Subgroups — #16018221

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125347	C₂	1	1	yes	maximal, characteristic, complemented
2	#13125265	C₂	1	1	yes	maximal, characteristic
3	#13117069	C₂	1	4	yes	maximal, complemented
4	#13117063	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117061	C₂	1	1	yes	maximal, characteristic, complemented
6	#13117060	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116542	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88173	—	2	1	no	—
11	C₂×#88173	C₂²	1	2	yes	—
12	#11719433	—	2	1	no	—
13	#11719430	—	2	1	no	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#494	C₂²	1	1	yes	characteristic
16	#11718852	—	2	1	no	—
17	#11718852	C₂²	1	2	yes	—
18	#11718746	—	2	1	no	—
19	#11718740	—	2	1	no	—
20	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
21	#11718541	—	2	1	no	—
22	#11718541	C₂²	1	2	yes	—
23	#11718539	—	2	1	no	—
24	#11718539	C₂²	1	2	yes	—
25	#11718356	C₂²	1	1	yes	characteristic
26	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
27	#11718343	C₂²	1	1	yes	characteristic
28	C₁₃×#358	C₂²	1	4	yes	—
29	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
30	#11718156	C₂²	1	4	yes	—
31	#11718101	C₂²	1	4	yes	—
32	C₁₆×Dic₁₃	C₂²	1	2	yes	—
33	#11718083	C₂²	1	4	yes	—
34	C₄×#6090	C₂²	1	1	yes	characteristic
35	C₂×#6131	—	2	2	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
38	C₁₃×C₄⋊Q₈	—	2	1	no	—
39	C₁₃×C₄⋊Q₈	—	2	1	no	—
40	#88201	—	2	2	no	—
41	#88200	—	4	1	no	—
42	Q₈×Dic₁₃	—	4	1	no	—
43	C₂×#6106	—	2	2	no	—
44	C₂×#6106	—	2	2	no	—
45	#88173	—	2	8	no	—
46	#88173	—	4	4	no	—
47	C₂×D₁₀₄	—	2	1	no	—
48	C₂×D₁₀₄	C₂³	1	2	yes	—
49	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
50	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
51	#88151	—	4	1	no	—
52	#88149	—	4	1	no	—
53	#88149	—	4	1	no	—
54	#88147	—	4	1	no	—
55	#88147	—	4	1	no	—
56	#88146	—	2	2	no	—
57	#88144	—	2	2	no	—
58	#88141	—	4	1	no	—
59	#88128	—	2	1	no	—
60	C₄²×D₁₃	D₄	1	1	yes	characteristic
61	#88123	—	2	1	no	—
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₁₃×#107	—	2	1	no	—
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#88058	—	2	1	no	—
67	#88058	C₂³	1	2	yes	—
68	C₈×Dic₁₃	D₄	1	1	yes	characteristic
69	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	2	yes	—
71	#88048	—	2	2	no	—
72	#88048	—	2	2	no	—
73	C₄×#1055	D₄	1	1	yes	characteristic
74	C₁₆×D₁₃	—	2	4	no	—
75	#33697	—	13	1	no	maximal, Sylow
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	4	1	no	—
78	#6131	—	4	4	no	—
79	#6131	—	4	4	no	—
80	C₂×D₅₂	—	2	2	no	—
81	C₂×D₅₂	—	4	2	no	—
82	C₂×D₅₂	—	4	1	no	—
83	C₂×D₅₂	—	4	1	no	—
84	C₂×C₄×D₁₃	—	4	2	no	—
85	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
86	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×Dic₂₆	—	4	1	no	—
89	C₁₃×Q₁₆	—	2	4	no	—
90	C₁₃×Q₁₆	—	4	2	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
98	#6106	—	4	4	no	—
99	#6106	—	4	4	no	—
100	#6103	—	4	2	no	—
101	#6103	—	4	2	no	—
102	#6103	—	4	2	no	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	4	1	no	—
105	C₄⋊Dic₁₃	—	4	2	no	—
106	#6101	—	4	2	no	—
107	C₄×Dic₁₃	—	4	2	no	—
108	C₄×Dic₁₃	D₈	1	1	yes	characteristic
