Subgroups — #16807199

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131595	C₂	1	2	yes	maximal, complemented
2	#13131234	C₂	1	4	yes	maximal, complemented
3	D₁₆×#261	C₂	1	4	yes	maximal, complemented
4	#13131187	C₂	1	1	yes	maximal, characteristic, complemented
5	#13131145	C₂	1	2	yes	maximal, complemented
6	#13131044	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
8	#11719623	—	2	2	no	—
9	C₂×D₈×D₁₃	C₂²	1	2	yes	—
10	#11719138	—	2	2	no	—
11	#11719102	—	2	4	no	—
12	D₈×#261	C₂²	1	8	yes	complemented
13	#11719092	C₂²	1	2	yes	—
14	#11719089	C₂²	1	1	yes	characteristic
15	D₁₃⋊M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×D₁₆	C₄	1	1	yes	characteristic, complemented
17	C₂×#88066	C₄	1	2	yes	complemented
18	D₁₃×D₁₆	C₄	1	2	yes	complemented
19	D₁₃×D₁₆	C₄	1	2	yes	complemented
20	D₁₃×D₁₆	C₂²	1	2	yes	—
21	D₁₃×D₁₆	C₂²	1	2	yes	—
22	C₂×D₂₀₈	C₄	1	1	yes	characteristic, complemented
23	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
24	#11718276	C₂²	1	8	yes	complemented
25	#11718222	C₂²	1	2	yes	complemented
26	#11718221	C₂²	1	2	yes	complemented
27	#11718216	C₂²	1	2	yes	complemented
28	C₁₆×#261	C₂²	1	2	yes	complemented
29	C₂×D₄×D₁₃	—	2	2	no	—
30	C₂×C₂²⋊#261	—	4	4	no	—
31	D₄×#261	—	2	8	no	—
32	D₄×#261	—	4	4	no	—
33	#88237	—	2	2	no	—
34	#88237	D₄	1	1	yes	characteristic
35	D₁₃⋊M₁₆	D₄	1	1	yes	characteristic
36	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	complemented
37	C₂×#6104	—	2	2	no	—
38	D₈×D₁₃	—	2	2	no	—
39	D₈×D₁₃	—	2	2	no	—
40	D₈×D₁₃	C₂×C₄	1	4	yes	complemented
41	D₈×D₁₃	C₂³	1	4	yes	—
42	C₂×D₁₀₄	C₂×C₄	1	2	yes	complemented
43	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
44	#88115	—	2	8	no	—
45	#88106	—	2	1	no	—
46	#88102	C₂³	1	4	yes	—
47	#88101	—	2	2	no	—
48	#88100	D₄	1	2	yes	—
49	C₈×#261	C₂³	1	2	yes	—
50	#88098	—	2	1	no	—
51	D₁₃⋊C₁₆	C₂³	1	2	yes	—
52	C₁₃×D₁₆	C₂×C₄	1	2	yes	complemented
53	C₁₃×D₁₆	C₂×C₄	1	2	yes	complemented
54	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic, complemented
55	#88066	C₂×C₄	1	4	yes	complemented
56	#88066	C₂×C₄	1	4	yes	complemented
57	C₂×#6090	C₂×C₄	1	1	yes	characteristic, complemented
58	D₂₀₈	C₂×C₄	1	2	yes	complemented
59	D₂₀₈	C₂×C₄	1	2	yes	complemented
60	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
61	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
62	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
63	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
64	#33456	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	—	4	2	no	—
66	C₂²×#261	—	4	8	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×C₂²⋊D₁₃	—	4	2	no	—
