Subgroups — #16807217

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131613	C₂	1	2	yes	maximal, complemented
2	#13131238	C₂	1	4	yes	maximal, complemented
3	#13131237	C₂	1	2	yes	maximal, complemented
4	#13131235	C₂	1	2	yes	maximal, complemented
5	C₂×#11718223	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131150	C₂	1	2	yes	maximal, complemented
7	#13131044	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃×#508	C₂	1	1	yes	maximal, characteristic
9	#11719629	—	2	2	no	—
10	D₁₃×#135	C₂²	1	2	yes	—
11	#11719166	—	2	2	no	—
12	#11719111	C₂²	1	4	yes	complemented
13	#11719108	—	2	4	no	—
14	#11719103	C₂²	1	4	yes	complemented
15	C₂×#88104	C₂²	1	2	yes	—
16	#11719089	C₂²	1	1	yes	characteristic
17	D₁₃⋊M₃₂	C₂²	1	1	yes	characteristic
18	C₁₃×#508	C₄	1	1	yes	characteristic, complemented
19	#11718855	C₄	1	2	yes	complemented
20	#11718580	C₂²	1	1	yes	characteristic
21	D₁₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
22	#11718577	C₂²	1	2	yes	—
23	D₁₃×SD₃₂	C₄	1	2	yes	complemented
24	#11718573	C₂²	1	1	yes	characteristic
25	D₁₃×D₁₆	C₄	1	1	yes	characteristic, complemented
26	#11718565	C₄	1	1	yes	characteristic, complemented
27	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
28	#11718281	C₂²	1	4	yes	complemented
29	#11718277	C₂²	1	4	yes	complemented
30	#11718223	C₂²	1	2	yes	complemented
31	#11718223	C₂²	1	2	yes	complemented
32	#11718216	C₂²	1	2	yes	complemented
33	C₁₆×#261	C₂²	1	2	yes	complemented
34	#41×D₁₃	—	2	2	no	—
35	#88240	—	2	4	no	—
36	#88238	—	2	4	no	—
37	#88238	—	4	4	no	—
38	#88237	D₄	1	1	yes	characteristic
39	D₁₃⋊M₁₆	—	4	2	no	—
40	D₁₃⋊M₁₆	D₄	1	1	yes	characteristic
41	C₂×#6118	—	2	2	no	—
42	C₂×#6118	—	4	2	no	—
43	C₁₃×#135	C₂×C₄	1	2	yes	—
44	#88204	—	2	2	no	—
45	#88173	C₂³	1	2	yes	—
46	Q₁₆×D₁₃	C₂×C₄	1	2	yes	—
47	#88170	—	2	2	no	—
48	SD₁₆×D₁₃	—	2	2	no	—
49	#88166	C₂³	1	2	yes	—
50	D₈×D₁₃	C₂×C₄	1	2	yes	—
51	#88158	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
53	#88118	—	2	4	no	—
54	#88116	—	2	4	no	—
55	#88106	—	2	1	no	—
56	#88104	—	2	2	no	—
57	#88104	C₂³	1	4	yes	—
58	#88100	D₄	1	2	yes	—
59	C₈×#261	C₂³	1	2	yes	—
60	#88098	—	2	1	no	—
61	D₁₃⋊C₁₆	C₂³	1	2	yes	—
62	C₁₃×Q₃₂	C₂×C₄	1	1	yes	characteristic
63	C₁₃×SD₃₂	C₂×C₄	1	2	yes	—
64	C₁₃×D₁₆	C₂×C₄	1	1	yes	characteristic
65	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
66	#88069	C₂×C₄	1	2	yes	—
67	#88068	C₂×C₄	1	2	yes	—
68	#88067	C₂×C₄	1	2	yes	—
69	#88066	C₂×C₄	1	2	yes	—
70	C₂×#6090	C₂×C₄	1	1	yes	characteristic
71	Dic₁₀₄	C₂×C₄	1	1	yes	characteristic
72	#88040	C₂×C₄	1	2	yes	—
73	D₂₀₈	C₂×C₄	1	1	yes	characteristic
74	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
75	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
76	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
77	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
