Subgroups — #16016030

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	#13116640	C₂	1	2	yes	maximal, complemented
3	#13116640	C₂	1	2	yes	maximal, complemented
4	#13116636	C₂	1	2	yes	maximal, complemented
5	#13116632	C₂	1	2	yes	maximal, complemented
6	#13116591	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718093	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116450	C₂	1	1	yes	maximal, characteristic
9	#13116435	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116432	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
13	#41×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×#88202	C₄	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	#11718783	C₄	1	1	yes	characteristic
18	#11718783	C₄	1	2	yes	—
19	M₁₆×Dic₁₃	C₄	1	2	yes	—
20	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
21	#11718762	C₂²	1	1	yes	characteristic
22	#11718762	C₂²	1	2	yes	—
23	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
25	C₂×#88052	C₂²	1	1	yes	characteristic
26	C₂²×#6090	C₂²	1	1	yes	characteristic
27	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₂²⋊#6090	C₂²	1	2	yes	—
30	C₂²⋊#6090	C₂²	1	2	yes	—
31	#11718094	C₂²	1	2	yes	—
32	#11718094	C₂²	1	2	yes	—
33	#11718093	C₂²	1	1	yes	characteristic
34	#11718093	C₂²	1	1	yes	characteristic
35	#11718093	C₂²	1	1	yes	characteristic
36	#11718093	C₂²	1	1	yes	characteristic
37	#11718093	C₂²	1	2	yes	—
38	#11718093	C₂²	1	2	yes	—
39	C₁₆×Dic₁₃	C₂²	1	2	yes	—
40	C₁₆×Dic₁₃	C₂²	1	2	yes	—
41	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
42	C₁₃×#131	—	2	2	no	—
43	C₁₃×#131	—	2	2	no	—
44	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	#88202	—	2	4	no	—
49	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
50	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
51	D₄×Dic₁₃	C₈	1	2	yes	—
52	D₄×Dic₁₃	C₈	1	1	yes	characteristic
53	D₄×Dic₁₃	C₈	1	2	yes	—
54	D₄×Dic₁₃	C₈	1	1	yes	characteristic
55	#88180	C₂×C₄	1	2	yes	—
56	#88180	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
59	C₂×#6099	C₂×C₄	1	2	yes	—
60	C₂×#6099	C₂×C₄	1	1	yes	characteristic
61	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
62	C₂²×#1055	C₂×C₄	1	2	yes	—
63	C₁₃×M₃₂	—	2	2	no	—
64	C₂×C₁₃×C₁₆	—	2	2	no	—
65	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
68	#88055	C₂×C₄	1	2	yes	—
69	#88055	C₂×C₄	1	2	yes	—
70	#88055	C₂×C₄	1	2	yes	—
71	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
73	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
74	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
76	C₈×Dic₁₃	C₂³	1	2	yes	—
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	#33423	—	13	1	no	maximal, Sylow
84	C₁₃×#41	—	2	2	no	—
85	C₁₃×#41	—	2	2	no	—
86	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
87	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
88	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
92	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
93	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
94	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
95	C₁₃×M₁₆	—	2	2	no	—
96	C₁₃×M₁₆	—	2	4	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	#41	1	2	yes	—
102	C₂×C₈×C₁₃	#41	1	2	yes	—
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
105	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
106	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
107	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
108	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
109	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
110	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
112	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
113	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
114	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
115	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
