Subgroups — C₁₃×#61880 #15748219

Order 3328.

2302 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#3391	C₂	1	8	yes	maximal, complemented
4	C₁₃×#3358	C₂	1	8	yes	maximal, complemented
5	C₁₃×#3205	C₂	1	6	yes	maximal, complemented
6	C₁₃×#3100	C₂	1	6	yes	maximal, complemented
7	C₁₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×2₋¹⁺⁴	—	2	1	no	—
9	C₂²×C₁₃×#41	—	2	1	no	—
10	C₁₃×#550	—	2	4	no	—
11	C₁₃×#550	C₂²	1	8	yes	—
12	C₁₃×#550	C₂²	1	4	yes	complemented
13	D₄×Q₈×C₁₃	C₂²	1	4	yes	complemented
14	C₁₃×#548	—	2	4	no	—
15	C₁₃×#537	—	2	3	no	—
16	C₁₃×#532	C₂²	1	6	yes	—
17	C₂×C₁₃×#127	C₂²	1	3	yes	—
18	C₂×C₁₃×#121	—	2	3	no	—
19	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₁₃	C₂²	1	3	yes	—
21	C₁₃×#493	C₂²	1	12	yes	complemented
22	C₁₃×#486	C₂²	1	12	yes	complemented
23	C₁₃×#480	C₂²	1	4	yes	—
24	C₁₃×#474	C₂²	1	12	yes	complemented
25	C₁₃×#459	C₂²	1	12	yes	complemented
26	C₁₃×#459	C₂²	1	24	yes	complemented
27	C₁₃×#447	C₂²	1	12	yes	—
28	C₈×Q₈×C₁₃	C₂²	1	8	yes	complemented
29	C₄×C₁₃×D₈	C₂²	1	24	yes	complemented
30	C₁₃×#431	C₂²	1	6	yes	—
31	C₁₃×2₋¹⁺⁴	—	4	8	no	—
32	C₂×C₁₃×#41	—	2	4	no	—
33	C₂×C₁₃×#41	—	2	8	no	—
34	C₂×C₁₃×#41	—	2	6	no	—
35	C₂×C₁₃×#41	—	4	8	no	—
36	C₂²×Q₈×C₁₃	—	2	4	no	—
37	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×D₄×C₁₃	—	2	3	no	—
39	C₂²×D₄×C₁₃	—	2	3	no	—
40	C₂²×D₄×C₁₃	C₂³	1	3	yes	—
41	C₂³×C₄×C₁₃	—	2	3	no	—
42	C₂×C₁₃×D₈	C₂³	1	24	yes	—
43	C₁₃×C₄⋊Q₈	C₂³	1	6	yes	—
44	C₁₃×#127	—	2	6	no	—
45	C₁₃×#127	C₂³	1	12	yes	—
46	C₁₃×#127	C₂³	1	6	yes	—
47	C₁₃×#124	—	2	6	no	—
48	C₁₃×C₂²⋊Q₈	—	2	12	no	—
49	C₁₃×#121	—	2	6	no	—
50	C₁₃×#121	—	2	24	no	—
51	C₁₃×#121	—	2	12	no	—
52	C₁₃×#121	C₂³	1	4	yes	—
53	C₄×Q₈×C₁₃	D₄	1	4	yes	—
54	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
55	C₄×D₄×C₁₃	—	2	6	no	—
56	C₄×D₄×C₁₃	—	2	6	no	—
57	C₄×D₄×C₁₃	C₂³	1	12	yes	—
58	C₄×D₄×C₁₃	C₂³	1	12	yes	—
59	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
60	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
61	C₂×C₁₃×C₂²⋊C₄	—	2	3	no	—
62	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
63	C₁₃×#107	C₂³	1	8	yes	—
64	C₁₃×C₄⋊C₈	C₂³	1	12	yes	—
65	C₁₃×#102	C₂³	1	8	yes	—
66	C₁₃×#102	C₂³	1	24	yes	—
67	C₁₃×C₂²⋊C₈	C₂³	1	4	yes	—
68	C₄×C₈×C₁₃	C₂³	1	12	yes	—
69	#61880	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂⁴×C₁₃	—	2	3	no	—
71	C₁₃×#41	—	4	32	no	—
72	C₁₃×#41	—	4	16	no	—
73	C₁₃×#41	—	4	24	no	—
74	C₂×Q₈×C₁₃	—	2	6	no	—
75	C₂×Q₈×C₁₃	—	4	16	no	—
76	C₂×Q₈×C₁₃	D₈	1	8	yes	complemented
77	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
78	C₂×D₄×C₁₃	—	2	4	no	—
79	C₂×D₄×C₁₃	—	2	3	no	—
80	C₂×D₄×C₁₃	—	2	6	no	—
81	C₂×D₄×C₁₃	—	2	12	no	—
82	C₂×D₄×C₁₃	—	2	6	no	—
83	C₂×D₄×C₁₃	—	2	12	no	—
84	C₂×D₄×C₁₃	—	4	6	no	—
