Subgroups — C₁₃×#12815 #15699154

Order 3328.

422 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#359	C₂	1	2	yes	maximal
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1382	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
7	C₄×C₁₃×#41	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
10	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
11	C₄×C₁₃×SD₁₆	—	2	1	no	—
12	C₄×C₁₃×SD₁₆	—	2	1	no	—
13	C₄×C₁₃×D₈	C₄	1	1	yes	characteristic
14	C₄×C₁₃×D₈	C₄	1	1	yes	characteristic
15	C₂×C₁₃×#108	—	2	1	no	—
16	C₂×C₁₃×#108	C₂²	1	2	yes	—
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₁₃×#423	—	2	1	no	—
19	C₁₃×#416	—	2	1	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×C₄⋊C₁₆	—	2	1	no	—
22	C₁₃×C₄⋊C₁₆	—	2	1	no	—
23	C₁₃×#359	—	2	4	no	—
24	C₂×C₁₃×#41	—	2	1	no	—
25	C₁₃×#135	—	2	1	no	—
26	C₁₃×#135	—	2	1	no	—
27	C₁₃×#135	D₄	1	2	yes	—
28	C₁₃×#135	Q₈	1	2	yes	—
29	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×SD₁₆	—	2	1	no	—
31	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×M₁₆	—	2	2	no	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₃	—	2	1	no	—
36	C₄×Q₈×C₁₃	—	2	1	no	—
37	C₄×D₄×C₁₃	—	2	1	no	—
38	C₄×D₄×C₁₃	—	2	1	no	—
39	C₄×D₄×C₁₃	—	4	1	no	—
40	C₄×D₄×C₁₃	—	4	1	no	—
41	C₁₃×#117	—	2	1	no	—
42	C₁₃×#117	—	4	1	no	—
43	C₂×C₄²×C₁₃	—	4	1	no	—
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₂×C₁₃×C₁₆	—	2	2	no	—
46	C₂×C₁₃×C₁₆	—	2	2	no	—
47	C₂×C₁₃×C₁₆	—	2	2	no	—
48	C₁₃×#108	—	2	2	no	—
49	C₁₃×#108	—	2	2	no	—
50	C₁₃×#108	—	4	2	no	—
51	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
52	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
53	C₁₃×#106	—	2	1	no	—
54	C₁₃×C₄⋊C₈	—	4	1	no	—
55	C₁₃×C₄⋊C₈	—	4	1	no	—
56	C₁₃×#103	—	2	1	no	—
57	C₁₃×#103	—	2	1	no	—
58	C₁₃×#102	—	2	1	no	—
59	C₁₃×#102	—	2	1	no	—
60	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	#12815	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₁₃×#41	—	2	2	no	—
66	C₁₃×#41	—	2	2	no	—
67	C₁₃×#41	—	4	1	no	—
68	C₁₃×#41	—	4	1	no	—
69	C₂×Q₈×C₁₃	—	2	1	no	—
70	C₂×D₄×C₁₃	—	2	1	no	—
71	C₂×D₄×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	—	4	1	no	—
74	C₂²×C₄×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₁₃×Q₁₆	—	2	1	no	—
79	C₁₃×Q₁₆	C₄⋊C₄	1	2	yes	—
80	C₁₃×SD₁₆	—	2	2	no	—
81	C₁₃×SD₁₆	—	2	2	no	—
82	C₁₃×D₈	—	2	1	no	—
83	C₁₃×D₈	C₄⋊C₄	1	2	yes	—
84	C₁₃×M₁₆	—	4	2	no	—
85	C₁₃×M₁₆	—	4	2	no	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	4	2	no	—
89	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	1	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₂²⋊C₄	—	4	1	no	—
102	C₁₃×C₂²⋊C₄	—	4	1	no	—
103	C₁₃×C₂²⋊C₄	—	4	1	no	—
104	C₄²×C₁₃	—	4	1	no	—
105	C₄²×C₁₃	—	4	1	no	—
106	C₄²×C₁₃	—	4	1	no	—
107	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
109	C₄²×C₁₃	D₈	1	1	yes	characteristic
110	C₄²×C₁₃	D₈	1	1	yes	characteristic
111	C₁₃×C₁₆	—	4	2	no	—
112	C₄×#135	C₂×C₁₃	1	1	yes	characteristic
113	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂×#359	C₂×C₁₃	1	2	yes	—
115	#1848	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#1382	C₂×C₁₃	1	2	yes	complemented
117	C₂³×C₁₃	—	4	1	no	—
118	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
119	Q₈×C₁₃	—	2	2	no	—
120	Q₈×C₁₃	—	4	1	no	—
121	D₄×C₁₃	—	2	2	no	—
122	D₄×C₁₃	—	4	2	no	—
123	D₄×C₁₃	—	4	2	no	—
124	D₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	—	4	1	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	8	2	no	—
133	C₂×C₄×C₁₃	—	8	2	no	—
134	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	8	2	no	—
147	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₈×C₁₃	#117	1	1	yes	characteristic
149	C₈×C₁₃	#121	1	1	yes	characteristic
150	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
152	C₄×#41	—	2	1	no	—
153	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
154	C₄×Q₁₆	C₄×C₁₃	1	1	yes	characteristic
155	C₄×Q₁₆	C₄×C₁₃	1	1	yes	characteristic
156	C₄×SD₁₆	—	2	1	no	—
157	C₄×SD₁₆	—	2	1	no	—
158	C₄×D₈	C₄×C₁₃	1	1	yes	characteristic
159	C₄×D₈	C₄×C₁₃	1	1	yes	characteristic
160	C₂×#108	—	2	1	no	—
161	C₂×#108	C₂²×C₁₃	1	2	yes	—
162	#427	C₂²×C₁₃	1	1	yes	characteristic
163	#423	—	2	1	no	—
164	#416	—	2	1	no	—
165	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
166	C₄⋊C₁₆	—	2	1	no	—
167	C₄⋊C₁₆	—	2	1	no	—
168	#359	—	2	4	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	4	2	no	—
172	C₂²×C₁₃	—	8	1	no	—
173	C₂²×C₁₃	—	8	1	no	—
