Subgroups — C₂×C₁₃×#2213 #15719931

Order 3328.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#509	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2213	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#361	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
10	C₂×C₁₃×#131	—	2	1	no	—
11	C₂×C₁₃×#127	—	2	1	no	—
12	C₁₃×#509	—	2	4	no	—
13	C₁₃×#509	C₂²	1	8	yes	complemented
14	C₂×C₁₃×SD₃₂	—	2	2	no	—
15	C₂×C₁₃×D₁₆	—	2	2	no	—
16	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
17	C₁₃×#493	—	2	2	no	—
18	C₁₃×#493	C₂²	1	4	yes	complemented
19	C₁₃×#443	—	2	2	no	—
20	C₁₃×#443	C₂²	1	4	yes	complemented
21	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#104	—	2	1	no	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#364	C₂²	1	4	yes	complemented
25	C₁₃×#361	C₂²	1	8	yes	complemented
26	C₂×C₁₃×#41	—	2	1	no	—
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂²×D₄×C₁₃	—	4	2	no	—
29	C₁₃×#135	—	2	1	no	—
30	C₁₃×#135	—	2	1	no	—
31	C₁₃×#135	D₄	1	2	yes	complemented
32	C₁₃×#135	C₂³	1	2	yes	—
33	C₂×C₁₃×Q₁₆	D₄	1	1	yes	characteristic, complemented
34	C₂×C₁₃×SD₁₆	—	2	1	no	—
35	C₂×C₁₃×D₈	—	2	2	no	—
36	C₂×C₁₃×D₈	—	2	2	no	—
37	C₂×C₁₃×D₈	—	2	2	no	—
38	C₂×C₁₃×D₈	—	2	4	no	—
39	C₂×C₁₃×D₈	—	2	2	no	—
40	C₂×C₁₃×D₈	—	2	2	no	—
41	C₂×C₁₃×D₈	—	4	1	no	—
42	C₂×C₁₃×D₈	—	4	1	no	—
43	C₂×C₁₃×D₈	—	4	2	no	—
44	C₂×C₁₃×D₈	D₄	1	1	yes	characteristic, complemented
45	C₂×C₁₃×D₈	C₂³	1	4	yes	—
46	C₁₃×#131	—	2	4	no	—
47	C₁₃×#131	—	4	2	no	—
48	C₂×C₁₃×M₁₆	—	2	1	no	—
49	C₂×C₁₃×M₁₆	—	4	1	no	—
50	C₂²×C₈×C₁₃	—	4	1	no	—
51	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×#127	—	2	4	no	—
54	C₁₃×#127	—	4	2	no	—
55	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
56	C₁₃×SD₃₂	—	2	8	no	—
57	C₁₃×D₁₆	—	2	8	no	—
58	C₁₃×M₃₂	—	2	2	no	—
59	C₁₃×M₃₂	C₂³	1	4	yes	—
60	C₂×C₁₃×C₁₆	—	2	2	no	—
61	C₁₃×#108	—	2	1	no	—
62	C₁₃×#108	C₂³	1	2	yes	—
63	C₁₃×#104	—	2	4	no	—
64	C₄×C₈×C₁₃	D₄	1	2	yes	—
65	C₄×C₈×C₁₃	C₂³	1	2	yes	—
66	C₂×#2213	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂⁴×C₁₃	—	4	2	no	—
68	C₁₃×#41	—	2	2	no	—
69	C₁₃×#41	—	2	2	no	—
70	C₁₃×#41	—	4	1	no	—
71	C₁₃×#41	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	2	1	no	—
73	C₂×D₄×C₁₃	—	2	1	no	—
74	C₂×D₄×C₁₃	—	2	4	no	—
75	C₂×D₄×C₁₃	—	4	8	no	—
76	C₂×D₄×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	4	2	no	—
78	C₂×D₄×C₁₃	—	4	2	no	—
79	C₂×D₄×C₁₃	—	4	2	no	—
80	C₂×D₄×C₁₃	—	4	2	no	—
81	C₂×D₄×C₁₃	—	4	2	no	—
82	C₂×D₄×C₁₃	—	8	8	no	—
83	C₂²×C₄×C₁₃	—	2	1	no	—
84	C₂²×C₄×C₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₁₃×Q₁₆	—	2	1	no	—
87	C₁₃×Q₁₆	C₂×D₄	1	2	yes	complemented
88	C₁₃×SD₁₆	—	2	2	no	—
89	C₁₃×SD₁₆	—	2	2	no	—
90	C₁₃×D₈	—	2	1	no	—
91	C₁₃×D₈	—	2	4	no	—
92	C₁₃×D₈	—	2	4	no	—
93	C₁₃×D₈	—	4	2	no	—
94	C₁₃×D₈	—	4	2	no	—
95	C₁₃×D₈	—	4	4	no	—
96	C₁₃×D₈	—	4	4	no	—
97	C₁₃×D₈	C₂×D₄	1	2	yes	complemented
98	C₁₃×M₁₆	—	2	2	no	—
99	C₁₃×M₁₆	—	4	1	no	—
100	C₁₃×M₁₆	—	4	4	no	—
101	C₂×C₈×C₁₃	—	4	1	no	—
102	C₂×C₈×C₁₃	—	4	4	no	—
103	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
104	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
105	C₂×C₈×C₁₃	D₈	1	2	yes	—
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
111	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₄²×C₁₃	—	2	1	no	—
114	C₄²×C₁₃	C₂×D₄	1	2	yes	—
115	C₁₃×C₁₆	—	2	4	no	—
116	C₂×#509	C₂×C₁₃	1	2	yes	complemented
117	C₂×#493	C₂×C₁₃	1	1	yes	characteristic, complemented
118	C₂×#443	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#2213	C₂×C₁₃	1	8	yes	direct product factor
120	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂×#361	C₂×C₁₃	1	2	yes	complemented
