Subgroups — #16020623

Order 3328.

484 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127846	C₂	1	1	yes	maximal, characteristic
2	#13127725	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117069	C₂	1	2	yes	maximal, complemented
4	#13117060	C₂	1	1	yes	maximal, characteristic
5	#13116996	C₂	1	2	yes	maximal, complemented
6	#13116978	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116900	C₂	1	2	yes	maximal, complemented
8	#13116876	C₂	1	1	yes	maximal, characteristic
9	C₁₆×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116685	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719290	—	2	1	no	—
13	#11719275	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	#11718852	C₂²	1	2	yes	—
18	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
19	C₂×#88068	C₂²	1	2	yes	—
20	#11718688	—	2	1	no	—
21	#11718681	—	2	1	no	—
22	#11718539	C₂²	1	2	yes	—
23	#11718533	C₂²	1	2	yes	—
24	#11718480	—	2	2	no	—
25	#11718457	—	2	2	no	—
26	#11718356	C₂²	1	1	yes	characteristic
27	#11718343	C₂²	1	1	yes	characteristic
28	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
29	C₁₃×#366	C₂²	1	1	yes	characteristic
30	C₁₃×#358	C₂²	1	1	yes	characteristic
31	C₁₃×#358	C₂²	1	1	yes	characteristic
32	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
33	#11718156	C₂²	1	2	yes	—
34	#11718101	C₂²	1	2	yes	—
35	#11718101	C₂²	1	2	yes	—
36	#11718093	C₂²	1	2	yes	—
37	C₁₆×Dic₁₃	C₂²	1	2	yes	—
38	#11718083	C₂²	1	2	yes	—
39	#11718082	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₁₃	—	2	1	no	—
44	C₄×Q₈×C₁₃	—	2	1	no	—
45	#88201	—	2	2	no	—
46	Q₈×Dic₁₃	—	2	2	no	—
47	C₂×#6106	—	2	2	no	—
48	C₂×D₁₀₄	—	2	1	no	—
49	C₂×D₁₀₄	C₂³	1	2	yes	—
50	#88148	—	4	2	no	—
51	#88147	—	2	2	no	—
52	#88144	—	2	2	no	—
53	#88144	—	4	2	no	—
54	#88129	—	4	1	no	—
55	#88128	—	2	1	no	—
56	C₄×D₅₂	—	4	1	no	—
57	#88123	—	2	1	no	—
58	#88123	—	4	1	no	—
59	C₄×Dic₂₆	—	4	1	no	—
60	C₄×Dic₂₆	D₄	1	1	yes	characteristic
61	C₁₃×Q₃₂	—	2	2	no	—
62	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₁₃×#107	C₂³	1	1	yes	characteristic
65	C₁₃×#107	C₂³	1	1	yes	characteristic
66	C₁₃×#103	—	2	1	no	—
67	C₁₃×#103	—	2	1	no	—
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	#88075	—	2	2	no	—
70	#88068	—	2	4	no	—
71	#88058	—	2	1	no	—
72	#88058	C₂³	1	2	yes	—
73	#88054	D₄	1	2	yes	—
74	C₈×Dic₁₃	C₂³	1	2	yes	—
75	C₂×#6090	C₂³	1	2	yes	—
76	#88049	—	2	2	no	—
77	#88048	—	2	2	no	—
78	C₄⋊#1055	D₄	1	1	yes	characteristic
79	#33932	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	2	1	no	—
82	C₂×D₅₂	—	2	2	no	—
83	C₂×D₅₂	—	4	1	no	—
84	C₂×D₅₂	—	4	1	no	—
85	C₂×C₄×D₁₃	—	4	2	no	—
86	C₂×Dic₂₆	—	4	1	no	—
87	C₂×Dic₂₆	D₈	1	1	yes	characteristic
88	C₁₃×Q₁₆	—	2	1	no	—
89	C₁₃×Q₁₆	—	2	1	no	—
90	C₁₃×Q₁₆	—	2	1	no	—
91	C₁₃×Q₁₆	#41	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	1	no	—
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₄²×C₁₃	—	4	1	no	—
99	C₄²×C₁₃	—	4	1	no	—
100	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
101	#6106	—	8	2	no	—
102	#6103	—	4	2	no	—
103	#6103	—	4	2	no	—
104	#6103	—	4	2	no	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	C₄⋊Dic₁₃	—	4	1	no	—
