Subgroups — #13123979

Order 1664.

354 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719233	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718660	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718501	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718500	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718498	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718494	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718391	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718390	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718390	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718386	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718367	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718362	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718349	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₃×#117	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
19	C₂×#6099	C₂²	1	1	yes	characteristic
20	C₂²×#1055	C₂²	1	1	yes	characteristic
21	C₂×#6094	C₂²	1	1	yes	characteristic
22	C₂×#6094	C₂²	1	1	yes	characteristic
23	C₂×#6094	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
25	#88152	C₄	1	1	yes	characteristic
26	#88150	C₄	1	1	yes	characteristic
27	#88148	C₄	1	1	yes	characteristic
28	#88143	C₄	1	1	yes	characteristic
29	#88139	C₄	1	1	yes	characteristic
30	#88138	C₄	1	1	yes	characteristic
31	#88137	C₄	1	1	yes	characteristic
32	#88133	C₄	1	1	yes	characteristic
33	C₄×D₅₂	C₂²	1	1	yes	characteristic
34	#88126	C₂²	1	1	yes	characteristic
35	C₄²×D₁₃	C₂²	1	1	yes	characteristic
36	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
37	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₃×#97	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₃	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
48	#88054	C₂²	1	1	yes	characteristic
49	#88054	C₂²	1	1	yes	characteristic
50	#88054	C₂²	1	1	yes	characteristic
51	#88054	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
55	C₄⋊#1055	C₂²	1	1	yes	characteristic
56	C₄⋊#1055	C₂²	1	1	yes	characteristic
57	C₄⋊#1055	C₂²	1	1	yes	characteristic
58	C₄⋊#1055	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
62	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
76	C₄²×C₁₃	C₂³	1	1	yes	characteristic
77	C₄²×C₁₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
80	#6103	C₂×C₄	1	1	yes	characteristic
81	#6103	C₂×C₄	1	1	yes	characteristic
82	#6103	C₂×C₄	1	1	yes	characteristic
83	#6103	C₂×C₄	1	1	yes	characteristic
84	#6103	C₂×C₄	1	1	yes	characteristic
85	#6103	C₂×C₄	1	1	yes	characteristic
86	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
87	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
88	#6101	C₂×C₄	1	1	yes	characteristic
89	#6101	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	C₂³	1	1	yes	characteristic
97	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
98	#6099	—	4	1	no	—
99	C₂×#1055	—	4	1	no	—
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	—	4	1	no	—
105	#6094	—	4	1	no	—
