Subgroups — #16015999

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	#13116637	C₂	1	4	yes	maximal, complemented
3	#13116633	C₂	1	4	yes	maximal, complemented
4	#13116591	C₂	1	2	yes	maximal, complemented
5	#13116446	C₂	1	2	yes	maximal
6	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	2	yes	—
16	C₂×C₄×#1055	C₄	1	1	yes	characteristic
17	C₄×C₈×D₁₃	C₄	1	2	yes	—
18	C₄×C₈×D₁₃	C₄	1	2	yes	—
19	C₄×C₈×D₁₃	C₂²	1	2	yes	—
20	C₄×C₈×D₁₃	C₂²	1	2	yes	—
21	C₁₃×#346	C₂²	1	2	yes	complemented
22	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
23	#11718099	C₂²	1	8	yes	—
24	#11718093	C₂²	1	8	yes	complemented
25	#11718053	C₂²	1	2	yes	complemented
26	C₄×#6090	C₂²	1	2	yes	complemented
27	C₂²×C₄×D₁₃	C₈	1	2	yes	—
28	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
33	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₁₃	—	2	4	no	—
37	C₂×C₈×D₁₃	—	2	4	no	—
38	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂³	1	2	yes	—
42	C₄²×D₁₃	C₈	1	2	yes	—
43	C₄²×D₁₃	C₈	1	2	yes	—
44	C₄²×D₁₃	C₂×C₄	1	2	yes	—
45	C₄²×D₁₃	C₂×C₄	1	2	yes	—
46	C₁₃×M₃₂	—	2	4	no	—
47	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₃	C₂³	1	2	yes	—
56	#88052	—	2	4	no	—
57	C₂×#6090	C₂³	1	4	yes	—
58	C₄×#1055	C₂×C₄	1	2	yes	—
59	C₄×#1055	C₂×C₄	1	2	yes	—
60	#33418	—	13	1	no	maximal, Sylow
61	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₁₃	—	2	2	no	—
67	C₂×C₄×D₁₃	—	2	2	no	—
68	C₂×C₄×D₁₃	—	2	8	no	—
69	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	#41	1	4	yes	—
80	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₄²×C₁₃	C₂×C₈	1	2	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
85	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
86	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
87	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
88	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
89	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
91	C₂×#1055	—	2	2	no	—
92	C₂×#1055	—	2	2	no	—
93	C₂×#1055	C₂²×C₄	1	2	yes	—
94	C₂×#1055	C₂²×C₄	1	2	yes	—
95	C₈×D₁₃	—	2	4	no	—
96	C₈×D₁₃	—	2	4	no	—
97	C₈×D₁₃	—	2	4	no	—
98	C₈×D₁₃	—	2	4	no	—
99	C₁₃×C₁₆	—	2	4	no	—
100	#6090	—	2	4	no	—
101	C₂²×C₄×C₈	—	13	1	no	—
102	#2163	—	13	4	no	—
103	#2162	—	13	4	no	—
104	#2150	—	13	2	no	—
105	#2112	—	13	2	no	—
106	C₄×M₃₂	—	13	1	no	—
107	C₄×M₃₂	D₁₃	1	1	yes	characteristic, complemented
108	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₁₃	—	2	2	no	—
110	C₂²×D₁₃	—	2	4	no	—
111	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	#117	1	2	yes	—
114	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	—	2	2	no	—
120	C₂×Dic₁₃	—	2	4	no	—
121	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
122	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
123	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
124	C₄×D₁₃	—	2	4	no	—
125	C₄×D₁₃	—	2	2	no	—
126	C₄×D₁₃	—	2	2	no	—
127	C₄×D₁₃	—	2	4	no	—
128	C₄×D₁₃	—	2	4	no	—
129	C₄×D₁₃	—	2	4	no	—
130	C₄×D₁₃	M₃₂	1	8	yes	complemented
131	C₈×C₁₃	#117	1	2	yes	—
132	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
133	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
134	C₈×C₁₃	C₂×#41	1	2	yes	—
135	#1055	—	2	2	no	—
136	#1055	—	2	2	no	—
137	C₂³×C₈	—	13	1	no	—
138	C₂³×C₈	—	13	1	no	—
139	C₂²×C₄²	—	13	1	no	—
140	C₂×M₃₂	—	13	2	no	—
141	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
142	C₂×C₄×C₈	—	13	1	no	—
143	C₂×C₄×C₈	—	13	1	no	—
144	C₂×C₄×C₈	—	13	1	no	—
145	C₂×C₄×C₈	—	13	2	no	—
146	C₂×C₄×C₈	—	13	2	no	—
147	C₂×C₄×C₈	—	13	2	no	—
148	C₂×C₄×C₈	—	13	2	no	—
149	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
150	C₄⋊C₁₆	—	13	8	no	—
151	C₂²⋊C₁₆	—	13	8	no	—
152	#346	—	13	2	no	—
153	#346	C₂×D₁₃	1	2	yes	complemented
154	C₄×C₁₆	—	13	2	no	—
155	C₄×C₁₆	C₂×D₁₃	1	2	yes	complemented
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	#431	1	2	yes	—
158	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
159	C₂×D₁₃	—	2	8	no	—
160	C₂×D₁₃	—	2	2	no	—
161	C₂×D₁₃	—	2	2	no	—
162	C₂×D₁₃	C₂×M₃₂	1	4	yes	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	#431	1	2	yes	—
165	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
