Subgroups — C₁₃×#61881 #15748220

Order 3328.

2038 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#550	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#3392	C₂	1	8	yes	maximal, complemented
4	C₁₃×#3362	C₂	1	8	yes	maximal, complemented
5	C₁₃×#3208	C₂	1	6	yes	maximal, complemented
6	C₁₃×#3101	C₂	1	6	yes	maximal, complemented
7	C₁₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×#41	—	2	1	no	—
9	Q₈²×C₁₃	C₂²	1	4	yes	complemented
10	C₁₃×#557	—	2	4	no	—
11	C₁₃×#557	C₂²	1	4	yes	complemented
12	C₁₃×#550	—	2	4	no	—
13	C₁₃×#550	C₂²	1	8	yes	—
14	C₁₃×#544	—	2	3	no	—
15	C₁₃×#533	C₂²	1	6	yes	—
16	C₂×C₁₃×#127	C₂²	1	3	yes	—
17	C₂×C₁₃×#121	—	2	3	no	—
18	C₁₃×#519	—	2	1	no	—
19	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₁₃	C₂²	1	3	yes	—
21	C₁₃×#492	C₂²	1	12	yes	complemented
22	C₁₃×#486	C₂²	1	12	yes	complemented
23	C₁₃×#483	C₂²	1	4	yes	—
24	C₁₃×#476	C₂²	1	12	yes	complemented
25	C₁₃×#461	C₂²	1	12	yes	complemented
26	C₁₃×#460	C₂²	1	24	yes	complemented
27	C₁₃×#450	C₂²	1	12	yes	—
28	C₈×Q₈×C₁₃	C₂²	1	8	yes	complemented
29	C₄×C₁₃×SD₁₆	C₂²	1	24	yes	complemented
30	C₁₃×#431	C₂²	1	6	yes	—
31	C₂×C₁₃×#41	—	2	8	no	—
32	C₂×C₁₃×#41	—	4	8	no	—
33	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×D₄×C₁₃	—	2	3	no	—
35	C₂²×D₄×C₁₃	—	2	3	no	—
36	C₂²×D₄×C₁₃	C₂³	1	3	yes	—
37	C₂³×C₄×C₁₃	—	2	3	no	—
38	C₂×C₁₃×SD₁₆	C₂³	1	24	yes	—
39	C₁₃×C₄⋊Q₈	—	2	12	no	—
40	C₁₃×C₄⋊Q₈	C₂³	1	6	yes	—
41	C₁₃×C₄⋊Q₈	C₂³	1	6	yes	—
42	C₁₃×#127	—	2	6	no	—
43	C₁₃×#127	C₂³	1	12	yes	—
44	C₁₃×#125	—	2	12	no	—
45	C₁₃×#125	—	2	6	no	—
46	C₁₃×C₂²⋊Q₈	—	2	6	no	—
47	C₁₃×C₂²⋊Q₈	C₂³	1	4	yes	—
48	C₁₃×#121	—	2	24	no	—
49	C₄×Q₈×C₁₃	—	2	4	no	—
50	C₄×Q₈×C₁₃	—	2	4	no	—
51	C₄×Q₈×C₁₃	—	2	6	no	—
52	C₄×Q₈×C₁₃	D₄	1	4	yes	—
53	C₄×Q₈×C₁₃	C₂³	1	12	yes	—
54	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
55	C₄×D₄×C₁₃	—	2	6	no	—
56	C₄×D₄×C₁₃	C₂³	1	12	yes	—
57	C₁₃×#117	—	2	6	no	—
58	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
59	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₂²⋊C₄	—	2	3	no	—
61	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
62	C₁₃×#106	C₂³	1	8	yes	—
63	C₁₃×C₄⋊C₈	C₂³	1	12	yes	—
64	C₁₃×#103	C₂³	1	8	yes	—
65	C₁₃×#102	C₂³	1	24	yes	—
66	C₁₃×C₂²⋊C₈	C₂³	1	4	yes	—
67	C₄×C₈×C₁₃	C₂³	1	12	yes	—
68	#61881	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂⁴×C₁₃	—	2	3	no	—
70	C₁₃×#41	—	4	32	no	—
71	C₂×Q₈×C₁₃	—	2	6	no	—
72	C₂×Q₈×C₁₃	SD₁₆	1	8	yes	complemented
73	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
74	C₂×Q₈×C₁₃	C₂⁴	1	4	yes	—
75	C₂×D₄×C₁₃	—	2	3	no	—
76	C₂×D₄×C₁₃	—	2	12	no	—