107	C₄⋊Dic₁₃	—	4	2	no	—
108	C₄⋊Dic₁₃	—	4	2	no	—
109	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
110	#6101	—	4	2	no	—
111	#6101	D₈	1	2	yes	—
112	C₄×Dic₁₃	—	4	2	no	—
113	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
114	C₂×#1055	C₂×D₄	1	2	yes	—
115	D₁₀₄	—	2	2	no	—
116	D₁₀₄	—	2	2	no	—
117	D₁₀₄	—	8	1	no	—
118	C₁₃×C₁₆	—	2	1	no	—
119	C₁₃×C₁₆	—	2	1	no	—
120	#6090	—	2	2	no	—
121	#3396	—	13	1	no	—
122	#3383	—	13	1	no	—
123	#2242	—	13	2	no	—
124	#2240	—	13	1	no	—
125	#2228	—	13	2	no	—
126	#2225	—	13	1	no	—
127	#2211	—	13	2	no	—
128	#2207	—	13	1	no	—
129	Q₈×C₁₆	—	13	1	no	—
130	C₄×Q₃₂	D₁₃	1	1	yes	characteristic, complemented
131	C₄×SD₃₂	—	13	2	no	—
132	C₂²×D₁₃	—	4	2	no	—
133	Q₈×C₁₃	—	2	2	no	—
134	Q₈×C₁₃	—	4	1	no	—
135	Q₈×C₁₃	—	4	1	no	—
136	Q₈×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₁₃	—	4	2	no	—
145	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
146	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
147	D₅₂	—	4	2	no	—
148	D₅₂	—	4	2	no	—
149	D₅₂	—	8	2	no	—
150	D₅₂	—	16	1	no	—
151	C₄×D₁₃	—	16	2	no	—
152	Dic₂₆	—	2	2	no	—
153	Dic₂₆	—	8	2	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
157	#1055	—	2	2	no	—
158	#559	—	26	1	no	—
159	#557	—	26	1	no	—
160	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
161	C₂×SD₃₂	—	13	2	no	—
162	#500	—	13	1	no	—
163	#499	—	13	2	no	—
164	#495	—	13	2	no	—
165	#493	—	13	1	no	—
166	#479	—	26	2	no	—
167	#475	—	26	1	no	—
168	#463	—	26	2	no	—
169	#460	—	26	1	no	—
170	C₈×Q₈	—	13	1	no	—
171	C₄×Q₁₆	—	13	2	no	—
172	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
173	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
174	C₄×D₈	—	13	2	no	—
175	#367	—	13	2	no	—
176	#366	C₂×D₁₃	1	1	yes	characteristic
177	C₄⋊C₁₆	—	13	2	no	—
178	C₄⋊C₁₆	—	13	1	no	—
179	#358	—	13	2	no	—
180	#358	C₂×D₁₃	1	1	yes	characteristic
181	#358	C₂×D₁₃	1	1	yes	characteristic
182	#357	—	13	2	no	—
183	#357	—	13	2	no	—
184	#357	—	13	2	no	—
185	C₄×C₁₆	—	13	2	no	—
186	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
187	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
188	C₂×D₁₃	—	8	2	no	—
189	C₂×D₁₃	—	8	2	no	—
190	C₂×D₁₃	—	8	2	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	4	2	no	—
193	C₄×C₁₃	—	8	1	no	—
194	C₄×C₁₃	—	8	1	no	—
195	C₄×C₁₃	—	8	1	no	—
196	C₄×C₁₃	#508	1	2	yes	—
197	C₄×C₁₃	#548	1	1	yes	characteristic
198	C₄×C₁₃	#577	1	1	yes	characteristic
199	Dic₁₃	—	2	2	no	—
200	Dic₁₃	—	2	2	no	—
201	Dic₁₃	—	8	2	no	—
202	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
203	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
204	C₂×SD₁₆	—	26	2	no	—
205	C₂×D₈	—	13	2	no	—
206	C₂×D₈	—	26	1	no	—
207	C₄⋊Q₈	—	26	1	no	—
208	C₄⋊Q₈	—	52	1	no	—
209	#127	—	26	1	no	—
210	#125	—	26	2	no	—
211	#125	—	52	2	no	—
212	#124	—	26	2	no	—
213	#124	—	52	1	no	—
214	#123	—	52	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	13	1	no	—
218	C₄×Q₈	—	26	2	no	—
219	C₄×Q₈	—	52	1	no	—
220	C₄×D₄	—	26	2	no	—
221	C₄×D₄	—	52	1	no	—
222	Q₃₂	—	2	2	no	—
223	SD₃₂	—	26	4	no	—
224	C₂×C₁₆	—	13	2	no	—
