Subgroups — C₁₃×#29358 #15715697

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2641	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2331	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2073	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2023	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1989	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1982	C₂	1	2	yes	maximal
7	C₁₃×#1936	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1934	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1768	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₃×#102	C₂²	1	2	yes	—
15	C₂×C₁₃×#102	C₂²	1	2	yes	—
16	C₂×C₁₃×#102	C₂²	1	2	yes	—
17	C₂×C₁₃×#102	C₂²	1	2	yes	—
18	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#394	—	2	2	no	—
21	C₁₃×#394	—	2	2	no	—
22	C₁₃×#391	—	2	2	no	—
23	C₁₃×#390	C₂²	1	2	yes	—
24	C₁₃×#390	C₂²	1	2	yes	—
25	C₁₃×#389	C₂²	1	2	yes	—
26	C₁₃×#387	—	2	2	no	—
27	C₁₃×#382	C₂²	1	1	yes	characteristic
28	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₁₃×#376	C₂²	1	1	yes	characteristic
30	C₁₃×#340	C₂²	1	2	yes	—
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	—	2	1	no	—
36	C₂²×D₄×C₁₃	—	2	2	no	—
37	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₁₃×C₄⋊Q₈	—	2	4	no	—
41	C₁₃×#127	—	2	4	no	—
42	C₁₃×#125	—	2	4	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
52	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
53	C₂×C₄²×C₁₃	—	2	2	no	—
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₁₃×C₄⋊C₈	—	2	4	no	—
58	C₁₃×#102	—	2	4	no	—
59	C₁₃×#102	—	2	4	no	—
60	C₁₃×#102	—	4	2	no	—
61	C₁₃×#102	—	4	2	no	—
62	C₁₃×#97	—	4	1	no	—
63	C₄×C₈×C₁₃	—	4	1	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	—	2	1	no	—
66	C₁₃×#95	—	2	1	no	—
67	C₁₃×#95	—	2	2	no	—
68	C₁₃×#95	D₄	1	2	yes	—
69	C₁₃×#95	D₄	1	2	yes	—
70	#29358	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂⁴×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	—	2	4	no	—
73	C₂×Q₈×C₁₃	—	8	2	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	4	4	no	—
81	C₂×D₄×C₁₃	—	8	2	no	—
82	C₂×D₄×C₁₃	—	8	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	—	2	2	no	—
85	C₂²×C₄×C₁₃	—	2	2	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	4	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	4	4	no	—
103	C₁₃×C₄⋊C₄	—	8	2	no	—
104	C₁₃×C₄⋊C₄	#41	1	4	yes	—
105	C₁₃×C₂²⋊C₄	—	8	2	no	—
106	C₁₃×C₂²⋊C₄	—	8	2	no	—
107	C₁₃×C₂²⋊C₄	—	8	2	no	—
108	C₁₃×C₂²⋊C₄	—	8	2	no	—
109	C₄²×C₁₃	—	2	2	no	—
110	C₄²×C₁₃	—	2	1	no	—
111	C₄²×C₁₃	—	2	1	no	—
112	C₄²×C₁₃	—	8	2	no	—
113	C₄²×C₁₃	#41	1	4	yes	—
114	#2641	C₂×C₁₃	1	1	yes	characteristic
115	#2331	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#2073	C₂×C₁₃	1	2	yes	complemented
117	#2023	C₂×C₁₃	1	2	yes	complemented
118	#1989	C₂×C₁₃	1	2	yes	complemented
119	#1982	C₂×C₁₃	1	2	yes	—
120	#1936	C₂×C₁₃	1	2	yes	complemented
121	#1934	C₂×C₁₃	1	2	yes	complemented
122	#1768	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₂³×C₁₃	—	4	2	no	—
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	—	4	2	no	—
126	C₂³×C₁₃	—	4	2	no	—
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	—	4	2	no	—
129	C₂³×C₁₃	—	4	2	no	—
130	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	Q₈×C₁₃	—	8	4	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	8	8	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	4	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	#124	1	4	yes	—
156	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
157	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
158	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
160	C₈×C₁₃	—	4	2	no	—
161	C₈×C₁₃	—	4	2	no	—
162	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
165	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
166	C₂×#102	C₂²×C₁₃	1	2	yes	—
167	C₂×#102	C₂²×C₁₃	1	2	yes	—
168	C₂×#102	C₂²×C₁₃	1	2	yes	—
169	C₂×#102	C₂²×C₁₃	1	2	yes	—
170	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
172	#394	—	2	2	no	—
173	#394	—	2	2	no	—
174	#391	—	2	2	no	—
175	#390	C₂²×C₁₃	1	2	yes	—
176	#390	C₂²×C₁₃	1	2	yes	—
177	#389	C₂²×C₁₃	1	2	yes	—
178	#387	—	2	2	no	—
179	#382	C₂²×C₁₃	1	1	yes	characteristic
180	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
181	#376	C₂²×C₁₃	1	1	yes	characteristic
