Subgroups — #15749503

Order 3328.

476 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114904	C₂	1	1	yes	maximal, characteristic
2	#13113063	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1728	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×M₁₆	—	2	1	no	—
5	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×#88071	—	2	1	no	—
7	C₂×#88071	—	2	1	no	—
8	C₂×#88071	C₄	1	2	yes	—
9	C₂×C₂²⋊#1055	—	2	1	no	—
10	C₁₃×#336	—	2	1	no	—
11	C₁₃×#336	C₄	1	2	yes	complemented
12	#11718116	—	2	1	no	—
13	#11718116	C₄	1	2	yes	complemented
14	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₂³×Dic₁₃	—	2	2	no	—
18	C₂×C₁₃×M₁₆	—	2	4	no	—
19	C₂×C₁₃×M₁₆	—	4	2	no	—
20	C₂×C₁₃×M₁₆	—	4	1	no	—
21	C₂×C₁₃×M₁₆	—	4	1	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂×C₄²×C₁₃	—	4	2	no	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
28	C₂²×#1055	—	2	2	no	—
29	#88071	—	2	4	no	—
30	#88071	—	4	1	no	—
31	#88071	—	4	1	no	—
32	#88071	—	4	2	no	—
33	C₂²⋊#1055	—	4	2	no	—
34	#7037	—	13	1	no	maximal, Sylow
35	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	—	2	2	no	—
37	C₂²×C₄×C₁₃	—	2	2	no	—
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	—	2	2	no	—
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	—	4	1	no	—
42	C₂²×C₄×C₁₃	—	4	1	no	—
43	C₂²×C₄×C₁₃	—	4	1	no	—
44	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
54	C₂²×Dic₁₃	—	2	4	no	—
55	C₂²×Dic₁₃	—	4	2	no	—
56	C₂²×Dic₁₃	—	4	2	no	—
57	C₂²×Dic₁₃	—	4	2	no	—
58	C₂²×Dic₁₃	—	4	2	no	—
59	C₂²×Dic₁₃	—	4	2	no	—
60	C₂²×Dic₁₃	—	4	2	no	—
61	C₁₃×M₁₆	—	4	4	no	—
62	C₁₃×M₁₆	—	4	4	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	—	4	2	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₄²×C₁₃	—	4	1	no	—
67	C₄²×C₁₃	—	4	1	no	—
68	C₄²×C₁₃	—	4	1	no	—
69	C₄²×C₁₃	—	4	1	no	—
70	C₂×#1055	—	4	2	no	—
71	C₂×#1055	—	4	2	no	—
72	C₂×#1055	—	4	2	no	—
73	#1817	—	13	1	no	—
74	#1728	D₁₃	1	1	yes	characteristic
75	#1434	—	13	1	no	—
76	C₂³×C₁₃	—	2	1	no	—
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	—	2	1	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	#95	1	1	yes	characteristic
81	C₂³×C₁₃	#99	1	2	yes	—
82	C₂³×C₁₃	#102	1	1	yes	characteristic
83	C₂³×C₁₃	#103	1	1	yes	characteristic
84	C₂³×C₁₃	#104	1	2	yes	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#104	1	2	yes	—
109	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
111	C₂×Dic₁₃	—	4	4	no	—
112	C₂×Dic₁₃	—	4	4	no	—
113	C₂×Dic₁₃	—	4	4	no	—
114	C₂×Dic₁₃	—	8	2	no	—
115	C₂×Dic₁₃	—	8	2	no	—
116	C₂×Dic₁₃	—	8	2	no	—
117	C₂×Dic₁₃	—	8	2	no	—
118	C₈×C₁₃	—	4	4	no	—
119	#1055	—	8	2	no	—
120	C₂²×M₁₆	—	2	1	no	—
121	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
122	C₂×C₂²⋊C₈	—	26	1	no	—
123	C₂×#95	—	13	2	no	—
124	C₂×#95	—	26	1	no	—
125	C₂×#95	—	26	1	no	—
126	#336	—	2	1	no	—
127	#336	—	13	2	no	—
128	#336	—	26	1	no	—
129	#336	Dic₁₃	1	2	yes	complemented
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	#327	1	2	yes	—
139	C₂²×C₁₃	#328	1	2	yes	—
140	C₂²×C₁₃	H(3,ℤ/4)	1	1	yes	characteristic
141	C₂²×C₁₃	#339	1	1	yes	characteristic
142	C₂²×C₁₃	#339	1	1	yes	characteristic
143	C₂²×C₁₃	#340	1	1	yes	characteristic
144	C₂²×C₁₃	#342	1	1	yes	characteristic
145	C₂²×C₁₃	#344	1	1	yes	characteristic
146	C₂²×C₁₃	#344	1	1	yes	characteristic
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	4	1	no	—
153	C₄×C₁₃	—	4	1	no	—
154	C₄×C₁₃	—	4	1	no	—
155	Dic₁₃	—	8	4	no	—
156	C₂³×C₄	—	26	2	no	—
157	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
158	C₂³×C₄	D₅₂	1	1	yes	characteristic
159	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
