Subgroups — #16016444

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#6131	C₂	1	1	yes	maximal, characteristic
2	#13116745	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116669	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116667	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116663	C₂	1	2	yes	maximal, complemented
6	#13116660	C₂	1	2	yes	maximal, complemented
7	#13116634	C₂	1	2	yes	maximal, complemented
8	#13116634	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116429	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#6131	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	Q₈×#1055	C₄	1	1	yes	characteristic
15	C₂×#88038	C₂²	1	2	yes	—
16	C₂×#88038	C₂²	1	2	yes	—
17	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
19	#11718495	C₄	1	2	yes	—
20	#11718495	C₄	1	1	yes	characteristic
21	#11718494	C₄	1	2	yes	—
22	#11718494	C₄	1	1	yes	characteristic
23	C₈×D₅₂	C₂²	1	2	yes	—
24	C₈×D₅₂	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
26	C₄×C₈×D₁₃	C₂²	1	2	yes	—
27	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₁₃×#346	C₂²	1	2	yes	—
30	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
31	#11718099	C₂²	1	2	yes	—
32	#11718099	C₂²	1	2	yes	—
33	#11718099	C₂²	1	2	yes	—
34	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₃	C₂²	1	2	yes	—
37	C₄⋊#6090	C₂²	1	2	yes	—
38	C₄⋊#6090	C₂²	1	1	yes	characteristic
39	#11718053	C₂²	1	2	yes	—
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×#6131	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
44	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂³	1	2	yes	—
48	#88147	C₈	1	2	yes	—
49	#88147	C₈	1	1	yes	characteristic
50	#88146	C₈	1	2	yes	—
51	#88146	C₈	1	1	yes	characteristic
52	C₄×D₅₂	C₂×C₄	1	2	yes	—
53	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
54	C₄²×D₁₃	C₂×C₄	1	2	yes	—
55	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	2	yes	—
63	#88060	C₂×C₄	1	2	yes	—
64	#88060	C₂×C₄	1	2	yes	—
65	#88060	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
68	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
69	C₂×#6090	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	2	yes	—
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	C₄⋊#1055	C₂×C₄	1	2	yes	—
73	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
74	C₄×#1055	C₂×C₄	1	1	yes	characteristic
75	C₄×#1055	C₂×C₄	1	2	yes	—
76	#88038	—	2	4	no	—
77	#88038	—	2	4	no	—
78	C₁₆×D₁₃	—	2	2	no	—
79	C₁₆×D₁₃	—	2	2	no	—
80	#33479	—	13	1	no	maximal, Sylow
81	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
82	#6131	—	2	4	no	—
83	C₂×D₅₂	C₂×C₈	1	2	yes	—
84	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
87	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
95	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	#6103	C₂×C₈	1	2	yes	—
99	#6103	C₂×C₈	1	2	yes	—
100	#6103	C₂×C₈	1	2	yes	—
101	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
102	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
104	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
106	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
109	C₂×#1055	C₂²×C₄	1	2	yes	—
110	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
111	C₈×D₁₃	—	2	1	no	—
112	C₈×D₁₃	—	2	1	no	—
113	C₈×D₁₃	—	2	2	no	—
114	C₈×D₁₃	—	2	2	no	—
115	C₈×D₁₃	—	2	2	no	—
116	C₈×D₁₃	—	2	4	no	—
117	C₁₃×C₁₆	—	2	1	no	—
118	C₁₃×C₁₆	—	2	2	no	—
119	C₁₃×C₁₆	#41	1	2	yes	complemented
120	#6090	—	2	1	no	—
121	#6090	—	2	2	no	—
