Subgroups — #15834664

Order 3328.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123959	C₂	1	2	yes	maximal, complemented
2	C₄×#88166	C₂	1	1	yes	maximal, characteristic
3	#13113746	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113744	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113672	C₂	1	2	yes	maximal, complemented
6	#13113669	C₂	1	2	yes	maximal, complemented
7	#13113660	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113590	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113272	C₂	1	2	yes	maximal, complemented
10	#13113122	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1580	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88166	C₄	1	1	yes	characteristic
13	C₄×#6129	C₂²	1	2	yes	—
14	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₁₃	C₂²	1	2	yes	—
16	D₈×Dic₁₃	C₄	1	1	yes	characteristic
17	#11718772	—	2	2	no	—
18	C₂×#88054	—	2	2	no	—
19	C₄×#6105	C₂²	1	2	yes	—
20	D₄×#1055	C₂²	1	2	yes	—
21	#11718541	C₄	1	1	yes	characteristic
22	#11718534	C₄	1	1	yes	characteristic
23	#11718498	C₂²	1	2	yes	—
24	#11718492	C₂²	1	2	yes	—
25	#11718433	C₄	1	2	yes	—
26	#11718419	C₄	1	2	yes	—
27	#11718391	—	2	2	no	—
28	#11718380	—	2	2	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	C₄×#6094	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×#334	C₂²	1	1	yes	characteristic
35	C₁₃×D₄⋊C₈	C₂²	1	2	yes	—
36	C₁₃×#322	C₂²	1	1	yes	characteristic
37	D₄⋊#1055	C₂²	1	2	yes	—
38	#11718076	C₂²	1	2	yes	—
39	#11718072	C₂²	1	2	yes	—
40	#11718049	C₂²	1	1	yes	characteristic
41	#11718048	C₂²	1	2	yes	—
42	#11718047	C₂²	1	1	yes	characteristic
43	#11718047	C₂²	1	1	yes	characteristic
44	#11718046	C₂²	1	1	yes	characteristic
45	C₂×#6129	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	—	2	2	no	—
48	C₄×D₄×C₁₃	C₂³	1	2	yes	—
49	D₄×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₂×#6105	C₂×C₄	1	2	yes	—
51	C₄×C₂²⋊D₁₃	—	2	2	no	—
52	C₂×C₄×Dic₁₃	—	2	2	no	—
53	C₂²×#1055	—	2	2	no	—
54	#88166	—	2	4	no	—
55	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
56	C₂×#6094	—	2	1	no	—
57	C₂×#6094	—	2	2	no	—
58	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
60	#88146	C₂×C₄	1	2	yes	—
61	#88141	C₂×C₄	1	2	yes	—
62	#88135	—	2	2	no	—
63	#88131	—	2	2	no	—
64	C₄²×D₁₃	C₂³	1	1	yes	characteristic
65	C₄×Dic₂₆	C₂³	1	2	yes	—
66	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
67	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
68	C₁₃×#102	C₂×C₄	1	2	yes	—
69	C₁₃×C₂²⋊C₈	—	2	2	no	—
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#88072	C₂×C₄	1	2	yes	—
74	C₂²⋊#1055	—	2	2	no	—
75	#88060	D₄	1	2	yes	—
76	#88058	C₂×C₄	1	1	yes	characteristic
77	#88056	C₂×C₄	1	2	yes	—
78	#88055	—	2	2	no	—
79	#88055	—	2	1	no	—
80	#88055	—	2	2	no	—
81	#88054	—	2	2	no	—
82	#88054	—	2	2	no	—
83	#88054	—	2	2	no	—
84	#88054	D₄	1	2	yes	—
