Subgroups — C₁₃×#33784 #15720123

Order 3328.

584 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×#3168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2251	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2245	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2209	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2208	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2188	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2185	C₂	1	2	yes	maximal
9	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×Q₁₆	—	2	1	no	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
13	C₁₃×#529	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
16	C₁₃×#499	C₂²	1	1	yes	characteristic
17	C₁₃×#495	C₂²	1	1	yes	characteristic
18	C₁₃×#494	—	2	1	no	—
19	C₁₃×#494	—	2	1	no	—
20	C₁₃×#494	C₂²	1	2	yes	—
21	C₁₃×#494	C₂²	1	2	yes	—
22	C₁₃×#488	—	2	1	no	—
23	C₁₃×#467	C₂²	1	1	yes	characteristic
24	C₁₃×#466	C₂²	1	1	yes	characteristic
25	C₁₃×#465	—	2	1	no	—
26	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₃×SD₁₆	—	2	1	no	—
28	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
30	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₁₃×#358	C₂²	1	4	yes	—
32	C₁₃×#358	C₂²	1	4	yes	—
33	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₁₃×#346	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	—	2	2	no	—
36	C₂×C₁₃×Q₁₆	—	2	2	no	—
37	C₂×C₁₃×Q₁₆	—	2	2	no	—
38	C₂×C₁₃×Q₁₆	—	2	2	no	—
39	C₂×C₁₃×Q₁₆	—	4	4	no	—
40	C₂×C₁₃×Q₁₆	—	4	4	no	—
41	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×SD₁₆	—	2	1	no	—
44	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	—	2	2	no	—
48	C₁₃×C₄⋊Q₈	—	2	2	no	—
49	C₁₃×C₄⋊Q₈	—	4	1	no	—
50	C₁₃×C₄⋊Q₈	—	4	1	no	—
51	C₁₃×#126	—	4	1	no	—
52	C₁₃×#125	—	2	1	no	—
53	C₁₃×#124	—	2	1	no	—
54	C₁₃×#123	—	4	1	no	—
55	C₁₃×C₂²⋊Q₈	—	2	1	no	—
56	C₁₃×#121	—	2	1	no	—
57	C₄×Q₈×C₁₃	—	2	1	no	—
58	C₄×D₄×C₁₃	—	2	1	no	—
59	C₄×D₄×C₁₃	—	4	1	no	—
60	C₁₃×#117	—	4	1	no	—
61	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
62	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
63	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
64	C₁₃×Q₃₂	—	4	4	no	—
65	C₁₃×SD₃₂	—	4	4	no	—
66	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#107	C₂³	1	1	yes	characteristic
69	C₁₃×#106	—	2	1	no	—
70	C₁₃×#103	—	2	1	no	—
71	C₁₃×#103	—	2	1	no	—
72	C₁₃×#102	—	2	1	no	—
73	C₁₃×#102	—	2	1	no	—
74	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
75	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	#33784	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×Q₈×C₁₃	—	2	1	no	—
80	C₂×Q₈×C₁₃	—	2	4	no	—
81	C₂×Q₈×C₁₃	—	4	4	no	—
82	C₂×Q₈×C₁₃	—	4	4	no	—
83	C₂×Q₈×C₁₃	—	4	2	no	—
84	C₂×D₄×C₁₃	—	2	1	no	—
85	C₂×D₄×C₁₃	—	4	1	no	—
86	C₂²×C₄×C₁₃	—	4	2	no	—
87	C₂²×C₄×C₁₃	—	4	1	no	—
88	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₁₃×Q₁₆	—	2	4	no	—
92	C₁₃×Q₁₆	—	2	2	no	—
93	C₁₃×Q₁₆	—	2	4	no	—
94	C₁₃×Q₁₆	—	4	4	no	—
95	C₁₃×Q₁₆	—	4	4	no	—
96	C₁₃×Q₁₆	—	4	4	no	—
97	C₁₃×SD₁₆	—	4	2	no	—
98	C₁₃×D₈	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	2	1	no	—
107	C₁₃×C₄⋊C₄	—	4	2	no	—
108	C₁₃×C₄⋊C₄	—	4	2	no	—
109	C₁₃×C₄⋊C₄	—	4	2	no	—
110	C₁₃×C₄⋊C₄	—	4	2	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₄⋊C₄	—	4	1	no	—
114	C₁₃×C₄⋊C₄	—	4	1	no	—
115	C₁₃×C₂²⋊C₄	—	4	1	no	—
116	C₁₃×C₂²⋊C₄	—	4	1	no	—
117	C₁₃×C₂²⋊C₄	—	4	1	no	—
