Subgroups — #16785728

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊#108	C₂	1	2	yes	maximal
2	#13130541	C₂	1	2	yes	maximal
3	C₂×#11718269	C₂	1	2	yes	maximal
4	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₃⋊M₁₆	C₂²	1	4	yes	—
6	C₂²×#6094	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88126	C₂²	1	1	yes	characteristic
9	D₁₃⋊#108	—	2	4	no	—
10	D₁₃⋊#108	C₄	1	8	yes	complemented
11	C₂×#88104	—	2	2	no	—
12	C₂×#88103	—	2	2	no	—
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
14	C₂×#88055	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
16	C₂×#88043	C₄	1	1	yes	characteristic
17	#11718350	—	2	2	no	—
18	#11718350	—	2	2	no	—
19	#11718291	—	2	4	no	—
20	#11718269	—	2	4	no	—
21	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
22	C₂×#6122	—	2	4	no	—
23	D₁₃⋊M₁₆	—	2	4	no	—
24	D₁₃⋊M₁₆	—	2	4	no	—
25	D₁₃⋊M₁₆	—	2	4	no	—
26	D₁₃⋊M₁₆	C₂×C₄	1	8	yes	—
27	C₂×#6118	—	2	4	no	—
28	C₂×D₁₃⋊C₈	—	2	4	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#6103	Q₈	1	2	yes	—
33	C₂×#6101	D₄	1	2	yes	—
34	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂×#6094	—	2	2	no	—
38	C₂×#6094	—	2	2	no	—
39	C₂×#6094	D₄	1	2	yes	—
40	C₂×#6094	Q₈	1	2	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₁₃	D₄	1	2	yes	—
43	C₂×C₈×D₁₃	D₄	1	2	yes	—
44	C₂×C₈×D₁₃	D₄	1	2	yes	—
45	C₂×C₈×D₁₃	Q₈	1	2	yes	—
46	#88126	—	2	2	no	—
47	#88126	—	2	2	no	—
48	#88104	—	2	8	no	—
49	#88103	—	2	8	no	—
50	C₄×C₈×C₁₃	—	2	1	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	#88055	—	2	2	no	—
53	C₈×Dic₁₃	—	2	1	no	—
54	C₈×Dic₁₃	—	2	1	no	—
55	#88043	—	2	2	no	—
56	#10867	—	13	1	no	maximal, Sylow
57	C₂³×D₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
61	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₄×D₁₃	#41	1	2	yes	—
69	C₂×C₄×D₁₃	#41	1	2	yes	—
70	#6122	—	2	8	no	—
71	#6122	—	4	4	no	—
72	C₂×#1057	—	4	4	no	—
73	#6118	—	2	8	no	—
74	#6118	—	4	4	no	—
75	D₁₃⋊C₈	—	2	8	no	—
76	D₁₃⋊C₈	—	4	4	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	C₄²	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	#6103	—	2	4	no	—
87	#6101	—	2	4	no	—
88	C₄×Dic₁₃	—	2	1	no	—
89	C₄×Dic₁₃	—	2	1	no	—
90	C₂×#1055	—	2	1	no	—
91	C₂×#1055	—	2	1	no	—
92	C₂×#1055	C₄²	1	2	yes	—
93	C₂×#1055	C₂²⋊C₄	1	2	yes	—
94	C₂×#1055	C₄⋊C₄	1	2	yes	—
95	C₂×#1055	C₄⋊C₄	1	2	yes	—
96	#6094	—	2	4	no	—
97	#6094	—	2	4	no	—
98	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
99	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
100	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
101	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
102	C₂²×#108	—	13	2	no	—
103	C₂×#405	—	13	1	no	—
104	#1742	—	13	2	no	—
105	C₂×#338	—	13	2	no	—
106	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×D₁₃	—	4	2	no	—
108	C₂²×D₁₃	C₄×Q₈	1	2	yes	—
109	C₂²×D₁₃	C₂²⋊Q₈	1	2	yes	—
110	C₂²×D₁₃	C₄⋊Q₈	1	1	yes	characteristic
