Subgroups — #16016014

Order 3328.

582 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	C₄×#88155	C₂	1	1	yes	maximal, characteristic
2	#13116642	C₂	1	2	yes	maximal, complemented
3	#13116640	C₂	1	2	yes	maximal, complemented
4	#13116639	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116637	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116637	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116593	C₂	1	2	yes	maximal, complemented
8	#13116448	C₂	1	2	yes	maximal
9	#13116429	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88155	C₄	1	1	yes	characteristic
13	C₂×#88155	C₂²	1	1	yes	characteristic
14	C₄×#6127	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718762	C₄	1	1	yes	characteristic
17	#11718762	C₂²	1	1	yes	characteristic
18	C₄×#6099	C₄	1	1	yes	characteristic
19	C₂×#88038	C₂²	1	2	yes	—
20	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
21	#11718350	C₄	1	1	yes	characteristic
22	#11718350	C₄	1	1	yes	characteristic
23	#11718350	C₂²	1	1	yes	characteristic
24	#11718350	C₂²	1	1	yes	characteristic
25	C₄×#6094	C₄	1	1	yes	characteristic
26	C₄×#6094	C₄	1	1	yes	characteristic
27	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
28	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
31	C₁₃×#346	C₂²	1	1	yes	characteristic
32	C₁₃×#346	C₂²	1	1	yes	characteristic
33	C₂²⋊#6090	C₂²	1	2	yes	—
34	#11718099	C₂²	1	2	yes	—
35	#11718099	C₂²	1	2	yes	—
36	#11718094	C₂²	1	2	yes	—
37	#11718093	C₂²	1	2	yes	—
38	#11718093	C₂²	1	2	yes	—
39	C₁₆×Dic₁₃	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	2	yes	—
41	#11718053	C₂²	1	1	yes	characteristic
42	#11718053	C₂²	1	1	yes	characteristic
43	C₂×#6127	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₂²⋊D₁₃	C₈	1	2	yes	—
48	C₄×C₂²⋊D₁₃	C₈	1	2	yes	—
49	#88180	C₂×C₄	1	1	yes	characteristic
50	C₂×#6099	C₂×C₄	1	1	yes	characteristic
51	C₂×#6099	C₂×C₄	1	1	yes	characteristic
52	#88155	—	2	4	no	—
53	#88155	—	2	4	no	—
54	C₂×#6094	C₂×C₄	1	1	yes	characteristic
55	C₂×#6094	C₂×C₄	1	1	yes	characteristic
56	C₂×#6094	C₂×C₄	1	1	yes	characteristic
57	C₂×#6094	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
62	C₄×D₅₂	C₈	1	2	yes	—
63	#88126	C₂×C₄	1	1	yes	characteristic
64	#88126	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
66	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₂₆	C₈	1	2	yes	—
68	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
69	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
70	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	#88055	C₂×C₄	1	1	yes	characteristic
75	#88055	C₂×C₄	1	1	yes	characteristic
76	#88055	C₂×C₄	1	1	yes	characteristic
77	#88055	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
81	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
82	C₂×#6090	C₂³	1	2	yes	—
83	C₂×#6090	C₂³	1	2	yes	—
84	#88043	C₂×C₄	1	1	yes	characteristic
85	#88043	C₂×C₄	1	1	yes	characteristic
86	C₄×#1055	C₂×C₄	1	1	yes	characteristic
87	C₄×#1055	C₂×C₄	1	1	yes	characteristic
88	#88038	—	2	4	no	—
89	C₁₆×D₁₃	—	2	4	no	—
90	#33421	—	13	1	no	maximal, Sylow
91	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₃	C₂×C₈	1	1	yes	characteristic
94	C₂×C₂²⋊D₁₃	C₂×C₈	1	1	yes	characteristic
95	#6127	—	2	2	no	—
96	#6127	—	2	2	no	—
97	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
99	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
100	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×C₈	1	2	yes	—
110	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
113	#6103	C₂×C₈	1	2	yes	—
114	#6103	C₂×C₈	1	2	yes	—
115	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
116	#6101	C₂×C₈	1	2	yes	—
117	#6101	C₂×C₈	1	2	yes	—
118	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
119	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
120	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
122	#6099	—	2	2	no	—
123	#6099	—	2	2	no	—
124	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
125	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
