Subgroups — #15846118

Order 3328.

710 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125982	C₂	1	1	yes	maximal, characteristic, complemented
2	#13125779	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718354	C₂	1	1	yes	maximal, characteristic
4	#13123770	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114309	C₂	1	4	yes	maximal, complemented
6	#13114185	C₂	1	4	yes	maximal, complemented
7	#13113507	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113503	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#6095	—	2	1	no	—
11	C₂²×#6095	—	2	1	no	—
12	C₂×#88128	C₂²	1	1	yes	characteristic
13	#11719175	C₂²	1	1	yes	characteristic
14	C₂×#88123	C₂²	1	1	yes	characteristic
15	C₁₃×#431	C₂²	1	1	yes	characteristic
16	C₁₃×#431	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	#11718763	—	2	1	no	—
19	#11718762	—	2	1	no	—
20	#11718585	C₂²	1	1	yes	characteristic
21	#11718512	C₂²	1	4	yes	complemented
22	#11718509	C₂²	1	4	yes	—
23	#11718392	—	2	2	no	—
24	#11718382	—	2	2	no	—
25	#11718355	—	2	1	no	—
26	#11718354	—	2	1	no	—
27	#11718354	—	2	1	no	—
28	#11718354	—	2	1	no	—
29	#11718354	C₂²	1	2	yes	—
30	#11718354	C₂²	1	2	yes	—
31	#11718342	—	2	1	no	—
32	#11718341	—	2	1	no	—
33	#11718340	—	2	1	no	—
34	#11718340	C₂²	1	2	yes	complemented
35	C₈×Dic₂₆	—	2	1	no	—
36	C₈×Dic₂₆	C₂²	1	2	yes	complemented
37	C₁₃×#335	C₂²	1	2	yes	complemented
38	C₁₃×#334	C₂²	1	2	yes	complemented
39	#11718078	C₂²	1	4	yes	complemented
40	#11718077	C₂²	1	4	yes	—
41	C₂²×D₅₂	—	2	1	no	—
42	C₂²×D₅₂	—	2	1	no	—
43	C₂²×D₅₂	—	2	1	no	—
44	C₂²×Dic₂₆	—	2	1	no	—
45	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	#88180	—	2	1	no	—
49	C₂×C₄⋊Dic₁₃	—	2	1	no	—
50	C₂×C₄⋊Dic₁₃	—	2	1	no	—
51	#88178	—	2	1	no	—
52	#88178	D₄	1	2	yes	—
53	C₂×#6099	—	2	1	no	—
54	C₂×#6095	—	2	2	no	—
55	C₂×#6095	—	2	2	no	—
56	C₂×#6095	—	2	2	no	—
57	C₂×#6095	—	2	2	no	—
58	C₂×#6095	—	2	4	no	—
59	C₂×#6095	—	4	4	no	—
60	C₂×#6095	—	4	4	no	—
61	#88128	—	2	1	no	—
62	#88128	—	2	1	no	—
63	#88128	—	2	1	no	—
64	#88128	C₂³	1	2	yes	—
65	#88124	—	2	1	no	—
66	#88123	—	2	1	no	—
67	#88123	—	2	1	no	—
68	#88123	—	2	1	no	—
69	#88123	D₄	1	1	yes	characteristic
70	#88123	D₄	1	1	yes	characteristic
71	#88123	C₂³	1	2	yes	—
72	C₄×Dic₂₆	—	2	1	no	—
73	C₄×Dic₂₆	C₂³	1	2	yes	—
74	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
75	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
76	C₁₃×C₂²⋊C₈	—	2	1	no	—
77	C₁₃×C₂²⋊C₈	—	2	1	no	—
78	C₄×C₈×C₁₃	—	2	1	no	—
79	C₄×C₈×C₁₃	—	2	1	no	—