85	C₂×D₄×C₁₃	—	4	12	no	—
86	C₂×D₄×C₁₃	—	8	6	no	—
87	C₂×D₄×C₁₃	C₂⁴	1	6	yes	—
88	C₂×D₄×C₁₃	C₂⁴	1	4	yes	—
89	C₂²×C₄×C₁₃	—	2	1	no	—
90	C₂²×C₄×C₁₃	—	2	12	no	—
91	C₂²×C₄×C₁₃	—	2	12	no	—
92	C₂²×C₄×C₁₃	—	4	12	no	—
93	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
95	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₁₃×D₈	—	4	24	no	—
97	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
98	C₁₃×C₄⋊C₄	—	2	1	no	—
99	C₁₃×C₄⋊C₄	—	2	6	no	—
100	C₁₃×C₄⋊C₄	C₂×D₄	1	12	yes	—
101	C₁₃×C₄⋊C₄	C₂⁴	1	4	yes	—
102	C₁₃×C₄⋊C₄	C₂⁴	1	2	yes	—
103	C₁₃×C₂²⋊C₄	—	2	12	no	—
104	C₁₃×C₂²⋊C₄	—	2	4	no	—
105	C₁₃×C₂²⋊C₄	—	2	12	no	—
106	C₄²×C₁₃	C₂×D₄	1	6	yes	—
107	C₄²×C₁₃	C₂⁴	1	6	yes	—
108	#3497	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂×#550	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#3391	C₂×C₁₃	1	8	yes	complemented
111	#3358	C₂×C₁₃	1	8	yes	complemented
112	#3205	C₂×C₁₃	1	6	yes	complemented
113	#3100	C₂×C₁₃	1	6	yes	complemented
114	#2982	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	—	2	4	no	—
116	C₂³×C₁₃	—	2	6	no	—
117	C₂³×C₁₃	—	4	6	no	—
118	C₂³×C₁₃	—	4	6	no	—
119	C₂³×C₁₃	—	4	6	no	—
120	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₁₃	—	4	12	no	—
122	Q₈×C₁₃	C₂×D₈	1	16	yes	—
123	D₄×C₁₃	—	2	8	no	—
124	D₄×C₁₃	—	2	12	no	—
125	D₄×C₁₃	—	4	6	no	—
126	D₄×C₁₃	—	4	8	no	—
127	D₄×C₁₃	—	4	24	no	—
128	D₄×C₁₃	—	4	12	no	—
129	D₄×C₁₃	—	8	6	no	—
130	D₄×C₁₃	—	8	12	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	4	3	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	24	no	—
137	C₂×C₄×C₁₃	—	4	24	no	—
138	C₂×C₄×C₁₃	C₂×D₈	1	12	yes	—
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
141	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
142	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
143	C₈×C₁₃	—	2	8	no	—
144	C₂×2₋¹⁺⁴	—	2	1	no	—
145	C₂²×#41	—	2	1	no	—
146	#550	—	2	4	no	—
147	#550	C₂²×C₁₃	1	8	yes	—
148	#550	C₂²×C₁₃	1	4	yes	complemented
149	D₄×Q₈	C₂²×C₁₃	1	4	yes	complemented
150	#548	—	2	4	no	—
151	#537	—	2	3	no	—
152	#532	C₂²×C₁₃	1	6	yes	—
153	C₂×#127	C₂²×C₁₃	1	3	yes	—
154	C₂×#121	—	2	3	no	—
155	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×C₄×D₄	C₂²×C₁₃	1	3	yes	—
157	#493	C₂²×C₁₃	1	12	yes	complemented
158	#486	C₂²×C₁₃	1	12	yes	complemented
159	#480	C₂²×C₁₃	1	4	yes	—
160	#474	C₂²×C₁₃	1	12	yes	complemented
161	#459	C₂²×C₁₃	1	12	yes	complemented
162	#459	C₂²×C₁₃	1	24	yes	complemented
163	#447	C₂²×C₁₃	1	12	yes	—
164	C₈×Q₈	C₂²×C₁₃	1	8	yes	complemented
165	C₄×D₈	C₂²×C₁₃	1	24	yes	complemented
166	#431	C₂²×C₁₃	1	6	yes	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	4	6	no	—
170	C₂²×C₁₃	—	4	4	no	—
171	C₂²×C₁₃	—	4	6	no	—
172	C₂²×C₁₃	—	4	4	no	—
173	C₂²×C₁₃	—	4	4	no	—
