Subgroups — #13125311

Order 1664.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88173	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88172	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719429	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718852	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718851	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718837	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718836	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718746	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718720	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718373	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×#6131	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
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, complemented
25	C₁₃×#124	C₂²	1	1	yes	characteristic
26	#88201	C₂²	1	1	yes	characteristic
27	#88201	C₂²	1	1	yes	characteristic, complemented
28	#88198	C₂²	1	1	yes	characteristic, complemented
29	C₂×#6107	C₂²	1	1	yes	characteristic, complemented
30	C₂×#6106	C₂²	1	1	yes	characteristic
31	C₂×#6106	C₂²	1	1	yes	characteristic
32	C₂×#6106	C₂²	1	1	yes	characteristic, complemented
33	C₂×#6106	C₂²	1	1	yes	characteristic, complemented
34	C₂×#6106	C₂²	1	1	yes	characteristic, complemented
35	#88194	C₂²	1	1	yes	characteristic
36	C₂×#6105	C₂²	1	1	yes	characteristic
37	C₂×#6104	C₂²	1	1	yes	characteristic
38	#88173	—	2	4	no	—
39	#88172	—	2	4	no	—
40	#88168	—	2	4	no	—
41	SD₁₆×D₁₃	—	2	4	no	—
42	C₂×D₁₀₄	C₂²	1	1	yes	characteristic
43	C₂×D₁₀₄	C₂²	1	1	yes	characteristic, complemented
44	C₂×#6095	C₂²	1	1	yes	characteristic
45	C₂×#6095	C₂²	1	1	yes	characteristic, complemented
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	#88138	—	2	2	no	—
51	#88136	—	2	2	no	—
52	#88129	C₂²	1	1	yes	characteristic, complemented
53	#88128	C₂²	1	1	yes	characteristic
54	#88126	C₂²	1	1	yes	characteristic
55	C₁₃×#97	C₂²	1	1	yes	characteristic, complemented
56	#88055	C₂²	1	1	yes	characteristic, complemented
57	C₈×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
58	C₄×#1055	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₃	—	2	1	no	—
60	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃	—	2	1	no	—
65	C₂×C₂²⋊D₁₃	—	2	2	no	—
66	#6131	—	2	2	no	—
67	#6131	—	2	2	no	—
68	#6131	—	2	2	no	—
69	#6131	—	2	2	no	—
70	Q₈×D₁₃	—	2	2	no	—
71	Q₈×D₁₃	—	2	2	no	—
72	D₄×D₁₃	—	2	2	no	—
73	D₄×D₁₃	—	2	2	no	—
74	C₂×D₅₂	—	2	1	no	—
75	C₂×D₅₂	—	2	1	no	—
76	C₂×D₅₂	—	2	2	no	—
77	C₂×D₅₂	C₂³	1	1	yes	characteristic
78	C₂×D₅₂	C₂³	1	1	yes	characteristic
79	C₂×D₅₂	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₁₃	—	2	1	no	—
81	C₂×C₄×D₁₃	—	2	1	no	—
82	C₂×C₄×D₁₃	—	2	1	no	—
83	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
84	C₂×Dic₂₆	—	2	1	no	—
85	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
86	C₁₃×Q₁₆	—	2	2	no	—
