Subgroups — #16792586

Order 3328.

914 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#6118	C₂	1	1	yes	maximal, characteristic
3	#13130041	C₂	1	4	yes	maximal, complemented
4	#13130038	C₂	1	4	yes	maximal, complemented
5	C₈×#6118	C₂	1	4	yes	maximal, complemented
6	D₁₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×#6118	C₄	1	2	yes	—
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	#11719125	C₂²	1	2	yes	—
10	D₁₃⋊(C₂²⋊C₈)	C₄	1	2	yes	—
11	C₂×#88110	C₄	1	2	yes	—
12	#11719119	C₄	1	2	yes	—
13	C₂×C₄×#1057	C₂²	1	2	yes	—
14	#11719051	C₂²	1	2	yes	—
15	D₁₃⋊(C₄⋊C₈)	C₄	1	2	yes	—
16	#11719049	C₄	1	2	yes	—
17	C₄×#6118	C₄	1	2	yes	—
18	C₄×#6118	C₄	1	2	yes	—
19	C₄×#6118	C₂²	1	2	yes	—
20	C₄×#6118	C₂²	1	2	yes	—
21	C₄×D₁₃⋊C₈	C₂²	1	2	yes	—
22	C₁₃×#431	C₄	1	1	yes	characteristic
23	#11718586	C₄	1	1	yes	characteristic
24	#11718494	C₂²	1	2	yes	—
25	D₁₃×C₄⋊C₈	C₄	1	2	yes	—
26	D₁₃×C₂²⋊C₈	C₄	1	2	yes	—
27	#11718379	C₂²	1	2	yes	—
28	#11718346	C₄	1	2	yes	—
29	C₄×C₈×D₁₃	C₂²	1	2	yes	—
30	#1057⋊C₈	C₂²	1	8	yes	complemented
31	C₈×#1057	C₂²	1	8	yes	complemented
32	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1057	C₂×C₄	1	4	yes	—
36	C₂×#6118	C₈	1	16	yes	complemented
37	C₂×#6118	C₂×C₄	1	4	yes	—
38	C₂×#6118	C₂×C₄	1	4	yes	—
39	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
45	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄²×D₁₃	C₂³	1	1	yes	characteristic
52	C₄²×D₁₃	C₂³	1	1	yes	characteristic
53	C₂²⋊#1057	C₂×C₄	1	4	yes	—
54	#88112	C₂×C₄	1	4	yes	—
55	#88111	C₂×C₄	1	4	yes	—
56	#88110	C₂×C₄	1	4	yes	—
57	#88110	C₂×C₄	1	4	yes	—
58	C₄⋊#1057	C₂×C₄	1	4	yes	—
59	C₄×#1057	C₂×C₄	1	4	yes	—
60	C₄×#1057	C₂×C₄	1	4	yes	—
61	C₄×#1057	C₂³	1	4	yes	—
62	C₄×#1057	C₂³	1	4	yes	—
63	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
64	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
66	C₂²⋊#1055	C₂×C₄	1	2	yes	—
67	#88060	C₂×C₄	1	4	yes	—
68	#88054	C₂×C₄	1	4	yes	—
69	C₈×Dic₁₃	C₂³	1	4	yes	—
70	C₄⋊#1055	C₂×C₄	1	2	yes	—
71	C₄×#1055	C₂×C₄	1	2	yes	—
72	#17314	—	13	1	no	maximal, Sylow
73	C₂³×D₁₃	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×D₁₃	—	4	2	no	—
81	C₂×C₄×D₁₃	—	4	2	no	—
82	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
84	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
88	C₂×C₄×D₁₃	C₂×C₈	1	8	yes	—
89	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×#1057	C₂×C₈	1	16	yes	complemented
96	C₂×#1057	C₂²×C₄	1	8	yes	—
97	C₂×#1057	C₂²×C₄	1	8	yes	—
98	#6118	C₂×C₈	1	16	yes	complemented