174	C₂²×C₁₃	—	4	6	no	—
175	C₂²×C₁₃	—	8	12	no	—
176	C₂²×C₁₃	C₂²×D₈	1	2	yes	—
177	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
178	C₄×C₁₃	—	2	6	no	—
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	4	2	no	—
181	C₄×C₁₃	—	4	4	no	—
182	C₄×C₁₃	C₂²×D₈	1	12	yes	—
183	C₄×C₁₃	#585	1	2	yes	—
184	2₋¹⁺⁴	—	4	8	no	—
185	C₂×#41	—	2	4	no	—
186	C₂×#41	—	2	8	no	—
187	C₂×#41	—	2	6	no	—
188	C₂×#41	—	4	8	no	—
189	C₂²×Q₈	—	2	4	no	—
190	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
191	C₂²×D₄	—	2	3	no	—
192	C₂²×D₄	—	2	3	no	—
193	C₂²×D₄	C₂³×C₁₃	1	3	yes	—
194	C₂³×C₄	—	2	3	no	—
195	C₂×D₈	C₂³×C₁₃	1	24	yes	—
196	C₄⋊Q₈	C₂³×C₁₃	1	6	yes	—
197	#127	—	2	6	no	—
198	#127	C₂³×C₁₃	1	12	yes	—
199	#127	C₂³×C₁₃	1	6	yes	—
200	#124	—	2	6	no	—
201	C₂²⋊Q₈	—	2	12	no	—
202	#121	—	2	6	no	—
203	#121	—	2	24	no	—
204	#121	—	2	12	no	—
205	#121	C₂³×C₁₃	1	4	yes	—
206	C₄×Q₈	D₄×C₁₃	1	4	yes	—
207	C₄×Q₈	C₂³×C₁₃	1	4	yes	—
208	C₄×D₄	—	2	6	no	—
209	C₄×D₄	—	2	6	no	—
210	C₄×D₄	C₂³×C₁₃	1	12	yes	—
211	C₄×D₄	C₂³×C₁₃	1	12	yes	—
212	C₂×C₄⋊C₄	D₄×C₁₃	1	3	yes	—
213	C₂×C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
214	C₂×C₂²⋊C₄	—	2	3	no	—
215	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
216	#107	C₂³×C₁₃	1	8	yes	—
217	C₄⋊C₈	C₂³×C₁₃	1	12	yes	—
218	#102	C₂³×C₁₃	1	8	yes	—
219	#102	C₂³×C₁₃	1	24	yes	—
220	C₂²⋊C₈	C₂³×C₁₃	1	4	yes	—
221	C₄×C₈	C₂³×C₁₃	1	12	yes	—
222	C₂×C₁₃	—	2	2	no	—
223	C₂×C₁₃	—	8	4	no	—
224	C₂×C₁₃	—	8	6	no	—
225	C₂×C₁₃	#3487	1	1	yes	characteristic
226	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
227	C₂×C₁₃	#3585	1	1	yes	characteristic
228	C₂⁴	—	2	3	no	—
229	#41	—	4	32	no	—
230	#41	—	4	16	no	—
231	#41	—	4	24	no	—
232	C₂×Q₈	—	2	6	no	—
233	C₂×Q₈	—	4	16	no	—
234	C₂×Q₈	C₁₃×D₈	1	8	yes	complemented
235	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	2	3	no	—
238	C₂×D₄	—	2	6	no	—
239	C₂×D₄	—	2	12	no	—
240	C₂×D₄	—	2	6	no	—
241	C₂×D₄	—	2	12	no	—
242	C₂×D₄	—	4	6	no	—
243	C₂×D₄	—	4	12	no	—
244	C₂×D₄	—	8	6	no	—
245	C₂×D₄	C₂⁴×C₁₃	1	6	yes	—
246	C₂×D₄	C₂⁴×C₁₃	1	4	yes	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	12	no	—
249	C₂²×C₄	—	2	12	no	—
250	C₂²×C₄	—	4	12	no	—
251	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
252	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
253	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
254	D₈	—	4	24	no	—
255	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	6	no	—
258	C₄⋊C₄	C₂×D₄×C₁₃	1	12	yes	—
259	C₄⋊C₄	C₂⁴×C₁₃	1	4	yes	—
260	C₄⋊C₄	C₂⁴×C₁₃	1	2	yes	—
261	C₂²⋊C₄	—	2	12	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₂²⋊C₄	—	2	12	no	—
264	C₄²	C₂×D₄×C₁₃	1	6	yes	—
265	C₄²	C₂⁴×C₁₃	1	6	yes	—
266	C₁₃	#61880	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³	—	2	4	no	—
