Subgroups — #16018451

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125410	C₂	1	2	yes	maximal, complemented
2	#13117110	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117084	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117058	C₂	1	2	yes	maximal, complemented
5	#13116922	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116920	C₂	1	2	yes	maximal, complemented
7	D₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116682	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116680	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2215	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719301	—	2	2	no	—
13	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#466	C₂²	1	2	yes	—
16	#11718812	C₂²	1	2	yes	—
17	D₈×Dic₁₃	C₂²	1	2	yes	—
18	C₂×#88066	C₂²	1	2	yes	—
19	#11718776	C₂²	1	2	yes	—
20	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂²×#6090	C₂²	1	1	yes	characteristic
22	#11718699	—	2	2	no	—
23	C₂×D₂₀₈	C₂²	1	1	yes	characteristic
24	#11718535	C₂²	1	2	yes	—
25	#11718533	C₂²	1	2	yes	—
26	#11718421	—	2	2	no	—
27	#11718417	—	2	2	no	—
28	C₁₃×#366	C₂²	1	1	yes	characteristic
29	C₁₃×#357	C₂²	1	2	yes	—
30	#11718154	C₂²	1	2	yes	—
31	#11718101	C₂²	1	2	yes	—
32	#11718096	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
37	#11718083	C₂²	1	2	yes	—
38	#11718081	C₂²	1	2	yes	—
39	C₂×#6129	—	4	2	no	—
40	C₂×C₁₃×D₈	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#121	—	2	2	no	—
43	#88194	—	2	2	no	—
44	#88193	—	4	2	no	—
45	D₄×Dic₁₃	—	2	2	no	—
46	C₂×#6104	—	2	2	no	—
47	#88184	—	2	2	no	—
48	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
49	C₂²×#1055	D₄	1	1	yes	characteristic
50	C₂×D₁₀₄	C₂³	1	2	yes	—
51	#88147	—	2	2	no	—
52	#88142	—	2	2	no	—
53	#88139	—	4	2	no	—
54	#88135	—	4	2	no	—
55	C₁₃×D₁₆	—	2	2	no	—
56	C₂×C₁₃×C₁₆	—	2	2	no	—
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#102	—	2	2	no	—
61	#88072	—	2	2	no	—
62	#88066	—	2	4	no	—
63	#88061	—	2	2	no	—
64	#88058	C₂³	1	2	yes	—
65	C₈×Dic₁₃	D₄	1	1	yes	characteristic
66	C₈×Dic₁₃	D₄	1	1	yes	characteristic
67	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
68	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
69	C₂×#6090	—	2	2	no	—
70	C₂×#6090	C₂³	1	1	yes	characteristic
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	#88049	—	2	2	no	—
73	#88047	—	2	2	no	—
74	D₂₀₈	—	2	2	no	—
75	#33733	—	13	1	no	maximal, Sylow
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	4	2	no	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₃	—	4	2	no	—
80	C₂×C₂²⋊D₁₃	—	4	2	no	—
81	C₂×C₂²⋊D₁₃	—	4	2	no	—
82	C₂²×Dic₁₃	—	4	2	no	—
83	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
84	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
85	#6129	—	8	8	no	—
86	C₂×D₅₂	—	2	2	no	—
87	C₂×C₄×D₁₃	—	4	2	no	—
88	C₂×Dic₂₆	—	4	2	no	—
89	C₁₃×D₈	—	2	2	no	—
90	C₁₃×D₈	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₂²⋊C₄	—	4	2	no	—
97	C₂²⋊Dic₁₃	—	4	2	no	—
98	#6104	—	4	4	no	—
99	#6103	—	4	2	no	—
100	#6103	—	4	2	no	—
101	C₄⋊Dic₁₃	—	2	2	no	—
102	#6101	—	4	2	no	—
103	C₄×Dic₁₃	D₈	1	1	yes	characteristic
104	C₄×Dic₁₃	D₈	1	1	yes	characteristic
105	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×#1055	—	2	2	no	—
108	C₂×#1055	C₂×D₄	1	1	yes	characteristic
109	C₂×#1055	C₂×D₄	1	1	yes	characteristic
110	D₁₀₄	—	2	2	no	—
