Subgroups — C₁₃×#3455 #13112513

Order 1664.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#542	C₂	1	4	yes	maximal, complemented
2	C₁₃×#541	C₂	1	2	yes	maximal, complemented
3	C₁₃×#540	C₂	1	2	yes	maximal, complemented
4	C₁₃×#539	C₂	1	4	yes	maximal, complemented
5	C₁₃×#538	C₂	1	4	yes	maximal, complemented
6	C₁₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#529	C₂	1	8	yes	maximal, complemented
9	C₁₃×#525	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₃×#41	C₂	1	2	yes	maximal, complemented
11	C₂×C₁₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
16	C₁₃×#127	C₂²	1	1	yes	characteristic, complemented
17	C₁₃×#126	C₂²	1	8	yes	—
18	C₁₃×#126	C₂²	1	8	yes	complemented
19	C₁₃×#125	C₂²	1	8	yes	complemented
20	C₁₃×#125	C₂²	1	2	yes	complemented
21	C₁₃×#124	C₂²	1	8	yes	—
22	C₁₃×#124	C₂²	1	2	yes	complemented
23	C₁₃×#123	C₂²	1	4	yes	—
24	C₁₃×#123	C₂²	1	8	yes	—
25	C₁₃×#123	C₂²	1	8	yes	complemented
26	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
27	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
28	C₁₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
29	C₁₃×#121	C₂²	1	4	yes	—
30	C₁₃×#121	C₂²	1	8	yes	—
31	C₁₃×#121	C₂²	1	4	yes	—
32	C₁₃×#120	C₂²	1	4	yes	—
33	C₄×Q₈×C₁₃	C₂²	1	4	yes	complemented
34	C₄×Q₈×C₁₃	C₂²	1	4	yes	complemented
35	C₄×D₄×C₁₃	C₂²	1	8	yes	—
36	C₄×D₄×C₁₃	C₂²	1	8	yes	—
37	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
38	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
39	C₁₃×#117	C₂²	1	4	yes	—
40	C₁₃×#117	C₂²	1	2	yes	—
41	C₁₃×#117	C₂²	1	2	yes	—
42	C₁₃×#117	C₂²	1	2	yes	—
43	C₁₃×#117	C₂²	1	4	yes	complemented
44	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
45	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂²	1	4	yes	complemented
49	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×#41	—	2	8	no	—
51	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
52	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
53	C₂×D₄×C₁₃	C₂³	1	4	yes	—
54	C₂×D₄×C₁₃	C₂³	1	2	yes	—
55	C₂×D₄×C₁₃	C₂³	1	8	yes	—
56	C₂×D₄×C₁₃	C₂³	1	4	yes	—
57	C₂²×C₄×C₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂³	1	8	yes	—
65	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
68	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
69	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
70	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
71	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
72	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
73	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
74	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
75	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
76	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
77	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
78	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
79	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
80	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
81	C₄²×C₁₃	C₂³	1	2	yes	—
82	C₄²×C₁₃	C₂³	1	1	yes	characteristic
83	C₄²×C₁₃	C₂³	1	1	yes	characteristic
84	C₄²×C₁₃	C₂³	1	8	yes	—
85	C₄²×C₁₃	C₂³	1	2	yes	—
86	C₄²×C₁₃	C₂³	1	2	yes	—
87	C₄²×C₁₃	C₂³	1	4	yes	—
88	#3455	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	4	2	no	—
91	C₂³×C₁₃	C₂⁴	1	2	yes	—
92	C₂³×C₁₃	C₂⁴	1	4	yes	—
93	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
94	Q₈×C₁₃	—	2	4	no	—
95	Q₈×C₁₃	—	4	4	no	—
96	D₄×C₁₃	—	2	8	no	—
97	D₄×C₁₃	—	2	4	no	—
98	D₄×C₁₃	—	4	4	no	—
99	D₄×C₁₃	—	4	8	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	8	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	8	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	#41	1	4	yes	—
108	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