80	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
81	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
82	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
84	#88061	—	2	4	no	—
85	#88057	—	2	4	no	—
86	#88056	—	2	4	no	—
87	#88055	—	2	2	no	—
88	#88054	—	2	2	no	—
89	#88054	—	2	2	no	—
90	C₈×Dic₁₃	—	2	2	no	—
91	C₄⋊#1055	—	2	1	no	—
92	C₄⋊#1055	C₂³	1	2	yes	—
93	#19202	—	13	1	no	maximal, Sylow
94	C₂³×D₁₃	—	4	1	no	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
98	C₂²×Dic₁₃	—	4	1	no	—
99	C₂×D₅₂	—	2	2	no	—
100	C₂×D₅₂	—	2	2	no	—
101	C₂×D₅₂	—	2	2	no	—
102	C₂×D₅₂	—	4	2	no	—
103	C₂×D₅₂	—	4	2	no	—
104	C₂×D₅₂	—	4	1	no	—
105	C₂×D₅₂	—	4	1	no	—
106	C₂×D₅₂	—	4	1	no	—
107	C₂×D₅₂	—	8	2	no	—
108	C₂×D₅₂	—	8	2	no	—
109	C₂×Dic₂₆	—	2	1	no	—
110	C₂×Dic₂₆	—	2	2	no	—
111	C₂×Dic₂₆	—	4	2	no	—
112	C₂×Dic₂₆	—	4	2	no	—
113	C₂×Dic₂₆	—	4	1	no	—
114	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₁₃	—	2	2	no	—
116	C₂×C₈×C₁₃	—	2	2	no	—
117	C₂×C₈×C₁₃	—	2	2	no	—
118	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
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	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₁₃	#41	1	1	yes	characteristic
126	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
127	C₂²⋊Dic₁₃	—	2	2	no	—
128	C₄⋊Dic₁₃	—	2	1	no	—
129	C₄⋊Dic₁₃	—	2	1	no	—
130	C₄⋊Dic₁₃	—	2	1	no	—
131	C₄⋊Dic₁₃	—	2	2	no	—
132	C₄⋊Dic₁₃	—	2	2	no	—
133	C₄⋊Dic₁₃	—	4	1	no	—
134	C₄⋊Dic₁₃	—	4	1	no	—
135	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
136	#6101	—	2	2	no	—
137	#6101	—	2	2	no	—
138	C₄×Dic₁₃	—	2	2	no	—
139	#6099	—	4	2	no	—
140	C₂×#1055	—	2	2	no	—
141	#6095	—	4	4	no	—
142	#6095	—	4	4	no	—
143	#6095	—	4	4	no	—
144	#6095	—	4	4	no	—
145	#6095	—	8	4	no	—
146	#3229	—	13	1	no	—
147	#3208	—	13	1	no	—
148	C₂×#492	—	13	1	no	—
149	#2960	—	13	1	no	—
150	#1686	—	13	4	no	—
151	#1666	—	13	4	no	—
152	#1564	D₁₃	1	1	yes	characteristic, complemented
153	#1546	—	13	1	no	—
154	#1545	—	13	1	no	—
155	C₂³×C₁₃	#121	1	1	yes	characteristic
156	C₂²×D₁₃	—	4	2	no	—
157	C₂²×D₁₃	—	8	2	no	—
158	C₂²×D₁₃	—	8	2	no	—
159	C₂²×D₁₃	—	8	2	no	—
160	C₂×C₄×C₁₃	—	2	2	no	—
161	C₂×C₄×C₁₃	—	2	2	no	—
162	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
163	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
164	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
166	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
170	C₂×Dic₁₃	—	2	2	no	—
171	C₂×Dic₁₃	—	2	2	no	—
172	C₂×Dic₁₃	—	4	2	no	—
173	C₂×Dic₁₃	—	8	2	no	—