111	D₁₀₄	—	2	2	no	—
112	C₁₃×C₁₆	—	2	1	no	—
113	C₁₃×C₁₆	#41	1	2	yes	complemented
114	#6090	—	2	1	no	—
115	#6090	#41	1	2	yes	complemented
116	#3169	—	13	2	no	—
117	#2252	—	13	1	no	—
118	#2246	—	13	1	no	—
119	#2240	—	13	2	no	—
120	#2215	—	13	1	no	—
121	#2215	—	13	2	no	—
122	#2215	D₁₃	1	1	yes	characteristic, complemented
123	C₄×D₁₆	—	13	1	no	—
124	C₄×D₁₆	—	13	1	no	—
125	C₄×D₁₆	—	13	2	no	—
126	C₂×C₄×C₁₆	—	13	1	no	—
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
129	C₂²×D₁₃	—	4	2	no	—
130	D₄×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	8	4	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂²⋊D₁₃	—	8	4	no	—
139	C₂²⋊D₁₃	—	8	4	no	—
140	C₂²⋊D₁₃	—	8	4	no	—
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	C₂×Dic₁₃	—	8	4	no	—
144	C₂×Dic₁₃	D₁₆	1	4	yes	complemented
145	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
149	D₅₂	—	4	2	no	—
150	D₅₂	—	4	2	no	—
151	C₄×D₁₃	—	8	4	no	—
152	Dic₂₆	—	8	4	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
156	#1055	—	2	1	no	—
157	#1055	#135	1	2	yes	—
158	#532	—	26	2	no	—
159	C₂×D₁₆	—	13	2	no	—
160	C₂×D₁₆	—	13	1	no	—
161	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic
162	C₂²×C₁₆	—	13	1	no	—
163	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
164	#500	—	13	2	no	—
165	#493	—	13	2	no	—
166	#486	—	26	2	no	—
167	#466	—	13	2	no	—
168	#466	—	26	2	no	—
169	#466	C₂×D₁₃	1	2	yes	—
170	C₄×D₈	—	13	2	no	—
171	C₄×D₈	—	13	2	no	—
172	C₄×D₈	—	26	2	no	—
173	C₂×C₄×C₈	—	13	1	no	—
174	#366	—	13	1	no	—
175	#366	—	13	2	no	—
176	#366	C₂×D₁₃	1	1	yes	characteristic
177	#357	—	13	2	no	—
178	#357	—	13	2	no	—
179	#357	—	13	2	no	—
180	#357	C₂×D₁₃	1	2	yes	—
181	C₄×C₁₆	—	13	1	no	—
182	C₄×C₁₆	—	13	1	no	—
183	C₄×C₁₆	—	13	1	no	—
184	C₄×C₁₆	—	13	1	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	8	2	no	—
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
191	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
192	C₂×D₁₃	—	8	2	no	—
193	C₂×D₁₃	—	8	2	no	—
194	C₂×D₁₃	—	8	2	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	#532	1	1	yes	characteristic
198	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
199	Dic₁₃	—	2	1	no	—
200	Dic₁₃	—	8	2	no	—
201	Dic₁₃	C₂×D₁₆	1	2	yes	—
202	Dic₁₃	C₂×D₁₆	1	2	yes	—
203	Dic₁₃	#508	1	2	yes	—
204	C₂×#41	—	52	2	no	—
205	C₂×D₈	—	13	2	no	—
206	C₂×D₈	—	26	2	no	—
207	C₂×D₈	C₂²×D₁₃	1	2	yes	—
208	C₂²×C₈	—	13	1	no	—
209	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
210	C₄⋊Q₈	—	26	2	no	—
211	#127	—	26	2	no	—
212	#124	—	52	2	no	—
213	#121	—	2	2	no	—
214	#121	—	26	2	no	—
215	#121	—	52	2	no	—
216	C₄×D₄	—	26	2	no	—
217	C₄×D₄	—	26	2	no	—
218	C₄×D₄	—	52	2	no	—
219	C₂×C₄²	—	13	1	no	—
220	D₁₆	—	2	2	no	—
221	D₁₆	—	26	4	no	—
222	D₁₆	—	26	2	no	—
223	C₂×C₁₆	—	2	2	no	—
224	C₂×C₁₆	—	13	1	no	—
225	C₂×C₁₆	—	13	1	no	—
226	C₂×C₁₆	—	26	2	no	—
227	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
228	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
229	#107	—	13	2	no	—
230	#107	—	26	2	no	—
231	#107	C₂²×D₁₃	1	2	yes	—
232	#102	—	2	2	no	—
233	#102	—	26	2	no	—
234	#102	—	26	2	no	—
235	#102	—	26	2	no	—
236	C₄×C₈	—	13	1	no	—
237	C₄×C₈	—	13	1	no	—
238	C₄×C₈	—	13	1	no	—
239	C₄×C₈	—	13	1	no	—
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	—	16	2	no	—
242	C₂×C₁₃	#3169	1	1	yes	characteristic
243	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
