Subgroups — #16018286

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718357	C₂	1	1	yes	maximal, characteristic
2	#13117075	C₂	1	4	yes	maximal, complemented
3	#13117070	C₂	1	4	yes	maximal, complemented
4	#13116555	C₂	1	2	yes	maximal, complemented
5	#13116551	C₂	1	2	yes	maximal
6	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#533	—	2	1	no	—
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
11	C₂²×#6095	—	2	1	no	—
12	C₂×#88129	—	2	1	no	—
13	C₁₃×#500	C₂²	1	2	yes	complemented
14	C₁₃×#499	—	2	1	no	—
15	C₁₃×#499	C₂²	1	2	yes	complemented
16	C₁₃×#498	—	2	1	no	—
17	C₁₃×#427	—	2	1	no	—
18	C₁₃×#427	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88052	C₂²	1	2	yes	—
21	#11718357	—	2	1	no	—
22	#11718357	—	2	1	no	—
23	#11718357	C₂²	1	2	yes	—
24	#11718357	C₂²	1	2	yes	—
25	#11718084	C₂²	1	8	yes	complemented
26	#11718083	C₂²	1	8	yes	—
27	#11718053	C₂²	1	2	yes	complemented
28	C₄×#6090	C₂²	1	2	yes	complemented
29	C₂²×D₅₂	—	2	1	no	—
30	C₂²×Dic₂₆	—	2	1	no	—
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×C₄⋊Q₈	—	2	2	no	—
34	C₁₃×#125	—	2	2	no	—
35	C₁₃×C₂²⋊Q₈	—	4	2	no	—
36	C₄×Q₈×C₁₃	—	4	2	no	—
37	C₁₃×#117	—	2	2	no	—
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×#6103	—	4	2	no	—
40	C₂×Dic₅₂	—	2	4	no	—
41	C₂×Dic₅₂	—	2	1	no	—
42	C₂×Dic₅₂	C₂³	1	2	yes	—
43	C₂×D₁₀₄	—	2	4	no	—
44	C₂×D₁₀₄	—	2	1	no	—
45	C₂×D₁₀₄	C₂³	1	2	yes	—
46	C₂×#6095	—	2	2	no	—
47	C₂×#6095	—	2	2	no	—
48	C₂×#6095	—	4	4	no	—
49	#88129	—	2	2	no	—
50	#88129	—	2	2	no	—
51	#88129	—	4	2	no	—
52	C₁₃×#107	—	2	2	no	—
53	C₁₃×#107	C₂³	1	4	yes	—
54	C₁₃×#106	—	2	2	no	—
55	C₁₃×#106	—	2	2	no	—
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	#88052	—	2	4	no	—
61	C₂×#6090	C₂³	1	4	yes	—
62	#33706	—	13	1	no	maximal, Sylow
63	C₂³×D₁₃	—	4	1	no	—
64	C₂×Q₈×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	D₈	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	—	4	1	no	—
68	C₂×D₅₂	—	2	2	no	—
69	C₂×D₅₂	—	4	2	no	—
70	C₂×D₅₂	—	4	2	no	—
71	C₂×D₅₂	—	4	1	no	—
72	C₂×Dic₂₆	—	2	2	no	—
73	C₂×Dic₂₆	—	4	2	no	—
74	C₂×Dic₂₆	—	4	2	no	—
75	C₂×Dic₂₆	—	4	1	no	—
76	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	#41	1	4	yes	—
81	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	4	4	no	—
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₁₃×C₂²⋊C₄	—	4	2	no	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	D₈	1	2	yes	—
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	#6103	—	4	4	no	—
91	#6103	—	4	4	no	—
92	Dic₅₂	—	2	4	no	—
93	Dic₅₂	—	4	2	no	—
94	D₁₀₄	—	2	4	no	—
95	D₁₀₄	—	4	2	no	—
96	#6095	—	4	4	no	—
97	#6095	—	4	4	no	—
98	#6090	—	2	4	no	—
99	#3160	D₁₃	1	1	yes	characteristic, complemented
100	C₂×#495	—	13	1	no	—
101	#2243	—	13	4	no	—
102	#2242	—	13	4	no	—
103	#2141	—	13	2	no	—
104	#2140	—	13	2	no	—
105	C₄×M₃₂	—	13	1	no	—
106	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₁₃	—	4	2	no	—
108	C₂²×D₁₃	—	8	4	no	—
109	C₂²×D₁₃	—	8	2	no	—
110	Q₈×C₁₃	—	8	4	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	#124	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃	—	4	2	no	—
120	C₂×Dic₁₃	—	8	2	no	—
121	D₅₂	—	4	2	no	—
122	D₅₂	—	4	2	no	—
123	D₅₂	—	8	2	no	—
124	Dic₂₆	—	4	2	no	—
125	Dic₂₆	—	4	2	no	—
126	Dic₂₆	—	8	2	no	—
127	C₈×C₁₃	#124	1	2	yes	—
128	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
129	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
130	C₈×C₁₃	C₂×#41	1	2	yes	—
131	C₂²×Q₁₆	—	13	1	no	—
132	C₂²×SD₁₆	—	26	1	no	—
133	C₂²×D₈	—	13	1	no	—
134	#533	—	2	1	no	—
135	C₂×#124	—	26	1	no	—
136	C₂×M₃₂	—	13	2	no	—
137	#500	C₂×D₁₃	1	2	yes	complemented
138	#499	—	2	1	no	—
139	#499	C₂×D₁₃	1	2	yes	complemented
140	#498	—	2	1	no	—
141	#495	—	13	2	no	—
142	#495	—	13	2	no	—
143	#495	—	26	1	no	—
144	#495	—	26	1	no	—
145	#427	—	2	1	no	—
146	#427	C₂×D₁₃	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
148	#358	—	13	8	no	—
149	#357	—	13	8	no	—
150	#346	—	13	2	no	—