174	D₅₂	—	4	2	no	—
175	D₅₂	—	4	2	no	—
176	D₅₂	—	4	4	no	—
177	D₅₂	—	8	2	no	—
178	D₅₂	—	8	2	no	—
179	D₅₂	—	8	2	no	—
180	D₅₂	—	8	2	no	—
181	Dic₂₆	—	2	4	no	—
182	Dic₂₆	—	4	2	no	—
183	Dic₂₆	—	4	2	no	—
184	Dic₂₆	—	4	2	no	—
185	Dic₂₆	—	8	2	no	—
186	C₈×C₁₃	—	2	1	no	—
187	C₈×C₁₃	—	2	1	no	—
188	C₈×C₁₃	—	4	2	no	—
189	C₈×C₁₃	—	4	2	no	—
190	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
191	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
192	#1055	—	4	2	no	—
193	C₂²×SD₁₆	—	26	1	no	—
194	C₂²×SD₁₆	—	26	1	no	—
195	#533	—	13	1	no	—
196	C₂×C₄⋊Q₈	—	13	1	no	—
197	C₂×#127	—	13	1	no	—
198	#498	—	26	1	no	—
199	#492	—	13	2	no	—
200	#492	—	13	2	no	—
201	#492	—	26	1	no	—
202	#492	—	26	1	no	—
203	#492	—	26	1	no	—
204	#487	—	26	1	no	—
205	#486	—	26	1	no	—
206	#483	—	26	2	no	—
207	#475	—	13	4	no	—
208	#460	—	13	4	no	—
209	#450	—	26	2	no	—
210	#446	—	13	2	no	—
211	#446	—	26	1	no	—
212	C₈×Q₈	—	13	2	no	—
213	C₈×Q₈	—	26	1	no	—
214	#431	C₂×D₁₃	1	1	yes	characteristic
215	#431	C₂×D₁₃	1	1	yes	characteristic
216	#424	—	26	1	no	—
217	#423	—	13	1	no	—
218	#405	—	26	1	no	—
219	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
220	#335	C₂×D₁₃	1	2	yes	complemented
221	#334	C₂×D₁₃	1	2	yes	complemented
222	#332	—	13	4	no	—
223	#331	—	13	4	no	—
224	C₂²×C₁₃	—	2	1	no	—
225	C₂²×C₁₃	—	2	1	no	—
226	C₂²×C₁₃	#460	1	2	yes	—
227	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
228	C₂×D₁₃	—	8	2	no	—
229	C₂×D₁₃	—	8	2	no	—
230	C₂×D₁₃	—	8	2	no	—
231	C₂×D₁₃	—	16	4	no	—
232	C₄×C₁₃	—	2	1	no	—
233	C₄×C₁₃	—	2	1	no	—
234	C₄×C₁₃	#460	1	2	yes	—
235	C₄×C₁₃	#471	1	2	yes	—
236	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
237	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₄×C₁₃	#577	1	2	yes	—
239	Dic₁₃	—	4	2	no	—
240	Dic₁₃	—	4	2	no	—
241	Dic₁₃	—	8	2	no	—
242	C₂²×Q₈	—	26	1	no	—
243	C₂²×D₄	—	26	1	no	—
244	C₂²×D₄	—	26	1	no	—
245	C₂²×D₄	—	26	1	no	—
246	C₂×SD₁₆	—	26	4	no	—
247	C₂×SD₁₆	—	26	2	no	—
248	C₂×SD₁₆	—	26	2	no	—
249	C₂×SD₁₆	—	26	2	no	—
250	C₂×SD₁₆	—	26	2	no	—
251	C₂×SD₁₆	—	52	4	no	—
252	C₂×SD₁₆	—	52	4	no	—
253	C₂×M₁₆	—	26	1	no	—
254	C₂²×C₈	D₅₂	1	1	yes	characteristic
255	C₂²×C₈	D₅₂	1	1	yes	characteristic
256	C₄⋊Q₈	—	13	1	no	—
257	C₄⋊Q₈	—	13	1	no	—
258	C₄⋊Q₈	—	13	2	no	—
259	C₄⋊Q₈	—	26	1	no	—
260	C₄⋊Q₈	—	26	1	no	—
261	C₄⋊Q₈	—	26	1	no	—
262	#127	—	13	2	no	—
263	#127	—	26	1	no	—
264	#127	—	26	1	no	—
265	#127	—	26	1	no	—
266	#125	—	26	1	no	—
267	C₂²⋊Q₈	—	13	2	no	—
268	C₂²⋊Q₈	—	26	1	no	—
269	C₄×Q₈	—	13	2	no	—