77	C₂×D₄×C₁₃	—	2	12	no	—
78	C₂×D₄×C₁₃	—	4	6	no	—
79	C₂×D₄×C₁₃	—	4	12	no	—
80	C₂×D₄×C₁₃	—	8	6	no	—
81	C₂×D₄×C₁₃	C₂⁴	1	6	yes	—
82	C₂²×C₄×C₁₃	—	2	1	no	—
83	C₂²×C₄×C₁₃	—	2	12	no	—
84	C₂²×C₄×C₁₃	—	4	12	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
87	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×SD₁₆	—	4	24	no	—
89	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
90	C₁₃×C₄⋊C₄	—	2	1	no	—
91	C₁₃×C₄⋊C₄	—	2	6	no	—
92	C₁₃×C₄⋊C₄	—	2	6	no	—
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	2	6	no	—
95	C₁₃×C₄⋊C₄	—	2	12	no	—
96	C₁₃×C₄⋊C₄	C₂×D₄	1	12	yes	—
97	C₁₃×C₄⋊C₄	C₂⁴	1	4	yes	—
98	C₁₃×C₄⋊C₄	C₂⁴	1	2	yes	—
99	C₁₃×C₂²⋊C₄	—	2	4	no	—
100	C₁₃×C₂²⋊C₄	—	2	12	no	—
101	C₄²×C₁₃	—	2	12	no	—
102	C₄²×C₁₃	C₂×D₄	1	6	yes	—
103	C₄²×C₁₃	C₂⁴	1	6	yes	—
104	#3502	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₂×#550	C₂×C₁₃	1	1	yes	characteristic
106	#3392	C₂×C₁₃	1	8	yes	complemented
107	#3362	C₂×C₁₃	1	8	yes	complemented
108	#3208	C₂×C₁₃	1	6	yes	complemented
109	#3101	C₂×C₁₃	1	6	yes	complemented
110	#2982	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂³×C₁₃	—	2	6	no	—
112	C₂³×C₁₃	—	4	6	no	—
113	C₂³×C₁₃	—	4	6	no	—
114	C₂³×C₁₃	—	4	6	no	—
115	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₁₃	—	2	8	no	—
117	Q₈×C₁₃	—	8	6	no	—
118	Q₈×C₁₃	C₂×SD₁₆	1	16	yes	—
119	D₄×C₁₃	—	2	12	no	—
120	D₄×C₁₃	—	4	6	no	—
121	D₄×C₁₃	—	4	24	no	—
122	D₄×C₁₃	—	8	12	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	4	3	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	24	no	—
130	C₂×C₄×C₁₃	C₂×SD₁₆	1	12	yes	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
133	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
134	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
135	C₈×C₁₃	—	2	8	no	—
136	C₂²×#41	—	2	1	no	—
137	Q₈²	C₂²×C₁₃	1	4	yes	complemented
138	#557	—	2	4	no	—
139	#557	C₂²×C₁₃	1	4	yes	complemented
140	#550	—	2	4	no	—
141	#550	C₂²×C₁₃	1	8	yes	—
142	#544	—	2	3	no	—
143	#533	C₂²×C₁₃	1	6	yes	—
144	C₂×#127	C₂²×C₁₃	1	3	yes	—
145	C₂×#121	—	2	3	no	—
146	#519	—	2	1	no	—
147	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×C₄×D₄	C₂²×C₁₃	1	3	yes	—
149	#492	C₂²×C₁₃	1	12	yes	complemented
150	#486	C₂²×C₁₃	1	12	yes	complemented
151	#483	C₂²×C₁₃	1	4	yes	—
152	#476	C₂²×C₁₃	1	12	yes	complemented
153	#461	C₂²×C₁₃	1	12	yes	complemented
154	#460	C₂²×C₁₃	1	24	yes	complemented
155	#450	C₂²×C₁₃	1	12	yes	—
156	C₈×Q₈	C₂²×C₁₃	1	8	yes	complemented
157	C₄×SD₁₆	C₂²×C₁₃	1	24	yes	complemented
158	#431	C₂²×C₁₃	1	6	yes	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	4	6	no	—
162	C₂²×C₁₃	—	4	6	no	—
