Subgroups — #15756093

Order 3328.

432 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	#13114856	C₂	1	2	yes	maximal
2	C₄×#88058	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113592	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112738	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2965	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
7	C₁₃×#446	C₄	1	1	yes	characteristic, complemented
8	C₁₃×#446	C₄	1	1	yes	characteristic, complemented
9	C₁₃×#436	C₄	1	1	yes	characteristic, complemented
10	C₁₃×#436	C₄	1	1	yes	characteristic, complemented
11	C₁₃×#435	—	2	1	no	—
12	C₁₃×#435	—	2	1	no	—
13	C₁₃×#431	—	2	1	no	—
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×M₁₆	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88077	—	2	2	no	—
18	C₂×#88058	—	2	1	no	—
19	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	#11718585	C₂²	1	2	yes	—
21	#11718143	—	2	1	no	—
22	#11718124	—	2	2	no	—
23	#11718049	—	2	1	no	—
24	#11718049	—	2	1	no	—
25	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	—	2	1	no	—
27	C₂×C₁₃×M₁₆	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	Q₈	1	1	yes	characteristic
32	C₄×D₄×C₁₃	—	2	1	no	—
33	C₄×D₄×C₁₃	—	2	1	no	—
34	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄×D₄×C₁₃	Q₈	1	1	yes	characteristic
36	C₁₃×#117	—	2	1	no	—
37	C₁₃×#117	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	—	2	1	no	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄×Dic₁₃	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₁₃×C₄⋊C₈	—	2	1	no	—
44	C₁₃×C₄⋊C₈	—	2	1	no	—
45	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₁₃×C₂²⋊C₈	—	2	1	no	—
48	C₁₃×C₂²⋊C₈	—	2	1	no	—
49	C₁₃×C₂²⋊C₈	—	2	1	no	—
50	C₁₃×#97	—	2	1	no	—
51	C₁₃×#97	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	#88077	—	4	4	no	—
58	#88071	—	2	2	no	—
59	#88058	—	2	4	no	—
60	C₄⋊#1055	—	2	2	no	—
61	C₄⋊#1055	—	2	2	no	—
62	#8668	—	13	1	no	maximal, Sylow
63	C₁₃×#41	—	2	2	no	—
64	C₁₃×#41	—	2	2	no	—
65	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂×D₄×C₁₃	—	2	1	no	—
67	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
74	C₂²×Dic₁₃	—	4	2	no	—
75	C₁₃×M₁₆	—	4	2	no	—
76	C₁₃×M₁₆	—	4	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	4	1	no	—
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	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₁₃×C₄⋊C₄	—	2	1	no	—
86	C₁₃×C₄⋊C₄	—	2	1	no	—
87	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
89	C₁₃×C₂²⋊C₄	—	2	1	no	—
90	C₁₃×C₂²⋊C₄	—	2	1	no	—
91	C₁₃×C₂²⋊C₄	—	2	1	no	—
92	C₄²×C₁₃	—	2	1	no	—
93	C₄²×C₁₃	—	2	1	no	—
94	C₄²×C₁₃	—	2	1	no	—
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₁₃	#41	1	1	yes	characteristic
99	C₄⋊Dic₁₃	—	2	4	no	—
100	C₄⋊Dic₁₃	—	4	2	no	—
101	C₄×Dic₁₃	—	4	2	no	—
102	C₄×Dic₁₃	—	4	2	no	—
103	#6099	—	4	2	no	—
104	#6099	—	4	2	no	—
105	C₂×#1055	—	4	2	no	—
106	#2965	D₁₃	1	1	yes	characteristic
107	#1806	—	13	2	no	—
108	C₄×#107	—	13	1	no	—
109	#1566	—	13	1	no	—
110	#1303	—	13	2	no	—
111	C₂³×C₁₃	—	2	1	no	—
112	C₂³×C₁₃	#121	1	1	yes	characteristic
113	Q₈×C₁₃	—	2	2	no	—
114	D₄×C₁₃	—	2	2	no	—
115	D₄×C₁₃	—	4	1	no	—
116	D₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	4	1	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₃	#137	1	2	yes	—
135	C₂×Dic₁₃	—	4	4	no	—
136	C₂×Dic₁₃	—	8	2	no	—
137	C₂×Dic₁₃	—	8	2	no	—
138	C₈×C₁₃	—	2	1	no	—
139	C₈×C₁₃	—	2	1	no	—
140	C₈×C₁₃	—	2	1	no	—
141	C₈×C₁₃	—	4	1	no	—
142	C₈×C₁₃	—	4	1	no	—
143	C₈×C₁₃	#104	1	2	yes	complemented
144	#1055	—	8	2	no	—
145	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
146	#446	Dic₁₃	1	1	yes	characteristic, complemented
147	#446	Dic₁₃	1	1	yes	characteristic, complemented
148	#436	Dic₁₃	1	1	yes	characteristic, complemented
149	#436	Dic₁₃	1	1	yes	characteristic, complemented
150	#435	—	2	1	no	—
151	#435	—	2	1	no	—
152	#431	—	2	1	no	—
153	#431	C₂×D₁₃	1	1	yes	characteristic
154	C₂×#107	—	26	1	no	—
155	#423	—	13	2	no	—
156	C₂×#104	—	26	2	no	—
157	C₄×M₁₆	—	2	1	no	—
158	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
159	C₄×C₄⋊C₄	—	13	2	no	—
160	#340	—	26	1	no	—
161	#339	—	26	2	no	—
162	#334	—	26	1	no	—
163	#334	—	26	1	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	2	1	no	—
166	C₂²×C₁₃	—	4	1	no	—
167	C₂²×C₁₃	—	4	1	no	—
168	C₂²×C₁₃	—	4	1	no	—
169	C₂²×C₁₃	#380	1	1	yes	characteristic
170	C₂²×C₁₃	#469	1	1	yes	characteristic
171	C₂²×C₁₃	#470	1	1	yes	characteristic