122	C₂³×C₁₃	—	4	1	no	—
123	C₂³×C₁₃	—	4	4	no	—
124	C₂³×C₁₃	—	8	8	no	—
125	C₂³×C₁₃	—	8	2	no	—
126	C₂³×C₁₃	—	8	2	no	—
127	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	Q₈×C₁₃	—	2	2	no	—
129	Q₈×C₁₃	—	4	1	no	—
130	D₄×C₁₃	—	2	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	4	no	—
134	D₄×C₁₃	—	4	4	no	—
135	D₄×C₁₃	—	4	1	no	—
136	D₄×C₁₃	—	8	16	no	—
137	D₄×C₁₃	—	8	2	no	—
138	D₄×C₁₃	—	8	2	no	—
139	C₂×C₄×C₁₃	—	2	4	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	4	1	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₁₃	—	4	2	no	—
151	C₈×C₁₃	#121	1	1	yes	characteristic
152	C₈×C₁₃	#121	1	1	yes	characteristic
153	C₈×C₁₃	C₂×D₈	1	2	yes	—
154	C₈×C₁₃	C₂×D₈	1	2	yes	—
155	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
156	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
158	C₂²×D₈	—	2	1	no	—
159	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
160	C₂×#131	—	2	1	no	—
161	C₂×#127	—	2	1	no	—
162	#509	—	2	4	no	—
163	#509	C₂²×C₁₃	1	8	yes	complemented
164	C₂×SD₃₂	—	2	2	no	—
165	C₂×D₁₆	—	2	2	no	—
166	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
167	#493	—	2	2	no	—
168	#493	C₂²×C₁₃	1	4	yes	complemented
169	#443	—	2	2	no	—
170	#443	C₂²×C₁₃	1	4	yes	complemented
171	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×#104	—	2	1	no	—
173	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
174	#364	C₂²×C₁₃	1	4	yes	complemented
175	#361	C₂²×C₁₃	1	8	yes	complemented
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	—	4	2	no	—
178	C₂²×C₁₃	—	8	8	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	—	8	4	no	—
181	C₂²×C₁₃	—	16	4	no	—
182	C₂²×C₁₃	—	16	4	no	—
183	C₂²×C₁₃	#459	1	1	yes	characteristic
184	C₂²×C₁₃	#459	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₁₃	—	2	4	no	—
187	C₄×C₁₃	—	4	2	no	—
188	C₄×C₁₃	#459	1	1	yes	characteristic
189	C₄×C₁₃	#459	1	1	yes	characteristic
190	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
191	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
192	C₂×#41	—	2	1	no	—
193	C₂²×D₄	—	2	2	no	—
194	C₂²×D₄	—	4	2	no	—
195	#135	—	2	1	no	—
196	#135	—	2	1	no	—
197	#135	D₄×C₁₃	1	2	yes	complemented
198	#135	C₂³×C₁₃	1	2	yes	—
199	C₂×Q₁₆	D₄×C₁₃	1	1	yes	characteristic, complemented
200	C₂×SD₁₆	—	2	1	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×D₈	—	2	2	no	—
203	C₂×D₈	—	2	4	no	—
204	C₂×D₈	—	2	2	no	—
205	C₂×D₈	—	2	2	no	—
206	C₂×D₈	—	2	2	no	—
207	C₂×D₈	—	4	1	no	—
208	C₂×D₈	—	4	1	no	—
209	C₂×D₈	—	4	2	no	—
210	C₂×D₈	D₄×C₁₃	1	1	yes	characteristic, complemented
211	C₂×D₈	C₂³×C₁₃	1	4	yes	—
212	#131	—	2	4	no	—
213	#131	—	4	2	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	4	1	no	—
216	C₂²×C₈	—	4	1	no	—
217	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
219	#127	—	2	4	no	—
220	#127	—	4	2	no	—
221	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
222	SD₃₂	—	2	8	no	—
223	D₁₆	—	2	8	no	—
224	M₃₂	—	2	2	no	—
225	M₃₂	C₂³×C₁₃	1	4	yes	—
226	C₂×C₁₆	—	2	2	no	—
227	#108	—	2	1	no	—
228	#108	C₂³×C₁₃	1	2	yes	—
229	#104	—	2	4	no	—
230	C₄×C₈	D₄×C₁₃	1	2	yes	—
231	C₄×C₈	C₂³×C₁₃	1	2	yes	—
232	C₂×C₁₃	—	2	1	no	—
233	C₂×C₁₃	—	2	1	no	—
234	C₂×C₁₃	—	8	2	no	—
235	C₂×C₁₃	—	16	4	no	—
236	C₂×C₁₃	#2213	1	2	yes	direct product factor
237	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
238	C₂⁴	—	4	2	no	—
239	#41	—	2	2	no	—
240	#41	—	2	2	no	—
241	#41	—	4	1	no	—
242	#41	—	4	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	4	8	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	8	8	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
257	Q₁₆	—	2	1	no	—
258	Q₁₆	C₂×D₄×C₁₃	1	2	yes	complemented