87	C₁₃×Q₁₆	—	2	2	no	—
88	C₁₃×SD₁₆	—	2	2	no	—
89	C₁₃×SD₁₆	—	2	2	no	—
90	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₂²⋊C₄	—	2	2	no	—
94	C₄²×C₁₃	C₂³	1	1	yes	characteristic
95	#6107	—	2	2	no	—
96	#6106	—	2	2	no	—
97	#6106	—	2	2	no	—
98	#6106	—	2	2	no	—
99	#6106	—	2	2	no	—
100	#6106	—	2	2	no	—
101	#6105	—	2	2	no	—
102	#6104	—	2	2	no	—
103	#6103	—	2	2	no	—
104	#6103	—	2	2	no	—
105	#6103	—	2	2	no	—
106	#6103	D₄	1	2	yes	—
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	D₁₀₄	—	2	2	no	—
113	D₁₀₄	—	2	2	no	—
114	#6095	—	2	2	no	—
115	#6095	—	2	2	no	—
116	#6094	D₄	1	4	yes	complemented
117	C₈×D₁₃	D₄	1	4	yes	complemented
118	#3154	—	13	1	no	maximal, Sylow
119	C₂³×C₁₃	—	2	1	no	—
120	C₂²×D₁₃	—	2	1	no	—
121	C₂²×D₁₃	—	2	1	no	—
122	C₂²×D₁₃	—	2	1	no	—
123	C₂²×D₁₃	—	4	2	no	—
124	C₂²×D₁₃	—	8	2	no	—
125	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
126	Q₈×C₁₃	—	2	1	no	—
127	Q₈×C₁₃	—	2	1	no	—
128	Q₈×C₁₃	—	2	1	no	—
129	Q₈×C₁₃	—	2	1	no	—
130	Q₈×C₁₃	—	2	1	no	—
131	Q₈×C₁₃	—	2	1	no	—
132	D₄×C₁₃	—	2	1	no	—
133	D₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
139	C₂²⋊D₁₃	—	4	2	no	—
140	C₂²⋊D₁₃	—	8	2	no	—
141	C₂×Dic₁₃	—	2	1	no	—
142	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
143	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
144	D₅₂	—	2	1	no	—
145	D₅₂	—	2	1	no	—
146	D₅₂	—	2	1	no	—
147	D₅₂	—	2	1	no	—
148	D₅₂	—	2	1	no	—
149	D₅₂	—	2	1	no	—
150	D₅₂	—	4	2	no	—
151	D₅₂	—	4	2	no	—
152	D₅₂	—	8	2	no	—
153	C₄×D₁₃	—	4	2	no	—
154	C₄×D₁₃	—	4	2	no	—
155	C₄×D₁₃	—	4	2	no	—
156	C₄×D₁₃	C₂×D₄	1	2	yes	—
157	C₄×D₁₃	C₂×D₄	1	2	yes	—
158	Dic₂₆	—	2	1	no	—
159	Dic₂₆	—	2	1	no	—
160	Dic₂₆	—	4	2	no	—
161	C₈×C₁₃	C₂×D₄	1	2	yes	—
162	C₈×C₁₃	C₂×D₄	1	2	yes	—
163	#1055	C₂×D₄	1	2	yes	—
164	#1055	C₂×D₄	1	2	yes	—
165	C₂×#137	—	13	1	no	—
166	C₂×AGL(1,ℤ/8)	—	13	1	no	—
167	C₂×#135	—	13	1	no	—
168	C₂²×SD₁₆	—	13	1	no	—
169	#536	—	13	1	no	—
170	#535	—	13	1	no	—
171	#497	—	13	1	no	—
172	#497	D₁₃	1	1	yes	characteristic, complemented
173	#496	—	13	1	no	—
174	#496	—	13	1	no	—
175	#495	—	13	1	no	—
176	#495	—	13	1	no	—
177	#492	—	13	1	no	—
178	#492	—	13	1	no	—
179	#405	—	13	1	no	—
180	C₂²×C₁₃	—	4	1	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
183	C₂×D₁₃	—	2	2	no	—
184	C₂×D₁₃	—	4	1	no	—
185	C₂×D₁₃	—	4	1	no	—
186	C₂×D₁₃	—	4	1	no	—
187	C₂×D₁₃	—	4	2	no	—
188	C₂×D₁₃	—	4	2	no	—
189	C₂×D₁₃	—	4	2	no	—
190	C₂×D₁₃	—	8	4	no	—
191	C₂×D₁₃	#127	1	2	yes	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	4	1	no	—
194	C₄×C₁₃	—	4	1	no	—
195	C₄×C₁₃	—	4	1	no	—
196	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
197	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
198	Dic₁₃	—	2	2	no	—
199	Dic₁₃	—	4	1	no	—
200	Dic₁₃	#127	1	2	yes	—
201	C₂×#41	—	13	1	no	—
202	C₂×#41	—	13	1	no	—
203	C₂²×Q₈	—	13	1	no	—
204	C₂²×D₄	—	13	1	no	—
205	#137	—	26	4	no	—
206	AGL(1,ℤ/8)	—	26	4	no	—
207	#135	—	26	4	no	—
