Subgroups — #15768617

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718783	C₂	1	1	yes	maximal, characteristic
2	#13114568	C₂	1	2	yes	maximal
3	#13114552	C₂	1	2	yes	maximal
4	C₂×#11718149	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₂²	1	2	yes	—
8	C₂×#88180	C₂²	1	2	yes	—
9	C₂×#88180	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	#11718783	—	2	4	no	—
12	C₂×#88055	C₄	1	2	yes	complemented
13	C₂×C₈×Dic₁₃	C₄	1	2	yes	complemented
14	C₁₃×H(3,ℤ/4)	C₄	1	4	yes	complemented
15	#11718149	C₄	1	4	yes	complemented
16	#11718121	—	2	4	no	—
17	#11718119	—	2	4	no	—
18	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₁₃×#117	—	2	2	no	—
25	C₁₃×#117	—	2	2	no	—
26	C₁₃×#117	—	2	2	no	—
27	C₁₃×#117	C₂×C₄	1	4	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
32	C₂×C₂²⋊Dic₁₃	D₄	1	2	yes	—
33	#88180	—	2	2	no	—
34	#88180	—	2	2	no	—
35	#88180	—	2	2	no	—
36	#88180	—	2	2	no	—
37	#88180	—	2	2	no	—
38	#88180	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊Dic₁₃	—	2	2	no	—
40	C₂×C₄⋊Dic₁₃	Q₈	1	2	yes	—
41	C₂×C₄×Dic₁₃	—	2	2	no	—
42	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
43	C₂²×#1055	—	2	2	no	—
44	C₂²×#1055	—	2	1	no	—
45	C₂²×#1055	—	2	1	no	—
46	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
47	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
48	C₂²×#1055	D₄	1	4	yes	—
49	#88055	—	2	4	no	—
50	C₈×Dic₁₃	—	2	4	no	—
51	#10814	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	2	yes	—
60	C₂²×C₄×C₁₃	#41	1	2	yes	—
61	C₂²×C₄×C₁₃	#41	1	2	yes	—
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₂²×Dic₁₃	—	4	2	no	—
65	C₂²×Dic₁₃	C₄⋊C₄	1	4	yes	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₄²	1	4	yes	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	4	2	no	—
73	C₁₃×C₂²⋊C₄	—	2	4	no	—
74	C₁₃×C₂²⋊C₄	—	4	2	no	—
75	C₄²×C₁₃	—	2	4	no	—
76	C₄²×C₁₃	—	4	2	no	—
77	C₂²⋊Dic₁₃	—	2	4	no	—
78	C₂²⋊Dic₁₃	—	2	4	no	—
79	C₂²⋊Dic₁₃	—	4	2	no	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄⋊Dic₁₃	—	2	4	no	—
82	C₄⋊Dic₁₃	—	4	2	no	—
83	C₄×Dic₁₃	—	2	2	no	—
84	C₄×Dic₁₃	—	2	2	no	—
85	C₄×Dic₁₃	—	2	4	no	—
86	C₄×Dic₁₃	—	4	2	no	—
87	C₂×#1055	—	2	1	no	—
88	C₂×#1055	—	2	1	no	—
89	C₂×#1055	—	2	1	no	—
90	C₂×#1055	—	2	1	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	C₂²⋊C₄	1	2	yes	—
94	C₂×#1055	C₂²⋊C₄	1	2	yes	—
95	C₂×#405	—	13	1	no	—
96	#1740	—	13	2	no	—
97	#1735	—	13	2	no	—
98	C₂×H(3,ℤ/4)	—	13	1	no	—
99	C₂×H(3,ℤ/4)	D₁₃	1	1	yes	characteristic
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
103	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
104	C₂³×C₁₃	#121	1	2	yes	—
105	C₂³×C₁₃	#124	1	2	yes	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	8	4	no	—
108	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#117	1	2	yes	—
113	C₂×C₄×C₁₃	#117	1	2	yes	—
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₁₃	#123	1	2	yes	—
122	C₂×C₄×C₁₃	#123	1	2	yes	—
123	C₂×C₄×C₁₃	#125	1	2	yes	—
124	C₂×C₄×C₁₃	#126	1	2	yes	—
125	C₂×C₄×C₁₃	#126	1	2	yes	—
126	C₂×Dic₁₃	—	2	4	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	8	4	no	—
130	C₈×C₁₃	—	2	4	no	—
131	#1055	—	2	2	no	—
132	#1055	—	2	2	no	—
133	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
134	C₂³×C₈	—	13	1	no	—
135	C₂×#117	—	13	1	no	—
136	C₂×#117	—	13	2	no	—
137	C₂×#117	C₂×D₁₃	1	2	yes	—
138	#405	—	26	4	no	—
139	C₂×#97	—	13	2	no	—
140	C₂×C₄×C₈	—	13	2	no	—
141	#342	—	26	4	no	—
142	#340	—	26	4	no	—
143	H(3,ℤ/4)	—	13	4	no	—
144	H(3,ℤ/4)	Dic₁₃	1	4	yes	complemented
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	2	4	no	—
147	C₂²×C₁₃	—	4	1	no	—
148	C₂²×C₁₃	—	4	1	no	—
149	C₂²×C₁₃	—	4	2	no	—
150	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₁₃	#380	1	1	yes	characteristic
152	C₂²×C₁₃	#382	1	1	yes	characteristic
153	C₂²×C₁₃	#388	1	2	yes	—
154	C₂²×C₁₃	#388	1	2	yes	—
155	C₄×C₁₃	—	8	4	no	—
156	C₄×C₁₃	#376	1	1	yes	characteristic
157	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₁₃	#381	1	1	yes	characteristic
159	C₄×C₁₃	#383	1	1	yes	characteristic
160	C₄×C₁₃	#387	1	2	yes	—
161	C₄×C₁₃	#387	1	2	yes	—
162	Dic₁₃	—	4	4	no	—
163	Dic₁₃	—	8	4	no	—
164	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
