Subgroups — #13127549

Order 1664.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719376	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#463	C₂	1	1	yes	maximal, characteristic
4	#11718825	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718823	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718814	C₂	1	1	yes	maximal, characteristic
7	D₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	#11718716	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718679	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718511	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718489	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718455	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718453	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718426	C₂	1	1	yes	maximal, characteristic, complemented
15	#11718420	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₂₆	—	2	1	no	—
17	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
19	C₁₃×#124	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
22	#88198	C₂²	1	1	yes	characteristic
23	#88190	C₂²	1	1	yes	characteristic
24	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
25	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×#6105	C₂²	1	1	yes	characteristic
28	#88178	—	2	1	no	—
29	#88142	C₂²	1	1	yes	characteristic
30	#88142	C₂²	1	1	yes	characteristic
31	#88133	—	2	1	no	—
32	#88133	—	2	1	no	—
33	#88132	—	2	1	no	—
34	#88132	—	2	1	no	—
35	#88132	—	2	1	no	—
36	#88131	—	2	1	no	—
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×#103	C₂²	1	1	yes	characteristic
42	C₁₃×#102	C₂²	1	1	yes	characteristic
43	#88075	C₂²	1	1	yes	characteristic
44	#88075	C₂²	1	1	yes	characteristic
45	#88072	C₂²	1	1	yes	characteristic
46	#88072	C₂²	1	1	yes	characteristic
47	#88072	C₂²	1	1	yes	characteristic
48	#88058	C₂²	1	1	yes	characteristic
49	#88058	C₂²	1	1	yes	characteristic
50	#88056	C₂²	1	1	yes	characteristic
51	#88056	C₂²	1	1	yes	characteristic
52	#88055	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	#88050	C₂²	1	1	yes	characteristic
57	#88047	C₂²	1	1	yes	characteristic
58	#88043	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×Dic₁₃	—	2	1	no	—
64	C₂²×Dic₁₃	—	2	1	no	—
65	C₂²×Dic₁₃	—	2	1	no	—
66	C₂×Dic₂₆	—	2	1	no	—
67	C₂×Dic₂₆	—	2	1	no	—
68	C₂×Dic₂₆	—	2	1	no	—
69	C₂×Dic₂₆	—	4	4	no	—
70	C₂×Dic₂₆	D₄	1	1	yes	characteristic
71	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
72	C₁₃×SD₁₆	—	2	2	no	—
73	C₁₃×D₈	—	2	2	no	—
74	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₁₃×C₂²⋊C₄	—	2	1	no	—
78	C₁₃×C₂²⋊C₄	—	2	1	no	—
79	C₁₃×C₂²⋊C₄	—	2	1	no	—
80	C₄²×C₁₃	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₁₃	—	2	1	no	—
82	C₂²⋊Dic₁₃	—	2	1	no	—
83	C₂²⋊Dic₁₃	—	2	1	no	—
84	C₂²⋊Dic₁₃	—	2	1	no	—
85	C₂²⋊Dic₁₃	—	2	1	no	—
86	#6105	—	2	2	no	—
87	C₄⋊Dic₁₃	—	2	1	no	—
88	C₄⋊Dic₁₃	—	2	1	no	—
89	C₄⋊Dic₁₃	D₄	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	D₄	1	1	yes	characteristic
