Subgroups — #16796543

Order 3328.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131582	C₂	1	2	yes	maximal, complemented
2	#13130468	C₂	1	2	yes	maximal, complemented
3	#13130436	C₂	1	4	yes	maximal, complemented
4	#13130417	C₂	1	2	yes	maximal, complemented
5	D₁₃⋊#335	C₂	1	1	yes	maximal, characteristic, complemented
6	#13130143	C₂	1	2	yes	maximal, complemented
7	C₈×#6118	C₂	1	1	yes	maximal, characteristic, complemented
8	#13125416	C₂	1	1	yes	maximal, characteristic
9	C₂×#88240	—	2	2	no	—
10	#11719430	C₂²	1	2	yes	—
11	#11719159	—	2	2	no	—
12	#11719076	C₂²	1	4	yes	—
13	#11719075	—	2	4	no	—
14	#11719073	C₂²	1	4	yes	—
15	#11719051	C₂²	1	1	yes	characteristic
16	D₁₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
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	#11718254	C₂²	1	1	yes	characteristic
28	#11718254	C₂²	1	1	yes	characteristic
29	#1057⋊C₈	C₂²	1	2	yes	—
30	C₈×#1057	C₂²	1	2	yes	—
31	#11718243	C₂²	1	4	yes	—
32	#11718243	C₂²	1	4	yes	—
33	C₂×#6131	—	2	2	no	—
34	#88240	—	4	8	no	—
35	C₂×#6118	—	2	4	no	—
36	C₂×#6118	D₄	1	1	yes	characteristic
37	C₂×#6118	D₄	1	1	yes	characteristic
38	C₂×D₁₃⋊C₈	—	2	2	no	—
39	C₂×D₁₃⋊C₈	—	2	4	no	—
40	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	#88201	—	2	2	no	—
42	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
43	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
46	#88149	—	2	2	no	—
47	#88149	C₂×C₄	1	4	yes	—
48	#88147	—	2	2	no	—
49	#88147	C₂³	1	4	yes	—
50	#88128	C₂×C₄	1	2	yes	—
51	C₄²×D₁₃	C₂³	1	1	yes	characteristic
52	#88112	—	2	2	no	—
53	#88111	—	2	1	no	—
54	#88111	—	2	4	no	—
55	#88111	—	2	4	no	—
56	#88110	D₄	1	2	yes	—
57	C₄⋊#1057	—	2	1	no	—
58	C₄⋊#1057	C₂³	1	2	yes	—
59	C₄⋊#1057	C₂³	1	2	yes	—
60	C₄×#1057	C₂³	1	2	yes	—
61	C₄×#1057	C₂³	1	2	yes	—
62	C₁₃×#107	C₂×C₄	1	2	yes	—
63	C₁₃×#107	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
66	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
67	#88049	C₂×C₄	1	4	yes	—
68	#88049	C₂×C₄	1	4	yes	—
69	C₄×#1055	C₂×C₄	1	1	yes	characteristic
70	#18066	—	13	1	no	maximal, Sylow
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	#6131	—	4	4	no	—
73	#6131	—	4	4	no	—
74	C₂×D₅₂	—	2	4	no	—
75	C₂×D₅₂	—	2	2	no	—
76	C₂×D₅₂	—	2	2	no	—
77	C₂×D₅₂	C₂²×C₄	1	4	yes	—
78	C₂×C₄×D₁₃	—	2	4	no	—
79	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
82	C₂×#1057	—	2	2	no	—
83	C₂×#1057	—	2	2	no	—
84	C₂×#1057	—	2	2	no	—
85	C₂×#1057	C₂×D₄	1	2	yes	—
86	C₂×#1057	C₂×D₄	1	2	yes	—
87	#6118	—	2	2	no	—
88	#6118	—	4	8	no	—
89	#6118	D₈	1	4	yes	complemented
90	D₁₃⋊C₈	—	4	4	no	—
91	D₁₃⋊C₈	—	4	8	no	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	#6103	—	2	4	no	—
98	#6103	—	2	4	no	—
99	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
100	C₄×Dic₁₃	#41	1	1	yes	characteristic
101	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
102	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
103	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
