Subgroups — #13123908

Order 1664.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719253	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#435	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718790	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718789	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718771	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718672	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718497	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718494	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718390	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718388	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718388	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718386	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718367	C₂	1	1	yes	maximal, characteristic, complemented
15	#11718366	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#6127	C₂²	1	1	yes	characteristic
17	C₂×#6127	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
21	#88194	C₄	1	1	yes	characteristic
22	#88192	C₄	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
25	C₂×#6099	C₂²	1	1	yes	characteristic
26	C₂²×#1055	C₂²	1	1	yes	characteristic
27	C₂×#6094	C₂²	1	1	yes	characteristic
28	C₂×#6094	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	#88149	C₄	1	1	yes	characteristic
32	#88142	C₄	1	1	yes	characteristic
33	#88139	C₄	1	1	yes	characteristic
34	#88139	C₄	1	1	yes	characteristic
35	#88138	C₄	1	1	yes	characteristic
36	#88138	C₄	1	1	yes	characteristic
37	C₄×D₅₂	C₂²	1	1	yes	characteristic
38	C₄²×D₁₃	C₂²	1	1	yes	characteristic
39	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₃×#97	C₂²	1	1	yes	characteristic
43	C₂²⋊#1055	C₂²	1	1	yes	characteristic
44	C₂²⋊#1055	C₂²	1	1	yes	characteristic
45	#88060	C₂²	1	1	yes	characteristic
46	#88060	C₂²	1	1	yes	characteristic
47	#88060	C₂²	1	1	yes	characteristic
48	#88060	C₂²	1	1	yes	characteristic
49	#88060	C₂²	1	1	yes	characteristic
50	#88060	C₂²	1	1	yes	characteristic
51	#88060	C₂²	1	1	yes	characteristic
52	#88060	C₂²	1	1	yes	characteristic
53	#88054	C₂²	1	1	yes	characteristic
54	#88054	C₂²	1	1	yes	characteristic
55	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
56	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
57	C₄⋊#1055	C₂²	1	1	yes	characteristic
58	#88043	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
68	#6127	—	4	1	no	—
69	#6127	—	4	1	no	—
70	#6127	—	4	1	no	—
71	#6127	—	4	1	no	—
72	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
73	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
74	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
80	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
81	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
82	C₁₃×M₁₆	—	4	1	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
89	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
90	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂³	1	1	yes	characteristic
92	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
93	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
94	#6103	C₂×C₄	1	1	yes	characteristic
95	#6103	C₂×C₄	1	1	yes	characteristic
96	#6103	C₂×C₄	1	1	yes	characteristic
97	#6103	C₂×C₄	1	1	yes	characteristic
98	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
99	#6101	C₂×C₄	1	1	yes	characteristic
100	#6101	C₂×C₄	1	1	yes	characteristic
101	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
102	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
103	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
104	#6099	—	4	1	no	—
105	C₂×#1055	—	4	1	no	—
106	C₂×#1055	C₂³	1	1	yes	characteristic
107	C₂×#1055	C₂³	1	1	yes	characteristic
108	C₂×#1055	C₂³	1	1	yes	characteristic
109	C₂×#1055	C₂³	1	1	yes	characteristic
