Subgroups — #13123829

Order 1664.

372 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	#11719285	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718685	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718501	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718501	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718497	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718497	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718495	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718494	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718369	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718367	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718367	C₂	1	1	yes	maximal, characteristic, complemented
13	D₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718350	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
17	#88201	C₄	1	1	yes	characteristic
18	#88200	C₄	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	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
25	#88152	C₄	1	1	yes	characteristic
26	#88152	C₄	1	1	yes	characteristic
27	#88149	C₄	1	1	yes	characteristic
28	#88148	C₄	1	1	yes	characteristic
29	#88148	C₄	1	1	yes	characteristic
30	#88144	C₄	1	1	yes	characteristic
31	C₄×D₅₂	C₂²	1	1	yes	characteristic
32	C₄×D₅₂	C₂²	1	1	yes	characteristic
33	C₄×D₅₂	C₂²	1	1	yes	characteristic
34	#88126	C₂²	1	1	yes	characteristic
35	#88126	C₂²	1	1	yes	characteristic
36	C₄²×D₁₃	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×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₃×#97	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	#88060	C₂²	1	1	yes	characteristic
48	#88060	C₂²	1	1	yes	characteristic
49	#88055	C₂²	1	1	yes	characteristic
50	#88055	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
53	C₄⋊#1055	C₂²	1	1	yes	characteristic
54	C₄⋊#1055	C₂²	1	1	yes	characteristic
55	C₄⋊#1055	C₂²	1	1	yes	characteristic
56	#88043	C₂²	1	1	yes	characteristic
57	#88043	C₂²	1	1	yes	characteristic
58	C₄×#1055	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
61	C₂×D₅₂	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×C₄×D₁₃	C₂³	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×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂³	1	1	yes	characteristic
76	C₄²×C₁₃	C₂³	1	1	yes	characteristic
77	C₄²×C₁₃	C₂³	1	1	yes	characteristic
78	#6103	C₂×C₄	1	1	yes	characteristic
79	#6103	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	#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	C₄⋊Dic₁₃	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	#6101	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	C₂³	1	1	yes	characteristic
98	C₂×#1055	C₂³	1	1	yes	characteristic
99	C₂×#1055	C₂³	1	1	yes	characteristic
100	C₂×#1055	C₂³	1	1	yes	characteristic
101	C₂×#1055	C₂³	1	1	yes	characteristic
102	#6094	—	2	2	no	—
103	#6094	—	2	2	no	—
104	C₈×D₁₃	—	2	2	no	—
105	C₈×D₁₃	—	2	2	no	—
106	C₈×D₁₃	—	2	2	no	—
107	C₈×D₁₃	—	2	2	no	—
108	#2968	—	13	1	no	maximal, Sylow
109	C₂²×D₁₃	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	Q₈×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
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	D₅₂	—	2	2	no	—
126	D₅₂	—	2	2	no	—
127	D₅₂	—	4	1	no	—
128	C₄×D₁₃	—	2	2	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	1	no	—
134	C₄×D₁₃	—	2	1	no	—
135	C₄×D₁₃	—	4	1	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	—	2	1	no	—
138	C₈×C₁₃	—	2	1	no	—
139	C₈×C₁₃	#41	1	2	yes	—
140	#1055	—	2	1	no	—
141	#1055	—	2	1	no	—
142	#1055	—	2	1	no	—
143	#1055	#41	1	2	yes	—
144	#529	—	13	1	no	—
145	#446	—	13	1	no	—
146	#446	D₁₃	1	1	yes	characteristic, complemented
147	#435	—	13	1	no	—
148	#435	—	13	1	no	—
149	#435	—	13	1	no	—
150	#435	—	13	1	no	—
151	C₈×D₄	—	13	1	no	—
152	C₈×D₄	—	13	1	no	—
153	#433	—	13	1	no	—
154	#433	—	13	1	no	—
155	#431	—	13	1	no	—
156	#405	—	13	1	no	—
157	#405	—	13	1	no	—
158	C₂×#97	—	13	1	no	—
159	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
160	C₂×D₁₃	—	2	1	no	—