117	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
118	#6099	—	2	4	no	—
119	#6099	M₁₆	1	4	yes	—
120	C₂×#1055	—	2	2	no	—
121	C₂×#1055	—	2	4	no	—
122	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
123	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
124	C₂×#1055	C₂²×C₄	1	2	yes	—
125	C₁₃×C₁₆	—	2	1	no	—
126	C₁₃×C₁₆	—	2	1	no	—
127	C₁₃×C₁₆	—	2	2	no	—
128	#6090	—	2	1	no	—
129	#6090	—	2	1	no	—
130	#6090	—	2	2	no	—
131	C₄×#131	—	13	1	no	—
132	#2164	—	13	2	no	—
133	#2164	—	13	2	no	—
134	#2163	—	13	2	no	—
135	#2162	—	13	2	no	—
136	#2150	—	13	1	no	—
137	C₂×C₄⋊C₁₆	—	13	1	no	—
138	#2113	—	13	1	no	—
139	#2113	D₁₃	1	1	yes	characteristic
140	#2108	—	13	1	no	—
141	#2108	—	13	1	no	—
142	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
143	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
144	Q₈×C₁₃	—	2	2	no	—
145	D₄×C₁₃	—	2	4	no	—
146	D₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	#117	1	2	yes	—
150	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
151	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
153	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×Dic₁₃	—	2	2	no	—
155	C₂×Dic₁₃	—	2	4	no	—
156	C₂×Dic₁₃	—	4	1	no	—
157	C₂×Dic₁₃	—	4	2	no	—
158	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
159	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
160	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
161	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
162	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
163	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
164	C₈×C₁₃	—	2	2	no	—
165	C₈×C₁₃	#117	1	2	yes	—
166	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
167	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
168	#1055	—	2	1	no	—
169	#1055	—	2	1	no	—
170	#1055	C₂×M₁₆	1	4	yes	—
171	C₂×#131	—	13	1	no	—
172	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
173	C₄×#41	—	13	1	no	—
174	C₂×M₃₂	—	13	1	no	—
175	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
176	C₂²×C₁₆	—	13	1	no	—
177	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
178	#405	—	13	1	no	—
179	#405	—	13	2	no	—
180	#405	—	13	1	no	—
181	#405	—	13	2	no	—
182	C₄×M₁₆	—	13	1	no	—
183	C₄×M₁₆	—	13	2	no	—
184	C₂×C₄×C₈	—	13	1	no	—
185	C₂×C₄×C₈	—	13	2	no	—
186	C₄⋊C₁₆	—	13	1	no	—
187	C₄⋊C₁₆	—	13	1	no	—
188	C₄⋊C₁₆	—	13	1	no	—
189	C₄⋊C₁₆	—	13	1	no	—
190	C₄⋊C₁₆	—	13	2	no	—
191	C₄⋊C₁₆	—	13	2	no	—
192	C₂²⋊C₁₆	—	13	2	no	—
193	C₂²⋊C₁₆	—	13	2	no	—
194	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
195	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
196	#346	—	13	2	no	—
197	#346	—	13	2	no	—
198	C₄×C₁₆	—	13	2	no	—
199	C₄×C₁₆	—	13	2	no	—
200	C₂²×C₁₃	—	2	1	no	—
201	C₂²×C₁₃	—	2	1	no	—
202	C₂²×C₁₃	—	2	2	no	—
203	C₂²×C₁₃	—	2	2	no	—
204	C₂²×C₁₃	—	2	2	no	—
205	C₂²×C₁₃	#431	1	2	yes	—
206	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
207	C₄×C₁₃	—	2	2	no	—
208	C₄×C₁₃	#431	1	2	yes	—
209	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
210	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
211	Dic₁₃	—	2	1	no	—
212	Dic₁₃	—	2	2	no	—
213	Dic₁₃	—	2	1	no	—
214	Dic₁₃	—	2	1	no	—
215	Dic₁₃	—	2	2	no	—
216	Dic₁₃	#504	1	2	yes	—
217	C₂×#41	C₄×D₁₃	1	1	yes	characteristic
218	#131	—	2	2	no	—
219	#131	—	2	2	no	—
220	#131	—	26	4	no	—
221	C₂×M₁₆	—	13	1	no	—
222	C₂×M₁₆	—	13	2	no	—
223	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
224	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
225	C₂²×C₈	—	13	2	no	—
226	C₂²×C₈	—	13	1	no	—
227	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
228	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
229	C₄×Q₈	—	13	1	no	—
230	C₄×Q₈	—	13	1	no	—
231	C₄×D₄	—	13	2	no	—
232	C₄×D₄	—	13	1	no	—
233	C₄×D₄	—	13	2	no	—
234	C₄×D₄	—	13	1	no	—
235	#117	—	13	2	no	—
236	#117	—	13	1	no	—
237	C₂×C₄²	—	13	2	no	—
238	C₂×C₄²	—	13	1	no	—