110	#6094	—	4	1	no	—
111	#6094	—	4	1	no	—
112	C₈×D₁₃	—	4	1	no	—
113	C₈×D₁₃	—	4	1	no	—
114	#2981	—	13	1	no	maximal, Sylow
115	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
120	D₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₃	—	2	2	no	—
131	C₂²⋊D₁₃	—	2	2	no	—
132	C₂²⋊D₁₃	—	4	1	no	—
133	C₂²⋊D₁₃	—	4	1	no	—
134	C₂²⋊D₁₃	—	4	1	no	—
135	C₂²⋊D₁₃	—	4	1	no	—
136	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
137	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
138	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
139	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
141	D₅₂	—	4	1	no	—
142	D₅₂	—	4	1	no	—
143	D₅₂	—	4	1	no	—
144	C₄×D₁₃	—	2	1	no	—
145	C₄×D₁₃	—	2	1	no	—
146	C₄×D₁₃	—	2	1	no	—
147	C₄×D₁₃	—	2	1	no	—
148	C₄×D₁₃	—	2	1	no	—
149	C₄×D₁₃	—	2	1	no	—
150	C₄×D₁₃	—	2	2	no	—
151	C₄×D₁₃	—	4	1	no	—
152	Dic₂₆	—	4	1	no	—
153	Dic₂₆	—	4	1	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	—	2	1	no	—
156	C₈×C₁₃	—	2	1	no	—
157	C₈×C₁₃	—	2	1	no	—
158	#1055	—	2	1	no	—
159	#1055	—	2	1	no	—
160	#1055	—	2	1	no	—
161	#1055	—	2	1	no	—
162	#532	—	13	1	no	—
163	#436	—	13	1	no	—
164	#436	—	13	1	no	—
165	#435	—	13	1	no	—
166	#435	—	13	1	no	—
167	#435	—	13	1	no	—
168	#435	D₁₃	1	1	yes	characteristic, complemented
169	C₈×D₄	—	13	1	no	—
170	C₈×D₄	—	13	1	no	—
171	#432	—	13	1	no	—
172	#431	—	13	1	no	—
173	#408	—	13	1	no	—
174	#408	—	13	1	no	—
175	#408	—	13	1	no	—
176	#408	—	13	1	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
184	C₂×D₁₃	—	2	1	no	—
185	C₂×D₁₃	—	2	1	no	—
186	C₂×D₁₃	—	2	1	no	—
187	C₂×D₁₃	—	2	1	no	—
188	C₂×D₁₃	—	2	1	no	—
189	C₂×D₁₃	—	2	1	no	—
190	C₂×D₁₃	—	2	1	no	—
191	C₂×D₁₃	—	2	1	no	—
192	C₂×D₁₃	—	2	1	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
198	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
199	Dic₁₃	—	2	1	no	—
200	Dic₁₃	—	2	1	no	—
201	Dic₁₃	—	2	1	no	—
202	Dic₁₃	—	2	1	no	—
203	Dic₁₃	#131	1	2	yes	—
204	C₂×#41	—	13	1	no	—
205	C₂×#41	—	13	1	no	—
206	C₂×M₁₆	—	13	1	no	—
207	C₂×M₁₆	—	13	1	no	—
208	C₂×M₁₆	—	13	1	no	—
209	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
210	C₂²×C₈	—	13	1	no	—
211	C₂²×C₈	—	13	1	no	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
214	C₄⋊Q₈	—	13	1	no	—
215	#127	—	13	1	no	—
216	#124	—	13	1	no	—
217	#124	—	13	1	no	—
218	#121	—	13	1	no	—
219	#121	—	13	1	no	—
220	#121	—	13	1	no	—
221	#121	—	13	1	no	—
222	C₄×D₄	—	13	1	no	—
223	C₄×D₄	—	13	1	no	—
224	C₄×D₄	—	13	1	no	—
225	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
226	C₂×C₄²	—	13	1	no	—
227	C₄⋊C₈	—	13	1	no	—
228	C₄⋊C₈	—	13	1	no	—
229	C₄⋊C₈	—	13	1	no	—
230	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
231	C₂²⋊C₈	—	13	1	no	—
232	C₂²⋊C₈	—	13	1	no	—
233	C₂²⋊C₈	—	13	1	no	—
234	C₂²⋊C₈	—	13	1	no	—
235	C₂²⋊C₈	—	13	1	no	—
236	C₂²⋊C₈	—	13	1	no	—
237	C₂²⋊C₈	—	13	1	no	—
238	C₂²⋊C₈	—	13	1	no	—
239	C₂²⋊C₈	—	13	1	no	—
240	C₂²⋊C₈	—	13	1	no	—
241	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
242	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
243	#97	—	13	1	no	—
244	#97	C₂×D₁₃	1	1	yes	characteristic
245	C₄×C₈	—	13	1	no	—
246	C₄×C₈	—	13	1	no	—
247	C₂×C₁₃	—	4	1	no	—
248	C₂×C₁₃	—	4	1	no	—
249	C₂×C₁₃	#518	1	1	yes	characteristic
250	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
251	C₂×C₁₃	#568	1	1	yes	characteristic
252	D₁₃	—	4	1	no	—
253	D₁₃	—	4	1	no	—
254	D₁₃	—	4	1	no	—
255	#41	—	52	1	no	—
256	#41	—	52	1	no	—
257	#41	—	52	1	no	—
258	#41	—	52	1	no	—
259	C₂×Q₈	—	13	1	no	—
260	C₂×Q₈	—	13	1	no	—
261	C₂×D₄	—	13	1	no	—
262	C₂×D₄	—	13	1	no	—
263	C₂×D₄	—	13	1	no	—
264	C₂×D₄	—	13	1	no	—
265	C₂×D₄	—	13	1	no	—
266	C₂×D₄	—	13	1	no	—
267	C₂×D₄	—	13	1	no	—
268	C₂×D₄	—	13	1	no	—
269	C₂×D₄	—	13	1	no	—
270	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
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	—	13	1	no	—