122	#6090	#41	1	2	yes	complemented
123	C₈×#41	—	13	1	no	—
124	#2183	—	13	1	no	—
125	#2183	D₁₃	1	1	yes	characteristic, complemented
126	#2169	—	13	1	no	—
127	#2169	—	13	1	no	—
128	#2168	—	13	2	no	—
129	#2168	—	13	2	no	—
130	#2162	—	13	2	no	—
131	#2162	—	13	1	no	—
132	C₄×M₃₂	—	13	2	no	—
133	C₂×C₄×C₁₆	—	13	1	no	—
134	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
135	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
136	Q₈×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
143	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
146	D₅₂	—	2	4	no	—
147	D₅₂	—	2	2	no	—
148	C₄×D₁₃	—	2	1	no	—
149	C₄×D₁₃	—	2	2	no	—
150	C₄×D₁₃	—	2	1	no	—
151	C₄×D₁₃	—	2	2	no	—
152	C₄×D₁₃	—	2	4	no	—
153	C₄×D₁₃	—	2	2	no	—
154	C₄×D₁₃	—	2	4	no	—
155	C₄×D₁₃	M₃₂	1	4	yes	complemented
156	C₈×C₁₃	—	2	1	no	—
157	C₈×C₁₃	—	2	2	no	—
158	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
160	#1055	—	2	1	no	—
161	#1055	—	2	2	no	—
162	#1055	#131	1	2	yes	—
163	C₄×#41	—	13	1	no	—
164	C₂×M₃₂	—	13	2	no	—
165	C₂×M₃₂	—	13	2	no	—
166	C₂²×C₁₆	—	13	1	no	—
167	C₂²×C₁₆	—	13	1	no	—
168	C₈×Q₈	—	13	1	no	—
169	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
170	C₈×D₄	—	13	1	no	—
171	C₈×D₄	—	13	2	no	—
172	C₈×D₄	—	13	2	no	—
173	C₈×D₄	—	13	1	no	—
174	#431	—	13	2	no	—
175	#431	—	13	1	no	—
176	C₂×C₄×C₈	—	13	1	no	—
177	C₂×C₄×C₈	—	13	2	no	—
178	C₄⋊C₁₆	—	13	2	no	—
179	C₄⋊C₁₆	—	13	1	no	—
180	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
181	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
182	C₂²⋊C₁₆	—	13	2	no	—
183	C₂²⋊C₁₆	—	13	2	no	—
184	C₂²⋊C₁₆	—	13	2	no	—
185	#346	—	13	2	no	—
186	#346	C₂×D₁₃	1	2	yes	—
187	C₄×C₁₆	—	13	1	no	—
188	C₄×C₁₆	—	13	1	no	—
189	C₄×C₁₆	—	13	2	no	—
190	C₄×C₁₆	—	13	1	no	—
191	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
192	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
193	C₂×D₁₃	—	2	1	no	—
194	C₂×D₁₃	—	2	1	no	—
195	C₂×D₁₃	—	2	2	no	—
196	C₂×D₁₃	—	2	2	no	—
197	C₂×D₁₃	—	2	2	no	—
198	C₂×D₁₃	C₂×M₃₂	1	2	yes	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	2	2	no	—
202	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
203	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
204	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
205	Dic₁₃	—	2	1	no	—
206	Dic₁₃	—	2	2	no	—
207	Dic₁₃	C₂×M₃₂	1	2	yes	—
208	Dic₁₃	#504	1	2	yes	—
209	C₂×#41	—	13	1	no	—
210	C₂²×C₈	—	13	1	no	—
211	C₂²×C₈	—	13	2	no	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	—	13	2	no	—
214	C₄×Q₈	—	13	1	no	—
215	C₄×Q₈	C₄×D₁₃	1	1	yes	characteristic
216	C₄×D₄	—	13	2	no	—
217	C₄×D₄	—	13	1	no	—
218	C₄×D₄	—	13	2	no	—
219	C₄×D₄	—	13	1	no	—
220	#117	—	13	1	no	—
221	#117	—	13	2	no	—
222	C₂×C₄²	—	13	1	no	—
223	C₂×C₄²	—	13	2	no	—
224	M₃₂	—	26	4	no	—
225	M₃₂	—	26	4	no	—
226	C₂×C₁₆	—	13	1	no	—
227	C₂×C₁₆	—	13	1	no	—
228	C₂×C₁₆	—	13	2	no	—
229	C₂×C₁₆	—	26	2	no	—
230	C₂×C₁₆	—	26	2	no	—
231	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
233	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
234	C₄⋊C₈	—	13	2	no	—
235	C₄⋊C₈	—	13	1	no	—
236	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
237	C₄⋊C₈	C₄×D₁₃	1	1	yes	characteristic
238	C₂²⋊C₈	—	13	2	no	—
239	C₂²⋊C₈	—	13	2	no	—
240	C₂²⋊C₈	—	13	2	no	—
241	C₄×C₈	—	13	1	no	—
242	C₄×C₈	—	13	2	no	—
243	C₄×C₈	—	13	1	no	—
244	C₄×C₈	—	13	1	no	—
245	C₄×C₈	—	13	2	no	—
246	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₁₃	1	2	yes	—
248	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