163	C₂²×C₁₃	—	4	6	no	—
164	C₂²×C₁₃	—	8	12	no	—
165	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
166	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
167	C₄×C₁₃	—	2	6	no	—
168	C₄×C₁₃	—	2	1	no	—
169	C₄×C₁₃	—	4	4	no	—
170	C₄×C₁₃	—	4	4	no	—
171	C₄×C₁₃	—	4	2	no	—
172	C₄×C₁₃	C₂²×SD₁₆	1	12	yes	—
173	C₄×C₁₃	#585	1	2	yes	—
174	C₂×#41	—	2	8	no	—
175	C₂×#41	—	4	8	no	—
176	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
177	C₂²×D₄	—	2	3	no	—
178	C₂²×D₄	—	2	3	no	—
179	C₂²×D₄	C₂³×C₁₃	1	3	yes	—
180	C₂³×C₄	—	2	3	no	—
181	C₂×SD₁₆	C₂³×C₁₃	1	24	yes	—
182	C₄⋊Q₈	—	2	12	no	—
183	C₄⋊Q₈	C₂³×C₁₃	1	6	yes	—
184	C₄⋊Q₈	C₂³×C₁₃	1	6	yes	—
185	#127	—	2	6	no	—
186	#127	C₂³×C₁₃	1	12	yes	—
187	#125	—	2	12	no	—
188	#125	—	2	6	no	—
189	C₂²⋊Q₈	—	2	6	no	—
190	C₂²⋊Q₈	C₂³×C₁₃	1	4	yes	—
191	#121	—	2	24	no	—
192	C₄×Q₈	—	2	4	no	—
193	C₄×Q₈	—	2	4	no	—
194	C₄×Q₈	—	2	6	no	—
195	C₄×Q₈	D₄×C₁₃	1	4	yes	—
196	C₄×Q₈	C₂³×C₁₃	1	12	yes	—
197	C₄×Q₈	C₂³×C₁₃	1	4	yes	—
198	C₄×D₄	—	2	6	no	—
199	C₄×D₄	C₂³×C₁₃	1	12	yes	—
200	#117	—	2	6	no	—
201	C₂×C₄⋊C₄	D₄×C₁₃	1	3	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	2	3	no	—
204	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
205	#106	C₂³×C₁₃	1	8	yes	—
206	C₄⋊C₈	C₂³×C₁₃	1	12	yes	—
207	#103	C₂³×C₁₃	1	8	yes	—
208	#102	C₂³×C₁₃	1	24	yes	—
209	C₂²⋊C₈	C₂³×C₁₃	1	4	yes	—
210	C₄×C₈	C₂³×C₁₃	1	12	yes	—
211	C₂×C₁₃	—	2	2	no	—
212	C₂×C₁₃	—	8	6	no	—
213	C₂×C₁₃	#3487	1	1	yes	characteristic
214	C₂×C₁₃	C₂³×SD₁₆	1	1	yes	characteristic
215	C₂×C₁₃	#3585	1	1	yes	characteristic
216	C₂⁴	—	2	3	no	—
217	#41	—	4	32	no	—
218	C₂×Q₈	—	2	6	no	—
219	C₂×Q₈	C₁₃×SD₁₆	1	8	yes	complemented
220	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
221	C₂×Q₈	C₂⁴×C₁₃	1	4	yes	—
222	C₂×D₄	—	2	3	no	—
223	C₂×D₄	—	2	12	no	—
224	C₂×D₄	—	2	12	no	—
225	C₂×D₄	—	4	6	no	—
226	C₂×D₄	—	4	12	no	—
227	C₂×D₄	—	8	6	no	—
228	C₂×D₄	C₂⁴×C₁₃	1	6	yes	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	12	no	—
231	C₂²×C₄	—	4	12	no	—
232	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
233	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
234	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
235	SD₁₆	—	4	24	no	—
236	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	6	no	—
239	C₄⋊C₄	—	2	6	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	6	no	—
242	C₄⋊C₄	—	2	12	no	—
243	C₄⋊C₄	C₂×D₄×C₁₃	1	12	yes	—
244	C₄⋊C₄	C₂⁴×C₁₃	1	4	yes	—
245	C₄⋊C₄	C₂⁴×C₁₃	1	2	yes	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₂²⋊C₄	—	2	12	no	—
248	C₄²	—	2	12	no	—
249	C₄²	C₂×D₄×C₁₃	1	6	yes	—
