Subgroups — #15828645

Order 3328.

814 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123800	C₂	1	2	yes	maximal
2	C₄×#88161	C₂	1	1	yes	maximal, characteristic
3	#13113140	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113139	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113130	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113124	C₂	1	2	yes	maximal, complemented
7	#13113121	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113117	C₂	1	2	yes	maximal, complemented
9	#13113115	C₂	1	2	yes	maximal, complemented
10	C₈²×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88161	C₄	1	2	yes	—
13	C₄×#6127	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	2	yes	—
15	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
16	#11718783	C₄	1	2	yes	—
17	C₄×#6099	C₂²	1	2	yes	—
18	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
19	#11718497	C₄	1	4	yes	—
20	#11718489	C₄	1	4	yes	—
21	#11718390	C₄	1	4	yes	—
22	#11718389	C₄	1	4	yes	—
23	#11718369	C₄	1	2	yes	—
24	#11718369	C₄	1	2	yes	—
25	C₄×#6094	C₂²	1	2	yes	—
26	C₄×#6094	C₂²	1	1	yes	characteristic
27	C₄×#6094	C₂²	1	1	yes	characteristic
28	#11718346	C₂²	1	2	yes	—
29	#11718346	C₂²	1	2	yes	—
30	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₁₃	C₂²	1	2	yes	—
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	C₁₃×#322	C₂²	1	2	yes	—
34	C₁₃×#322	C₂²	1	1	yes	characteristic
35	C₈²×C₁₃	C₂²	1	1	yes	characteristic
36	#11718047	C₂²	1	1	yes	characteristic
37	#11718046	C₂²	1	2	yes	—
38	#11718046	C₂²	1	2	yes	—
39	#11718046	C₂²	1	2	yes	—
40	C₈×#1055	C₂²	1	2	yes	—
41	C₈×#1055	C₂²	1	1	yes	characteristic
42	C₈×#1055	C₂²	1	2	yes	—
43	C₂×#6127	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
47	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
48	#88180	C₂×C₄	1	1	yes	characteristic
49	C₂×#6099	C₂×C₄	1	4	yes	—
50	C₂²×#1055	C₂×C₄	1	2	yes	—
51	#88161	—	2	8	no	—
52	C₂×#6094	C₂×C₄	1	2	yes	—
53	C₂×#6094	C₂×C₄	1	2	yes	—
54	C₂×#6094	C₂×C₄	1	4	yes	—
55	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
57	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
58	C₄×D₅₂	C₂×C₄	1	2	yes	—
59	#88126	C₂×C₄	1	1	yes	characteristic
60	#88126	C₂×C₄	1	1	yes	characteristic
61	C₄²×D₁₃	C₂³	1	1	yes	characteristic
62	C₄²×D₁₃	C₂³	1	1	yes	characteristic
63	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
64	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
65	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
66	C₁₃×#97	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	2	yes	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	#88060	C₂×C₄	1	4	yes	—
72	#88060	C₂×C₄	1	4	yes	—
73	#88055	C₂×C₄	1	4	yes	—
74	#88055	C₂×C₄	1	2	yes	—
75	#88055	C₂×C₄	1	2	yes	—
76	#88054	C₂×C₄	1	4	yes	—
77	#88054	C₂×C₄	1	4	yes	—
