Subgroups — C₁₃×#33771 #15720110

Order 3328.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2246	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2240	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2206	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2184	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
10	C₁₃×#532	—	2	1	no	—
11	C₂×C₁₃×#127	—	2	1	no	—
12	C₂×C₁₃×D₁₆	C₂²	1	8	yes	complemented
13	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#493	—	2	1	no	—
15	C₁₃×#493	—	2	1	no	—
16	C₁₃×#493	C₂²	1	2	yes	—
17	C₁₃×#493	C₂²	1	2	yes	—
18	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
19	C₁₃×#486	—	2	1	no	—
20	C₁₃×#466	—	2	1	no	—
21	C₁₃×#466	C₂²	1	2	yes	—
22	C₄×C₁₃×D₈	—	2	1	no	—
23	C₄×C₁₃×D₈	C₂²	1	2	yes	complemented
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
26	C₁₃×#357	C₂²	1	8	yes	complemented
27	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
29	C₂×C₁₃×#41	—	4	1	no	—
30	C₂²×D₄×C₁₃	—	2	2	no	—
31	C₂²×D₄×C₁₃	—	4	1	no	—
32	C₂²×D₄×C₁₃	—	4	1	no	—
33	C₂×C₁₃×D₈	—	2	1	no	—
34	C₂×C₁₃×D₈	—	2	2	no	—
35	C₂×C₁₃×D₈	—	2	2	no	—
36	C₂×C₁₃×D₈	—	2	2	no	—
37	C₂×C₁₃×D₈	—	4	4	no	—
38	C₂×C₁₃×D₈	—	4	4	no	—
39	C₂×C₁₃×D₈	C₂³	1	4	yes	—
40	C₂×C₁₃×D₈	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×#127	—	2	1	no	—
45	C₁₃×#127	—	2	2	no	—
46	C₁₃×#127	—	2	2	no	—
47	C₁₃×#127	—	4	1	no	—
48	C₁₃×#127	—	4	1	no	—
49	C₁₃×#124	—	4	1	no	—
50	C₁₃×#121	—	2	2	no	—
51	C₁₃×#121	—	4	1	no	—
52	C₄×D₄×C₁₃	—	2	2	no	—
53	C₄×D₄×C₁₃	—	4	1	no	—
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₁₃×D₁₆	—	4	8	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
57	C₁₃×#107	—	2	1	no	—
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₁₃×#102	—	2	2	no	—
60	C₁₃×#102	—	2	2	no	—
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	#33771	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂⁴×C₁₃	—	4	2	no	—
67	C₁₃×#41	—	8	4	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	2	2	no	—
70	C₂×D₄×C₁₃	—	2	4	no	—
71	C₂×D₄×C₁₃	—	4	4	no	—
72	C₂×D₄×C₁₃	—	4	4	no	—
73	C₂×D₄×C₁₃	—	4	2	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂×D₄×C₁₃	—	4	2	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	—	8	4	no	—
82	C₂×D₄×C₁₃	—	8	4	no	—
83	C₂²×C₄×C₁₃	—	4	2	no	—
84	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₁₃×D₈	—	2	4	no	—
88	C₁₃×D₈	—	2	8	no	—
89	C₁₃×D₈	—	4	4	no	—
90	C₁₃×D₈	—	4	4	no	—
91	C₁₃×D₈	—	4	2	no	—
92	C₁₃×D₈	—	4	4	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₂²⋊C₄	—	4	2	no	—
102	C₁₃×C₂²⋊C₄	—	4	2	no	—
103	C₄²×C₁₃	D₈	1	1	yes	characteristic
104	C₄²×C₁₃	D₈	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₁₃×C₁₆	—	2	4	no	—
108	#3169	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂×#493	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#2246	C₂×C₁₃	1	2	yes	complemented
111	#2240	C₂×C₁₃	1	2	yes	complemented
112	#2206	C₂×C₁₃	1	4	yes	complemented
113	#2184	C₂×C₁₃	1	4	yes	complemented
114	#2162	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	—	4	2	no	—
116	C₂³×C₁₃	—	4	4	no	—
117	C₂³×C₁₃	—	8	4	no	—
118	C₂³×C₁₃	—	8	4	no	—
119	C₂³×C₁₃	—	8	4	no	—
120	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₁₃	—	8	2	no	—
122	D₄×C₁₃	—	4	4	no	—
123	D₄×C₁₃	—	4	2	no	—
124	D₄×C₁₃	—	4	4	no	—
125	D₄×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	8	4	no	—
127	D₄×C₁₃	—	8	4	no	—
128	D₄×C₁₃	—	8	4	no	—
129	D₄×C₁₃	—	8	4	no	—
130	D₄×C₁₃	—	8	4	no	—
131	D₄×C₁₃	—	8	2	no	—
132	D₄×C₁₃	—	8	2	no	—
133	D₄×C₁₃	—	8	4	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	8	4	no	—
139	C₂×C₄×C₁₃	D₁₆	1	4	yes	complemented
140	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
149	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
150	C₂²×D₈	—	2	1	no	—
151	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
152	#532	—	2	1	no	—
153	C₂×#127	—	2	1	no	—
154	C₂×D₁₆	C₂²×C₁₃	1	8	yes	complemented
155	#500	C₂²×C₁₃	1	1	yes	characteristic, complemented
156	#493	—	2	1	no	—
157	#493	—	2	1	no	—
158	#493	C₂²×C₁₃	1	2	yes	—
159	#493	C₂²×C₁₃	1	2	yes	—
160	#493	C₂²×C₁₃	1	1	yes	characteristic, complemented
161	#486	—	2	1	no	—
162	#466	—	2	1	no	—
163	#466	C₂²×C₁₃	1	2	yes	—
164	C₄×D₈	—	2	1	no	—
165	C₄×D₈	C₂²×C₁₃	1	2	yes	complemented
166	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
167	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	complemented
168	#357	C₂²×C₁₃	1	8	yes	complemented
169	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
170	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	8	2	no	—
174	C₂²×C₁₃	—	8	4	no	—