111	C₂²×D₁₃	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₄⋊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	#117	1	1	yes	characteristic
120	C₂×Dic₁₃	—	2	2	no	—
121	C₂×Dic₁₃	—	4	2	no	—
122	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
123	C₂×Dic₁₃	#121	1	2	yes	—
124	C₂×Dic₁₃	#127	1	1	yes	characteristic
125	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
126	C₄×D₁₃	C₂×C₂²⋊C₄	1	2	yes	—
127	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₄×D₁₃	C₄×D₄	1	2	yes	—
129	C₄×D₁₃	C₄×Q₈	1	2	yes	—
130	C₄×D₁₃	#121	1	2	yes	—
131	C₄×D₁₃	C₂²⋊Q₈	1	2	yes	—
132	C₄×D₁₃	#124	1	2	yes	—
133	C₄×D₁₃	#125	1	2	yes	—
134	#1057	—	4	8	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	#95	1	2	yes	—
137	C₈×C₁₃	#95	1	2	yes	—
138	#1055	—	2	2	no	—
139	#1055	#95	1	4	yes	—
140	C₂²×M₁₆	—	13	1	no	—
141	C₂²×M₁₆	—	13	4	no	—
142	C₂³×C₈	—	13	1	no	—
143	C₂×#117	—	13	1	no	—
144	C₂×#108	—	13	8	no	—
145	C₂×#108	—	26	2	no	—
146	C₂×#108	—	26	2	no	—
147	C₂×#108	—	26	4	no	—
148	#405	—	26	2	no	—
149	#405	—	26	2	no	—
150	C₂×#97	—	13	1	no	—
151	C₂×#97	—	13	1	no	—
152	C₂×C₄×C₈	—	13	1	no	—
153	C₂×C₄×C₈	#261	1	1	yes	characteristic
154	#343	—	26	4	no	—
155	#338	—	26	4	no	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₃	#380	1	1	yes	characteristic
159	C₂²×C₁₃	#384	1	1	yes	characteristic
160	C₂×D₁₃	—	2	4	no	—
161	C₂×D₁₃	—	4	2	no	—
162	C₂×D₁₃	—	4	1	no	—
163	C₂×D₁₃	—	4	1	no	—
164	C₂×D₁₃	#389	1	2	yes	—
165	C₂×D₁₃	#391	1	2	yes	—
166	C₄×C₁₃	—	4	2	no	—
167	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
168	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₁₃	#380	1	1	yes	characteristic
170	C₄×C₁₃	#382	1	1	yes	characteristic
171	Dic₁₃	—	4	2	no	—
172	Dic₁₃	#389	1	2	yes	—
173	Dic₁₃	#390	1	2	yes	—
174	C₂³×C₄	—	13	1	no	—
175	C₂×M₁₆	—	13	2	no	—
176	C₂×M₁₆	—	13	2	no	—
177	C₂×M₁₆	—	13	8	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	—	26	2	no	—
180	C₂×M₁₆	—	26	4	no	—
181	C₂×M₁₆	—	26	4	no	—
182	C₂×M₁₆	—	26	4	no	—
183	C₂×M₁₆	—	26	4	no	—
184	C₂×M₁₆	—	26	4	no	—
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	13	4	no	—
187	C₂²×C₈	—	13	2	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	13	2	no	—
191	C₂²×C₈	—	13	1	no	—
192	C₂²×C₈	—	26	4	no	—
193	C₂²×C₈	C₂×#261	1	1	yes	characteristic
194	C₂²×C₈	C₂×#261	1	1	yes	characteristic
195	#117	—	26	2	no	—
196	#117	—	26	2	no	—
197	C₂×C₄⋊C₄	—	13	2	no	—
198	C₂×C₂²⋊C₄	—	13	2	no	—
199	C₂×C₄²	—	13	1	no	—
200	C₂×C₄²	C₂×#261	1	1	yes	characteristic
201	#108	—	26	8	no	—
202	#108	—	26	8	no	—
203	#97	—	26	2	no	—
204	#97	—	26	2	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	26	1	no	—
208	C₄×C₈	—	26	1	no	—
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
212	C₂×C₁₃	#1813	1	2	yes	—
213	D₁₃	—	2	2	no	—
214	D₁₃	—	2	2	no	—
215	C₂⁴	—	13	1	no	—
216	C₂²×C₄	—	13	1	no	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	13	1	no	—