78	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
79	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
80	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
81	#88043	C₂×C₄	1	4	yes	—
82	C₄×#1055	C₂×C₄	1	2	yes	—
83	C₄×#1055	C₂³	1	1	yes	characteristic
84	C₄×#1055	C₂³	1	2	yes	—
85	C₄×#1055	C₂³	1	2	yes	—
86	C₄×#1055	C₂³	1	1	yes	characteristic
87	#17335	—	13	1	no	maximal, Sylow
88	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₁₃	C₄²	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₃	C₄²	1	1	yes	characteristic
92	#6127	—	2	4	no	—
93	C₂×D₅₂	C₄²	1	1	yes	characteristic
94	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₂₆	C₄²	1	1	yes	characteristic
99	C₁₃×M₁₆	—	2	4	no	—
100	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₁₃	C₄²	1	2	yes	—
108	#6103	C₄²	1	2	yes	—
109	#6103	C₄²	1	2	yes	—
110	C₄⋊Dic₁₃	C₄²	1	2	yes	—
111	#6101	C₄²	1	2	yes	—
112	#6101	C₄²	1	2	yes	—
113	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
114	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
115	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
117	#6099	—	2	8	no	—
118	C₂×#1055	—	2	4	no	—
119	C₂×#1055	C₂²×C₄	1	2	yes	—
120	C₂×#1055	C₂²×C₄	1	4	yes	—
121	C₂×#1055	C₂²×C₄	1	2	yes	—
122	C₂×#1055	C₂²×C₄	1	4	yes	—
123	#6094	—	2	4	no	—
124	#6094	—	2	4	no	—
125	#6094	—	2	8	no	—
126	C₈×D₁₃	—	2	4	no	—
127	C₈×D₁₃	—	2	8	no	—
128	C₈×D₁₃	M₁₆	1	8	yes	complemented
129	#2965	—	13	2	no	—
130	C₄×#131	—	13	1	no	—
131	#1455	—	13	1	no	—
132	#1455	—	13	1	no	—
133	#1453	—	13	1	no	—
134	#1453	D₁₃	1	1	yes	characteristic, complemented
135	#1451	—	13	2	no	—
136	#1450	—	13	1	no	—
137	C₈×M₁₆	—	13	2	no	—
138	C₈×M₁₆	—	13	2	no	—
139	C₂×C₈²	—	13	1	no	—
140	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
141	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
142	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	4	no	—
145	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
146	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
149	C₂²⋊D₁₃	—	2	2	no	—
150	C₂²⋊D₁₃	—	2	2	no	—
151	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
152	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
153	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
154	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
155	D₅₂	—	2	2	no	—
156	C₄×D₁₃	—	2	2	no	—
157	C₄×D₁₃	—	2	4	no	—
158	C₄×D₁₃	—	2	4	no	—
159	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
160	Dic₂₆	—	2	2	no	—
161	C₈×C₁₃	—	2	2	no	—
162	C₈×C₁₃	—	2	4	no	—
163	C₈×C₁₃	—	2	4	no	—
164	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
165	#1055	—	2	2	no	—
166	#1055	—	2	4	no	—
167	#1055	C₂×M₁₆	1	4	yes	—
168	#1055	#131	1	8	yes	complemented
169	C₂×#131	—	13	2	no	—
170	C₄×#41	—	13	1	no	—
171	#446	—	13	4	no	—
172	#436	—	13	4	no	—
173	#435	—	13	4	no	—