99	#6118	C₂×C₈	1	16	yes	complemented
100	D₁₃⋊C₈	—	4	4	no	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
102	C₄²×C₁₃	C₄²	1	1	yes	characteristic
103	C₄²×C₁₃	C₄²	1	1	yes	characteristic
104	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
107	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
108	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
109	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
118	C₂×#1055	C₂²×C₄	1	4	yes	—
119	C₈×D₁₃	—	4	4	no	—
120	C₂×#431	—	13	1	no	—
121	C₂×#431	—	13	1	no	—
122	C₂×C₄×M₁₆	—	13	1	no	—
123	#1450	—	13	4	no	—
124	C₈×M₁₆	—	13	4	no	—
125	C₈×M₁₆	—	13	4	no	—
126	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
127	C₂²×D₁₃	—	4	1	no	—
128	C₂²×D₁₃	—	4	1	no	—
129	C₂²×D₁₃	C₄×C₈	1	4	yes	—
130	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
131	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
135	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	—	4	1	no	—
143	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
148	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
150	C₄×D₁₃	—	2	2	no	—
151	C₄×D₁₃	—	2	2	no	—
152	C₄×D₁₃	—	2	2	no	—
153	C₄×D₁₃	—	2	2	no	—
154	C₄×D₁₃	—	4	1	no	—
155	C₄×D₁₃	—	4	1	no	—
156	C₄×D₁₃	—	4	1	no	—
157	C₄×D₁₃	—	4	1	no	—
158	C₄×D₁₃	C₄×C₈	1	4	yes	—
159	C₄×D₁₃	C₄×C₈	1	4	yes	—
160	C₄×D₁₃	C₂²×C₈	1	4	yes	—
161	C₄×D₁₃	C₂²×C₈	1	4	yes	—
162	#1057	—	2	8	no	—
163	#1057	C₂²×C₈	1	16	yes	—
164	C₈×C₁₃	—	2	4	no	—
165	#1055	—	2	4	no	—
166	C₂²×M₁₆	—	13	2	no	—
167	C₂²×C₄²	—	13	1	no	—
168	#431	—	13	2	no	—
169	#431	—	13	2	no	—
170	#431	—	13	2	no	—
171	#431	—	13	2	no	—
172	#431	—	13	1	no	—
173	#431	—	13	2	no	—
174	#431	—	13	2	no	—
175	#431	—	13	2	no	—
176	#431	#261	1	1	yes	characteristic
177	C₂×C₄⋊C₈	—	13	2	no	—
178	C₂×C₄⋊C₈	—	13	2	no	—
179	C₂×C₂²⋊C₈	—	13	2	no	—
180	C₂×C₂²⋊C₈	—	13	2	no	—
181	C₄×M₁₆	—	13	2	no	—
182	C₄×M₁₆	—	13	2	no	—
183	C₄×M₁₆	—	13	2	no	—
184	C₄×M₁₆	—	13	2	no	—
185	C₂×#97	—	13	2	no	—
186	C₂×C₄×C₈	—	13	2	no	—
187	C₂×C₄×C₈	—	13	2	no	—
188	C₂×C₄×C₈	—	13	2	no	—
189	#322	—	13	8	no	—
190	C₈²	—	13	8	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
194	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
195	C₂×D₁₃	—	2	1	no	—
196	C₂×D₁₃	—	2	1	no	—
197	C₂×D₁₃	—	2	1	no	—
198	C₂×D₁₃	—	2	1	no	—
199	C₂×D₁₃	—	4	4	no	—
200	C₂×D₁₃	C₂×C₄×C₈	1	2	yes	—
201	C₂×D₁₃	C₂×C₄×C₈	1	2	yes	—
202	C₄×C₁₃	—	2	1	no	—
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	—	2	1	no	—
205	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
