Subgroups — C₁₃×#13237 #15699576

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1970	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1969	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×SD₃₂	—	2	2	no	—
11	C₂×C₁₃×SD₃₂	—	2	2	no	—
12	C₂×C₁₃×SD₃₂	C₄	1	4	yes	complemented
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#103	—	2	1	no	—
18	C₂×C₁₃×#102	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#391	—	2	1	no	—
21	C₁₃×#390	—	2	1	no	—
22	C₁₃×#367	—	2	2	no	—
23	C₁₃×#358	—	2	2	no	—
24	C₁₃×#357	—	2	2	no	—
25	C₄×C₁₃×C₁₆	—	2	2	no	—
26	C₂²×Q₈×C₁₃	—	2	1	no	—
27	C₂²×D₄×C₁₃	—	2	1	no	—
28	C₂×C₁₃×Q₁₆	—	2	1	no	—
29	C₂×C₁₃×Q₁₆	—	2	1	no	—
30	C₂×C₁₃×Q₁₆	—	2	1	no	—
31	C₂×C₁₃×Q₁₆	—	2	1	no	—
32	C₂×C₁₃×Q₁₆	—	2	1	no	—
33	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×D₈	—	2	1	no	—
35	C₂×C₁₃×D₈	—	2	1	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	—	2	1	no	—
39	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
45	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
46	C₁₃×SD₃₂	—	2	8	no	—
47	C₂×C₁₃×C₁₆	—	2	1	no	—
48	C₂×C₁₃×C₁₆	—	2	1	no	—
49	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
50	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
51	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
52	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
53	C₁₃×#107	—	2	1	no	—
54	C₁₃×#107	—	2	1	no	—
55	C₁₃×#107	—	2	1	no	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#103	—	4	2	no	—
58	C₁₃×#102	—	4	2	no	—
59	C₄×C₈×C₁₃	—	2	1	no	—
60	C₄×C₈×C₁₃	—	2	1	no	—
61	C₁₃×#95	—	4	2	no	—
62	#13237	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	—	4	1	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	—	4	1	no	—
66	C₂×Q₈×C₁₃	—	4	1	no	—
67	C₂×Q₈×C₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×Q₈×C₁₃	—	4	1	no	—
70	C₂×D₄×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	4	1	no	—
72	C₂×D₄×C₁₃	—	4	1	no	—
73	C₂×D₄×C₁₃	—	4	1	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	D₈	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₁₃×Q₁₆	—	2	2	no	—
82	C₁₃×Q₁₆	—	2	2	no	—
83	C₁₃×Q₁₆	—	4	1	no	—
84	C₁₃×Q₁₆	—	4	1	no	—
85	C₁₃×D₈	—	2	2	no	—
86	C₁₃×D₈	—	2	2	no	—
87	C₁₃×D₈	—	4	1	no	—
88	C₁₃×D₈	—	4	1	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	D₈	1	2	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
97	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₂²⋊C₄	—	8	4	no	—
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₁₃×C₁₆	—	2	2	no	—
106	C₁₃×C₁₆	C₂²⋊C₄	1	4	yes	complemented
107	C₂²×SD₃₂	C₂×C₁₃	1	1	yes	characteristic
108	C₂×#367	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂×#358	C₂×C₁₃	1	1	yes	characteristic, complemented
110	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
111	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#1970	C₂×C₁₃	1	1	yes	characteristic
113	#1969	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂³×C₁₃	—	4	2	no	—
115	C₂³×C₁₃	—	8	1	no	—
116	C₂³×C₁₃	—	8	1	no	—
117	C₂³×C₁₃	—	8	2	no	—
118	C₂³×C₁₃	—	8	2	no	—
119	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₁₃	—	4	2	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	8	1	no	—
123	Q₈×C₁₃	—	8	1	no	—
124	D₄×C₁₃	—	4	2	no	—
125	D₄×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	8	1	no	—
127	D₄×C₁₃	—	8	1	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	8	1	no	—
132	C₂×C₄×C₁₃	—	8	1	no	—
133	C₂×C₄×C₁₃	—	8	1	no	—
134	C₂×C₄×C₁₃	—	8	1	no	—
135	C₂×C₄×C₁₃	—	8	2	no	—
136	C₂×C₄×C₁₃	—	8	2	no	—
137	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
138	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
144	C₈×C₁₃	—	2	2	no	—
145	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₈×C₁₃	#121	1	1	yes	characteristic
148	C₈×C₁₃	#124	1	1	yes	characteristic
149	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
150	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×SD₃₂	—	2	2	no	—
152	C₂×SD₃₂	—	2	2	no	—
153	C₂×SD₃₂	C₄×C₁₃	1	4	yes	complemented
154	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
155	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#103	—	2	1	no	—
159	C₂×#102	—	2	1	no	—
160	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
161	#391	—	2	1	no	—
162	#390	—	2	1	no	—
163	#367	—	2	2	no	—
164	#358	—	2	2	no	—
165	#357	—	2	2	no	—
166	C₄×C₁₆	—	2	2	no	—
167	C₂²×C₁₃	—	8	4	no	—
168	C₂²×C₁₃	—	8	2	no	—
169	C₂²×C₁₃	—	16	2	no	—
170	C₂²×C₁₃	—	16	1	no	—
171	C₂²×C₁₃	—	16	1	no	—
172	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
