Subgroups — #15756085

Order 3328.

454 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×#6127	C₂	1	1	yes	maximal, characteristic
2	#13114853	C₂	1	2	yes	maximal
3	C₄×#88058	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112737	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1567	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×#88058	—	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×#335	—	2	1	no	—
22	C₁₃×#335	—	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	#88058	—	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	#8667	—	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	1	no	—
69	#6127	—	2	1	no	—
70	#6127	D₈	1	2	yes	—
71	#6127	Q₁₆	1	2	yes	—
72	C₂×D₅₂	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×D₁₃	—	2	1	no	—
74	C₂×C₄×D₁₃	—	2	1	no	—
75	C₂×C₄×D₁₃	—	2	1	no	—
76	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
77	C₁₃×M₁₆	—	4	2	no	—
78	C₁₃×M₁₆	—	4	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	4	1	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
89	C₄²×C₁₃	#41	1	1	yes	characteristic
90	C₂²⋊Dic₁₃	—	2	1	no	—
91	#6103	—	2	1	no	—
92	#6103	—	2	1	no	—
93	C₄⋊Dic₁₃	—	2	4	no	—
94	C₄⋊Dic₁₃	—	4	2	no	—
95	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
97	#6101	—	2	1	no	—
98	#6101	—	2	1	no	—
99	C₄×Dic₁₃	—	2	1	no	—
100	C₄×Dic₁₃	—	2	1	no	—
101	C₄×Dic₁₃	—	2	1	no	—
102	C₄×Dic₁₃	—	4	2	no	—
103	C₄×Dic₁₃	—	4	2	no	—
104	C₂×#1055	—	2	1	no	—
105	C₂×#1055	—	2	1	no	—
106	C₈×D₁₃	—	4	2	no	—
107	C₈×D₁₃	—	4	2	no	—
108	C₈×#41	—	13	1	no	—
109	#1806	—	13	2	no	—
110	C₄×#107	—	13	1	no	—
111	#1567	D₁₃	1	1	yes	characteristic, complemented
112	#1303	—	13	2	no	—
113	C₂³×C₁₃	#121	1	1	yes	characteristic
114	C₂²×D₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
128	C₂²⋊D₁₃	—	2	2	no	—
129	C₂²⋊D₁₃	—	2	2	no	—
130	C₂×Dic₁₃	—	2	1	no	—
131	C₂×Dic₁₃	—	2	1	no	—
132	C₂×Dic₁₃	—	2	1	no	—
133	C₂×Dic₁₃	—	4	4	no	—
134	C₂×Dic₁₃	—	8	2	no	—
135	C₂×Dic₁₃	—	8	2	no	—
136	D₅₂	—	2	1	no	—
137	D₅₂	#107	1	2	yes	complemented
138	C₄×D₁₃	—	2	2	no	—
139	C₄×D₁₃	—	2	2	no	—
140	C₄×D₁₃	—	4	2	no	—
141	C₄×D₁₃	—	4	1	no	—
142	C₄×D₁₃	—	4	1	no	—
143	Dic₂₆	—	2	1	no	—
144	Dic₂₆	#107	1	2	yes	complemented
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	8	2	no	—
149	C₈×C₁₃	#104	1	2	yes	complemented
150	#1055	—	4	1	no	—
151	#1055	—	4	1	no	—
152	C₄×#41	—	13	1	no	—
153	C₈×Q₈	—	13	1	no	—
154	C₈×Q₈	—	13	1	no	—
155	C₈×D₄	—	13	1	no	—
156	C₈×D₄	—	13	1	no	—
157	C₈×D₄	—	26	1	no	—
158	C₈×D₄	—	26	1	no	—
159	#431	—	13	1	no	—
160	#431	—	26	1	no	—
161	C₂×#107	—	26	1	no	—
162	#423	C₂×D₁₃	1	2	yes	—
163	C₂×#104	—	26	2	no	—
164	C₂×C₄×C₈	—	26	1	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	C₄×C₄⋊C₄	—	13	2	no	—
167	#340	—	26	1	no	—
168	#339	—	26	2	no	—
169	#335	—	2	1	no	—
