Subgroups — C₂×C₁₃×#1668 #15705451

Order 3328.

870 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#460	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1668	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#331	C₂	1	2	yes	maximal, complemented
7	C₂²×C₁₃×SD₁₆	—	2	2	no	—
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂²×C₁₃×D₈	—	2	1	no	—
10	C₂×C₁₃×#127	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#493	—	2	2	no	—
13	C₁₃×#493	C₂²	1	4	yes	—
14	C₁₃×#460	—	2	4	no	—
15	C₁₃×#460	C₂²	1	8	yes	complemented
16	C₈×Q₈×C₁₃	—	2	2	no	—
17	C₈×Q₈×C₁₃	C₂²	1	4	yes	complemented
18	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
19	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#102	—	2	2	no	—
22	C₂×C₄×C₈×C₁₃	—	2	1	no	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#335	C₂²	1	4	yes	complemented
25	C₁₃×#331	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂²×D₄×C₁₃	—	2	2	no	—
29	C₂²×D₄×C₁₃	—	2	2	no	—
30	C₂×C₁₃×SD₁₆	—	2	8	no	—
31	C₂×C₁₃×SD₁₆	—	4	8	no	—
32	C₂×C₁₃×D₈	—	2	4	no	—
33	C₂×C₁₃×D₈	—	2	4	no	—
34	C₂×C₁₃×D₈	—	4	4	no	—
35	C₂×C₁₃×D₈	—	4	4	no	—
36	C₂²×C₈×C₁₃	—	2	1	no	—
37	C₂²×C₈×C₁₃	—	2	2	no	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₁₃×#127	—	2	2	no	—
41	C₁₃×#127	—	2	2	no	—
42	C₁₃×#127	—	2	2	no	—
43	C₁₃×#127	C₂³	1	4	yes	—
44	C₄×Q₈×C₁₃	—	2	1	no	—
45	C₄×Q₈×C₁₃	—	2	1	no	—
46	C₄×Q₈×C₁₃	D₄	1	2	yes	—
47	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
49	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	—	2	1	no	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	—	2	4	no	—
53	C₁₃×C₄⋊C₈	—	2	1	no	—
54	C₁₃×C₄⋊C₈	—	2	2	no	—
55	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
56	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
57	C₁₃×#102	—	2	8	no	—
58	C₄×C₈×C₁₃	—	2	4	no	—
59	C₄×C₈×C₁₃	D₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂³	1	2	yes	—
61	C₂×#1668	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	—	4	2	no	—
63	C₂×Q₈×C₁₃	—	2	1	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	D₈	1	4	yes	complemented
66	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×D₄×C₁₃	—	2	4	no	—
68	C₂×D₄×C₁₃	—	2	4	no	—
69	C₂×D₄×C₁₃	—	2	4	no	—
70	C₂×D₄×C₁₃	—	4	2	no	—
71	C₂×D₄×C₁₃	—	4	2	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂×D₄×C₁₃	—	4	4	no	—
74	C₂×D₄×C₁₃	—	4	4	no	—
75	C₂×D₄×C₁₃	—	4	4	no	—
76	C₂×D₄×C₁₃	—	4	4	no	—
77	C₂×D₄×C₁₃	—	8	4	no	—
78	C₂²×C₄×C₁₃	—	2	1	no	—
79	C₂²×C₄×C₁₃	—	2	1	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
83	C₁₃×SD₁₆	—	4	16	no	—
84	C₁₃×D₈	—	4	8	no	—
85	C₁₃×D₈	—	4	8	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	—	4	4	no	—
90	C₂×C₈×C₁₃	—	4	2	no	—
91	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
92	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	#41	1	1	yes	characteristic
103	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₂×#493	C₂×C₁₃	1	1	yes	characteristic
105	C₂×#460	C₂×C₁₃	1	2	yes	complemented
106	C₂×C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#1668	C₂×C₁₃	1	8	yes	direct product factor
108	C₂×#335	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂×#331	C₂×C₁₃	1	2	yes	complemented
110	C₂³×C₁₃	—	4	4	no	—
111	C₂³×C₁₃	—	8	4	no	—