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	#88048	C₂×C₄	1	2	yes	—
89	C₄⋊#1055	C₂³	1	2	yes	—
90	C₄×#1055	C₂³	1	1	yes	characteristic
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	#18065	—	13	1	no	maximal, Sylow
94	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₁₃	—	2	2	no	—
96	C₂×C₂²⋊D₁₃	—	2	2	no	—
97	C₂²×Dic₁₃	—	2	2	no	—
98	C₂²×Dic₁₃	—	2	2	no	—
99	#6129	—	2	4	no	—
100	#6129	—	4	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	C₂²×C₄	1	2	yes	—
105	C₁₃×D₈	M₁₆	1	4	yes	complemented
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₂×C₈×C₁₃	—	2	1	no	—
108	C₂×C₈×C₁₃	—	8	2	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	2	yes	—
112	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
113	C₁₃×C₂²⋊C₄	—	2	2	no	—
114	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
115	C₂²⋊Dic₁₃	—	2	2	no	—
116	#6105	—	2	4	no	—
117	#6103	—	2	2	no	—
118	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
119	#6101	—	2	2	no	—
120	#6101	—	2	2	no	—
121	C₄×Dic₁₃	—	2	2	no	—
122	C₄×Dic₁₃	—	2	2	no	—
123	C₄×Dic₁₃	—	2	2	no	—
124	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
125	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
126	C₄×Dic₁₃	#41	1	1	yes	characteristic
127	C₂×#1055	—	2	2	no	—
128	C₂×#1055	—	2	1	no	—
129	C₂×#1055	—	8	2	no	—
130	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
131	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
132	C₂×#1055	C₂×D₄	1	2	yes	—
133	Dic₅₂	—	2	2	no	—
134	#6094	—	4	2	no	—
135	#6094	—	8	2	no	—
136	C₈×D₁₃	—	2	2	no	—
137	C₈×D₁₃	—	4	1	no	—
138	#2990	—	13	2	no	—
139	C₄×#135	—	13	1	no	—
140	#1593	—	13	1	no	—
141	#1593	—	13	1	no	—
142	#1582	—	13	2	no	—
143	#1582	—	13	2	no	—
144	#1580	—	13	1	no	—
145	#1580	D₁₃	1	1	yes	characteristic, complemented
146	#1566	—	13	1	no	—
147	#1489	—	13	2	no	—
148	#1451	—	13	1	no	—
149	C₂³×C₁₃	—	2	2	no	—
150	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
151	D₄×C₁₃	—	2	2	no	—
152	D₄×C₁₃	C₂×M₁₆	1	4	yes	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₂²⋊D₁₃	—	4	4	no	—
160	C₂×Dic₁₃	—	2	2	no	—
161	C₂×Dic₁₃	—	2	2	no	—
162	C₂×Dic₁₃	—	4	4	no	—
163	C₂×Dic₁₃	—	4	4	no	—
164	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
165	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
166	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
167	C₄×D₁₃	—	2	1	no	—
168	C₄×D₁₃	—	2	1	no	—
169	C₄×D₁₃	—	2	2	no	—
170	C₄×D₁₃	—	2	2	no	—
171	Dic₂₆	—	2	2	no	—
172	Dic₂₆	—	2	2	no	—
173	C₈×C₁₃	—	2	1	no	—
174	C₈×C₁₃	—	2	2	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	2	no	—
181	#1055	—	4	1	no	—
182	#1055	—	4	2	no	—
183	C₂×#135	—	13	1	no	—
184	C₄×#41	—	13	2	no	—
185	#446	—	13	2	no	—
186	#446	—	13	2	no	—
187	C₄×Q₁₆	—	13	1	no	—
188	C₄×Q₁₆	—	13	1	no	—
189	C₄×SD₁₆	—	13	2	no	—
190	C₄×SD₁₆	—	13	2	no	—
191	C₄×D₈	—	13	1	no	—
192	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