270	C₄×Q₈	—	26	1	no	—
271	#117	—	26	1	no	—
272	C₂×C₄⋊C₄	—	26	1	no	—
273	C₂×C₄⋊C₄	—	26	1	no	—
274	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
275	#106	—	26	4	no	—
276	C₄⋊C₈	—	13	2	no	—
277	C₄⋊C₈	—	26	2	no	—
278	C₄⋊C₈	—	26	2	no	—
279	C₄⋊C₈	—	26	1	no	—
280	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
281	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
282	#103	—	26	4	no	—
283	#102	—	26	4	no	—
284	C₂²⋊C₈	—	2	1	no	—
285	C₂²⋊C₈	—	2	1	no	—
286	#97	—	26	2	no	—
287	C₄×C₈	—	2	1	no	—
288	C₄×C₈	—	2	1	no	—
289	C₄×C₈	—	26	2	no	—
290	C₄×C₈	D₅₂	1	1	yes	characteristic
291	C₄×C₈	D₅₂	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
293	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
294	C₂×C₁₃	—	2	2	no	—
295	C₂×C₁₃	C₂×#460	1	1	yes	characteristic
296	C₂×C₁₃	#3057	1	1	yes	characteristic
297	C₂×C₁₃	C₂×#471	1	1	yes	characteristic
298	D₁₃	—	16	2	no	—
299	C₂⁴	—	52	1	no	—
300	C₂×Q₈	—	13	2	no	—
301	C₂×Q₈	—	26	1	no	—
302	C₂×Q₈	—	26	2	no	—
303	C₂×Q₈	—	52	1	no	—
304	C₂×Q₈	—	52	2	no	—
305	C₂×Q₈	—	52	2	no	—
306	C₂×D₄	—	26	2	no	—
307	C₂×D₄	—	26	2	no	—
308	C₂×D₄	—	26	2	no	—
309	C₂×D₄	—	52	1	no	—
310	C₂×D₄	—	52	1	no	—
311	C₂×D₄	—	52	1	no	—
312	C₂×D₄	—	52	2	no	—
313	C₂×D₄	—	52	2	no	—
314	C₂×D₄	—	104	2	no	—
315	C₂×D₄	—	104	2	no	—
316	C₂²×C₄	—	52	1	no	—
317	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
318	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
319	C₂²×C₄	#6131	1	1	yes	characteristic
320	SD₁₆	—	52	4	no	—
321	SD₁₆	—	52	4	no	—
322	SD₁₆	—	52	4	no	—
323	SD₁₆	—	52	4	no	—
324	SD₁₆	—	104	4	no	—
325	M₁₆	—	52	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	26	2	no	—
330	C₂×C₈	#6095	1	4	yes	complemented
331	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
332	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
333	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
334	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
335	C₄⋊C₄	—	13	2	no	—
336	C₄⋊C₄	—	26	1	no	—
337	C₄⋊C₄	—	26	1	no	—
338	C₄⋊C₄	—	26	2	no	—
339	C₄⋊C₄	—	26	1	no	—
340	C₄⋊C₄	—	26	2	no	—
341	C₄⋊C₄	—	26	2	no	—
342	C₄⋊C₄	—	26	2	no	—
343	C₄⋊C₄	—	52	1	no	—
344	C₄⋊C₄	—	52	1	no	—
345	C₂²⋊C₄	—	26	2	no	—
346	C₄²	—	26	2	no	—
347	C₄²	C₂×D₅₂	1	1	yes	characteristic
348	C₄²	D₄×D₁₃	1	1	yes	characteristic
349	C₄²	#6131	1	1	yes	characteristic
350	C₄²	C₂³×D₁₃	1	1	yes	characteristic
351	C₁₃	#19202	1	1	yes	characteristic, complemented, Sylow
352	C₂³	—	52	2	no	—
353	C₂³	—	104	2	no	—
354	C₂³	—	104	2	no	—