250	C₄²	C₂⁴×C₁₃	1	6	yes	—
251	C₁₃	#61881	1	1	yes	characteristic, direct product factor, Sylow
252	C₂³	—	2	6	no	—
253	C₂³	—	4	6	no	—
254	C₂³	—	4	6	no	—
255	C₂³	—	4	6	no	—
256	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
257	Q₈	—	2	8	no	—
258	Q₈	—	8	6	no	—
259	Q₈	C₂×C₁₃×SD₁₆	1	16	yes	—
260	D₄	—	2	12	no	—
261	D₄	—	4	6	no	—
262	D₄	—	4	24	no	—
263	D₄	—	8	12	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	3	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	24	no	—
271	C₂×C₄	C₂×C₁₃×SD₁₆	1	12	yes	—
272	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
274	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
275	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
276	C₈	—	2	8	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	6	no	—
280	C₂²	—	4	6	no	—
281	C₂²	—	4	6	no	—
282	C₂²	—	8	12	no	—
283	C₂²	C₂²×C₁₃×SD₁₆	1	2	yes	—
284	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
285	C₄	—	2	6	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	4	no	—
290	C₄	C₂²×C₁₃×SD₁₆	1	12	yes	—
291	C₄	C₁₃×#585	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	8	6	no	—
294	C₂	C₁₃×#3487	1	1	yes	characteristic
295	C₂	C₂³×C₁₃×SD₁₆	1	1	yes	characteristic
296	C₂	C₁₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
prime radical	C₂×C₁₃×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3502	C₂	semidirect
3. C₁₃×#3392	C₂	semidirect
4. C₁₃×#3362	C₂	semidirect
5. C₁₃×#3208	C₂	semidirect
6. C₁₃×#3101	C₂	semidirect
7. C₁₃×#2982	C₂	semidirect
9. Q₈²×C₁₃	C₂²	semidirect
11. C₁₃×#557	C₂²	semidirect
21. C₁₃×#492	C₂²	semidirect
22. C₁₃×#486	C₂²	semidirect
24. C₁₃×#476	C₂²	semidirect
25. C₁₃×#461	C₂²	semidirect
26. C₁₃×#460	C₂²	semidirect
28. C₈×Q₈×C₁₃	C₂²	semidirect
29. C₄×C₁₃×SD₁₆	C₂²	semidirect
68. #61881	C₁₃	direct (N × H)
72. C₂×Q₈×C₁₃	SD₁₆	semidirect
104. #3502	C₂×C₁₃	semidirect
106. #3392	C₂×C₁₃	semidirect
107. #3362	C₂×C₁₃	semidirect
108. #3208	C₂×C₁₃	semidirect
109. #3101	C₂×C₁₃	semidirect
110. #2982	C₂×C₁₃	semidirect
137. Q₈²	C₂²×C₁₃	semidirect
139. #557	C₂²×C₁₃	semidirect
149. #492	C₂²×C₁₃	semidirect
150. #486	C₂²×C₁₃	semidirect
152. #476	C₂²×C₁₃	semidirect
153. #461	C₂²×C₁₃	semidirect
154. #460	C₂²×C₁₃	semidirect
156. C₈×Q₈	C₂²×C₁₃	semidirect
157. C₄×SD₁₆	C₂²×C₁₃	semidirect
219. C₂×Q₈	C₁₃×SD₁₆	semidirect
251. C₁₃	#61881	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×#3502	2
C₂×C₁₃×#550	2
C₁₃×#3392	2
C₁₃×#3362	2
C₁₃×#3208	2
C₁₃×#3101	2
C₁₃×#2982	2
#61881	13

Subgroups — C13×#61881 #15748220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#61881 #15748220