Subgroups — #15768638

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	#13114553	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718147	C₂	1	2	yes	maximal
4	#13114534	C₂	1	2	yes	maximal
5	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	—	2	4	no	—
12	C₂×C₈×Dic₁₃	C₄	1	4	yes	complemented
13	C₁₃×#341	C₄	1	4	yes	complemented
14	#11718147	—	2	4	no	—
15	#11718143	C₄	1	4	yes	complemented
16	#11718142	—	2	4	no	—
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂³×Dic₁₃	—	2	1	no	—
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
32	#88180	—	2	2	no	—
33	#88180	—	2	2	no	—
34	#88180	—	2	2	no	—
35	#88180	—	2	2	no	—
36	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₁₃	—	2	2	no	—
38	C₂×C₄×Dic₁₃	—	2	1	no	—
39	C₂×C₄×Dic₁₃	—	2	2	no	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₁₃	D₄	1	4	yes	—
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	2	no	—
44	C₂²×#1055	C₂×C₄	1	2	yes	—
45	C₂²×#1055	D₄	1	2	yes	—
46	C₂²×#1055	Q₈	1	2	yes	—
47	C₈×Dic₁₃	—	2	8	no	—
48	#10818	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	#41	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	#41	1	2	yes	—
57	C₂²×C₄×C₁₃	#41	1	2	yes	—
58	C₂²×C₄×C₁₃	#41	1	2	yes	—
59	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
60	C₂²×Dic₁₃	—	2	8	no	—
61	C₂²×Dic₁₃	—	2	2	no	—
62	C₂²×Dic₁₃	—	2	4	no	—
63	C₂²×Dic₁₃	—	2	4	no	—
64	C₁₃×M₁₆	—	2	4	no	—
65	C₁₃×M₁₆	—	2	4	no	—
66	C₁₃×M₁₆	—	4	2	no	—
67	C₁₃×M₁₆	—	4	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	4	2	no	—
74	C₂×C₈×C₁₃	—	4	2	no	—
75	C₂×C₈×C₁₃	C₄²	1	4	yes	—
76	C₂²⋊Dic₁₃	—	4	2	no	—
77	C₂²⋊Dic₁₃	—	4	2	no	—
78	C₄⋊Dic₁₃	—	2	4	no	—
79	C₄⋊Dic₁₃	—	2	4	no	—
80	C₄×Dic₁₃	—	2	2	no	—
81	C₄×Dic₁₃	—	2	4	no	—
82	C₄×Dic₁₃	—	4	2	no	—
83	C₄×Dic₁₃	—	4	2	no	—
84	C₄×Dic₁₃	C₂²⋊C₄	1	4	yes	—
85	C₂×#1055	—	2	2	no	—
86	C₂×#1055	—	2	2	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#1055	C₄⋊C₄	1	4	yes	—
90	C₂²×C₄×C₈	—	13	1	no	—
91	C₂×#341	D₁₃	1	1	yes	characteristic
92	#1735	—	13	1	no	—
93	C₂×#339	—	13	2	no	—
94	#1729	—	13	2	no	—
95	C₂³×C₁₃	—	2	4	no	—
96	C₂³×C₁₃	#117	1	1	yes	characteristic
97	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
99	C₂³×C₁₃	#123	1	2	yes	—
100	C₂³×C₁₃	#126	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#117	1	2	yes	—
107	C₂×C₄×C₁₃	#117	1	2	yes	—
108	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#121	1	2	yes	—
114	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₃	#123	1	2	yes	—
116	C₂×C₄×C₁₃	#124	1	2	yes	—
117	C₂×C₄×C₁₃	#125	1	2	yes	—
118	C₂×C₄×C₁₃	#126	1	2	yes	—
119	C₂×Dic₁₃	—	2	16	no	—
120	C₂×Dic₁₃	—	2	8	no	—
121	C₂×Dic₁₃	—	2	4	no	—
122	C₂×Dic₁₃	—	4	4	no	—
123	C₂×Dic₁₃	—	4	8	no	—
124	C₈×C₁₃	—	2	4	no	—
125	C₈×C₁₃	—	4	4	no	—
126	#1055	—	2	4	no	—
127	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
128	C₂³×C₈	—	13	1	no	—
129	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
130	C₂×#117	—	13	2	no	—
131	C₂²×C₄²	—	13	1	no	—
132	C₂×C₄×C₈	—	13	4	no	—
133	C₂×C₄×C₈	—	26	4	no	—
134	#341	Dic₁₃	1	4	yes	complemented
135	#340	—	13	4	no	—
136	#339	—	26	4	no	—
137	#336	—	26	4	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₃	#381	1	1	yes	characteristic
144	C₂²×C₁₃	#383	1	1	yes	characteristic
145	C₂²×C₁₃	#387	1	2	yes	—
146	C₂²×C₁₃	#388	1	2	yes	—
147	C₂²×C₁₃	#443	1	4	yes	—
148	C₄×C₁₃	#376	1	1	yes	characteristic
149	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₁₃	#380	1	1	yes	characteristic
151	C₄×C₁₃	#382	1	1	yes	characteristic
152	C₄×C₁₃	#387	1	2	yes	—
153	C₄×C₁₃	#388	1	2	yes	—
154	Dic₁₃	—	2	8	no	—
155	Dic₁₃	—	8	4	no	—
156	C₂³×C₄	—	26	1	no	—
157	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	13	2	no	—
168	C₂²×C₈	—	13	2	no	—