118	C₁₃×C₂²⋊C₄	—	4	1	no	—
119	C₁₃×C₂²⋊C₄	—	4	1	no	—
120	C₄²×C₁₃	—	4	1	no	—
121	C₄²×C₁₃	D₈	1	1	yes	characteristic
122	C₄²×C₁₃	D₈	1	1	yes	characteristic
123	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
125	C₁₃×C₁₆	—	2	4	no	—
126	#3168	C₂×C₁₃	1	1	yes	characteristic
127	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#2251	C₂×C₁₃	1	2	yes	complemented
129	#2245	C₂×C₁₃	1	2	yes	complemented
130	#2209	C₂×C₁₃	1	2	yes	complemented
131	#2208	C₂×C₁₃	1	2	yes	complemented
132	#2188	C₂×C₁₃	1	2	yes	complemented
133	#2185	C₂×C₁₃	1	2	yes	—
134	#2163	C₂×C₁₃	1	1	yes	characteristic, complemented
135	C₂³×C₁₃	—	4	1	no	—
136	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
137	Q₈×C₁₃	—	4	4	no	—
138	Q₈×C₁₃	—	4	1	no	—
139	Q₈×C₁₃	—	4	1	no	—
140	Q₈×C₁₃	—	4	4	no	—
141	Q₈×C₁₃	—	8	4	no	—
142	D₄×C₁₃	—	4	1	no	—
143	D₄×C₁₃	—	4	1	no	—
144	D₄×C₁₃	—	8	2	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	1	no	—
147	C₂×C₄×C₁₃	—	2	1	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	—	4	1	no	—
151	C₂×C₄×C₁₃	—	4	1	no	—
152	C₂×C₄×C₁₃	—	4	4	no	—
153	C₂×C₄×C₁₃	—	8	4	no	—
154	C₂×C₄×C₁₃	—	8	2	no	—
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
164	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
165	C₂²×Q₁₆	—	2	1	no	—
166	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
167	C₂×C₄⋊Q₈	—	2	1	no	—
168	#529	—	2	1	no	—
169	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	—
170	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
171	#499	C₂²×C₁₃	1	1	yes	characteristic
172	#495	C₂²×C₁₃	1	1	yes	characteristic
173	#494	—	2	1	no	—
174	#494	—	2	1	no	—
175	#494	C₂²×C₁₃	1	2	yes	—
176	#494	C₂²×C₁₃	1	2	yes	—
177	#488	—	2	1	no	—
178	#467	C₂²×C₁₃	1	1	yes	characteristic
179	#466	C₂²×C₁₃	1	1	yes	characteristic
180	#465	—	2	1	no	—
181	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
182	C₄×SD₁₆	—	2	1	no	—
183	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
185	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
186	#358	C₂²×C₁₃	1	4	yes	—
187	#358	C₂²×C₁₃	1	4	yes	—
188	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
189	#346	C₂²×C₁₃	1	2	yes	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	8	1	no	—
193	C₂²×C₁₃	—	8	1	no	—
194	C₂²×C₁₃	—	8	1	no	—
195	C₂²×C₁₃	#510	1	2	yes	—
196	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	—	8	1	no	—
203	C₄×C₁₃	—	8	4	no	—
204	C₄×C₁₃	—	8	1	no	—
205	C₄×C₁₃	#510	1	2	yes	—
206	C₄×C₁₃	#535	1	1	yes	characteristic
207	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₂²×Q₈	—	2	2	no	—
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×Q₁₆	—	2	2	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	4	4	no	—
213	C₂×Q₁₆	—	4	4	no	—
214	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
215	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
216	C₂×SD₁₆	—	2	1	no	—
217	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
218	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
220	C₄⋊Q₈	—	2	2	no	—
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	—	4	1	no	—
223	C₄⋊Q₈	—	4	1	no	—
224	#126	—	4	1	no	—
225	#125	—	2	1	no	—
226	#124	—	2	1	no	—
227	#123	—	4	1	no	—
228	C₂²⋊Q₈	—	2	1	no	—
229	#121	—	2	1	no	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×D₄	—	2	1	no	—
232	C₄×D₄	—	4	1	no	—
233	#117	—	4	1	no	—
234	C₂×C₄⋊C₄	—	4	1	no	—
