Subgroups — #15753960

Order 3328.

444 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123811	C₂	1	1	yes	maximal, characteristic
2	#13114522	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113425	C₂	1	1	yes	maximal, characteristic
4	#13113323	C₂	1	1	yes	maximal, characteristic
5	#13112736	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112730	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719184	C₂²	1	1	yes	characteristic
9	C₁₃×#431	C₂²	1	1	yes	characteristic
10	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718772	—	2	1	no	—
13	C₈×C₂²⋊D₁₃	—	2	1	no	—
14	C₂×#88055	—	2	1	no	—
15	C₂×C₈×Dic₁₃	—	2	1	no	—
16	#11718593	C₂²	1	1	yes	characteristic
17	#11718587	C₂²	1	1	yes	characteristic
18	C₂×#88043	C₂²	1	1	yes	characteristic
19	#11718351	—	2	1	no	—
20	#11718350	—	2	1	no	—
21	#11718349	C₄	1	1	yes	characteristic, complemented
22	C₈×D₅₂	C₄	1	1	yes	characteristic, complemented
23	#11718340	C₄	1	1	yes	characteristic, complemented
24	C₈×Dic₂₆	C₄	1	1	yes	characteristic, complemented
25	C₁₃×#322	—	2	1	no	—
26	C₈²×C₁₃	—	2	1	no	—
27	#11718062	C₄	1	2	yes	—
28	#11718059	C₄	1	2	yes	—
29	#11718057	C₄	1	2	yes	complemented
30	#11718056	C₄	1	2	yes	complemented
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	#88184	C₂×C₄	1	1	yes	characteristic
36	#88183	—	2	1	no	—
37	C₄×C₂²⋊D₁₃	—	2	1	no	—
38	#88178	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	—	2	1	no	—
40	C₂²×#1055	—	2	1	no	—
41	C₂×#6094	—	2	1	no	—
42	C₂×C₈×D₁₃	—	2	1	no	—
43	#88130	—	2	1	no	—
44	#88129	C₂×C₄	1	1	yes	characteristic
45	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
46	#88126	—	2	1	no	—
47	#88124	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
49	C₁₃×C₄⋊C₈	—	2	1	no	—
50	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
51	C₁₃×#97	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
58	#88071	—	2	1	no	—
59	#88071	C₂×C₄	1	2	yes	—
60	C₂²⋊#1055	—	2	1	no	—
61	#88060	—	2	1	no	—
62	#88060	—	2	1	no	—
63	#88055	—	2	4	no	—
64	#88055	—	2	1	no	—
65	#88054	—	2	1	no	—
66	#88054	—	2	1	no	—
67	C₈×Dic₁₃	—	2	4	no	—
68	C₈×Dic₁₃	—	2	1	no	—
69	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
70	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
71	#88043	—	2	1	no	—
72	#88043	—	2	1	no	—
73	#88043	C₂×C₄	1	2	yes	—
74	#8247	—	13	1	no	maximal, Sylow
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₃	—	2	1	no	—
79	C₂²×Dic₁₃	—	2	2	no	—
80	C₂×D₅₂	C₄²	1	1	yes	characteristic
81	C₂×C₄×D₁₃	—	2	1	no	—
82	C₂×Dic₂₆	C₄²	1	1	yes	characteristic
83	C₁₃×M₁₆	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₃	#41	1	1	yes	characteristic
96	C₂²⋊Dic₁₃	—	2	1	no	—
97	#6103	—	2	1	no	—
98	#6103	—	2	1	no	—
99	#6103	—	2	1	no	—
100	#6103	—	2	1	no	—
101	C₄⋊Dic₁₃	C₄²	1	1	yes	characteristic
102	C₄⋊Dic₁₃	C₄²	1	1	yes	characteristic
103	#6101	—	2	1	no	—
104	#6101	—	2	1	no	—
105	#6101	—	2	1	no	—
106	#6101	—	2	1	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	—	2	2	no	—
109	C₄×Dic₁₃	—	2	1	no	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	—	2	2	no	—
113	C₂×#1055	—	2	1	no	—
114	C₂×#1055	—	2	1	no	—
115	#6094	—	4	2	no	—
116	C₈×D₁₃	—	4	2	no	—