112	C₂³×C₁₃	—	8	4	no	—
113	C₂³×C₁₃	—	8	4	no	—
114	C₂³×C₁₃	#121	1	1	yes	characteristic
115	Q₈×C₁₃	—	2	2	no	—
116	Q₈×C₁₃	—	2	4	no	—
117	Q₈×C₁₃	C₂×D₈	1	4	yes	complemented
118	D₄×C₁₃	—	4	4	no	—
119	D₄×C₁₃	—	4	4	no	—
120	D₄×C₁₃	—	4	4	no	—
121	D₄×C₁₃	—	4	4	no	—
122	D₄×C₁₃	—	8	4	no	—
123	D₄×C₁₃	—	8	4	no	—
124	D₄×C₁₃	—	8	4	no	—
125	D₄×C₁₃	—	8	4	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₁₃	#135	1	2	yes	—
137	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	—	4	4	no	—
143	C₈×C₁₃	—	4	2	no	—
144	C₈×C₁₃	C₂×SD₁₆	1	4	yes	complemented
145	C₂²×SD₁₆	—	2	2	no	—
146	C₂²×D₈	—	2	1	no	—
147	C₂²×D₈	—	2	1	no	—
148	C₂×#127	C₂²×C₁₃	1	2	yes	—
149	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
150	#493	—	2	2	no	—
151	#493	C₂²×C₁₃	1	4	yes	—
152	#460	—	2	4	no	—
153	#460	C₂²×C₁₃	1	8	yes	complemented
154	C₈×Q₈	—	2	2	no	—
155	C₈×Q₈	C₂²×C₁₃	1	4	yes	complemented
156	C₂×C₄⋊C₈	—	2	1	no	—
157	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
159	C₂×#102	—	2	2	no	—
160	C₂×C₄×C₈	—	2	1	no	—
161	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
162	#335	C₂²×C₁₃	1	4	yes	complemented
163	#331	C₂²×C₁₃	1	8	yes	complemented
164	C₂²×C₁₃	—	8	4	no	—
165	C₂²×C₁₃	—	8	4	no	—
166	C₂²×C₁₃	—	8	4	no	—
167	C₂²×C₁₃	—	16	8	no	—
168	C₂²×C₁₃	#460	1	2	yes	—
169	C₂²×C₁₃	#466	1	2	yes	—
170	C₂²×C₁₃	#472	1	2	yes	—
171	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
172	C₄×C₁₃	—	2	4	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	4	2	no	—
176	C₄×C₁₃	#460	1	2	yes	—
177	C₄×C₁₃	#466	1	2	yes	—
178	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
179	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
180	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
181	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
183	C₂²×D₄	—	2	2	no	—
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	—	2	2	no	—
186	C₂×SD₁₆	—	2	8	no	—
187	C₂×SD₁₆	—	4	8	no	—
188	C₂×D₈	—	2	4	no	—
189	C₂×D₈	—	2	4	no	—
190	C₂×D₈	—	4	4	no	—
191	C₂×D₈	—	4	4	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
195	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
196	#127	—	2	2	no	—
197	#127	—	2	2	no	—
198	#127	—	2	2	no	—
199	#127	C₂³×C₁₃	1	4	yes	—
200	C₄×Q₈	—	2	1	no	—
201	C₄×Q₈	—	2	1	no	—
202	C₄×Q₈	D₄×C₁₃	1	2	yes	—
203	C₄×Q₈	C₂³×C₁₃	1	2	yes	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
208	C₄⋊C₈	—	2	4	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	C₂³×C₁₃	1	2	yes	—
212	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
213	#102	—	2	8	no	—
214	C₄×C₈	—	2	4	no	—
215	C₄×C₈	D₄×C₁₃	1	2	yes	—
216	C₄×C₈	C₂³×C₁₃	1	2	yes	—
217	C₂×C₁₃	—	16	4	no	—
218	C₂×C₁₃	#1668	1	4	yes	direct product factor
219	C₂×C₁₃	C₂×#460	1	1	yes	characteristic
220	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
221	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
222	C₂⁴	—	4	2	no	—
223	C₂×Q₈	—	2	1	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	C₁₃×D₈	1	4	yes	complemented
226	C₂×Q₈	C₂×D₄×C₁₃	1	2	yes	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	—	4	4	no	—
235	C₂×D₄	—	4	4	no	—
236	C₂×D₄	—	4	4	no	—
237	C₂×D₄	—	8	4	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
243	SD₁₆	—	4	16	no	—
