Subgroups — #13131594

Order 1664.

470 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719631	C₂	1	1	yes	maximal, characteristic
2	#11719627	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₃×#137	C₂	1	1	yes	maximal, characteristic
4	#11719165	C₂	1	1	yes	maximal, characteristic
5	C₂×#88117	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719110	C₂	1	2	yes	maximal, complemented
7	Q₁₆×#261	C₂	1	2	yes	maximal, complemented
8	#11719106	C₂	1	2	yes	maximal, complemented
9	SD₁₆×#261	C₂	1	2	yes	maximal, complemented
10	#11719099	C₂	1	1	yes	maximal, characteristic, complemented
11	#11719098	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
14	Q₈×#261	—	2	1	no	—
15	Q₈×#261	—	2	1	no	—
16	Q₈×#261	C₂²	1	2	yes	—
17	Q₈×#261	C₂²	1	2	yes	—
18	Q₈×#261	C₂²	1	2	yes	—
19	D₄×#261	—	2	2	no	—
20	D₄×#261	C₂²	1	2	yes	—
21	#88237	—	2	2	no	—
22	#88237	—	2	1	no	—
23	#88237	C₂²	1	1	yes	characteristic
24	#88237	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#261	—	2	1	no	—
26	C₂×C₄⋊#261	C₂²	1	1	yes	characteristic
27	C₂×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₃×#137	C₄	1	1	yes	characteristic, complemented
29	#88205	C₄	1	1	yes	characteristic, complemented
30	#88196	C₄	1	1	yes	characteristic, complemented
31	#88172	C₄	1	2	yes	complemented
32	Q₁₆×D₁₃	C₂²	1	2	yes	—
33	#88169	C₄	1	2	yes	complemented
34	SD₁₆×D₁₃	C₂²	1	2	yes	—
35	#88163	C₄	1	1	yes	characteristic, complemented
36	M₁₆×D₁₃	C₂²	1	1	yes	characteristic
37	#88117	C₂²	1	2	yes	—
38	#88117	C₂²	1	2	yes	—
39	#88117	C₂²	1	1	yes	characteristic
40	#88117	C₂²	1	1	yes	characteristic
41	#88115	C₂²	1	1	yes	characteristic
42	#88115	C₂²	1	1	yes	characteristic
43	#88102	C₂²	1	2	yes	—
44	#88101	C₂²	1	2	yes	—
45	#88100	C₂²	1	2	yes	—
46	C₈×#261	C₂²	1	2	yes	—
47	C₂²×#261	—	2	1	no	—
48	C₂²×#261	—	2	2	no	—
49	C₁₃×#41	C₂×C₄	1	1	yes	characteristic
50	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
51	#6131	C₂×C₄	1	1	yes	characteristic
52	Q₈×D₁₃	—	2	1	no	—
53	Q₈×D₁₃	—	2	1	no	—
54	Q₈×D₁₃	C₂×C₄	1	2	yes	complemented
55	Q₈×D₁₃	C₂³	1	2	yes	—
56	Q₈×D₁₃	C₂³	1	1	yes	characteristic
57	#6129	—	2	1	no	—
58	#6129	C₂×C₄	1	1	yes	characteristic
59	D₄×D₁₃	—	2	1	no	—
60	D₄×D₁₃	C₂³	1	1	yes	characteristic
61	#6127	—	2	1	no	—
62	#6127	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₃	—	2	1	no	—
64	C₂×C₄×D₁₃	—	2	1	no	—
65	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
66	C₂×Dic₂₆	—	2	1	no	—
67	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic, complemented
68	C₂²⋊#261	—	2	1	no	—
69	C₂²⋊#261	—	2	2	no	—
70	C₂²⋊#261	D₄	1	2	yes	—
71	C₂×#1057	—	2	1	no	—
72	C₄⋊#261	—	2	2	no	—
73	C₄⋊#261	—	2	2	no	—
74	C₄⋊#261	—	2	2	no	—
75	C₄⋊#261	—	2	1	no	—
76	C₄⋊#261	D₄	1	1	yes	characteristic
77	C₄⋊#261	D₄	1	1	yes	characteristic
78	C₄⋊#261	C₂³	1	1	yes	characteristic