104	D₁₀₄	—	2	4	no	—
105	D₁₀₄	—	4	2	no	—
106	C₈×D₁₃	—	2	2	no	—
107	C₈×D₁₃	—	4	1	no	—
108	#3169	—	13	1	no	—
109	#2935	—	13	2	no	—
110	#1589	—	13	2	no	—
111	#1581	—	13	4	no	—
112	C₈×D₈	—	13	2	no	—
113	C₂×#335	—	13	1	no	—
114	#1480	—	13	2	no	—
115	C₈×M₁₆	—	13	1	no	—
116	C₂²×D₁₃	—	2	4	no	—
117	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
126	D₅₂	—	2	4	no	—
127	D₅₂	—	2	4	no	—
128	D₅₂	—	4	8	no	—
129	D₅₂	—	4	4	no	—
130	D₅₂	—	8	2	no	—
131	C₄×D₁₃	—	2	1	no	—
132	C₄×D₁₃	—	2	1	no	—
133	C₄×D₁₃	—	4	1	no	—
134	C₄×D₁₃	—	4	4	no	—
135	C₄×D₁₃	—	4	4	no	—
136	C₄×D₁₃	C₂×D₈	1	2	yes	—
137	C₄×D₁₃	#135	1	2	yes	—
138	#1057	—	2	2	no	—
139	#1057	—	4	2	no	—
140	#1057	—	4	2	no	—
141	#1057	—	4	2	no	—
142	#1057	C₂×D₈	1	4	yes	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	#131	1	2	yes	complemented
145	#1055	—	2	1	no	—
146	#1055	—	2	1	no	—
147	C₂×#131	—	26	2	no	—
148	#532	—	13	2	no	—
149	#500	#261	1	1	yes	characteristic
150	#493	—	13	1	no	—
151	#486	—	13	2	no	—
152	#466	—	13	2	no	—
153	#466	—	13	2	no	—
154	C₄×D₈	—	13	2	no	—
155	C₄×D₈	—	13	2	no	—
156	#436	—	13	4	no	—
157	#435	—	26	4	no	—
158	C₈×D₄	—	13	4	no	—
159	#431	—	13	1	no	—
160	C₂×C₄⋊C₈	—	13	2	no	—
161	#408	—	26	2	no	—
162	C₄×M₁₆	—	13	1	no	—
163	C₂×C₄×C₈	—	13	1	no	—
164	#335	—	13	1	no	—
165	#335	—	13	2	no	—
166	#335	—	13	1	no	—
167	D₄⋊C₈	—	13	4	no	—
168	D₄⋊C₈	—	13	4	no	—
169	#322	—	13	2	no	—
170	C₈²	—	13	2	no	—
171	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₁₃	—	2	1	no	—
173	C₂×D₁₃	—	2	1	no	—
174	C₂×D₁₃	—	4	4	no	—
175	C₂×D₁₃	—	4	4	no	—
176	C₂×D₁₃	—	4	4	no	—
177	C₂×D₁₃	C₄×D₈	1	2	yes	—
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	—	4	4	no	—
180	C₄×C₁₃	C₄×D₈	1	2	yes	—
181	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
182	C₄×C₁₃	#568	1	1	yes	characteristic
183	Dic₁₃	—	2	1	no	—
184	Dic₁₃	—	2	1	no	—
185	Dic₁₃	C₂²×D₈	1	1	yes	characteristic
186	Dic₁₃	C₂×#135	1	1	yes	characteristic
187	C₂×#41	—	26	2	no	—
188	C₂×D₈	—	13	2	no	—
189	C₂×D₈	—	13	2	no	—
190	#131	—	52	8	no	—
191	C₂×M₁₆	—	13	1	no	—
192	C₂×M₁₆	—	13	1	no	—
193	C₂×M₁₆	—	26	4	no	—
194	C₂²×C₈	—	13	1	no	—
195	C₂²×C₈	—	13	1	no	—
196	C₂²×C₈	—	26	4	no	—
197	C₂²×C₈	—	26	2	no	—
198	C₄⋊Q₈	C₂×#261	1	2	yes	—
199	#127	—	13	2	no	—
200	#124	—	26	2	no	—
201	#121	—	13	4	no	—
202	#121	—	26	2	no	—
203	C₄×D₄	—	13	4	no	—
204	C₄×D₄	—	26	2	no	—
205	C₂×C₄²	—	13	1	no	—
206	#107	C₂×#261	1	2	yes	—
207	#107	C₂×#261	1	2	yes	—
208	C₄⋊C₈	—	13	2	no	—
209	C₄⋊C₈	—	13	2	no	—
210	C₄⋊C₈	—	26	1	no	—
211	C₄⋊C₈	—	26	2	no	—
212	#102	—	13	4	no	—
213	#102	—	13	4	no	—
214	C₂²⋊C₈	—	26	4	no	—
215	C₂²⋊C₈	—	26	4	no	—
216	C₂²⋊C₈	—	26	1	no	—
217	#97	—	13	2	no	—
218	C₄×C₈	—	13	2	no	—
219	C₄×C₈	—	13	1	no	—
220	C₄×C₈	—	13	1	no	—
221	C₄×C₈	—	13	2	no	—
222	C₄×C₈	—	13	1	no	—
223	C₄×C₈	C₂×#261	1	1	yes	characteristic
224	C₂×C₁₃	#2935	1	1	yes	characteristic
225	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
