂×C₁₃×C₄⋊C₄	C₂²	semidirect
30. C₁₃×#107	C₂²	semidirect
31. C₁₃×#107	C₂²	semidirect
32. C₁₃×C₄⋊C₈	C₂²	semidirect
35. C₁₃×#102	C₂²	semidirect
36. C₁₃×#102	C₂²	semidirect
37. C₁₃×#102	C₂²	semidirect
38. C₁₃×#102	C₂²	semidirect
39. C₁₃×C₂²⋊C₈	C₂²	semidirect
40. C₄×C₈×C₁₃	C₂²	semidirect
78. #3294	C₁₃	direct (N × H)
97. D₄×C₁₃	D₈	semidirect
98. D₄×C₁₃	#41	semidirect
114. #547	C₂×C₁₃	semidirect
115. D₄²	C₂×C₁₃	semidirect
116. #486	C₂×C₁₃	semidirect
117. #480	C₂×C₁₃	semidirect
118. #474	C₂×C₁₃	semidirect
119. #466	C₂×C₁₃	semidirect
120. #459	C₂×C₁₃	semidirect
121. #447	C₂×C₁₃	semidirect
122. C₄×D₈	C₂×C₁₃	semidirect
123. C₈×D₄	C₂×C₁₃	semidirect
124. C₂×#102	C₂×C₁₃	semidirect
151. C₂×D₈	C₂²×C₁₃	semidirect
152. C₂²×C₈	C₂²×C₁₃	semidirect
153. C₄⋊Q₈	C₂²×C₁₃	semidirect
159. #121	C₂²×C₁₃	semidirect
163. C₄×D₄	C₂²×C₁₃	semidirect
164. C₂×C₄⋊C₄	C₂²×C₁₃	semidirect
165. #107	C₂²×C₁₃	semidirect
166. #107	C₂²×C₁₃	semidirect
167. C₄⋊C₈	C₂²×C₁₃	semidirect
170. #102	C₂²×C₁₃	semidirect
171. #102	C₂²×C₁₃	semidirect
172. #102	C₂²×C₁₃	semidirect
173. #102	C₂²×C₁₃	semidirect
174. C₂²⋊C₈	C₂²×C₁₃	semidirect
175. C₄×C₈	C₂²×C₁₃	semidirect
220. C₁₃	#3294	direct (N × H)
239. D₄	C₁₃×D₈	semidirect
240. D₄	C₁₃×#41	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#547	2
D₄²×C₁₃	2
C₁₃×#486	2
C₁₃×#480	2
C₁₃×#474	2
C₁₃×#466	2
C₁₃×#459	2
C₁₃×#447	2
C₄×C₁₃×D₈	2
C₈×D₄×C₁₃	2
C₂×C₁₃×#102	2
#3294	13

Subgroups — C13×#3294 #13112352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3294 #13112352