Subgroups — #16807086

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131202	C₂	1	4	yes	maximal, complemented
2	#13131198	C₂	1	2	yes	maximal, complemented
3	#13131194	C₂	1	2	yes	maximal, complemented
4	#13131170	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718260	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131066	C₂	1	2	yes	maximal
7	#13131047	C₂	1	2	yes	maximal, complemented
8	#131×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
10	#41×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×#88202	C₂²	1	1	yes	characteristic
12	C₂×#88106	C₂²	1	2	yes	—
13	C₂²×#6092	C₂²	1	2	yes	—
14	#11718783	C₂²	1	1	yes	characteristic
15	#11718783	C₂²	1	2	yes	—
16	M₁₆×Dic₁₃	C₄	1	2	yes	—
17	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
18	#11718762	C₄	1	1	yes	characteristic
19	#11718762	C₄	1	2	yes	—
20	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
22	C₂²⋊#6092	C₂²	1	4	yes	—
23	C₂²⋊#6092	C₂²	1	4	yes	—
24	#11718260	C₂²	1	1	yes	characteristic
25	#11718260	C₂²	1	1	yes	characteristic
26	#11718260	C₂²	1	2	yes	—
27	#11718260	C₂²	1	2	yes	—
28	#11718260	C₂²	1	2	yes	—
29	#11718258	C₂²	1	4	yes	—
30	#11718257	C₂²	1	4	yes	—
31	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
32	C₁₃×#131	—	2	4	no	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	#88202	—	2	2	no	—
38	#88202	—	2	2	no	—
39	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
40	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
41	D₄×Dic₁₃	C₈	1	2	yes	—
42	D₄×Dic₁₃	C₈	1	1	yes	characteristic
43	D₄×Dic₁₃	C₈	1	2	yes	—
44	D₄×Dic₁₃	C₈	1	1	yes	characteristic
45	#88180	C₂×C₄	1	2	yes	—
46	#88180	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×#6099	C₂×C₄	1	1	yes	characteristic
50	C₂×#6099	C₂×C₄	1	2	yes	—
51	C₂²×#1055	C₂³	1	2	yes	—
52	C₂²×#1055	C₂³	1	1	yes	characteristic
53	#88106	—	2	4	no	—
54	C₂×#6092	—	2	4	no	—
55	C₂×#6092	C₂³	1	2	yes	—
56	C₂×#6092	C₂³	1	4	yes	—
57	C₂×#6092	C₂³	1	2	yes	—
58	#88055	C₂×C₄	1	2	yes	—
59	#88055	C₂×C₄	1	2	yes	—
60	#88055	C₂×C₄	1	2	yes	—
61	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
62	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
64	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
66	C₈×Dic₁₃	C₂³	1	2	yes	—
67	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
68	#33423	—	13	1	no	maximal, Sylow
69	C₁₃×#41	—	2	2	no	—
70	C₁₃×#41	—	2	2	no	—
71	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
72	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
73	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
77	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
79	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
80	C₁₃×M₁₆	—	2	4	no	—
81	C₁₃×M₁₆	M₁₆	1	4	yes	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
88	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
89	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
90	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
91	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
93	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
95	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
97	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