69	D₄×D₁₃	—	2	4	no	—
70	D₄×D₁₃	—	2	4	no	—
71	D₄×D₁₃	—	4	2	no	—
72	D₄×D₁₃	—	4	2	no	—
73	C₂×D₅₂	—	2	2	no	—
74	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²⋊#261	—	4	2	no	—
76	C₂²⋊#261	—	4	8	no	—
77	C₂²⋊#261	D₈	1	2	yes	—
78	#6122	—	2	1	no	—
79	C₄⋊#261	—	2	4	no	—
80	C₄⋊#261	D₈	1	2	yes	—
81	C₄×#261	—	4	2	no	—
82	C₄×#261	C₂×D₄	1	2	yes	—
83	#6118	—	2	1	no	—
84	D₁₃⋊C₈	C₂×D₄	1	2	yes	—
85	C₁₃×D₈	—	2	2	no	—
86	C₁₃×D₈	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	#6104	—	2	4	no	—
89	#6104	—	2	4	no	—
90	C₂×#1055	#41	1	1	yes	characteristic
91	D₁₀₄	—	2	2	no	—
92	D₁₀₄	C₂²×C₄	1	4	yes	—
93	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
95	C₈×D₁₃	#41	1	1	yes	characteristic
96	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
97	#6092	—	2	2	no	—
98	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
99	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
100	#6090	C₂²×C₄	1	1	yes	characteristic
101	#6090	C₂²×C₄	1	1	yes	characteristic
102	C₂²×D₁₆	—	13	1	no	—
103	#2948	—	13	2	no	—
104	#2177	—	13	4	no	—
105	C₄×D₁₆	—	13	4	no	—
106	#2158	—	13	1	no	—
107	#2140	—	13	2	no	—
108	#2108	—	13	1	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂²×D₁₃	—	4	2	no	—
111	C₂²×D₁₃	—	4	4	no	—
112	C₂²×D₁₃	—	4	4	no	—
113	C₂²×D₁₃	—	16	4	no	—
114	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂×#261	—	4	4	no	—
116	C₂×#261	—	16	8	no	—
117	C₂×#261	C₂×D₈	1	4	yes	—
118	D₄×C₁₃	—	2	4	no	—
119	D₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂²⋊D₁₃	—	4	4	no	—
122	C₂²⋊D₁₃	—	4	4	no	—
123	C₂×Dic₁₃	#135	1	1	yes	characteristic
124	D₅₂	—	2	4	no	—
125	D₅₂	—	4	2	no	—
126	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
127	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
128	C₄×D₁₃	#135	1	1	yes	characteristic
129	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
130	#1057	—	2	2	no	—
131	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	#1055	C₄×D₄	1	1	yes	characteristic
134	#1055	C₂×#41	1	1	yes	characteristic
135	C₂²×D₈	—	13	2	no	—
136	#518	—	26	2	no	—
137	C₂×D₁₆	—	13	2	no	—
138	C₂×D₁₆	—	13	2	no	—
139	C₂×D₁₆	—	13	2	no	—
140	C₂×D₁₆	—	13	2	no	—
141	C₂×D₁₆	—	13	2	no	—
142	C₂×D₁₆	—	13	1	no	—
143	C₂×D₁₆	#261	1	1	yes	characteristic, complemented
144	C₂×M₃₂	—	13	1	no	—
145	C₂²×C₁₆	—	13	1	no	—
146	#442	—	26	4	no	—
147	C₄×D₈	—	13	8	no	—
148	#427	—	13	2	no	—
149	#418	—	26	2	no	—
150	#405	—	13	1	no	—
151	#367	—	13	2	no	—
152	#366	—	13	2	no	—
153	#357	—	13	8	no	—
154	#346	—	13	2	no	—
155	C₄×C₁₆	—	13	2	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	—	16	2	no	—
