Subgroups — #13123492

Order 1664.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718661	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718500	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718500	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718496	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718494	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718387	C₂	1	1	yes	maximal, characteristic
9	D₁₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
10	#11718381	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718381	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718369	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718366	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718350	C₂	1	1	yes	maximal, characteristic, complemented
15	#11718346	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#117	C₂²	1	1	yes	characteristic
18	#88180	C₂²	1	1	yes	characteristic
19	C₂×#6094	C₂²	1	1	yes	characteristic
20	C₂×#6094	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
23	#88146	C₄	1	1	yes	characteristic
24	#88146	C₄	1	1	yes	characteristic
25	C₄⋊C₄×D₁₃	C₄	1	1	yes	characteristic
26	C₄⋊C₄×D₁₃	C₄	1	1	yes	characteristic
27	C₂²⋊C₄×D₁₃	C₄	1	1	yes	characteristic
28	C₂²⋊C₄×D₁₃	C₄	1	1	yes	characteristic
29	#88131	C₄	1	1	yes	characteristic
30	#88131	C₄	1	1	yes	characteristic
31	#88126	C₂²	1	1	yes	characteristic
32	#88126	C₂²	1	1	yes	characteristic
33	C₄²×D₁₃	C₂²	1	1	yes	characteristic
34	C₄²×D₁₃	C₂²	1	1	yes	characteristic
35	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₃×#97	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
41	C₂²⋊#1055	C₂²	1	1	yes	characteristic
42	C₂²⋊#1055	C₂²	1	1	yes	characteristic
43	#88060	C₂²	1	1	yes	characteristic
44	#88060	C₂²	1	1	yes	characteristic
45	#88060	C₂²	1	1	yes	characteristic
46	#88060	C₂²	1	1	yes	characteristic
47	#88055	C₂²	1	1	yes	characteristic, complemented
48	#88055	C₂²	1	1	yes	characteristic, complemented
49	#88054	C₂²	1	1	yes	characteristic, complemented
50	#88054	C₂²	1	1	yes	characteristic, complemented
51	#88054	C₂²	1	1	yes	characteristic, complemented
52	#88054	C₂²	1	1	yes	characteristic, complemented
53	C₈×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
54	C₈×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
55	C₄⋊#1055	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊#1055	C₂²	1	1	yes	characteristic, complemented
57	#88043	C₂²	1	1	yes	characteristic, complemented
58	C₄×#1055	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₃	—	2	2	no	—
63	C₂×C₄×D₁₃	—	2	2	no	—
64	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
78	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
79	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂³	1	1	yes	characteristic
81	C₄²×C₁₃	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
84	#6103	C₂×C₄	1	1	yes	characteristic
85	#6103	C₂×C₄	1	1	yes	characteristic
86	#6103	C₂×C₄	1	1	yes	characteristic
87	#6103	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
89	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
90	#6101	C₂×C₄	1	1	yes	characteristic
91	#6101	C₂×C₄	1	1	yes	characteristic
92	#6101	C₂×C₄	1	1	yes	characteristic
93	#6101	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
100	C₂×#1055	C₂³	1	1	yes	characteristic
101	C₂×#1055	C₂³	1	1	yes	characteristic
102	C₂×#1055	C₂³	1	1	yes	characteristic
103	C₂×#1055	C₂³	1	1	yes	characteristic
104	#6094	—	2	2	no	—
105	#6094	—	2	2	no	—