109	C₄×Dic₁₃	D₈	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×#1055	C₂×D₄	1	1	yes	characteristic
112	C₂×#1055	C₂×D₄	1	1	yes	characteristic
113	D₁₀₄	—	2	4	no	—
114	D₁₀₄	—	4	2	no	—
115	C₈×D₁₃	SD₁₆	1	4	yes	—
116	C₈×D₁₃	#41	1	4	yes	—
117	C₁₃×C₁₆	—	2	2	no	—
118	#6090	—	2	2	no	—
119	#3160	—	13	1	no	—
120	#3147	—	13	1	no	—
121	#2242	—	13	4	no	—
122	#2241	—	13	1	no	—
123	#2241	D₁₃	1	1	yes	characteristic, complemented
124	#2240	—	13	1	no	—
125	#2240	—	13	1	no	—
126	#2138	—	13	4	no	—
127	C₂×C₄×C₁₆	—	13	1	no	—
128	C₂²×D₁₃	—	4	2	no	—
129	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
130	Q₈×C₁₃	—	4	2	no	—
131	Q₈×C₁₃	—	4	2	no	—
132	Q₈×C₁₃	—	8	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₁₃	—	4	2	no	—
139	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
142	D₅₂	—	4	2	no	—
143	D₅₂	—	4	2	no	—
144	D₅₂	—	8	4	no	—
145	D₅₂	—	8	2	no	—
146	D₅₂	—	8	2	no	—
147	C₄×D₁₃	—	2	2	no	—
148	C₄×D₁₃	—	2	2	no	—
149	C₄×D₁₃	—	8	4	no	—
150	C₄×D₁₃	#124	1	2	yes	—
151	C₄×D₁₃	C₂×SD₁₆	1	2	yes	—
152	Dic₂₆	—	8	2	no	—
153	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
154	C₈×C₁₃	C₂×#41	1	2	yes	—
155	#1055	#124	1	2	yes	—
156	#1055	C₂×SD₁₆	1	2	yes	—
157	C₂×#135	—	13	2	no	—
158	C₂×#135	—	26	1	no	—
159	#533	—	26	1	no	—
160	#532	—	26	1	no	—
161	C₂²×C₁₆	—	13	2	no	—
162	#500	—	13	1	no	—
163	#500	C₂×D₁₃	1	1	yes	characteristic
164	#499	—	13	2	no	—
165	#499	—	26	1	no	—
166	#498	—	26	1	no	—
167	#495	—	13	2	no	—
168	#495	—	26	1	no	—
169	#494	C₂×D₁₃	1	1	yes	characteristic
170	#493	—	13	1	no	—
171	#492	—	26	1	no	—
172	#427	—	13	2	no	—
173	#427	—	26	1	no	—
174	C₂×C₄×C₈	—	13	1	no	—
175	#358	—	13	4	no	—
176	#358	C₂×D₁₃	1	4	yes	—
177	#357	—	13	4	no	—
178	#357	—	13	4	no	—
179	C₄×C₁₆	—	13	2	no	—
180	C₄×C₁₆	—	13	1	no	—
181	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
182	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
183	C₂×D₁₃	—	2	1	no	—
184	C₂×D₁₃	—	2	1	no	—
185	C₂×D₁₃	—	8	2	no	—
186	C₂×D₁₃	—	8	2	no	—
187	C₂×D₁₃	—	8	2	no	—
188	C₂×D₁₃	#486	1	2	yes	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	#508	1	2	yes	—
192	C₄×C₁₃	#508	1	2	yes	—
193	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
194	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
195	Dic₁₃	—	2	1	no	—
196	Dic₁₃	—	2	1	no	—
197	Dic₁₃	—	8	2	no	—
198	Dic₁₃	#486	1	2	yes	—
199	C₂×#41	—	26	2	no	—
200	#135	—	26	8	no	—
201	#135	—	52	4	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
204	C₂×SD₁₆	—	26	2	no	—
205	C₂×SD₁₆	—	26	2	no	—
206	C₂×D₈	—	13	2	no	—
207	C₂×D₈	—	26	1	no	—
208	C₂²×C₈	—	13	1	no	—
209	C₂²×C₈	—	13	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	26	1	no	—
213	#127	—	26	1	no	—
214	#125	—	26	2	no	—
215	#124	—	26	2	no	—
216	C₂²⋊Q₈	—	52	1	no	—
217	C₂²⋊Q₈	—	52	1	no	—
218	#121	—	52	1	no	—
219	#121	—	52	1	no	—
220	C₄×Q₈	—	52	1	no	—
221	C₄×Q₈	—	52	1	no	—
222	C₄×D₄	—	52	1	no	—
223	C₄×D₄	—	52	1	no	—
224	#117	—	26	2	no	—
225	C₂×C₄²	—	13	1	no	—
226	C₂×C₁₆	—	13	2	no	—
227	C₂×C₁₆	—	26	4	no	—
228	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
229	#107	—	2	1	no	—
230	#107	—	13	2	no	—
231	#107	—	26	1	no	—
232	#107	C₂²×D₁₃	1	2	yes	—
233	#106	—	26	2	no	—
234	#106	—	26	2	no	—