170	#335	—	2	1	no	—
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	#380	1	1	yes	characteristic
174	C₂²×C₁₃	#466	1	1	yes	characteristic
175	C₂²×C₁₃	#467	1	1	yes	characteristic
176	C₂×D₁₃	—	2	2	no	—
177	C₂×D₁₃	—	4	1	no	—
178	C₂×D₁₃	—	4	1	no	—
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
183	C₄×C₁₃	#421	1	1	yes	characteristic
184	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
185	C₄×C₁₃	#427	1	1	yes	characteristic
186	C₄×C₁₃	#466	1	1	yes	characteristic
187	C₄×C₁₃	#467	1	1	yes	characteristic
188	Dic₁₃	—	2	2	no	—
189	Dic₁₃	—	4	1	no	—
190	Dic₁₃	—	4	1	no	—
191	Dic₁₃	—	8	4	no	—
192	C₂×#41	—	13	1	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂²×C₈	—	26	1	no	—
195	C₂²×C₈	—	26	1	no	—
196	C₂²×C₈	D₅₂	1	1	yes	characteristic
197	C₂²×C₈	D₅₂	1	1	yes	characteristic
198	C₄×Q₈	—	13	1	no	—
199	C₄×Q₈	—	13	1	no	—
200	C₄×D₄	—	13	1	no	—
201	C₄×D₄	—	13	1	no	—
202	C₄×D₄	—	26	1	no	—
203	C₄×D₄	—	26	1	no	—
204	#117	—	13	1	no	—
205	#117	—	26	1	no	—
206	C₂×C₄⋊C₄	—	26	2	no	—
207	C₂×C₄²	—	26	1	no	—
208	C₂×C₄²	—	26	2	no	—
209	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
210	#107	—	26	4	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	13	1	no	—
214	C₄⋊C₈	—	13	1	no	—
215	C₄⋊C₈	—	26	1	no	—
216	C₄⋊C₈	—	26	1	no	—
217	#104	—	52	4	no	—
218	C₂²⋊C₈	—	26	1	no	—
219	C₂²⋊C₈	—	26	1	no	—
220	C₂²⋊C₈	—	26	1	no	—
221	C₄×C₈	—	26	1	no	—
222	C₄×C₈	—	26	1	no	—
223	C₄×C₈	—	26	1	no	—
224	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
225	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
226	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
227	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
228	#95	—	26	2	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	—	2	1	no	—
231	C₂×C₁₃	#1806	1	1	yes	characteristic
232	C₂×C₁₃	#1808	1	1	yes	characteristic
233	C₂×C₁₃	#1814	1	1	yes	characteristic
234	D₁₃	—	4	2	no	—
235	#41	—	13	2	no	—
236	#41	—	13	2	no	—
237	#41	—	26	1	no	—
238	#41	—	26	1	no	—
239	C₂×Q₈	—	13	1	no	—
240	C₂×D₄	—	13	1	no	—
241	C₂×D₄	—	26	1	no	—
242	C₂²×C₄	—	26	1	no	—
243	C₂²×C₄	—	26	1	no	—
244	C₂²×C₄	—	26	1	no	—
245	C₂²×C₄	—	52	2	no	—
246	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
247	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
248	C₂²×C₄	#6131	1	1	yes	characteristic
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	26	1	no	—
255	C₂×C₈	—	26	1	no	—
256	C₂×C₈	—	52	2	no	—
257	C₂×C₈	—	52	2	no	—
258	C₂×C₈	#6103	1	1	yes	characteristic
259	C₂×C₈	#6103	1	1	yes	characteristic
260	C₂×C₈	#6103	1	1	yes	characteristic
261	C₂×C₈	#6103	1	1	yes	characteristic
262	C₄⋊C₄	—	13	1	no	—
263	C₄⋊C₄	—	13	1	no	—
264	C₄⋊C₄	—	26	4	no	—
265	C₄⋊C₄	—	26	1	no	—
266	C₄⋊C₄	—	26	1	no	—
267	C₄⋊C₄	—	52	2	no	—
268	C₂²⋊C₄	—	26	1	no	—
269	C₂²⋊C₄	—	26	1	no	—
