Subgroups — C₁₃×#33780 #15720119

Order 3328.

632 conjugacy classes of subgroups.

The subgroup lattice has 358 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	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2250	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2243	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2208	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2206	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2186	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	—	2	1	no	—
10	C₂×C₁₃×#135	C₂²	1	2	yes	—
11	C₁₃×#532	—	2	1	no	—
12	C₁₃×#532	—	2	1	no	—
13	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
14	C₂×C₁₃×D₁₆	C₂²	1	4	yes	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#495	—	2	1	no	—
17	C₁₃×#495	C₂²	1	2	yes	—
18	C₁₃×#494	C₂²	1	1	yes	characteristic
19	C₁₃×#493	C₂²	1	1	yes	characteristic
20	C₁₃×#493	C₂²	1	1	yes	characteristic
21	C₁₃×#492	—	2	1	no	—
22	C₁₃×#486	—	2	1	no	—
23	C₁₃×#466	—	2	1	no	—
24	C₁₃×#466	C₂²	1	2	yes	—
25	C₄×C₁₃×D₈	—	2	1	no	—
26	C₄×C₁₃×D₈	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₁₃×#358	C₂²	1	4	yes	—
30	C₁₃×#357	C₂²	1	4	yes	—
31	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₁₃×#346	C₂²	1	2	yes	—
33	C₂×C₁₃×#41	—	2	2	no	—
34	C₂×C₁₃×#41	—	4	1	no	—
35	C₁₃×#135	—	4	4	no	—
36	C₁₃×#135	—	4	4	no	—
37	C₂×C₁₃×Q₁₆	—	2	1	no	—
38	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
39	C₂×C₁₃×SD₁₆	—	2	2	no	—
40	C₂×C₁₃×SD₁₆	—	2	2	no	—
41	C₂×C₁₃×D₈	—	2	1	no	—
42	C₂×C₁₃×D₈	—	2	1	no	—
43	C₂×C₁₃×D₈	C₂³	1	2	yes	—
44	C₂×C₁₃×D₈	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×#127	—	2	1	no	—
50	C₁₃×#127	—	2	1	no	—
51	C₁₃×#124	—	2	2	no	—
52	C₁₃×#124	—	4	1	no	—
53	C₁₃×#121	—	2	2	no	—
54	C₁₃×#121	—	4	1	no	—
55	C₁₃×#121	—	4	1	no	—
56	C₁₃×#121	—	4	1	no	—
57	C₄×D₄×C₁₃	—	2	2	no	—
58	C₄×D₄×C₁₃	—	4	1	no	—
59	C₄×D₄×C₁₃	—	4	1	no	—
60	C₄×D₄×C₁₃	—	4	1	no	—
61	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
62	C₁₃×SD₃₂	—	4	4	no	—
63	C₁₃×D₁₆	—	4	4	no	—
64	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	C₂³	1	2	yes	—
67	C₁₃×#102	—	2	2	no	—
68	C₁₃×#102	—	2	2	no	—
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#33780	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₁₃×#41	—	4	4	no	—
75	C₁₃×#41	—	4	4	no	—
76	C₁₃×#41	—	8	4	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	2	2	no	—
81	C₂×D₄×C₁₃	—	4	2	no	—
82	C₂×D₄×C₁₃	—	4	2	no	—
83	C₂×D₄×C₁₃	—	4	1	no	—
84	C₂×D₄×C₁₃	—	4	1	no	—
85	C₂×D₄×C₁₃	—	4	1	no	—
86	C₂×D₄×C₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₃	—	4	2	no	—
88	C₂²×C₄×C₁₃	—	4	2	no	—
89	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₁₃×Q₁₆	—	2	4	no	—
93	C₁₃×Q₁₆	—	4	2	no	—
94	C₁₃×SD₁₆	—	4	4	no	—
95	C₁₃×SD₁₆	—	4	4	no	—
96	C₁₃×D₈	—	2	2	no	—
97	C₁₃×D₈	—	2	4	no	—
98	C₁₃×D₈	—	2	2	no	—
99	C₁₃×D₈	—	4	2	no	—
100	C₁₃×D₈	—	4	2	no	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	1	no	—
111	C₁₃×C₂²⋊C₄	—	4	2	no	—
112	C₁₃×C₂²⋊C₄	—	4	2	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₄²×C₁₃	D₈	1	1	yes	characteristic
116	C₄²×C₁₃	D₈	1	1	yes	characteristic
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₁₃×C₁₆	—	2	4	no	—
120	#3169	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#3147	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#2250	C₂×C₁₃	1	2	yes	complemented
123	#2243	C₂×C₁₃	1	2	yes	complemented
124	#2208	C₂×C₁₃	1	2	yes	complemented
125	#2206	C₂×C₁₃	1	2	yes	complemented
126	#2186	C₂×C₁₃	1	4	yes	complemented
127	#2163	C₂×C₁₃	1	1	yes	characteristic, complemented
128	C₂³×C₁₃	—	4	2	no	—
129	C₂³×C₁₃	—	4	2	no	—
130	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
131	Q₈×C₁₃	—	4	2	no	—
132	Q₈×C₁₃	—	4	2	no	—
133	Q₈×C₁₃	—	8	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	8	2	no	—
139	D₄×C₁₃	—	8	2	no	—
140	D₄×C₁₃	—	8	2	no	—
141	D₄×C₁₃	—	8	4	no	—
142	D₄×C₁₃	—	8	4	no	—
143	D₄×C₁₃	—	8	2	no	—
144	C₂×C₄×C₁₃	—	2	1	no	—
145	C₂×C₄×C₁₃	—	2	1	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	4	1	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	8	4	no	—
151	C₂×C₄×C₁₃	—	8	4	no	—
152	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₈×C₁₃	—	2	1	no	—
159	C₈×C₁₃	—	2	1	no	—
160	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
161	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
162	C₂×#135	—	2	1	no	—
163	C₂×#135	C₂²×C₁₃	1	2	yes	—
164	#532	—	2	1	no	—
