Subgroups — #15854510

Order 3328.

692 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127811	C₂	1	1	yes	maximal, characteristic, complemented
2	#13127165	C₂	1	2	yes	maximal, complemented
3	#13126558	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114442	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114398	C₂	1	2	yes	maximal, complemented
6	#13114206	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114191	C₂	1	1	yes	maximal, characteristic, complemented
8	#13114039	C₂	1	2	yes	maximal, complemented
9	#13113928	C₂	1	2	yes	maximal, complemented
10	#13113722	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88158	—	2	1	no	—
13	#11719282	C₂²	1	2	yes	—
14	#11719256	C₂²	1	2	yes	—
15	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
17	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
18	#11718833	—	2	2	no	—
19	#11718777	—	2	1	no	—
20	#11718681	C₂²	1	2	yes	—
21	#11718676	C₂²	1	2	yes	—
22	#11718524	—	2	2	no	—
23	#11718504	—	2	2	no	—
24	#11718504	C₂²	1	2	yes	—
25	#11718502	C₂²	1	2	yes	—
26	#11718464	—	2	2	no	—
27	#11718443	—	2	2	no	—
28	#11718394	—	2	2	no	—
29	#11718360	C₂²	1	1	yes	characteristic
30	#11718356	—	2	1	no	—
31	#11718356	C₂²	1	1	yes	characteristic
32	#11718355	C₂²	1	2	yes	—
33	#11718355	C₂²	1	2	yes	—
34	C₄×D₁₀₄	—	2	1	no	—
35	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
36	#11718349	C₂²	1	2	yes	—
37	C₁₃×#334	C₂²	1	1	yes	characteristic
38	C₁₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₈²×C₁₃	C₂²	1	1	yes	characteristic
41	#11718077	C₂²	1	2	yes	—
42	#11718074	C₂²	1	2	yes	—
43	#11718051	C₂²	1	2	yes	—
44	#11718050	C₂²	1	1	yes	characteristic
45	C₂×#6131	—	2	2	no	—
46	C₂×#6127	—	2	2	no	—
47	C₂×C₁₃×SD₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	—	2	1	no	—
49	C₄×Q₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×D₄×C₁₃	C₂³	1	1	yes	characteristic
51	#88192	—	2	2	no	—
52	C₂×#6104	—	2	2	no	—
53	#88178	—	2	2	no	—
54	C₂×Dic₅₂	—	2	1	no	—
55	C₂×Dic₅₂	—	2	1	no	—
56	#88158	—	4	4	no	—
57	C₂×D₁₀₄	—	2	1	no	—
58	C₂×D₁₀₄	—	2	2	no	—
59	C₂×D₁₀₄	—	2	1	no	—
60	C₂×D₁₀₄	D₄	1	1	yes	characteristic
61	C₂×#6095	—	2	2	no	—
62	C₂×#6094	—	2	2	no	—
63	#88149	—	2	2	no	—
64	#88149	—	4	2	no	—
65	C₄⋊C₄×D₁₃	—	2	2	no	—
66	#88137	—	4	2	no	—
67	#88128	—	2	2	no	—
68	#88128	C₂³	1	2	yes	—
69	C₄×D₅₂	—	2	2	no	—
70	C₄×D₅₂	C₂³	1	2	yes	—
71	#88123	C₂³	1	2	yes	—
72	C₁₃×#106	D₄	1	1	yes	characteristic
73	C₁₃×C₄⋊C₈	—	2	1	no	—
74	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