98	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
101	#6099	—	2	2	no	—
102	#6099	—	2	4	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	—	2	2	no	—
105	C₂×#1055	C₂²×C₄	1	2	yes	—
106	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
107	C₂×#1055	#41	1	2	yes	—
108	C₂×#1055	#41	1	2	yes	—
109	C₂×#1055	C₂⁴	1	1	yes	characteristic
110	#6092	—	2	2	no	—
111	#6092	—	2	2	no	—
112	#6092	—	2	4	no	—
113	C₄×#131	—	13	1	no	—
114	#2164	—	13	4	no	—
115	#2163	—	13	2	no	—
116	#2162	—	13	2	no	—
117	#2150	—	13	1	no	—
118	C₂×C₄⋊C₁₆	—	13	1	no	—
119	#2113	—	13	2	no	—
120	#2108	—	13	2	no	—
121	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
122	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₁₃	—	2	2	no	—
124	D₄×C₁₃	—	2	4	no	—
125	D₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	#117	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×Dic₁₃	—	2	2	no	—
134	C₂×Dic₁₃	—	2	4	no	—
135	C₂×Dic₁₃	—	4	1	no	—
136	C₂×Dic₁₃	—	4	2	no	—
137	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
138	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
139	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
141	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
146	#1055	—	2	2	no	—
147	#1055	#117	1	2	yes	—
148	#1055	C₂×#41	1	1	yes	characteristic
149	#1055	C₂×#41	1	1	yes	characteristic
150	C₂×#131	—	13	1	no	—
151	C₂×#131	#261	1	1	yes	characteristic
152	C₄×#41	—	13	1	no	—
153	C₂×M₃₂	—	13	2	no	—
154	C₂²×C₁₆	—	13	2	no	—
155	#405	—	13	1	no	—
156	#405	—	13	2	no	—
157	#405	—	13	2	no	—
158	#405	—	13	1	no	—
159	C₄×M₁₆	—	13	2	no	—
160	C₄×M₁₆	—	13	1	no	—
161	C₂×C₄×C₈	—	13	1	no	—
162	C₂×C₄×C₈	—	13	2	no	—
163	C₄⋊C₁₆	—	13	1	no	—
164	C₄⋊C₁₆	—	13	1	no	—
165	C₄⋊C₁₆	—	13	2	no	—
166	C₄⋊C₁₆	—	13	2	no	—
167	C₄⋊C₁₆	—	13	2	no	—
168	C₂²⋊C₁₆	—	13	4	no	—
169	C₂²⋊C₁₆	—	13	4	no	—
170	#346	—	13	4	no	—
171	C₄×C₁₆	—	13	4	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	2	2	no	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	#431	1	2	yes	—
178	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
179	C₄×C₁₃	—	2	2	no	—
180	C₄×C₁₃	#431	1	2	yes	—
181	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
182	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
183	Dic₁₃	—	2	1	no	—
184	Dic₁₃	—	2	2	no	—
185	Dic₁₃	—	2	1	no	—
186	Dic₁₃	—	2	1	no	—
187	Dic₁₃	—	2	2	no	—
188	Dic₁₃	#504	1	2	yes	—
189	C₂×#41	C₂×#261	1	1	yes	characteristic
190	#131	—	2	4	no	—
191	#131	—	26	2	no	—
192	#131	—	26	2	no	—
193	C₂×M₁₆	—	13	2	no	—
194	C₂×M₁₆	—	13	1	no	—
195	C₂×M₁₆	C₂×#261	1	2	yes	—
196	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
197	C₂²×C₈	—	13	1	no	—
198	C₂²×C₈	—	13	2	no	—
199	C₂²×C₈	C₂×#261	1	1	yes	characteristic
200	C₂²×C₈	C₂×#261	1	2	yes	—
201	C₄×Q₈	—	13	1	no	—
202	C₄×Q₈	—	13	1	no	—
203	C₄×D₄	—	13	2	no	—
204	C₄×D₄	—	13	1	no	—
205	C₄×D₄	—	13	2	no	—
206	C₄×D₄	—	13	1	no	—
207	#117	—	13	2	no	—
208	#117	—	13	1	no	—
209	C₂×C₄²	—	13	2	no	—
210	C₂×C₄²	—	13	1	no	—
211	M₃₂	—	26	4	no	—
212	C₂×C₁₆	—	13	2	no	—
213	C₂×C₁₆	—	13	4	no	—
214	C₂×C₁₆	—	13	2	no	—
215	C₂×C₁₆	—	26	4	no	—
216	#97	—	13	2	no	—
217	#97	—	13	2	no	—
218	#97	—	13	2	no	—
219	C₄×C₈	—	13	2	no	—