78	#33465	—	13	1	no	maximal, Sylow
79	C₁₃×#41	—	2	2	no	—
80	#6131	—	2	2	no	—
81	Q₈×D₁₃	—	2	2	no	—
82	#6129	—	2	2	no	—
83	#6129	—	4	2	no	—
84	D₄×D₁₃	—	2	2	no	—
85	D₄×D₁₃	—	4	2	no	—
86	#6127	—	2	2	no	—
87	#6127	—	4	2	no	—
88	C₂×C₄×D₁₃	—	4	2	no	—
89	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²⋊#261	—	2	1	no	—
91	#6122	—	4	4	no	—
92	#6122	D₈	1	2	yes	—
93	C₂×#1057	—	4	2	no	—
94	C₂×#1057	—	4	4	no	—
95	C₄⋊#261	—	2	1	no	—
96	C₄×#261	C₂×D₄	1	2	yes	—
97	#6118	—	2	4	no	—
98	#6118	—	4	2	no	—
99	#6118	—	4	4	no	—
100	#6118	D₈	1	1	yes	characteristic
101	#6118	D₈	1	1	yes	characteristic
102	D₁₃⋊C₈	—	4	4	no	—
103	D₁₃⋊C₈	C₂×D₄	1	2	yes	—
104	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
105	C₁₃×SD₁₆	—	2	2	no	—
106	C₁₃×D₈	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	#6107	—	2	2	no	—
109	#6106	—	2	2	no	—
110	#6105	—	2	2	no	—
111	#6104	—	2	2	no	—
112	C₂×#1055	#41	1	1	yes	characteristic
113	Dic₅₂	C₂²×C₄	1	2	yes	—
114	D₁₀₄	C₂²×C₄	1	2	yes	—
115	#6095	—	2	2	no	—
116	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
118	C₈×D₁₃	#41	1	1	yes	characteristic
119	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
120	#6092	—	2	2	no	—
121	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
122	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
123	#6090	C₂²×C₄	1	1	yes	characteristic
124	#6090	C₂²×C₄	1	1	yes	characteristic
125	C₂×#508	—	13	1	no	—
126	#2957	—	13	2	no	—
127	#2179	—	13	4	no	—
128	#2178	—	13	2	no	—
129	#2178	—	13	2	no	—
130	C₂×#368	—	13	1	no	—
131	#2143	—	13	2	no	—
132	#2108	—	13	1	no	—
133	C₂²×D₁₃	—	4	2	no	—
134	C₂²×D₁₃	#135	1	1	yes	characteristic
135	C₂×#261	—	2	2	no	—
136	Q₈×C₁₃	—	2	2	no	—
137	D₄×C₁₃	—	2	2	no	—
138	D₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
141	C₂²⋊D₁₃	—	4	2	no	—
142	C₂²⋊D₁₃	—	4	2	no	—
143	C₂²⋊D₁₃	—	4	2	no	—
144	C₂×Dic₁₃	—	4	2	no	—
145	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
146	D₅₂	—	2	2	no	—
147	D₅₂	—	4	2	no	—
148	C₄×D₁₃	—	4	2	no	—
149	C₄×D₁₃	—	4	2	no	—
150	C₄×D₁₃	—	4	2	no	—
151	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
152	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
153	C₄×D₁₃	#135	1	1	yes	characteristic
154	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
155	Dic₂₆	—	2	2	no	—
156	Dic₂₆	—	4	2	no	—
157	#1057	—	4	4	no	—
158	#1057	C₂×D₈	1	2	yes	—
159	#1057	C₂×D₈	1	2	yes	—
160	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
161	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
162	#1055	C₄×D₄	1	1	yes	characteristic
163	#1055	C₂×#41	1	1	yes	characteristic
164	C₂×#135	—	13	2	no	—
165	#568	—	26	2	no	—
166	#508	—	13	2	no	—
167	#508	—	13	1	no	—
168	#508	—	13	2	no	—
169	#508	—	13	1	no	—
170	#508	—	13	1	no	—
171	#508	#261	1	1	yes	characteristic, complemented
