Subgroups — #15846799

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125422	C₂	1	1	yes	maximal, characteristic
2	#13124967	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
4	#13114267	C₂	1	4	yes	maximal, complemented
5	#13114231	C₂	1	2	yes	maximal, complemented
6	#13114207	C₂	1	2	yes	maximal, complemented
7	#13113537	C₂	1	2	yes	maximal
8	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic
9	#11719175	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₁₃×#432	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718777	—	2	1	no	—
14	#11718777	—	2	1	no	—
15	C₂×#88054	—	2	1	no	—
16	C₂×C₈×Dic₁₃	—	2	1	no	—
17	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
18	#11718512	C₂²	1	4	yes	—
19	#11718510	C₂²	1	4	yes	—
20	#11718400	—	2	2	no	—
21	#11718382	—	2	2	no	—
22	#11718377	—	2	4	no	—
23	#11718360	C₂²	1	1	yes	characteristic
24	C₄×Dic₅₂	—	2	1	no	—
25	C₄×Dic₅₂	—	2	1	no	—
26	#11718343	C₂²	1	1	yes	characteristic
27	#11718342	C₂²	1	2	yes	—
28	C₈×Dic₂₆	—	2	1	no	—
29	C₈×Dic₂₆	—	2	1	no	—
30	C₈×Dic₂₆	C₂²	1	2	yes	—
31	C₈×Dic₂₆	C₂²	1	2	yes	—
32	C₁₃×#335	C₂²	1	2	yes	—
33	C₁₃×#334	C₂²	1	2	yes	—
34	#11718078	C₂²	1	4	yes	—
35	#11718074	C₂²	1	4	yes	—
36	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
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	2	no	—
41	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
42	#88178	—	2	2	no	—
43	#88178	—	2	2	no	—
44	C₂×#6101	—	2	1	no	—
45	C₂×C₄×Dic₁₃	—	2	1	no	—
46	C₂²×#1055	—	2	1	no	—
47	C₂×Dic₅₂	—	2	4	no	—
48	C₂×Dic₅₂	—	2	1	no	—
49	C₂×Dic₅₂	—	2	1	no	—
50	#88124	—	2	2	no	—
51	#88123	C₂³	1	2	yes	—
52	#88123	C₂³	1	2	yes	—
53	C₄×Dic₂₆	—	2	1	no	—
54	C₄×Dic₂₆	—	2	1	no	—
55	C₄×Dic₂₆	—	2	2	no	—
56	C₄×Dic₂₆	—	2	2	no	—
57	C₄×Dic₂₆	D₄	1	2	yes	—
58	C₄×Dic₂₆	C₂³	1	2	yes	—
59	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
60	C₁₃×C₂²⋊C₈	—	2	2	no	—
61	C₁₃×#97	—	2	2	no	—
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#88058	—	2	1	no	—
67	#88058	—	2	1	no	—
68	#88057	—	2	4	no	—
69	#88056	—	2	4	no	—
70	#88056	—	2	2	no	—
71	#88056	—	2	2	no	—
72	#88056	—	2	4	no	—
73	#88054	—	2	2	no	—
74	#88054	—	2	2	no	—
75	C₈×Dic₁₃	—	2	2	no	—
76	C₈×Dic₁₃	—	2	2	no	—
77	C₄⋊#1055	—	2	1	no	—
78	C₄⋊#1055	C₂³	1	2	yes	—
79	#19265	—	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	C₂²×Dic₁₃	—	2	1	no	—
84	C₂²×Dic₁₃	—	2	1	no	—
85	C₂×Dic₂₆	—	2	1	no	—
86	C₂×Dic₂₆	—	2	4	no	—
87	C₂×Dic₂₆	—	2	2	no	—
88	C₂×Dic₂₆	—	2	2	no	—
89	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	4	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	#41	1	1	yes	characteristic
100	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊Dic₁₃	—	4	2	no	—
102	C₄⋊Dic₁₃	—	2	1	no	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	C₄⋊Dic₁₃	—	2	2	no	—
107	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
108	#6101	—	2	2	no	—
109	#6101	—	2	2	no	—
110	#6101	—	4	2	no	—
111	#6101	—	4	2	no	—
112	C₄×Dic₁₃	—	2	2	no	—