220	C₄×C₈	—	13	1	no	—
221	C₄×C₈	—	13	1	no	—
222	C₄×C₈	—	13	1	no	—
223	C₄×C₈	—	13	2	no	—
224	C₄×C₈	—	13	1	no	—
225	C₄×C₈	—	13	2	no	—
226	C₂×C₁₃	—	2	2	no	—
227	C₂×C₁₃	—	4	2	no	—
228	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
229	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
230	C₂×C₁₃	#3407	1	1	yes	characteristic
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
234	C₂×D₄	D₁₃⋊C₈	1	2	yes	—
235	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
236	C₂²×C₄	—	13	2	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	2	no	—
239	C₂²×C₄	—	13	1	no	—
240	C₂²×C₄	C₂²×#261	1	2	yes	—
241	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
242	M₁₆	—	2	4	no	—
243	M₁₆	—	26	4	no	—
244	M₁₆	—	26	2	no	—
245	M₁₆	#6122	1	4	yes	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	13	2	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	13	1	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	26	2	no	—
254	C₂×C₈	—	26	2	no	—
255	C₂×C₈	C₂²×#261	1	1	yes	characteristic
256	C₂×C₈	C₂²×#261	1	1	yes	characteristic
257	C₂×C₈	C₂²×#261	1	2	yes	—
258	C₄⋊C₄	—	13	2	no	—
259	C₄⋊C₄	—	13	2	no	—
260	C₄⋊C₄	—	13	1	no	—
261	C₄⋊C₄	—	13	1	no	—
262	C₂²⋊C₄	—	13	2	no	—
263	C₂²⋊C₄	—	13	2	no	—
264	C₂²⋊C₄	—	13	2	no	—
265	C₄²	—	13	2	no	—
266	C₄²	—	13	1	no	—
267	C₄²	—	13	2	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	13	1	no	—
270	C₄²	—	13	2	no	—
271	C₄²	—	13	1	no	—
272	C₁₆	—	26	2	no	—
273	C₁₆	—	26	2	no	—
274	C₁₆	—	26	4	no	—
275	C₁₃	#33423	1	1	yes	characteristic, complemented, Sylow
276	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
277	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	13	1	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	26	2	no	—
290	C₂×C₄	—	26	4	no	—
291	C₂×C₄	—	52	1	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
294	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
295	C₂×C₄	#88237	1	2	yes	—
296	C₂×C₄	C₂×#6122	1	2	yes	—
297	C₂×C₄	C₂³×#261	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	13	1	no	—
301	C₈	—	13	1	no	—
302	C₈	—	13	2	no	—
303	C₈	—	26	2	no	—
304	C₈	C₂×#6122	1	4	yes	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#11719125	1	2	yes	—
311	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	13	2	no	—
314	C₄	—	26	1	no	—
315	C₄	—	26	2	no	—
316	C₄	—	26	1	no	—
317	C₄	—	26	2	no	—
318	C₄	—	26	1	no	—
319	C₄	#11719125	1	2	yes	—
320	C₄	C₂×#88237	1	1	yes	characteristic
321	C₄	C₂²×#6122	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#11719125	1	1	yes	characteristic
325	C₂	D₁₃⋊#504	1	1	yes	characteristic
326	C₂	#13131804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₁₃×#131	C₄
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#131	#261
hypocenter	C₁₃	#33423
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #13131202	C₂	semidirect
2. #13131198	C₂	semidirect
3. #13131194	C₂	semidirect
4. #13131170	C₂	semidirect
5. C₂×#11718260	C₂	semidirect
7. #13131047	C₂	semidirect
275. C₁₃	#33423	semidirect

Maximal subgroups

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

subgroup	index
#13131202	2
#13131198	2
#13131194	2
#13131170	2
C₂×#11718260	2
#13131066	2
#13131047	2
#131×Dic₁₃	2
#33423	13