175	C₂²×C₁₃	—	8	4	no	—
176	C₂²×C₁₃	—	8	2	no	—
177	C₂²×C₁₃	—	8	4	no	—
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	—	16	8	no	—
180	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
181	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
186	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
187	C₄×C₁₃	#509	1	2	yes	—
188	C₄×C₁₃	#535	1	1	yes	characteristic
189	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×#41	—	4	1	no	—
191	C₂²×D₄	—	2	2	no	—
192	C₂²×D₄	—	4	1	no	—
193	C₂²×D₄	—	4	1	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	—	4	4	no	—
199	C₂×D₈	—	4	4	no	—
200	C₂×D₈	C₂³×C₁₃	1	4	yes	—
201	C₂×D₈	C₂³×C₁₃	1	2	yes	—
202	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₄⋊Q₈	—	2	1	no	—
205	#127	—	2	2	no	—
206	#127	—	2	2	no	—
207	#127	—	2	1	no	—
208	#127	—	4	1	no	—
209	#127	—	4	1	no	—
210	#124	—	4	1	no	—
211	#121	—	2	2	no	—
212	#121	—	4	1	no	—
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	—	4	1	no	—
215	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
216	D₁₆	—	4	8	no	—
217	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
218	#107	—	2	1	no	—
219	#107	C₂³×C₁₃	1	2	yes	—
220	#102	—	2	2	no	—
221	#102	—	2	2	no	—
222	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
223	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
226	C₂×C₁₃	—	2	2	no	—
227	C₂×C₁₃	—	16	2	no	—
228	C₂×C₁₃	—	16	4	no	—
229	C₂×C₁₃	#3205	1	1	yes	characteristic
230	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
231	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
232	C₂⁴	—	4	2	no	—
233	#41	—	8	4	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	8	4	no	—
248	C₂×D₄	—	8	4	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
251	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
253	D₈	—	2	4	no	—
254	D₈	—	2	8	no	—
255	D₈	—	4	4	no	—
256	D₈	—	4	4	no	—
257	D₈	—	4	4	no	—
258	D₈	—	4	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₄²	C₁₃×D₈	1	1	yes	characteristic
270	C₄²	C₁₃×D₈	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₁₆	—	2	4	no	—
274	C₁₃	#33771	1	1	yes	characteristic, direct product factor, Sylow
275	C₂³	—	4	2	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	8	4	no	—
278	C₂³	—	8	4	no	—
279	C₂³	—	8	4	no	—
280	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
281	Q₈	—	8	2	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	8	4	no	—
287	D₄	—	8	4	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	4	no	—
291	D₄	—	8	2	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	C₁₃×D₁₆	1	4	yes	complemented
300	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
304	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
309	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	4	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	4	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	16	8	no	—
319	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
320	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	C₂×C₁₃×D₁₆	1	2	yes	—
325	C₄	C₂×C₁₃×D₁₆	1	2	yes	—
326	C₄	C₁₃×#509	1	2	yes	—
327	C₄	C₁₃×#535	1	1	yes	characteristic
328	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	16	4	no	—
331	C₂	—	16	2	no	—
332	C₂	C₁₃×#3205	1	1	yes	characteristic
333	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
334	C₂	C₂×C₁₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3169	C₂	semidirect
2. C₂×C₁₃×#493	C₂	semidirect
3. C₁₃×#2246	C₂	semidirect
4. C₁₃×#2240	C₂	semidirect
5. C₁₃×#2206	C₂	semidirect
6. C₁₃×#2184	C₂	semidirect
7. C₁₃×#2162	C₂	semidirect
12. C₂×C₁₃×D₁₆	C₂²	semidirect
13. C₁₃×#500	C₂²	semidirect
18. C₁₃×#493	C₂²	semidirect
23. C₄×C₁₃×D₈	C₂²	semidirect
25. C₁₃×C₄⋊C₁₆	C₂²	semidirect
26. C₁₃×#357	C₂²	semidirect
28. C₄×C₁₃×C₁₆	C₂²	semidirect
65. #33771	C₁₃	direct (N × H)
108. #3169	C₂×C₁₃	semidirect
109. C₂×#493	C₂×C₁₃	semidirect
110. #2246	C₂×C₁₃	semidirect
111. #2240	C₂×C₁₃	semidirect
112. #2206	C₂×C₁₃	semidirect
113. #2184	C₂×C₁₃	semidirect
114. #2162	C₂×C₁₃	semidirect
139. C₂×C₄×C₁₃	D₁₆	semidirect
154. C₂×D₁₆	C₂²×C₁₃	semidirect
155. #500	C₂²×C₁₃	semidirect
160. #493	C₂²×C₁₃	semidirect
165. C₄×D₈	C₂²×C₁₃	semidirect
167. C₄⋊C₁₆	C₂²×C₁₃	semidirect
168. #357	C₂²×C₁₃	semidirect
170. C₄×C₁₆	C₂²×C₁₃	semidirect
274. C₁₃	#33771	direct (N × H)
299. C₂×C₄	C₁₃×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#3169	2
C₂×C₁₃×#493	2
C₁₃×#2246	2
C₁₃×#2240	2
C₁₃×#2206	2
C₁₃×#2184	2
C₁₃×#2162	2
#33771	13

Subgroups — C13×#33771 #15720110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33771 #15720110