126	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
127	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
128	#6094	—	2	2	no	—
129	#6094	—	2	2	no	—
130	#6094	—	2	2	no	—
131	#6094	—	2	2	no	—
132	C₈×D₁₃	—	2	2	no	—
133	C₈×D₁₃	—	2	2	no	—
134	C₈×D₁₃	M₁₆	1	4	yes	—
135	C₈×D₁₃	#41	1	4	yes	—
136	C₁₃×C₁₆	—	2	2	no	—
137	C₁₃×C₁₆	—	2	2	no	—
138	#6090	—	2	2	no	—
139	#6090	—	2	2	no	—
140	C₄×#131	—	13	1	no	—
141	#2164	—	13	2	no	—
142	#2164	—	13	2	no	—
143	#2163	—	13	1	no	—
144	#2163	—	13	1	no	—
145	#2163	—	13	1	no	—
146	#2163	D₁₃	1	1	yes	characteristic, complemented
147	#2151	—	13	2	no	—
148	#2113	—	13	2	no	—
149	C₄×M₃₂	—	13	1	no	—
150	C₂×#346	—	13	1	no	—
151	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
152	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
153	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	—	2	1	no	—
155	C₂×C₄×C₁₃	—	2	1	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
158	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
159	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
161	C₂²⋊D₁₃	—	2	2	no	—
162	C₂²⋊D₁₃	—	2	2	no	—
163	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
164	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
165	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
166	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
167	D₅₂	—	2	2	no	—
168	C₄×D₁₃	—	2	1	no	—
169	C₄×D₁₃	—	2	1	no	—
170	C₄×D₁₃	—	4	2	no	—
171	C₄×D₁₃	—	4	2	no	—
172	C₄×D₁₃	#117	1	2	yes	—
173	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
174	Dic₂₆	—	2	2	no	—
175	C₈×C₁₃	—	2	1	no	—
176	C₈×C₁₃	—	2	1	no	—
177	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
178	C₈×C₁₃	C₂×#41	1	2	yes	—
179	#1055	—	2	1	no	—
180	#1055	—	2	1	no	—
181	#1055	#117	1	2	yes	—
182	#1055	C₂×M₁₆	1	2	yes	—
183	C₂×#131	—	13	1	no	—
184	C₂×#131	—	13	1	no	—
185	C₄×#41	—	13	1	no	—
186	C₂×M₃₂	—	13	2	no	—
187	C₂²×C₁₆	—	13	2	no	—
188	#405	—	13	1	no	—
189	#405	—	13	1	no	—
190	#405	—	13	1	no	—
191	#405	—	13	1	no	—
192	#405	—	13	1	no	—
193	#405	—	13	1	no	—
194	C₄×M₁₆	—	13	1	no	—
195	C₄×M₁₆	—	13	1	no	—
196	C₄×M₁₆	—	13	1	no	—
197	C₂×C₄×C₈	—	13	1	no	—
198	C₂×C₄×C₈	—	13	1	no	—
199	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
200	C₄⋊C₁₆	—	13	2	no	—
201	C₄⋊C₁₆	—	13	2	no	—
202	C₄⋊C₁₆	—	13	2	no	—
203	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
204	C₂²⋊C₁₆	—	13	2	no	—
205	C₂²⋊C₁₆	—	13	2	no	—
206	C₂²⋊C₁₆	—	13	2	no	—
207	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
208	#346	—	13	2	no	—
209	#346	—	13	1	no	—
210	#346	—	13	1	no	—
211	#346	C₂×D₁₃	1	1	yes	characteristic
212	#346	C₂×D₁₃	1	1	yes	characteristic
213	C₄×C₁₆	—	13	2	no	—
214	C₂²×C₁₃	—	2	2	no	—
215	C₂²×C₁₃	—	2	1	no	—
216	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
217	C₂×D₁₃	—	2	2	no	—
218	C₂×D₁₃	—	2	2	no	—
219	C₂×D₁₃	—	2	1	no	—
220	C₂×D₁₃	#431	1	2	yes	—
221	C₄×C₁₃	—	2	2	no	—
222	C₄×C₁₃	—	2	2	no	—
223	C₄×C₁₃	—	2	1	no	—
224	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
225	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
226	Dic₁₃	—	2	2	no	—
227	Dic₁₃	—	2	2	no	—
228	Dic₁₃	—	2	1	no	—
229	Dic₁₃	#431	1	2	yes	—
230	C₂×#41	—	13	1	no	—
231	#131	—	26	4	no	—
232	#131	—	26	4	no	—
233	C₂×M₁₆	—	13	1	no	—
234	C₂×M₁₆	—	13	1	no	—
235	C₂×M₁₆	—	13	1	no	—
236	C₂×M₁₆	—	13	1	no	—
237	C₂×M₁₆	—	13	1	no	—
238	C₂×M₁₆	—	13	1	no	—
239	C₂²×C₈	—	13	1	no	—
240	C₂²×C₈	—	13	1	no	—
241	C₂²×C₈	—	13	1	no	—
242	C₂²×C₈	—	13	1	no	—
243	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
244	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
245	C₄×Q₈	—	13	2	no	—
246	C₄×D₄	—	13	2	no	—
247	C₄×D₄	—	13	2	no	—
248	C₄×D₄	—	13	2	no	—
249	#117	—	13	1	no	—
250	#117	—	13	1	no	—
251	#117	—	13	1	no	—
252	C₂×C₄²	—	13	1	no	—
253	C₂×C₄²	—	13	1	no	—
254	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
255	M₃₂	—	26	4	no	—
256	C₂×C₁₆	—	13	2	no	—
257	C₂×C₁₆	—	13	2	no	—
258	C₂×C₁₆	—	26	4	no	—
259	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