235	C₄×C₈	—	13	1	no	—
236	C₄×C₈	—	13	1	no	—
237	C₄×C₈	—	13	1	no	—
238	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
240	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
241	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
242	D₁₃	—	2	2	no	—
243	D₁₃	—	16	2	no	—
244	#41	—	52	4	no	—
245	#41	—	52	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	52	1	no	—
249	C₂×D₄	—	26	2	no	—
250	C₂×D₄	—	52	2	no	—
251	C₂×D₄	—	52	1	no	—
252	C₂×D₄	—	52	1	no	—
253	C₂²×C₄	—	13	1	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	1	no	—
256	C₂²×C₄	—	52	2	no	—
257	Q₁₆	—	2	4	no	—
258	Q₁₆	—	4	2	no	—
259	SD₁₆	—	52	4	no	—
260	SD₁₆	—	52	4	no	—
261	D₈	—	26	4	no	—
262	D₈	—	52	2	no	—
263	C₂×C₈	—	13	1	no	—
264	C₂×C₈	—	13	1	no	—
265	C₂×C₈	—	13	4	no	—
266	C₂×C₈	—	13	4	no	—
267	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	52	2	no	—
275	C₄⋊C₄	—	52	2	no	—
276	C₄⋊C₄	—	52	1	no	—
277	C₂²⋊C₄	—	52	2	no	—
278	C₂²⋊C₄	—	52	2	no	—
279	C₂²⋊C₄	—	52	2	no	—
280	C₄²	—	13	1	no	—
281	C₄²	—	13	1	no	—
282	C₄²	—	13	1	no	—
283	C₄²	—	52	2	no	—
284	C₄²	D₄×D₁₃	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	26	2	no	—
287	C₁₃	#33697	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	13	1	no	—
289	C₂³	—	52	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	8	2	no	—
293	Q₈	—	104	2	no	—
294	D₄	—	52	2	no	—
295	D₄	—	52	2	no	—
296	D₄	—	104	4	no	—
297	D₄	—	104	2	no	—
298	D₄	—	104	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	13	1	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	13	2	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	13	1	no	—
306	C₂×C₄	—	26	2	no	—
307	C₂×C₄	—	26	2	no	—
308	C₂×C₄	—	52	2	no	—
309	C₂×C₄	—	104	4	no	—
310	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
311	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
313	C₈	—	13	2	no	—
314	C₈	—	13	2	no	—
315	C₈	SD₁₆×D₁₃	1	2	yes	—
316	C₈	#41×D₁₃	1	2	yes	—
317	C₂²	—	13	2	no	—
318	C₂²	—	26	1	no	—
319	C₂²	—	26	1	no	—
320	C₂²	—	104	2	no	—
321	C₂²	—	104	2	no	—
322	C₂²	—	104	2	no	—
323	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	13	2	no	—
327	C₄	—	26	1	no	—
328	C₄	—	26	1	no	—
329	C₄	—	104	2	no	—
330	C₄	#11718577	1	2	yes	—
331	C₄	#11718580	1	2	yes	—
332	C₄	D₁₃×#124	1	1	yes	characteristic
333	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
334	C₂	—	26	2	no	—
335	C₂	—	208	2	no	—
336	C₂	D₁₃×#486	1	1	yes	characteristic
337	C₂	C₂×#11718577	1	1	yes	characteristic
338	C₂	C₂×#11718580	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2241	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13125265	C₂
hypercenter	#2241	D₁₃
hypocenter	C₁₃	#33697
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
1. #13125347	C₂	semidirect
3. #13117069	C₂	semidirect
4. #13117063	C₂	semidirect
5. #13117061	C₂	semidirect
6. #13117060	C₂	semidirect
7. #13116542	C₂	semidirect
8. C₄×C₁₆×D₁₃	C₂	semidirect
9. C₁₃×#2241	C₂	semidirect
123. #2241	D₁₃	semidirect
287. C₁₃	#33697	semidirect

Maximal subgroups

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

subgroup	index
#13125347	2
#13125265	2
#13117069	2
#13117063	2
#13117061	2
#13117060	2
#13116542	2
C₄×C₁₆×D₁₃	2
C₁₃×#2241	2
#33697	13