206	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
207	C₄×C₁₃	#568	1	2	yes	—
208	Dic₁₃	—	2	1	no	—
209	Dic₁₃	—	2	1	no	—
210	Dic₁₃	—	2	1	no	—
211	Dic₁₃	C₂×C₄×C₈	1	2	yes	—
212	Dic₁₃	C₂³×C₈	1	2	yes	—
213	Dic₁₃	#568	1	2	yes	—
214	C₂³×C₄	—	13	1	no	—
215	C₂³×C₄	—	13	1	no	—
216	C₂³×C₄	—	13	1	no	—
217	C₂×M₁₆	—	13	4	no	—
218	C₂×M₁₆	—	13	4	no	—
219	C₂×M₁₆	—	13	16	no	—
220	C₂²×C₈	—	13	4	no	—
221	C₂²×C₈	—	13	4	no	—
222	C₂²×C₈	—	13	4	no	—
223	C₂×C₄²	—	13	1	no	—
224	C₂×C₄²	—	13	1	no	—
225	C₂×C₄²	—	13	1	no	—
226	C₂×C₄²	—	13	1	no	—
227	C₂×C₄²	—	13	1	no	—
228	C₂×C₄²	—	13	1	no	—
229	C₂×C₄²	—	13	1	no	—
230	C₂×C₄²	—	13	1	no	—
231	C₂×C₄²	—	13	1	no	—
232	C₂×C₄²	—	13	1	no	—
233	C₂×C₄²	—	13	1	no	—
234	C₂×C₄²	C₂×#261	1	1	yes	characteristic
235	C₄⋊C₈	—	13	4	no	—
236	C₄⋊C₈	—	13	4	no	—
237	C₄⋊C₈	—	13	4	no	—
238	C₄⋊C₈	—	13	2	no	—
239	C₄⋊C₈	C₂×#261	1	2	yes	—
240	C₂²⋊C₈	—	13	4	no	—
241	C₂²⋊C₈	—	13	4	no	—
242	C₂²⋊C₈	—	13	2	no	—
243	C₂²⋊C₈	—	13	4	no	—
244	C₂²⋊C₈	C₂×#261	1	2	yes	—
245	#97	—	13	4	no	—
246	#97	—	13	4	no	—
247	C₄×C₈	—	13	4	no	—
248	C₄×C₈	—	13	4	no	—
249	C₄×C₈	—	13	4	no	—
250	C₄×C₈	—	13	4	no	—
251	C₄×C₈	—	13	4	no	—
252	C₄×C₈	—	13	2	no	—
253	C₄×C₈	C₂×#261	1	2	yes	—
254	C₂×C₁₃	—	2	2	no	—
255	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
256	C₂×C₁₃	#2873	1	1	yes	characteristic
257	C₂×C₁₃	#2873	1	1	yes	characteristic
258	D₁₃	—	2	2	no	—
259	D₁₃	—	2	2	no	—
260	C₂⁴	—	13	1	no	—
261	C₂²×C₄	—	13	1	no	—
262	C₂²×C₄	—	13	8	no	—
263	C₂²×C₄	—	13	1	no	—
264	C₂²×C₄	—	13	1	no	—
265	C₂²×C₄	—	13	1	no	—
266	C₂²×C₄	—	13	1	no	—
267	C₂²×C₄	—	13	1	no	—
268	C₂²×C₄	—	13	1	no	—
269	C₂²×C₄	—	13	1	no	—
270	C₂²×C₄	—	13	1	no	—
271	C₂²×C₄	—	13	1	no	—
272	C₂²×C₄	—	13	1	no	—
273	C₂²×C₄	—	13	1	no	—
274	C₂²×C₄	—	13	1	no	—
275	C₂²×C₄	—	13	1	no	—
276	C₂²×C₄	—	13	1	no	—
277	C₂²×C₄	—	52	2	no	—
278	C₂²×C₄	—	52	2	no	—
279	C₂²×C₄	C₄×#261	1	1	yes	characteristic
280	C₂²×C₄	C₄×#261	1	1	yes	characteristic
281	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
282	M₁₆	—	13	16	no	—
283	M₁₆	—	13	16	no	—
284	C₂×C₈	—	13	8	no	—
285	C₂×C₈	—	13	16	no	—
286	C₂×C₈	—	13	8	no	—
287	C₂×C₈	—	13	4	no	—
288	C₂×C₈	—	52	4	no	—
289	C₂×C₈	—	52	4	no	—
290	C₂×C₈	C₂²×#261	1	4	yes	—
291	C₄²	—	13	1	no	—
292	C₄²	—	13	1	no	—
293	C₄²	—	13	1	no	—
294	C₄²	—	13	1	no	—
295	C₄²	—	13	1	no	—
296	C₄²	—	13	1	no	—
297	C₄²	—	13	1	no	—
298	C₄²	—	13	1	no	—
299	C₄²	—	13	1	no	—
300	C₄²	—	13	1	no	—
301	C₄²	—	13	1	no	—
302	C₄²	—	13	1	no	—
303	C₄²	C₄×#261	1	1	yes	characteristic
