Subgroups — #16018445

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125349	C₂	1	1	yes	maximal, characteristic
2	D₁₃×#499	C₂	1	1	yes	maximal, characteristic
3	#13117121	C₂	1	2	yes	maximal, complemented
4	#13117120	C₂	1	2	yes	maximal, complemented
5	#13117115	C₂	1	1	yes	maximal, characteristic, complemented
6	#13117112	C₂	1	1	yes	maximal, characteristic, complemented
7	#13117112	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116623	C₂	1	2	yes	maximal
9	#13116621	C₂	1	2	yes	maximal
10	C₄×#88038	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719393	—	2	1	no	—
13	D₁₃×#125	—	2	1	no	—
14	C₁₃×#499	C₂²	1	1	yes	characteristic
15	C₁₃×#499	C₂²	1	1	yes	characteristic
16	#11718725	—	2	1	no	—
17	#11718725	—	2	1	no	—
18	C₂×#88038	C₂²	1	1	yes	characteristic
19	C₂×#88038	C₂²	1	1	yes	characteristic
20	#11718541	C₂²	1	2	yes	—
21	D₁₃×#107	C₂²	1	2	yes	—
22	#11718539	C₂²	1	2	yes	—
23	#11718535	C₂²	1	2	yes	—
24	#11718521	—	2	1	no	—
25	D₁₃×#106	—	2	1	no	—
26	#11718519	—	2	1	no	—
27	#11718516	—	2	1	no	—
28	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
29	#11718344	C₂²	1	1	yes	characteristic
30	#11718344	C₂²	1	1	yes	characteristic
31	C₁₃×#367	C₂²	1	2	yes	—
32	C₁₃×#366	C₂²	1	2	yes	—
33	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
34	#11718097	C₂²	1	2	yes	—
35	#11718096	C₂²	1	2	yes	—
36	#11718094	C₂²	1	1	yes	characteristic
37	#11718094	C₂²	1	1	yes	characteristic
38	#11718082	C₂²	1	2	yes	—
39	#11718082	C₂²	1	2	yes	—
40	#11718081	C₂²	1	2	yes	—
41	#11718081	C₂²	1	2	yes	—
42	C₄×#6090	C₂²	1	1	yes	characteristic
43	C₁₃×#125	—	2	1	no	—
44	C₁₃×#125	—	2	1	no	—
45	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
46	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
47	#88150	—	4	2	no	—
48	#88146	—	2	2	no	—
49	C₄⋊C₄×D₁₃	—	2	2	no	—
50	C₄⋊C₄×D₁₃	—	4	2	no	—
51	#88144	—	2	2	no	—
52	#88143	—	4	2	no	—
53	#88142	—	2	2	no	—
54	#88141	—	4	2	no	—
55	C₄²×D₁₃	D₄	1	1	yes	characteristic
56	#88124	—	2	1	no	—
57	#88124	—	2	1	no	—
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₁₃×#107	C₂³	1	2	yes	—
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#106	—	2	1	no	—
63	C₁₃×#106	—	2	1	no	—
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	#88058	C₂³	1	2	yes	—
66	#88058	C₂³	1	2	yes	—
67	#88057	—	2	1	no	—
68	#88057	—	2	1	no	—
69	C₈×Dic₁₃	D₄	1	1	yes	characteristic
70	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	C₂×#6090	C₂³	1	1	yes	characteristic
73	#88048	—	2	2	no	—
74	#88048	—	2	2	no	—
75	#88047	—	2	2	no	—
76	#88047	—	2	2	no	—
77	C₄×#1055	D₄	1	1	yes	characteristic
78	#88038	Q₈	1	4	yes	complemented
79	#88038	Q₈	1	4	yes	complemented
80	#33732	—	13	1	no	maximal, Sylow
81	C₂×C₄×D₁₃	—	4	2	no	—
82	C₂×C₄×D₁₃	D₈	1	2	yes	—
83	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×Dic₂₆	—	4	2	no	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	4	2	no	—
90	C₁₃×C₄⋊C₄	—	4	2	no	—
