Subgroups — #13125328

Order 1664.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88172	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×Q₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88170	C₂	1	1	yes	maximal, characteristic
4	C₂×#88169	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719429	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718851	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718842	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718829	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718828	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718753	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718720	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718376	C₂	1	1	yes	maximal, characteristic, complemented
15	#11718369	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#6131	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
19	C₂×#6129	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
24	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₁₃×#124	C₂²	1	1	yes	characteristic
26	#88201	C₂²	1	1	yes	characteristic
27	#88198	C₂²	1	1	yes	characteristic
28	#88198	C₂²	1	1	yes	characteristic
29	C₂×#6107	C₂²	1	1	yes	characteristic
30	C₂×#6107	C₂²	1	1	yes	characteristic
31	C₂×#6107	C₂²	1	1	yes	characteristic
32	C₂×#6107	C₂²	1	1	yes	characteristic
33	C₂×#6107	C₂²	1	1	yes	characteristic
34	C₂×#6106	C₂²	1	1	yes	characteristic
35	#88190	C₂²	1	1	yes	characteristic
36	C₂×#6105	C₂²	1	1	yes	characteristic
37	C₂×#6104	C₂²	1	1	yes	characteristic
38	#88172	—	2	4	no	—
39	Q₁₆×D₁₃	—	2	4	no	—
40	#88170	—	2	4	no	—
41	#88169	—	2	4	no	—
42	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
43	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
44	C₂×#6095	C₂²	1	1	yes	characteristic
45	C₂×#6095	C₂²	1	1	yes	characteristic
46	C₂×#6094	C₂²	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
48	#88150	—	2	2	no	—
49	#88148	—	2	2	no	—
50	#88137	—	2	2	no	—
51	#88132	—	2	2	no	—
52	#88129	C₂²	1	1	yes	characteristic
53	#88126	C₂²	1	1	yes	characteristic
54	#88123	C₂²	1	1	yes	characteristic
55	C₁₃×#97	C₂²	1	1	yes	characteristic
56	#88055	C₂²	1	1	yes	characteristic
57	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
58	C₄×#1055	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃	—	2	1	no	—
64	C₂²×Dic₁₃	—	2	1	no	—
65	#6131	—	2	2	no	—
66	#6131	—	2	2	no	—
67	Q₈×D₁₃	—	2	2	no	—
68	Q₈×D₁₃	—	2	2	no	—
69	Q₈×D₁₃	—	2	2	no	—
70	Q₈×D₁₃	—	2	2	no	—
71	#6129	—	2	2	no	—
72	#6129	—	2	2	no	—
73	C₂×D₅₂	—	2	1	no	—
74	C₂×D₅₂	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₃	—	2	1	no	—
76	C₂×C₄×D₁₃	—	2	1	no	—
77	C₂×C₄×D₁₃	—	2	1	no	—
78	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₂₆	—	2	1	no	—
80	C₂×Dic₂₆	—	2	1	no	—
81	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
82	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
83	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₁₃×Q₁₆	—	2	2	no	—
86	C₁₃×SD₁₆	—	2	2	no	—
87	C₁₃×SD₁₆	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₂²⋊C₄	—	2	2	no	—
92	C₄²×C₁₃	C₂³	1	1	yes	characteristic
93	C₂²⋊Dic₁₃	—	2	2	no	—
94	#6107	—	2	2	no	—
95	#6107	—	2	2	no	—
96	#6107	—	2	2	no	—
97	#6107	—	2	2	no	—
98	#6107	—	2	2	no	—
99	#6106	—	2	2	no	—
100	#6105	—	2	2	no	—
101	#6104	—	2	2	no	—
102	#6103	—	2	2	no	—
103	#6103	—	2	2	no	—
104	#6103	D₄	1	2	yes	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	#6101	—	2	2	no	—
107	#6101	—	2	2	no	—
108	#6101	D₄	1	2	yes	—
109	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
110	C₂×#1055	C₂³	1	1	yes	characteristic