166	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
167	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
168	Dic₁₃	—	2	2	no	—
169	Dic₁₃	—	2	2	no	—
170	Dic₁₃	C₂×M₃₂	1	4	yes	—
171	C₂³×C₄	—	13	2	no	—
172	C₂³×C₄	—	13	1	no	—
173	C₂²×C₈	—	13	1	no	—
174	C₂²×C₈	—	13	1	no	—
175	C₂²×C₈	—	13	2	no	—
176	C₂²×C₈	—	13	2	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	—	26	4	no	—
180	C₂²×C₈	—	26	4	no	—
181	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	C₂×C₄²	—	13	2	no	—
184	C₂×C₄²	—	13	1	no	—
185	C₂×C₄²	—	13	2	no	—
186	C₂×C₄²	—	13	2	no	—
187	C₂×C₄²	—	13	2	no	—
188	C₂×C₄²	—	13	2	no	—
189	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
190	M₃₂	—	2	4	no	—
191	M₃₂	—	26	4	no	—
192	C₂×C₁₆	—	13	4	no	—
193	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
194	C₄×C₈	—	13	2	no	—
195	C₄×C₈	—	13	2	no	—
196	C₄×C₈	—	13	2	no	—
197	C₄×C₈	—	13	2	no	—
198	C₄×C₈	—	13	2	no	—
199	C₄×C₈	—	13	2	no	—
200	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
201	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
206	C₂×C₁₃	C₂²×M₃₂	1	2	yes	—
207	D₁₃	—	2	4	no	—
208	C₂⁴	—	13	1	no	—
209	C₂²×C₄	—	13	2	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	—	13	1	no	—
212	C₂²×C₄	—	13	2	no	—
213	C₂²×C₄	—	13	2	no	—
214	C₂²×C₄	—	13	2	no	—
215	C₂²×C₄	—	13	2	no	—
216	C₂²×C₄	—	13	2	no	—
217	C₂²×C₄	—	26	8	no	—
218	C₂²×C₄	—	26	2	no	—
219	C₂²×C₄	—	26	2	no	—
220	C₂²×C₄	C₈×D₁₃	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
222	C₂×C₈	—	13	2	no	—
223	C₂×C₈	—	13	2	no	—
224	C₂×C₈	—	26	2	no	—
225	C₂×C₈	—	26	2	no	—
226	C₂×C₈	—	26	4	no	—
227	C₂×C₈	—	26	4	no	—
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	#6094	1	4	yes	—
231	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
234	C₂×C₈	#6127	1	4	yes	—
235	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
236	C₄²	—	13	1	no	—
237	C₄²	—	13	2	no	—
238	C₄²	—	13	2	no	—
239	C₄²	—	13	2	no	—
240	C₄²	—	13	2	no	—
241	C₄²	—	13	1	no	—
242	C₄²	—	13	2	no	—
243	C₄²	C₈×D₁₃	1	2	yes	—
244	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
246	C₁₆	—	2	4	no	—
247	C₁₆	—	26	4	no	—
248	C₁₃	#33418	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	13	2	no	—
250	C₂³	—	26	4	no	—
251	C₂³	—	26	2	no	—
252	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	13	2	no	—
255	C₂×C₄	—	13	8	no	—
256	C₂×C₄	—	13	2	no	—
257	C₂×C₄	—	13	2	no	—
258	C₂×C₄	—	26	4	no	—
259	C₂×C₄	—	26	4	no	—
260	C₂×C₄	—	26	4	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	—	26	2	no	—
263	C₂×C₄	—	26	2	no	—
264	C₂×C₄	—	26	4	no	—
265	C₂×C₄	—	26	4	no	—
266	C₂×C₄	#88126	1	2	yes	—
267	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
268	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
269	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
270	C₂×C₄	C₂×#6094	1	2	yes	—
271	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
272	C₈	—	26	2	no	—
273	C₈	—	26	2	no	—
274	C₈	#88126	1	2	yes	—
275	C₈	C₂×#6094	1	2	yes	—
276	C₈	C₂×#6094	1	2	yes	—
277	C₈	C₂×#6127	1	2	yes	—
278	C₂²	—	2	2	no	—
279	C₂²	—	13	4	no	—
280	C₂²	—	26	8	no	—
281	C₂²	—	26	2	no	—
282	C₂²	—	26	2	no	—
283	C₂²	#11718346	1	2	yes	—
284	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	13	4	no	—
287	C₄	—	26	2	no	—
288	C₄	—	26	2	no	—
289	C₄	#11718346	1	2	yes	—
290	C₄	D₁₃×M₃₂	1	4	yes	—
291	C₄	C₂×#88126	1	1	yes	characteristic
292	C₄	C₂²×#6094	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	26	4	no	—
295	C₂	C₂×#11718346	1	1	yes	characteristic
296	C₂	C₂×D₁₃×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#33418
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #13116637	C₂	semidirect
3. #13116633	C₂	semidirect
4. #13116591	C₂	semidirect
6. C₄×#88052	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
21. C₁₃×#346	C₂²	semidirect
22. C₄×C₁₃×C₁₆	C₂²	semidirect
24. #11718093	C₂²	semidirect
25. #11718053	C₂²	semidirect
26. C₄×#6090	C₂²	semidirect
107. C₄×M₃₂	D₁₃	semidirect
130. C₄×D₁₃	M₃₂	semidirect
153. #346	C₂×D₁₃	semidirect
155. C₄×C₁₆	C₂×D₁₃	semidirect
248. C₁₃	#33418	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₁₃	2
#13116637	2
#13116633	2
#13116591	2
#13116446	2
C₄×#88052	2
C₄×C₁₃×M₃₂	2
#33418	13