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	#6103	—	4	2	no	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	—	2	2	no	—
95	C₄⋊Dic₁₃	—	4	2	no	—
96	#6101	—	4	2	no	—
97	#6101	—	4	2	no	—
98	#6101	—	4	2	no	—
99	#6101	—	4	2	no	—
100	#6101	—	4	2	no	—
101	C₄×Dic₁₃	—	4	2	no	—
102	C₄×Dic₁₃	D₈	1	2	yes	—
103	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×#1055	C₂×D₄	1	1	yes	characteristic
105	C₂×#1055	C₂×D₄	1	1	yes	characteristic
106	C₈×D₁₃	Q₁₆	1	4	yes	—
107	C₈×D₁₃	C₂×Q₈	1	2	yes	—
108	C₈×D₁₃	C₂×Q₈	1	2	yes	—
109	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
110	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
111	#6090	C₂×Q₈	1	2	yes	—
112	#6090	C₂×Q₈	1	2	yes	—
113	#3160	—	13	1	no	—
114	C₂×#499	—	13	1	no	—
115	#2255	—	13	2	no	—
116	#2255	—	13	2	no	—
117	#2253	—	13	1	no	—
118	#2253	—	13	1	no	—
119	#2253	—	13	1	no	—
120	#2253	D₁₃	1	1	yes	characteristic, complemented
121	#2159	—	13	2	no	—
122	#2159	—	13	2	no	—
123	C₄×M₃₂	—	13	1	no	—
124	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₁₃	—	4	2	no	—
130	C₂×Dic₁₃	—	4	2	no	—
131	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
132	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
133	C₄×D₁₃	—	4	2	no	—
134	C₄×D₁₃	—	8	4	no	—
135	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
136	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
137	Dic₂₆	—	8	4	no	—
138	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
139	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
140	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
141	#1055	C₄⋊Q₈	1	2	yes	—
142	#1055	C₂×Q₁₆	1	2	yes	—
143	#533	—	26	1	no	—
144	C₂×#125	—	26	1	no	—
145	C₂×M₃₂	—	13	1	no	—
146	C₂×M₃₂	—	13	1	no	—
147	#500	—	13	2	no	—
148	#499	—	13	2	no	—
149	#499	—	13	1	no	—
150	#499	—	13	1	no	—
151	#499	—	26	1	no	—
152	#499	—	26	1	no	—
153	#499	—	26	1	no	—
154	#499	C₂×D₁₃	1	1	yes	characteristic
155	#499	C₂×D₁₃	1	1	yes	characteristic
156	#498	—	26	1	no	—
157	#427	—	13	2	no	—
158	#427	—	26	1	no	—
159	C₂×#107	—	13	2	no	—
160	C₂×#106	—	26	1	no	—
161	C₂×C₄×C₈	—	13	1	no	—
162	#367	—	13	2	no	—
163	#367	—	13	2	no	—
164	#367	—	13	2	no	—
165	#367	C₂×D₁₃	1	2	yes	—
166	#366	—	13	2	no	—
167	#366	—	13	2	no	—
168	#366	—	13	2	no	—
169	#366	C₂×D₁₃	1	2	yes	—
170	#346	—	13	1	no	—
171	#346	—	13	1	no	—
172	C₄×C₁₆	—	13	1	no	—
173	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
174	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₁₃	—	2	2	no	—
176	C₂×D₁₃	#500	1	2	yes	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
182	Dic₁₃	—	2	2	no	—
183	Dic₁₃	—	8	2	no	—
184	Dic₁₃	—	8	2	no	—
185	Dic₁₃	#500	1	2	yes	—
186	C₂²×C₈	—	13	1	no	—
187	C₂²×C₈	—	13	1	no	—
188	C₄⋊Q₈	—	26	2	no	—
189	#125	—	2	1	no	—
190	#125	—	2	1	no	—
191	#125	—	26	2	no	—
192	#125	—	26	1	no	—
193	#125	—	26	1	no	—
194	#125	—	52	2	no	—
195	C₂²⋊Q₈	—	52	2	no	—
196	C₄×Q₈	—	52	2	no	—
197	#117	—	26	2	no	—
198	C₂×C₄⋊C₄	—	26	2	no	—
199	C₂×C₄⋊C₄	—	52	2	no	—