268	C₂³	—	2	6	no	—
269	C₂³	—	4	6	no	—
270	C₂³	—	4	6	no	—
271	C₂³	—	4	6	no	—
272	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
273	Q₈	—	4	12	no	—
274	Q₈	C₂×C₁₃×D₈	1	16	yes	—
275	D₄	—	2	8	no	—
276	D₄	—	2	12	no	—
277	D₄	—	4	6	no	—
278	D₄	—	4	8	no	—
279	D₄	—	4	24	no	—
280	D₄	—	4	12	no	—
281	D₄	—	8	12	no	—
282	D₄	—	8	6	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	3	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	24	no	—
289	C₂×C₄	—	4	24	no	—
290	C₂×C₄	C₂×C₁₃×D₈	1	12	yes	—
291	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
293	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
294	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
295	C₈	—	2	8	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	6	no	—
300	C₂²	—	4	6	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	6	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	8	12	no	—
305	C₂²	C₂²×C₁₃×D₈	1	2	yes	—
306	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
307	C₄	—	2	6	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	4	no	—
311	C₄	C₂²×C₁₃×D₈	1	12	yes	—
312	C₄	C₁₃×#585	1	2	yes	—
313	C₂	—	2	2	no	—
314	C₂	—	8	6	no	—
315	C₂	—	8	4	no	—
316	C₂	C₁₃×#3487	1	1	yes	characteristic
317	C₂	C₂³×C₁₃×D₈	1	1	yes	characteristic
318	C₂	C₁₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
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×#3497	C₂	semidirect
2. C₂×C₁₃×#550	C₂	semidirect
3. C₁₃×#3391	C₂	semidirect
4. C₁₃×#3358	C₂	semidirect
5. C₁₃×#3205	C₂	semidirect
6. C₁₃×#3100	C₂	semidirect
7. C₁₃×#2982	C₂	semidirect
12. C₁₃×#550	C₂²	semidirect
13. D₄×Q₈×C₁₃	C₂²	semidirect
21. C₁₃×#493	C₂²	semidirect
22. C₁₃×#486	C₂²	semidirect
24. C₁₃×#474	C₂²	semidirect
25. C₁₃×#459	C₂²	semidirect
26. C₁₃×#459	C₂²	semidirect
28. C₈×Q₈×C₁₃	C₂²	semidirect
29. C₄×C₁₃×D₈	C₂²	semidirect
69. #61880	C₁₃	direct (N × H)
76. C₂×Q₈×C₁₃	D₈	semidirect
108. #3497	C₂×C₁₃	semidirect
109. C₂×#550	C₂×C₁₃	semidirect
110. #3391	C₂×C₁₃	semidirect
111. #3358	C₂×C₁₃	semidirect
112. #3205	C₂×C₁₃	semidirect
113. #3100	C₂×C₁₃	semidirect
114. #2982	C₂×C₁₃	semidirect
148. #550	C₂²×C₁₃	semidirect
149. D₄×Q₈	C₂²×C₁₃	semidirect
157. #493	C₂²×C₁₃	semidirect
158. #486	C₂²×C₁₃	semidirect
160. #474	C₂²×C₁₃	semidirect
161. #459	C₂²×C₁₃	semidirect
162. #459	C₂²×C₁₃	semidirect
164. C₈×Q₈	C₂²×C₁₃	semidirect
165. C₄×D₈	C₂²×C₁₃	semidirect
234. C₂×Q₈	C₁₃×D₈	semidirect
266. C₁₃	#61880	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×#3497	2
C₂×C₁₃×#550	2
C₁₃×#3391	2
C₁₃×#3358	2
C₁₃×#3205	2
C₁₃×#3100	2
C₁₃×#2982	2
#61880	13

Subgroups — C13×#61880 #15748219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#61880 #15748219