Subgroups — C₁₃×#33588 #15719927

Order 3328.

680 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#510	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#3156	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2211	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2209	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2138	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#137	—	2	1	no	—
11	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×#536	—	2	1	no	—
15	C₁₃×#519	—	2	1	no	—
16	C₁₃×#510	—	2	4	no	—
17	C₂×C₁₃×Q₃₂	—	2	4	no	—
18	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	—
19	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
20	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
21	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
22	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
23	C₁₃×#497	—	2	2	no	—
24	C₁₃×#495	C₂²	1	2	yes	—
25	C₁₃×#494	C₂²	1	2	yes	—
26	C₁₃×#441	—	2	2	no	—
27	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	—
28	C₁₃×#427	C₂²	1	1	yes	characteristic
29	C₁₃×#419	—	2	1	no	—
30	C₁₃×#405	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
32	C₁₃×#358	C₂²	1	2	yes	—
33	C₁₃×#358	C₂²	1	4	yes	—
34	C₁₃×#357	C₂²	1	2	yes	—
35	C₂×C₁₃×#41	—	2	1	no	—
36	C₂²×Q₈×C₁₃	—	2	1	no	—
37	C₂²×Q₈×C₁₃	—	2	1	no	—
38	C₁₃×#137	—	4	4	no	—
39	C₁₃×#135	—	2	2	no	—
40	C₁₃×#135	—	2	2	no	—
41	C₂×C₁₃×Q₁₆	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	—	2	2	no	—
43	C₂×C₁₃×Q₁₆	—	2	2	no	—
44	C₂×C₁₃×Q₁₆	—	2	1	no	—
45	C₂×C₁₃×Q₁₆	—	2	2	no	—
46	C₂×C₁₃×Q₁₆	—	2	2	no	—
47	C₂×C₁₃×Q₁₆	D₄	1	4	yes	—
48	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
50	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
51	C₂×C₁₃×SD₁₆	—	2	2	no	—
52	C₂×C₁₃×SD₁₆	—	2	1	no	—
53	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
54	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₁₃×C₄⋊Q₈	—	2	2	no	—
57	C₁₃×#124	—	2	2	no	—
58	C₁₃×#123	—	4	2	no	—
59	C₁₃×C₂²⋊Q₈	—	4	2	no	—
60	C₄×Q₈×C₁₃	—	2	4	no	—
61	C₄×Q₈×C₁₃	—	4	2	no	—
62	C₁₃×#117	—	2	1	no	—
63	C₁₃×#117	—	4	2	no	—
64	C₁₃×#117	D₄	1	1	yes	characteristic
65	C₁₃×Q₃₂	—	2	8	no	—
66	C₁₃×Q₃₂	—	2	4	no	—
67	C₁₃×SD₃₂	—	2	4	no	—
68	C₁₃×M₃₂	—	2	2	no	—
69	C₂×C₁₃×C₁₆	—	2	2	no	—
70	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
71	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
72	C₁₃×#107	C₂³	1	2	yes	—
73	C₁₃×#106	—	2	1	no	—
74	C₁₃×#103	—	2	4	no	—
75	C₁₃×#97	D₄	1	2	yes	—
76	C₄×C₈×C₁₃	C₂³	1	2	yes	—
77	#33588	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₁₃×#41	—	4	2	no	—
79	C₁₃×#41	—	4	2	no	—
80	C₂×Q₈×C₁₃	—	2	4	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×Q₈×C₁₃	—	2	2	no	—
83	C₂×Q₈×C₁₃	—	2	1	no	—
84	C₂×Q₈×C₁₃	—	4	1	no	—
85	C₂×Q₈×C₁₃	—	4	1	no	—
86	C₂×Q₈×C₁₃	—	4	2	no	—
87	C₂×Q₈×C₁₃	—	4	2	no	—
88	C₂×Q₈×C₁₃	—	4	2	no	—
89	C₂×D₄×C₁₃	—	2	1	no	—
90	C₂×D₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	—	4	1	no	—
92	C₂²×C₄×C₁₃	—	4	1	no	—
93	C₂²×C₄×C₁₃	—	4	1	no	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₁₃×Q₁₆	—	2	4	no	—
96	C₁₃×Q₁₆	—	2	2	no	—
97	C₁₃×Q₁₆	—	2	1	no	—
98	C₁₃×Q₁₆	—	2	2	no	—
99	C₁₃×Q₁₆	—	2	2	no	—
100	C₁₃×Q₁₆	—	2	1	no	—
101	C₁₃×Q₁₆	—	4	4	no	—
102	C₁₃×Q₁₆	—	4	2	no	—
103	C₁₃×Q₁₆	C₂×D₄	1	4	yes	—
104	C₁₃×SD₁₆	—	4	4	no	—
105	C₁₃×SD₁₆	—	4	2	no	—
106	C₁₃×D₈	—	2	1	no	—
107	C₁₃×D₈	—	2	1	no	—
108	C₁₃×M₁₆	—	2	2	no	—