208	C₂×Q₁₆	—	13	1	no	—
209	C₂×Q₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
210	C₂×Q₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
211	C₂×SD₁₆	—	13	1	no	—
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	—	13	1	no	—
217	C₂×SD₁₆	—	13	1	no	—
218	C₂×SD₁₆	—	13	1	no	—
219	C₂×SD₁₆	—	26	4	no	—
220	C₂×SD₁₆	C₂×D₁₃	1	1	yes	characteristic
221	C₂×SD₁₆	C₂×D₁₃	1	1	yes	characteristic
222	C₂×D₈	—	13	1	no	—
223	C₂×D₈	—	13	1	no	—
224	C₂×D₈	—	13	1	no	—
225	C₂×M₁₆	—	13	1	no	—
226	C₂²×C₈	—	13	1	no	—
227	C₄⋊Q₈	—	13	1	no	—
228	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
229	#127	—	13	1	no	—
230	#127	—	13	1	no	—
231	#124	—	13	1	no	—
232	#124	—	13	1	no	—
233	#124	—	13	1	no	—
234	#124	C₂×D₁₃	1	1	yes	characteristic
235	#123	—	26	2	no	—
236	C₂²⋊Q₈	—	26	2	no	—
237	#121	—	26	2	no	—
238	#120	—	26	2	no	—
239	#117	—	13	1	no	—
240	#97	—	13	1	no	—
241	#97	C₂×D₁₃	1	1	yes	characteristic, complemented
242	C₄×C₈	—	13	1	no	—
243	C₄×C₈	—	13	1	no	—
244	C₂×C₁₃	—	8	1	no	—
245	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
246	C₂×C₁₃	#577	1	2	yes	—
247	D₁₃	—	2	2	no	—
248	D₁₃	—	8	1	no	—
249	D₁₃	—	8	1	no	—
250	D₁₃	—	8	1	no	—
251	C₂⁴	—	26	1	no	—
252	#41	—	26	2	no	—
253	#41	—	26	2	no	—
254	#41	—	26	2	no	—
255	#41	—	26	2	no	—
256	C₂×Q₈	—	13	1	no	—
257	C₂×Q₈	—	26	1	no	—
258	C₂×Q₈	—	26	2	no	—
259	C₂×Q₈	—	26	2	no	—
260	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
261	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
262	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
263	C₂×D₄	—	13	1	no	—
264	C₂×D₄	—	13	1	no	—
265	C₂×D₄	—	13	1	no	—
266	C₂×D₄	—	26	1	no	—
267	C₂×D₄	—	26	1	no	—
268	C₂×D₄	—	26	2	no	—
269	C₂×D₄	—	26	2	no	—
270	C₂×D₄	—	26	1	no	—
271	C₂×D₄	—	26	2	no	—
272	C₂×D₄	—	26	2	no	—
273	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
274	C₂²×C₄	—	13	1	no	—
275	C₂²×C₄	—	26	1	no	—
276	C₂²×C₄	—	26	1	no	—
277	C₂²×C₄	—	26	1	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	26	2	no	—
281	SD₁₆	—	2	2	no	—
282	SD₁₆	—	2	2	no	—
283	SD₁₆	—	26	2	no	—
284	SD₁₆	—	26	2	no	—
285	SD₁₆	—	26	2	no	—
286	SD₁₆	—	26	2	no	—
287	SD₁₆	—	26	2	no	—
288	SD₁₆	—	26	2	no	—
289	SD₁₆	—	26	2	no	—
290	SD₁₆	—	26	2	no	—
291	D₈	—	26	2	no	—
292	D₈	—	26	2	no	—
293	D₈	—	26	2	no	—
294	M₁₆	—	13	4	no	—
295	C₂×C₈	—	13	1	no	—
296	C₂×C₈	—	13	1	no	—
297	C₂×C₈	—	13	4	no	—
298	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
299	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	13	2	no	—
302	C₄⋊C₄	—	26	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	13	2	no	—
305	C₂²⋊C₄	—	26	2	no	—
306	C₂²⋊C₄	—	26	2	no	—
307	C₂²⋊C₄	—	26	2	no	—
308	C₄²	—	13	1	no	—
309	C₄²	C₂²×D₁₃	1	1	yes	characteristic
310	C₁₃	#3154	1	1	yes	characteristic, complemented, Sylow
311	C₂³	—	2	1	no	—
312	C₂³	—	13	1	no	—
313	C₂³	—	26	1	no	—
314	C₂³	—	26	1	no	—
315	C₂³	—	26	1	no	—
316	C₂³	—	52	2	no	—