117	#2966	—	13	1	no	—
118	#1726	—	13	1	no	—
119	#1524	—	13	1	no	—
120	#1500	—	13	1	no	—
121	C₈×M₁₆	D₁₃	1	1	yes	characteristic, complemented
122	#1302	—	13	1	no	—
123	#1300	—	13	1	no	—
124	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
125	C₂²×D₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	4	1	no	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
136	C₂²⋊D₁₃	—	4	2	no	—
137	C₂×Dic₁₃	—	2	4	no	—
138	C₂×Dic₁₃	—	2	1	no	—
139	C₂×Dic₁₃	—	2	1	no	—
140	C₂×Dic₁₃	—	2	1	no	—
141	C₂×Dic₁₃	—	4	2	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	D₅₂	—	4	1	no	—
144	C₄×D₁₃	—	4	1	no	—
145	C₄×D₁₃	—	4	1	no	—
146	Dic₂₆	—	4	1	no	—
147	C₈×C₁₃	—	2	4	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	4	2	no	—
151	C₈×C₁₃	#104	1	2	yes	—
152	C₈×C₁₃	#104	1	2	yes	—
153	#1055	—	2	4	no	—
154	#1055	—	4	1	no	—
155	#1055	—	4	1	no	—
156	#529	—	13	1	no	—
157	#446	—	13	1	no	—
158	C₈×Q₈	—	13	1	no	—
159	#436	—	13	1	no	—
160	#435	—	26	1	no	—
161	C₈×D₄	—	13	1	no	—
162	C₈×D₄	—	26	1	no	—
163	#433	—	26	1	no	—
164	#432	—	13	1	no	—
165	#431	C₂×D₁₃	1	1	yes	characteristic
166	#405	—	26	1	no	—
167	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
168	C₂×#97	—	13	1	no	—
169	C₂×#97	—	26	1	no	—
170	C₂×C₄×C₈	—	26	1	no	—
171	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
172	#376	—	13	1	no	—
173	#330	—	13	2	no	—
174	#329	—	13	2	no	—
175	#328	—	13	2	no	—
176	#327	—	13	2	no	—
177	#322	—	2	1	no	—
178	C₈²	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₁₃	#443	1	2	yes	—
183	C₂×D₁₃	—	4	1	no	—
184	C₂×D₁₃	—	4	1	no	—
185	C₂×D₁₃	—	4	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
192	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
193	Dic₁₃	—	4	4	no	—
194	Dic₁₃	—	4	1	no	—
195	Dic₁₃	—	4	1	no	—
196	Dic₁₃	—	4	1	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	26	1	no	—
199	C₂²×C₈	—	26	1	no	—
200	C₂²×C₈	—	26	1	no	—
201	C₂²×C₈	D₅₂	1	1	yes	characteristic
202	C₂²×C₈	D₅₂	1	1	yes	characteristic
203	#126	—	26	1	no	—
204	#125	—	13	1	no	—
205	#124	—	13	1	no	—
206	#123	—	26	1	no	—
207	C₂²⋊Q₈	—	13	1	no	—
208	#121	—	13	1	no	—
209	C₄×Q₈	—	13	1	no	—
210	C₄×D₄	—	13	1	no	—
211	C₄×D₄	—	26	1	no	—
212	#117	—	26	1	no	—
213	C₂×C₄²	—	26	1	no	—
214	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	13	1	no	—
217	C₄⋊C₈	—	13	1	no	—
218	C₄⋊C₈	—	26	1	no	—
219	C₄⋊C₈	—	26	1	no	—
220	C₂²⋊C₈	—	26	1	no	—
221	C₂²⋊C₈	—	26	1	no	—
222	C₂²⋊C₈	—	26	1	no	—
223	C₂²⋊C₈	C₄×D₁₃	1	2	yes	—
224	#97	—	13	2	no	—
225	#97	—	26	4	no	—
226	#97	—	26	1	no	—
227	#97	—	26	1	no	—
228	#97	—	26	1	no	—
229	#97	C₄×D₁₃	1	2	yes	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	26	4	no	—
233	C₄×C₈	—	26	1	no	—
234	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
235	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
236	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
238	#95	—	13	2	no	—
239	#95	—	26	1	no	—
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	#1757	1	1	yes	characteristic
242	C₂×C₁₃	C₄×#104	1	1	yes	characteristic
243	C₂×C₁₃	#1764	1	1	yes	characteristic
244	D₁₃	—	8	1	no	—
245	C₂×Q₈	—	13	1	no	—
246	C₂×D₄	—	13	1	no	—
247	C₂×D₄	—	26	1	no	—