226	C₂×C₁₃	#2957	1	1	yes	characteristic
227	D₁₃	—	2	2	no	—
228	D₁₃	—	8	4	no	—
229	#41	—	52	4	no	—
230	#41	—	52	4	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×D₄	—	13	4	no	—
233	C₂×D₄	—	26	4	no	—
234	C₂×D₄	—	26	2	no	—
235	C₂×D₄	—	26	2	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	1	no	—
239	C₂²×C₄	—	26	4	no	—
240	D₈	—	26	4	no	—
241	D₈	—	52	2	no	—
242	M₁₆	—	13	4	no	—
243	M₁₆	—	26	2	no	—
244	M₁₆	—	52	8	no	—
245	C₂×C₈	—	13	2	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	13	1	no	—
248	C₂×C₈	—	13	1	no	—
249	C₂×C₈	—	26	2	no	—
250	C₂×C₈	—	26	2	no	—
251	C₂×C₈	—	26	2	no	—
252	C₂×C₈	—	26	2	no	—
253	C₂×C₈	—	52	1	no	—
254	C₂×C₈	—	52	8	no	—
255	C₂×C₈	—	52	4	no	—
256	C₂×C₈	C₂²×#261	1	1	yes	characteristic
257	C₂×C₈	C₂²×#261	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	C₂²×#261	1	4	yes	—
260	C₂²⋊C₄	—	26	4	no	—
261	C₂²⋊C₄	—	26	4	no	—
262	C₄²	—	13	1	no	—
263	C₄²	—	13	1	no	—
264	C₄²	—	13	1	no	—
265	C₄²	C₂²×#261	1	1	yes	characteristic
266	C₁₃	#18066	1	1	yes	characteristic, complemented, Sylow
267	C₂³	—	13	1	no	—
268	C₂³	—	26	4	no	—
269	Q₈	—	4	4	no	—
270	D₄	—	26	4	no	—
271	D₄	—	26	4	no	—
272	D₄	—	52	8	no	—
273	D₄	—	52	4	no	—
274	D₄	—	104	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	13	1	no	—
277	C₂×C₄	—	13	2	no	—
278	C₂×C₄	—	13	2	no	—
279	C₂×C₄	—	13	1	no	—
280	C₂×C₄	—	13	1	no	—
281	C₂×C₄	—	26	1	no	—
282	C₂×C₄	—	26	1	no	—
283	C₂×C₄	—	52	1	no	—
284	C₂×C₄	—	52	4	no	—
285	C₂×C₄	—	52	4	no	—
286	C₂×C₄	D₄×#261	1	1	yes	characteristic
287	C₂×C₄	D₄×#261	1	1	yes	characteristic
288	C₂×C₄	C₂³×#261	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	13	4	no	—
291	C₈	—	26	2	no	—
292	C₈	—	26	1	no	—
293	C₈	—	26	1	no	—
294	C₈	—	52	2	no	—
295	C₈	—	52	2	no	—
296	C₈	—	52	2	no	—
297	C₈	#88240	1	2	yes	complemented
298	C₂²	—	13	2	no	—
299	C₂²	—	26	1	no	—
300	C₂²	—	26	1	no	—
301	C₂²	—	52	4	no	—
302	C₂²	—	52	4	no	—
303	C₂²	—	52	4	no	—
304	C₂²	C₂×D₄×#261	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	4	4	no	—
307	C₄	—	13	1	no	—
308	C₄	—	13	1	no	—
309	C₄	—	26	1	no	—
310	C₄	—	26	1	no	—
311	C₄	D₈×#261	1	2	yes	—
312	C₄	C₂×#88240	1	1	yes	characteristic
313	C₄	#11719625	1	1	yes	characteristic
314	C₂	—	26	2	no	—
315	C₂	—	104	4	no	—
316	C₂	#13131582	1	1	yes	characteristic
317	C₂	C₂×D₈×#261	1	1	yes	characteristic
318	C₂	#13131616	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	#18066
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. #13131582	C₂	semidirect
2. #13130468	C₂	semidirect
3. #13130436	C₂	semidirect
4. #13130417	C₂	semidirect
5. D₁₃⋊#335	C₂	semidirect
6. #13130143	C₂	semidirect
7. C₈×#6118	C₂	semidirect
89. #6118	D₈	semidirect
144. C₈×C₁₃	#131	semidirect
266. C₁₃	#18066	semidirect
297. C₈	#88240	semidirect

Maximal subgroups

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

subgroup	index
#13131582	2
#13130468	2
#13130436	2
#13130417	2
D₁₃⋊#335	2
#13130143	2
C₈×#6118	2
#13125416	2
#18066	13