Subgroups — #15768543

Order 3328.

650 conjugacy classes of subgroups.

The subgroup lattice has 302 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	C₂×#11718149	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718149	C₂	1	2	yes	maximal
4	#13114534	C₂	1	2	yes	maximal
5	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
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×#338	—	2	2	no	—
15	#11718149	—	2	2	no	—
16	#11718149	—	2	4	no	—
17	#11718142	C₄	1	8	yes	complemented
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	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
28	C₂×C₁₃×M₁₆	Q₈	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
32	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
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	D₄	1	4	yes	—
39	C₂×C₄⋊Dic₁₃	—	2	1	no	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
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	C₂×C₄	1	4	yes	—
49	#88055	—	2	4	no	—
50	C₈×Dic₁₃	—	2	4	no	—
51	#10799	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	#41	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	#41	1	2	yes	—
59	C₂²×C₄×C₁₃	#41	1	2	yes	—
60	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	2	2	no	—
64	C₂²×Dic₁₃	—	2	4	no	—
65	C₁₃×M₁₆	—	2	4	no	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	—	4	2	no	—
68	C₁₃×M₁₆	—	4	2	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
77	C₂²⋊Dic₁₃	—	2	4	no	—
78	C₂²⋊Dic₁₃	—	4	2	no	—
79	C₂²⋊Dic₁₃	—	4	2	no	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄⋊Dic₁₃	—	4	2	no	—
82	C₄⋊Dic₁₃	—	4	2	no	—
83	C₄×Dic₁₃	—	2	2	no	—
84	C₄×Dic₁₃	—	2	4	no	—
85	C₄×Dic₁₃	—	2	4	no	—
86	C₄×Dic₁₃	C₂²⋊C₄	1	4	yes	—
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	C₄²	1	2	yes	—
94	C₂×#1055	C₄²	1	2	yes	—
95	C₂×#405	—	13	1	no	—
96	C₂×#338	D₁₃	1	1	yes	characteristic
97	C₂×H(3,ℤ/4)	—	13	2	no	—
98	C₂×H(3,ℤ/4)	—	13	1	no	—
99	#1729	—	13	2	no	—
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	#117	1	1	yes	characteristic
102	C₂³×C₁₃	#117	1	2	yes	—
103	C₂³×C₁₃	#123	1	2	yes	—
104	C₂³×C₁₃	#124	1	1	yes	characteristic
105	C₂³×C₁₃	#125	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#117	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	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₁₃	#121	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₃	#123	1	2	yes	—
120	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#126	1	2	yes	—
123	C₂×C₄×C₁₃	#126	1	2	yes	—
124	C₂×Dic₁₃	—	2	4	no	—
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	—	4	8	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₈×C₁₃	—	2	4	no	—
129	C₈×C₁₃	—	4	4	no	—
130	#1055	—	2	2	no	—
131	#1055	—	2	2	no	—
132	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
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	#405	—	26	4	no	—
138	C₂×#97	—	13	2	no	—
139	C₂×C₄×C₈	—	13	2	no	—
140	#338	—	2	2	no	—
141	H(3,ℤ/4)	—	26	4	no	—
142	H(3,ℤ/4)	—	26	2	no	—
143	#336	—	13	8	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	4	no	—
146	C₂²×C₁₃	—	4	2	no	—
147	C₂²×C₁₃	—	4	1	no	—
148	C₂²×C₁₃	—	4	1	no	—
149	C₂²×C₁₃	#376	1	1	yes	characteristic
150	C₂²×C₁₃	#381	1	1	yes	characteristic
151	C₂²×C₁₃	#382	1	1	yes	characteristic
152	C₂²×C₁₃	#387	1	2	yes	—
153	C₂²×C₁₃	#388	1	2	yes	—
154	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₁₃	#380	1	1	yes	characteristic
157	C₄×C₁₃	#383	1	1	yes	characteristic
158	C₄×C₁₃	#387	1	2	yes	—
159	C₄×C₁₃	#388	1	2	yes	—
160	Dic₁₃	—	4	4	no	—
161	Dic₁₃	—	8	4	no	—
162	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	Dic₂₆	1	2	yes	—
172	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
173	C₂²×C₈	—	13	4	no	—