106	C₈×D₁₃	—	2	2	no	—
107	C₈×D₁₃	—	2	2	no	—
108	#2924	—	13	1	no	maximal, Sylow
109	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₁₃	—	2	2	no	—
111	C₂²×D₁₃	—	2	2	no	—
112	C₂²×D₁₃	—	4	1	no	—
113	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₁₃	—	2	2	no	—
124	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
130	C₄×D₁₃	—	2	2	no	—
131	C₄×D₁₃	—	2	2	no	—
132	C₄×D₁₃	—	2	2	no	—
133	C₄×D₁₃	—	2	1	no	—
134	C₄×D₁₃	—	2	1	no	—
135	C₄×D₁₃	—	2	2	no	—
136	C₄×D₁₃	—	4	1	no	—
137	C₄×D₁₃	—	4	1	no	—
138	C₄×D₁₃	#41	1	2	yes	—
139	C₄×D₁₃	#41	1	2	yes	—
140	C₈×C₁₃	—	2	1	no	—
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	—	2	1	no	—
144	#1055	—	2	1	no	—
145	#1055	—	2	1	no	—
146	#1055	—	2	1	no	—
147	#1055	—	2	1	no	—
148	C₂×#117	—	13	1	no	—
149	#433	—	13	1	no	—
150	#433	—	13	1	no	—
151	#433	—	13	1	no	—
152	#433	D₁₃	1	1	yes	characteristic, complemented
153	#432	—	13	1	no	—
154	#432	—	13	1	no	—
155	#431	—	13	1	no	—
156	#431	—	13	1	no	—
157	#424	—	13	1	no	—
158	#424	—	13	1	no	—
159	#407	—	13	1	no	—
160	C₂×C₂²⋊C₈	—	13	1	no	—
161	#405	—	13	1	no	—
162	#405	—	13	1	no	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	2	1	no	—
166	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
167	C₂×D₁₃	—	2	1	no	—
168	C₂×D₁₃	—	2	1	no	—
169	C₂×D₁₃	—	2	1	no	—
170	C₂×D₁₃	—	2	1	no	—
171	C₂×D₁₃	—	4	4	no	—
172	C₂×D₁₃	#117	1	2	yes	—
173	C₂×D₁₃	#131	1	2	yes	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
180	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
181	Dic₁₃	—	2	1	no	—
182	Dic₁₃	—	2	1	no	—
183	Dic₁₃	—	2	1	no	—
184	Dic₁₃	—	2	1	no	—
185	Dic₁₃	—	2	1	no	—
186	Dic₁₃	#117	1	2	yes	—
187	C₂³×C₄	—	13	1	no	—
188	C₂×M₁₆	—	13	1	no	—
189	C₂×M₁₆	—	13	1	no	—
190	C₂²×C₈	—	13	1	no	—
191	C₂²×C₈	—	13	1	no	—
192	#117	—	13	1	no	—
193	#117	—	13	1	no	—
194	#117	—	13	1	no	—
195	#117	—	13	1	no	—
196	#117	—	13	1	no	—
197	#117	—	13	1	no	—
198	#117	—	13	1	no	—
199	#117	C₂×D₁₃	1	1	yes	characteristic
200	C₂×C₄⋊C₄	—	13	1	no	—
201	C₂×C₄⋊C₄	—	13	1	no	—
202	C₂×C₂²⋊C₄	—	13	1	no	—
203	C₂×C₂²⋊C₄	—	13	1	no	—
204	C₂×C₄²	—	13	1	no	—
205	C₂×C₄²	—	13	1	no	—
206	C₄⋊C₈	—	13	1	no	—
207	C₄⋊C₈	—	13	1	no	—
208	C₄⋊C₈	—	13	1	no	—
209	C₄⋊C₈	—	13	1	no	—
210	C₄⋊C₈	—	13	1	no	—
211	C₄⋊C₈	—	13	1	no	—
212	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
213	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
214	C₂²⋊C₈	—	13	1	no	—
215	C₂²⋊C₈	—	13	1	no	—
216	C₂²⋊C₈	—	13	1	no	—
217	C₂²⋊C₈	—	13	1	no	—
218	C₂²⋊C₈	—	13	1	no	—
219	C₂²⋊C₈	—	13	1	no	—
220	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
221	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
222	#97	—	13	1	no	—
223	#97	—	13	1	no	—
224	#97	—	13	1	no	—
225	#97	C₂×D₁₃	1	1	yes	characteristic, complemented
226	C₄×C₈	—	13	1	no	—
227	C₄×C₈	—	13	1	no	—
228	C₄×C₈	—	13	1	no	—
229	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
230	C₂×C₁₃	—	4	1	no	—
231	C₂×C₁₃	C₂×#117	1	1	yes	characteristic
232	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
233	C₂×C₁₃	#568	1	1	yes	characteristic
234	D₁₃	—	2	2	no	—
235	D₁₃	—	2	2	no	—
236	C₂⁴	—	13	1	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	1	no	—
239	C₂²×C₄	—	13	1	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	1	no	—
243	C₂²×C₄	—	13	1	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	26	2	no	—
247	C₂²×C₄	—	26	2	no	—
248	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
249	M₁₆	—	26	2	no	—
250	M₁₆	—	26	2	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	1	no	—