193	#435	—	26	2	no	—
194	#435	—	26	2	no	—
195	C₈×D₄	—	13	2	no	—
196	C₈×D₄	C₂×D₁₃	1	2	yes	—
197	#432	—	26	2	no	—
198	#431	—	13	1	no	—
199	#427	—	13	1	no	—
200	#423	—	13	2	no	—
201	C₂×C₄⋊C₈	—	26	2	no	—
202	#416	—	13	2	no	—
203	C₄×M₁₆	—	13	1	no	—
204	C₂×C₄×C₈	—	13	1	no	—
205	#334	—	13	2	no	—
206	#334	—	13	1	no	—
207	#334	C₂×D₁₃	1	1	yes	characteristic
208	Q₈⋊C₈	—	13	2	no	—
209	Q₈⋊C₈	—	13	2	no	—
210	D₄⋊C₈	—	13	2	no	—
211	D₄⋊C₈	C₂×D₁₃	1	2	yes	—
212	#322	—	13	1	no	—
213	#322	—	13	1	no	—
214	#322	—	13	1	no	—
215	#322	C₂×D₁₃	1	1	yes	characteristic
216	C₂²×C₁₃	—	2	4	no	—
217	C₂²×C₁₃	—	4	2	no	—
218	C₂²×C₁₃	—	4	2	no	—
219	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂×D₁₃	—	2	1	no	—
221	C₂×D₁₃	—	2	1	no	—
222	C₂×D₁₃	—	2	1	no	—
223	C₄×C₁₃	—	2	1	no	—
224	C₄×C₁₃	—	4	2	no	—
225	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
226	C₄×C₁₃	#568	1	1	yes	characteristic
227	C₄×C₁₃	#577	1	2	yes	—
228	Dic₁₃	—	2	1	no	—
229	Dic₁₃	—	2	1	no	—
230	Dic₁₃	—	4	2	no	—
231	Dic₁₃	—	4	2	no	—
232	Dic₁₃	#444	1	2	yes	—
233	C₂×#41	—	13	2	no	—
234	#135	—	26	4	no	—
235	C₂×Q₁₆	—	13	1	no	—
236	C₂×SD₁₆	—	13	2	no	—
237	C₂×D₈	C₄×D₁₃	1	1	yes	characteristic
238	C₂×M₁₆	—	26	1	no	—
239	C₂×M₁₆	—	26	2	no	—
240	C₂²×C₈	—	2	2	no	—
241	C₂²×C₈	—	13	1	no	—
242	C₂²×C₈	—	13	1	no	—
243	C₂²×C₈	—	26	2	no	—
244	C₄×Q₈	—	13	2	no	—
245	C₄×Q₈	—	13	2	no	—
246	C₄×D₄	—	13	2	no	—
247	C₄×D₄	—	26	2	no	—
248	C₄×D₄	—	26	2	no	—
249	C₄×D₄	C₂²×D₁₃	1	2	yes	—
250	#117	—	13	2	no	—
251	#117	—	26	2	no	—
252	C₂×C₄²	—	13	1	no	—
253	C₂×C₄²	—	26	2	no	—
254	#107	—	13	1	no	—
255	#107	C₄×D₁₃	1	1	yes	characteristic
256	#106	—	13	2	no	—
257	C₄⋊C₈	—	13	2	no	—
258	C₄⋊C₈	—	13	2	no	—
259	C₄⋊C₈	—	26	2	no	—
260	C₄⋊C₈	—	26	2	no	—
261	C₄⋊C₈	—	26	2	no	—
262	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
263	#103	—	13	2	no	—
264	#103	—	13	2	no	—
265	#102	—	13	2	no	—
266	#102	C₄×D₁₃	1	2	yes	—
267	C₂²⋊C₈	—	2	2	no	—
268	C₂²⋊C₈	—	13	2	no	—
269	C₂²⋊C₈	—	26	2	no	—
270	#97	—	26	1	no	—
271	#97	—	26	2	no	—
272	#97	—	26	2	no	—
273	C₄×C₈	—	13	1	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	C₂²×D₁₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₂×C₁₃	—	4	4	no	—
282	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
283	C₂×C₁₃	#2952	1	1	yes	characteristic
284	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
285	D₁₃	—	4	1	no	—
286	#41	—	26	4	no	—
287	#41	—	52	2	no	—
288	C₂×Q₈	—	13	2	no	—
289	C₂×D₄	—	26	2	no	—
290	C₂×D₄	C₂×C₄×D₁₃	1	2	yes	—
291	C₂²×C₄	—	2	2	no	—
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	—	26	2	no	—
298	SD₁₆	—	26	4	no	—
299	D₈	#6094	1	4	yes	complemented
300	M₁₆	—	52	2	no	—
301	M₁₆	—	104	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	8	2	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	13	2	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	—	26	2	no	—