79	C₄⋊#261	C₂³	1	2	yes	—
80	C₄⋊#261	C₂³	1	1	yes	characteristic
81	C₄×#261	—	2	2	no	—
82	C₄×#261	—	2	2	no	—
83	C₄×#261	—	2	2	no	—
84	C₄×#261	—	2	1	no	—
85	C₄×#261	C₂³	1	2	yes	—
86	D₁₃⋊C₈	—	2	1	no	—
87	D₁₃⋊C₈	C₂³	1	1	yes	characteristic
88	D₁₃⋊C₈	C₂³	1	1	yes	characteristic
89	C₁₃×Q₁₆	C₂×C₄	1	2	yes	complemented
90	C₁₃×SD₁₆	C₂×C₄	1	2	yes	complemented
91	C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic, complemented
92	#6107	C₂×C₄	1	2	yes	complemented
93	#6107	C₂×C₄	1	2	yes	complemented
94	#6106	C₂×C₄	1	2	yes	complemented
95	#6105	C₂×C₄	1	2	yes	complemented
96	#6099	C₂×C₄	1	1	yes	characteristic, complemented
97	Dic₅₂	C₂×C₄	1	2	yes	complemented
98	#6095	C₂×C₄	1	2	yes	complemented
99	#6094	C₂×C₄	1	2	yes	complemented
100	C₈×D₁₃	C₂³	1	2	yes	—
101	#2947	—	13	1	no	maximal, Sylow
102	C₂²×D₁₃	—	2	1	no	—
103	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×#261	—	2	1	no	—
105	C₂×#261	—	2	1	no	—
106	C₂×#261	—	2	2	no	—
107	C₂×#261	—	8	2	no	—
108	C₂×#261	—	8	1	no	—
109	C₂×#261	C₂×D₄	1	2	yes	—
110	C₂×#261	C₂×D₄	1	2	yes	—
111	Q₈×C₁₃	—	2	1	no	—
112	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
113	Q₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	D₄×C₁₃	—	2	1	no	—
115	D₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²⋊D₁₃	—	2	1	no	—
120	C₂²⋊D₁₃	—	2	1	no	—
121	C₂²⋊D₁₃	—	2	1	no	—
122	C₂×Dic₁₃	—	2	1	no	—
123	C₂×Dic₁₃	—	2	1	no	—
124	C₂×Dic₁₃	#41	1	1	yes	characteristic
125	D₅₂	—	2	1	no	—
126	D₅₂	C₂²×C₄	1	1	yes	characteristic
127	C₄×D₁₃	—	2	2	no	—
128	C₄×D₁₃	—	2	1	no	—
129	C₄×D₁₃	—	2	2	no	—
130	C₄×D₁₃	—	2	1	no	—
131	C₄×D₁₃	—	2	1	no	—
132	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
133	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
134	C₄×D₁₃	#41	1	1	yes	characteristic
135	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
136	Dic₂₆	—	2	1	no	—
137	Dic₂₆	—	2	2	no	—
138	Dic₂₆	—	2	2	no	—
139	Dic₂₆	—	2	1	no	—
140	Dic₂₆	C₂²×C₄	1	2	yes	—
141	Dic₂₆	C₂²×C₄	1	1	yes	characteristic
142	#1057	—	2	1	no	—
143	#1057	—	2	1	no	—
144	C₈×C₁₃	C₂²×C₄	1	2	yes	—
145	#1055	C₂²×C₄	1	2	yes	—
146	C₂×#137	—	13	1	no	—
147	#520	—	13	1	no	—
148	#519	—	13	1	no	—
149	#441	—	13	2	no	—
150	#440	—	13	2	no	—
151	C₄×Q₁₆	—	13	2	no	—
152	C₄×SD₁₆	—	13	2	no	—
153	#428	—	13	1	no	—
154	#416	—	13	1	no	—
155	C₂×#103	—	13	1	no	—
156	#405	—	13	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
159	C₂×D₁₃	—	2	1	no	—
160	C₂×D₁₃	—	2	1	no	—
161	C₂×D₁₃	—	2	1	no	—
162	C₂×D₁₃	—	4	1	no	—
163	C₂×D₁₃	—	4	1	no	—
164	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
165	C₂×D₁₃	C₂²×D₄	1	1	yes	characteristic
166	#261	—	2	2	no	—
167	#261	—	2	2	no	—
168	#261	—	4	1	no	—
169	#261	—	4	2	no	—
170	#261	—	4	1	no	—
171	C₄×C₁₃	—	2	2	no	—
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