75	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
76	C₁₃×#103	D₄	1	1	yes	characteristic
77	C₁₃×#102	D₄	1	1	yes	characteristic
78	C₁₃×C₂²⋊C₈	—	2	1	no	—
79	C₄×C₈×C₁₃	—	2	1	no	—
80	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
83	#88075	—	2	2	no	—
84	#88061	—	2	2	no	—
85	#88061	—	2	2	no	—
86	#88061	—	2	2	no	—
87	#88061	—	4	2	no	—
88	#88061	D₄	1	2	yes	—
89	#88060	—	2	2	no	—
90	#88058	D₄	1	1	yes	characteristic
91	#88056	—	2	2	no	—
92	#88049	—	2	2	no	—
93	#88048	—	2	2	no	—
94	C₄⋊#1055	C₂³	1	2	yes	—
95	#19908	—	13	1	no	maximal, Sylow
96	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×D₄×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₃	—	2	1	no	—
99	C₂×C₂²⋊D₁₃	—	4	2	no	—
100	#6131	—	4	4	no	—
101	#6131	—	8	2	no	—
102	#6127	—	4	4	no	—
103	#6127	—	8	2	no	—
104	C₂×D₅₂	—	2	2	no	—
105	C₂×D₅₂	—	2	2	no	—
106	C₂×D₅₂	—	2	2	no	—
107	C₂×D₅₂	—	4	2	no	—
108	C₂×D₅₂	—	4	2	no	—
109	C₂×D₅₂	C₂×D₄	1	2	yes	—
110	C₂×C₄×D₁₃	—	2	2	no	—
111	C₂×C₄×D₁₃	—	4	2	no	—
112	C₂×C₄×D₁₃	—	4	2	no	—
113	C₂×Dic₂₆	—	2	2	no	—
114	C₁₃×SD₁₆	—	2	2	no	—
115	C₂×C₈×C₁₃	—	2	1	no	—
116	C₂×C₈×C₁₃	—	2	1	no	—
117	C₂×C₈×C₁₃	—	2	1	no	—
118	C₂×C₈×C₁₃	—	4	2	no	—
119	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
123	C₁₃×C₄⋊C₄	—	2	1	no	—
124	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
125	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
126	C₁₃×C₂²⋊C₄	—	2	1	no	—
127	C₄²×C₁₃	—	2	1	no	—
128	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
129	C₂²⋊Dic₁₃	—	4	2	no	—
130	#6104	—	8	2	no	—
131	#6103	—	2	2	no	—
132	#6103	—	4	2	no	—
133	C₄⋊Dic₁₃	—	2	2	no	—
134	C₄⋊Dic₁₃	—	2	2	no	—
135	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
136	#6101	—	4	2	no	—
137	C₂×#1055	—	2	2	no	—
138	Dic₅₂	—	4	2	no	—
139	Dic₅₂	—	8	1	no	—
140	D₁₀₄	—	2	2	no	—
141	D₁₀₄	—	4	2	no	—
142	D₁₀₄	—	8	1	no	—
143	D₁₀₄	—	8	2	no	—
144	#6095	—	4	4	no	—
145	#6094	—	8	2	no	—
146	#3391	—	13	1	no	—
147	#3332	—	13	2	no	—
148	#3283	—	13	1	no	—
149	#1709	—	13	1	no	—
150	#1701	—	13	2	no	—
151	#1670	—	13	1	no	—
152	#1667	—	13	1	no	—
153	#1642	—	13	2	no	—
154	#1625	—	13	2	no	—
155	#1589	—	13	1	no	—
156	C₈×SD₁₆	D₁₃	1	1	yes	characteristic, complemented
157	C₂³×C₁₃	—	2	1	no	—
158	C₂²×D₁₃	—	2	2	no	—
159	C₂²×D₁₃	—	4	2	no	—
160	Q₈×C₁₃	—	2	1	no	—
161	Q₈×C₁₃	#135	1	2	yes	complemented
162	D₄×C₁₃	—	2	1	no	—
163	D₄×C₁₃	—	2	1	no	—
164	C₂×C₄×C₁₃	—	2	1	no	—
165	C₂×C₄×C₁₃	—	2	1	no	—
166	C₂×C₄×C₁₃	—	2	1	no	—
167	C₂×C₄×C₁₃	—	4	1	no	—
168	C₂×C₄×C₁₃	—	4	1	no	—
169	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂²⋊D₁₃	—	8	4	no	—