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	4	1	no	—
176	C₄×C₁₃	—	4	1	no	—
177	C₄×C₁₃	—	4	1	no	—
178	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
179	C₄×C₁₃	#421	1	1	yes	characteristic
180	C₄×C₁₃	#428	1	2	yes	—
181	C₄×C₁₃	#468	1	2	yes	—
182	Dic₁₃	—	8	4	no	—
183	C₂×#41	D₅₂	1	1	yes	characteristic
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	—	26	2	no	—
187	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
189	C₄×Q₈	Dic₂₆	1	1	yes	characteristic
190	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	Dic₂₆	1	1	yes	characteristic
194	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
195	#117	—	2	1	no	—
196	#117	D₅₂	1	1	yes	characteristic
197	C₂×C₄⋊C₄	—	26	2	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	26	2	no	—
200	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
201	#107	—	26	4	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	—	2	1	no	—
204	C₄⋊C₈	—	26	2	no	—
205	C₄⋊C₈	—	26	2	no	—
206	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
207	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
208	#104	—	52	4	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	2	1	no	—
212	#97	—	2	1	no	—
213	#97	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
216	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
218	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
219	#95	—	26	2	no	—
220	C₂×C₁₃	—	2	1	no	—
221	C₂×C₁₃	—	2	1	no	—
222	C₂×C₁₃	—	8	1	no	—
223	C₂×C₁₃	#1806	1	1	yes	characteristic
224	C₂×C₁₃	#1809	1	1	yes	characteristic
225	C₂×C₁₃	#1814	1	1	yes	characteristic
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	C₄⋊Dic₁₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	52	2	no	—
235	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
236	C₂²×C₄	#6127	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
238	M₁₆	—	4	2	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	52	2	no	—
241	M₁₆	—	52	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	52	2	no	—
247	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	26	4	no	—
254	C₄⋊C₄	—	52	2	no	—
255	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
256	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	52	2	no	—
264	C₄²	—	52	2	no	—
265	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
266	C₄²	#6127	1	1	yes	characteristic
267	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
269	C₁₃	#8668	1	1	yes	characteristic, complemented, Sylow
270	C₂³	—	2	1	no	—
271	C₂³	#88184	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	D₄	—	2	2	no	—
274	D₄	—	4	1	no	—
275	D₄	—	4	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	52	4	no	—
287	C₂×C₄	—	104	2	no	—
288	C₂×C₄	—	104	2	no	—
289	C₂×C₄	#88162	1	2	yes	—
290	C₂×C₄	#88163	1	2	yes	—
291	C₂×C₄	#88178	1	1	yes	characteristic
292	C₂×C₄	#88178	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
294	C₂×C₄	#88180	1	1	yes	characteristic
295	C₂×C₄	#88184	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	104	2	no	—
303	C₈	#88077	1	2	yes	complemented
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	#11718796	1	1	yes	characteristic
310	C₂²	#11718798	1	1	yes	characteristic
311	C₂²	#11718873	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	104	4	no	—
319	C₄	#11718782	1	2	yes	—
320	C₄	#11718795	1	2	yes	—
321	C₄	C₂×#88077	1	1	yes	characteristic
322	C₄	#11718897	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	8	1	no	—
326	C₂	#13114856	1	1	yes	characteristic
327	C₂	#13114869	1	1	yes	characteristic
328	C₂	#13114890	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2965	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2965	D₁₃
hypocenter	C₁₃	#8668
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2965

Complements

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

N	H ≅ G/N	type
2. C₄×#88058	C₂	semidirect
3. #13113592	C₂	semidirect
4. #13112738	C₂	semidirect
7. C₁₃×#446	C₄	semidirect
8. C₁₃×#446	C₄	semidirect
9. C₁₃×#436	C₄	semidirect
10. C₁₃×#436	C₄	semidirect
143. C₈×C₁₃	#104	semidirect
146. #446	Dic₁₃	semidirect
147. #446	Dic₁₃	semidirect
148. #436	Dic₁₃	semidirect
149. #436	Dic₁₃	semidirect
269. C₁₃	#8668	semidirect
303. C₈	#88077	semidirect

Maximal subgroups

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

subgroup	index
#13114856	2
C₄×#88058	2
#13113592	2
#13112738	2
C₁₃×#2965	2
#8668	13