Subgroups — #13127293

Order 1664.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719248	C₂	1	1	yes	maximal, characteristic
2	#11719246	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#435	C₂	1	1	yes	maximal, characteristic
4	#11718798	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718782	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718777	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88057	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718512	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718490	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718384	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718383	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718382	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718382	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718377	C₂	1	1	yes	maximal, characteristic, complemented
15	#11718363	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
19	#88189	—	2	1	no	—
20	D₄×Dic₁₃	—	2	1	no	—
21	#88180	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	#88178	C₂²	1	1	yes	characteristic
26	#88178	C₂²	1	1	yes	characteristic
27	#88178	C₂²	1	1	yes	characteristic
28	#88178	C₂²	1	1	yes	characteristic
29	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
30	#88144	—	2	1	no	—
31	#88143	—	2	1	no	—
32	#88133	—	2	1	no	—
33	#88132	—	2	1	no	—
34	#88132	—	2	1	no	—
35	#88131	—	2	1	no	—
36	#88124	C₂²	1	1	yes	characteristic
37	#88123	C₂²	1	1	yes	characteristic
38	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
39	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
40	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₃×#97	C₂²	1	1	yes	characteristic
44	#88058	C₂²	1	1	yes	characteristic
45	#88058	C₂²	1	1	yes	characteristic
46	#88058	C₂²	1	1	yes	characteristic
47	#88057	C₂²	1	1	yes	characteristic
48	#88057	C₂²	1	1	yes	characteristic
49	#88057	C₂²	1	1	yes	characteristic
50	#88057	C₂²	1	1	yes	characteristic
51	#88057	C₂²	1	1	yes	characteristic
52	#88057	C₂²	1	1	yes	characteristic
53	#88056	C₂²	1	1	yes	characteristic
54	#88056	C₂²	1	1	yes	characteristic
55	#88056	C₂²	1	1	yes	characteristic
56	#88056	C₂²	1	1	yes	characteristic
57	#88056	C₂²	1	1	yes	characteristic
58	#88056	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₃	—	2	1	no	—
63	C₂²×Dic₁₃	—	2	1	no	—
64	C₂²×Dic₁₃	—	2	1	no	—
65	C₂×Dic₂₆	—	2	1	no	—
66	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
68	C₁₃×M₁₆	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
75	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
76	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
77	C₄²×C₁₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₁₃	—	2	1	no	—
79	C₂²⋊Dic₁₃	—	2	1	no	—
80	C₂²⋊Dic₁₃	—	2	1	no	—
81	C₂²⋊Dic₁₃	—	2	1	no	—
82	C₂²⋊Dic₁₃	—	2	1	no	—
83	C₄⋊Dic₁₃	—	2	1	no	—
84	C₄⋊Dic₁₃	—	2	1	no	—
85	C₄⋊Dic₁₃	—	2	1	no	—
86	C₄⋊Dic₁₃	—	2	1	no	—
87	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
93	#6101	—	2	1	no	—
94	#6101	—	2	1	no	—
