Subgroups — C₁₃×#9197 #15695536

Order 3328.

624 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2905	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2619	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1545	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1458	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1286	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×M₁₆	—	2	1	no	—
7	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#121	—	2	1	no	—
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₁₃×#431	C₂²	1	2	yes	—
11	C₁₃×#424	—	2	4	no	—
12	C₁₃×#423	—	2	1	no	—
13	C₁₃×#423	C₂²	1	2	yes	—
14	C₁₃×#394	—	2	1	no	—
15	C₁₃×#394	C₄	1	2	yes	—
16	C₁₃×#392	—	2	1	no	—
17	C₁₃×#392	C₄	1	2	yes	—
18	C₁₃×#390	—	2	1	no	—
19	C₁₃×#382	C₂²	1	1	yes	characteristic
20	C₁₃×#331	—	2	4	no	—
21	C₁₃×#324	—	2	1	no	—
22	C₁₃×#323	—	2	1	no	—
23	C₂²×D₄×C₁₃	—	2	2	no	—
24	C₂²×D₄×C₁₃	—	2	1	no	—
25	C₂²×D₄×C₁₃	—	2	1	no	—
26	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	2	no	—
30	C₂×C₁₃×M₁₆	—	4	2	no	—
31	C₂×C₁₃×M₁₆	—	4	2	no	—
32	C₂²×C₈×C₁₃	—	2	2	no	—
33	C₁₃×#127	—	2	2	no	—
34	C₁₃×#127	—	2	2	no	—
35	C₁₃×#121	—	4	2	no	—
36	C₁₃×#121	—	4	2	no	—
37	C₁₃×#117	—	2	2	no	—
38	C₁₃×#117	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
42	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊C₈	—	2	4	no	—
50	C₁₃×C₄⋊C₈	—	2	2	no	—
51	C₁₃×C₄⋊C₈	—	2	2	no	—
52	C₁₃×C₄⋊C₈	D₄	1	4	yes	complemented
53	C₁₃×C₂²⋊C₈	—	2	2	no	—
54	C₄×C₈×C₁₃	—	2	2	no	—
55	C₁₃×#95	—	2	1	no	—
56	C₁₃×#95	C₂×C₄	1	2	yes	—
57	#9197	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	—	2	2	no	—
59	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₄×C₁₃	—	2	4	no	—
61	C₂×D₄×C₁₃	—	4	4	no	—
62	C₂×D₄×C₁₃	—	4	1	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	4	1	no	—
65	C₂×D₄×C₁₃	—	4	1	no	—
66	C₂×D₄×C₁₃	—	4	1	no	—
67	C₂×D₄×C₁₃	—	4	1	no	—
68	C₂×D₄×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	8	2	no	—
70	C₂×D₄×C₁₃	—	8	2	no	—
71	C₂×D₄×C₁₃	—	8	2	no	—
72	C₂×D₄×C₁₃	—	8	2	no	—
73	C₂²×C₄×C₁₃	—	2	1	no	—
74	C₂²×C₄×C₁₃	—	2	1	no	—
75	C₂²×C₄×C₁₃	—	2	1	no	—
76	C₂²×C₄×C₁₃	—	2	1	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	D₈	1	2	yes	—
82	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₁₃×M₁₆	—	4	4	no	—
87	C₁₃×M₁₆	—	4	2	no	—
88	C₁₃×M₁₆	—	4	2	no	—
89	C₂×C₈×C₁₃	—	2	4	no	—
90	C₂×C₈×C₁₃	—	4	4	no	—
91	C₂×C₈×C₁₃	—	4	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₂²⋊C₄	—	2	2	no	—
97	C₁₃×C₂²⋊C₄	—	2	2	no	—
98	C₁₃×C₂²⋊C₄	—	4	4	no	—
99	C₁₃×C₂²⋊C₄	—	8	2	no	—
100	C₁₃×C₂²⋊C₄	—	8	2	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	C₂×D₄	1	2	yes	—
104	C₄²×C₁₃	#41	1	2	yes	—
105	#2905	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#2619	C₂×C₁₃	1	1	yes	characteristic
107	#1545	C₂×C₁₃	1	2	yes	complemented
108	#1458	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#1286	C₂×C₁₃	1	2	yes	complemented
110	C₂³×C₁₃	—	2	1	no	—
111	C₂³×C₁₃	—	2	1	no	—
112	C₂³×C₁₃	—	4	1	no	—
113	C₂³×C₁₃	—	4	1	no	—
114	C₂³×C₁₃	—	4	2	no	—
115	C₂³×C₁₃	—	4	2	no	—
116	C₂³×C₁₃	—	4	2	no	—
117	C₂³×C₁₃	—	4	2	no	—
118	C₂³×C₁₃	—	4	2	no	—
119	C₂³×C₁₃	—	4	2	no	—
120	C₂³×C₁₃	—	4	2	no	—
121	C₂³×C₁₃	#102	1	2	yes	—
122	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	D₄×C₁₃	—	8	4	no	—
124	D₄×C₁₃	—	8	4	no	—
125	D₄×C₁₃	—	8	8	no	—
126	D₄×C₁₃	—	8	1	no	—
127	D₄×C₁₃	—	8	1	no	—
128	D₄×C₁₃	—	8	1	no	—
129	D₄×C₁₃	—	8	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	4	1	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	#102	1	2	yes	—
146	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#121	1	2	yes	—
149	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
152	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
154	C₈×C₁₃	—	4	4	no	—
155	C₈×C₁₃	—	8	2	no	—
156	C₂²×M₁₆	—	2	1	no	—
157	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#121	—	2	1	no	—
159	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
160	#431	C₂²×C₁₃	1	2	yes	—
161	#424	—	2	4	no	—
162	#423	—	2	1	no	—
163	#423	C₂²×C₁₃	1	2	yes	—
164	#394	—	2	1	no	—
165	#394	C₄×C₁₃	1	2	yes	—
166	#392	—	2	1	no	—
167	#392	C₄×C₁₃	1	2	yes	—
168	#390	—	2	1	no	—
169	#382	C₂²×C₁₃	1	1	yes	characteristic
170	#331	—	2	4	no	—
171	#324	—	2	1	no	—
172	#323	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	2	no	—
176	C₂²×C₁₃	—	4	2	no	—
177	C₂²×C₁₃	—	4	1	no	—
178	C₂²×C₁₃	—	4	1	no	—
179	C₂²×C₁₃	—	8	1	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	8	2	no	—