172	C₂×Q₃₂	—	13	1	no	—
173	C₂×SD₃₂	—	13	2	no	—
174	C₂×D₁₆	—	13	1	no	—
175	C₂×M₃₂	—	13	1	no	—
176	C₂²×C₁₆	—	13	1	no	—
177	#444	—	26	4	no	—
178	#443	—	13	4	no	—
179	#443	—	13	4	no	—
180	C₂×#108	—	13	2	no	—
181	#421	—	26	2	no	—
182	#405	—	13	1	no	—
183	#368	—	13	2	no	—
184	#368	—	13	2	no	—
185	#359	—	13	4	no	—
186	#359	—	13	4	no	—
187	#346	—	13	2	no	—
188	C₄×C₁₆	—	13	2	no	—
189	C₂²×C₁₃	—	4	2	no	—
190	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
191	C₂×D₁₃	—	4	1	no	—
192	C₂×D₁₃	—	4	2	no	—
193	C₂×D₁₃	—	16	2	no	—
194	C₂×D₁₃	C₄×D₈	1	1	yes	characteristic
195	C₂×D₁₃	C₂×#135	1	1	yes	characteristic
196	#261	—	4	2	no	—
197	C₄×C₁₃	—	4	2	no	—
198	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
199	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
200	Dic₁₃	—	4	2	no	—
201	Dic₁₃	C₄×D₈	1	1	yes	characteristic
202	Dic₁₃	C₂²×D₈	1	1	yes	characteristic
203	C₂×#41	—	26	2	no	—
204	#135	—	13	2	no	—
205	#135	—	13	2	no	—
206	#135	—	13	2	no	—
207	#135	—	26	2	no	—
208	#135	—	26	2	no	—
209	#135	C₂×#261	1	2	yes	—
210	C₂×Q₁₆	—	13	2	no	—
211	C₂×SD₁₆	—	26	2	no	—
212	C₂×D₈	—	13	2	no	—
213	#131	—	26	4	no	—
214	#131	—	26	4	no	—
215	#131	—	52	4	no	—
216	C₂×M₁₆	—	13	1	no	—
217	C₂×M₁₆	—	26	2	no	—
218	C₂×M₁₆	—	52	2	no	—
219	C₂×M₁₆	—	52	2	no	—
220	C₂²×C₈	—	13	1	no	—
221	#117	—	13	1	no	—
222	Q₃₂	—	13	2	no	—
223	Q₃₂	—	13	1	no	—
224	Q₃₂	C₂×#261	1	1	yes	characteristic
225	SD₃₂	—	13	2	no	—
226	SD₃₂	—	13	2	no	—
227	SD₃₂	—	13	2	no	—
228	SD₃₂	C₂×#261	1	2	yes	—
229	D₁₆	—	13	2	no	—
230	D₁₆	—	13	1	no	—
231	D₁₆	C₂×#261	1	1	yes	characteristic
232	M₃₂	—	26	1	no	—
233	M₃₂	—	26	1	no	—
234	C₂×C₁₆	—	13	1	no	—
235	C₂×C₁₆	—	13	1	no	—
236	C₂×C₁₆	—	13	1	no	—
237	C₂×C₁₆	—	13	1	no	—
238	C₂×C₁₆	—	13	1	no	—
239	C₂×C₁₆	—	13	2	no	—
240	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
241	#108	—	13	4	no	—
242	#108	—	26	2	no	—
243	#104	—	26	4	no	—
244	#104	—	26	4	no	—
245	#97	—	13	2	no	—
246	C₄×C₈	—	13	2	no	—
247	C₂×C₁₃	—	2	1	no	—
248	C₂×C₁₃	—	8	2	no	—
249	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
250	D₁₃	—	2	1	no	—
251	D₁₃	—	2	1	no	—
252	D₁₃	—	8	2	no	—
253	#41	—	2	2	no	—
254	#41	—	26	2	no	—
255	#41	—	26	2	no	—
256	#41	—	26	2	no	—
257	#41	—	52	2	no	—
258	#41	—	52	2	no	—
259	C₂×Q₈	—	26	2	no	—
260	C₂×D₄	—	26	2	no	—
261	C₂×D₄	—	52	2	no	—
262	C₂²×C₄	—	13	1	no	—
263	C₂²×C₄	—	52	2	no	—
264	Q₁₆	—	13	2	no	—
265	Q₁₆	—	26	2	no	—
266	Q₁₆	C₂²×#261	1	2	yes	—
267	SD₁₆	—	2	2	no	—
268	SD₁₆	—	26	2	no	—
269	SD₁₆	—	26	2	no	—
270	SD₁₆	—	26	2	no	—
271	D₈	—	13	2	no	—
272	D₈	—	26	2	no	—
273	D₈	C₂²×#261	1	2	yes	—
274	M₁₆	—	13	1	no	—
275	M₁₆	—	13	1	no	—
276	M₁₆	—	13	2	no	—
277	M₁₆	—	26	4	no	—
278	M₁₆	—	52	4	no	—
279	M₁₆	—	52	4	no	—
280	M₁₆	—	52	2	no	—
281	C₂×C₈	—	13	2	no	—
282	C₂×C₈	—	13	1	no	—
283	C₂×C₈	—	13	1	no	—
284	C₂×C₈	—	13	1	no	—
