Subgroups — #16806730

Order 3328.

826 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88098	C₂	1	1	yes	maximal, characteristic
2	C₂×D₁₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic
3	#13131047	C₂	1	2	yes	maximal, complemented
4	#13131046	C₂	1	4	yes	maximal, complemented
5	#13131045	C₂	1	2	yes	maximal, complemented
6	#13131043	C₂	1	2	yes	maximal, complemented
7	C₄×#88098	C₂	1	2	yes	maximal, complemented
8	D₁₃×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×D₁₃	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
11	D₁₃×#117	C₄	1	1	yes	characteristic
12	C₂×#88106	C₂²	1	2	yes	—
13	C₂²×#6092	C₂²	1	2	yes	—
14	D₁₃⋊M₃₂	C₄	1	8	yes	complemented
15	C₂×#88098	C₄	1	8	yes	complemented
16	C₂×#88098	C₂²	1	2	yes	—
17	C₂×#88098	C₂²	1	1	yes	characteristic
18	C₂×#88098	C₂²	1	1	yes	characteristic
19	C₂×#88098	C₂²	1	2	yes	—
20	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
21	C₁₃×#405	C₄	1	1	yes	characteristic
22	#11718783	C₂²	1	1	yes	characteristic
23	#11718762	C₄	1	1	yes	characteristic
24	#11718658	C₂²	1	1	yes	characteristic
25	#11718369	C₂²	1	2	yes	—
26	D₁₃×#97	C₄	1	2	yes	—
27	#11718350	C₄	1	2	yes	—
28	C₄×C₈×D₁₃	C₂²	1	2	yes	—
29	#11718258	C₂²	1	4	yes	complemented
30	#11718257	C₂²	1	4	yes	complemented
31	#11718231	C₂²	1	4	yes	complemented
32	C₄×#6092	C₂²	1	4	yes	complemented
33	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
37	#88180	C₂×C₄	1	1	yes	characteristic
38	C₂×#6099	C₂×C₄	1	1	yes	characteristic
39	C₂²×#1055	C₂³	1	1	yes	characteristic
40	M₁₆×D₁₃	C₈	1	8	yes	—
41	C₂×#6094	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
45	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
46	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
49	#88146	C₈	1	2	yes	—
50	C₄⋊C₄×D₁₃	C₈	1	2	yes	—
51	C₂²⋊C₄×D₁₃	C₈	1	2	yes	—
52	#88131	C₈	1	2	yes	—
53	#88126	C₂×C₄	1	2	yes	—
54	C₄²×D₁₃	C₂×C₄	1	2	yes	—
55	#88106	C₂×C₄	1	8	yes	complemented
56	C₂×#6092	C₂×C₄	1	8	yes	complemented
57	C₂×#6092	C₂³	1	2	yes	—
58	C₂×#6092	C₂³	1	4	yes	—
59	C₂×#6092	C₂³	1	2	yes	—
60	#88098	C₂×C₄	1	4	yes	—
61	#88098	C₂×C₄	1	4	yes	—
62	#88098	C₂×C₄	1	8	yes	—
63	#88098	C₂×C₄	1	8	yes	complemented
64	D₁₃⋊C₁₆	C₂×C₄	1	8	yes	complemented
65	C₁₃×#97	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
67	#88055	C₂×C₄	1	2	yes	—
68	#88055	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
70	C₈×Dic₁₃	C₂³	1	2	yes	—
71	#88043	C₂×C₄	1	2	yes	—
72	C₄×#1055	C₂³	1	2	yes	—
73	#33324	—	13	1	no	maximal, Sylow
74	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₁₃	C₄²	1	4	yes	—
78	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
79	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
82	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
85	C₂×C₈×C₁₃	C₄²	1	4	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
90	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
93	#6103	C₂×C₈	1	4	yes	—
94	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
95	#6101	C₂×C₈	1	4	yes	—
96	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
97	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
98	#6099	C₂×C₈	1	4	yes	—
99	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
100	C₂×#1055	C₂²×C₄	1	2	yes	—
101	C₂×#1055	C₂²×C₄	1	2	yes	—
102	C₂×#1055	C₂²×C₄	1	2	yes	—
103	C₂×#1055	C₂⁴	1	1	yes	characteristic
104	#6094	C₂×C₈	1	8	yes	—
105	C₈×D₁₃	C₄²	1	4	yes	—
106	C₈×D₁₃	C₄²	1	4	yes	—
107	C₈×D₁₃	C₂×C₈	1	8	yes	—