106	#6094	—	4	1	no	—
107	C₈×D₁₃	—	4	1	no	—
108	#2993	—	13	1	no	maximal, Sylow
109	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₁₃	—	2	2	no	—
121	C₂²⋊D₁₃	—	4	1	no	—
122	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
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	D₅₂	—	4	1	no	—
129	C₄×D₁₃	—	2	1	no	—
130	C₄×D₁₃	—	2	1	no	—
131	C₄×D₁₃	—	2	1	no	—
132	C₄×D₁₃	—	2	1	no	—
133	C₄×D₁₃	—	2	2	no	—
134	C₄×D₁₃	—	4	1	no	—
135	Dic₂₆	—	2	2	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	—	2	1	no	—
138	C₈×C₁₃	—	2	1	no	—
139	C₈×C₁₃	—	2	1	no	—
140	#1055	—	2	1	no	—
141	#1055	—	2	1	no	—
142	#1055	—	2	1	no	—
143	#1055	—	2	1	no	—
144	#529	—	13	1	no	—
145	#446	—	13	1	no	—
146	C₈×Q₈	—	13	1	no	—
147	#436	—	13	1	no	—
148	#436	—	13	1	no	—
149	#436	—	13	1	no	—
150	#435	—	13	1	no	—
151	#435	—	13	1	no	—
152	C₈×D₄	—	13	1	no	—
153	#433	—	13	1	no	—
154	#433	D₁₃	1	1	yes	characteristic, complemented
155	#431	—	13	1	no	—
156	#424	—	13	1	no	—
157	#424	—	13	1	no	—
158	#423	—	13	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
163	C₂×D₁₃	—	2	1	no	—
164	C₂×D₁₃	—	2	1	no	—
165	C₂×D₁₃	—	2	1	no	—
166	C₂×D₁₃	—	2	1	no	—
167	C₂×D₁₃	—	2	1	no	—
168	C₂×D₁₃	—	2	1	no	—
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
175	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
176	Dic₁₃	—	2	1	no	—
177	Dic₁₃	—	2	1	no	—
178	Dic₁₃	—	2	1	no	—
179	Dic₁₃	—	2	1	no	—
180	Dic₁₃	—	2	1	no	—
181	Dic₁₃	#131	1	2	yes	—
182	C₂×M₁₆	—	13	1	no	—
183	C₂×M₁₆	—	13	1	no	—
184	C₂×M₁₆	—	13	1	no	—
185	C₂×M₁₆	—	13	1	no	—
186	C₂²×C₈	—	13	1	no	—
187	C₂²×C₈	—	13	1	no	—
188	#126	—	13	1	no	—
189	#126	—	13	1	no	—
190	#125	—	13	1	no	—
191	#124	—	13	1	no	—
192	#123	—	13	1	no	—
193	#123	—	13	1	no	—
194	C₂²⋊Q₈	—	13	1	no	—
195	#121	—	13	1	no	—
196	C₄×Q₈	—	13	1	no	—
197	C₄×D₄	—	13	1	no	—
198	C₄×D₄	—	13	1	no	—
199	C₄×D₄	—	13	1	no	—
200	#117	—	13	1	no	—
201	#117	C₂×D₁₃	1	1	yes	characteristic
202	C₂×C₄²	—	13	1	no	—
203	C₄⋊C₈	—	13	1	no	—
204	C₄⋊C₈	—	13	1	no	—
205	C₄⋊C₈	—	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	C₂×D₁₃	1	1	yes	characteristic
212	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
213	C₂²⋊C₈	—	13	1	no	—
214	C₂²⋊C₈	—	13	1	no	—
215	C₂²⋊C₈	—	13	1	no	—
216	C₂²⋊C₈	—	13	1	no	—
217	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
218	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
219	#97	—	13	1	no	—
220	#97	C₂×D₁₃	1	1	yes	characteristic
221	C₄×C₈	—	13	1	no	—
222	C₄×C₈	—	13	1	no	—
223	C₄×C₈	—	13	1	no	—
224	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
225	C₂×C₁₃	—	4	1	no	—
226	C₂×C₁₃	#520	1	1	yes	characteristic
227	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
228	C₂×C₁₃	#568	1	1	yes	characteristic
229	D₁₃	—	4	1	no	—
230	D₁₃	—	4	1	no	—
231	C₂×Q₈	—	13	1	no	—
232	C₂×D₄	—	13	1	no	—
233	C₂×D₄	—	13	1	no	—
234	C₂×D₄	—	13	1	no	—
235	C₂²×C₄	—	13	1	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	1	no	—
239	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
240	M₁₆	—	52	1	no	—
241	M₁₆	—	52	1	no	—
242	M₁₆	—	52	1	no	—