173	C₂×Dic₁₃	—	2	2	no	—
174	C₂×Dic₁₃	—	4	2	no	—
175	D₅₂	—	2	2	no	—
176	D₅₂	—	2	2	no	—
177	D₅₂	—	4	2	no	—
178	D₅₂	—	4	2	no	—
179	D₅₂	—	4	2	no	—
180	D₅₂	—	4	2	no	—
181	D₅₂	—	8	4	no	—
182	D₅₂	—	8	2	no	—
183	D₅₂	—	16	2	no	—
184	C₄×D₁₃	—	4	2	no	—
185	C₄×D₁₃	—	4	2	no	—
186	C₄×D₁₃	—	8	4	no	—
187	C₄×D₁₃	—	8	4	no	—
188	Dic₂₆	—	4	2	no	—
189	Dic₂₆	—	4	2	no	—
190	C₈×C₁₃	—	2	1	no	—
191	C₈×C₁₃	—	2	1	no	—
192	C₈×C₁₃	—	2	1	no	—
193	C₈×C₁₃	—	4	1	no	—
194	C₈×C₁₃	—	4	1	no	—
195	C₈×C₁₃	—	4	1	no	—
196	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
197	#1055	—	4	2	no	—
198	C₂×#135	—	26	1	no	—
199	#550	—	13	2	no	—
200	#547	—	13	2	no	—
201	#494	—	13	1	no	—
202	#493	—	13	1	no	—
203	#493	—	26	1	no	—
204	#486	—	13	2	no	—
205	#486	—	13	2	no	—
206	#481	—	26	2	no	—
207	#475	—	13	2	no	—
208	#469	—	26	2	no	—
209	#467	—	26	1	no	—
210	#461	—	13	2	no	—
211	#459	—	13	2	no	—
212	#459	—	13	2	no	—
213	#459	—	26	2	no	—
214	#452	—	26	2	no	—
215	#449	—	26	2	no	—
216	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
217	C₄×SD₁₆	C₂×D₁₃	1	1	yes	characteristic
218	C₄×D₈	—	13	1	no	—
219	C₄×D₈	—	26	1	no	—
220	#436	—	13	2	no	—
221	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
222	#417	—	26	2	no	—
223	#335	—	13	1	no	—
224	#334	C₂×D₁₃	1	1	yes	characteristic
225	#332	—	13	2	no	—
226	#331	—	13	2	no	—
227	Q₈⋊C₈	C₂×D₁₃	1	1	yes	characteristic
228	D₄⋊C₈	—	13	2	no	—
229	D₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
230	C₈²	C₂×D₁₃	1	1	yes	characteristic
231	C₂²×C₁₃	—	4	1	no	—
232	C₂²×C₁₃	—	4	1	no	—
233	C₂²×C₁₃	—	4	1	no	—
234	C₂²×C₁₃	D₄²	1	1	yes	characteristic
235	C₂×D₁₃	—	4	2	no	—
236	C₂×D₁₃	—	4	2	no	—
237	C₂×D₁₃	—	4	2	no	—
238	C₂×D₁₃	—	8	2	no	—
239	C₂×D₁₃	—	8	2	no	—
240	C₂×D₁₃	—	8	2	no	—
241	C₄×C₁₃	—	2	2	no	—
242	C₄×C₁₃	—	2	1	no	—
243	C₄×C₁₃	—	4	1	no	—
244	C₄×C₁₃	—	4	1	no	—
245	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
246	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₄×C₁₃	#576	1	1	yes	characteristic
248	C₄×C₁₃	#576	1	1	yes	characteristic
249	Dic₁₃	—	4	2	no	—
250	Dic₁₃	—	8	2	no	—
251	C₂×#41	—	26	2	no	—
252	C₂×#41	—	26	2	no	—
253	#135	—	52	4	no	—
254	C₂×Q₁₆	—	26	1	no	—
255	C₂×Q₁₆	—	26	1	no	—
256	C₂×SD₁₆	—	26	2	no	—
257	C₂×SD₁₆	D₅₂	1	1	yes	characteristic
258	C₂×D₈	—	13	1	no	—
259	C₂×D₈	—	26	2	no	—
260	C₂×D₈	—	26	2	no	—
261	C₂×D₈	—	26	1	no	—
262	C₂×D₈	—	26	1	no	—
263	C₂×M₁₆	—	26	2	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₄⋊Q₈	—	13	2	no	—
266	#127	—	13	2	no	—
267	#127	—	26	2	no	—
268	#123	—	52	2	no	—
