Subgroups — #16796702

Order 3328.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131719	C₂	1	2	yes	maximal, complemented
2	#13130485	C₂	1	2	yes	maximal, complemented
3	#13130436	C₂	1	4	yes	maximal, complemented
4	#13130434	C₂	1	2	yes	maximal, complemented
5	#13130384	C₂	1	1	yes	maximal, characteristic, complemented
6	#13130157	C₂	1	2	yes	maximal, complemented
7	#13130046	C₂	1	1	yes	maximal, characteristic, complemented
8	#13125416	C₂	1	1	yes	maximal, characteristic
9	#11719430	C₂²	1	2	yes	—
10	#11719159	—	2	4	no	—
11	#11719076	—	2	2	no	—
12	#11719076	C₂²	1	4	yes	—
13	#11719073	—	2	2	no	—
14	#11719073	C₂²	1	4	yes	—
15	#11719051	C₂²	1	2	yes	—
16	#11719051	C₂²	1	1	yes	characteristic
17	C₄×#6118	C₂²	1	1	yes	characteristic
18	C₁₃×#500	C₄	1	1	yes	characteristic
19	#11718749	C₄	1	2	yes	—
20	#11718544	C₄	1	2	yes	—
21	#11718544	C₄	1	2	yes	—
22	#11718533	C₂²	1	2	yes	—
23	#11718533	C₂²	1	2	yes	—
24	#11718356	C₄	1	1	yes	characteristic
25	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
26	#11718256	C₂²	1	2	yes	—
27	#11718253	C₂²	1	1	yes	characteristic
28	#11718253	C₂²	1	1	yes	characteristic
29	#1057⋊C₈	C₂²	1	2	yes	—
30	#11718251	C₂²	1	2	yes	—
31	#11718243	C₂²	1	4	yes	—
32	#11718243	C₂²	1	4	yes	—
33	C₂×#6131	—	2	2	no	—
34	C₂×#6118	—	2	1	no	—
35	C₂×#6118	—	2	4	no	—
36	C₂×D₁₃⋊C₈	—	2	4	no	—
37	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	#88201	—	2	2	no	—
39	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
40	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
43	#88149	—	2	2	no	—
44	#88149	C₂×C₄	1	4	yes	—
45	#88147	—	2	2	no	—
46	#88147	C₂³	1	4	yes	—
47	#88128	C₂×C₄	1	2	yes	—
48	C₄²×D₁₃	C₂³	1	1	yes	characteristic
49	#88111	—	2	2	no	—
50	#88111	—	2	4	no	—
51	#88111	—	2	4	no	—
52	#88111	D₄	1	2	yes	—
53	#88110	—	2	1	no	—
54	C₄⋊#1057	D₄	1	2	yes	—
55	C₄⋊#1057	C₂³	1	2	yes	—
56	C₄⋊#1057	C₂³	1	2	yes	—
57	C₄×#1057	—	2	2	no	—
58	C₄×#1057	C₂³	1	2	yes	—
59	C₄×#1057	C₂³	1	2	yes	—
60	C₁₃×#107	C₂×C₄	1	2	yes	—
61	C₁₃×#107	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
64	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
65	#88049	C₂×C₄	1	4	yes	—
66	#88049	C₂×C₄	1	4	yes	—
67	C₄×#1055	C₂×C₄	1	1	yes	characteristic
68	#18091	—	13	1	no	maximal, Sylow
69	C₂×Q₈×C₁₃	—	2	2	no	—
70	#6131	—	8	2	no	—
71	#6131	—	8	2	no	—
72	C₂×D₅₂	—	2	2	no	—
73	C₂×D₅₂	—	2	4	no	—
74	C₂×D₅₂	—	2	2	no	—
75	C₂×D₅₂	C₂²×C₄	1	4	yes	—
76	C₂×C₄×D₁₃	—	2	4	no	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
80	C₂×#1057	—	2	2	no	—
81	C₂×#1057	—	2	2	no	—
82	C₂×#1057	—	2	2	no	—
83	C₂×#1057	C₂×D₄	1	4	yes	—
84	#6118	—	4	2	no	—
85	#6118	—	8	4	no	—
86	D₁₃⋊C₈	—	8	4	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	#6103	—	2	4	no	—
93	#6103	—	2	4	no	—
94	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
95	C₄×Dic₁₃	#41	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