113	C₄×Dic₁₃	—	2	2	no	—
114	C₄×Dic₁₃	—	4	2	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	—	4	2	no	—
117	Dic₅₂	—	4	2	no	—
118	Dic₅₂	—	4	8	no	—
119	Dic₅₂	—	4	2	no	—
120	#3170	—	13	1	no	—
121	#3113	—	13	2	no	—
122	C₂×C₈×Q₈	—	13	1	no	—
123	#1680	—	13	4	no	—
124	#1674	—	13	2	no	—
125	#1670	—	13	2	no	—
126	#1568	D₁₃	1	1	yes	characteristic
127	#1552	—	13	2	no	—
128	C₂³×C₁₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₁₃	—	2	4	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	C₂×Dic₁₃	—	4	2	no	—
144	C₂×Dic₁₃	—	4	2	no	—
145	Dic₂₆	—	2	4	no	—
146	Dic₂₆	—	2	2	no	—
147	Dic₂₆	—	4	4	no	—
148	Dic₂₆	—	4	2	no	—
149	Dic₂₆	—	4	2	no	—
150	Dic₂₆	#135	1	4	yes	—
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	—	2	1	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	—	4	4	no	—
155	C₈×C₁₃	#137	1	2	yes	—
156	#1055	—	4	2	no	—
157	#533	—	13	2	no	—
158	C₂×C₄×Q₈	—	13	1	no	—
159	#500	—	13	1	no	—
160	#494	—	13	1	no	—
161	#487	—	13	2	no	—
162	#483	—	26	2	no	—
163	#475	—	13	4	no	—
164	#467	—	26	1	no	—
165	#467	—	26	1	no	—
166	#462	—	13	4	no	—
167	#451	—	26	2	no	—
168	C₈×Q₈	—	13	2	no	—
169	C₈×Q₈	—	13	2	no	—
170	C₈×Q₈	—	26	1	no	—
171	C₈×Q₈	—	26	1	no	—
172	#441	—	26	4	no	—
173	C₄×Q₁₆	—	26	1	no	—
174	C₄×Q₁₆	—	26	1	no	—
175	#432	C₂×D₁₃	1	2	yes	—
176	C₂×C₄⋊C₈	—	13	1	no	—
177	C₂×C₄⋊C₈	—	26	1	no	—
178	C₂×C₄×C₈	—	26	1	no	—
179	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
180	#335	C₂×D₁₃	1	2	yes	—
181	#334	C₂×D₁₃	1	2	yes	—
182	#332	—	13	4	no	—
183	#332	—	13	4	no	—
184	C₂²×C₁₃	—	2	1	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	#473	1	2	yes	—
187	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	C₂×#135	1	2	yes	—
193	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
194	C₄×C₁₃	#575	1	1	yes	characteristic
195	Dic₁₃	—	2	4	no	—
196	Dic₁₃	—	4	2	no	—
197	Dic₁₃	—	8	2	no	—
198	Dic₁₃	—	8	2	no	—
199	C₂²×Q₈	—	13	1	no	—
200	C₂×Q₁₆	—	26	4	no	—
201	C₂×Q₁₆	—	26	1	no	—
202	C₂×Q₁₆	—	26	1	no	—
203	C₂²×C₈	—	26	1	no	—
204	C₂²×C₈	D₅₂	1	1	yes	characteristic
205	C₂²×C₈	D₅₂	1	1	yes	characteristic
206	C₄⋊Q₈	—	13	2	no	—
207	C₄⋊Q₈	—	13	2	no	—
208	#125	—	26	2	no	—
209	C₂²⋊Q₈	—	26	2	no	—
210	C₂²⋊Q₈	—	26	2	no	—
211	C₄×Q₈	—	13	2	no	—
212	C₄×Q₈	—	13	2	no	—
213	C₄×Q₈	—	26	1	no	—
214	C₄×Q₈	—	26	1	no	—
215	C₄×Q₈	—	26	2	no	—
216	C₄×Q₈	—	26	2	no	—
217	#117	—	26	2	no	—
218	C₂×C₄⋊C₄	—	13	1	no	—
219	C₂×C₄⋊C₄	—	26	1	no	—
220	C₂×C₄²	—	26	1	no	—
221	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
222	#107	—	26	1	no	—
223	#107	—	26	1	no	—
224	#106	—	26	4	no	—
225	C₄⋊C₈	—	13	2	no	—
226	C₄⋊C₈	—	26	2	no	—
227	C₄⋊C₈	—	26	2	no	—
228	C₄⋊C₈	—	26	1	no	—
229	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
230	#103	—	26	4	no	—
231	#103	—	26	4	no	—
232	#103	—	26	2	no	—
233	#103	—	26	2	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	#97	—	2	2	no	—
236	C₄×C₈	—	26	2	no	—
237	C₄×C₈	—	26	2	no	—
238	C₄×C₈	D₅₂	1	1	yes	characteristic