174	C₂²×C₈	—	13	1	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₂²×C₈	—	26	2	no	—
177	C₂²×C₈	—	26	1	no	—
178	C₂²×C₈	—	26	1	no	—
179	C₂²×C₈	C₄×D₁₃	1	4	yes	—
180	C₂²×C₈	C₄×D₁₃	1	2	yes	—
181	#117	—	13	4	no	—
182	#117	—	26	2	no	—
183	#117	—	26	2	no	—
184	#117	—	26	2	no	—
185	#117	—	26	2	no	—
186	#117	—	26	2	no	—
187	C₂×C₄⋊C₄	—	26	1	no	—
188	C₂×C₄⋊C₄	—	26	2	no	—
189	C₂×C₂²⋊C₄	—	26	1	no	—
190	C₂×C₂²⋊C₄	—	26	2	no	—
191	C₂×C₄²	—	13	2	no	—
192	C₂×C₄²	—	13	4	no	—
193	#97	—	26	4	no	—
194	C₄×C₈	—	26	4	no	—
195	C₂×C₁₃	—	2	1	no	—
196	C₂×C₁₃	—	2	2	no	—
197	C₂×C₁₃	—	2	1	no	—
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	#1434	1	1	yes	characteristic
200	C₂×C₁₃	#1747	1	2	yes	—
201	C₂⁴	#6131	1	1	yes	characteristic
202	C₂²×C₄	—	26	2	no	—
203	C₂²×C₄	—	26	4	no	—
204	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
206	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	#6127	1	2	yes	—
208	C₂²×C₄	D₄×D₁₃	1	2	yes	—
209	C₂²×C₄	#6129	1	2	yes	—
210	C₂²×C₄	#6129	1	1	yes	characteristic
211	C₂²×C₄	#6129	1	1	yes	characteristic
212	C₂²×C₄	#6131	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	26	1	no	—
228	C₂×C₈	—	26	1	no	—
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	—	26	1	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	#6101	1	4	yes	—
234	C₄⋊C₄	—	26	4	no	—
235	C₄⋊C₄	—	52	2	no	—
236	C₄⋊C₄	—	52	2	no	—
237	C₂²⋊C₄	—	26	4	no	—
238	C₂²⋊C₄	—	52	2	no	—
239	C₂²⋊C₄	—	52	2	no	—
240	C₄²	—	13	4	no	—
241	C₄²	—	26	2	no	—
242	C₄²	—	26	4	no	—
243	C₄²	—	26	4	no	—
244	C₁₃	#10799	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	4	2	no	—
246	C₂³	#88144	1	1	yes	characteristic
247	C₂³	#88146	1	2	yes	—
248	C₂³	#88146	1	1	yes	characteristic
249	C₂³	#88148	1	2	yes	—
250	C₂³	#88201	1	1	yes	characteristic
251	C₂×C₄	—	26	4	no	—
252	C₂×C₄	—	52	4	no	—
253	C₂×C₄	—	52	8	no	—
254	C₂×C₄	—	52	4	no	—
255	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
256	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
257	C₂×C₄	#88126	1	1	yes	characteristic
258	C₂×C₄	#88131	1	1	yes	characteristic
259	C₂×C₄	#88131	1	2	yes	—
260	C₂×C₄	#88132	1	2	yes	—
261	C₂×C₄	#88133	1	2	yes	—
262	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
263	C₂×C₄	#88135	1	2	yes	—
264	C₂×C₄	#88137	1	2	yes	—
265	C₂×C₄	#88144	1	1	yes	characteristic
266	C₂×C₄	#88146	1	1	yes	characteristic
267	C₂×C₄	#88147	1	2	yes	—
268	C₂×C₄	#88152	1	2	yes	—
269	C₂×C₄	C₂×#6101	1	1	yes	characteristic
270	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
271	C₂×C₄	#88190	1	1	yes	characteristic
272	C₂×C₄	#88193	1	2	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	4	4	no	—
275	C₈	—	26	2	no	—
276	C₈	—	26	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	#11718309	1	1	yes	characteristic
283	C₂²	#11718316	1	2	yes	—
284	C₂²	#11718325	1	1	yes	characteristic
285	C₂²	#11718638	1	1	yes	characteristic
286	C₂²	#11718651	1	2	yes	—
287	C₄	—	52	4	no	—
288	C₄	—	104	4	no	—
289	C₄	#11718312	1	1	yes	characteristic
290	C₄	#11718315	1	2	yes	—
291	C₄	#11718327	1	1	yes	characteristic
292	C₄	C₄×#6101	1	1	yes	characteristic
293	C₄	#11718603	1	1	yes	characteristic
294	C₄	#11718606	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	#13113062	1	1	yes	characteristic
300	C₂	#13114592	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#338	C₂
socle	C₂²×C₁₃	#376
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#338	D₁₃
hypocenter	C₁₃	#10799
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#338

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
17. #11718142	C₄	semidirect
244. C₁₃	#10799	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
C₂×#11718149	2
#13114534	2
C₂×C₁₃×#338	2
#10799	13