304	C₄²	C₄×#261	1	1	yes	characteristic
305	C₄²	C₂²×#261	1	1	yes	characteristic
306	C₄²	C₂²×#261	1	1	yes	characteristic
307	C₁₃	#17314	1	1	yes	characteristic, complemented, Sylow
308	C₂³	—	13	1	no	—
309	C₂³	—	13	1	no	—
310	C₂³	—	13	4	no	—
311	C₂³	—	52	1	no	—
312	C₂³	—	52	1	no	—
313	C₂³	C₂×C₄×#261	1	1	yes	characteristic
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	13	4	no	—
317	C₂×C₄	—	13	1	no	—
318	C₂×C₄	—	13	1	no	—
319	C₂×C₄	—	13	1	no	—
320	C₂×C₄	—	13	4	no	—
321	C₂×C₄	—	13	4	no	—
322	C₂×C₄	—	13	4	no	—
323	C₂×C₄	—	13	4	no	—
324	C₂×C₄	—	13	1	no	—
325	C₂×C₄	—	13	1	no	—
326	C₂×C₄	—	13	1	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	26	2	no	—
330	C₂×C₄	—	26	2	no	—
331	C₂×C₄	—	26	2	no	—
332	C₂×C₄	—	52	1	no	—
333	C₂×C₄	—	52	1	no	—
334	C₂×C₄	—	52	1	no	—
335	C₂×C₄	—	52	1	no	—
336	C₂×C₄	—	52	1	no	—
337	C₂×C₄	C₈×#261	1	4	yes	—
338	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
341	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
342	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
343	C₂×C₄	C₂³×#261	1	1	yes	characteristic
344	C₈	—	2	4	no	—
345	C₈	—	13	16	no	—
346	C₈	—	26	8	no	—
347	C₈	—	26	4	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	13	2	no	—
351	C₂²	—	13	2	no	—
352	C₂²	—	26	1	no	—
353	C₂²	—	26	1	no	—
354	C₂²	—	26	1	no	—
355	C₂²	—	26	1	no	—
356	C₂²	—	52	4	no	—
357	C₂²	C₂×C₈×#261	1	2	yes	—
358	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	13	2	no	—
363	C₄	—	13	2	no	—
364	C₄	—	13	2	no	—
365	C₄	—	26	1	no	—
366	C₄	—	26	1	no	—
367	C₄	—	26	1	no	—
368	C₄	C₂×C₈×#261	1	2	yes	—
369	C₄	C₂×C₈×#261	1	2	yes	—
370	C₄	#11719629	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	26	2	no	—
373	C₂	—	26	2	no	—
374	C₂	C₂²×C₈×#261	1	1	yes	characteristic
375	C₂	#13131399	1	1	yes	characteristic
376	C₂	#13131401	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₁₃×#431	C₄
socle	C₂²×C₁₃	C₂²×C₄²
prime radical	C₂³×D₁₃	C₄²
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#431	#261
hypocenter	C₁₃	#17314
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
3. #13130041	C₂	semidirect
4. #13130038	C₂	semidirect
5. C₈×#6118	C₂	semidirect
30. #1057⋊C₈	C₂²	semidirect
31. C₈×#1057	C₂²	semidirect
36. C₂×#6118	C₈	semidirect
95. C₂×#1057	C₂×C₈	semidirect
98. #6118	C₂×C₈	semidirect
99. #6118	C₂×C₈	semidirect
307. C₁₃	#17314	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊#431	2
C₂×C₄×#6118	2
#13130041	2
#13130038	2
C₈×#6118	2
D₁₃×#431	2
#17314	13