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	2	2	no	—
261	D₈	—	2	1	no	—
262	D₈	—	2	4	no	—
263	D₈	—	2	4	no	—
264	D₈	—	4	2	no	—
265	D₈	—	4	2	no	—
266	D₈	—	4	4	no	—
267	D₈	—	4	4	no	—
268	D₈	C₂×D₄×C₁₃	1	2	yes	complemented
269	M₁₆	—	2	2	no	—
270	M₁₆	—	4	1	no	—
271	M₁₆	—	4	4	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	4	no	—
274	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
275	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
276	C₂×C₈	C₁₃×D₈	1	2	yes	—
277	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
278	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
281	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
282	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
283	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
284	C₄²	—	2	1	no	—
285	C₄²	C₂×D₄×C₁₃	1	2	yes	—
286	C₁₆	—	2	4	no	—
287	C₁₃	C₂×#2213	1	1	yes	characteristic, direct product factor, Sylow
288	C₂³	—	4	1	no	—
289	C₂³	—	4	4	no	—
290	C₂³	—	8	8	no	—
291	C₂³	—	8	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	Q₈	—	4	1	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	4	no	—
301	D₄	—	4	4	no	—
302	D₄	—	8	16	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
311	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
312	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
313	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
314	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	C₁₃×#121	1	1	yes	characteristic
318	C₈	C₁₃×#121	1	1	yes	characteristic
319	C₈	C₂×C₁₃×D₈	1	2	yes	—
320	C₈	C₂×C₁₃×D₈	1	2	yes	—
321	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
322	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	8	8	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	4	no	—
328	C₂²	—	16	4	no	—
329	C₂²	—	16	4	no	—
330	C₂²	C₁₃×#459	1	1	yes	characteristic
331	C₂²	C₁₃×#459	1	1	yes	characteristic
332	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	C₁₃×#459	1	1	yes	characteristic
336	C₄	C₁₃×#459	1	1	yes	characteristic
337	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
338	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	—	16	4	no	—
343	C₂	C₁₃×#2213	1	2	yes	direct product factor
344	C₂	C₂×C₁₃×#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#509	C₂	semidirect
2. C₂×C₁₃×#493	C₂	semidirect
3. C₂×C₁₃×#443	C₂	semidirect
4. C₁₃×#2213	C₂	direct (N × H)
5. C₂×C₁₃×#364	C₂	semidirect
6. C₂×C₁₃×#361	C₂	semidirect
13. C₁₃×#509	C₂²	semidirect
18. C₁₃×#493	C₂²	semidirect
20. C₁₃×#443	C₂²	semidirect
24. C₁₃×#364	C₂²	semidirect
25. C₁₃×#361	C₂²	semidirect
31. C₁₃×#135	D₄	semidirect
33. C₂×C₁₃×Q₁₆	D₄	semidirect
44. C₂×C₁₃×D₈	D₄	semidirect
66. C₂×#2213	C₁₃	direct (N × H)
87. C₁₃×Q₁₆	C₂×D₄	semidirect
97. C₁₃×D₈	C₂×D₄	semidirect
116. C₂×#509	C₂×C₁₃	semidirect
117. C₂×#493	C₂×C₁₃	semidirect
118. C₂×#443	C₂×C₁₃	semidirect
119. #2213	C₂×C₁₃	direct (N × H)
120. C₂×#364	C₂×C₁₃	semidirect
121. C₂×#361	C₂×C₁₃	semidirect
163. #509	C₂²×C₁₃	semidirect
168. #493	C₂²×C₁₃	semidirect
170. #443	C₂²×C₁₃	semidirect
174. #364	C₂²×C₁₃	semidirect
175. #361	C₂²×C₁₃	semidirect
197. #135	D₄×C₁₃	semidirect
199. C₂×Q₁₆	D₄×C₁₃	semidirect
210. C₂×D₈	D₄×C₁₃	semidirect
236. C₂×C₁₃	#2213	direct (N × H)
258. Q₁₆	C₂×D₄×C₁₃	semidirect
268. D₈	C₂×D₄×C₁₃	semidirect
287. C₁₃	C₂×#2213	direct (N × H)
343. C₂	C₁₃×#2213	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×#509	2
C₂×C₁₃×#493	2
C₂×C₁₃×#443	2
C₁₃×#2213	2
C₂×C₁₃×#364	2
C₂×C₁₃×#361	2
C₂×#2213	13

Subgroups — C2×C13×#2213 #15719931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2213 #15719931