95	#6101	—	2	1	no	—
96	#6101	—	2	1	no	—
97	#6101	—	2	1	no	—
98	#6101	—	2	1	no	—
99	#6101	—	2	1	no	—
100	#6101	—	2	1	no	—
101	C₄×Dic₁₃	—	2	1	no	—
102	C₄×Dic₁₃	—	2	1	no	—
103	C₄×Dic₁₃	—	2	1	no	—
104	Dic₅₂	—	2	2	no	—
105	#3344	—	13	1	no	maximal, Sylow
106	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
108	D₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₁₃	—	2	1	no	—
119	C₂×Dic₁₃	—	2	1	no	—
120	C₂×Dic₁₃	—	2	1	no	—
121	C₂×Dic₁₃	—	2	1	no	—
122	C₂×Dic₁₃	—	2	1	no	—
123	C₂×Dic₁₃	—	2	1	no	—
124	C₂×Dic₁₃	—	2	1	no	—
125	C₂×Dic₁₃	—	2	1	no	—
126	C₂×Dic₁₃	—	4	2	no	—
127	C₂×Dic₁₃	—	8	1	no	—
128	C₂×Dic₁₃	—	8	1	no	—
129	Dic₂₆	—	2	1	no	—
130	Dic₂₆	—	2	1	no	—
131	Dic₂₆	—	2	1	no	—
132	Dic₂₆	—	2	1	no	—
133	Dic₂₆	—	8	1	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	#41	1	2	yes	—
138	#554	—	13	1	no	—
139	#552	—	13	1	no	—
140	#501	—	13	1	no	—
141	#485	—	13	1	no	—
142	#484	—	13	1	no	—
143	#483	—	13	1	no	—
144	#483	—	13	1	no	—
145	#479	—	13	1	no	—
146	#475	—	13	1	no	—
147	#470	—	13	1	no	—
148	#467	—	13	1	no	—
149	#441	—	13	1	no	—
150	#435	D₁₃	1	1	yes	characteristic
151	#428	—	13	1	no	—
152	C₂×#106	—	13	1	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	2	1	no	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	#137	1	2	yes	—
159	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
166	Dic₁₃	—	4	1	no	—
167	Dic₁₃	—	4	1	no	—
168	Dic₁₃	—	4	1	no	—
169	Dic₁₃	—	4	1	no	—
170	Dic₁₃	—	4	1	no	—
171	Dic₁₃	—	4	1	no	—
172	Dic₁₃	—	4	1	no	—
173	Dic₁₃	—	4	1	no	—
174	C₂×Q₁₆	—	13	1	no	—
175	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
177	C₄⋊Q₈	—	13	1	no	—
178	#126	—	26	1	no	—
179	#125	—	13	1	no	—
180	#125	—	26	1	no	—
181	#125	—	26	1	no	—
182	#123	—	26	1	no	—
183	C₂²⋊Q₈	—	13	1	no	—
184	C₂²⋊Q₈	—	13	1	no	—
185	C₂²⋊Q₈	—	13	1	no	—
186	C₂²⋊Q₈	—	13	1	no	—
187	C₂²⋊Q₈	—	26	1	no	—
188	C₂²⋊Q₈	—	26	1	no	—
189	C₄×Q₈	—	13	1	no	—
190	C₄×Q₈	—	13	1	no	—
191	C₄×D₄	—	26	1	no	—
192	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
193	#117	—	13	1	no	—
194	#117	—	26	1	no	—
195	C₂×C₄⋊C₄	—	13	1	no	—
196	C₂×C₄⋊C₄	—	13	1	no	—
197	C₂×C₄⋊C₄	—	13	1	no	—
198	#107	—	13	1	no	—
199	#107	—	13	1	no	—
200	#107	—	13	1	no	—
201	#106	—	13	1	no	—
202	#106	—	13	1	no	—
203	#106	—	13	1	no	—
204	#106	—	13	1	no	—
205	#106	—	13	1	no	—
206	#106	—	13	1	no	—
207	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
208	#103	—	13	1	no	—
209	#103	—	13	1	no	—
210	#103	—	13	1	no	—
211	#103	—	13	1	no	—
212	#103	—	13	1	no	—
213	#103	—	13	1	no	—
214	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
215	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
216	#97	C₂×D₁₃	1	1	yes	characteristic
217	C₂×C₁₃	—	2	2	no	—
218	C₂×C₁₃	—	4	1	no	—
219	C₂×C₁₃	#547	1	1	yes	characteristic
220	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
221	C₂×C₁₃	#577	1	1	yes	characteristic
222	C₂×Q₈	—	13	1	no	—
223	C₂×Q₈	—	13	1	no	—
224	C₂×Q₈	—	26	1	no	—
225	C₂×D₄	D₅₂	1	1	yes	characteristic
226	C₂²×C₄	—	26	1	no	—
227	C₂²×C₄	—	26	1	no	—
228	C₂²×C₄	—	26	1	no	—