200	C₂×C₄²	—	13	1	no	—
201	M₃₂	—	13	4	no	—
202	M₃₂	—	13	4	no	—
203	C₂×C₁₆	—	13	1	no	—
204	C₂×C₁₆	—	13	1	no	—
205	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
206	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
207	#107	—	13	2	no	—
208	#107	—	13	2	no	—
209	#107	—	26	2	no	—
210	#107	—	26	2	no	—
211	#107	C₂²×D₁₃	1	2	yes	—
212	#107	C₂²×D₁₃	1	2	yes	—
213	#106	—	2	1	no	—
214	#106	—	2	1	no	—
215	#106	—	26	2	no	—
216	#106	—	26	2	no	—
217	#106	—	26	1	no	—
218	#106	—	26	1	no	—
219	C₄×C₈	—	13	1	no	—
220	C₄×C₈	—	13	1	no	—
221	C₄×C₈	—	13	1	no	—
222	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
224	C₂×C₁₃	#3414	1	2	yes	—
225	D₁₃	—	2	2	no	—
226	C₂×Q₈	—	52	2	no	—
227	C₂²×C₄	—	13	1	no	—
228	C₂²×C₄	—	13	2	no	—
229	C₂²×C₄	—	52	2	no	—
230	C₂×C₈	—	13	1	no	—
231	C₂×C₈	—	13	1	no	—
232	C₂×C₈	—	13	4	no	—
233	C₂×C₈	—	13	2	no	—
234	C₂×C₈	—	13	2	no	—
235	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	26	2	no	—
242	C₄⋊C₄	—	26	2	no	—
243	C₄⋊C₄	—	52	2	no	—
244	C₄⋊C₄	—	52	2	no	—
245	C₄⋊C₄	—	52	2	no	—
246	C₄⋊C₄	—	52	2	no	—
247	C₄⋊C₄	—	52	2	no	—
248	C₄⋊C₄	—	52	2	no	—
249	C₂²⋊C₄	—	52	2	no	—
250	C₄²	—	13	2	no	—
251	C₄²	—	13	1	no	—
252	C₄²	—	52	2	no	—
253	C₄²	D₄×D₁₃	1	1	yes	characteristic
254	C₁₆	—	13	2	no	—
255	C₁₆	—	13	2	no	—
256	C₁₆	Q₈×D₁₃	1	2	yes	—
257	C₁₆	Q₈×D₁₃	1	2	yes	—
258	C₁₃	#33732	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	13	1	no	—
260	Q₈	—	104	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	13	2	no	—
265	C₂×C₄	—	13	2	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	52	2	no	—
268	C₂×C₄	—	52	2	no	—
269	C₂×C₄	—	52	2	no	—
270	C₂×C₄	—	104	4	no	—
271	C₂×C₄	D₈×D₁₃	1	2	yes	—
272	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
273	C₈	—	13	2	no	—
274	C₈	—	13	2	no	—
275	C₈	Q₁₆×D₁₃	1	2	yes	—
276	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
277	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
278	C₂²	—	13	2	no	—
279	C₂²	—	26	2	no	—
280	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	13	2	no	—
285	C₄	—	26	2	no	—
286	C₄	—	104	2	no	—
287	C₄	—	104	2	no	—
288	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
290	C₂	—	26	2	no	—
291	C₂	D₁₃×#500	1	1	yes	characteristic
292	C₂	#13127969	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2253	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
hypercenter	#2253	D₁₃
hypocenter	C₁₃	#33732
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
3. #13117121	C₂	semidirect
4. #13117120	C₂	semidirect
5. #13117115	C₂	semidirect
6. #13117112	C₂	semidirect
7. #13117112	C₂	semidirect
11. C₁₃×#2253	C₂	semidirect
78. #88038	Q₈	semidirect
79. #88038	Q₈	semidirect
120. #2253	D₁₃	semidirect
258. C₁₃	#33732	semidirect

Maximal subgroups

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

subgroup	index
#13125349	2
D₁₃×#499	2
#13117121	2
#13117120	2
#13117115	2
#13117112	2
#13116623	2
#13116621	2
C₄×#88038	2
C₁₃×#2253	2
#33732	13