Subgroups — #15832759

Order 3328.

614 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	#13123918	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
3	#13113743	C₂	1	2	yes	maximal, complemented
4	#13113742	C₂	1	2	yes	maximal, complemented
5	#13113734	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113733	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113732	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113583	C₂	1	1	yes	maximal, characteristic
9	#13113229	C₂	1	2	yes	maximal
10	#13113118	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₁₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₁₃	C₂²	1	2	yes	—
14	C₁₃×#446	C₂²	1	2	yes	—
15	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×Dic₁₃	C₄	1	1	yes	characteristic
17	C₄×#6107	C₂²	1	2	yes	—
18	#11718685	C₂²	1	2	yes	—
19	D₁₃×#107	C₄	1	1	yes	characteristic
20	#11718534	C₄	1	1	yes	characteristic
21	#11718496	—	2	2	no	—
22	#11718494	C₂²	1	2	yes	—
23	#11718492	C₂²	1	2	yes	—
24	#11718489	—	2	2	no	—
25	D₁₃×#103	C₄	1	2	yes	—
26	#11718451	C₄	1	2	yes	—
27	#11718366	—	2	2	no	—
28	#11718362	—	2	2	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	#11718346	C₂²	1	1	yes	characteristic
31	#11718346	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	#11718340	C₂²	1	2	yes	—
34	C₁₃×#335	C₂²	1	1	yes	characteristic
35	C₁₃×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₈²×C₁₃	C₂²	1	1	yes	characteristic
37	Q₈⋊#1055	C₂²	1	2	yes	—
38	#11718076	C₂²	1	2	yes	—
39	#11718072	C₂²	1	2	yes	—
40	#11718050	C₂²	1	1	yes	characteristic
41	#11718048	C₂²	1	2	yes	—
42	#11718046	C₂²	1	1	yes	characteristic
43	#11718046	C₂²	1	1	yes	characteristic
44	C₈×#1055	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
48	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₂×#6107	C₂×C₄	1	2	yes	—
50	Q₁₆×D₁₃	—	2	4	no	—
51	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₄×D₁₃	—	2	2	no	—
55	C₄⋊C₄×D₁₃	C₂×C₄	1	2	yes	—
56	#88141	—	2	2	no	—
57	#88141	C₂×C₄	1	2	yes	—
58	C₄²×D₁₃	—	2	2	no	—
59	C₄²×D₁₃	C₂³	1	1	yes	characteristic
60	C₄×Dic₂₆	—	2	2	no	—
61	C₄×Dic₂₆	C₂³	1	2	yes	—
62	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₈	—	2	2	no	—
64	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
65	C₁₃×#103	C₂×C₄	1	2	yes	—
66	C₁₃×#97	—	2	2	no	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#88075	C₂×C₄	1	2	yes	—
71	#88060	—	2	2	no	—
72	#88060	—	2	1	no	—
73	#88060	D₄	1	1	yes	characteristic
74	#88060	D₄	1	1	yes	characteristic
75	#88058	C₂×C₄	1	1	yes	characteristic
76	#88056	C₂×C₄	1	2	yes	—
77	#88055	—	2	2	no	—
78	#88055	—	2	2	no	—
79	#88054	—	2	2	no	—
80	#88054	—	2	1	no	—
81	#88054	—	2	2	no	—
82	#88054	D₄	1	1	yes	characteristic
83	#88054	D₄	1	1	yes	characteristic
84	C₈×Dic₁₃	—	2	2	no	—
85	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
87	#88050	C₂×C₄	1	2	yes	—
88	#88047	C₂×C₄	1	2	yes	—