285	C₂×C₈	—	13	1	no	—
286	C₂×C₈	—	13	1	no	—
287	C₂×C₈	—	52	4	no	—
288	C₂×C₈	—	52	4	no	—
289	C₂×C₈	—	52	2	no	—
290	C₂×C₈	C₂²×#261	1	1	yes	characteristic
291	C₄⋊C₄	—	26	1	no	—
292	C₂²⋊C₄	—	26	1	no	—
293	C₄²	—	13	2	no	—
294	C₁₆	—	13	1	no	—
295	C₁₆	—	13	1	no	—
296	C₁₆	—	26	2	no	—
297	C₁₆	C₂²×#261	1	1	yes	characteristic
298	C₁₆	C₂²×#261	1	1	yes	characteristic
299	C₁₃	#33465	1	1	yes	characteristic, complemented, Sylow
300	C₂³	—	13	1	no	—
301	C₂³	—	52	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	26	2	no	—
304	Q₈	—	52	2	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	26	2	no	—
308	D₄	—	52	2	no	—
309	D₄	—	52	2	no	—
310	D₄	—	52	2	no	—
311	D₄	—	52	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	13	1	no	—
314	C₂×C₄	—	13	1	no	—
315	C₂×C₄	—	13	1	no	—
316	C₂×C₄	—	13	1	no	—
317	C₂×C₄	—	13	1	no	—
318	C₂×C₄	—	26	2	no	—
319	C₂×C₄	—	52	2	no	—
320	C₂×C₄	—	52	2	no	—
321	C₂×C₄	—	52	2	no	—
322	C₂×C₄	—	52	2	no	—
323	C₂×C₄	D₄×#261	1	1	yes	characteristic
324	C₈	—	13	2	no	—
325	C₈	—	13	2	no	—
326	C₈	—	13	1	no	—
327	C₈	—	13	1	no	—
328	C₈	—	52	4	no	—
329	C₈	D₄×#261	1	1	yes	characteristic
330	C₈	C₂³×#261	1	1	yes	characteristic
331	C₂²	—	4	2	no	—
332	C₂²	—	13	1	no	—
333	C₂²	—	13	1	no	—
334	C₂²	—	52	1	no	—
335	C₂²	—	52	2	no	—
336	C₂²	—	208	2	no	—
337	C₂²	D₈×#261	1	1	yes	characteristic
338	C₄	—	4	2	no	—
339	C₄	—	13	1	no	—
340	C₄	—	13	1	no	—
341	C₄	—	52	2	no	—
342	C₄	—	52	2	no	—
343	C₄	D₈×#261	1	1	yes	characteristic
344	C₄	C₂×D₄×#261	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	8	2	no	—
347	C₂	—	26	1	no	—
348	C₂	—	26	1	no	—
349	C₂	—	104	2	no	—
350	C₂	C₂×D₈×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#261
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₁₃×#508	C₄
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	D₁₃×#508	C₂
cosocle	C₈×D₁₃	C₂⁴
hypercenter	#508	#261
hypocenter	C₁₃	#33465
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #13131613	C₂	semidirect
2. #13131238	C₂	semidirect
3. #13131237	C₂	semidirect
4. #13131235	C₂	semidirect
5. C₂×#11718223	C₂	semidirect
6. #13131150	C₂	semidirect
7. #13131044	C₂	semidirect
12. #11719111	C₂²	semidirect
14. #11719103	C₂²	semidirect
18. C₁₃×#508	C₄	semidirect
19. #11718855	C₄	semidirect
21. D₁₃×Q₃₂	C₄	semidirect
23. D₁₃×SD₃₂	C₄	semidirect
25. D₁₃×D₁₆	C₄	semidirect
26. #11718565	C₄	semidirect
28. #11718281	C₂²	semidirect
29. #11718277	C₂²	semidirect
30. #11718223	C₂²	semidirect
31. #11718223	C₂²	semidirect
32. #11718216	C₂²	semidirect
33. C₁₆×#261	C₂²	semidirect
171. #508	#261	semidirect
299. C₁₃	#33465	semidirect

Maximal subgroups

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

subgroup	index
#13131613	2
#13131238	2
#13131237	2
#13131235	2
C₂×#11718223	2
#13131150	2
#13131044	2
D₁₃×#508	2
#33465	13