109	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
110	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
111	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
112	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
113	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
114	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
116	#542	C₂×C₁₃	1	4	yes	complemented
117	#541	C₂×C₁₃	1	2	yes	complemented
118	#540	C₂×C₁₃	1	2	yes	complemented
119	#539	C₂×C₁₃	1	4	yes	complemented
120	#538	C₂×C₁₃	1	4	yes	complemented
121	#533	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#532	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#529	C₂×C₁₃	1	8	yes	complemented
124	#525	C₂×C₁₃	1	2	yes	complemented
125	C₄×#41	C₂×C₁₃	1	2	yes	complemented
126	C₂×#117	C₂×C₁₃	1	1	yes	characteristic, complemented
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	4	no	—
130	C₂²×C₁₃	—	2	8	no	—
131	C₂²×C₁₃	—	2	4	no	—
132	C₂²×C₁₃	—	4	4	no	—
133	C₂²×C₁₃	C₂×#41	1	2	yes	—
134	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	—	2	4	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	8	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	C₂×#41	1	2	yes	—
142	C₄×C₁₃	C₂×#41	1	2	yes	—
143	C₂×#41	C₂²×C₁₃	1	2	yes	—
144	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
145	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
146	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	complemented
147	#127	C₂²×C₁₃	1	1	yes	characteristic, complemented
148	#126	C₂²×C₁₃	1	8	yes	—
149	#126	C₂²×C₁₃	1	8	yes	complemented
150	#125	C₂²×C₁₃	1	8	yes	complemented
151	#125	C₂²×C₁₃	1	2	yes	complemented
152	#124	C₂²×C₁₃	1	8	yes	—
153	#124	C₂²×C₁₃	1	2	yes	complemented
154	#123	C₂²×C₁₃	1	4	yes	—
155	#123	C₂²×C₁₃	1	8	yes	—
156	#123	C₂²×C₁₃	1	8	yes	complemented
157	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
158	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
159	C₂²⋊Q₈	C₂²×C₁₃	1	8	yes	complemented
160	#121	C₂²×C₁₃	1	4	yes	—
161	#121	C₂²×C₁₃	1	8	yes	—
162	#121	C₂²×C₁₃	1	4	yes	—
163	#120	C₂²×C₁₃	1	4	yes	—
164	C₄×Q₈	C₂²×C₁₃	1	4	yes	complemented
165	C₄×Q₈	C₂²×C₁₃	1	4	yes	complemented
166	C₄×D₄	C₂²×C₁₃	1	8	yes	—
167	C₄×D₄	C₂²×C₁₃	1	8	yes	—
168	C₄×D₄	C₂²×C₁₃	1	4	yes	complemented
169	C₄×D₄	C₂²×C₁₃	1	4	yes	complemented
170	#117	C₂²×C₁₃	1	4	yes	—
171	#117	C₂²×C₁₃	1	2	yes	—
172	#117	C₂²×C₁₃	1	2	yes	—
173	#117	C₂²×C₁₃	1	2	yes	—
174	#117	C₂²×C₁₃	1	4	yes	complemented
175	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
177	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²×C₁₃	1	4	yes	complemented
180	C₂×C₁₃	—	2	2	no	—
181	C₂×C₁₃	—	4	4	no	—
182	C₂×C₁₃	—	4	2	no	—
183	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
184	C₂×C₁₃	#585	1	2	yes	—
185	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
186	#41	—	2	8	no	—
187	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
188	C₂×Q₈	C₂³×C₁₃	1	4	yes	—
189	C₂×D₄	C₂³×C₁₃	1	4	yes	—
190	C₂×D₄	C₂³×C₁₃	1	2	yes	—
191	C₂×D₄	C₂³×C₁₃	1	8	yes	—
192	C₂×D₄	C₂³×C₁₃	1	4	yes	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
195	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
197	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
198	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
199	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
200	C₂²×C₄	C₂³×C₁₃	1	8	yes	—
201	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
203	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
204	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
205	C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
206	C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
207	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