89	C₄⋊#1055	—	2	2	no	—
90	C₄⋊#1055	C₂³	1	2	yes	—
91	#88043	—	2	2	no	—
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	C₄×#1055	C₂³	1	1	yes	characteristic
94	C₄×#1055	C₂³	1	1	yes	characteristic
95	#17897	—	13	1	no	maximal, Sylow
96	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
97	Q₈×D₁₃	—	2	4	no	—
98	Q₈×D₁₃	—	2	4	no	—
99	C₂×C₄×D₁₃	—	2	2	no	—
100	C₂×C₄×D₁₃	—	2	2	no	—
101	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
104	C₂×Dic₂₆	—	2	2	no	—
105	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
106	C₁₃×Q₁₆	—	2	2	no	—
107	C₂×C₈×C₁₃	—	2	2	no	—
108	C₂×C₈×C₁₃	—	2	1	no	—
109	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₁₃×C₄⋊C₄	—	2	2	no	—
114	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
115	C₄²×C₁₃	—	2	2	no	—
116	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
117	#6107	—	2	4	no	—
118	C₄⋊Dic₁₃	—	2	2	no	—
119	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
120	#6101	—	2	2	no	—
121	#6101	—	2	2	no	—
122	#6101	—	2	2	no	—
123	C₄×Dic₁₃	—	2	2	no	—
124	C₄×Dic₁₃	—	2	2	no	—
125	C₄×Dic₁₃	—	2	2	no	—
126	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
128	C₄×Dic₁₃	#41	1	1	yes	characteristic
129	C₂×#1055	—	2	2	no	—
130	C₂×#1055	—	2	1	no	—
131	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
132	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
133	C₂×#1055	C₂×D₄	1	1	yes	characteristic
134	C₂×#1055	C₂×D₄	1	1	yes	characteristic
135	Dic₅₂	—	2	2	no	—
136	C₈×D₁₃	—	2	2	no	—
137	C₈×D₁₃	M₁₆	1	4	yes	—
138	#2983	—	13	2	no	—
139	C₂×C₄×Q₁₆	—	13	1	no	—
140	#1593	—	13	2	no	—
141	#1593	—	13	2	no	—
142	#1591	—	13	1	no	—
143	#1591	—	13	1	no	—
144	#1591	—	13	1	no	—
145	#1591	D₁₃	1	1	yes	characteristic, complemented
146	#1564	—	13	1	no	—
147	#1479	—	13	2	no	—
148	#1450	—	13	1	no	—
149	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
150	Q₈×C₁₃	—	2	2	no	—
151	Q₈×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂×Dic₁₃	—	2	2	no	—
158	C₂×Dic₁₃	—	2	2	no	—
159	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
160	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
161	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
162	C₄×D₁₃	—	2	1	no	—
163	C₄×D₁₃	—	2	1	no	—
164	C₄×D₁₃	—	4	1	no	—
165	C₄×D₁₃	—	4	4	no	—
166	C₄×D₁₃	—	8	2	no	—
167	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
168	C₄×D₁₃	#131	1	2	yes	—
169	Dic₂₆	—	2	2	no	—
170	Dic₂₆	—	2	2	no	—
171	Dic₂₆	—	4	4	no	—
172	C₈×C₁₃	—	2	1	no	—
173	C₈×C₁₃	—	2	1	no	—
174	C₈×C₁₃	—	2	1	no	—
175	C₈×C₁₃	—	4	1	no	—
176	C₈×C₁₃	—	4	2	no	—
177	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
178	#1055	—	2	1	no	—
179	#1055	—	2	1	no	—
180	#1055	—	2	1	no	—
181	#1055	—	4	1	no	—
182	#1055	—	4	2	no	—
183	#1055	C₂×M₁₆	1	2	yes	—
184	C₂²×Q₁₆	—	13	1	no	—
185	C₂×C₄×Q₈	—	13	2	no	—
186	#446	—	13	2	no	—
187	#446	—	13	2	no	—
188	#446	—	13	2	no	—
189	#446	—	26	2	no	—
190	#446	—	26	2	no	—