186	C₂²×C₁₃	—	8	2	no	—
187	C₂²×C₁₃	#386	1	1	yes	characteristic
188	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
189	C₂²×C₁₃	#417	1	1	yes	characteristic
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	4	1	no	—
194	C₄×C₁₃	—	4	1	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	#447	1	1	yes	characteristic
197	C₄×C₁₃	#448	1	1	yes	characteristic
198	C₄×C₁₃	#480	1	1	yes	characteristic
199	C₄×C₁₃	#481	1	1	yes	characteristic
200	C₂²×D₄	—	2	2	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	—	2	1	no	—
203	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
204	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	4	2	no	—
208	C₂×M₁₆	—	4	2	no	—
209	C₂²×C₈	—	2	2	no	—
210	#127	—	2	2	no	—
211	#127	—	2	2	no	—
212	#121	—	4	2	no	—
213	#121	—	4	2	no	—
214	#117	—	2	2	no	—
215	#117	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
218	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
224	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
226	C₄⋊C₈	—	2	4	no	—
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	D₄×C₁₃	1	4	yes	complemented
230	C₂²⋊C₈	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	#95	—	2	1	no	—
233	#95	C₂×C₄×C₁₃	1	2	yes	—
234	C₂×C₁₃	—	2	1	no	—
235	C₂×C₁₃	—	2	1	no	—
236	C₂×C₁₃	—	4	2	no	—
237	C₂×C₁₃	—	16	2	no	—
238	C₂×C₁₃	#1874	1	1	yes	characteristic
239	C₂×C₁₃	#1910	1	1	yes	characteristic
240	C₂×C₁₃	#1914	1	1	yes	characteristic
241	C₂⁴	—	2	2	no	—
242	C₂⁴	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	8	2	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂×D₄	—	8	2	no	—
255	C₂×D₄	—	8	2	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
262	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
263	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
264	C₂²×C₄	C₁₃×D₈	1	2	yes	—
265	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
266	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
269	M₁₆	—	4	4	no	—
270	M₁₆	—	4	2	no	—
271	M₁₆	—	4	2	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	4	4	no	—
274	C₂×C₈	—	4	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	8	2	no	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	C₂×D₄×C₁₃	1	2	yes	—
287	C₄²	C₁₃×#41	1	2	yes	—
288	C₁₃	#9197	1	1	yes	characteristic, direct product factor, Sylow
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	C₁₃×#102	1	2	yes	—
301	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
302	D₄	—	8	4	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	8	no	—
305	D₄	—	8	1	no	—
306	D₄	—	8	1	no	—
307	D₄	—	8	1	no	—
308	D₄	—	8	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	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	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	C₁₃×#102	1	2	yes	—
325	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
326	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
327	C₂×C₄	C₁₃×#121	1	2	yes	—
328	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
329	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
330	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
331	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
332	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
333	C₈	—	4	4	no	—
334	C₈	—	8	2	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	C₁₃×#386	1	1	yes	characteristic
350	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
351	C₂²	C₁₃×#417	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	8	2	no	—
358	C₄	C₁₃×#447	1	1	yes	characteristic
359	C₄	C₁₃×#448	1	1	yes	characteristic
360	C₄	C₁₃×#480	1	1	yes	characteristic
361	C₄	C₁₃×#481	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	4	2	no	—
365	C₂	—	16	2	no	—
366	C₂	C₁₃×#1874	1	1	yes	characteristic
367	C₂	C₁₃×#1910	1	1	yes	characteristic
368	C₂	C₁₃×#1914	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₁₃×#2619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2905	C₂	semidirect
3. C₁₃×#1545	C₂	semidirect
4. C₁₃×#1458	C₂	semidirect
5. C₁₃×#1286	C₂	semidirect
52. C₁₃×C₄⋊C₈	D₄	semidirect
57. #9197	C₁₃	direct (N × H)
105. #2905	C₂×C₁₃	semidirect
107. #1545	C₂×C₁₃	semidirect
108. #1458	C₂×C₁₃	semidirect
109. #1286	C₂×C₁₃	semidirect
229. C₄⋊C₈	D₄×C₁₃	semidirect
288. C₁₃	#9197	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×#2905	2
C₁₃×#2619	2
C₁₃×#1545	2
C₁₃×#1458	2
C₁₃×#1286	2
#9197	13

Subgroups — C13×#9197 #15695536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9197 #15695536