151	C₄×C₁₆	—	13	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	#486	1	2	yes	—
154	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₁₃	—	8	2	no	—
156	C₂×D₁₃	—	8	4	no	—
157	C₂×D₁₃	—	16	4	no	—
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	8	4	no	—
161	C₄×C₁₃	#486	1	2	yes	—
162	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
163	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
164	Dic₁₃	—	8	2	no	—
165	C₂²×Q₈	—	26	1	no	—
166	C₂²×D₄	—	26	1	no	—
167	C₂×Q₁₆	—	13	2	no	—
168	C₂×Q₁₆	—	26	4	no	—
169	C₂×Q₁₆	—	26	1	no	—
170	C₂×SD₁₆	—	26	2	no	—
171	C₂×SD₁₆	—	26	2	no	—
172	C₂×SD₁₆	—	52	4	no	—
173	C₂×D₈	—	13	2	no	—
174	C₂×D₈	—	26	4	no	—
175	C₂×D₈	—	26	1	no	—
176	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	2	no	—
179	#125	—	2	2	no	—
180	#124	—	26	2	no	—
181	#124	—	26	2	no	—
182	#124	—	52	2	no	—
183	C₂²⋊Q₈	—	4	2	no	—
184	C₄×Q₈	—	4	2	no	—
185	#117	—	2	2	no	—
186	C₂×C₂²⋊C₄	—	52	2	no	—
187	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
188	M₃₂	—	26	4	no	—
189	C₂×C₁₆	—	13	4	no	—
190	#107	—	2	2	no	—
191	#107	C₂²×D₁₃	1	4	yes	—
192	#106	—	2	2	no	—
193	#106	—	2	2	no	—
194	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
196	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
197	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
200	C₂×C₁₃	#3414	1	2	yes	—
201	D₁₃	—	16	2	no	—
202	C₂⁴	—	52	1	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂×Q₈	—	26	2	no	—
205	C₂×Q₈	—	52	1	no	—
206	C₂×Q₈	—	52	2	no	—
207	C₂×Q₈	—	52	2	no	—
208	C₂×D₄	—	26	2	no	—
209	C₂×D₄	—	52	1	no	—
210	C₂×D₄	—	52	2	no	—
211	C₂×D₄	—	52	2	no	—
212	C₂²×C₄	—	52	1	no	—
213	C₂²×C₄	#6104	1	2	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
215	Q₁₆	—	26	4	no	—
216	Q₁₆	—	52	2	no	—
217	SD₁₆	—	52	4	no	—
218	SD₁₆	—	52	4	no	—
219	D₈	—	26	4	no	—
220	D₈	—	52	2	no	—
221	C₂×C₈	#6106	1	4	yes	—
222	C₂×C₈	#6131	1	4	yes	—
223	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	52	4	no	—
232	C₂²⋊C₄	—	52	4	no	—
233	C₄²	—	4	2	no	—
234	C₄²	#6104	1	2	yes	—
235	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
236	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
237	C₁₆	—	26	4	no	—
238	C₁₃	#33706	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	52	2	no	—
240	C₂³	—	104	4	no	—
241	C₂³	—	104	2	no	—
242	C₂³	C₂×#6104	1	1	yes	characteristic
243	Q₈	—	8	4	no	—
244	Q₈	—	52	2	no	—
245	Q₈	—	52	2	no	—
246	Q₈	—	104	2	no	—
247	D₄	—	52	2	no	—
248	D₄	—	52	2	no	—
249	D₄	—	104	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	52	2	no	—
253	C₂×C₄	—	104	2	no	—
254	C₂×C₄	C₂×#6104	1	2	yes	—
255	C₂×C₄	C₂×#6104	1	2	yes	—
256	C₂×C₄	C₂×#6104	1	1	yes	characteristic
257	C₂×C₄	C₂×#6106	1	2	yes	—
258	C₂×C₄	#88201	1	2	yes	—
259	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₈	C₂×#6106	1	2	yes	—
261	C₈	C₂×#6106	1	2	yes	—
262	C₈	#88201	1	2	yes	—
263	C₈	C₂×#6131	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	104	2	no	—
266	C₂²	—	104	4	no	—
267	C₂²	—	208	4	no	—
268	C₂²	#11718749	1	2	yes	—
269	C₂²	C₂²×#6104	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	8	4	no	—
273	C₄	—	104	2	no	—
274	C₄	#11718749	1	2	yes	—
275	C₄	C₂²×#6106	1	1	yes	characteristic
276	C₄	C₂×#88201	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	208	2	no	—
279	C₂	C₂×#11718749	1	1	yes	characteristic
280	C₂	#13127970	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#33706
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #13117075	C₂	semidirect
3. #13117070	C₂	semidirect
4. #13116555	C₂	semidirect
6. C₄×#88052	C₂	semidirect
7. C₁₃×#3160	C₂	semidirect
13. C₁₃×#500	C₂²	semidirect
15. C₁₃×#499	C₂²	semidirect
25. #11718084	C₂²	semidirect
27. #11718053	C₂²	semidirect
28. C₄×#6090	C₂²	semidirect
99. #3160	D₁₃	semidirect
137. #500	C₂×D₁₃	semidirect
139. #499	C₂×D₁₃	semidirect
238. C₁₃	#33706	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718357	2
#13117075	2
#13117070	2
#13116555	2
#13116551	2
C₄×#88052	2
C₁₃×#3160	2
#33706	13