97	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
98	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
99	D₁₀₄	—	2	4	no	—
100	D₁₀₄	—	4	2	no	—
101	C₈×D₁₃	—	2	2	no	—
102	C₈×D₁₃	—	4	1	no	—
103	#3169	—	13	1	no	—
104	#2979	—	13	2	no	—
105	#1594	—	13	2	no	—
106	#1581	—	13	4	no	—
107	#1580	—	13	2	no	—
108	#1564	—	13	1	no	—
109	#1484	—	13	2	no	—
110	#1451	—	13	1	no	—
111	C₂²×D₁₃	—	2	4	no	—
112	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
113	Q₈×C₁₃	—	8	2	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
121	D₅₂	—	2	4	no	—
122	D₅₂	—	2	4	no	—
123	D₅₂	—	4	4	no	—
124	D₅₂	—	8	2	no	—
125	D₅₂	—	8	4	no	—
126	C₄×D₁₃	—	2	1	no	—
127	C₄×D₁₃	—	2	1	no	—
128	C₄×D₁₃	—	2	2	no	—
129	C₄×D₁₃	—	2	2	no	—
130	C₄×D₁₃	—	4	4	no	—
131	C₄×D₁₃	—	4	4	no	—
132	#1057	—	2	4	no	—
133	#1057	—	4	2	no	—
134	#1057	—	4	2	no	—
135	#1057	—	4	2	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	#131	1	2	yes	—
138	#1055	—	2	1	no	—
139	#1055	—	2	1	no	—
140	#532	—	13	2	no	—
141	#500	#261	1	1	yes	characteristic
142	#493	—	13	1	no	—
143	#486	—	13	2	no	—
144	#466	—	13	2	no	—
145	#466	—	13	2	no	—
146	C₄×D₈	—	13	2	no	—
147	C₄×D₈	—	13	2	no	—
148	#436	—	13	4	no	—
149	#436	—	26	2	no	—
150	C₈×D₄	—	13	4	no	—
151	C₈×D₄	—	26	2	no	—
152	#431	—	13	1	no	—
153	#431	—	13	2	no	—
154	#408	—	26	4	no	—
155	C₄×M₁₆	—	13	1	no	—
156	C₂×C₄×C₈	—	13	1	no	—
157	#335	—	13	2	no	—
158	#334	—	13	1	no	—
159	#334	—	13	1	no	—
160	D₄⋊C₈	—	13	4	no	—
161	D₄⋊C₈	—	13	4	no	—
162	#322	—	13	2	no	—
163	#322	—	13	2	no	—
164	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×D₁₃	—	2	1	no	—
166	C₂×D₁₃	—	2	1	no	—
167	C₂×D₁₃	—	2	1	no	—
168	C₂×D₁₃	—	4	4	no	—
169	C₂×D₁₃	—	4	4	no	—
170	C₂×D₁₃	—	4	4	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	4	4	no	—
173	C₄×C₁₃	#444	1	2	yes	—
174	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
175	C₄×C₁₃	#568	1	1	yes	characteristic
176	Dic₁₃	—	2	1	no	—
177	Dic₁₃	—	2	1	no	—
178	Dic₁₃	#576	1	2	yes	—
179	C₂×#41	—	26	2	no	—
180	C₂×D₈	—	13	2	no	—
181	C₂×D₈	—	13	2	no	—
182	C₂×M₁₆	—	26	4	no	—
183	C₂×M₁₆	—	26	1	no	—
184	C₂²×C₈	—	13	1	no	—
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	26	4	no	—
187	C₄⋊Q₈	C₂×#261	1	2	yes	—
188	#127	—	13	2	no	—
189	#124	—	26	2	no	—
190	#121	—	13	4	no	—
191	#121	—	26	2	no	—
192	C₄×D₄	—	13	4	no	—
193	C₄×D₄	—	26	2	no	—
194	C₂×C₄²	—	13	1	no	—
195	#107	C₂×#261	1	2	yes	—
196	#107	C₂×#261	1	2	yes	—
197	C₄⋊C₈	—	13	2	no	—
198	C₄⋊C₈	—	13	2	no	—
199	C₄⋊C₈	—	13	2	no	—
200	#102	—	13	4	no	—
201	#102	—	13	4	no	—
202	C₂²⋊C₈	—	13	2	no	—
203	C₂²⋊C₈	—	26	4	no	—
204	C₂²⋊C₈	—	26	4	no	—
205	C₂²⋊C₈	—	26	2	no	—
206	#97	—	26	1	no	—
207	C₄×C₈	—	13	2	no	—
208	C₄×C₈	—	13	1	no	—
209	C₄×C₈	—	13	1	no	—
210	C₄×C₈	—	13	2	no	—
211	C₄×C₈	—	13	1	no	—
212	C₄×C₈	—	26	2	no	—
213	C₄×C₈	C₂×#261	1	1	yes	characteristic
214	C₂×C₁₃	#2935	1	1	yes	characteristic
215	C₂×C₁₃	#2948	1	1	yes	characteristic