235	C₂×C₄⋊C₄	—	4	1	no	—
236	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
237	Q₃₂	—	4	4	no	—
238	SD₃₂	—	4	4	no	—
239	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
240	#107	C₂³×C₁₃	1	1	yes	characteristic
241	#107	C₂³×C₁₃	1	1	yes	characteristic
242	#106	—	2	1	no	—
243	#103	—	2	1	no	—
244	#103	—	2	1	no	—
245	#102	—	2	1	no	—
246	#102	—	2	1	no	—
247	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
248	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
251	C₂×C₁₃	—	2	2	no	—
252	C₂×C₁₃	—	16	1	no	—
253	C₂×C₁₃	#3205	1	1	yes	characteristic
254	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
255	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	4	no	—
258	C₂×Q₈	—	4	4	no	—
259	C₂×Q₈	—	4	4	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
266	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
268	Q₁₆	—	2	4	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	4	no	—
271	Q₁₆	—	4	4	no	—
272	Q₁₆	—	4	4	no	—
273	Q₁₆	—	4	4	no	—
274	SD₁₆	—	4	2	no	—
275	D₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
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	1	yes	characteristic
281	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₂²⋊C₄	—	4	1	no	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	1	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₄²	—	4	1	no	—
298	C₄²	C₁₃×D₈	1	1	yes	characteristic
299	C₄²	C₁₃×D₈	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
302	C₁₆	—	2	4	no	—
303	C₁₃	#33784	1	1	yes	characteristic, direct product factor, Sylow
304	C₂³	—	4	1	no	—
305	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
306	Q₈	—	4	4	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	4	4	no	—
310	Q₈	—	8	4	no	—
311	D₄	—	4	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	8	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	8	4	no	—
323	C₂×C₄	—	8	2	no	—
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	C₂×C₁₃×D₈	1	1	yes	characteristic
328	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
329	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
333	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₁₃×#510	1	2	yes	—
340	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	4	no	—
348	C₄	—	8	1	no	—
349	C₄	C₁₃×#510	1	2	yes	—
350	C₄	C₁₃×#535	1	1	yes	characteristic
351	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	16	1	no	—
354	C₂	C₁₃×#3205	1	1	yes	characteristic
355	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
356	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#466	C₂²
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
2. C₂×C₁₃×#494	C₂	semidirect
3. C₁₃×#2251	C₂	semidirect
4. C₁₃×#2245	C₂	semidirect
5. C₁₃×#2209	C₂	semidirect
6. C₁₃×#2208	C₂	semidirect
7. C₁₃×#2188	C₂	semidirect
9. C₁₃×#2163	C₂	semidirect
78. #33784	C₁₃	direct (N × H)
127. C₂×#494	C₂×C₁₃	semidirect
128. #2251	C₂×C₁₃	semidirect
129. #2245	C₂×C₁₃	semidirect
130. #2209	C₂×C₁₃	semidirect
131. #2208	C₂×C₁₃	semidirect
132. #2188	C₂×C₁₃	semidirect
134. #2163	C₂×C₁₃	semidirect
303. C₁₃	#33784	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×#3168	2
C₂×C₁₃×#494	2
C₁₃×#2251	2
C₁₃×#2245	2
C₁₃×#2209	2
C₁₃×#2208	2
C₁₃×#2188	2
C₁₃×#2185	2
C₁₃×#2163	2
#33784	13

Subgroups — C13×#33784 #15720123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33784 #15720123