174	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
175	C₂²×C₁₃	#461	1	1	yes	characteristic
176	C₂²×C₁₃	#476	1	1	yes	characteristic
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	4	2	no	—
181	C₄×C₁₃	—	8	1	no	—
182	C₄×C₁₃	—	8	1	no	—
183	C₄×C₁₃	#359	1	2	yes	complemented
184	C₄×C₁₃	#380	1	1	yes	characteristic
185	C₄×C₁₃	#417	1	1	yes	characteristic
186	C₄×C₁₃	#459	1	1	yes	characteristic
187	C₄×C₁₃	#474	1	1	yes	characteristic
188	C₂×#41	—	2	1	no	—
189	#135	—	2	1	no	—
190	#135	—	2	1	no	—
191	#135	D₄×C₁₃	1	2	yes	—
192	#135	Q₈×C₁₃	1	2	yes	—
193	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
194	C₂×SD₁₆	—	2	1	no	—
195	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
196	C₂×M₁₆	—	2	2	no	—
197	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
199	C₄×Q₈	—	2	1	no	—
200	C₄×Q₈	—	2	1	no	—
201	C₄×D₄	—	2	1	no	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	4	1	no	—
204	C₄×D₄	—	4	1	no	—
205	#117	—	2	1	no	—
206	#117	—	4	1	no	—
207	C₂×C₄²	—	4	1	no	—
208	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₁₆	—	2	2	no	—
210	C₂×C₁₆	—	2	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	#108	—	2	2	no	—
213	#108	—	2	2	no	—
214	#108	—	4	2	no	—
215	#107	C₂×C₄×C₁₃	1	1	yes	characteristic
216	#107	C₂×C₄×C₁₃	1	1	yes	characteristic
217	#106	—	2	1	no	—
218	C₄⋊C₈	—	4	1	no	—
219	C₄⋊C₈	—	4	1	no	—
220	#103	—	2	1	no	—
221	#103	—	2	1	no	—
222	#102	—	2	1	no	—
223	#102	—	2	1	no	—
224	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
226	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
227	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	—	8	2	no	—
231	C₂×C₁₃	#1802	1	1	yes	characteristic
232	C₂×C₁₃	C₂×#359	1	1	yes	characteristic
233	C₂×C₁₃	#2143	1	1	yes	characteristic
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	#41	—	4	1	no	—
237	#41	—	4	1	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
245	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	Q₁₆	—	2	1	no	—
248	Q₁₆	C₁₃×C₄⋊C₄	1	2	yes	—
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	D₈	—	2	1	no	—
252	D₈	C₁₃×C₄⋊C₄	1	2	yes	—
253	M₁₆	—	4	2	no	—
254	M₁₆	—	4	2	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
264	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
265	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
277	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
278	C₄²	C₁₃×D₈	1	1	yes	characteristic
279	C₄²	C₁₃×D₈	1	1	yes	characteristic
280	C₁₆	—	4	2	no	—
281	C₁₃	#12815	1	1	yes	characteristic, direct product factor, Sylow
282	C₂³	—	4	1	no	—
283	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	4	1	no	—
286	D₄	—	2	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
300	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
301	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
302	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
304	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
305	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
306	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
307	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	8	2	no	—
312	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
313	C₈	C₁₃×#117	1	1	yes	characteristic
314	C₈	C₁₃×#121	1	1	yes	characteristic
315	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
322	C₂²	C₁₃×#461	1	1	yes	characteristic
323	C₂²	C₁₃×#476	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	1	no	—
330	C₄	C₁₃×#359	1	2	yes	complemented
331	C₄	C₁₃×#380	1	1	yes	characteristic
332	C₄	C₁₃×#417	1	1	yes	characteristic
333	C₄	C₁₃×#459	1	1	yes	characteristic
334	C₄	C₁₃×#474	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	8	2	no	—
338	C₂	C₁₃×#1802	1	1	yes	characteristic
339	C₂	C₂×C₁₃×#359	1	1	yes	characteristic
340	C₂	C₁₃×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×C₄⋊C₁₆	C₂	semidirect
4. C₁₃×#1848	C₂	semidirect
5. C₁₃×#1382	C₂	semidirect
64. #12815	C₁₃	direct (N × H)
113. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
115. #1848	C₂×C₁₃	semidirect
116. #1382	C₂×C₁₃	semidirect
183. C₄×C₁₃	#359	semidirect
281. C₁₃	#12815	direct (N × H)
330. C₄	C₁₃×#359	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#135	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#359	2
C₁₃×#1848	2
C₁₃×#1382	2
#12815	13

Subgroups — C13×#12815 #15699154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12815 #15699154