Subgroups — #15756080

Order 3328.

454 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#6127	C₂	1	1	yes	maximal, characteristic
2	#13114853	C₂	1	2	yes	maximal
3	C₄×#88057	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112737	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#6127	C₂²	1	1	yes	characteristic
7	C₁₃×#423	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88065	—	2	2	no	—
10	C₈×C₂²⋊D₁₃	—	2	1	no	—
11	C₈×C₂²⋊D₁₃	—	2	1	no	—
12	C₂×#88057	—	2	1	no	—
13	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	#11718586	C₂²	1	1	yes	characteristic
15	C₈×D₅₂	C₄	1	1	yes	characteristic, complemented
16	C₈×D₅₂	C₄	1	1	yes	characteristic, complemented
17	#11718346	—	2	1	no	—
18	C₄×C₈×D₁₃	—	2	1	no	—
19	C₈×Dic₂₆	C₄	1	1	yes	characteristic, complemented
20	C₈×Dic₂₆	C₄	1	1	yes	characteristic, complemented
21	C₁₃×#334	—	2	1	no	—
22	C₁₃×#334	—	2	1	no	—
23	#11718147	—	2	2	no	—
24	#11718143	—	2	1	no	—
25	C₂×#6127	D₄	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₃	—	2	1	no	—
31	C₄×C₂²⋊D₁₃	—	2	1	no	—
32	#88180	D₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₃	—	2	2	no	—
34	C₂×C₄×Dic₁₃	—	2	2	no	—
35	C₂×C₈×D₁₃	—	2	1	no	—
36	C₂×C₈×D₁₃	—	2	1	no	—
37	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
38	C₄×D₅₂	Q₈	1	1	yes	characteristic
39	#88126	—	2	1	no	—
40	C₄²×D₁₃	—	2	1	no	—
41	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₂₆	Q₈	1	1	yes	characteristic
43	C₁₃×C₄⋊C₈	—	2	2	no	—
44	C₁₃×C₄⋊C₈	—	2	2	no	—
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	#88071	—	2	2	no	—
50	C₂²⋊#1055	—	2	1	no	—
51	#88065	—	4	4	no	—
52	#88060	—	2	1	no	—
53	#88060	—	2	1	no	—
54	#88057	—	2	4	no	—
55	#88054	—	2	1	no	—
56	#88054	—	2	1	no	—
57	C₈×Dic₁₃	—	2	1	no	—
58	C₈×Dic₁₃	—	2	1	no	—
59	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
60	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
61	C₄×#1055	—	2	1	no	—
62	#8666	—	13	1	no	maximal, Sylow
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₃	—	2	1	no	—
67	C₂²×Dic₁₃	—	4	2	no	—
68	#6127	—	2	2	no	—
69	#6127	SD₁₆	1	4	yes	—
70	C₂×D₅₂	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×D₁₃	—	2	1	no	—
72	C₂×C₄×D₁₃	—	2	1	no	—
73	C₂×C₄×D₁₃	—	2	1	no	—
74	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
75	C₁₃×M₁₆	—	4	2	no	—
76	C₁₃×M₁₆	—	4	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	4	1	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
87	C₄²×C₁₃	#41	1	1	yes	characteristic
88	C₂²⋊Dic₁₃	—	2	1	no	—
89	#6103	—	2	1	no	—
90	#6103	—	2	1	no	—
91	C₄⋊Dic₁₃	—	2	4	no	—
92	C₄⋊Dic₁₃	—	4	2	no	—
93	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
94	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
95	#6101	—	2	1	no	—
96	#6101	—	2	1	no	—
97	C₄×Dic₁₃	—	2	1	no	—
98	C₄×Dic₁₃	—	2	1	no	—
99	C₄×Dic₁₃	—	2	1	no	—
100	C₄×Dic₁₃	—	4	2	no	—
101	C₄×Dic₁₃	—	4	2	no	—
102	C₂×#1055	—	2	1	no	—
103	C₂×#1055	—	2	1	no	—
104	C₈×D₁₃	—	4	2	no	—
105	C₈×D₁₃	—	4	2	no	—
106	C₈×#41	—	13	1	no	—
107	#1806	—	13	2	no	—
108	C₄×#106	—	13	1	no	—
109	#1566	D₁₃	1	1	yes	characteristic, complemented
110	#1303	—	13	2	no	—
111	C₂³×C₁₃	#121	1	1	yes	characteristic
112	C₂²×D₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₁₃	#135	1	2	yes	—