160	C₂×M₁₆	—	2	4	no	—
161	C₂×M₁₆	—	4	2	no	—
162	C₂×M₁₆	—	4	1	no	—
163	C₂×M₁₆	—	4	1	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	26	2	no	—
166	C₂×C₄²	—	4	2	no	—
167	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
168	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
171	C₂²⋊C₈	—	52	2	no	—
172	#95	—	26	4	no	—
173	#95	—	52	2	no	—
174	#95	—	52	1	no	—
175	#95	—	52	1	no	—
176	C₂×C₁₃	—	2	2	no	—
177	C₂×C₁₃	—	2	2	no	—
178	C₂×C₁₃	#1284	1	1	yes	characteristic
179	C₂×C₁₃	#1286	1	1	yes	characteristic
180	C₂×C₁₃	#1288	1	1	yes	characteristic
181	C₂×C₁₃	#1294	1	1	yes	characteristic
182	C₂×C₁₃	#1300	1	1	yes	characteristic
183	C₂×C₁₃	#1303	1	1	yes	characteristic
184	C₂×C₁₃	#1306	1	1	yes	characteristic
185	C₂⁴	#6103	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	26	4	no	—
195	C₂²×C₄	—	52	2	no	—
196	C₂²×C₄	—	52	2	no	—
197	C₂²×C₄	—	52	2	no	—
198	C₂²×C₄	—	52	2	no	—
199	C₂²×C₄	—	52	2	no	—
200	C₂²×C₄	—	52	2	no	—
201	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
202	C₂²×C₄	#6101	1	1	yes	characteristic
203	C₂²×C₄	#6101	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
205	C₂²×C₄	#6103	1	1	yes	characteristic
206	C₂²×C₄	#6104	1	1	yes	characteristic
207	C₂²×C₄	#6105	1	1	yes	characteristic
208	C₂²×C₄	#6106	1	1	yes	characteristic
209	C₂²×C₄	#6107	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
211	M₁₆	—	4	4	no	—
212	M₁₆	—	4	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	52	2	no	—
217	C₂×C₈	—	52	2	no	—
218	C₂×C₈	—	52	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₁₃	#7037	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	#88045	1	2	yes	—
229	C₂³	#88046	1	2	yes	—
230	C₂³	#88049	1	1	yes	characteristic
231	C₂³	#88050	1	1	yes	characteristic
232	C₂³	#88071	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	52	4	no	—
255	C₂×C₄	—	52	4	no	—
256	C₂×C₄	—	52	4	no	—
257	C₂×C₄	—	104	2	no	—
258	C₂×C₄	—	104	2	no	—
259	C₂×C₄	—	104	2	no	—
260	C₂×C₄	—	104	2	no	—
261	C₂×C₄	#88047	1	1	yes	characteristic
262	C₂×C₄	#88048	1	1	yes	characteristic
263	C₂×C₄	#88065	1	2	yes	—
264	C₂×C₄	#88072	1	1	yes	characteristic
265	C₂×C₄	#88075	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	104	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#11718043	1	2	yes	—
277	C₂²	#11718044	1	2	yes	—
278	C₂²	#11718115	1	1	yes	characteristic
279	C₂²	#11718119	1	1	yes	characteristic
280	C₂²	#11718121	1	1	yes	characteristic
281	C₂²	#11718147	1	1	yes	characteristic
282	C₂²	#11718148	1	1	yes	characteristic
283	C₂²	#11718149	1	1	yes	characteristic
284	C₂²	#11718151	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	104	4	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	#13112690	1	1	yes	characteristic
297	C₂	#13112696	1	1	yes	characteristic
298	C₂	#13112701	1	1	yes	characteristic
299	C₂	#13112715	1	1	yes	characteristic
300	C₂	#13112730	1	1	yes	characteristic
301	C₂	#13112737	1	1	yes	characteristic
302	C₂	#13112745	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88071
commutator	C₂²×C₄×C₁₃	C₄²
Frattini	C₂²×C₄²	C₂×D₁₃
Fitting	C₁₃×#1728	C₂
socle	C₂³×C₁₃	#95
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
cosocle	C₂²×C₄²×C₁₃	C₂²
hypercenter	#1728	D₁₃
hypocenter	C₁₃	#7037
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
11. C₁₃×#336	C₄	semidirect
13. #11718116	C₄	semidirect
129. #336	Dic₁₃	semidirect
223. C₁₃	#7037	semidirect

Maximal subgroups

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

subgroup	index
#13114904	2
#13113063	2
C₁₃×#1728	2
#7037	13