169	C₂²×C₈	—	13	2	no	—
170	C₂²×C₈	—	26	2	no	—
171	C₂²×C₈	—	26	2	no	—
172	C₂²×C₈	C₄×D₁₃	1	2	yes	—
173	C₂²×C₈	C₄×D₁₃	1	2	yes	—
174	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
175	#117	—	26	2	no	—
176	#117	—	26	2	no	—
177	#117	—	26	2	no	—
178	#117	—	26	2	no	—
179	C₂×C₄⋊C₄	—	13	4	no	—
180	C₂×C₂²⋊C₄	—	26	2	no	—
181	C₂×C₄²	—	13	2	no	—
182	C₂×C₄²	—	13	4	no	—
183	C₂×C₄²	—	26	2	no	—
184	C₂×C₄²	—	26	1	no	—
185	C₂×C₄²	—	26	2	no	—
186	C₄×C₈	—	26	8	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	#1434	1	1	yes	characteristic
191	C₂×C₁₃	#1772	1	2	yes	—
192	C₂×C₁₃	#1937	1	4	yes	—
193	C₂⁴	#6131	1	1	yes	characteristic
194	C₂²×C₄	—	26	8	no	—
195	C₂²×C₄	—	26	2	no	—
196	C₂²×C₄	—	26	4	no	—
197	C₂²×C₄	—	26	4	no	—
198	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
200	C₂²×C₄	#6127	1	2	yes	—
201	C₂²×C₄	#6127	1	2	yes	—
202	C₂²×C₄	D₄×D₁₃	1	2	yes	—
203	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	#6129	1	2	yes	—
205	C₂²×C₄	#6129	1	1	yes	characteristic
206	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	13	4	no	—
220	C₂×C₈	—	26	2	no	—
221	C₂×C₈	—	26	2	no	—
222	C₂×C₈	—	26	4	no	—
223	C₂×C₈	—	26	4	no	—
224	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
225	C₄⋊C₄	—	26	4	no	—
226	C₄⋊C₄	—	26	4	no	—
227	C₂²⋊C₄	—	52	2	no	—
228	C₂²⋊C₄	—	52	2	no	—
229	C₄²	—	13	4	no	—
230	C₄²	—	26	2	no	—
231	C₄²	—	26	4	no	—
232	C₄²	—	52	2	no	—
233	C₄²	—	52	2	no	—
234	C₁₃	#10818	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	2	4	no	—
236	C₂³	#88146	1	1	yes	characteristic
237	C₂³	#88147	1	1	yes	characteristic
238	C₂³	#88148	1	2	yes	—
239	C₂³	#88152	1	2	yes	—
240	C₂³	Q₈×Dic₁₃	1	1	yes	characteristic
241	C₂×C₄	—	26	8	no	—
242	C₂×C₄	—	26	16	no	—
243	C₂×C₄	—	26	4	no	—
244	C₂×C₄	—	52	8	no	—
245	C₂×C₄	—	52	4	no	—
246	C₂×C₄	#88126	1	1	yes	characteristic
247	C₂×C₄	#88126	1	2	yes	—
248	C₂×C₄	#88126	1	1	yes	characteristic
249	C₂×C₄	#88131	1	1	yes	characteristic
250	C₂×C₄	#88133	1	2	yes	—
251	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
252	C₂×C₄	#88135	1	2	yes	—
253	C₂×C₄	#88137	1	2	yes	—
254	C₂×C₄	#88138	1	2	yes	—
255	C₂×C₄	#88139	1	2	yes	—
256	C₂×C₄	#88141	1	1	yes	characteristic
257	C₂×C₄	#88143	1	2	yes	—
258	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
259	C₂×C₄	#88150	1	2	yes	—
260	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
261	C₂×C₄	#88180	1	2	yes	—
262	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
263	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
264	C₈	—	2	4	no	—
265	C₈	—	4	4	no	—
266	C₈	—	26	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	#11718312	1	1	yes	characteristic
272	C₂²	#11718325	1	1	yes	characteristic
273	C₂²	#11718330	1	2	yes	—
274	C₂²	#11718556	1	4	yes	—
275	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
276	C₂²	#11718642	1	2	yes	—
277	C₄	—	26	8	no	—
278	C₄	—	104	4	no	—
279	C₄	#11718311	1	1	yes	characteristic
280	C₄	#11718313	1	2	yes	—
281	C₄	#11718328	1	1	yes	characteristic
282	C₄	#11718593	1	1	yes	characteristic
283	C₄	C₂²⋊C₄×Dic₁₃	1	1	yes	characteristic
284	C₄	#11718612	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	2	no	—
288	C₂	#13113061	1	1	yes	characteristic
289	C₂	#13114718	1	2	yes	—
290	C₂	#13115538	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#341	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#341	D₁₃
hypocenter	C₁₃	#10818
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
12. C₂×C₈×Dic₁₃	C₄	semidirect
13. C₁₃×#341	C₄	semidirect
15. #11718143	C₄	semidirect
134. #341	Dic₁₃	semidirect
234. C₁₃	#10818	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₁₃	2
#13114553	2
C₂×#11718147	2
#13114534	2
C₂×C₁₃×#341	2
#10818	13