Subgroups — #15846189

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13126181	C₂	1	2	yes	maximal
2	#13125350	C₂	1	1	yes	maximal, characteristic
3	#13123770	C₂	1	1	yes	maximal, characteristic
4	#13114343	C₂	1	4	yes	maximal, complemented
5	#13114241	C₂	1	2	yes	maximal, complemented
6	#13114231	C₂	1	2	yes	maximal, complemented
7	#13113524	C₂	1	2	yes	maximal
8	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic
9	#11719175	C₂²	1	2	yes	—
10	#11719175	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718777	—	2	4	no	—
14	#11718767	—	2	1	no	—
15	#11718767	—	2	1	no	—
16	#11718763	—	2	1	no	—
17	#11718762	—	2	1	no	—
18	#11718585	C₂²	1	1	yes	characteristic
19	#11718511	—	2	2	no	—
20	#11718511	C₂²	1	4	yes	—
21	#11718510	—	2	2	no	—
22	#11718510	C₂²	1	4	yes	—
23	#11718344	—	2	1	no	—
24	#11718344	—	2	1	no	—
25	#11718343	C₂²	1	1	yes	characteristic
26	#11718343	C₂²	1	1	yes	characteristic
27	#11718341	C₂²	1	2	yes	—
28	#11718340	—	2	1	no	—
29	#11718340	C₂²	1	2	yes	—
30	C₈×Dic₂₆	—	2	1	no	—
31	C₈×Dic₂₆	C₂²	1	2	yes	—
32	C₁₃×#335	C₂²	1	2	yes	—
33	C₁₃×#335	C₂²	1	2	yes	—
34	#11718074	C₂²	1	4	yes	—
35	#11718073	C₂²	1	4	yes	—
36	C₂²×C₈×C₁₃	—	2	2	no	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	#88180	—	2	1	no	—
41	#88180	—	2	2	no	—
42	#88178	—	2	1	no	—
43	#88178	—	2	2	no	—
44	#88178	—	2	2	no	—
45	#88178	D₄	1	2	yes	—
46	C₂×#6099	—	2	1	no	—
47	C₂×Dic₅₂	—	2	4	no	—
48	#88124	—	2	1	no	—
49	#88124	—	2	2	no	—
50	#88123	D₄	1	1	yes	characteristic
51	#88123	D₄	1	1	yes	characteristic
52	#88123	C₂³	1	2	yes	—
53	#88123	C₂³	1	2	yes	—
54	C₄×Dic₂₆	—	2	1	no	—
55	C₄×Dic₂₆	—	2	2	no	—
56	C₄×Dic₂₆	—	2	2	no	—
57	C₄×Dic₂₆	C₂³	1	2	yes	—
58	C₁₃×C₄⋊C₈	—	2	2	no	—
59	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#88058	—	2	1	no	—
65	#88058	—	2	1	no	—
66	#88058	—	2	1	no	—
67	#88058	—	2	1	no	—
68	#88058	—	2	4	no	—
69	#88057	—	2	2	no	—
70	#88057	—	2	2	no	—
71	#88056	—	2	4	no	—
72	#88056	—	2	4	no	—
73	#88055	—	2	2	no	—
74	#88054	—	2	2	no	—
75	#88054	—	2	2	no	—
76	C₈×Dic₁₃	—	2	2	no	—
77	C₄⋊#1055	—	2	1	no	—
78	C₄⋊#1055	C₂³	1	2	yes	—
79	#19211	—	13	1	no	maximal, Sylow
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
83	C₂×Dic₂₆	—	2	1	no	—
84	C₂×Dic₂₆	—	2	2	no	—
85	C₂×Dic₂₆	—	2	2	no	—
86	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	—	8	2	no	—
90	C₂×C₈×C₁₃	Q₁₆	1	4	yes	complemented
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃	#41	1	1	yes	characteristic
98	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₂²⋊Dic₁₃	—	2	2	no	—
100	C₂²⋊Dic₁₃	—	4	2	no	—
101	C₄⋊Dic₁₃	—	2	1	no	—
102	C₄⋊Dic₁₃	—	2	1	no	—
103	C₄⋊Dic₁₃	—	2	1	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	C₄⋊Dic₁₃	—	2	2	no	—
107	C₄⋊Dic₁₃	—	2	2	no	—
108	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
109	#6101	—	2	2	no	—
110	#6101	—	2	2	no	—
111	#6101	—	4	2	no	—
112	#6101	—	4	2	no	—
113	C₄×Dic₁₃	—	2	2	no	—
114	C₄×Dic₁₃	—	4	2	no	—
115	#6099	—	4	2	no	—
116	C₂×#1055	—	2	2	no	—
117	Dic₅₂	—	8	4	no	—
118	#3249	—	13	2	no	—
119	#3160	—	13	1	no	—
120	#2960	—	13	1	no	—