208	C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
209	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
210	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
211	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
212	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
213	C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
214	C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
215	C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
216	C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
217	C₄²	C₂³×C₁₃	1	2	yes	—
218	C₄²	C₂³×C₁₃	1	2	yes	—
219	C₄²	C₂³×C₁₃	1	2	yes	—
220	C₄²	C₂³×C₁₃	1	1	yes	characteristic
221	C₄²	C₂³×C₁₃	1	1	yes	characteristic
222	C₄²	C₂³×C₁₃	1	4	yes	—
223	C₄²	C₂³×C₁₃	1	8	yes	—
224	C₁₃	#3455	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂⁴×C₁₃	1	2	yes	—
228	C₂³	C₂⁴×C₁₃	1	4	yes	—
229	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	4	4	no	—
232	D₄	—	2	8	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	C₁₃×#41	1	4	yes	—
244	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
245	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
246	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
247	C₂×C₄	C₂⁴×C₁₃	1	8	yes	—
248	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
249	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
250	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
251	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	4	4	no	—
258	C₂²	C₂×C₁₃×#41	1	2	yes	—
259	C₂²	C₂⁵×C₁₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	8	no	—
265	C₄	—	2	2	no	—
266	C₄	C₂×C₁₃×#41	1	2	yes	—
267	C₄	C₂×C₁₃×#41	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	4	no	—
271	C₂	C₂²×C₁₃×#41	1	1	yes	characteristic
272	C₂	C₁₃×#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₃
socle	C₂²×C₁₃	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#542	C₂	semidirect
2. C₁₃×#541	C₂	semidirect
3. C₁₃×#540	C₂	semidirect
4. C₁₃×#539	C₂	semidirect
5. C₁₃×#538	C₂	semidirect
6. C₁₃×#533	C₂	semidirect
7. C₁₃×#532	C₂	semidirect
8. C₁₃×#529	C₂	semidirect
9. C₁₃×#525	C₂	semidirect
10. C₄×C₁₃×#41	C₂	semidirect
11. C₂×C₁₃×#117	C₂	semidirect
14. C₁₃×C₄⋊Q₈	C₂²	semidirect
15. C₁₃×C₄⋊Q₈	C₂²	semidirect
16. C₁₃×#127	C₂²	semidirect
18. C₁₃×#126	C₂²	semidirect
19. C₁₃×#125	C₂²	semidirect
20. C₁₃×#125	C₂²	semidirect
22. C₁₃×#124	C₂²	semidirect
25. C₁₃×#123	C₂²	semidirect
28. C₁₃×C₂²⋊Q₈	C₂²	semidirect
33. C₄×Q₈×C₁₃	C₂²	semidirect
34. C₄×Q₈×C₁₃	C₂²	semidirect
37. C₄×D₄×C₁₃	C₂²	semidirect
38. C₄×D₄×C₁₃	C₂²	semidirect
43. C₁₃×#117	C₂²	semidirect
48. C₂×C₄²×C₁₃	C₂²	semidirect
88. #3455	C₁₃	direct (N × H)
116. #542	C₂×C₁₃	semidirect
117. #541	C₂×C₁₃	semidirect
118. #540	C₂×C₁₃	semidirect
119. #539	C₂×C₁₃	semidirect
120. #538	C₂×C₁₃	semidirect
121. #533	C₂×C₁₃	semidirect
122. #532	C₂×C₁₃	semidirect
123. #529	C₂×C₁₃	semidirect
124. #525	C₂×C₁₃	semidirect
125. C₄×#41	C₂×C₁₃	semidirect
126. C₂×#117	C₂×C₁₃	semidirect
145. C₄⋊Q₈	C₂²×C₁₃	semidirect
146. C₄⋊Q₈	C₂²×C₁₃	semidirect
147. #127	C₂²×C₁₃	semidirect
149. #126	C₂²×C₁₃	semidirect
150. #125	C₂²×C₁₃	semidirect
151. #125	C₂²×C₁₃	semidirect
153. #124	C₂²×C₁₃	semidirect
156. #123	C₂²×C₁₃	semidirect
159. C₂²⋊Q₈	C₂²×C₁₃	semidirect
164. C₄×Q₈	C₂²×C₁₃	semidirect
165. C₄×Q₈	C₂²×C₁₃	semidirect
168. C₄×D₄	C₂²×C₁₃	semidirect
169. C₄×D₄	C₂²×C₁₃	semidirect
174. #117	C₂²×C₁₃	semidirect
179. C₂×C₄²	C₂²×C₁₃	semidirect
224. C₁₃	#3455	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×#542	2
C₁₃×#541	2
C₁₃×#540	2
C₁₃×#539	2
C₁₃×#538	2
C₁₃×#533	2
C₁₃×#532	2
C₁₃×#529	2
C₁₃×#525	2
C₄×C₁₃×#41	2
C₂×C₁₃×#117	2
#3455	13

Subgroups — C13×#3455 #13112513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3455 #13112513