99	#6101	D₄	1	1	yes	characteristic
100	C₄×Dic₁₃	—	2	1	no	—
101	C₄×Dic₁₃	—	2	1	no	—
102	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
103	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
104	C₂×#1055	C₂³	1	1	yes	characteristic
105	C₂×#1055	C₂³	1	1	yes	characteristic
106	#3367	—	13	1	no	maximal, Sylow
107	C₂³×C₁₃	—	2	1	no	—
108	C₂³×C₁₃	—	2	1	no	—
109	Q₈×C₁₃	—	2	1	no	—
110	Q₈×C₁₃	—	2	1	no	—
111	D₄×C₁₃	—	2	1	no	—
112	D₄×C₁₃	—	2	1	no	—
113	D₄×C₁₃	—	2	1	no	—
114	D₄×C₁₃	#41	1	2	yes	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
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	—	4	2	no	—
126	C₂×Dic₁₃	—	4	2	no	—
127	C₂×Dic₁₃	—	4	2	no	—
128	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
132	Dic₂₆	—	2	2	no	—
133	Dic₂₆	—	2	1	no	—
134	Dic₂₆	—	2	1	no	—
135	Dic₂₆	—	4	2	no	—
136	Dic₂₆	—	4	2	no	—
137	Dic₂₆	—	4	2	no	—
138	C₈×C₁₃	—	2	1	no	—
139	C₈×C₁₃	—	2	1	no	—
140	#1055	—	2	1	no	—
141	#1055	—	2	1	no	—
142	#556	—	13	1	no	—
143	D₄×Q₈	—	13	1	no	—
144	#489	—	13	1	no	—
145	#489	—	13	1	no	—
146	#487	—	13	1	no	—
147	#477	—	13	1	no	—
148	#477	—	13	1	no	—
149	#474	—	13	1	no	—
150	#463	—	13	1	no	—
151	#463	D₁₃	1	1	yes	characteristic
152	#461	—	13	1	no	—
153	#446	—	13	1	no	—
154	#440	—	13	1	no	—
155	#440	—	13	1	no	—
156	C₄×D₈	—	13	1	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	4	1	no	—
159	C₂²×C₁₃	—	4	1	no	—
160	C₂²×C₁₃	—	4	1	no	—
161	C₂²×C₁₃	—	4	1	no	—
162	C₂²×C₁₃	—	4	1	no	—
163	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	4	1	no	—
167	C₄×C₁₃	—	4	1	no	—
168	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
169	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
170	Dic₁₃	—	2	1	no	—
171	Dic₁₃	—	2	1	no	—
172	Dic₁₃	—	2	1	no	—
173	Dic₁₃	—	4	1	no	—
174	Dic₁₃	—	4	1	no	—
175	Dic₁₃	—	4	1	no	—
176	Dic₁₃	—	4	1	no	—
177	Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
178	C₂²×Q₈	—	26	1	no	—
179	C₂×SD₁₆	—	13	1	no	—
180	C₂×SD₁₆	C₂×D₁₃	1	1	yes	characteristic
181	C₂×D₈	C₂×D₁₃	1	1	yes	characteristic
182	C₄⋊Q₈	—	13	1	no	—
183	C₄⋊Q₈	—	13	1	no	—
184	C₄⋊Q₈	—	13	1	no	—
185	C₄⋊Q₈	—	13	1	no	—
186	#126	—	26	1	no	—
187	#126	—	26	1	no	—
188	#124	—	13	1	no	—
189	#124	C₂×D₁₃	1	1	yes	characteristic
190	C₂²⋊Q₈	—	26	1	no	—
191	C₂²⋊Q₈	—	26	1	no	—
192	C₂²⋊Q₈	—	26	1	no	—
193	C₂²⋊Q₈	—	26	1	no	—
194	C₄×Q₈	—	13	1	no	—
195	C₄×Q₈	—	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	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
201	#117	—	26	1	no	—
202	#107	—	13	1	no	—
203	#107	—	13	1	no	—
204	#107	—	13	1	no	—
205	C₄⋊C₈	—	13	1	no	—
206	C₄⋊C₈	—	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	C₂×D₁₃	1	1	yes	characteristic
214	#102	—	13	1	no	—
215	#102	—	13	1	no	—
216	#102	—	13	1	no	—
217	#102	C₂×D₁₃	1	1	yes	characteristic
218	#97	—	13	1	no	—
219	#97	—	13	1	no	—
220	C₄×C₈	—	13	1	no	—
221	C₂×C₁₃	—	4	2	no	—
222	C₂×C₁₃	—	8	1	no	—
223	C₂×C₁₃	#548	1	1	yes	characteristic
224	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
