Subgroups — #16015910

Order 3328.

706 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718568	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718562	C₂	1	1	yes	maximal, characteristic
3	#13123258	C₂	1	1	yes	maximal, characteristic
4	#13116595	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116593	C₂	1	2	yes	maximal, complemented
6	#13116593	C₂	1	2	yes	maximal, complemented
7	#13116591	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116590	C₂	1	2	yes	maximal, complemented
9	C₂×#11718093	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₃×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
11	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88161	C₄	1	1	yes	characteristic
13	C₂×#88155	C₂²	1	1	yes	characteristic
14	#11719227	C₄	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#405	C₂²	1	1	yes	characteristic
18	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88052	C₂²	1	1	yes	characteristic
21	C₂²×#6090	C₂²	1	1	yes	characteristic
22	#11718658	C₄	1	1	yes	characteristic
23	#11718568	—	2	4	no	—
24	#11718562	—	2	4	no	—
25	C₂×#88038	C₂²	1	2	yes	—
26	C₂×#88038	C₂²	1	2	yes	—
27	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
28	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
29	#11718369	C₄	1	2	yes	—
30	#11718368	C₄	1	2	yes	—
31	#11718350	C₂²	1	2	yes	—
32	C₄×C₈×D₁₃	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
35	#11718093	C₂²	1	1	yes	characteristic
36	#11718093	C₂²	1	2	yes	—
37	#11718093	C₂²	1	1	yes	characteristic
38	#11718093	C₂²	1	1	yes	characteristic
39	#11718093	C₂²	1	1	yes	characteristic
40	#11718093	C₂²	1	2	yes	—
41	C₄⋊#6090	C₂²	1	2	yes	—
42	C₄⋊#6090	C₂²	1	2	yes	—
43	C₂×#6127	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×#6099	C₂×C₄	1	1	yes	characteristic
49	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
50	#88161	—	2	4	no	—
51	#88155	D₄	1	4	yes	—
52	#88155	Q₈	1	4	yes	—
53	C₂×#6094	C₂×C₄	1	2	yes	—
54	C₂×#6094	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₁₃	C₂³	1	2	yes	—
57	#88147	C₈	1	1	yes	characteristic
58	#88147	C₈	1	1	yes	characteristic
59	#88141	C₈	1	1	yes	characteristic
60	#88141	C₈	1	1	yes	characteristic
61	#88135	C₈	1	2	yes	—
62	#88135	C₈	1	2	yes	—
63	#88126	C₂×C₄	1	2	yes	—
64	C₄²×D₁₃	C₂×C₄	1	2	yes	—
65	C₁₃×M₃₂	—	2	2	no	—
66	C₂×C₁₃×C₁₆	—	2	2	no	—
67	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
70	C₁₃×#97	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₁₃	C₂³	1	2	yes	—
72	#88055	C₂×C₄	1	2	yes	—
73	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
76	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
77	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
78	#88052	—	2	2	no	—
79	C₂×#6090	—	2	2	no	—
80	C₂×#6090	C₂³	1	1	yes	characteristic
81	C₂×#6090	C₂³	1	1	yes	characteristic
82	C₂×#6090	C₂³	1	2	yes	—
83	#88043	C₂×C₄	1	2	yes	—
84	C₄×#1055	C₂×C₄	1	2	yes	—
85	#88038	—	2	4	no	—
86	#88038	—	2	4	no	—
87	C₁₆×D₁₃	—	2	4	no	—
88	C₁₆×D₁₃	—	2	4	no	—
89	#33398	—	13	1	no	maximal, Sylow
90	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
92	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
93	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
94	#6127	C₄⋊C₄	1	4	yes	—
95	#6127	M₁₆	1	4	yes	—
96	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
98	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
99	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
100	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
101	C₁₃×M₁₆	—	2	2	no	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
106	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
108	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
109	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
110	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
111	#6103	C₂×C₈	1	2	yes	—
112	#6103	C₂×C₈	1	2	yes	—
113	#6101	C₂×C₈	1	2	yes	—
114	#6101	C₂×C₈	1	2	yes	—
115	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
116	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
117	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
118	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
119	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
121	#6099	C₄⋊C₄	1	4	yes	—