243	M₁₆	—	52	1	no	—
244	C₂×C₈	—	13	1	no	—
245	C₂×C₈	—	13	1	no	—
246	C₂×C₈	—	13	1	no	—
247	C₂×C₈	—	13	1	no	—
248	C₂×C₈	—	52	1	no	—
249	C₂×C₈	—	52	1	no	—
250	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
254	C₄⋊C₄	—	13	1	no	—
255	C₄⋊C₄	—	13	1	no	—
256	C₄⋊C₄	—	13	1	no	—
257	C₄⋊C₄	—	13	1	no	—
258	C₄⋊C₄	—	13	1	no	—
259	C₄⋊C₄	—	13	1	no	—
260	C₄⋊C₄	—	13	1	no	—
261	C₄⋊C₄	—	13	1	no	—
262	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
263	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
264	C₂²⋊C₄	—	13	1	no	—
265	C₂²⋊C₄	—	13	1	no	—
266	C₂²⋊C₄	—	13	1	no	—
267	C₂²⋊C₄	—	13	1	no	—
268	C₂²⋊C₄	—	13	1	no	—
269	C₂²⋊C₄	—	13	1	no	—
270	C₂²⋊C₄	—	13	1	no	—
271	C₂²⋊C₄	—	13	1	no	—
272	C₂²⋊C₄	C₄×D₁₃	1	1	yes	characteristic
273	C₂²⋊C₄	C₄×D₁₃	1	1	yes	characteristic
274	C₄²	—	13	1	no	—
275	C₄²	—	13	1	no	—
276	C₄²	—	13	1	no	—
277	C₄²	—	13	1	no	—
278	C₄²	C₂²×D₁₃	1	1	yes	characteristic
279	C₄²	C₂²×D₁₃	1	1	yes	characteristic
280	C₁₃	#2993	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	13	1	no	—
282	C₂³	—	13	1	no	—
283	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
284	Q₈	—	26	2	no	—
285	D₄	—	26	2	no	—
286	D₄	—	52	1	no	—
287	D₄	—	52	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	13	1	no	—
292	C₂×C₄	—	13	1	no	—
293	C₂×C₄	—	13	1	no	—
294	C₂×C₄	—	13	1	no	—
295	C₂×C₄	—	13	1	no	—
296	C₂×C₄	—	26	2	no	—
297	C₂×C₄	—	26	1	no	—
298	C₂×C₄	—	26	1	no	—
299	C₂×C₄	—	26	1	no	—
300	C₂×C₄	—	26	1	no	—
301	C₂×C₄	—	52	1	no	—
302	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	26	1	no	—
313	C₈	—	26	1	no	—
314	C₈	—	26	1	no	—
315	C₈	—	26	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	26	1	no	—
320	C₂²	—	26	1	no	—
321	C₂²	—	26	1	no	—
322	C₂²	—	26	1	no	—
323	C₂²	—	26	1	no	—
324	C₂²	—	26	1	no	—
325	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	13	2	no	—
332	C₄	—	26	1	no	—
333	C₄	—	26	1	no	—
334	C₄	—	26	1	no	—
335	C₄	—	26	1	no	—
336	C₄	—	26	1	no	—
337	C₄	#88256	1	1	yes	characteristic
338	C₄	#88257	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	52	1	no	—
341	C₂	—	52	1	no	—
342	C₂	#11719228	1	1	yes	characteristic
343	C₂	D₁₃×#131	1	1	yes	characteristic
344	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	#88138	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#433	D₁₃
hypocenter	C₁₃	#2993
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. #11718660	C₂	semidirect
4. #11718501	C₂	semidirect
5. #11718500	C₂	semidirect
6. #11718498	C₂	semidirect
7. #11718494	C₂	semidirect
8. #11718391	C₂	semidirect
9. #11718390	C₂	semidirect
10. #11718390	C₂	semidirect
11. #11718386	C₂	semidirect
12. #11718367	C₂	semidirect
13. #11718362	C₂	semidirect
14. #11718349	C₂	semidirect
15. C₈×Dic₂₆	C₂	semidirect
154. #433	D₁₃	semidirect
280. C₁₃	#2993	semidirect

Maximal subgroups

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

subgroup	index
#11719233	2
C₁₃×#433	2
#11718660	2
#11718501	2
#11718500	2
#11718498	2
#11718494	2
#11718391	2
#11718390	2
#11718386	2
#11718367	2
#11718362	2
#11718349	2
C₈×Dic₂₆	2
#2993	13