124	C₂²⋊D₁₃	—	2	4	no	—
125	C₂×Dic₁₃	—	2	1	no	—
126	C₂×Dic₁₃	—	2	1	no	—
127	C₂×Dic₁₃	—	2	1	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	8	2	no	—
130	C₂×Dic₁₃	—	8	2	no	—
131	D₅₂	—	2	1	no	—
132	D₅₂	#106	1	2	yes	complemented
133	C₄×D₁₃	—	2	4	no	—
134	C₄×D₁₃	—	4	2	no	—
135	C₄×D₁₃	—	4	1	no	—
136	C₄×D₁₃	—	4	1	no	—
137	Dic₂₆	—	2	1	no	—
138	Dic₂₆	#106	1	2	yes	complemented
139	C₈×C₁₃	—	2	1	no	—
140	C₈×C₁₃	—	2	1	no	—
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	8	2	no	—
143	C₈×C₁₃	#104	1	2	yes	complemented
144	#1055	—	4	1	no	—
145	#1055	—	4	1	no	—
146	C₄×#41	—	13	1	no	—
147	C₈×Q₈	—	13	1	no	—
148	C₈×Q₈	—	13	1	no	—
149	C₈×D₄	—	13	1	no	—
150	C₈×D₄	—	13	1	no	—
151	C₈×D₄	—	26	1	no	—
152	C₈×D₄	—	26	1	no	—
153	#431	—	13	1	no	—
154	#431	—	26	1	no	—
155	C₂×#106	—	26	1	no	—
156	#423	C₂×D₁₃	1	2	yes	—
157	C₂×#104	—	26	2	no	—
158	C₂×C₄×C₈	—	26	1	no	—
159	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
160	C₄×C₄⋊C₄	—	13	2	no	—
161	#340	—	26	1	no	—
162	#339	—	26	2	no	—
163	#334	—	2	1	no	—
164	#334	—	2	1	no	—
165	C₂²×C₁₃	—	2	1	no	—
166	C₂²×C₁₃	—	2	1	no	—
167	C₂²×C₁₃	#380	1	1	yes	characteristic
168	C₂²×C₁₃	#465	1	2	yes	—
169	C₂×D₁₃	—	2	2	no	—
170	C₂×D₁₃	—	4	1	no	—
171	C₂×D₁₃	—	4	1	no	—
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
176	C₄×C₁₃	#421	1	1	yes	characteristic
177	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
178	C₄×C₁₃	#427	1	1	yes	characteristic
179	C₄×C₁₃	#465	1	2	yes	—
180	Dic₁₃	—	2	2	no	—
181	Dic₁₃	—	4	1	no	—
182	Dic₁₃	—	4	1	no	—
183	Dic₁₃	—	8	4	no	—
184	C₂×#41	—	13	1	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂²×C₈	—	26	1	no	—
187	C₂²×C₈	—	26	1	no	—
188	C₂²×C₈	D₅₂	1	1	yes	characteristic
189	C₂²×C₈	D₅₂	1	1	yes	characteristic
190	C₄×Q₈	—	13	1	no	—
191	C₄×Q₈	—	13	1	no	—
192	C₄×D₄	—	13	1	no	—
193	C₄×D₄	—	13	1	no	—
194	C₄×D₄	—	26	1	no	—
195	C₄×D₄	—	26	1	no	—
196	#117	—	13	1	no	—
197	#117	—	26	1	no	—
198	C₂×C₄⋊C₄	—	26	2	no	—
199	C₂×C₄²	—	26	1	no	—
200	C₂×C₄²	—	26	2	no	—
201	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
202	#106	—	26	4	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	13	1	no	—
206	C₄⋊C₈	—	13	1	no	—
207	C₄⋊C₈	—	26	1	no	—
208	C₄⋊C₈	—	26	1	no	—
209	#104	—	52	4	no	—
210	C₂²⋊C₈	—	26	1	no	—
211	C₂²⋊C₈	—	26	1	no	—
212	C₂²⋊C₈	—	26	1	no	—
213	C₄×C₈	—	26	1	no	—
214	C₄×C₈	—	26	1	no	—
215	C₄×C₈	—	26	1	no	—
216	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
217	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
218	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
219	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
220	#95	—	26	2	no	—
221	C₂×C₁₃	—	2	2	no	—
222	C₂×C₁₃	#1806	1	1	yes	characteristic
223	C₂×C₁₃	#1807	1	1	yes	characteristic
224	C₂×C₁₃	#1814	1	1	yes	characteristic
225	D₁₃	—	4	2	no	—
226	#41	—	13	4	no	—
227	#41	—	26	2	no	—
228	C₂×Q₈	—	13	1	no	—
229	C₂×D₄	—	13	1	no	—
230	C₂×D₄	—	26	1	no	—
231	C₂²×C₄	—	26	1	no	—
232	C₂²×C₄	—	26	1	no	—
233	C₂²×C₄	—	26	1	no	—
234	C₂²×C₄	—	52	2	no	—
235	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
236	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
237	C₂²×C₄	#6131	1	1	yes	characteristic
238	M₁₆	—	4	2	no	—