109	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₁₃	#41	1	2	yes	—
113	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
114	C₁₃×C₄⋊C₄	—	2	2	no	—
115	C₁₃×C₄⋊C₄	—	4	4	no	—
116	C₁₃×C₄⋊C₄	—	4	4	no	—
117	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
118	C₁₃×C₂²⋊C₄	—	4	1	no	—
119	C₁₃×C₂²⋊C₄	—	4	4	no	—
120	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
121	C₄²×C₁₃	—	4	1	no	—
122	C₄²×C₁₃	—	4	4	no	—
123	C₄²×C₁₃	C₂×D₄	1	2	yes	—
124	C₁₃×C₁₆	—	2	2	no	—
125	C₁₃×C₁₆	—	2	2	no	—
126	C₂×#510	C₂×C₁₃	1	1	yes	characteristic
127	C₂²×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#3156	C₂×C₁₃	1	1	yes	characteristic
129	#2950	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2211	C₂×C₁₃	1	4	yes	complemented
131	#2209	C₂×C₁₃	1	4	yes	complemented
132	#2151	C₂×C₁₃	1	1	yes	characteristic, complemented
133	#2141	C₂×C₁₃	1	1	yes	characteristic, complemented
134	#2138	C₂×C₁₃	1	1	yes	characteristic
135	C₂³×C₁₃	—	4	1	no	—
136	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
137	Q₈×C₁₃	—	2	4	no	—
138	Q₈×C₁₃	—	4	2	no	—
139	Q₈×C₁₃	—	4	4	no	—
140	Q₈×C₁₃	—	4	1	no	—
141	Q₈×C₁₃	—	4	2	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	Q₈×C₁₃	—	8	2	no	—
145	D₄×C₁₃	—	4	1	no	—
146	D₄×C₁₃	—	4	1	no	—
147	D₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	4	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	8	2	no	—
154	C₂×C₄×C₁₃	—	8	2	no	—
155	C₂×C₄×C₁₃	#121	1	2	yes	—
156	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
157	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₈×C₁₃	—	2	2	no	—
161	C₈×C₁₃	#121	1	2	yes	—
162	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
164	C₂×#137	—	2	1	no	—
165	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
166	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
167	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
168	#536	—	2	1	no	—
169	#519	—	2	1	no	—
170	#510	—	2	4	no	—
171	C₂×Q₃₂	—	2	4	no	—
172	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	—
173	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
174	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
175	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
176	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
177	#497	—	2	2	no	—
178	#495	C₂²×C₁₃	1	2	yes	—
179	#494	C₂²×C₁₃	1	2	yes	—
180	#441	—	2	2	no	—
181	C₄×Q₁₆	C₂²×C₁₃	1	4	yes	—
182	#427	C₂²×C₁₃	1	1	yes	characteristic
183	#419	—	2	1	no	—
184	#405	C₂²×C₁₃	1	1	yes	characteristic
185	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	—
186	#358	C₂²×C₁₃	1	2	yes	—
187	#358	C₂²×C₁₃	1	4	yes	—
188	#357	C₂²×C₁₃	1	2	yes	—
189	C₂²×C₁₃	—	2	2	no	—
190	C₂²×C₁₃	—	8	2	no	—
191	C₂²×C₁₃	—	8	1	no	—
192	C₂²×C₁₃	#459	1	2	yes	—
193	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
194	C₄×C₁₃	—	2	4	no	—
195	C₄×C₁₃	—	4	4	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	1	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	#459	1	2	yes	—
200	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
201	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₂×#41	—	2	1	no	—
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×Q₈	—	2	1	no	—
205	#137	—	4	4	no	—
206	#135	—	2	2	no	—
207	#135	—	2	2	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×Q₁₆	—	2	2	no	—
214	C₂×Q₁₆	D₄×C₁₃	1	4	yes	—
215	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
216	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