239	M₃₂	—	2	2	no	—
240	M₃₂	—	26	2	no	—
241	C₂×C₁₆	—	2	2	no	—
242	C₂×C₁₆	—	13	1	no	—
243	C₂×C₁₆	—	13	2	no	—
244	C₂×C₁₆	—	13	1	no	—
245	C₂×C₁₆	—	26	2	no	—
246	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
247	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
248	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
249	#97	—	13	2	no	—
250	#97	—	13	2	no	—
251	#97	—	13	2	no	—
252	C₄×C₈	—	13	2	no	—
253	C₄×C₈	—	13	1	no	—
254	C₄×C₈	—	13	2	no	—
255	C₄×C₈	—	13	1	no	—
256	C₄×C₈	—	13	2	no	—
257	C₄×C₈	—	13	1	no	—
258	C₄×C₈	—	13	1	no	—
259	C₂×C₁₃	—	2	2	no	—
260	C₂×C₁₃	—	4	2	no	—
261	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
262	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
263	C₂×C₁₃	#3407	1	1	yes	characteristic
264	#41	—	2	2	no	—
265	#41	—	2	2	no	—
266	C₂×Q₈	C₈×D₁₃	1	1	yes	characteristic
267	C₂×D₄	C₈×D₁₃	1	2	yes	—
268	C₂×D₄	C₈×D₁₃	1	1	yes	characteristic
269	C₂²×C₄	—	13	2	no	—
270	C₂²×C₄	—	13	1	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	2	no	—
276	M₁₆	—	2	4	no	—
277	M₁₆	—	13	4	no	—
278	M₁₆	—	26	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	13	2	no	—
282	C₂×C₈	—	13	1	no	—
283	C₂×C₈	—	13	1	no	—
284	C₂×C₈	—	26	4	no	—
285	C₂×C₈	—	26	2	no	—
286	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
287	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
288	C₂×C₈	#6129	1	2	yes	—
289	C₂×C₈	#6129	1	2	yes	—
290	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
291	C₄⋊C₄	—	13	2	no	—
292	C₄⋊C₄	—	13	2	no	—
293	C₄⋊C₄	—	13	1	no	—
294	C₄⋊C₄	—	13	1	no	—
295	C₂²⋊C₄	—	13	2	no	—
296	C₂²⋊C₄	—	13	2	no	—
297	C₂²⋊C₄	—	13	2	no	—
298	C₄²	—	13	2	no	—
299	C₄²	—	13	1	no	—
300	C₄²	—	13	2	no	—
301	C₄²	—	13	1	no	—
302	C₄²	—	13	1	no	—
303	C₄²	—	13	2	no	—
304	C₄²	—	13	1	no	—
305	C₁₆	—	2	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	2	no	—
308	C₁₆	—	26	1	no	—
309	C₁₆	—	26	1	no	—
310	C₁₆	—	26	2	no	—
311	C₁₃	#33423	1	1	yes	characteristic, complemented, Sylow
312	C₂³	C₂×C₈×D₁₃	1	2	yes	—
313	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	D₄	—	2	4	no	—
316	D₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	13	1	no	—
320	C₂×C₄	—	13	1	no	—
321	C₂×C₄	—	13	2	no	—
322	C₂×C₄	—	13	1	no	—
323	C₂×C₄	—	13	2	no	—
324	C₂×C₄	—	13	1	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	26	4	no	—
327	C₂×C₄	—	52	1	no	—
328	C₂×C₄	—	52	2	no	—
329	C₂×C₄	#88131	1	2	yes	—
330	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
331	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
332	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	13	4	no	—
336	C₈	—	26	1	no	—
337	C₈	—	26	1	no	—
338	C₈	#88131	1	2	yes	—
339	C₈	C₂×#6129	1	1	yes	characteristic
340	C₈	C₂×#6129	1	1	yes	characteristic
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	#11718379	1	2	yes	—
347	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	13	2	no	—
350	C₄	—	26	1	no	—
351	C₄	—	26	2	no	—
352	C₄	—	26	1	no	—
353	C₄	—	26	1	no	—
354	C₄	—	26	2	no	—
355	C₄	#11718379	1	2	yes	—
356	C₄	C₂×#88131	1	1	yes	characteristic
357	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	4	2	no	—
360	C₂	C₂×#11718379	1	1	yes	characteristic
361	C₂	D₁₃×#504	1	1	yes	characteristic
362	C₂	#13127937	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2113	C₂
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2113	D₁₃
hypocenter	C₁₃	#33423
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
2. #13116640	C₂	semidirect
3. #13116640	C₂	semidirect
4. #13116636	C₂	semidirect
5. #13116632	C₂	semidirect
6. #13116591	C₂	semidirect
7. C₂×#11718093	C₂	semidirect
9. #13116435	C₂	semidirect
10. #13116432	C₂	semidirect
311. C₁₃	#33423	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₁₃	2
#13116640	2
#13116636	2
#13116632	2
#13116591	2
C₂×#11718093	2
#13116450	2
#13116435	2
#13116432	2
C₁₃×#2113	2
#33423	13