216	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
217	D₁₃	—	4	1	no	—
218	D₁₃	—	8	4	no	—
219	#41	—	104	2	no	—
220	#41	—	104	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×D₄	—	13	4	no	—
223	C₂×D₄	—	26	4	no	—
224	C₂×D₄	—	26	2	no	—
225	C₂×D₄	—	26	2	no	—
226	C₂²×C₄	—	13	1	no	—
227	C₂²×C₄	—	13	1	no	—
228	C₂²×C₄	—	13	1	no	—
229	C₂²×C₄	—	26	4	no	—
230	D₈	—	26	4	no	—
231	D₈	—	52	2	no	—
232	M₁₆	—	52	2	no	—
233	M₁₆	—	104	4	no	—
234	C₂×C₈	—	13	4	no	—
235	C₂×C₈	—	13	1	no	—
236	C₂×C₈	—	13	1	no	—
237	C₂×C₈	—	26	2	no	—
238	C₂×C₈	—	26	2	no	—
239	C₂×C₈	—	26	2	no	—
240	C₂×C₈	—	26	2	no	—
241	C₂×C₈	—	52	1	no	—
242	C₂×C₈	—	104	4	no	—
243	C₂×C₈	C₂²×#261	1	1	yes	characteristic
244	C₂×C₈	C₂²×#261	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	C₂²×#261	1	4	yes	—
247	C₂²⋊C₄	—	26	4	no	—
248	C₂²⋊C₄	—	26	4	no	—
249	C₄²	—	13	1	no	—
250	C₄²	—	13	1	no	—
251	C₄²	—	13	1	no	—
252	C₄²	C₂²×#261	1	1	yes	characteristic
253	C₁₃	#18091	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	13	1	no	—
255	C₂³	—	26	4	no	—
256	Q₈	—	8	2	no	—
257	D₄	—	26	4	no	—
258	D₄	—	26	4	no	—
259	D₄	—	52	4	no	—
260	D₄	—	104	2	no	—
261	D₄	—	104	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	13	1	no	—
265	C₂×C₄	—	13	1	no	—
266	C₂×C₄	—	26	2	no	—
267	C₂×C₄	—	26	2	no	—
268	C₂×C₄	—	26	1	no	—
269	C₂×C₄	—	26	1	no	—
270	C₂×C₄	—	52	4	no	—
271	C₂×C₄	—	52	4	no	—
272	C₂×C₄	D₄×#261	1	1	yes	characteristic
273	C₂×C₄	D₄×#261	1	1	yes	characteristic
274	C₂×C₄	C₂³×#261	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	26	4	no	—
277	C₈	—	26	1	no	—
278	C₈	—	26	1	no	—
279	C₈	—	52	2	no	—
280	C₈	—	52	2	no	—
281	C₈	—	52	2	no	—
282	C₈	#88240	1	2	yes	—
283	C₂²	—	26	1	no	—
284	C₂²	—	26	1	no	—
285	C₂²	—	26	1	no	—
286	C₂²	—	52	4	no	—
287	C₂²	—	52	4	no	—
288	C₂²	—	52	4	no	—
289	C₂²	C₂×D₄×#261	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	4	4	no	—
292	C₄	—	13	2	no	—
293	C₄	—	26	1	no	—
294	C₄	—	26	1	no	—
295	C₄	#11719112	1	2	yes	—
296	C₄	C₂×#88240	1	1	yes	characteristic
297	C₄	#11719625	1	1	yes	characteristic
298	C₂	—	52	1	no	—
299	C₂	—	104	4	no	—
300	C₂	#13131582	1	1	yes	characteristic
301	C₂	#13131595	1	1	yes	characteristic
302	C₂	C₂×#11719112	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#500	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125416	C₂
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#500	#261
hypocenter	C₁₃	#18091
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
1. #13131719	C₂	semidirect
2. #13130485	C₂	semidirect
3. #13130436	C₂	semidirect
4. #13130434	C₂	semidirect
5. #13130384	C₂	semidirect
6. #13130157	C₂	semidirect
7. #13130046	C₂	semidirect
253. C₁₃	#18091	semidirect

Maximal subgroups

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

subgroup	index
#13131719	2
#13130485	2
#13130436	2
#13130434	2
#13130384	2
#13130157	2
#13130046	2
#13125416	2
#18091	13