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	1	no	—
221	C₂²×C₄	—	13	1	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	2	no	—
225	C₂²×C₄	—	13	2	no	—
226	C₂²×C₄	C₄⋊#261	1	2	yes	—
227	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
228	M₁₆	—	26	4	no	—
229	M₁₆	—	26	4	no	—
230	M₁₆	—	26	8	no	—
231	M₁₆	—	26	8	no	—
232	M₁₆	—	52	4	no	—
233	M₁₆	—	52	4	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	13	2	no	—
239	C₂×C₈	—	13	4	no	—
240	C₂×C₈	—	13	4	no	—
241	C₂×C₈	—	13	4	no	—
242	C₂×C₈	—	13	4	no	—
243	C₂×C₈	—	13	2	no	—
244	C₂×C₈	—	26	1	no	—
245	C₂×C₈	—	26	1	no	—
246	C₂×C₈	—	26	8	no	—
247	C₂×C₈	—	52	4	no	—
248	C₂×C₈	—	52	4	no	—
249	C₂×C₈	C₄×#261	1	2	yes	—
250	C₂×C₈	C₄⋊#261	1	2	yes	—
251	C₂×C₈	C₄⋊#261	1	2	yes	—
252	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
253	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
254	C₄⋊C₄	—	26	4	no	—
255	C₂²⋊C₄	—	26	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	26	1	no	—
259	C₄²	—	26	1	no	—
260	C₁₃	#10867	1	1	yes	characteristic, complemented, Sylow
261	C₂³	—	13	2	no	—
262	C₂³	—	13	2	no	—
263	C₂³	—	13	1	no	—
264	C₂³	—	13	1	no	—
265	C₂³	—	52	2	no	—
266	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	13	1	no	—
271	C₂×C₄	—	13	1	no	—
272	C₂×C₄	—	13	2	no	—
273	C₂×C₄	—	13	2	no	—
274	C₂×C₄	—	13	2	no	—
275	C₂×C₄	—	13	2	no	—
276	C₂×C₄	—	13	2	no	—
277	C₂×C₄	—	13	2	no	—
278	C₂×C₄	—	13	2	no	—
279	C₂×C₄	—	13	2	no	—
280	C₂×C₄	—	13	2	no	—
281	C₂×C₄	—	26	2	no	—
282	C₂×C₄	—	52	2	no	—
283	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
286	C₂×C₄	#88237	1	1	yes	characteristic
287	C₂×C₄	#88237	1	1	yes	characteristic
288	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	13	4	no	—
291	C₈	—	26	2	no	—
292	C₈	—	52	8	no	—
293	C₈	(C₂×#261)⋊C₄	1	2	yes	—
294	C₈	(C₂×#261)⋊C₄	1	2	yes	—
295	C₂²	—	2	2	no	—
296	C₂²	—	13	2	no	—
297	C₂²	—	13	2	no	—
298	C₂²	—	26	4	no	—
299	C₂²	—	52	2	no	—
300	C₂²	—	52	1	no	—
301	C₂²	—	52	1	no	—
302	C₂²	C₄×C₄⋊#261	1	1	yes	characteristic
303	C₂²	#11719059	1	1	yes	characteristic
304	C₂²	#11719136	1	1	yes	characteristic
305	C₄	—	4	2	no	—
306	C₄	—	13	2	no	—
307	C₄	—	13	2	no	—
308	C₄	—	52	2	no	—
309	C₄	C₄×C₄⋊#261	1	1	yes	characteristic
310	C₄	#11719060	1	1	yes	characteristic
311	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
312	C₄	#11719136	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	26	2	no	—
316	C₂	—	26	2	no	—
317	C₂	#13130014	1	1	yes	characteristic
318	C₂	#13130720	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88237
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	#380
prime radical	C₂³×D₁₃	C₂×Q₈
cosocle	C₂²×C₄×D₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#10867
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. D₁₃⋊#108	C₄	semidirect
260. C₁₃	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃⋊#108	2
#13130541	2
C₂×#11718269	2
C₂×#11718350	2
#10867	13