217	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
218	C₂×SD₁₆	—	2	1	no	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
221	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
223	C₄⋊Q₈	—	2	2	no	—
224	#124	—	2	2	no	—
225	#123	—	4	2	no	—
226	C₂²⋊Q₈	—	4	2	no	—
227	C₄×Q₈	—	2	4	no	—
228	C₄×Q₈	—	4	2	no	—
229	#117	—	2	1	no	—
230	#117	—	4	2	no	—
231	#117	D₄×C₁₃	1	1	yes	characteristic
232	Q₃₂	—	2	8	no	—
233	Q₃₂	—	2	4	no	—
234	SD₃₂	—	2	4	no	—
235	M₃₂	—	2	2	no	—
236	C₂×C₁₆	—	2	2	no	—
237	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
238	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
239	#107	C₂³×C₁₃	1	2	yes	—
240	#106	—	2	1	no	—
241	#103	—	2	4	no	—
242	#97	D₄×C₁₃	1	2	yes	—
243	C₄×C₈	C₂³×C₁₃	1	2	yes	—
244	C₂×C₁₃	—	2	2	no	—
245	C₂×C₁₃	—	16	1	no	—
246	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
247	C₂×C₁₃	#3417	1	2	yes	—
248	#41	—	4	2	no	—
249	#41	—	4	2	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	4	4	no	—
272	Q₁₆	—	4	2	no	—
273	Q₁₆	C₂×D₄×C₁₃	1	4	yes	—
274	SD₁₆	—	4	4	no	—
275	SD₁₆	—	4	2	no	—
276	D₈	—	2	1	no	—
277	D₈	—	2	1	no	—
278	M₁₆	—	2	2	no	—
279	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
282	C₂×C₈	C₁₃×#41	1	2	yes	—
283	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	4	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
291	C₄²	—	4	1	no	—
292	C₄²	—	4	4	no	—
293	C₄²	C₂×D₄×C₁₃	1	2	yes	—
294	C₁₆	—	2	2	no	—
295	C₁₆	—	2	2	no	—
296	C₁₃	#33588	1	1	yes	characteristic, direct product factor, Sylow
297	C₂³	—	4	1	no	—
298	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
299	Q₈	—	2	4	no	—
300	Q₈	—	4	4	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	1	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	8	2	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	C₁₃×#121	1	2	yes	—
318	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
319	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
320	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
321	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	C₁₃×#121	1	2	yes	—
324	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
325	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
326	C₂²	—	2	2	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₁₃×#459	1	2	yes	—
330	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
331	C₄	—	2	4	no	—
332	C₄	—	4	4	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	C₁₃×#459	1	2	yes	—
337	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
338	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	16	1	no	—
341	C₂	C₂×C₁₃×#459	1	1	yes	characteristic
342	C₂	C₁₃×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₃×Q₃₂	C₂	semidirect
4. C₁₃×#2950	C₂	semidirect
5. C₁₃×#2211	C₂	semidirect
6. C₁₃×#2209	C₂	semidirect
7. C₁₃×#2151	C₂	semidirect
8. C₁₃×#2141	C₂	semidirect
77. #33588	C₁₃	direct (N × H)
127. C₂²×Q₃₂	C₂×C₁₃	semidirect
129. #2950	C₂×C₁₃	semidirect
130. #2211	C₂×C₁₃	semidirect
131. #2209	C₂×C₁₃	semidirect
132. #2151	C₂×C₁₃	semidirect
133. #2141	C₂×C₁₃	semidirect
296. C₁₃	#33588	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×#510	2
C₂²×C₁₃×Q₃₂	2
C₁₃×#3156	2
C₁₃×#2950	2
C₁₃×#2211	2
C₁₃×#2209	2
C₁₃×#2151	2
C₁₃×#2141	2
C₁₃×#2138	2
#33588	13

Subgroups — C13×#33588 #15719927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33588 #15719927