225	C₂×C₁₃	#578	1	1	yes	characteristic
226	C₂×Q₈	—	13	1	no	—
227	C₂×Q₈	—	13	1	no	—
228	C₂×Q₈	—	26	1	no	—
229	C₂×Q₈	—	26	1	no	—
230	C₂×Q₈	—	26	1	no	—
231	C₂×Q₈	—	52	4	no	—
232	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
234	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	26	1	no	—
237	C₂²×C₄	—	26	1	no	—
238	C₂²×C₄	—	26	1	no	—
239	SD₁₆	—	2	2	no	—
240	SD₁₆	—	26	2	no	—
241	D₈	—	2	2	no	—
242	C₂×C₈	—	13	1	no	—
243	C₂×C₈	—	13	1	no	—
244	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
246	C₄⋊C₄	—	13	1	no	—
247	C₄⋊C₄	—	13	1	no	—
248	C₄⋊C₄	—	13	1	no	—
249	C₄⋊C₄	—	13	1	no	—
250	C₄⋊C₄	—	13	1	no	—
251	C₄⋊C₄	—	13	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	—	26	1	no	—
257	C₄⋊C₄	—	26	1	no	—
258	C₄⋊C₄	—	26	1	no	—
259	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	26	1	no	—
264	C₂²⋊C₄	—	26	1	no	—
265	C₂²⋊C₄	—	26	1	no	—
266	C₂²⋊C₄	—	26	1	no	—
267	C₂²⋊C₄	—	26	1	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	13	1	no	—
270	C₄²	—	26	1	no	—
271	C₄²	—	26	1	no	—
272	C₄²	C₂²×D₁₃	1	1	yes	characteristic
273	C₁₃	#3367	1	1	yes	characteristic, complemented, Sylow
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	26	2	no	—
279	Q₈	—	26	1	no	—
280	Q₈	—	26	1	no	—
281	Q₈	—	52	2	no	—
282	Q₈	—	52	2	no	—
283	Q₈	—	52	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	#6129	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	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	—	26	1	no	—
296	C₂×C₄	—	26	1	no	—
297	C₂×C₄	—	26	1	no	—
298	C₂×C₄	—	26	1	no	—
299	C₂×C₄	—	52	2	no	—
300	C₂×C₄	—	52	2	no	—
301	C₂×C₄	—	52	2	no	—
302	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
303	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	26	1	no	—
308	C₈	—	26	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	13	2	no	—
321	C₄	—	26	1	no	—
322	C₄	—	26	1	no	—
323	C₄	—	26	1	no	—
324	C₄	—	52	1	no	—
325	C₄	—	52	1	no	—
326	C₄	—	52	1	no	—
327	C₄	—	52	1	no	—
328	C₄	C₂×#6129	1	1	yes	characteristic
329	C₄	#88257	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	#11719303	1	1	yes	characteristic
333	C₂	#11719476	1	1	yes	characteristic
334	C₂	D₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#463	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×C₁₃×D₈	C₂²
hypercenter	#463	D₁₃
hypocenter	C₁₃	#3367
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #11719376	C₂	semidirect
2. D₄×Dic₂₆	C₂	semidirect
4. #11718825	C₂	semidirect
5. #11718823	C₂	semidirect
8. #11718716	C₂	semidirect
9. #11718679	C₂	semidirect
10. #11718511	C₂	semidirect
11. #11718489	C₂	semidirect
12. #11718455	C₂	semidirect
13. #11718453	C₂	semidirect
14. #11718426	C₂	semidirect
15. #11718420	C₂	semidirect
273. C₁₃	#3367	semidirect

Maximal subgroups

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

subgroup	index
#11719376	2
D₄×Dic₂₆	2
C₁₃×#463	2
#11718825	2
#11718823	2
#11718814	2
D₈×Dic₁₃	2
#11718716	2
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#3367	13