174	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
175	Dic₁₃	—	2	2	no	—
176	Dic₁₃	—	2	1	no	—
177	Dic₁₃	C₄×D₄	1	1	yes	characteristic
178	Dic₁₃	C₂×#41	1	1	yes	characteristic
179	C₂×#41	—	13	1	no	—
180	C₂²×Q₈	—	13	1	no	—
181	#137	—	13	1	no	—
182	#137	—	13	1	no	—
183	#137	—	13	2	no	—
184	#137	—	13	2	no	—
185	#137	—	13	1	no	—
186	#137	#261	1	1	yes	characteristic, complemented
187	C₂×Q₁₆	—	13	2	no	—
188	C₂×SD₁₆	—	13	2	no	—
189	C₂×M₁₆	—	13	1	no	—
190	C₂²×C₈	—	13	1	no	—
191	C₄×Q₈	—	13	2	no	—
192	C₄×Q₈	—	13	2	no	—
193	C₄×Q₈	—	13	2	no	—
194	C₄×Q₈	—	26	1	no	—
195	C₄×Q₈	—	26	1	no	—
196	C₄×D₄	—	13	2	no	—
197	C₄×D₄	—	26	2	no	—
198	#117	—	13	1	no	—
199	#117	—	13	1	no	—
200	#117	—	26	2	no	—
201	#117	—	26	1	no	—
202	C₂×C₄⋊C₄	—	13	1	no	—
203	C₂×C₄⋊C₄	—	26	1	no	—
204	#107	—	13	2	no	—
205	#106	—	13	2	no	—
206	#103	—	13	2	no	—
207	#103	—	13	2	no	—
208	#103	—	13	1	no	—
209	#103	—	13	1	no	—
210	#102	—	13	1	no	—
211	#102	—	13	1	no	—
212	#97	—	13	2	no	—
213	C₄×C₈	—	13	2	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	4	1	no	—
216	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
217	D₁₃	—	2	1	no	—
218	D₁₃	—	4	1	no	—
219	D₁₃	#577	1	1	yes	characteristic
220	D₁₃	#578	1	1	yes	characteristic
221	#41	—	13	1	no	—
222	#41	—	13	1	no	—
223	#41	—	13	1	no	—
224	#41	—	26	1	no	—
225	#41	—	26	1	no	—
226	#41	C₂×#261	1	1	yes	characteristic
227	C₂×Q₈	—	13	2	no	—
228	C₂×Q₈	—	13	1	no	—
229	C₂×Q₈	—	13	2	no	—
230	C₂×Q₈	—	13	1	no	—
231	C₂×Q₈	—	26	1	no	—
232	C₂×Q₈	—	26	1	no	—
233	C₂×Q₈	—	26	1	no	—
234	C₂×Q₈	C₂×#261	1	1	yes	characteristic, complemented
235	C₂×D₄	—	13	1	no	—
236	C₂×D₄	—	26	1	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	26	1	no	—
239	C₂²×C₄	—	26	1	no	—
240	C₂²×C₄	—	26	2	no	—
241	C₂²×C₄	—	26	1	no	—
242	Q₁₆	—	13	2	no	—
243	Q₁₆	—	13	2	no	—
244	Q₁₆	—	13	2	no	—
245	Q₁₆	C₂×#261	1	2	yes	complemented
246	SD₁₆	—	13	2	no	—
247	SD₁₆	—	13	2	no	—
248	SD₁₆	—	13	2	no	—
249	SD₁₆	C₂×#261	1	2	yes	complemented
250	M₁₆	—	13	1	no	—
251	M₁₆	—	13	2	no	—
252	M₁₆	C₂×#261	1	1	yes	characteristic, complemented
253	C₂×C₈	—	13	2	no	—
254	C₂×C₈	—	13	1	no	—
255	C₂×C₈	—	13	1	no	—
256	C₂×C₈	—	26	1	no	—
257	C₂×C₈	—	26	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	—	13	2	no	—
263	C₄⋊C₄	—	26	1	no	—
264	C₄⋊C₄	—	26	2	no	—
265	C₄⋊C₄	—	26	2	no	—
266	C₄⋊C₄	—	26	2	no	—
267	C₂²⋊C₄	—	13	2	no	—
268	C₂²⋊C₄	—	26	2	no	—
269	C₂²⋊C₄	—	26	1	no	—
270	C₄²	—	13	2	no	—
271	C₄²	—	26	1	no	—
272	C₄²	—	26	2	no	—
273	C₄²	—	26	2	no	—
274	C₄²	—	26	2	no	—
275	C₁₃	#2947	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	13	1	no	—
277	C₂³	—	26	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	13	2	no	—
280	Q₈	—	13	1	no	—
281	Q₈	—	26	2	no	—
282	Q₈	—	26	2	no	—
283	Q₈	—	26	1	no	—
284	Q₈	—	26	1	no	—
285	Q₈	C₂²×#261	1	2	yes	—
286	Q₈	C₂²×#261	1	1	yes	characteristic