260	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
261	#97	—	13	1	no	—
262	#97	—	13	1	no	—
263	#97	—	13	1	no	—
264	#97	—	13	1	no	—
265	#97	—	13	1	no	—
266	#97	—	13	1	no	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	13	1	no	—
269	C₄×C₈	—	13	1	no	—
270	C₄×C₈	—	13	1	no	—
271	C₄×C₈	—	13	1	no	—
272	C₄×C₈	—	13	1	no	—
273	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
274	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
277	C₂×C₁₃	—	4	1	no	—
278	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
279	C₂×C₁₃	#3407	1	2	yes	—
280	D₁₃	—	2	2	no	—
281	D₁₃	—	4	1	no	—
282	#41	—	26	2	no	—
283	#41	—	26	2	no	—
284	C₂×Q₈	—	13	1	no	—
285	C₂×D₄	—	13	1	no	—
286	C₂×D₄	—	13	1	no	—
287	C₂×D₄	—	13	1	no	—
288	C₂²×C₄	—	13	1	no	—
289	C₂²×C₄	—	13	1	no	—
290	C₂²×C₄	—	13	2	no	—
291	C₂²×C₄	—	13	2	no	—
292	C₂²×C₄	C₈×D₁₃	1	2	yes	—
293	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
294	M₁₆	—	26	2	no	—
295	M₁₆	—	26	2	no	—
296	M₁₆	—	26	2	no	—
297	M₁₆	—	26	2	no	—
298	M₁₆	—	26	2	no	—
299	M₁₆	—	26	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	13	1	no	—
303	C₂×C₈	—	13	1	no	—
304	C₂×C₈	—	13	4	no	—
305	C₂×C₈	—	13	4	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	13	1	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	—	26	2	no	—
310	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
311	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
313	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
314	C₄⋊C₄	—	13	2	no	—
315	C₄⋊C₄	—	13	2	no	—
316	C₄⋊C₄	—	13	2	no	—
317	C₂²⋊C₄	—	13	2	no	—
318	C₂²⋊C₄	—	13	2	no	—
319	C₂²⋊C₄	—	13	2	no	—
320	C₄²	—	13	2	no	—
321	C₄²	—	13	2	no	—
322	C₄²	—	13	1	no	—
323	C₄²	—	13	1	no	—
324	C₄²	C₈×D₁₃	1	2	yes	—
325	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
326	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
327	C₁₆	—	2	2	no	—
328	C₁₆	—	2	2	no	—
329	C₁₆	—	26	2	no	—
330	C₁₆	—	26	2	no	—
331	C₁₃	#33421	1	1	yes	characteristic, complemented, Sylow
332	C₂³	—	13	1	no	—
333	C₂³	—	13	1	no	—
334	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
335	Q₈	—	26	2	no	—
336	D₄	—	26	2	no	—
337	D₄	—	26	2	no	—
338	D₄	—	26	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	13	1	no	—
343	C₂×C₄	—	13	2	no	—
344	C₂×C₄	—	13	2	no	—
345	C₂×C₄	—	13	2	no	—
346	C₂×C₄	—	13	2	no	—
347	C₂×C₄	—	13	1	no	—
348	C₂×C₄	—	26	1	no	—
349	C₂×C₄	—	26	1	no	—
350	C₂×C₄	—	52	2	no	—
351	C₂×C₄	—	52	2	no	—
352	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
353	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
354	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
355	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	13	2	no	—
359	C₈	—	13	2	no	—
360	C₈	—	26	1	no	—
361	C₈	—	26	1	no	—
362	C₈	M₁₆×D₁₃	1	2	yes	—
363	C₈	#41×D₁₃	1	2	yes	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	13	2	no	—
367	C₂²	—	26	2	no	—
368	C₂²	—	26	2	no	—
369	C₂²	—	26	1	no	—
370	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
371	C₄	—	2	2	no	—
372	C₄	—	2	2	no	—
373	C₄	—	2	1	no	—
374	C₄	—	13	2	no	—
375	C₄	—	26	2	no	—
376	C₄	—	26	2	no	—
377	C₄	—	26	1	no	—
378	C₄	D₁₃×#117	1	1	yes	characteristic
379	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
380	C₂	—	4	1	no	—
381	C₂	—	26	2	no	—
382	C₂	—	52	1	no	—
383	C₂	D₁₃×#431	1	1	yes	characteristic
384	C₂	#13127938	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2163	C₂
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2163	D₁₃
hypocenter	C₁₃	#33421
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #13116642	C₂	semidirect
3. #13116640	C₂	semidirect
4. #13116639	C₂	semidirect
5. #13116637	C₂	semidirect
6. #13116637	C₂	semidirect
7. #13116593	C₂	semidirect
9. #13116429	C₂	semidirect
10. D₁₃×#346	C₂	semidirect
11. C₁₃×#2163	C₂	semidirect
146. #2163	D₁₃	semidirect
331. C₁₃	#33421	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88155	2
#13116642	2
#13116640	2
#13116639	2
#13116637	2
#13116593	2
#13116448	2
#13116429	2
D₁₃×#346	2
C₁₃×#2163	2
#33421	13