122	C₂×#1055	—	2	2	no	—
123	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
124	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
125	C₂×#1055	C₂²×C₄	1	2	yes	—
126	#6094	—	2	4	no	—
127	#6094	C₄⋊C₄	1	8	yes	—
128	C₈×D₁₃	—	2	4	no	—
129	C₈×D₁₃	C₂×D₄	1	4	yes	—
130	C₈×D₁₃	C₂×Q₈	1	4	yes	—
131	C₁₃×C₁₆	—	2	1	no	—
132	C₁₃×C₁₆	—	2	1	no	—
133	C₁₃×C₁₆	—	2	2	no	—
134	#6090	—	2	1	no	—
135	#6090	—	2	2	no	—
136	#6090	—	2	1	no	—
137	C₂×#504	—	13	1	no	—
138	C₂×#504	—	13	1	no	—
139	C₄×#131	—	13	1	no	—
140	#2151	—	13	1	no	—
141	#2151	—	13	2	no	—
142	#2151	—	13	2	no	—
143	#2151	D₁₃	1	1	yes	characteristic, complemented
144	#2150	—	13	1	no	—
145	#2150	—	13	2	no	—
146	C₂×C₄⋊C₁₆	—	13	1	no	—
147	C₂×C₄⋊C₁₆	—	13	2	no	—
148	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
149	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
150	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
153	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
155	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
156	C₂²⋊D₁₃	C₄⋊C₈	1	8	yes	—
157	C₂×Dic₁₃	—	2	2	no	—
158	C₂×Dic₁₃	—	4	1	no	—
159	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
160	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
161	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
162	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
163	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
164	D₅₂	C₄⋊C₈	1	4	yes	—
165	C₄×D₁₃	—	2	4	no	—
166	C₄×D₁₃	—	4	2	no	—
167	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
168	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
169	Dic₂₆	C₄⋊C₈	1	4	yes	—
170	C₈×C₁₃	—	2	2	no	—
171	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
172	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
173	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
174	#1055	—	2	1	no	—
175	#1055	—	2	1	no	—
176	#1055	C₂×C₄⋊C₄	1	4	yes	—
177	C₂×#131	—	13	1	no	—
178	C₂×#131	—	13	1	no	—
179	C₄×#41	—	13	1	no	—
180	#504	—	26	4	no	—
181	#504	—	26	4	no	—
182	C₂×M₃₂	—	13	1	no	—
183	C₂×M₃₂	—	13	2	no	—
184	C₂×M₃₂	—	13	2	no	—
185	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
186	C₂²×C₁₆	—	13	1	no	—
187	C₂²×C₁₆	—	13	2	no	—
188	C₂²×C₁₆	—	13	2	no	—
189	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
190	#405	—	13	1	no	—
191	#405	—	13	2	no	—
192	#405	—	13	2	no	—
193	#405	C₂×D₁₃	1	1	yes	characteristic
194	C₄×M₁₆	—	13	1	no	—
195	C₄×M₁₆	—	13	2	no	—
196	C₂×C₄×C₈	—	13	1	no	—
197	C₂×C₄×C₈	—	13	2	no	—
198	C₄⋊C₁₆	—	13	1	no	—
199	C₄⋊C₁₆	—	13	2	no	—
200	C₄⋊C₁₆	—	13	1	no	—
201	C₄⋊C₁₆	—	13	1	no	—
202	C₄⋊C₁₆	—	13	1	no	—
203	C₄⋊C₁₆	—	13	2	no	—
204	C₄⋊C₁₆	—	13	2	no	—
205	C₄⋊C₁₆	—	13	2	no	—
206	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
207	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
208	C₂²×C₁₃	—	2	1	no	—
209	C₂²×C₁₃	—	2	1	no	—
210	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
211	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
212	C₂×D₁₃	—	2	2	no	—
213	C₂×D₁₃	—	2	2	no	—
214	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
215	C₄×C₁₃	—	2	2	no	—
216	C₄×C₁₃	—	2	2	no	—
217	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
218	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
219	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
220	Dic₁₃	—	2	1	no	—
221	Dic₁₃	—	2	1	no	—
222	Dic₁₃	—	2	1	no	—
223	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
224	Dic₁₃	#504	1	2	yes	complemented
225	C₂×#41	—	13	1	no	—
226	#131	—	13	4	no	—
227	#131	—	13	4	no	—
228	#131	—	26	4	no	—
229	C₂×M₁₆	—	13	1	no	—
230	C₂×M₁₆	—	13	2	no	—
231	C₂×M₁₆	—	13	2	no	—
232	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
233	C₂²×C₈	—	13	2	no	—
234	C₂²×C₈	—	13	1	no	—
235	C₂²×C₈	—	13	2	no	—
236	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
237	C₄×Q₈	—	13	1	no	—
238	C₄×Q₈	—	13	1	no	—
239	C₄×D₄	—	13	1	no	—
240	C₄×D₄	—	13	1	no	—
241	C₄×D₄	—	13	2	no	—
242	C₄×D₄	—	13	2	no	—
243	#117	—	13	2	no	—
244	#117	C₄×D₁₃	1	1	yes	characteristic
245	C₂×C₄²	—	13	1	no	—
246	C₂×C₄²	—	13	2	no	—
247	M₃₂	—	2	2	no	—
248	M₃₂	—	26	4	no	—
249	M₃₂	—	26	4	no	—
250	M₃₂	—	26	2	no	—
251	C₂×C₁₆	—	2	2	no	—
252	C₂×C₁₆	—	13	1	no	—
253	C₂×C₁₆	—	13	1	no	—
254	C₂×C₁₆	—	13	2	no	—