191	#446	C₂×D₁₃	1	2	yes	—
192	C₄×Q₁₆	—	13	1	no	—
193	C₄×Q₁₆	—	13	2	no	—
194	C₄×Q₁₆	—	13	1	no	—
195	C₄×Q₁₆	—	13	2	no	—
196	C₄×Q₁₆	—	13	1	no	—
197	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
198	#432	—	26	2	no	—
199	#432	—	26	2	no	—
200	#431	—	13	1	no	—
201	#431	—	13	1	no	—
202	#431	—	13	2	no	—
203	C₂×#107	—	13	1	no	—
204	C₂×#103	—	13	2	no	—
205	C₂×C₄×C₈	—	13	1	no	—
206	#335	—	13	1	no	—
207	#335	C₂×D₁₃	1	1	yes	characteristic
208	#334	—	13	2	no	—
209	Q₈⋊C₈	—	13	2	no	—
210	Q₈⋊C₈	—	13	2	no	—
211	Q₈⋊C₈	—	13	2	no	—
212	Q₈⋊C₈	C₂×D₁₃	1	2	yes	—
213	#322	—	13	1	no	—
214	#322	—	13	1	no	—
215	C₈²	—	13	1	no	—
216	C₈²	C₂×D₁₃	1	1	yes	characteristic
217	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂×D₁₃	—	2	1	no	—
219	C₂×D₁₃	—	2	1	no	—
220	C₂×D₁₃	#436	1	2	yes	—
221	C₄×C₁₃	—	2	1	no	—
222	C₄×C₁₃	—	4	2	no	—
223	C₄×C₁₃	—	4	2	no	—
224	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
225	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
226	C₄×C₁₃	#578	1	1	yes	characteristic
227	C₄×C₁₃	#578	1	1	yes	characteristic
228	Dic₁₃	—	2	1	no	—
229	Dic₁₃	—	2	1	no	—
230	Dic₁₃	—	4	2	no	—
231	Dic₁₃	—	4	2	no	—
232	Dic₁₃	#436	1	2	yes	—
233	C₂²×Q₈	—	13	2	no	—
234	C₂×Q₁₆	—	13	2	no	—
235	C₂×Q₁₆	—	13	1	no	—
236	C₂×Q₁₆	—	26	4	no	—
237	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
238	C₂²×C₈	—	13	1	no	—
239	C₂²×C₈	—	13	1	no	—
240	C₄×Q₈	—	13	2	no	—
241	C₄×Q₈	—	13	2	no	—
242	C₄×Q₈	—	13	2	no	—
243	C₄×Q₈	—	26	2	no	—
244	C₄×Q₈	—	26	2	no	—
245	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
246	C₂×C₄⋊C₄	—	13	2	no	—
247	C₂×C₄⋊C₄	—	26	2	no	—
248	C₂×C₄²	—	13	1	no	—
249	C₂×C₄²	—	26	2	no	—
250	#107	—	13	1	no	—
251	#107	—	13	2	no	—
252	#107	C₄×D₁₃	1	1	yes	characteristic
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	13	1	no	—
255	C₄⋊C₈	—	13	1	no	—
256	C₄⋊C₈	—	13	2	no	—
257	C₄⋊C₈	—	26	2	no	—
258	C₄⋊C₈	—	26	1	no	—
259	C₄⋊C₈	—	26	2	no	—
260	C₄⋊C₈	—	26	2	no	—
261	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
262	#103	—	13	2	no	—
263	#103	—	13	2	no	—
264	#103	—	13	2	no	—
265	#103	C₄×D₁₃	1	2	yes	—
266	C₂²⋊C₈	—	13	1	no	—
267	C₂²⋊C₈	—	13	1	no	—
268	C₂²⋊C₈	—	26	2	no	—
269	C₂²⋊C₈	—	26	1	no	—
270	#97	—	2	2	no	—
271	#97	—	26	2	no	—
272	#97	—	26	2	no	—
273	#97	—	26	2	no	—
274	C₄×C₈	—	13	1	no	—
275	C₄×C₈	—	13	1	no	—
276	C₄×C₈	—	13	1	no	—
277	C₄×C₈	—	13	1	no	—
278	C₄×C₈	—	13	1	no	—
279	C₄×C₈	—	26	2	no	—
280	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
283	C₂×C₁₃	C₂×#436	1	1	yes	characteristic
284	C₂×C₁₃	#2950	1	1	yes	characteristic
285	C₂×C₁₃	#2957	1	1	yes	characteristic
286	D₁₃	—	2	2	no	—
287	C₂×Q₈	—	13	2	no	—
288	C₂×Q₈	—	26	2	no	—
289	C₂×Q₈	—	26	4	no	—
290	C₂×Q₈	—	26	4	no	—
291	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
292	C₂²×C₄	—	13	1	no	—
293	C₂²×C₄	—	13	1	no	—
294	C₂²×C₄	—	13	1	no	—
295	C₂²×C₄	—	26	2	no	—
296	C₂²×C₄	—	26	2	no	—