244	D₈	—	4	8	no	—
245	D₈	—	4	8	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	4	4	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	C₁₃×SD₁₆	1	4	yes	complemented
252	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
253	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	C₂×D₄×C₁₃	1	2	yes	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₄²	C₁₃×#41	1	1	yes	characteristic
263	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
264	C₁₃	C₂×#1668	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³	—	4	4	no	—
266	C₂³	—	8	4	no	—
267	C₂³	—	8	4	no	—
268	C₂³	—	8	4	no	—
269	C₂³	C₁₃×#121	1	1	yes	characteristic
270	Q₈	—	2	4	no	—
271	Q₈	—	2	2	no	—
272	Q₈	C₂×C₁₃×D₈	1	4	yes	complemented
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	4	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	4	no	—
280	D₄	—	8	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
287	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
288	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
289	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
290	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
291	C₂×C₄	C₁₃×#135	1	2	yes	—
292	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	4	4	no	—
298	C₈	—	4	2	no	—
299	C₈	C₂×C₁₃×SD₁₆	1	4	yes	complemented
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	4	no	—
303	C₂²	—	16	8	no	—
304	C₂²	C₁₃×#460	1	2	yes	—
305	C₂²	C₁₃×#466	1	2	yes	—
306	C₂²	C₁₃×#472	1	2	yes	—
307	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	C₁₃×#460	1	2	yes	—
313	C₄	C₁₃×#466	1	2	yes	—
314	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
315	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
316	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
317	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₂	—	16	4	no	—
319	C₂	C₁₃×#1668	1	4	yes	direct product factor
320	C₂	C₂×C₁₃×#460	1	1	yes	characteristic
321	C₂	C₂×C₁₃×#466	1	1	yes	characteristic
322	C₂	C₂×C₁₃×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#460	C₂	semidirect
3. C₂×C₈×Q₈×C₁₃	C₂	semidirect
4. C₁₃×#1668	C₂	direct (N × H)
5. C₂×C₁₃×#335	C₂	semidirect
6. C₂×C₁₃×#331	C₂	semidirect
15. C₁₃×#460	C₂²	semidirect
17. C₈×Q₈×C₁₃	C₂²	semidirect
24. C₁₃×#335	C₂²	semidirect
25. C₁₃×#331	C₂²	semidirect
61. C₂×#1668	C₁₃	direct (N × H)
65. C₂×Q₈×C₁₃	D₈	semidirect
91. C₂×C₈×C₁₃	SD₁₆	semidirect
105. C₂×#460	C₂×C₁₃	semidirect
106. C₂×C₈×Q₈	C₂×C₁₃	semidirect
107. #1668	C₂×C₁₃	direct (N × H)
108. C₂×#335	C₂×C₁₃	semidirect
109. C₂×#331	C₂×C₁₃	semidirect
117. Q₈×C₁₃	C₂×D₈	semidirect
144. C₈×C₁₃	C₂×SD₁₆	semidirect
153. #460	C₂²×C₁₃	semidirect
155. C₈×Q₈	C₂²×C₁₃	semidirect
162. #335	C₂²×C₁₃	semidirect
163. #331	C₂²×C₁₃	semidirect
218. C₂×C₁₃	#1668	direct (N × H)
225. C₂×Q₈	C₁₃×D₈	semidirect
251. C₂×C₈	C₁₃×SD₁₆	semidirect
264. C₁₃	C₂×#1668	direct (N × H)
272. Q₈	C₂×C₁₃×D₈	semidirect
299. C₈	C₂×C₁₃×SD₁₆	semidirect
319. C₂	C₁₃×#1668	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#493	2
C₂×C₁₃×#460	2
C₂×C₈×Q₈×C₁₃	2
C₁₃×#1668	2
C₂×C₁₃×#335	2
C₂×C₁₃×#331	2
C₂×#1668	13

Subgroups — C2×C13×#1668 #15705451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#1668 #15705451