229	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
231	Q₁₆	—	26	2	no	—
232	M₁₆	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
238	C₄⋊C₄	—	13	1	no	—
239	C₄⋊C₄	—	13	1	no	—
240	C₄⋊C₄	—	13	1	no	—
241	C₄⋊C₄	—	13	1	no	—
242	C₄⋊C₄	—	13	1	no	—
243	C₄⋊C₄	—	13	1	no	—
244	C₄⋊C₄	—	26	1	no	—
245	C₄⋊C₄	—	26	1	no	—
246	C₄⋊C₄	—	26	1	no	—
247	C₄⋊C₄	—	26	1	no	—
248	C₄⋊C₄	—	26	1	no	—
249	C₄⋊C₄	—	26	1	no	—
250	C₄⋊C₄	—	26	1	no	—
251	C₄⋊C₄	—	26	1	no	—
252	C₄⋊C₄	—	26	1	no	—
253	C₄⋊C₄	—	26	1	no	—
254	C₄⋊C₄	—	26	1	no	—
255	C₄⋊C₄	—	26	1	no	—
256	C₄⋊C₄	D₅₂	1	1	yes	characteristic
257	C₂²⋊C₄	—	26	1	no	—
258	C₂²⋊C₄	—	26	1	no	—
259	C₂²⋊C₄	—	26	1	no	—
260	C₂²⋊C₄	—	26	1	no	—
261	C₂²⋊C₄	—	26	1	no	—
262	C₂²⋊C₄	D₅₂	1	1	yes	characteristic
263	C₂²⋊C₄	D₅₂	1	1	yes	characteristic
264	C₄²	—	26	1	no	—
265	C₄²	—	26	1	no	—
266	C₄²	—	26	1	no	—
267	C₄²	C₂²×D₁₃	1	1	yes	characteristic
268	C₁₃	#3344	1	1	yes	characteristic, complemented, Sylow
269	C₂³	C₂×D₅₂	1	1	yes	characteristic
270	C₂³	C₂×D₅₂	1	1	yes	characteristic
271	Q₈	—	26	1	no	—
272	Q₈	—	26	1	no	—
273	Q₈	—	26	1	no	—
274	Q₈	—	26	1	no	—
275	Q₈	—	104	1	no	—
276	D₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	26	1	no	—
282	C₂×C₄	—	26	1	no	—
283	C₂×C₄	—	26	1	no	—
284	C₂×C₄	—	26	1	no	—
285	C₂×C₄	—	26	1	no	—
286	C₂×C₄	—	26	1	no	—
287	C₂×C₄	—	26	1	no	—
288	C₂×C₄	—	26	1	no	—
289	C₂×C₄	—	52	2	no	—
290	C₂×C₄	—	104	1	no	—
291	C₂×C₄	—	104	1	no	—
292	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
293	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
294	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
295	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	#6129	1	2	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	#88163	1	2	yes	—
307	C₂²	C₂²×D₅₂	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	52	1	no	—
313	C₄	—	52	1	no	—
314	C₄	—	52	1	no	—
315	C₄	—	52	1	no	—
316	C₄	—	52	1	no	—
317	C₄	—	52	1	no	—
318	C₄	—	52	1	no	—
319	C₄	—	52	1	no	—
320	C₄	C₂×#6129	1	1	yes	characteristic
321	C₄	#88257	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	1	no	—
324	C₂	#11719260	1	1	yes	characteristic
325	C₂	C₂×#88163	1	1	yes	characteristic
326	C₂	#11719456	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#435	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×D₄×C₁₃	D₄
hypercenter	#435	D₁₃
hypocenter	C₁₃	#3344
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
4. #11718798	C₂	semidirect
5. #11718782	C₂	semidirect
6. #11718777	C₂	semidirect
7. C₂×#88057	C₂	semidirect
8. #11718512	C₂	semidirect
9. #11718490	C₂	semidirect
10. #11718384	C₂	semidirect
11. #11718383	C₂	semidirect
12. #11718382	C₂	semidirect
13. #11718382	C₂	semidirect
14. #11718377	C₂	semidirect
15. #11718363	C₂	semidirect
268. C₁₃	#3344	semidirect

Maximal subgroups

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

subgroup	index
#11719248	2
#11719246	2
C₁₃×#435	2
#11718798	2
#11718782	2
#11718777	2
C₂×#88057	2
#11718512	2
#11718490	2
#11718384	2
#11718383	2
#11718382	2
#11718377	2
#11718363	2
#3344	13