174	#435	—	13	4	no	—
175	#431	—	13	2	no	—
176	#431	—	13	2	no	—
177	#431	C₂×D₁₃	1	2	yes	—
178	#405	—	13	2	no	—
179	#405	—	13	2	no	—
180	#405	—	13	2	no	—
181	C₄×M₁₆	—	13	2	no	—
182	C₄×M₁₆	—	13	1	no	—
183	C₄×M₁₆	—	13	1	no	—
184	C₄×M₁₆	—	13	2	no	—
185	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
186	C₂×C₄×C₈	—	13	1	no	—
187	C₂×C₄×C₈	—	13	1	no	—
188	C₂×C₄×C₈	—	13	1	no	—
189	C₂×C₄×C₈	—	13	2	no	—
190	#322	—	13	2	no	—
191	#322	—	13	2	no	—
192	#322	—	13	2	no	—
193	#322	—	13	1	no	—
194	#322	C₂×D₁₃	1	2	yes	—
195	#322	C₂×D₁₃	1	1	yes	characteristic
196	C₈²	—	13	2	no	—
197	C₈²	—	13	1	no	—
198	C₈²	—	13	2	no	—
199	C₈²	C₂×D₁₃	1	1	yes	characteristic
200	C₂²×C₁₃	—	2	1	no	—
201	C₂²×C₁₃	—	2	2	no	—
202	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
203	C₂×D₁₃	—	2	2	no	—
204	C₂×D₁₃	—	2	1	no	—
205	C₂×D₁₃	—	2	2	no	—
206	C₂×D₁₃	C₄×M₁₆	1	2	yes	—
207	C₄×C₁₃	—	2	1	no	—
208	C₄×C₁₃	—	2	2	no	—
209	C₄×C₁₃	C₂²×M₁₆	1	2	yes	—
210	C₄×C₁₃	C₂×#131	1	4	yes	—
211	Dic₁₃	—	2	1	no	—
212	Dic₁₃	—	2	2	no	—
213	Dic₁₃	—	2	2	no	—
214	Dic₁₃	C₄×M₁₆	1	2	yes	—
215	C₂×#41	—	13	1	no	—
216	#131	—	26	8	no	—
217	C₂×M₁₆	—	13	4	no	—
218	C₂×M₁₆	—	13	2	no	—
219	C₂×M₁₆	—	13	2	no	—
220	C₂×M₁₆	—	13	4	no	—
221	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
222	C₂²×C₈	—	13	2	no	—
223	C₂²×C₈	—	13	2	no	—
224	C₂²×C₈	—	13	2	no	—
225	C₂²×C₈	—	13	4	no	—
226	C₄×Q₈	—	13	2	no	—
227	C₄×D₄	—	13	2	no	—
228	C₄×D₄	—	13	2	no	—
229	C₄×D₄	—	13	2	no	—
230	#117	—	13	1	no	—
231	#117	—	13	1	no	—
232	#117	—	13	1	no	—
233	C₂×C₄²	—	13	1	no	—
234	C₂×C₄²	—	13	1	no	—
235	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
236	C₄⋊C₈	—	13	4	no	—
237	C₄⋊C₈	—	13	4	no	—
238	C₄⋊C₈	C₄×D₁₃	1	4	yes	—
239	C₂²⋊C₈	—	13	4	no	—
240	C₂²⋊C₈	—	13	4	no	—
241	C₂²⋊C₈	C₄×D₁₃	1	4	yes	—
242	#97	—	13	2	no	—
243	#97	—	13	2	no	—
244	#97	—	13	4	no	—
245	#97	—	13	4	no	—
246	#97	C₄×D₁₃	1	2	yes	—
247	C₄×C₈	—	13	2	no	—
248	C₄×C₈	—	13	2	no	—
249	C₄×C₈	—	13	4	no	—
250	C₄×C₈	—	13	2	no	—
251	C₄×C₈	—	13	1	no	—
252	C₄×C₈	—	13	2	no	—
253	C₄×C₈	—	13	2	no	—
254	C₄×C₈	—	13	1	no	—
255	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
256	C₄×C₈	C₂²×D₁₃	1	2	yes	—
257	C₄×C₈	C₂²×D₁₃	1	2	yes	—
258	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
259	C₂×C₁₃	—	4	1	no	—
260	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
261	C₂×C₁₃	C₄×#131	1	2	yes	—
262	D₁₃	—	2	2	no	—
263	D₁₃	—	4	1	no	—
264	#41	—	26	4	no	—
265	C₂×Q₈	—	13	1	no	—
266	C₂×D₄	—	13	1	no	—
267	C₂×D₄	—	13	1	no	—
268	C₂×D₄	—	13	1	no	—
269	C₂²×C₄	—	13	1	no	—
270	C₂²×C₄	—	13	2	no	—
271	C₂²×C₄	—	13	2	no	—
272	C₂²×C₄	—	13	1	no	—
273	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
274	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
275	M₁₆	—	2	4	no	—
276	M₁₆	—	26	8	no	—
277	M₁₆	—	26	4	no	—
278	M₁₆	—	26	8	no	—
279	M₁₆	—	26	4	no	—