248	C₂²×C₄	—	26	2	no	—
249	C₂²×C₄	—	26	1	no	—
250	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
252	C₂²×C₄	#6127	1	1	yes	characteristic
253	M₁₆	—	2	4	no	—
254	M₁₆	—	52	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	26	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	—	26	1	no	—
263	C₂×C₈	—	26	1	no	—
264	C₂×C₈	—	52	2	no	—
265	C₂×C₈	#6103	1	1	yes	characteristic
266	C₂×C₈	#6103	1	1	yes	characteristic
267	C₂×C₈	#6103	1	1	yes	characteristic
268	C₂×C₈	#6103	1	1	yes	characteristic
269	C₄⋊C₄	—	13	1	no	—
270	C₄⋊C₄	—	13	1	no	—
271	C₄⋊C₄	—	26	1	no	—
272	C₄⋊C₄	—	26	1	no	—
273	C₄⋊C₄	—	26	1	no	—
274	C₄⋊C₄	—	26	1	no	—
275	C₂²⋊C₄	—	26	1	no	—
276	C₂²⋊C₄	—	26	1	no	—
277	C₂²⋊C₄	—	26	1	no	—
278	C₂²⋊C₄	—	26	1	no	—
279	C₂²⋊C₄	—	26	1	no	—
280	C₄²	—	26	2	no	—
281	C₄²	—	26	2	no	—
282	C₄²	—	26	1	no	—
283	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
284	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
285	C₄²	#6127	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₁₃	#8247	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	26	1	no	—
289	C₂³	C₄×D₅₂	1	1	yes	characteristic
290	Q₈	—	52	1	no	—
291	D₄	—	52	1	no	—
292	D₄	—	52	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	26	4	no	—
298	C₂×C₄	—	26	1	no	—
299	C₂×C₄	—	26	1	no	—
300	C₂×C₄	—	26	1	no	—
301	C₂×C₄	—	52	2	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	—	52	1	no	—
304	C₂×C₄	—	52	1	no	—
305	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
306	C₂×C₄	#88126	1	1	yes	characteristic
307	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
308	C₂×C₄	C₂×#6103	1	1	yes	characteristic
309	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
310	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	4	2	no	—
315	C₈	—	26	4	no	—
316	C₈	—	52	1	no	—
317	C₈	—	52	1	no	—
318	C₈	#88065	1	2	yes	—
319	C₈	#88065	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	52	1	no	—
323	C₂²	—	52	1	no	—
324	C₂²	—	52	1	no	—
325	C₂²	#11718361	1	2	yes	—
326	C₂²	C₄×#6103	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	52	4	no	—
333	C₄	—	52	1	no	—
334	C₄	—	52	1	no	—
335	C₄	—	52	1	no	—
336	C₄	C₂×#88065	1	1	yes	characteristic
337	C₄	C₂×#88065	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	104	1	no	—
340	C₂	#13114637	1	1	yes	characteristic
341	C₂	C₄×#88065	1	1	yes	characteristic
342	C₂	#13114673	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#6103
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₄×C₂²⋊C₄
prime radical	#88184	C₂×C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#8247
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13114522	C₂	semidirect
5. #13112736	C₂	semidirect
6. #13112730	C₂	semidirect
7. C₈×C₁₃×M₁₆	C₂	semidirect
21. #11718349	C₄	semidirect
22. C₈×D₅₂	C₄	semidirect
23. #11718340	C₄	semidirect
24. C₈×Dic₂₆	C₄	semidirect
29. #11718057	C₄	semidirect
30. #11718056	C₄	semidirect
121. C₈×M₁₆	D₁₃	semidirect
287. C₁₃	#8247	semidirect

Maximal subgroups

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

subgroup	index
#13123811	2
#13114522	2
#13113425	2
#13113323	2
#13112736	2
#13112730	2
C₈×C₁₃×M₁₆	2
#8247	13