182	#340	C₂²×C₁₃	1	2	yes	—
183	#340	C₂²×C₁₃	1	2	yes	—
184	#340	C₂²×C₁₃	1	2	yes	—
185	#340	C₂²×C₁₃	1	2	yes	—
186	#336	C₂²×C₁₃	1	2	yes	—
187	C₂²×C₁₃	—	8	2	no	—
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	—	8	2	no	—
191	C₂²×C₁₃	—	8	2	no	—
192	C₂²×C₁₃	—	8	2	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	#525	1	1	yes	characteristic
195	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
196	C₂²×C₁₃	#537	1	1	yes	characteristic
197	C₂²×C₁₃	#576	1	2	yes	—
198	C₂²×C₁₃	#577	1	2	yes	—
199	C₄×C₁₃	—	2	4	no	—
200	C₄×C₁₃	—	4	1	no	—
201	C₄×C₁₃	—	4	4	no	—
202	C₄×C₁₃	—	4	1	no	—
203	C₄×C₁₃	—	4	2	no	—
204	C₄×C₁₃	—	8	2	no	—
205	C₄×C₁₃	—	8	2	no	—
206	C₄×C₁₃	#556	1	2	yes	—
207	C₄×C₁₃	#559	1	2	yes	—
208	C₂²×Q₈	—	2	1	no	—
209	C₂²×D₄	—	2	2	no	—
210	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
211	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
212	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
213	C₄⋊Q₈	—	2	4	no	—
214	#127	—	2	4	no	—
215	#125	—	2	4	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
222	C₂×C₂²⋊C₄	—	2	2	no	—
223	C₂×C₂²⋊C₄	—	2	2	no	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
228	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
229	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
230	C₄⋊C₈	—	2	4	no	—
231	#102	—	2	4	no	—
232	#102	—	2	4	no	—
233	#102	—	4	2	no	—
234	#102	—	4	2	no	—
235	#97	—	4	1	no	—
236	C₄×C₈	—	4	1	no	—
237	#95	—	2	2	no	—
238	#95	—	2	1	no	—
239	#95	—	2	1	no	—
240	#95	—	2	2	no	—
241	#95	D₄×C₁₃	1	2	yes	—
242	#95	D₄×C₁₃	1	2	yes	—
243	C₂×C₁₃	—	16	2	no	—
244	C₂×C₁₃	#2592	1	1	yes	characteristic
245	C₂×C₁₃	#3122	1	1	yes	characteristic
246	C₂×C₁₃	#3126	1	1	yes	characteristic
247	C₂×C₁₃	#3136	1	2	yes	—
248	C₂×C₁₃	#3257	1	2	yes	—
249	C₂⁴	—	2	2	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	—	8	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	4	no	—
259	C₂×D₄	—	8	2	no	—
260	C₂×D₄	—	8	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
268	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	4	no	—
281	C₄⋊C₄	—	8	2	no	—
282	C₄⋊C₄	C₁₃×#41	1	4	yes	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₂²⋊C₄	—	8	2	no	—
285	C₂²⋊C₄	—	8	2	no	—
286	C₂²⋊C₄	—	8	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	8	2	no	—
291	C₄²	C₁₃×#41	1	4	yes	—
292	C₁₃	#29358	1	1	yes	characteristic, direct product factor, Sylow
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	C₂²×D₄×C₁₃	1	1	yes	characteristic
301	Q₈	—	8	4	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	8	8	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	C₁₃×#124	1	4	yes	—
326	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
327	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
328	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
329	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
330	C₈	—	4	2	no	—
331	C₈	—	4	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₁₃×#525	1	1	yes	characteristic
340	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
341	C₂²	C₁₃×#537	1	1	yes	characteristic
342	C₂²	C₁₃×#576	1	2	yes	—
343	C₂²	C₁₃×#577	1	2	yes	—
344	C₄	—	2	4	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	4	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	2	no	—
351	C₄	C₁₃×#556	1	2	yes	—
352	C₄	C₁₃×#559	1	2	yes	—
353	C₂	—	16	2	no	—
354	C₂	C₁₃×#2592	1	1	yes	characteristic
355	C₂	C₁₃×#3122	1	1	yes	characteristic
356	C₂	C₁₃×#3126	1	1	yes	characteristic
357	C₂	C₁₃×#3136	1	2	yes	—
358	C₂	C₁₃×#3257	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2331	C₂	semidirect
3. C₁₃×#2073	C₂	semidirect
4. C₁₃×#2023	C₂	semidirect
5. C₁₃×#1989	C₂	semidirect
7. C₁₃×#1936	C₂	semidirect
8. C₁₃×#1934	C₂	semidirect
9. C₁₃×#1768	C₂	semidirect
70. #29358	C₁₃	direct (N × H)
115. #2331	C₂×C₁₃	semidirect
116. #2073	C₂×C₁₃	semidirect
117. #2023	C₂×C₁₃	semidirect
118. #1989	C₂×C₁₃	semidirect
120. #1936	C₂×C₁₃	semidirect
121. #1934	C₂×C₁₃	semidirect
122. #1768	C₂×C₁₃	semidirect
292. C₁₃	#29358	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×#2641	2
C₁₃×#2331	2
C₁₃×#2073	2
C₁₃×#2023	2
C₁₃×#1989	2
C₁₃×#1982	2
C₁₃×#1936	2
C₁₃×#1934	2
C₁₃×#1768	2
#29358	13

Subgroups — C13×#29358 #15715697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29358 #15715697