Subgroups — C₁₃×#2000 #13111058

Order 1664.

444 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#121	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₁₃	—	2	1	no	—
10	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×SD₁₆	—	2	2	no	—
13	C₂×C₁₃×SD₁₆	—	2	2	no	—
14	C₂×C₁₃×SD₁₆	—	2	2	no	—
15	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×C₂²⋊Q₈	—	2	2	no	—
18	C₁₃×C₂²⋊Q₈	—	2	2	no	—
19	C₁₃×#121	—	2	2	no	—
20	C₁₃×#121	—	2	2	no	—
21	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
25	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
26	C₁₃×#103	—	2	2	no	—
27	C₁₃×#102	—	2	2	no	—
28	C₁₃×C₂²⋊C₈	—	2	2	no	—
29	C₂⁴×C₁₃	—	2	1	no	—
30	C₂⁴×C₁₃	D₄	1	1	yes	characteristic
31	C₂×Q₈×C₁₃	—	2	1	no	—
32	C₂×Q₈×C₁₃	—	2	1	no	—
33	C₂×Q₈×C₁₃	—	2	1	no	—
34	C₂×Q₈×C₁₃	—	2	1	no	—
35	C₂×Q₈×C₁₃	—	2	1	no	—
36	C₂×Q₈×C₁₃	D₄	1	2	yes	complemented
37	C₂×D₄×C₁₃	—	2	4	no	—
38	C₂×D₄×C₁₃	—	2	1	no	—
39	C₂×D₄×C₁₃	—	2	1	no	—
40	C₂×D₄×C₁₃	—	2	1	no	—
41	C₂×D₄×C₁₃	—	2	1	no	—
42	C₂×D₄×C₁₃	—	2	1	no	—
43	C₂×D₄×C₁₃	—	4	2	no	—
44	C₂×D₄×C₁₃	—	4	2	no	—
45	C₂×D₄×C₁₃	D₄	1	2	yes	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₁₃×SD₁₆	—	2	8	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	D₄	1	2	yes	complemented
64	C₁₃×C₄⋊C₄	—	2	4	no	—
65	C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×C₄⋊C₄	—	2	1	no	—
67	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
68	C₁₃×C₄⋊C₄	D₄	1	2	yes	complemented
69	C₁₃×C₂²⋊C₄	—	2	4	no	—
70	C₁₃×C₂²⋊C₄	—	2	4	no	—
71	#2000	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₁₃	—	2	1	no	—
73	C₂³×C₁₃	—	2	1	no	—
74	C₂³×C₁₃	—	2	1	no	—
75	C₂³×C₁₃	—	2	1	no	—
76	C₂³×C₁₃	—	2	1	no	—
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	—	2	2	no	—
79	C₂³×C₁₃	—	4	1	no	—
80	C₂³×C₁₃	—	4	1	no	—
81	C₂³×C₁₃	—	4	1	no	—
82	C₂³×C₁₃	—	4	1	no	—
83	C₂³×C₁₃	—	4	1	no	—
84	C₂³×C₁₃	—	4	1	no	—
85	C₂³×C₁₃	SD₁₆	1	2	yes	complemented
86	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₁₃	—	2	2	no	—
88	Q₈×C₁₃	—	2	2	no	—
89	Q₈×C₁₃	—	4	1	no	—
90	Q₈×C₁₃	—	4	1	no	—
91	D₄×C₁₃	—	2	2	no	—
92	D₄×C₁₃	—	2	2	no	—
93	D₄×C₁₃	—	4	8	no	—
94	D₄×C₁₃	—	4	1	no	—
95	D₄×C₁₃	—	4	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	4	1	no	—
103	C₂×C₄×C₁₃	—	4	1	no	—
104	C₂×C₄×C₁₃	—	4	1	no	—
105	C₂×C₄×C₁₃	—	4	1	no	—
106	C₂×C₄×C₁₃	—	4	1	no	—
107	C₂×C₄×C₁₃	—	4	1	no	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	—	4	1	no	—
120	C₂²×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂×C₂²⋊Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂×#121	C₂×C₁₃	1	1	yes	characteristic
123	C₂×#103	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂×#102	C₂×C₁₃	1	1	yes	characteristic, complemented
125	C₂×C₂²⋊C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#340	C₂×C₁₃	1	1	yes	characteristic, complemented
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	4	1	no	—
131	C₂²×C₁₃	—	4	1	no	—
132	C₂²×C₁₃	—	4	1	no	—
133	C₂²×C₁₃	—	4	1	no	—
134	C₂²×C₁₃	—	4	2	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	—	4	2	no	—
137	C₂²×C₁₃	—	8	1	no	—
138	C₂²×C₁₃	—	8	1	no	—
139	C₂²×C₁₃	—	8	1	no	—
140	C₂²×C₁₃	—	8	1	no	—
141	C₂²×C₁₃	#120	1	1	yes	characteristic
142	C₂²×C₁₃	#120	1	1	yes	characteristic
143	C₂²×C₁₃	#121	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
145	C₂²×C₁₃	#135	1	1	yes	characteristic
146	C₂²×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂²×C₁₃	#137	1	1	yes	characteristic
148	C₄×C₁₃	—	4	1	no	—
149	C₄×C₁₃	—	4	2	no	—
150	C₄×C₁₃	—	4	2	no	—
151	C₄×C₁₃	—	4	2	no	—
152	C₄×C₁₃	#120	1	1	yes	characteristic
153	C₄×C₁₃	#121	1	1	yes	characteristic
154	C₄×C₁₃	#121	1	1	yes	characteristic
155	C₄×C₁₃	#127	1	1	yes	characteristic
156	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
157	C₂²×D₄	—	2	1	no	—