121	#1692	—	13	4	no	—
122	#1676	—	13	2	no	—
123	#1674	—	13	2	no	—
124	C₂×#335	D₁₃	1	1	yes	characteristic
125	#1550	—	13	2	no	—
126	C₂³×C₁₃	#121	1	1	yes	characteristic
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	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₁₃	#137	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₁₃	—	2	2	no	—
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	C₂×Dic₁₃	—	4	2	no	—
142	Dic₂₆	—	2	2	no	—
143	Dic₂₆	—	2	2	no	—
144	Dic₂₆	—	4	2	no	—
145	Dic₂₆	—	4	2	no	—
146	Dic₂₆	—	4	2	no	—
147	Dic₂₆	—	8	2	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	4	4	no	—
151	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
152	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
153	#1055	—	4	2	no	—
154	#533	—	13	1	no	—
155	#533	—	13	2	no	—
156	#500	—	13	1	no	—
157	#500	—	13	1	no	—
158	#499	—	26	1	no	—
159	#499	—	26	1	no	—
160	#498	—	13	2	no	—
161	#477	—	13	4	no	—
162	#477	—	26	2	no	—
163	#467	—	26	4	no	—
164	#462	—	13	4	no	—
165	#462	—	26	2	no	—
166	#446	—	13	2	no	—
167	#446	—	26	1	no	—
168	C₈×Q₈	—	13	2	no	—
169	C₈×Q₈	—	26	1	no	—
170	#427	—	26	1	no	—
171	#427	—	26	1	no	—
172	#424	—	26	1	no	—
173	#423	—	13	1	no	—
174	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
175	#405	—	26	1	no	—
176	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
177	#335	C₂×D₁₃	1	2	yes	—
178	#335	C₂×D₁₃	1	2	yes	—
179	#333	—	13	4	no	—
180	#332	—	13	4	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	#462	1	2	yes	—
184	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	#462	1	2	yes	—
189	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
190	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
191	C₄×C₁₃	#576	1	1	yes	characteristic
192	C₄×C₁₃	#578	1	1	yes	characteristic
193	Dic₁₃	—	4	2	no	—
194	Dic₁₃	—	4	2	no	—
195	Dic₁₃	—	8	2	no	—
196	Dic₁₃	—	8	2	no	—
197	C₂×Q₁₆	—	26	4	no	—
198	C₂×M₁₆	—	26	1	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	D₅₂	1	1	yes	characteristic
201	C₂²×C₈	D₅₂	1	1	yes	characteristic
202	C₄⋊Q₈	—	13	1	no	—
203	C₄⋊Q₈	—	13	1	no	—
204	C₄⋊Q₈	—	13	2	no	—
205	C₄⋊Q₈	—	13	2	no	—
206	#125	—	26	2	no	—
207	#125	—	26	1	no	—
208	C₂²⋊Q₈	—	13	2	no	—
209	C₂²⋊Q₈	—	26	1	no	—
210	C₂²⋊Q₈	—	26	2	no	—
211	C₂²⋊Q₈	—	26	2	no	—
212	C₄×Q₈	—	13	2	no	—
213	C₄×Q₈	—	26	1	no	—
214	C₄×Q₈	—	26	2	no	—
215	C₄×Q₈	—	26	2	no	—
216	#117	—	26	1	no	—
217	#117	—	26	2	no	—
218	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
219	#107	—	26	4	no	—
220	#107	—	26	1	no	—
221	#107	—	26	1	no	—
222	#107	—	26	1	no	—
223	#107	—	26	1	no	—
224	#106	—	26	2	no	—
225	#106	—	26	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	13	2	no	—
228	C₄⋊C₈	—	26	2	no	—
229	C₄⋊C₈	—	26	2	no	—
230	C₄⋊C₈	—	26	1	no	—
231	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
232	#103	—	26	4	no	—
233	#103	—	26	4	no	—
234	#97	—	26	2	no	—
235	C₄×C₈	—	26	2	no	—
236	C₄×C₈	D₅₂	1	1	yes	characteristic
237	C₄×C₈	D₅₂	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	C₂×#462	1	1	yes	characteristic
242	C₂×C₁₃	#3061	1	1	yes	characteristic