173	C₂²×C₁₃	#466	1	1	yes	characteristic
174	C₂²×C₁₃	#493	1	1	yes	characteristic
175	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
176	C₂²×C₁₃	#508	1	2	yes	—
177	C₄×C₁₃	—	2	4	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	—	16	1	no	—
180	C₄×C₁₃	—	16	1	no	—
181	C₄×C₁₃	#390	1	1	yes	characteristic
182	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
183	C₄×C₁₃	#466	1	1	yes	characteristic
184	C₄×C₁₃	#495	1	1	yes	characteristic
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
199	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₂²⋊C₄	—	4	2	no	—
204	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
205	SD₃₂	—	2	8	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
209	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
210	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
211	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
212	#107	—	2	1	no	—
213	#107	—	2	1	no	—
214	#107	—	2	1	no	—
215	#107	—	2	1	no	—
216	#103	—	4	2	no	—
217	#102	—	4	2	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	#95	—	4	2	no	—
221	C₂×C₁₃	—	16	2	no	—
222	C₂×C₁₃	#1970	1	1	yes	characteristic
223	C₂×C₁₃	C₄×SD₃₂	1	2	yes	—
224	C₂×C₁₃	#2217	1	2	yes	—
225	C₂×C₁₃	#2248	1	1	yes	characteristic
226	C₂×C₁₃	#2249	1	1	yes	characteristic
227	C₂⁴	—	4	1	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	C₁₃×D₈	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
245	Q₁₆	—	2	2	no	—
246	Q₁₆	—	2	2	no	—
247	Q₁₆	—	4	1	no	—
248	Q₁₆	—	4	1	no	—
249	D₈	—	2	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	4	1	no	—
252	D₈	—	4	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	C₁₃×D₈	1	2	yes	—
256	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
261	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	8	4	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₁₆	—	2	2	no	—
270	C₁₆	C₁₃×C₂²⋊C₄	1	4	yes	complemented
271	C₁₃	#13237	1	1	yes	characteristic, direct product factor, Sylow
272	C₂³	—	4	2	no	—
273	C₂³	—	8	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	—	8	2	no	—
276	C₂³	—	8	2	no	—
277	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	8	1	no	—
281	Q₈	—	8	1	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	1	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	C₁₃×SD₃₂	1	4	yes	complemented
296	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
297	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
298	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
299	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
300	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
301	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
304	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
305	C₈	C₁₃×#121	1	1	yes	characteristic
306	C₈	C₁₃×#124	1	1	yes	characteristic
307	C₂²	—	8	4	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	16	1	no	—
310	C₂²	—	16	1	no	—
311	C₂²	—	16	2	no	—
312	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
313	C₂²	C₁₃×#466	1	1	yes	characteristic
314	C₂²	C₁₃×#493	1	1	yes	characteristic
315	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
316	C₂²	C₁₃×#508	1	2	yes	—
317	C₄	—	2	4	no	—
318	C₄	—	8	2	no	—
319	C₄	—	16	1	no	—
320	C₄	—	16	1	no	—
321	C₄	C₁₃×#390	1	1	yes	characteristic
322	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
323	C₄	C₁₃×#466	1	1	yes	characteristic
324	C₄	C₁₃×#495	1	1	yes	characteristic
325	C₂	—	16	2	no	—
326	C₂	C₁₃×#1970	1	1	yes	characteristic
327	C₂	C₄×C₁₃×SD₃₂	1	2	yes	—
328	C₂	C₁₃×#2217	1	2	yes	—
329	C₂	C₁₃×#2248	1	1	yes	characteristic
330	C₂	C₁₃×#2249	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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×#367	C₂	semidirect
3. C₂×C₁₃×#358	C₂	semidirect
5. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
7. C₁₃×#1969	C₂	semidirect
12. C₂×C₁₃×SD₃₂	C₄	semidirect
62. #13237	C₁₃	direct (N × H)
106. C₁₃×C₁₆	C₂²⋊C₄	semidirect
108. C₂×#367	C₂×C₁₃	semidirect
109. C₂×#358	C₂×C₁₃	semidirect
111. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
113. #1969	C₂×C₁₃	semidirect
137. C₂×C₄×C₁₃	SD₃₂	semidirect
153. C₂×SD₃₂	C₄×C₁₃	semidirect
270. C₁₆	C₁₃×C₂²⋊C₄	semidirect
271. C₁₃	#13237	direct (N × H)
295. C₂×C₄	C₁₃×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×SD₃₂	2
C₂×C₁₃×#367	2
C₂×C₁₃×#358	2
C₂×C₁₃×#357	2
C₂×C₄×C₁₃×C₁₆	2
C₁₃×#1970	2
C₁₃×#1969	2
#13237	13

Subgroups — C13×#13237 #15699576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13237 #15699576