270	C₂²⋊C₄	—	26	1	no	—
271	C₄²	—	26	1	no	—
272	C₄²	—	26	1	no	—
273	C₄²	—	26	1	no	—
274	C₄²	—	52	2	no	—
275	C₄²	—	52	2	no	—
276	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
277	C₄²	#6129	1	1	yes	characteristic
278	C₄²	Q₈×D₁₃	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
280	C₁₃	#8667	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	26	1	no	—
282	C₂³	#88149	1	1	yes	characteristic
283	Q₈	—	13	2	no	—
284	Q₈	—	26	1	no	—
285	D₄	—	13	2	no	—
286	D₄	—	26	2	no	—
287	D₄	—	26	2	no	—
288	D₄	—	26	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	26	1	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	26	1	no	—
296	C₂×C₄	—	26	1	no	—
297	C₂×C₄	—	52	2	no	—
298	C₂×C₄	—	52	1	no	—
299	C₂×C₄	—	52	1	no	—
300	C₂×C₄	—	52	4	no	—
301	C₂×C₄	—	104	2	no	—
302	C₂×C₄	—	104	2	no	—
303	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
304	C₂×C₄	#88146	1	1	yes	characteristic
305	C₂×C₄	#88151	1	1	yes	characteristic
306	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
307	C₂×C₄	#88166	1	1	yes	characteristic
308	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
309	C₂×C₄	#88173	1	1	yes	characteristic
310	C₂×C₄	C₂×#6103	1	1	yes	characteristic
311	C₂×C₄	#88192	1	1	yes	characteristic
312	C₂×C₄	#88200	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	8	2	no	—
317	C₈	—	52	1	no	—
318	C₈	—	52	1	no	—
319	C₈	#88065	1	2	yes	complemented
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	26	2	no	—
323	C₂²	—	52	1	no	—
324	C₂²	—	52	1	no	—
325	C₂²	#11718544	1	1	yes	characteristic
326	C₂²	#11718548	1	1	yes	characteristic
327	C₂²	#11718648	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	26	2	no	—
332	C₄	—	52	1	no	—
333	C₄	—	52	1	no	—
334	C₄	—	104	4	no	—
335	C₄	D₁₃×#107	1	1	yes	characteristic
336	C₄	#11718541	1	1	yes	characteristic
337	C₄	C₂×#88065	1	1	yes	characteristic
338	C₄	#11718800	1	1	yes	characteristic
339	C₄	#11718818	1	1	yes	characteristic
340	C₄	#11718849	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	52	2	no	—
344	C₂	#13114853	1	1	yes	characteristic
345	C₂	#13114862	1	1	yes	characteristic
346	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×#1567	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂×#6127	D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1567	D₁₃
hypocenter	C₁₃	#8667
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. C₄×#88058	C₂	semidirect
4. #13112737	C₂	semidirect
5. C₁₃×#1567	C₂	semidirect
15. C₈×D₅₂	C₄	semidirect
16. C₈×D₅₂	C₄	semidirect
19. C₈×Dic₂₆	C₄	semidirect
20. C₈×Dic₂₆	C₄	semidirect
111. #1567	D₁₃	semidirect
137. D₅₂	#107	semidirect
144. Dic₂₆	#107	semidirect
149. C₈×C₁₃	#104	semidirect
280. C₁₃	#8667	semidirect
319. 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×#88058	2
#13112737	2
C₁₃×#1567	2
#8667	13