161	C₂×D₁₃	—	2	1	no	—
162	C₂×D₁₃	—	2	1	no	—
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	#131	1	2	yes	—
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	—	2	1	no	—
175	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
176	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
177	Dic₁₃	—	2	1	no	—
178	Dic₁₃	—	2	1	no	—
179	Dic₁₃	—	2	1	no	—
180	Dic₁₃	—	2	1	no	—
181	Dic₁₃	—	2	1	no	—
182	C₂×M₁₆	—	13	1	no	—
183	C₂×M₁₆	—	13	1	no	—
184	C₂²×C₈	—	13	1	no	—
185	C₂²×C₈	—	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	C₂×D₁₃	1	1	yes	characteristic
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	—	13	1	no	—
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	C₂×D₁₃	1	1	yes	characteristic
207	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
208	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
209	C₂²⋊C₈	—	13	1	no	—
210	C₂²⋊C₈	—	13	1	no	—
211	C₂²⋊C₈	—	13	1	no	—
212	C₂²⋊C₈	—	13	1	no	—
213	C₂²⋊C₈	—	13	1	no	—
214	C₂²⋊C₈	—	13	1	no	—
215	#97	—	13	1	no	—
216	#97	—	13	1	no	—
217	#97	—	13	1	no	—
218	#97	—	13	1	no	—
219	#97	C₂×D₁₃	1	1	yes	characteristic
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	C₄×#41	1	1	yes	characteristic
226	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
227	C₂×C₁₃	#568	1	1	yes	characteristic
228	D₁₃	—	2	2	no	—
229	D₁₃	—	4	1	no	—
230	D₁₃	—	4	1	no	—
231	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
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	—	13	1	no	—
240	M₁₆	—	26	2	no	—
241	M₁₆	—	26	2	no	—
242	C₂×C₈	—	13	1	no	—
243	C₂×C₈	—	13	1	no	—
244	C₂×C₈	—	13	1	no	—
245	C₂×C₈	—	13	1	no	—
246	C₂×C₈	—	26	2	no	—
247	C₂×C₈	—	26	2	no	—
248	C₂×C₈	—	26	2	no	—
249	C₂×C₈	—	26	2	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	C₄×D₁₃	1	1	yes	characteristic
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	—	13	1	no	—
273	C₂²⋊C₄	—	13	1	no	—
274	C₄²	—	13	1	no	—
275	C₄²	—	13	1	no	—
276	C₄²	—	13	1	no	—
277	C₄²	C₂²×D₁₃	1	1	yes	characteristic
278	C₄²	C₂²×D₁₃	1	1	yes	characteristic
279	C₄²	C₂²×D₁₃	1	1	yes	characteristic
280	C₁₃	#2968	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	13	1	no	—
282	C₂³	—	13	1	no	—
283	C₂³	—	13	1	no	—
284	Q₈	—	4	1	no	—
285	D₄	—	26	2	no	—
286	D₄	—	26	2	no	—
287	D₄	—	52	1	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	13	1	no	—
292	C₂×C₄	—	13	1	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	1	no	—
295	C₂×C₄	—	26	1	no	—
296	C₂×C₄	—	26	1	no	—
297	C₂×C₄	—	26	1	no	—
298	C₂×C₄	—	26	1	no	—
299	C₂×C₄	—	26	1	no	—
300	C₂×C₄	—	52	1	no	—
301	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
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	—	13	2	no	—
312	C₈	—	26	1	no	—
313	C₈	—	26	1	no	—
314	C₈	—	26	1	no	—
315	C₈	#6131	1	2	yes	—
316	C₂²	—	13	2	no	—
317	C₂²	—	26	1	no	—
318	C₂²	—	26	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	—	2	1	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	C₂×#6131	1	1	yes	characteristic
338	C₄	#41×D₁₃	1	1	yes	characteristic
339	C₂	—	26	2	no	—
340	C₂	—	52	1	no	—
341	C₂	—	52	1	no	—
342	C₂	C₄×#6131	1	1	yes	characteristic
343	C₂	#11719450	1	1	yes	characteristic
344	C₂	D₁₃×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#446	C₂
prime radical	#88149	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#446	D₁₃
hypocenter	C₁₃	#2968
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₁₃×#446	C₂	semidirect
3. #11718685	C₂	semidirect
4. #11718501	C₂	semidirect
5. #11718501	C₂	semidirect
6. #11718497	C₂	semidirect
7. #11718497	C₂	semidirect
8. #11718495	C₂	semidirect
9. #11718494	C₂	semidirect
10. #11718369	C₂	semidirect
11. #11718367	C₂	semidirect
12. #11718367	C₂	semidirect
13. D₁₃×#97	C₂	semidirect
14. #11718350	C₂	semidirect
15. C₈×D₅₂	C₂	semidirect
146. #446	D₁₃	semidirect
280. C₁₃	#2968	semidirect

Maximal subgroups

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

subgroup	index
#11719285	2
C₁₃×#446	2
#11718685	2
#11718501	2
#11718497	2
#11718495	2
#11718494	2
#11718369	2
#11718367	2
D₁₃×#97	2
#11718350	2
C₈×D₅₂	2
#2968	13