168	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
169	C₂²×C₈	—	13	1	no	—
170	C₂²×C₈	—	13	1	no	—
171	C₂²×C₈	—	13	4	no	—
172	C₂²×C₈	—	26	2	no	—
173	C₂²×C₈	—	26	1	no	—
174	C₂²×C₈	—	26	1	no	—
175	C₂²×C₈	C₄×D₁₃	1	2	yes	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	13	4	no	—
180	#117	—	26	2	no	—
181	#117	—	26	2	no	—
182	#117	—	26	2	no	—
183	#117	—	26	2	no	—
184	#117	—	26	2	no	—
185	#117	C₂×Dic₁₃	1	4	yes	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	13	2	no	—
188	C₂×C₄⋊C₄	—	26	2	no	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₂²⋊C₄	—	13	2	no	—
191	C₂×C₂²⋊C₄	—	26	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	13	2	no	—
194	C₂×C₄²	—	26	2	no	—
195	#97	—	26	4	no	—
196	C₄×C₈	—	26	4	no	—
197	C₂×C₁₃	—	2	2	no	—
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	#1434	1	1	yes	characteristic
202	C₂×C₁₃	#1812	1	2	yes	—
203	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	13	4	no	—
206	C₂²×C₄	—	52	2	no	—
207	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
209	C₂²×C₄	#6127	1	2	yes	—
210	C₂²×C₄	#6127	1	2	yes	—
211	C₂²×C₄	D₄×D₁₃	1	2	yes	—
212	C₂²×C₄	#6129	1	2	yes	—
213	C₂²×C₄	#6129	1	1	yes	characteristic
214	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
215	C₂²×C₄	#6131	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	13	2	no	—
222	C₂×C₈	—	13	2	no	—
223	C₂×C₈	—	26	1	no	—
224	C₂×C₈	—	26	1	no	—
225	C₂×C₈	—	26	4	no	—
226	C₂×C₈	—	26	1	no	—
227	C₂×C₈	—	26	1	no	—
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	26	4	no	—
233	C₄⋊C₄	—	26	4	no	—
234	C₄⋊C₄	—	52	2	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	26	4	no	—
238	C₂²⋊C₄	—	26	4	no	—
239	C₂²⋊C₄	—	52	2	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	26	2	no	—
243	C₄²	—	26	2	no	—
244	C₄²	—	26	4	no	—
245	C₄²	—	52	2	no	—
246	C₁₃	#10814	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	4	2	no	—
248	C₂³	#88138	1	2	yes	—
249	C₂³	#88139	1	2	yes	—
250	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
251	C₂³	#88147	1	1	yes	characteristic
252	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	—	52	4	no	—
257	C₂×C₄	—	52	4	no	—
258	C₂×C₄	—	104	4	no	—
259	C₂×C₄	#88126	1	2	yes	—
260	C₂×C₄	#88126	1	1	yes	characteristic
261	C₂×C₄	#88126	1	1	yes	characteristic
262	C₂×C₄	#88131	1	1	yes	characteristic
263	C₂×C₄	#88133	1	2	yes	—
264	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
265	C₂×C₄	#88135	1	2	yes	—
266	C₂×C₄	#88137	1	2	yes	—
267	C₂×C₄	#88141	1	1	yes	characteristic
268	C₂×C₄	#88143	1	2	yes	—
269	C₂×C₄	#88146	1	1	yes	characteristic
270	C₂×C₄	#88148	1	2	yes	—
271	C₂×C₄	#88150	1	2	yes	—
272	C₂×C₄	#88152	1	2	yes	—
273	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
274	C₂×C₄	#88180	1	2	yes	—
275	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
276	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
277	C₈	—	2	4	no	—
278	C₈	—	26	2	no	—
279	C₈	—	26	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	#11718311	1	1	yes	characteristic
286	C₂²	#11718328	1	1	yes	characteristic
287	C₂²	#11718330	1	2	yes	—
288	C₂²	#11718612	1	2	yes	—
289	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
290	C₄	—	8	4	no	—
291	C₄	—	52	4	no	—
292	C₄	—	104	4	no	—
293	C₄	#11718312	1	1	yes	characteristic
294	C₄	#11718313	1	2	yes	—
295	C₄	#11718325	1	1	yes	characteristic
296	C₄	#11718593	1	1	yes	characteristic
297	C₄	C₂²⋊C₄×Dic₁₃	1	1	yes	characteristic
298	C₄	#11718642	1	2	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	#13113061	1	1	yes	characteristic
304	C₂	#13114881	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88126
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×H(3,ℤ/4)	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₁₃
hypocenter	C₁₃	#10814
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
12. C₂×#88055	C₄	semidirect
13. C₂×C₈×Dic₁₃	C₄	semidirect
14. C₁₃×H(3,ℤ/4)	C₄	semidirect
15. #11718149	C₄	semidirect
144. H(3,ℤ/4)	Dic₁₃	semidirect
246. C₁₃	#10814	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718783	2
#13114568	2
#13114552	2
C₂×#11718149	2
C₂×C₁₃×H(3,ℤ/4)	2
#10814	13