287	D₄	—	2	1	no	—
288	D₄	—	13	1	no	—
289	D₄	—	26	1	no	—
290	D₄	—	26	1	no	—
291	D₄	—	26	1	no	—
292	D₄	—	26	1	no	—
293	D₄	C₂²×#261	1	1	yes	characteristic
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	13	1	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	13	1	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	26	1	no	—
304	C₂×C₄	—	26	1	no	—
305	C₂×C₄	—	26	2	no	—
306	C₂×C₄	—	26	2	no	—
307	C₂×C₄	—	26	1	no	—
308	C₂×C₄	—	26	1	no	—
309	C₂×C₄	—	26	1	no	—
310	C₂×C₄	—	26	1	no	—
311	C₂×C₄	—	26	1	no	—
312	C₂×C₄	—	26	2	no	—
313	C₂×C₄	—	104	2	no	—
314	C₂×C₄	—	104	1	no	—
315	C₂×C₄	C₂²×#261	1	1	yes	characteristic
316	C₈	—	13	2	no	—
317	C₈	—	26	1	no	—
318	C₈	—	26	1	no	—
319	C₈	C₂²×#261	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	13	1	no	—
322	C₂²	—	13	1	no	—
323	C₂²	—	26	1	no	—
324	C₂²	—	26	1	no	—
325	C₂²	—	26	1	no	—
326	C₂²	—	52	1	no	—
327	C₂²	—	52	1	no	—
328	C₂²	D₄×#261	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	13	1	no	—
332	C₄	—	13	1	no	—
333	C₄	—	26	2	no	—
334	C₄	—	26	2	no	—
335	C₄	—	26	1	no	—
336	C₄	—	26	2	no	—
337	C₄	—	52	1	no	—
338	C₄	—	52	1	no	—
339	C₄	—	52	2	no	—
340	C₄	D₄×#261	1	1	yes	characteristic
341	C₄	C₂³×#261	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	4	1	no	—
344	C₂	—	13	1	no	—
345	C₂	—	13	1	no	—
346	C₂	—	26	1	no	—
347	C₂	—	52	1	no	—
348	C₂	C₂×D₄×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₄	C₂³×#261
Fitting	C₁₃×#137	C₄
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	#41×D₁₃	C₂²
cosocle	C₄×D₁₃	C₂⁴
hypercenter	#137	#261
hypocenter	C₁₃	#2947
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. #11719627	C₂	semidirect
5. C₂×#88117	C₂	semidirect
6. #11719110	C₂	semidirect
7. Q₁₆×#261	C₂	semidirect
8. #11719106	C₂	semidirect
9. SD₁₆×#261	C₂	semidirect
10. #11719099	C₂	semidirect
11. #11719098	C₂	semidirect
28. C₁₃×#137	C₄	semidirect
29. #88205	C₄	semidirect
30. #88196	C₄	semidirect
31. #88172	C₄	semidirect
33. #88169	C₄	semidirect
35. #88163	C₄	semidirect
50. C₂×Q₈×C₁₃	C₂×C₄	semidirect
54. Q₈×D₁₃	C₂×C₄	semidirect
67. C₂×Dic₂₆	C₂×C₄	semidirect
89. C₁₃×Q₁₆	C₂×C₄	semidirect
90. C₁₃×SD₁₆	C₂×C₄	semidirect
91. C₁₃×M₁₆	C₂×C₄	semidirect
92. #6107	C₂×C₄	semidirect
93. #6107	C₂×C₄	semidirect
94. #6106	C₂×C₄	semidirect
95. #6105	C₂×C₄	semidirect
96. #6099	C₂×C₄	semidirect
97. Dic₅₂	C₂×C₄	semidirect
98. #6095	C₂×C₄	semidirect
99. #6094	C₂×C₄	semidirect
186. #137	#261	semidirect
234. C₂×Q₈	C₂×#261	semidirect
245. Q₁₆	C₂×#261	semidirect
249. SD₁₆	C₂×#261	semidirect
252. M₁₆	C₂×#261	semidirect
275. C₁₃	#2947	semidirect

Maximal subgroups

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

subgroup	index
#11719631	2
#11719627	2
D₁₃×#137	2
#11719165	2
C₂×#88117	2
#11719110	2
Q₁₆×#261	2
#11719106	2
SD₁₆×#261	2
#11719099	2
#11719098	2
#2947	13