111	C₂×#1055	C₂³	1	1	yes	characteristic
112	Dic₅₂	—	2	2	no	—
113	Dic₅₂	—	2	2	no	—
114	#6095	—	2	2	no	—
115	#6095	—	2	2	no	—
116	#6094	D₄	1	4	yes	—
117	C₈×D₁₃	D₄	1	4	yes	—
118	#3156	—	13	1	no	maximal, Sylow
119	C₂³×C₁₃	—	2	1	no	—
120	C₂²×D₁₃	—	2	1	no	—
121	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₁₃	—	2	1	no	—
123	Q₈×C₁₃	—	2	1	no	—
124	Q₈×C₁₃	—	2	1	no	—
125	Q₈×C₁₃	—	2	1	no	—
126	Q₈×C₁₃	—	2	1	no	—
127	Q₈×C₁₃	—	2	1	no	—
128	D₄×C₁₃	—	2	1	no	—
129	D₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
135	C₂²⋊D₁₃	—	4	2	no	—
136	C₂×Dic₁₃	—	2	1	no	—
137	C₂×Dic₁₃	—	2	1	no	—
138	C₂×Dic₁₃	—	2	1	no	—
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
141	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
142	D₅₂	—	2	1	no	—
143	D₅₂	—	2	1	no	—
144	D₅₂	—	4	2	no	—
145	C₄×D₁₃	—	4	2	no	—
146	C₄×D₁₃	—	4	2	no	—
147	C₄×D₁₃	—	4	2	no	—
148	C₄×D₁₃	C₂×D₄	1	2	yes	—
149	C₄×D₁₃	C₂×D₄	1	2	yes	—
150	Dic₂₆	—	2	1	no	—
151	Dic₂₆	—	2	1	no	—
152	Dic₂₆	—	2	1	no	—
153	Dic₂₆	—	2	1	no	—
154	Dic₂₆	—	2	1	no	—
155	Dic₂₆	—	2	1	no	—
156	Dic₂₆	—	4	2	no	—
157	Dic₂₆	—	4	2	no	—
158	C₈×C₁₃	C₂×D₄	1	2	yes	—
159	C₈×C₁₃	C₂×D₄	1	2	yes	—
160	#1055	C₂×D₄	1	2	yes	—
161	#1055	C₂×D₄	1	2	yes	—
162	C₂×#137	—	13	1	no	—
163	C₂×#137	—	13	1	no	—
164	C₂×#135	—	13	1	no	—
165	C₂²×Q₁₆	—	13	1	no	—
166	#536	—	13	1	no	—
167	#536	—	13	1	no	—
168	#497	—	13	1	no	—
169	#497	—	13	1	no	—
170	#497	—	13	1	no	—
171	#497	D₁₃	1	1	yes	characteristic, complemented
172	#495	—	13	1	no	—
173	#495	—	13	1	no	—
174	#494	—	13	1	no	—
175	#494	—	13	1	no	—
176	#405	—	13	1	no	—
177	C₂²×C₁₃	—	4	1	no	—
178	C₂²×C₁₃	—	4	2	no	—
179	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×D₁₃	—	2	2	no	—
181	C₂×D₁₃	—	4	1	no	—
182	C₂×D₁₃	—	4	2	no	—
183	C₂×D₁₃	#127	1	2	yes	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	—	4	1	no	—
186	C₄×C₁₃	—	4	1	no	—
187	C₄×C₁₃	—	4	1	no	—
188	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
189	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
190	Dic₁₃	—	2	2	no	—
191	Dic₁₃	—	4	1	no	—
192	Dic₁₃	—	4	1	no	—
193	Dic₁₃	—	4	1	no	—
194	Dic₁₃	#127	1	2	yes	—
195	C₂×#41	—	13	1	no	—
196	C₂×#41	—	13	1	no	—
197	C₂²×Q₈	—	13	1	no	—
198	C₂²×Q₈	—	13	1	no	—
199	#137	—	26	4	no	—
200	#137	—	26	4	no	—
201	#135	—	26	4	no	—
202	C₂×Q₁₆	—	13	1	no	—
203	C₂×Q₁₆	—	13	1	no	—
204	C₂×Q₁₆	—	13	1	no	—
205	C₂×Q₁₆	—	13	1	no	—
206	C₂×Q₁₆	—	13	1	no	—
207	C₂×Q₁₆	—	13	1	no	—
208	C₂×Q₁₆	—	13	1	no	—
209	C₂×Q₁₆	—	26	4	no	—
210	C₂×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
211	C₂×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
212	C₂×SD₁₆	—	13	1	no	—
213	C₂×SD₁₆	—	13	1	no	—
214	C₂×SD₁₆	—	13	1	no	—
215	C₂×SD₁₆	—	13	1	no	—
216	C₂×SD₁₆	C₂×D₁₃	1	1	yes	characteristic
217	C₂×SD₁₆	C₂×D₁₃	1	1	yes	characteristic
218	C₂×D₈	—	13	1	no	—
219	C₂×M₁₆	—	13	1	no	—
220	C₂²×C₈	—	13	1	no	—
221	C₄⋊Q₈	—	13	1	no	—
222	C₄⋊Q₈	—	13	1	no	—
223	C₄⋊Q₈	—	13	1	no	—
224	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
225	#124	—	13	1	no	—
226	#124	—	13	1	no	—
227	#124	—	13	1	no	—
228	#124	C₂×D₁₃	1	1	yes	characteristic
229	#123	—	26	2	no	—
230	#123	—	26	2	no	—
231	C₂²⋊Q₈	—	26	2	no	—
232	C₂²⋊Q₈	—	26	2	no	—
233	#117	—	13	1	no	—
234	#97	—	13	1	no	—
235	#97	C₂×D₁₃	1	1	yes	characteristic
236	C₄×C₈	—	13	1	no	—
237	C₄×C₈	—	13	1	no	—
238	C₂×C₁₃	—	8	1	no	—
239	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
240	C₂×C₁₃	#578	1	2	yes	—
241	D₁₃	—	2	2	no	—
242	D₁₃	—	8	1	no	—
243	#41	—	26	2	no	—
244	#41	—	26	2	no	—
245	#41	—	26	2	no	—
246	#41	—	26	2	no	—
247	C₂×Q₈	—	13	1	no	—
248	C₂×Q₈	—	13	1	no	—