239	M₁₆	—	4	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	26	1	no	—
244	C₂×C₈	—	26	1	no	—
245	C₂×C₈	—	52	2	no	—
246	C₂×C₈	—	52	2	no	—
247	C₂×C₈	#6103	1	1	yes	characteristic
248	C₂×C₈	#6103	1	1	yes	characteristic
249	C₂×C₈	#6103	1	1	yes	characteristic
250	C₂×C₈	#6103	1	1	yes	characteristic
251	C₄⋊C₄	—	13	1	no	—
252	C₄⋊C₄	—	13	1	no	—
253	C₄⋊C₄	—	26	4	no	—
254	C₄⋊C₄	—	26	1	no	—
255	C₄⋊C₄	—	26	1	no	—
256	C₄⋊C₄	—	52	2	no	—
257	C₂²⋊C₄	—	26	1	no	—
258	C₂²⋊C₄	—	26	1	no	—
259	C₂²⋊C₄	—	26	1	no	—
260	C₄²	—	26	1	no	—
261	C₄²	—	26	1	no	—
262	C₄²	—	26	1	no	—
263	C₄²	—	52	2	no	—
264	C₄²	—	52	2	no	—
265	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
266	C₄²	#6129	1	1	yes	characteristic
267	C₄²	Q₈×D₁₃	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
269	C₁₃	#8666	1	1	yes	characteristic, complemented, Sylow
270	C₂³	—	26	1	no	—
271	C₂³	#88149	1	1	yes	characteristic
272	Q₈	—	13	2	no	—
273	Q₈	—	26	1	no	—
274	D₄	—	13	2	no	—
275	D₄	—	26	4	no	—
276	D₄	—	26	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	26	1	no	—
281	C₂×C₄	—	26	4	no	—
282	C₂×C₄	—	26	1	no	—
283	C₂×C₄	—	26	1	no	—
284	C₂×C₄	—	52	2	no	—
285	C₂×C₄	—	52	1	no	—
286	C₂×C₄	—	52	1	no	—
287	C₂×C₄	—	52	4	no	—
288	C₂×C₄	—	104	2	no	—
289	C₂×C₄	—	104	2	no	—
290	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
291	C₂×C₄	#88146	1	1	yes	characteristic
292	C₂×C₄	#88151	1	1	yes	characteristic
293	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
294	C₂×C₄	#88170	1	2	yes	—
295	C₂×C₄	C₂×#6103	1	1	yes	characteristic
296	C₂×C₄	#88192	1	1	yes	characteristic
297	C₂×C₄	#88200	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	8	2	no	—
302	C₈	—	52	1	no	—
303	C₈	—	52	1	no	—
304	C₈	#88065	1	2	yes	complemented
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	26	2	no	—
308	C₂²	—	52	1	no	—
309	C₂²	—	52	1	no	—
310	C₂²	#11718525	1	2	yes	—
311	C₂²	#11718648	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	26	2	no	—
316	C₄	—	52	1	no	—
317	C₄	—	52	1	no	—
318	C₄	—	104	4	no	—
319	C₄	D₁₃×#106	1	1	yes	characteristic
320	C₄	#11718521	1	1	yes	characteristic
321	C₄	C₂×#88065	1	1	yes	characteristic
322	C₄	#11718800	1	1	yes	characteristic
323	C₄	#11718832	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	52	2	no	—
326	C₂	#13114853	1	1	yes	characteristic
327	C₂	#13114858	1	1	yes	characteristic
328	C₂	#13114891	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88192
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1566	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂×#6127	D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1566	D₁₃
hypocenter	C₁₃	#8666
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. C₄×#88057	C₂	semidirect
4. #13112737	C₂	semidirect
5. C₁₃×#1566	C₂	semidirect
15. C₈×D₅₂	C₄	semidirect
16. C₈×D₅₂	C₄	semidirect
19. C₈×Dic₂₆	C₄	semidirect
20. C₈×Dic₂₆	C₄	semidirect
109. #1566	D₁₃	semidirect
132. D₅₂	#106	semidirect
138. Dic₂₆	#106	semidirect
143. C₈×C₁₃	#104	semidirect
269. C₁₃	#8666	semidirect
304. C₈	#88065	semidirect

Maximal subgroups

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

subgroup	index
C₈×#6127	2
#13114853	2
C₄×#88057	2
#13112737	2
C₁₃×#1566	2
#8666	13