108	C₈×D₁₃	C₂²×C₄	1	2	yes	—
109	C₈×D₁₃	C₂²×C₄	1	2	yes	—
110	C₈×D₁₃	C₂²×C₄	1	2	yes	—
111	C₈×D₁₃	C₂²×C₄	1	2	yes	—
112	#6092	C₂²×C₄	1	8	yes	—
113	#6092	C₂²×C₄	1	4	yes	—
114	#6092	C₂²×C₄	1	4	yes	—
115	C₂²×M₃₂	—	13	1	no	—
116	C₂²×M₃₂	—	13	1	no	—
117	C₂×#405	—	13	1	no	—
118	#2108	—	13	2	no	—
119	#2108	—	13	4	no	—
120	#2108	—	13	2	no	—
121	C₄×M₃₂	—	13	2	no	—
122	C₄×M₃₂	—	13	2	no	—
123	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
124	C₂²×D₁₃	—	4	2	no	—
125	C₂²×D₁₃	C₄×C₈	1	4	yes	—
126	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
127	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×Dic₁₃	C₄×C₈	1	4	yes	—
134	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
135	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
137	C₄×D₁₃	—	4	4	no	—
138	C₄×D₁₃	C₄×C₈	1	4	yes	—
139	C₄×D₁₃	C₄×C₈	1	4	yes	—
140	C₄×D₁₃	C₂×C₄²	1	2	yes	—
141	C₄×D₁₃	C₂×C₄²	1	2	yes	—
142	C₄×D₁₃	C₂²×C₈	1	2	yes	—
143	C₄×D₁₃	C₂²×C₈	1	2	yes	—
144	C₈×C₁₃	C₂×C₄²	1	2	yes	—
145	C₈×C₁₃	C₂×C₄²	1	2	yes	—
146	C₈×C₁₃	C₂²×C₈	1	4	yes	—
147	#1055	C₂×C₄²	1	2	yes	—
148	#1055	C₂²×C₈	1	4	yes	—
149	#1055	C₂³×C₄	1	1	yes	characteristic
150	#1055	C₂³×C₄	1	1	yes	characteristic
151	C₂²×M₁₆	—	13	1	no	—
152	C₂³×C₈	—	13	1	no	—
153	C₂×#117	—	13	1	no	—
154	C₂×M₃₂	—	13	1	no	—
155	C₂×M₃₂	—	13	2	no	—
156	C₂×M₃₂	—	13	1	no	—
157	C₂×M₃₂	—	13	2	no	—
158	C₂×M₃₂	—	13	2	no	—
159	C₂×M₃₂	—	13	8	no	—
160	C₂×M₃₂	—	13	8	no	—
161	C₂²×C₁₆	—	13	2	no	—
162	C₂²×C₁₆	—	13	2	no	—
163	#405	—	13	1	no	—
164	#405	—	13	1	no	—
165	#405	—	13	1	no	—
166	#405	—	13	2	no	—
167	#405	—	13	2	no	—
168	#405	#261	1	1	yes	characteristic
169	C₂×#97	—	13	2	no	—
170	C₂×C₄×C₈	—	13	2	no	—
171	#346	—	13	4	no	—
172	#346	—	13	4	no	—
173	C₄×C₁₆	—	13	4	no	—
174	C₄×C₁₆	—	13	4	no	—
175	C₂²×C₁₃	—	2	2	no	—
176	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
177	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
178	C₂×D₁₃	—	2	2	no	—
179	C₂×D₁₃	—	2	2	no	—
180	C₂×D₁₃	—	4	4	no	—
181	C₂×D₁₃	C₂×C₄×C₈	1	2	yes	—
182	C₂×D₁₃	C₂×C₄×C₈	1	2	yes	—
183	C₄×C₁₃	—	2	4	no	—
184	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
185	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
186	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
187	Dic₁₃	—	2	4	no	—
188	Dic₁₃	C₂×C₄×C₈	1	2	yes	—
189	Dic₁₃	C₂×C₄×C₈	1	2	yes	—
190	C₂³×C₄	—	13	1	no	—
191	C₂×M₁₆	—	13	1	no	—
192	C₂×M₁₆	—	13	8	no	—
193	C₂×M₁₆	—	13	2	no	—
194	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
195	C₂²×C₈	—	13	1	no	—
196	C₂²×C₈	—	13	4	no	—
197	C₂²×C₈	—	13	4	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	13	1	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	13	2	no	—
203	C₂²×C₈	C₂×#261	1	1	yes	characteristic
204	#117	—	13	1	no	—
205	#117	—	13	2	no	—
206	#117	—	13	2	no	—
207	#117	—	13	2	no	—
208	#117	C₂×#261	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	13	2	no	—
210	C₂×C₂²⋊C₄	—	13	2	no	—
211	C₂×C₄²	—	13	2	no	—
212	M₃₂	—	13	4	no	—
213	M₃₂	—	13	4	no	—
214	M₃₂	—	13	8	no	—
215	M₃₂	—	13	8	no	—
216	M₃₂	—	13	8	no	—
217	C₂×C₁₆	—	13	8	no	—
218	C₂×C₁₆	—	13	8	no	—
219	C₂×C₁₆	—	13	2	no	—
220	C₂×C₁₆	—	13	4	no	—
221	C₂×C₁₆	—	13	2	no	—
222	#97	—	13	2	no	—
223	#97	—	13	2	no	—
224	#97	—	13	2	no	—
225	#97	C₂×#261	1	2	yes	—
226	C₄×C₈	—	13	2	no	—
227	C₄×C₈	—	13	2	no	—
228	C₄×C₈	—	13	2	no	—
229	C₄×C₈	C₂×#261	1	2	yes	—
230	C₂×C₁₃	—	2	2	no	—
231	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