239	C₄×C₈	D₅₂	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₂×C₁₃	—	2	2	no	—
243	C₂×C₁₃	#3037	1	1	yes	characteristic
244	C₂×C₁₃	#3050	1	1	yes	characteristic
245	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
246	C₂×Q₈	—	13	2	no	—
247	C₂×Q₈	—	26	4	no	—
248	C₂×Q₈	—	26	1	no	—
249	C₂×Q₈	—	26	2	no	—
250	C₂×Q₈	—	26	2	no	—
251	C₂²×C₄	—	26	1	no	—
252	C₂²×C₄	—	26	1	no	—
253	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	#6131	1	1	yes	characteristic
256	Q₁₆	—	52	2	no	—
257	Q₁₆	—	52	8	no	—
258	Q₁₆	—	52	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	—	52	2	no	—
264	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
265	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
266	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
267	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
268	C₄⋊C₄	—	13	2	no	—
269	C₄⋊C₄	—	26	1	no	—
270	C₄⋊C₄	—	26	2	no	—
271	C₄⋊C₄	—	26	2	no	—
272	C₄⋊C₄	—	26	2	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	26	2	no	—
275	C₄⋊C₄	—	26	2	no	—
276	C₄⋊C₄	—	52	2	no	—
277	C₄⋊C₄	—	52	2	no	—
278	C₂²⋊C₄	—	52	2	no	—
279	C₄²	—	26	2	no	—
280	C₄²	—	26	2	no	—
281	C₄²	—	52	2	no	—
282	C₄²	C₂×D₅₂	1	1	yes	characteristic
283	C₄²	D₄×D₁₃	1	1	yes	characteristic
284	C₄²	#6131	1	1	yes	characteristic
285	C₄²	C₂³×D₁₃	1	1	yes	characteristic
286	C₁₃	#19265	1	1	yes	characteristic, complemented, Sylow
287	C₂³	#88149	1	1	yes	characteristic
288	Q₈	—	13	4	no	—
289	Q₈	—	26	4	no	—
290	Q₈	—	26	2	no	—
291	Q₈	—	52	2	no	—
292	Q₈	—	52	2	no	—
293	Q₈	—	52	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	26	4	no	—
299	C₂×C₄	—	26	2	no	—
300	C₂×C₄	—	26	2	no	—
301	C₂×C₄	—	52	2	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	—	52	2	no	—
304	C₂×C₄	#88149	1	1	yes	characteristic
305	C₂×C₄	#88149	1	1	yes	characteristic
306	C₂×C₄	#88149	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#6131	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	4	4	no	—
314	C₈	—	52	2	no	—
315	C₈	#88163	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	#11718557	1	2	yes	—
319	C₂²	C₂×#88149	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	26	4	no	—
325	C₄	—	52	2	no	—
326	C₄	—	104	2	no	—
327	C₄	—	104	2	no	—
328	C₄	C₂×#88163	1	1	yes	characteristic
329	C₄	#11719453	1	1	yes	characteristic
330	C₄	D₁₃×#135	1	2	yes	—
331	C₂	—	2	2	no	—
332	C₂	#13124310	1	1	yes	characteristic
333	C₂	#13124418	1	1	yes	characteristic
334	C₂	C₂×#11718557	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88149
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1568	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂³×C₁₃	#121
hypercenter	#1568	D₁₃
hypocenter	C₁₃	#19265
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
4. #13114267	C₂	semidirect
5. #13114231	C₂	semidirect
6. #13114207	C₂	semidirect
286. C₁₃	#19265	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13124967	2
C₂×C₈×Dic₂₆	2
#13114267	2
#13114231	2
#13114207	2
#13113537	2
C₁₃×#1568	2
#19265	13