158	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
159	C₂×D₁₃	—	2	2	no	—
160	C₂×D₁₃	—	4	4	no	—
161	C₂×D₁₃	—	8	8	no	—
162	C₂×D₁₃	—	16	4	no	—
163	C₂×D₁₃	—	16	2	no	—
164	C₂×D₁₃	C₄×D₈	1	1	yes	characteristic
165	C₂×D₁₃	C₂²×D₈	1	1	yes	characteristic
166	#261	—	2	4	no	—
167	#261	—	8	4	no	—
168	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
169	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
170	Dic₁₃	C₄×D₈	1	1	yes	characteristic
171	Dic₁₃	C₂×#135	1	1	yes	characteristic
172	C₂²×D₄	—	26	2	no	—
173	C₂×D₈	—	13	4	no	—
174	C₂×D₈	—	13	4	no	—
175	C₂×D₈	—	13	2	no	—
176	C₂×D₈	—	26	2	no	—
177	C₂×D₈	—	26	2	no	—
178	C₂×D₈	—	26	2	no	—
179	C₂×D₈	C₂×#261	1	2	yes	complemented
180	C₂×M₁₆	—	13	1	no	—
181	C₂²×C₈	—	13	1	no	—
182	C₄×D₄	—	26	8	no	—
183	C₄×D₄	—	52	4	no	—
184	#117	—	13	1	no	—
185	#117	—	26	2	no	—
186	C₂×C₂²⋊C₄	—	52	4	no	—
187	D₁₆	—	13	4	no	—
188	D₁₆	—	13	4	no	—
189	D₁₆	—	13	2	no	—
190	D₁₆	—	13	2	no	—
191	D₁₆	C₂×#261	1	2	yes	complemented
192	D₁₆	C₂×#261	1	2	yes	complemented
193	M₃₂	—	26	1	no	—
194	M₃₂	—	26	1	no	—
195	C₂×C₁₆	—	13	1	no	—
196	C₂×C₁₆	—	13	1	no	—
197	C₂×C₁₆	—	13	1	no	—
198	C₂×C₁₆	—	13	1	no	—
199	C₂×C₁₆	—	13	1	no	—
200	C₂×C₁₆	—	13	2	no	—
201	C₂×C₁₆	C₂×#261	1	1	yes	characteristic, complemented
202	#107	—	13	4	no	—
203	#106	—	26	2	no	—
204	#102	—	26	8	no	—
205	#97	—	13	2	no	—
206	C₄×C₈	—	13	2	no	—
207	C₂×C₁₃	—	2	1	no	—
208	C₂×C₁₃	—	8	4	no	—
209	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
210	D₁₃	—	2	1	no	—
211	D₁₃	—	8	4	no	—
212	D₁₃	#3415	1	2	yes	—
213	C₂⁴	—	52	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	26	4	no	—
216	C₂×D₄	—	26	4	no	—
217	C₂×D₄	—	26	2	no	—
218	C₂×D₄	—	52	2	no	—
219	C₂×D₄	—	52	2	no	—
220	C₂×D₄	—	52	2	no	—
221	C₂²×C₄	—	13	1	no	—
222	C₂²×C₄	—	52	8	no	—
223	D₈	—	2	2	no	—
224	D₈	—	13	4	no	—
225	D₈	—	26	4	no	—
226	D₈	—	26	4	no	—
227	D₈	—	26	2	no	—
228	D₈	C₂²×#261	1	4	yes	—
229	M₁₆	—	26	1	no	—
230	M₁₆	—	26	1	no	—
231	C₂×C₈	—	13	1	no	—
232	C₂×C₈	—	13	1	no	—
233	C₂×C₈	—	13	1	no	—
234	C₂×C₈	—	13	1	no	—
235	C₂×C₈	—	13	1	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	C₂²×#261	1	1	yes	characteristic
238	C₄⋊C₄	—	13	2	no	—
239	C₄⋊C₄	—	26	4	no	—
240	C₂²⋊C₄	—	13	2	no	—
241	C₂²⋊C₄	—	52	8	no	—
242	C₂²⋊C₄	—	52	2	no	—
243	C₄²	—	13	2	no	—
244	C₄²	—	52	2	no	—
245	C₁₆	—	13	1	no	—
246	C₁₆	—	13	1	no	—
247	C₁₆	—	26	2	no	—
248	C₁₆	C₂²×#261	1	1	yes	characteristic
249	C₁₆	C₂²×#261	1	1	yes	characteristic
250	C₁₃	#33456	1	1	yes	characteristic, complemented, Sylow
251	C₂³	—	4	2	no	—
252	C₂³	—	13	1	no	—
253	C₂³	—	52	4	no	—
254	C₂³	—	52	4	no	—