158	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
159	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
165	C₂²⋊Q₈	—	2	2	no	—
166	C₂²⋊Q₈	—	2	2	no	—
167	#121	—	2	2	no	—
168	#121	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	#103	—	2	2	no	—
175	#102	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂×C₁₃	—	4	2	no	—
178	C₂×C₁₃	—	8	2	no	—
179	C₂×C₁₃	#392	1	1	yes	characteristic
180	C₂×C₁₃	#448	1	1	yes	characteristic
181	C₂×C₁₃	#449	1	1	yes	characteristic
182	C₂×C₁₃	#450	1	1	yes	characteristic
183	C₂×C₁₃	#452	1	1	yes	characteristic
184	C₂×C₁₃	#465	1	1	yes	characteristic
185	C₂×C₁₃	#468	1	1	yes	characteristic
186	C₂⁴	—	2	1	no	—
187	C₂⁴	D₄×C₁₃	1	1	yes	characteristic
188	C₂×Q₈	—	2	1	no	—
189	C₂×Q₈	—	2	1	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	D₄×C₁₃	1	2	yes	complemented
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	D₄×C₁₃	1	2	yes	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
214	SD₁₆	—	2	8	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	D₄×C₁₃	1	2	yes	complemented
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
225	C₄⋊C₄	D₄×C₁₃	1	2	yes	complemented
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₁₃	#2000	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	—	4	1	no	—
239	C₂³	—	4	1	no	—
240	C₂³	—	4	1	no	—
241	C₂³	—	4	1	no	—
242	C₂³	C₁₃×SD₁₆	1	2	yes	complemented
243	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	2	2	no	—
246	Q₈	—	4	1	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	2	2	no	—
249	D₄	—	2	2	no	—
250	D₄	—	4	8	no	—
251	D₄	—	4	1	no	—
252	D₄	—	4	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
274	C₂×C₄	C₁₃×#41	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	4	1	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	8	1	no	—
288	C₂²	—	8	1	no	—
289	C₂²	—	8	1	no	—
290	C₂²	—	8	1	no	—
291	C₂²	C₁₃×#120	1	1	yes	characteristic
292	C₂²	C₁₃×#120	1	1	yes	characteristic
293	C₂²	C₁₃×#121	1	1	yes	characteristic
294	C₂²	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
295	C₂²	C₁₃×#135	1	1	yes	characteristic
296	C₂²	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂²	C₁₃×#137	1	1	yes	characteristic
298	C₄	—	4	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	C₁₃×#120	1	1	yes	characteristic
303	C₄	C₁₃×#121	1	1	yes	characteristic
304	C₄	C₁₃×#121	1	1	yes	characteristic
305	C₄	C₁₃×#127	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	—	8	2	no	—
308	C₂	C₁₃×#392	1	1	yes	characteristic
309	C₂	C₁₃×#448	1	1	yes	characteristic
310	C₂	C₁₃×#449	1	1	yes	characteristic
311	C₂	C₁₃×#450	1	1	yes	characteristic
312	C₂	C₁₃×#452	1	1	yes	characteristic
313	C₂	C₁₃×#465	1	1	yes	characteristic
314	C₂	C₁₃×#468	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₃
prime radical	C₂×C₁₃×#121	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×SD₁₆	C₂	semidirect
2. C₂×C₁₃×C₂²⋊Q₈	C₂	semidirect
4. C₂×C₁₃×#103	C₂	semidirect
5. C₂×C₁₃×#102	C₂	semidirect
6. C₂×C₁₃×C₂²⋊C₈	C₂	semidirect
7. C₁₃×#340	C₂	semidirect
36. C₂×Q₈×C₁₃	D₄	semidirect
63. C₂×C₈×C₁₃	D₄	semidirect
68. C₁₃×C₄⋊C₄	D₄	semidirect
71. #2000	C₁₃	direct (N × H)
85. C₂³×C₁₃	SD₁₆	semidirect
120. C₂²×SD₁₆	C₂×C₁₃	semidirect
121. C₂×C₂²⋊Q₈	C₂×C₁₃	semidirect
123. C₂×#103	C₂×C₁₃	semidirect
124. C₂×#102	C₂×C₁₃	semidirect
125. C₂×C₂²⋊C₈	C₂×C₁₃	semidirect
126. #340	C₂×C₁₃	semidirect
193. C₂×Q₈	D₄×C₁₃	semidirect
220. C₂×C₈	D₄×C₁₃	semidirect
225. C₄⋊C₄	D₄×C₁₃	semidirect
228. C₁₃	#2000	direct (N × H)
242. C₂³	C₁₃×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×SD₁₆	2
C₂×C₁₃×C₂²⋊Q₈	2
C₂×C₁₃×#121	2
C₂×C₁₃×#103	2
C₂×C₁₃×#102	2
C₂×C₁₃×C₂²⋊C₈	2
C₁₃×#340	2
#2000	13

Subgroups — C13×#2000 #13111058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2000 #13111058