317	C₂³	—	104	2	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	—	26	1	no	—
325	Q₈	—	26	1	no	—
326	Q₈	—	52	2	no	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	26	1	no	—
330	D₄	—	26	1	no	—
331	D₄	—	26	1	no	—
332	D₄	—	26	1	no	—
333	D₄	—	26	1	no	—
334	D₄	—	26	1	no	—
335	D₄	—	52	2	no	—
336	D₄	—	52	2	no	—
337	D₄	—	52	2	no	—
338	D₄	—	104	2	no	—
339	D₄	—	104	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	13	1	no	—
344	C₂×C₄	—	13	2	no	—
345	C₂×C₄	—	13	2	no	—
346	C₂×C₄	—	13	2	no	—
347	C₂×C₄	—	26	1	no	—
348	C₂×C₄	—	52	2	no	—
349	C₂×C₄	—	52	2	no	—
350	C₂×C₄	—	52	2	no	—
351	C₂×C₄	D₄×D₁₃	1	2	yes	—
352	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
353	C₈	—	13	2	no	—
354	C₈	—	13	2	no	—
355	C₈	D₄×D₁₃	1	2	yes	—
356	C₈	D₄×D₁₃	1	2	yes	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	13	2	no	—
360	C₂²	—	26	2	no	—
361	C₂²	—	52	2	no	—
362	C₂²	—	52	2	no	—
363	C₂²	—	52	2	no	—
364	C₂²	—	52	1	no	—
365	C₂²	—	52	1	no	—
366	C₂²	—	52	1	no	—
367	C₂²	—	104	4	no	—
368	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	1	no	—
373	C₄	—	13	2	no	—
374	C₄	—	26	2	no	—
375	C₄	—	52	1	no	—
376	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
377	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
378	C₂	—	8	1	no	—
379	C₂	—	26	2	no	—
380	C₂	—	104	1	no	—
381	C₂	—	104	1	no	—
382	C₂	—	104	1	no	—
383	C₂	D₁₃×#127	1	1	yes	characteristic
384	C₂	#11719483	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
hypercenter	#497	D₁₃
hypocenter	C₁₃	#3154
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×#88173	C₂	semidirect
2. C₂×#88172	C₂	semidirect
3. C₂×#88168	C₂	semidirect
4. C₂×SD₁₆×D₁₃	C₂	semidirect
5. #11719429	C₂	semidirect
6. #11719379	C₂	semidirect
7. C₁₃×#497	C₂	semidirect
8. #11718852	C₂	semidirect
9. #11718851	C₂	semidirect
10. #11718837	C₂	semidirect
11. #11718836	C₂	semidirect
12. #11718746	C₂	semidirect
13. #11718720	C₂	semidirect
14. #11718373	C₂	semidirect
15. #11718369	C₂	semidirect
20. C₂×C₁₃×Q₁₆	C₂²	semidirect
21. C₂×C₁₃×Q₁₆	C₂²	semidirect
24. C₁₃×C₄⋊Q₈	C₂²	semidirect
27. #88201	C₂²	semidirect
28. #88198	C₂²	semidirect
29. C₂×#6107	C₂²	semidirect
32. C₂×#6106	C₂²	semidirect
33. C₂×#6106	C₂²	semidirect
34. C₂×#6106	C₂²	semidirect
43. C₂×D₁₀₄	C₂²	semidirect
45. C₂×#6095	C₂²	semidirect
52. #88129	C₂²	semidirect
55. C₁₃×#97	C₂²	semidirect
56. #88055	C₂²	semidirect
57. C₈×Dic₁₃	C₂²	semidirect
58. C₄×#1055	C₂²	semidirect
116. #6094	D₄	semidirect
117. C₈×D₁₃	D₄	semidirect
172. #497	D₁₃	semidirect
209. C₂×Q₁₆	C₂×D₁₃	semidirect
210. C₂×Q₁₆	C₂×D₁₃	semidirect
228. C₄⋊Q₈	C₂×D₁₃	semidirect
241. #97	C₂×D₁₃	semidirect
310. C₁₃	#3154	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88173	2
C₂×#88172	2
C₂×#88168	2
C₂×SD₁₆×D₁₃	2
#11719429	2
#11719379	2
C₁₃×#497	2
#11718852	2
#11718851	2
#11718837	2
#11718836	2
#11718746	2
#11718720	2
#11718373	2
#11718369	2
#3154	13