255	C₂³	—	52	2	no	—
256	C₂³	—	208	4	no	—
257	D₄	—	2	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	26	4	no	—
260	D₄	—	52	4	no	—
261	D₄	—	52	4	no	—
262	D₄	—	52	2	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	13	1	no	—
265	C₂×C₄	—	13	1	no	—
266	C₂×C₄	—	13	1	no	—
267	C₂×C₄	—	13	1	no	—
268	C₂×C₄	—	13	4	no	—
269	C₂×C₄	—	52	4	no	—
270	C₂×C₄	—	208	8	no	—
271	C₂×C₄	D₄×#261	1	1	yes	characteristic
272	C₈	—	13	1	no	—
273	C₈	—	13	1	no	—
274	C₈	—	26	2	no	—
275	C₈	D₄×#261	1	1	yes	characteristic
276	C₈	C₂³×#261	1	1	yes	characteristic
277	C₂²	—	4	4	no	—
278	C₂²	—	13	1	no	—
279	C₂²	—	13	1	no	—
280	C₂²	—	16	2	no	—
281	C₂²	—	26	2	no	—
282	C₂²	—	52	4	no	—
283	C₂²	—	104	8	no	—
284	C₂²	—	208	2	no	—
285	C₂²	—	208	4	no	—
286	C₂²	D₈×#261	1	1	yes	characteristic
287	C₄	—	13	1	no	—
288	C₄	—	13	1	no	—
289	C₄	—	26	4	no	—
290	C₄	—	104	4	no	—
291	C₄	D₈×#261	1	1	yes	characteristic
292	C₄	C₂×D₄×#261	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	8	4	no	—
295	C₂	—	13	2	no	—
296	C₂	—	26	1	no	—
297	C₂	—	104	4	no	—
298	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	C₂×C₁₃×D₁₆	C₄
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	C₂×D₁₃×D₁₆	C₂
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂×D₁₆	#261
hypocenter	C₁₃	#33456
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. #13131595	C₂	semidirect
2. #13131234	C₂	semidirect
3. D₁₆×#261	C₂	semidirect
4. #13131187	C₂	semidirect
5. #13131145	C₂	semidirect
6. #13131044	C₂	semidirect
12. D₈×#261	C₂²	semidirect
16. C₂×C₁₃×D₁₆	C₄	semidirect
17. C₂×#88066	C₄	semidirect
18. D₁₃×D₁₆	C₄	semidirect
19. D₁₃×D₁₆	C₄	semidirect
22. C₂×D₂₀₈	C₄	semidirect
24. #11718276	C₂²	semidirect
25. #11718222	C₂²	semidirect
26. #11718221	C₂²	semidirect
27. #11718216	C₂²	semidirect
28. C₁₆×#261	C₂²	semidirect
36. C₂×C₁₃×D₈	C₂×C₄	semidirect
40. D₈×D₁₃	C₂×C₄	semidirect
42. C₂×D₁₀₄	C₂×C₄	semidirect
52. C₁₃×D₁₆	C₂×C₄	semidirect
53. C₁₃×D₁₆	C₂×C₄	semidirect
54. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
55. #88066	C₂×C₄	semidirect
56. #88066	C₂×C₄	semidirect
57. C₂×#6090	C₂×C₄	semidirect
58. D₂₀₈	C₂×C₄	semidirect
59. D₂₀₈	C₂×C₄	semidirect
60. C₁₆×D₁₃	C₂×C₄	semidirect
61. C₁₆×D₁₃	C₂×C₄	semidirect
143. C₂×D₁₆	#261	semidirect
179. C₂×D₈	C₂×#261	semidirect
191. D₁₆	C₂×#261	semidirect
192. D₁₆	C₂×#261	semidirect
201. C₂×C₁₆	C₂×#261	semidirect
250. C₁₃	#33456	semidirect

Maximal subgroups

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

subgroup	index
#13131595	2
#13131234	2
D₁₆×#261	2
#13131187	2
#13131145	2
#13131044	2
C₂×D₁₃×D₁₆	2
#33456	13