355	C₂³	—	104	2	no	—
356	C₂³	#88149	1	1	yes	characteristic
357	Q₈	—	26	4	no	—
358	Q₈	—	52	2	no	—
359	Q₈	—	52	2	no	—
360	Q₈	—	52	2	no	—
361	Q₈	—	104	2	no	—
362	D₄	—	52	2	no	—
363	D₄	—	52	2	no	—
364	D₄	—	52	4	no	—
365	D₄	—	104	2	no	—
366	D₄	—	104	2	no	—
367	D₄	—	104	2	no	—
368	D₄	—	104	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	26	2	no	—
372	C₂×C₄	—	26	2	no	—
373	C₂×C₄	—	52	2	no	—
374	C₂×C₄	—	104	2	no	—
375	C₂×C₄	#88149	1	1	yes	characteristic
376	C₂×C₄	#88149	1	1	yes	characteristic
377	C₂×C₄	#88149	1	1	yes	characteristic
378	C₂×C₄	C₂×#6095	1	2	yes	—
379	C₂×C₄	#88162	1	2	yes	—
380	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
381	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
382	C₂×C₄	C₂×#6131	1	1	yes	characteristic
383	C₈	—	2	1	no	—
384	C₈	—	2	1	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₈	—	52	2	no	—
388	C₈	C₂×#6095	1	2	yes	—
389	C₈	C₂×#6095	1	2	yes	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	104	2	no	—
393	C₂²	—	104	2	no	—
394	C₂²	—	104	2	no	—
395	C₂²	—	208	4	no	—
396	C₂²	#11718509	1	2	yes	—
397	C₂²	C₂×#88149	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	52	2	no	—
401	C₄	—	52	2	no	—
402	C₄	—	104	2	no	—
403	C₄	#11718509	1	2	yes	—
404	C₄	#11718559	1	2	yes	—
405	C₄	C₂²×#6095	1	1	yes	characteristic
406	C₄	C₂×#88162	1	1	yes	characteristic
407	C₄	#11719475	1	2	yes	—
408	C₂	—	2	2	no	—
409	C₂	—	208	2	no	—
410	C₂	C₂×#11718509	1	1	yes	characteristic
411	C₂	#13124476	1	1	yes	characteristic
412	C₂	C₂×#11718559	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88149
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂×#88128	C₂²
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#19202
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #13125982	C₂	semidirect
4. #13123770	C₂	semidirect
5. #13114309	C₂	semidirect
6. #13114185	C₂	semidirect
7. #13113507	C₂	semidirect
9. C₁₃×#1564	C₂	semidirect
21. #11718512	C₂²	semidirect
34. #11718340	C₂²	semidirect
36. C₈×Dic₂₆	C₂²	semidirect
37. C₁₃×#335	C₂²	semidirect
38. C₁₃×#334	C₂²	semidirect
39. #11718078	C₂²	semidirect
118. C₂×C₈×C₁₃	SD₁₆	semidirect
152. #1564	D₁₃	semidirect
220. #335	C₂×D₁₃	semidirect
221. #334	C₂×D₁₃	semidirect
330. C₂×C₈	#6095	semidirect
351. C₁₃	#19202	semidirect

Maximal subgroups

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

subgroup	index
#13125982	2
#13125779	2
C₂×#11718354	2
#13123770	2
#13114309	2
#13114185	2
#13113507	2
#13113503	2
C₁₃×#1564	2
#19202	13