269	C₂²⋊Q₈	—	26	2	no	—
270	#121	—	26	2	no	—
271	#121	—	26	2	no	—
272	#121	—	52	2	no	—
273	C₄×Q₈	C₂²×D₁₃	1	1	yes	characteristic
274	C₄×D₄	—	13	2	no	—
275	C₄×D₄	—	26	2	no	—
276	C₄×D₄	C₂²×D₁₃	1	1	yes	characteristic
277	C₂×C₄⋊C₄	—	26	2	no	—
278	#107	—	13	1	no	—
279	#106	—	26	2	no	—
280	#106	D₅₂	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	13	2	no	—
283	C₄⋊C₈	C₂²×D₁₃	1	1	yes	characteristic
284	C₄⋊C₈	C₂²×D₁₃	1	1	yes	characteristic
285	#103	—	26	2	no	—
286	#103	—	26	2	no	—
287	#103	D₅₂	1	1	yes	characteristic
288	#102	—	13	2	no	—
289	#102	—	26	2	no	—
290	#102	—	26	2	no	—
291	#102	—	26	2	no	—
292	#102	—	26	2	no	—
293	#102	—	52	2	no	—
294	#102	D₅₂	1	1	yes	characteristic
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	26	2	no	—
297	C₄×C₈	—	2	1	no	—
298	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
299	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
300	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
301	C₂×C₁₃	—	8	1	no	—
302	C₂×C₁₃	#3274	1	1	yes	characteristic
303	C₂×C₁₃	#3280	1	1	yes	characteristic
304	C₂×C₁₃	#3290	1	1	yes	characteristic
305	D₁₃	—	8	2	no	—
306	D₁₃	—	16	2	no	—
307	#41	—	52	4	no	—
308	#41	—	52	4	no	—
309	#41	—	104	2	no	—
310	#41	—	104	2	no	—
311	C₂×Q₈	—	26	2	no	—
312	C₂×Q₈	C₂×D₅₂	1	1	yes	characteristic
313	C₂×D₄	—	13	2	no	—
314	C₂×D₄	—	26	2	no	—
315	C₂×D₄	—	26	2	no	—
316	C₂×D₄	—	26	2	no	—
317	C₂×D₄	—	52	2	no	—
318	C₂×D₄	—	52	2	no	—
319	C₂×D₄	—	52	2	no	—
320	C₂×D₄	C₂×D₅₂	1	1	yes	characteristic
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	26	2	no	—
323	C₂²×C₄	—	52	2	no	—
324	C₂²×C₄	—	52	2	no	—
325	Q₁₆	—	52	2	no	—
326	Q₁₆	—	104	1	no	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	52	4	no	—
329	D₈	—	26	2	no	—
330	D₈	—	52	2	no	—
331	D₈	—	104	1	no	—
332	D₈	—	104	2	no	—
333	D₈	—	104	2	no	—
334	M₁₆	—	104	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	4	2	no	—
339	C₂×C₈	—	26	2	no	—
340	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
341	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
342	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
343	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	13	2	no	—
346	C₄⋊C₄	—	26	2	no	—
347	C₄⋊C₄	—	26	2	no	—
348	C₄⋊C₄	—	52	2	no	—
349	C₄⋊C₄	C₂×D₅₂	1	1	yes	characteristic
350	C₄⋊C₄	C₂×D₅₂	1	1	yes	characteristic
351	C₂²⋊C₄	—	2	1	no	—
352	C₂²⋊C₄	—	26	2	no	—
353	C₂²⋊C₄	—	52	2	no	—
354	C₂²⋊C₄	—	52	2	no	—
355	C₄²	—	2	1	no	—
356	C₄²	C₂³×D₁₃	1	1	yes	characteristic
357	C₁₃	#19908	1	1	yes	characteristic, complemented, Sylow