276	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
277	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
278	M₁₆	—	4	1	no	—
279	M₁₆	—	52	1	no	—
280	M₁₆	—	52	1	no	—
281	M₁₆	—	52	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	13	1	no	—
284	C₂×C₈	—	13	1	no	—
285	C₂×C₈	—	13	1	no	—
286	C₂×C₈	—	13	1	no	—
287	C₂×C₈	—	52	1	no	—
288	C₂×C₈	—	52	1	no	—
289	C₂×C₈	—	52	1	no	—
290	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
291	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
292	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
293	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
294	C₄⋊C₄	—	13	1	no	—
295	C₄⋊C₄	—	13	1	no	—
296	C₄⋊C₄	—	13	1	no	—
297	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
298	C₂²⋊C₄	—	13	1	no	—
299	C₂²⋊C₄	—	13	1	no	—
300	C₂²⋊C₄	—	13	1	no	—
301	C₂²⋊C₄	—	13	1	no	—
302	C₂²⋊C₄	—	13	1	no	—
303	C₂²⋊C₄	—	13	1	no	—
304	C₂²⋊C₄	C₄×D₁₃	1	1	yes	characteristic
305	C₂²⋊C₄	C₄×D₁₃	1	1	yes	characteristic
306	C₄²	—	13	1	no	—
307	C₄²	—	13	1	no	—
308	C₄²	—	13	1	no	—
309	C₄²	C₂²×D₁₃	1	1	yes	characteristic
310	C₁₃	#2981	1	1	yes	characteristic, complemented, Sylow
311	C₂³	—	13	1	no	—
312	C₂³	—	13	1	no	—
313	C₂³	—	13	1	no	—
314	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
315	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
316	Q₈	—	52	1	no	—
317	Q₈	—	52	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	26	2	no	—
320	D₄	—	26	2	no	—
321	D₄	—	52	1	no	—
322	D₄	—	52	1	no	—
323	D₄	—	52	1	no	—
324	D₄	—	52	1	no	—
325	D₄	—	52	1	no	—
326	D₄	—	52	1	no	—
327	D₄	—	52	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	13	1	no	—
333	C₂×C₄	—	13	1	no	—
334	C₂×C₄	—	13	1	no	—
335	C₂×C₄	—	13	1	no	—
336	C₂×C₄	—	13	1	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	26	1	no	—
339	C₂×C₄	—	26	1	no	—
340	C₂×C₄	—	26	1	no	—
341	C₂×C₄	—	26	1	no	—
342	C₂×C₄	—	26	1	no	—
343	C₂×C₄	—	26	1	no	—
344	C₂×C₄	—	52	1	no	—
345	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
346	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
347	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
349	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	26	1	no	—
355	C₈	—	26	1	no	—
356	C₈	—	26	1	no	—
357	C₈	—	26	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	26	1	no	—
365	C₂²	—	26	1	no	—
366	C₂²	—	26	1	no	—
367	C₂²	—	26	1	no	—
368	C₂²	—	26	1	no	—
369	C₂²	—	26	1	no	—
370	C₂²	—	26	1	no	—
371	C₂²	—	26	1	no	—
372	C₂²	—	26	1	no	—
373	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	13	2	no	—
379	C₄	—	26	1	no	—
380	C₄	—	26	1	no	—
381	C₄	—	26	1	no	—
382	C₄	—	26	1	no	—
383	C₄	#88251	1	1	yes	characteristic
384	C₄	#88256	1	1	yes	characteristic
385	C₂	—	4	1	no	—
386	C₂	—	4	1	no	—
387	C₂	—	52	1	no	—
388	C₂	—	52	1	no	—
389	C₂	—	52	1	no	—
390	C₂	#11719250	1	1	yes	characteristic
391	C₂	#11719450	1	1	yes	characteristic
392	C₂	D₁₃×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#435	C₂
prime radical	#11719253	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#435	D₁₃
hypocenter	C₁₃	#2981
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. C₁₃×#435	C₂	semidirect
3. #11718790	C₂	semidirect
4. #11718789	C₂	semidirect
5. #11718771	C₂	semidirect
6. C₈×C₂²⋊D₁₃	C₂	semidirect
7. #11718672	C₂	semidirect
8. #11718497	C₂	semidirect
9. #11718494	C₂	semidirect
10. #11718390	C₂	semidirect
11. #11718388	C₂	semidirect
12. #11718388	C₂	semidirect
13. #11718386	C₂	semidirect
14. #11718367	C₂	semidirect
15. #11718366	C₂	semidirect
168. #435	D₁₃	semidirect
310. C₁₃	#2981	semidirect

Maximal subgroups

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

subgroup	index
#11719253	2
C₁₃×#435	2
#11718790	2
#11718789	2
#11718771	2
C₈×C₂²⋊D₁₃	2
#11718672	2
#11718497	2
#11718494	2
#11718390	2
#11718388	2
#11718386	2
#11718367	2
#11718366	2
#2981	13