232	C₂×C₁₃	#3407	1	2	yes	—
233	D₁₃	—	2	2	no	—
234	D₁₃	—	2	2	no	—
235	C₂⁴	—	13	1	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	1	no	—
239	C₂²×C₄	—	13	4	no	—
240	C₂²×C₄	—	13	4	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	1	no	—
243	C₂²×C₄	—	13	4	no	—
244	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
245	M₁₆	—	13	4	no	—
246	M₁₆	—	13	8	no	—
247	M₁₆	D₁₃⋊C₈	1	4	yes	—
248	C₂×C₈	—	13	2	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	13	1	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	13	1	no	—
253	C₂×C₈	—	13	8	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	2	no	—
258	C₂×C₈	—	13	4	no	—
259	C₂×C₈	—	13	4	no	—
260	C₂×C₈	C₄×#261	1	4	yes	—
261	C₂×C₈	C₂²×#261	1	2	yes	—
262	C₂×C₈	C₂²×#261	1	1	yes	characteristic
263	C₂×C₈	C₂²×#261	1	1	yes	characteristic
264	C₄⋊C₄	—	13	4	no	—
265	C₄⋊C₄	—	13	2	no	—
266	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
267	C₂²⋊C₄	—	13	2	no	—
268	C₂²⋊C₄	—	13	4	no	—
269	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
270	C₄²	—	13	4	no	—
271	C₄²	—	13	2	no	—
272	C₄²	C₂²×#261	1	2	yes	—
273	C₁₆	—	13	8	no	—
274	C₁₆	—	13	4	no	—
275	C₁₆	—	13	4	no	—
276	C₁₃	#33324	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	13	1	no	—
278	C₂³	—	13	1	no	—
279	C₂³	—	13	4	no	—
280	C₂³	—	52	2	no	—
281	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
282	C₂×C₄	—	13	4	no	—
283	C₂×C₄	—	13	1	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	13	4	no	—
286	C₂×C₄	—	13	4	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	2	no	—
289	C₂×C₄	—	13	2	no	—
290	C₂×C₄	—	13	2	no	—
291	C₂×C₄	—	13	4	no	—
292	C₂×C₄	—	52	4	no	—
293	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
294	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
295	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
296	C₂×C₄	C₂×C₄×#261	1	2	yes	—
297	C₂×C₄	C₂³×#261	1	1	yes	characteristic
298	C₈	—	13	4	no	—
299	C₈	—	13	1	no	—
300	C₈	—	13	1	no	—
301	C₈	—	13	2	no	—
302	C₈	C₂×D₁₃⋊C₈	1	4	yes	—
303	C₈	C₂×C₄×#261	1	2	yes	—
304	C₈	C₂×C₄×#261	1	2	yes	—
305	C₂²	—	2	2	no	—
306	C₂²	—	13	2	no	—
307	C₂²	—	13	2	no	—
308	C₂²	—	26	2	no	—
309	C₂²	—	26	2	no	—
310	C₂²	—	52	4	no	—
311	C₂²	C₄×D₁₃⋊C₈	1	2	yes	—
312	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
313	C₄	—	2	4	no	—
314	C₄	—	13	2	no	—
315	C₄	—	13	2	no	—
316	C₄	—	26	4	no	—
317	C₄	C₄×D₁₃⋊C₈	1	2	yes	—
318	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
319	C₄	C₂²×C₄×#261	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	26	2	no	—
322	C₂	—	26	2	no	—
323	C₂	C₂×C₄×D₁₃⋊C₈	1	1	yes	characteristic
324	C₂	#13131804	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₁₃×#405	C₄
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#405	#261
hypocenter	C₁₃	#33324
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
3. #13131047	C₂	semidirect
4. #13131046	C₂	semidirect
5. #13131045	C₂	semidirect
6. #13131043	C₂	semidirect
7. C₄×#88098	C₂	semidirect
14. D₁₃⋊M₃₂	C₄	semidirect
15. C₂×#88098	C₄	semidirect
29. #11718258	C₂²	semidirect
30. #11718257	C₂²	semidirect
31. #11718231	C₂²	semidirect
32. C₄×#6092	C₂²	semidirect
55. #88106	C₂×C₄	semidirect
56. C₂×#6092	C₂×C₄	semidirect
63. #88098	C₂×C₄	semidirect
64. D₁₃⋊C₁₆	C₂×C₄	semidirect
276. C₁₃	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88098	2
C₂×D₁₃⋊M₃₂	2
#13131047	2
#13131046	2
#13131045	2
#13131043	2
C₄×#88098	2
D₁₃×#405	2
#33324	13