225	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
227	#107	—	13	2	no	—
228	#107	—	26	1	no	—
229	#107	C₂²×D₁₃	1	1	yes	characteristic
230	#107	C₂²×D₁₃	1	1	yes	characteristic
231	#106	—	26	2	no	—
232	C₄⋊C₈	—	13	2	no	—
233	C₄⋊C₈	—	13	1	no	—
234	#103	—	2	1	no	—
235	#103	—	2	1	no	—
236	#103	—	26	2	no	—
237	#102	—	26	2	no	—
238	C₄×C₈	—	13	2	no	—
239	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₂×C₁₃	#3358	1	1	yes	characteristic
241	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
242	C₂×C₁₃	#3416	1	1	yes	characteristic
243	D₁₃	—	16	2	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	13	1	no	—
247	C₂×Q₈	—	52	1	no	—
248	C₂×D₄	—	26	2	no	—
249	C₂×D₄	—	52	1	no	—
250	C₂×D₄	—	52	1	no	—
251	C₂²×C₄	—	52	2	no	—
252	Q₁₆	—	2	1	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	#6127	1	2	yes	—
256	SD₁₆	—	104	2	no	—
257	D₈	—	26	2	no	—
258	D₈	—	26	2	no	—
259	D₈	—	104	1	no	—
260	C₂×C₈	—	13	2	no	—
261	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	13	1	no	—
268	C₄⋊C₄	—	13	2	no	—
269	C₄⋊C₄	—	26	2	no	—
270	C₄⋊C₄	—	52	2	no	—
271	C₄⋊C₄	—	52	2	no	—
272	C₄⋊C₄	—	52	2	no	—
273	C₄⋊C₄	—	52	1	no	—
274	C₂²⋊C₄	—	52	2	no	—
275	C₂²⋊C₄	—	52	2	no	—
276	C₂²⋊C₄	—	52	2	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	13	2	no	—
280	C₄²	—	52	2	no	—
281	C₄²	D₄×D₁₃	1	1	yes	characteristic
282	C₁₆	—	2	1	no	—
283	C₁₆	—	2	1	no	—
284	C₁₆	—	26	2	no	—
285	C₁₃	#33932	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	52	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	26	2	no	—
292	Q₈	—	104	2	no	—
293	D₄	—	52	2	no	—
294	D₄	—	52	2	no	—
295	D₄	—	104	2	no	—
296	D₄	—	208	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	52	2	no	—
304	C₂×C₄	—	208	2	no	—
305	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
306	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	26	2	no	—
310	C₈	C₂×#6127	1	1	yes	characteristic
311	C₈	#88257	1	1	yes	characteristic
312	C₂²	—	104	2	no	—
313	C₂²	—	104	2	no	—
314	C₂²	—	104	2	no	—
315	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	1	no	—
321	C₄	—	26	2	no	—
322	C₄	—	26	2	no	—
323	C₄	—	104	2	no	—
324	C₄	#11718580	1	2	yes	—
325	C₄	#11719257	1	1	yes	characteristic
326	C₄	#11719475	1	1	yes	characteristic
327	C₂	—	208	2	no	—
328	C₂	#13127469	1	1	yes	characteristic
329	C₂	C₂×#11718580	1	1	yes	characteristic
330	C₂	#13127979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×Q₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#11718356	C₂²
hypercenter	C₄×Q₃₂	D₁₃
hypocenter	C₁₃	#33932
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #13127725	C₂	semidirect
3. #13117069	C₂	semidirect
5. #13116996	C₂	semidirect
6. #13116978	C₂	semidirect
7. #13116900	C₂	semidirect
9. C₁₆×Dic₂₆	C₂	semidirect
10. #13116685	C₂	semidirect
11. C₄×C₁₃×Q₃₂	C₂	semidirect
130. C₄×Q₃₂	D₁₃	semidirect
285. C₁₃	#33932	semidirect

Maximal subgroups

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

subgroup	index
#13127846	2
#13127725	2
#13117069	2
#13117060	2
#13116996	2
#13116978	2
#13116900	2
#13116876	2
C₁₆×Dic₂₆	2
#13116685	2
C₄×C₁₃×Q₃₂	2
#33932	13