244	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
245	D₁₃	—	16	2	no	—
246	#41	—	104	8	no	—
247	C₂×Q₈	—	52	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	26	2	no	—
251	C₂×D₄	—	52	2	no	—
252	C₂×D₄	—	52	2	no	—
253	C₂×D₄	—	52	2	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	1	no	—
256	C₂²×C₄	—	52	2	no	—
257	C₂²×C₄	—	52	2	no	—
258	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
259	D₈	—	2	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	26	2	no	—
262	D₈	—	26	2	no	—
263	D₈	—	52	4	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	13	1	no	—
267	C₂×C₈	—	13	1	no	—
268	C₂×C₈	—	26	2	no	—
269	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	26	2	no	—
273	C₄⋊C₄	—	52	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	52	2	no	—
276	C₂²⋊C₄	—	52	2	no	—
277	C₂²⋊C₄	—	52	2	no	—
278	C₄²	—	13	1	no	—
279	C₄²	—	13	1	no	—
280	C₄²	—	13	1	no	—
281	C₄²	—	13	1	no	—
282	C₁₆	—	2	1	no	—
283	C₁₆	—	13	2	no	—
284	C₁₆	—	26	1	no	—
285	C₁₆	#6129	1	2	yes	complemented
286	C₁₃	#33733	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	4	2	no	—
288	C₂³	—	52	2	no	—
289	C₂³	D₈×D₁₃	1	1	yes	characteristic
290	Q₈	—	104	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	4	no	—
294	D₄	—	52	2	no	—
295	D₄	—	52	2	no	—
296	D₄	—	104	4	no	—
297	D₄	—	104	4	no	—
298	D₄	—	104	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	13	1	no	—
303	C₂×C₄	—	13	1	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	13	4	no	—
306	C₂×C₄	—	13	1	no	—
307	C₂×C₄	—	26	2	no	—
308	C₂×C₄	—	52	2	no	—
309	C₂×C₄	—	104	4	no	—
310	C₂×C₄	—	104	4	no	—
311	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	13	2	no	—
315	C₈	—	26	1	no	—
316	C₈	C₂×#6129	1	1	yes	characteristic
317	C₈	#41×D₁₃	1	1	yes	characteristic
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	104	2	no	—
324	C₂²	—	104	2	no	—
325	C₂²	—	104	2	no	—
326	C₂²	D₁₃×D₁₆	1	2	yes	—
327	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	13	2	no	—
331	C₄	—	13	2	no	—
332	C₄	—	13	2	no	—
333	C₄	—	26	1	no	—
334	C₄	—	104	2	no	—
335	C₄	#11719301	1	1	yes	characteristic
336	C₄	D₁₃×#135	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	16	2	no	—
339	C₂	—	208	2	no	—
340	C₂	#13125410	1	1	yes	characteristic
341	C₂	C₂×D₁₃×D₁₆	1	1	yes	characteristic
342	C₂	D₁₃×#508	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×#2215	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#2215	D₁₃
hypocenter	C₁₃	#33733
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2215

Complements

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

N	H ≅ G/N	type
1. #13125410	C₂	semidirect
2. #13117110	C₂	semidirect
3. #13117084	C₂	semidirect
4. #13117058	C₂	semidirect
5. #13116922	C₂	semidirect
6. #13116920	C₂	semidirect
7. D₁₆×Dic₁₃	C₂	semidirect
8. #13116682	C₂	semidirect
9. #13116680	C₂	semidirect
10. C₂×C₁₆×Dic₁₃	C₂	semidirect
11. C₁₃×#2215	C₂	semidirect
113. C₁₃×C₁₆	#41	semidirect
115. #6090	#41	semidirect
122. #2215	D₁₃	semidirect
144. C₂×Dic₁₃	D₁₆	semidirect
285. C₁₆	#6129	semidirect
286. C₁₃	#33733	semidirect

Maximal subgroups

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

subgroup	index
#13125410	2
#13117110	2
#13117084	2
#13117058	2
#13116922	2
#13116920	2
D₁₆×Dic₁₃	2
#13116682	2
#13116680	2
C₂×C₁₆×Dic₁₃	2
C₁₃×#2215	2
#33733	13