255	C₂×C₁₆	—	26	4	no	—
256	C₂×C₁₆	—	26	4	no	—
257	C₂×C₁₆	—	26	2	no	—
258	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
259	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
260	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
261	#97	—	13	2	no	—
262	#97	—	13	2	no	—
263	#97	C₄×D₁₃	1	2	yes	—
264	C₄×C₈	—	13	2	no	—
265	C₄×C₈	—	13	1	no	—
266	C₄×C₈	—	13	1	no	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	13	1	no	—
269	C₄×C₈	—	13	2	no	—
270	C₄×C₈	C₂²×D₁₃	1	2	yes	—
271	C₂×C₁₃	—	2	2	no	—
272	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
273	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
274	C₂×C₁₃	#3407	1	1	yes	characteristic
275	D₁₃	—	2	4	no	—
276	#41	—	13	4	no	—
277	#41	—	13	4	no	—
278	C₂×Q₈	—	13	1	no	—
279	C₂×D₄	—	13	2	no	—
280	C₂×D₄	—	13	1	no	—
281	C₂²×C₄	—	13	2	no	—
282	C₂²×C₄	—	13	1	no	—
283	C₂²×C₄	—	13	1	no	—
284	C₂²×C₄	—	13	2	no	—
285	C₂²×C₄	—	13	2	no	—
286	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
287	M₁₆	—	2	2	no	—
288	M₁₆	—	13	4	no	—
289	M₁₆	—	13	8	no	—
290	M₁₆	—	26	4	no	—
291	C₂×C₈	—	13	2	no	—
292	C₂×C₈	—	13	4	no	—
293	C₂×C₈	—	13	4	no	—
294	C₂×C₈	—	13	1	no	—
295	C₂×C₈	—	13	1	no	—
296	C₂×C₈	—	26	4	no	—
297	C₂×C₈	—	26	2	no	—
298	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
299	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
300	C₂×C₈	D₄×D₁₃	1	2	yes	—
301	C₂×C₈	Q₈×D₁₃	1	2	yes	—
302	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
303	C₄⋊C₄	—	13	2	no	—
304	C₄⋊C₄	—	13	2	no	—
305	C₄⋊C₄	C₈×D₁₃	1	1	yes	characteristic
306	C₄⋊C₄	C₈×D₁₃	1	1	yes	characteristic
307	C₂²⋊C₄	—	13	2	no	—
308	C₂²⋊C₄	—	13	2	no	—
309	C₂²⋊C₄	C₈×D₁₃	1	2	yes	—
310	C₄²	—	13	2	no	—
311	C₄²	—	13	1	no	—
312	C₄²	—	13	2	no	—
313	C₄²	—	13	1	no	—
314	C₄²	—	13	1	no	—
315	C₄²	—	13	1	no	—
316	C₄²	C₂×C₄×D₁₃	1	2	yes	—
317	C₁₆	—	2	1	no	—
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	2	no	—
320	C₁₆	—	26	1	no	—
321	C₁₆	—	26	2	no	—
322	C₁₆	—	26	1	no	—
323	C₁₃	#33398	1	1	yes	characteristic, complemented, Sylow
324	C₂³	—	13	2	no	—
325	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
326	Q₈	—	13	4	no	—
327	D₄	—	13	8	no	—
328	D₄	—	13	4	no	—
329	C₂×C₄	—	13	1	no	—
330	C₂×C₄	—	13	2	no	—
331	C₂×C₄	—	13	4	no	—
332	C₂×C₄	—	13	4	no	—
333	C₂×C₄	—	13	1	no	—
334	C₂×C₄	—	13	1	no	—
335	C₂×C₄	—	13	1	no	—
336	C₂×C₄	—	26	4	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	52	2	no	—
339	C₂×C₄	—	52	1	no	—
340	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
341	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
342	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
343	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
344	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
345	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	13	4	no	—
348	C₈	—	26	1	no	—
349	C₈	—	26	1	no	—
350	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
351	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
352	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	13	4	no	—
356	C₂²	—	26	2	no	—
357	C₂²	—	26	2	no	—
358	C₂²	D₁₃×C₄⋊C₈	1	2	yes	—
359	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	13	4	no	—
363	C₄	—	13	2	no	—
364	C₄	—	26	1	no	—
365	C₄	—	26	1	no	—
366	C₄	—	26	1	no	—
367	C₄	D₁₃×C₄⋊C₈	1	2	yes	—
368	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
369	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
370	C₂	—	2	2	no	—
371	C₂	—	26	4	no	—
372	C₂	C₂×D₁₃×C₄⋊C₈	1	1	yes	characteristic
373	C₂	D₁₃×#504	1	1	yes	characteristic
374	C₂	#13127937	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2151	C₂
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2151	D₁₃
hypocenter	C₁₃	#33398
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
4. #13116595	C₂	semidirect
5. #13116593	C₂	semidirect
6. #13116593	C₂	semidirect
7. #13116591	C₂	semidirect
8. #13116590	C₂	semidirect
9. C₂×#11718093	C₂	semidirect
10. D₁₃×C₄⋊C₁₆	C₂	semidirect
11. C₁₃×#2151	C₂	semidirect
143. #2151	D₁₃	semidirect
224. Dic₁₃	#504	semidirect
323. C₁₃	#33398	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718568	2
C₂×#11718562	2
#13123258	2
#13116595	2
#13116593	2
#13116591	2
#13116590	2
C₂×#11718093	2
D₁₃×C₄⋊C₁₆	2
C₁₃×#2151	2
#33398	13