280	C₂×C₈	—	13	2	no	—
281	C₂×C₈	—	13	4	no	—
282	C₂×C₈	—	13	8	no	—
283	C₂×C₈	—	13	2	no	—
284	C₂×C₈	—	13	4	no	—
285	C₂×C₈	—	26	4	no	—
286	C₂×C₈	—	26	8	no	—
287	C₂×C₈	—	26	4	no	—
288	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
290	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
291	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
292	C₄⋊C₄	—	13	2	no	—
293	C₄⋊C₄	—	13	2	no	—
294	C₄⋊C₄	—	13	2	no	—
295	C₂²⋊C₄	—	13	2	no	—
296	C₂²⋊C₄	—	13	2	no	—
297	C₂²⋊C₄	—	13	2	no	—
298	C₄²	—	13	1	no	—
299	C₄²	—	13	1	no	—
300	C₄²	—	13	2	no	—
301	C₄²	—	13	2	no	—
302	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₁₃	1	2	yes	—
304	C₄²	C₂³×D₁₃	1	1	yes	characteristic
305	C₁₃	#17335	1	1	yes	characteristic, complemented, Sylow
306	C₂³	—	13	1	no	—
307	C₂³	—	13	1	no	—
308	C₂³	C₄²×D₁₃	1	1	yes	characteristic
309	Q₈	—	26	2	no	—
310	D₄	—	26	2	no	—
311	D₄	—	26	2	no	—
312	D₄	—	26	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	13	1	no	—
316	C₂×C₄	—	13	4	no	—
317	C₂×C₄	—	13	2	no	—
318	C₂×C₄	—	13	2	no	—
319	C₂×C₄	—	13	1	no	—
320	C₂×C₄	—	26	2	no	—
321	C₂×C₄	—	26	4	no	—
322	C₂×C₄	—	26	4	no	—
323	C₂×C₄	C₄²×D₁₃	1	2	yes	—
324	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
326	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
327	C₈	—	2	2	no	—
328	C₈	—	2	4	no	—
329	C₈	—	2	4	no	—
330	C₈	—	13	4	no	—
331	C₈	—	13	8	no	—
332	C₈	—	26	2	no	—
333	C₈	—	26	4	no	—
334	C₈	M₁₆×D₁₃	1	4	yes	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	13	2	no	—
338	C₂²	—	26	2	no	—
339	C₂²	—	26	1	no	—
340	C₂²	—	26	2	no	—
341	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	13	2	no	—
345	C₄	—	26	1	no	—
346	C₄	—	26	2	no	—
347	C₄	—	26	2	no	—
348	C₄	C₂×M₁₆×D₁₃	1	2	yes	—
349	C₄	D₁₃×#131	1	4	yes	—
350	C₂	—	4	1	no	—
351	C₂	—	26	2	no	—
352	C₂	—	52	1	no	—
353	C₂	C₄×M₁₆×D₁₃	1	1	yes	characteristic
354	C₂	#13123258	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Fitting	C₁₃×#1453	C₂
prime radical	C₂×#6127	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1453	D₁₃
hypocenter	C₁₃	#17335
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ #1453

Complements

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

N	H ≅ G/N	type
3. #13113140	C₂	semidirect
4. #13113139	C₂	semidirect
5. #13113130	C₂	semidirect
6. #13113124	C₂	semidirect
7. #13113121	C₂	semidirect
8. #13113117	C₂	semidirect
9. #13113115	C₂	semidirect
10. C₈²×D₁₃	C₂	semidirect
11. C₁₃×#1453	C₂	semidirect
128. C₈×D₁₃	M₁₆	semidirect
134. #1453	D₁₃	semidirect
168. #1055	#131	semidirect
305. C₁₃	#17335	semidirect

Maximal subgroups

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

subgroup	index
#13123800	2
C₄×#88161	2
#13113140	2
#13113139	2
#13113130	2
#13113124	2
#13113121	2
#13113117	2
#13113115	2
C₈²×D₁₃	2
C₁₃×#1453	2
#17335	13