165	#532	—	2	1	no	—
166	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
167	C₂×D₁₆	C₂²×C₁₃	1	4	yes	—
168	#500	C₂²×C₁₃	1	1	yes	characteristic
169	#495	—	2	1	no	—
170	#495	C₂²×C₁₃	1	2	yes	—
171	#494	C₂²×C₁₃	1	1	yes	characteristic
172	#493	C₂²×C₁₃	1	1	yes	characteristic
173	#493	C₂²×C₁₃	1	1	yes	characteristic
174	#492	—	2	1	no	—
175	#486	—	2	1	no	—
176	#466	—	2	1	no	—
177	#466	C₂²×C₁₃	1	2	yes	—
178	C₄×D₈	—	2	1	no	—
179	C₄×D₈	C₂²×C₁₃	1	2	yes	—
180	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
181	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
182	#358	C₂²×C₁₃	1	4	yes	—
183	#357	C₂²×C₁₃	1	4	yes	—
184	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
185	#346	C₂²×C₁₃	1	2	yes	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	2	1	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	—	8	2	no	—
191	C₂²×C₁₃	—	8	2	no	—
192	C₂²×C₁₃	—	8	2	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	8	2	no	—
201	C₄×C₁₃	—	8	2	no	—
202	C₄×C₁₃	#509	1	2	yes	—
203	C₄×C₁₃	#535	1	1	yes	characteristic
204	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂×#41	—	2	2	no	—
206	C₂×#41	—	4	1	no	—
207	#135	—	4	4	no	—
208	#135	—	4	4	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	C₂³×C₁₃	1	2	yes	—
216	C₂×D₈	C₂³×C₁₃	1	2	yes	—
217	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	#127	—	2	1	no	—
222	#127	—	2	1	no	—
223	#124	—	2	2	no	—
224	#124	—	4	1	no	—
225	#121	—	2	2	no	—
226	#121	—	4	1	no	—
227	#121	—	4	1	no	—
228	#121	—	4	1	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	4	1	no	—
231	C₄×D₄	—	4	1	no	—
232	C₄×D₄	—	4	1	no	—
233	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
234	SD₃₂	—	4	4	no	—
235	D₁₆	—	4	4	no	—
236	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
237	#107	—	2	1	no	—
238	#107	C₂³×C₁₃	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
242	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
245	C₂×C₁₃	—	4	1	no	—
246	C₂×C₁₃	—	16	2	no	—
247	C₂×C₁₃	—	16	2	no	—
248	C₂×C₁₃	#3205	1	1	yes	characteristic
249	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
250	C₂×C₁₃	#3414	1	1	yes	characteristic
251	#41	—	4	4	no	—
252	#41	—	4	4	no	—
253	#41	—	8	4	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
267	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
269	Q₁₆	—	2	4	no	—
270	Q₁₆	—	4	2	no	—
271	SD₁₆	—	4	4	no	—
272	SD₁₆	—	4	4	no	—
273	D₈	—	2	2	no	—
274	D₈	—	2	4	no	—
275	D₈	—	2	2	no	—
276	D₈	—	4	2	no	—
277	D₈	—	4	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
283	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₄²	C₁₃×D₈	1	1	yes	characteristic
293	C₄²	C₁₃×D₈	1	1	yes	characteristic
294	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
296	C₁₆	—	2	4	no	—
297	C₁₃	#33780	1	1	yes	characteristic, direct product factor, Sylow
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	2	no	—
311	D₄	—	8	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	—	8	4	no	—
322	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
323	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
324	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
325	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
326	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
331	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₁₃×#509	1	2	yes	—
349	C₄	C₁₃×#535	1	1	yes	characteristic
350	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	16	2	no	—
353	C₂	—	16	2	no	—
354	C₂	C₁₃×#3205	1	1	yes	characteristic
355	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
356	C₂	C₁₃×#3414	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. C₁₃×#3147	C₂	semidirect
3. C₁₃×#2250	C₂	semidirect
4. C₁₃×#2243	C₂	semidirect
5. C₁₃×#2208	C₂	semidirect
6. C₁₃×#2206	C₂	semidirect
7. C₁₃×#2186	C₂	semidirect
8. C₁₃×#2163	C₂	semidirect
73. #33780	C₁₃	direct (N × H)
120. #3169	C₂×C₁₃	semidirect
121. #3147	C₂×C₁₃	semidirect
122. #2250	C₂×C₁₃	semidirect
123. #2243	C₂×C₁₃	semidirect
124. #2208	C₂×C₁₃	semidirect
125. #2206	C₂×C₁₃	semidirect
126. #2186	C₂×C₁₃	semidirect
127. #2163	C₂×C₁₃	semidirect
297. C₁₃	#33780	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×#3169	2
C₁₃×#3147	2
C₁₃×#2250	2
C₁₃×#2243	2
C₁₃×#2208	2
C₁₃×#2206	2
C₁₃×#2186	2
C₁₃×#2163	2
#33780	13

Subgroups — C13×#33780 #15720119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33780 #15720119