249	C₂×Q₈	—	13	1	no	—
250	C₂×Q₈	—	26	1	no	—
251	C₂×Q₈	—	26	1	no	—
252	C₂×Q₈	—	26	2	no	—
253	C₂×Q₈	—	26	2	no	—
254	C₂×Q₈	—	26	2	no	—
255	C₂×Q₈	—	26	2	no	—
256	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
257	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
259	C₂×D₄	—	13	1	no	—
260	C₂×D₄	—	26	1	no	—
261	C₂×D₄	—	26	1	no	—
262	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
263	C₂²×C₄	—	13	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	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	26	2	no	—
271	Q₁₆	—	26	2	no	—
272	Q₁₆	—	26	2	no	—
273	Q₁₆	—	26	2	no	—
274	Q₁₆	—	26	2	no	—
275	Q₁₆	—	26	2	no	—
276	Q₁₆	—	26	2	no	—
277	SD₁₆	—	2	2	no	—
278	SD₁₆	—	2	2	no	—
279	SD₁₆	—	26	2	no	—
280	SD₁₆	—	26	2	no	—
281	SD₁₆	—	26	2	no	—
282	SD₁₆	—	26	2	no	—
283	D₈	—	26	2	no	—
284	M₁₆	—	13	4	no	—
285	C₂×C₈	—	13	1	no	—
286	C₂×C₈	—	13	1	no	—
287	C₂×C₈	—	13	4	no	—
288	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
289	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	13	2	no	—
292	C₄⋊C₄	—	26	2	no	—
293	C₄⋊C₄	—	26	2	no	—
294	C₄⋊C₄	—	26	2	no	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	13	2	no	—
297	C₂²⋊C₄	—	26	2	no	—
298	C₂²⋊C₄	—	26	2	no	—
299	C₂²⋊C₄	—	26	2	no	—
300	C₄²	—	13	1	no	—
301	C₄²	C₂²×D₁₃	1	1	yes	characteristic
302	C₁₃	#3156	1	1	yes	characteristic, complemented, Sylow
303	C₂³	—	2	1	no	—
304	C₂³	—	13	1	no	—
305	C₂³	—	26	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	26	1	no	—
313	Q₈	—	26	1	no	—
314	Q₈	—	26	1	no	—
315	Q₈	—	26	1	no	—
316	Q₈	—	26	1	no	—
317	Q₈	—	26	1	no	—
318	Q₈	—	52	2	no	—
319	Q₈	—	52	2	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	26	1	no	—
323	D₄	—	26	1	no	—
324	D₄	—	52	2	no	—
325	D₄	—	52	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	13	1	no	—
330	C₂×C₄	—	13	2	no	—
331	C₂×C₄	—	13	2	no	—
332	C₂×C₄	—	13	2	no	—
333	C₂×C₄	—	26	1	no	—
334	C₂×C₄	—	26	1	no	—
335	C₂×C₄	—	26	1	no	—
336	C₂×C₄	—	52	2	no	—
337	C₂×C₄	—	52	2	no	—
338	C₂×C₄	—	52	2	no	—
339	C₂×C₄	—	52	2	no	—
340	C₂×C₄	D₄×D₁₃	1	2	yes	—
341	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
342	C₈	—	13	2	no	—
343	C₈	—	13	2	no	—
344	C₈	D₄×D₁₃	1	2	yes	—
345	C₈	D₄×D₁₃	1	2	yes	—
346	C₂²	—	4	2	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	13	2	no	—
349	C₂²	—	26	2	no	—
350	C₂²	—	52	2	no	—
351	C₂²	—	52	1	no	—
352	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	13	2	no	—
358	C₄	—	26	2	no	—
359	C₄	—	52	1	no	—
360	C₄	—	52	1	no	—
361	C₄	—	52	1	no	—
362	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
363	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	26	2	no	—
366	C₂	—	104	1	no	—
367	C₂	D₁₃×#127	1	1	yes	characteristic
368	C₂	#11719484	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#497	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×#88170	C₂
hypercenter	#497	D₁₃
hypocenter	C₁₃	#3156
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. C₂×#88172	C₂	semidirect
2. C₂×Q₁₆×D₁₃	C₂	semidirect
4. C₂×#88169	C₂	semidirect
5. #11719429	C₂	semidirect
6. #11719380	C₂	semidirect
7. C₁₃×#497	C₂	semidirect
8. #11718851	C₂	semidirect
9. #11718842	C₂	semidirect
10. #11718829	C₂	semidirect
11. #11718828	C₂	semidirect
12. #11718753	C₂	semidirect
13. #11718720	C₂	semidirect
14. #11718376	C₂	semidirect
15. #11718369	C₂	semidirect
171. #497	D₁₃	semidirect
302. C₁₃	#3156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88172	2
C₂×Q₁₆×D₁₃	2
C₂×#88170	2
C₂×#88169	2
#11719429	2
#11719380	2
C₁₃×#497	2
#11718851	2
#11718842	2
#11718829	2
#11718828	2
#11718753	2
#11718720	2
#11718376	2
#11718369	2
#3156	13