358	C₂³	—	2	1	no	—
359	C₂³	—	26	2	no	—
360	C₂³	—	52	2	no	—
361	Q₈	—	2	1	no	—
362	Q₈	—	52	2	no	—
363	Q₈	—	52	2	no	—
364	Q₈	#88158	1	2	yes	complemented
365	D₄	—	2	1	no	—
366	D₄	—	2	1	no	—
367	D₄	—	26	2	no	—
368	D₄	—	26	2	no	—
369	D₄	—	52	2	no	—
370	D₄	—	52	2	no	—
371	D₄	—	52	2	no	—
372	D₄	—	52	2	no	—
373	D₄	—	104	4	no	—
374	D₄	—	104	2	no	—
375	D₄	—	104	4	no	—
376	D₄	—	208	2	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	26	2	no	—
383	C₂×C₄	—	52	2	no	—
384	C₂×C₄	—	52	2	no	—
385	C₂×C₄	—	52	2	no	—
386	C₂×C₄	—	104	4	no	—
387	C₂×C₄	—	104	4	no	—
388	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
389	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
390	C₂×C₄	#88256	1	1	yes	characteristic
391	C₈	—	2	1	no	—
392	C₈	—	2	1	no	—
393	C₈	—	2	1	no	—
394	C₈	—	4	1	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	1	no	—
397	C₈	—	52	2	no	—
398	C₈	#88168	1	2	yes	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	1	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	52	2	no	—
403	C₂²	—	52	2	no	—
404	C₂²	—	52	2	no	—
405	C₂²	—	104	2	no	—
406	C₂²	—	104	2	no	—
407	C₂²	—	104	2	no	—
408	C₂²	D₄×D₅₂	1	1	yes	characteristic
409	C₄	—	2	2	no	—
410	C₄	—	2	1	no	—
411	C₄	—	4	1	no	—
412	C₄	—	4	1	no	—
413	C₄	—	52	2	no	—
414	C₄	—	104	2	no	—
415	C₄	C₂×#88158	1	1	yes	characteristic
416	C₄	#11719457	1	1	yes	characteristic
417	C₄	C₂×#88168	1	1	yes	characteristic
418	C₄	#11719474	1	1	yes	characteristic
419	C₂	—	8	1	no	—
420	C₂	—	104	2	no	—
421	C₂	—	208	2	no	—
422	C₂	#13126457	1	1	yes	characteristic
423	C₂	#13126523	1	1	yes	characteristic
424	C₂	#13126623	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅₂
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×SD₁₆	C₂
socle	C₂²×C₁₃	D₄²
hypercenter	C₈×SD₁₆	D₁₃
hypocenter	C₁₃	#19908
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #13127811	C₂	semidirect
2. #13127165	C₂	semidirect
3. #13126558	C₂	semidirect
4. #13114442	C₂	semidirect
5. #13114398	C₂	semidirect
6. #13114206	C₂	semidirect
7. #13114191	C₂	semidirect
8. #13114039	C₂	semidirect
9. #13113928	C₂	semidirect
10. #13113722	C₂	semidirect
11. C₈×C₁₃×SD₁₆	C₂	semidirect
156. C₈×SD₁₆	D₁₃	semidirect
161. Q₈×C₁₃	#135	semidirect
357. C₁₃	#19908	semidirect
364. Q₈	#88158	semidirect

Maximal subgroups

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

subgroup	index
#13127811	2
#13127165	2
#13126558	2
#13114442	2
#13114398	2
#13114206	2
#13114191	2
#13114039	2
#13113928	2
#13113722	2
C₈×C₁₃×SD₁₆	2
#19908	13