249	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
250	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
251	D₁₃	—	2	2	no	—
252	D₁₃	—	4	2	no	—
253	#41	—	26	4	no	—
254	C₂×Q₈	C₈×D₁₃	1	1	yes	characteristic
255	C₂×D₄	—	13	2	no	—
256	C₂×D₄	—	13	1	no	—
257	C₂²×C₄	—	13	1	no	—
258	C₂²×C₄	—	13	2	no	—
259	C₂²×C₄	—	13	1	no	—
260	C₂²×C₄	—	13	2	no	—
261	C₂²×C₄	—	13	1	no	—
262	C₂²×C₄	—	13	2	no	—
263	C₂×C₈	—	13	1	no	—
264	C₂×C₈	—	13	1	no	—
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	26	2	no	—
267	C₂×C₈	—	26	4	no	—
268	C₂×C₈	—	26	1	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	—	26	1	no	—
271	C₂×C₈	—	26	2	no	—
272	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
274	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
275	C₄⋊C₄	—	13	2	no	—
276	C₄⋊C₄	—	13	1	no	—
277	C₄⋊C₄	C₈×D₁₃	1	2	yes	—
278	C₄⋊C₄	C₈×D₁₃	1	1	yes	characteristic
279	C₂²⋊C₄	—	13	2	no	—
280	C₂²⋊C₄	—	13	2	no	—
281	C₂²⋊C₄	—	13	2	no	—
282	C₄²	—	13	1	no	—
283	C₄²	—	13	2	no	—
284	C₄²	—	13	2	no	—
285	C₄²	—	13	1	no	—
286	C₄²	—	13	1	no	—
287	C₄²	C₂×C₄×D₁₃	1	2	yes	—
288	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
289	C₁₆	—	2	1	no	—
290	C₁₆	—	2	2	no	—
291	C₁₆	—	13	2	no	—
292	C₁₆	—	26	1	no	—
293	C₁₆	—	26	2	no	—
294	C₁₆	#6131	1	2	yes	complemented
295	C₁₃	#33479	1	1	yes	characteristic, complemented, Sylow
296	C₂³	—	13	1	no	—
297	C₂³	—	13	2	no	—
298	Q₈	—	2	2	no	—
299	D₄	—	26	4	no	—
300	D₄	—	26	2	no	—
301	C₂×C₄	—	13	1	no	—
302	C₂×C₄	—	13	1	no	—
303	C₂×C₄	—	13	4	no	—
304	C₂×C₄	—	13	2	no	—
305	C₂×C₄	—	13	1	no	—
306	C₂×C₄	—	26	2	no	—
307	C₂×C₄	—	26	4	no	—
308	C₂×C₄	—	26	4	no	—
309	C₂×C₄	—	26	1	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	26	1	no	—
312	C₂×C₄	—	26	2	no	—
313	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
314	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
316	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	13	2	no	—
321	C₈	—	26	1	no	—
322	C₈	—	26	2	no	—
323	C₈	C₂×#6131	1	1	yes	characteristic
324	C₈	#41×D₁₃	1	1	yes	characteristic
325	C₂²	—	13	2	no	—
326	C₂²	—	26	1	no	—
327	C₂²	—	26	1	no	—
328	C₂²	—	26	2	no	—
329	C₂²	—	26	2	no	—
330	C₂²	—	26	2	no	—
331	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	2	no	—
335	C₄	—	13	2	no	—
336	C₄	—	13	2	no	—
337	C₄	—	26	1	no	—
338	C₄	—	26	2	no	—
339	C₄	D₁₃×M₃₂	1	2	yes	—
340	C₄	C₄×#6131	1	1	yes	characteristic
341	C₄	D₁₃×#131	1	1	yes	characteristic
342	C₂	—	26	2	no	—
343	C₂	—	52	2	no	—
344	C₂	C₈×#6131	1	1	yes	characteristic
345	C₂	C₂×D₁₃×M₃₂	1	1	yes	characteristic
346	C₂	D₁₃×#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2183	C₂
prime radical	C₂×#6131	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2183	D₁₃
hypocenter	C₁₃	#33479
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #13116745	C₂	semidirect
3. #13116669	C₂	semidirect
4. #13116667	C₂	semidirect
5. #13116663	C₂	semidirect
6. #13116660	C₂	semidirect
7. #13116634	C₂	semidirect
8. #13116634	C₂	semidirect
9. #13116429	C₂	semidirect
10. C₄×C₁₆×D₁₃	C₂	semidirect
11. C₁₃×#2183	C₂	semidirect
119. C₁₃×C₁₆	#41	semidirect
122. #6090	#41	semidirect
125. #2183	D₁₃	semidirect
155. C₄×D₁₃	M₃₂	semidirect
294. C₁₆	#6131	semidirect
295. C₁₃	#33479	semidirect

Maximal subgroups

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

subgroup	index
C₈×#6131	2
#13116745	2
#13116669	2
#13116667	2
#13116663	2
#13116660	2
#13116634	2
#13116429	2
C₄×C₁₆×D₁₃	2
C₁₃×#2183	2
#33479	13