310	C₂×C₈	—	26	1	no	—
311	C₂×C₈	—	52	1	no	—
312	C₂×C₈	—	104	2	no	—
313	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
314	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
315	C₂×C₈	D₄×D₁₃	1	2	yes	—
316	C₄⋊C₄	—	13	2	no	—
317	C₄⋊C₄	—	26	2	no	—
318	C₄⋊C₄	—	26	2	no	—
319	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	26	2	no	—
322	C₂²⋊C₄	—	26	2	no	—
323	C₄²	—	13	1	no	—
324	C₄²	—	13	1	no	—
325	C₄²	—	13	1	no	—
326	C₄²	—	26	2	no	—
327	C₄²	—	26	2	no	—
328	C₄²	—	26	2	no	—
329	C₄²	C₂³×D₁₃	1	1	yes	characteristic
330	C₁₃	#18065	1	1	yes	characteristic, complemented, Sylow
331	C₂³	—	2	2	no	—
332	C₂³	—	13	1	no	—
333	Q₈	—	26	2	no	—
334	Q₈	—	26	2	no	—
335	D₄	—	2	2	no	—
336	D₄	—	52	4	no	—
337	D₄	C₂×#6094	1	4	yes	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	13	1	no	—
342	C₂×C₄	—	13	1	no	—
343	C₂×C₄	—	13	1	no	—
344	C₂×C₄	—	26	2	no	—
345	C₂×C₄	—	26	1	no	—
346	C₂×C₄	—	26	1	no	—
347	C₂×C₄	—	26	2	no	—
348	C₂×C₄	—	26	2	no	—
349	C₂×C₄	—	26	2	no	—
350	C₂×C₄	—	52	4	no	—
351	C₂×C₄	—	52	4	no	—
352	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
354	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	2	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	2	no	—
359	C₈	—	26	1	no	—
360	C₈	—	26	1	no	—
361	C₈	—	26	2	no	—
362	C₈	—	52	1	no	—
363	C₈	—	52	2	no	—
364	C₈	C₂×#6094	1	2	yes	—
365	C₂²	—	2	4	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	26	1	no	—
369	C₂²	—	26	1	no	—
370	C₂²	—	26	1	no	—
371	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	4	2	no	—
374	C₄	—	13	2	no	—
375	C₄	—	26	1	no	—
376	C₄	—	26	1	no	—
377	C₄	—	52	2	no	—
378	C₄	—	52	2	no	—
379	C₄	C₂²×#6094	1	1	yes	characteristic
380	C₄	#11719455	1	1	yes	characteristic
381	C₄	#11719480	1	2	yes	—
382	C₂	—	4	4	no	—
383	C₂	—	52	1	no	—
384	C₂	D₄×#6094	1	1	yes	characteristic
385	C₂	#13123718	1	1	yes	characteristic
386	C₂	D₁₃×#444	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×#1580	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88166	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1580	D₁₃
hypocenter	C₁₃	#18065
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1580

Complements

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

N	H ≅ G/N	type
1. #13123959	C₂	semidirect
3. #13113746	C₂	semidirect
4. #13113744	C₂	semidirect
5. #13113672	C₂	semidirect
6. #13113669	C₂	semidirect
7. #13113660	C₂	semidirect
8. #13113590	C₂	semidirect
9. #13113272	C₂	semidirect
10. #13113122	C₂	semidirect
11. C₁₃×#1580	C₂	semidirect
105. C₁₃×D₈	M₁₆	semidirect
145. #1580	D₁₃	semidirect
299. D₈	#6094	semidirect
330. C₁₃	#18065	semidirect

Maximal subgroups

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

subgroup	index
#13123959	2
C₄×#88166	2
#13113746	2
#13113744	2
#13113672	2
#13113669	2
#13113660	2
#13113590	2
#13113272	2
#13113122	2
C₁₃×#1580	2
#18065	13