243	C₂×C₁₃	#3067	1	1	yes	characteristic
244	C₂×Q₈	—	13	2	no	—
245	C₂×Q₈	—	26	1	no	—
246	C₂×Q₈	—	26	2	no	—
247	C₂×Q₈	—	26	2	no	—
248	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
250	C₂²×C₄	#6131	1	1	yes	characteristic
251	Q₁₆	—	104	4	no	—
252	M₁₆	—	52	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	8	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	Dic₅₂	1	4	yes	complemented
258	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
259	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
260	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
261	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
262	C₄⋊C₄	—	13	2	no	—
263	C₄⋊C₄	—	26	1	no	—
264	C₄⋊C₄	—	26	1	no	—
265	C₄⋊C₄	—	26	2	no	—
266	C₄⋊C₄	—	26	1	no	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	—	26	2	no	—
270	C₄⋊C₄	—	26	2	no	—
271	C₄⋊C₄	—	26	2	no	—
272	C₄⋊C₄	—	52	2	no	—
273	C₄⋊C₄	—	52	2	no	—
274	C₂²⋊C₄	—	26	2	no	—
275	C₂²⋊C₄	—	52	2	no	—
276	C₄²	—	26	2	no	—
277	C₄²	—	52	2	no	—
278	C₄²	C₂×D₅₂	1	1	yes	characteristic
279	C₄²	D₄×D₁₃	1	1	yes	characteristic
280	C₄²	#6131	1	1	yes	characteristic
281	C₄²	C₂³×D₁₃	1	1	yes	characteristic
282	C₁₃	#19211	1	1	yes	characteristic, complemented, Sylow
283	C₂³	#88149	1	1	yes	characteristic
284	Q₈	—	26	2	no	—
285	Q₈	—	26	2	no	—
286	Q₈	—	52	2	no	—
287	Q₈	—	52	2	no	—
288	Q₈	—	52	2	no	—
289	Q₈	—	104	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	52	2	no	—
296	C₂×C₄	—	52	2	no	—
297	C₂×C₄	#88149	1	1	yes	characteristic
298	C₂×C₄	#88149	1	1	yes	characteristic
299	C₂×C₄	#88149	1	1	yes	characteristic
300	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
301	C₂×C₄	#88163	1	2	yes	—
302	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
304	C₂×C₄	C₂×#6131	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	4	4	no	—
308	C₈	—	52	2	no	—
309	C₈	C₂×Dic₅₂	1	2	yes	—
310	C₈	C₂×Dic₅₂	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	#11718510	1	2	yes	—
314	C₂²	C₂×#88149	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	52	2	no	—
319	C₄	—	52	2	no	—
320	C₄	—	104	2	no	—
321	C₄	—	104	2	no	—
322	C₄	#11718510	1	2	yes	—
323	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
324	C₄	C₂×#88163	1	1	yes	characteristic
325	C₄	#11719462	1	1	yes	characteristic
326	C₄	#11719471	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	C₂×#11718510	1	1	yes	characteristic
329	C₂	#13124508	1	1	yes	characteristic
330	C₂	#13124557	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88149
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂³×C₁₃	#121
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	#19211
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #13114343	C₂	semidirect
5. #13114241	C₂	semidirect
6. #13114231	C₂	semidirect
90. C₂×C₈×C₁₃	Q₁₆	semidirect
257. C₂×C₈	Dic₅₂	semidirect
282. C₁₃	#19211	semidirect

Maximal subgroups

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

subgroup	index
#13126181	2
#13125350	2
#13123770	2
#13114343	2
#13114241	2
#13114231	2
#13113524	2
C₂×C₁₃×#335	2
#19211	13