297	Q₁₆	—	2	2	no	—
298	Q₁₆	—	26	4	no	—
299	Q₁₆	—	26	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	13	1	no	—
303	C₂×C₈	—	13	4	no	—
304	C₂×C₈	—	13	1	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	26	2	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	—	26	1	no	—
310	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
311	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
312	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
313	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	13	2	no	—
316	C₄⋊C₄	—	26	2	no	—
317	C₄⋊C₄	—	26	2	no	—
318	C₄⋊C₄	—	26	2	no	—
319	C₄⋊C₄	—	26	2	no	—
320	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
321	C₄²	—	2	2	no	—
322	C₄²	—	13	1	no	—
323	C₄²	—	13	1	no	—
324	C₄²	—	13	1	no	—
325	C₄²	—	26	2	no	—
326	C₄²	—	26	2	no	—
327	C₄²	—	26	2	no	—
328	C₄²	C₂³×D₁₃	1	1	yes	characteristic
329	C₁₃	#17897	1	1	yes	characteristic, complemented, Sylow
330	C₂³	—	13	1	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	26	2	no	—
334	Q₈	—	26	2	no	—
335	Q₈	—	52	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	13	1	no	—
339	C₂×C₄	—	13	2	no	—
340	C₂×C₄	—	13	2	no	—
341	C₂×C₄	—	13	1	no	—
342	C₂×C₄	—	13	1	no	—
343	C₂×C₄	—	26	1	no	—
344	C₂×C₄	—	26	1	no	—
345	C₂×C₄	—	26	2	no	—
346	C₂×C₄	—	26	2	no	—
347	C₂×C₄	—	52	1	no	—
348	C₂×C₄	—	52	4	no	—
349	C₂×C₄	—	104	2	no	—
350	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
351	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
352	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	1	no	—
357	C₈	—	4	2	no	—
358	C₈	—	13	2	no	—
359	C₈	—	26	1	no	—
360	C₈	—	26	1	no	—
361	C₈	—	26	1	no	—
362	C₈	—	52	1	no	—
363	C₈	—	52	2	no	—
364	C₈	M₁₆×D₁₃	1	2	yes	—
365	C₂²	—	13	2	no	—
366	C₂²	—	26	1	no	—
367	C₂²	—	26	1	no	—
368	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	13	2	no	—
373	C₄	—	26	1	no	—
374	C₄	—	26	1	no	—
375	C₄	—	52	2	no	—
376	C₄	—	52	2	no	—
377	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
378	C₄	D₁₃×#131	1	1	yes	characteristic
379	C₄	#11719471	1	1	yes	characteristic
380	C₄	#11719476	1	1	yes	characteristic
381	C₂	—	26	2	no	—
382	C₂	D₁₃×#436	1	1	yes	characteristic
383	C₂	#13123699	1	1	yes	characteristic
384	C₂	#13123748	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1591	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1591	D₁₃
hypocenter	C₁₃	#17897
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #13113743	C₂	semidirect
4. #13113742	C₂	semidirect
5. #13113734	C₂	semidirect
6. #13113733	C₂	semidirect
7. #13113732	C₂	semidirect
11. C₁₃×#1591	C₂	semidirect
145. #1591	D₁₃	semidirect
329. C₁₃	#17897	semidirect

Maximal subgroups

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

subgroup	index
#13123918	2
C₄×Q₁₆×D₁₃	2
#13113743	2
#13113742	2
#13113734	2
#13113733	2
#13113732	2
#13113583	2
#13113229	2
#13113118	2
C₁₃×#1591	2
#17897	13