253	C₂×C₈	—	13	1	no	—
254	C₂×C₈	—	13	1	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	13	1	no	—
262	C₄⋊C₄	—	13	1	no	—
263	C₄⋊C₄	—	13	1	no	—
264	C₄⋊C₄	—	13	1	no	—
265	C₄⋊C₄	—	13	1	no	—
266	C₄⋊C₄	—	13	1	no	—
267	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
268	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
269	C₂²⋊C₄	—	13	1	no	—
270	C₂²⋊C₄	—	13	1	no	—
271	C₂²⋊C₄	—	13	1	no	—
272	C₂²⋊C₄	—	13	1	no	—
273	C₂²⋊C₄	—	13	1	no	—
274	C₂²⋊C₄	—	13	1	no	—
275	C₂²⋊C₄	C₄×D₁₃	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₁₃	1	1	yes	characteristic
277	C₄²	—	13	1	no	—
278	C₄²	—	13	1	no	—
279	C₄²	—	13	1	no	—
280	C₄²	—	13	1	no	—
281	C₄²	—	13	1	no	—
282	C₄²	—	13	1	no	—
283	C₄²	C₂²×D₁₃	1	1	yes	characteristic
284	C₄²	C₂²×D₁₃	1	1	yes	characteristic
285	C₁₃	#2924	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	13	1	no	—
287	C₂³	—	13	1	no	—
288	C₂³	—	26	2	no	—
289	C₂³	—	26	2	no	—
290	C₂³	—	52	1	no	—
291	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	13	1	no	—
295	C₂×C₄	—	13	2	no	—
296	C₂×C₄	—	13	2	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	13	1	no	—
301	C₂×C₄	—	13	1	no	—
302	C₂×C₄	—	26	2	no	—
303	C₂×C₄	—	26	2	no	—
304	C₂×C₄	—	26	2	no	—
305	C₂×C₄	—	26	2	no	—
306	C₂×C₄	—	26	2	no	—
307	C₂×C₄	—	26	1	no	—
308	C₂×C₄	—	26	1	no	—
309	C₂×C₄	—	52	1	no	—
310	C₂×C₄	—	52	1	no	—
311	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	26	1	no	—
322	C₈	—	26	1	no	—
323	C₈	—	26	1	no	—
324	C₈	—	26	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	13	2	no	—
329	C₂²	—	13	2	no	—
330	C₂²	—	26	1	no	—
331	C₂²	—	26	1	no	—
332	C₂²	—	26	1	no	—
333	C₂²	—	26	1	no	—
334	C₂²	—	52	4	no	—
335	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	13	2	no	—
342	C₄	—	26	1	no	—
343	C₄	—	26	1	no	—
344	C₄	—	26	1	no	—
345	C₄	—	26	1	no	—
346	C₄	—	26	1	no	—
347	C₄	#41×D₁₃	1	1	yes	characteristic
348	C₄	#41×D₁₃	1	1	yes	characteristic
349	C₂	—	4	1	no	—
350	C₂	—	26	2	no	—
351	C₂	—	26	2	no	—
352	C₂	D₁₃×#117	1	1	yes	characteristic
353	C₂	D₁₃×#131	1	1	yes	characteristic
354	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#433	C₂
prime radical	C₂³×D₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#433	D₁₃
hypocenter	C₁₃	#2924
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. C₁₃×#433	C₂	semidirect
3. #11718661	C₂	semidirect
4. #11718500	C₂	semidirect
5. #11718500	C₂	semidirect
6. #11718496	C₂	semidirect
7. #11718494	C₂	semidirect
10. #11718381	C₂	semidirect
11. #11718381	C₂	semidirect
12. #11718369	C₂	semidirect
13. #11718366	C₂	semidirect
14. #11718350	C₂	semidirect
15. #11718346	C₂	semidirect
35. C₁₃×C₄⋊C₈	C₂²	semidirect
36. C₁₃×C₄⋊C₈	C₂²	semidirect
39. C₁₃×#97	C₂²	semidirect
40. C₄×C₈×C₁₃	C₂²	semidirect
47. #88055	C₂²	semidirect
48. #88055	C₂²	semidirect
49. #88054	C₂²	semidirect
50. #88054	C₂²	semidirect
51. #88054	C₂²	semidirect
52. #88054	C₂²	semidirect
53. C₈×Dic₁₃	C₂²	semidirect
54. C₈×Dic₁₃	C₂²	semidirect
55. C₄⋊#1055	C₂²	semidirect
56. C₄⋊#1055	C₂²	semidirect
57. #88043	C₂²	semidirect
58. C₄×#1055	C₂²	semidirect
152. #433	D₁₃	semidirect
212. C₄⋊C₈	C₂×D₁₃	semidirect
213. C₄⋊C₈	C₂×D₁₃	semidirect
225. #97	C₂×D₁₃	semidirect
229. C₄×C₈	C₂×D₁₃	semidirect
285. C₁₃	#2924	semidirect

Maximal subgroups

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

subgroup	index
D₁₃×#117	2
C₁₃×#433	2
#11718661	2
#11718500	2
#11718496	2
#11718494	2
#11718387	2
D₁₃×C₂²⋊C₈	2
#11718381	2
#11718369	2
#11718366	2
#11718350	2
#11718346	2
#2924	13