Subgroups — #16807081

Order 3328.

600 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131203	C₂	1	4	yes	maximal, complemented
2	#13131199	C₂	1	2	yes	maximal, complemented
3	#13131195	C₂	1	2	yes	maximal, complemented
4	#13131173	C₂	1	2	yes	maximal, complemented
5	#13131065	C₂	1	1	yes	maximal, characteristic
6	C₂×#11718259	C₂	1	1	yes	maximal, characteristic
7	#13131045	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	C₁₃×#405	C₄	1	1	yes	characteristic
15	#11718783	C₂²	1	1	yes	characteristic
16	#11718762	C₄	1	1	yes	characteristic
17	#11718658	C₂²	1	1	yes	characteristic
18	#11718369	C₂²	1	2	yes	—
19	D₁₃×#97	C₄	1	2	yes	—
20	#11718350	C₄	1	2	yes	—
21	C₄×C₈×D₁₃	C₂²	1	2	yes	—
22	#11718260	C₂²	1	4	yes	complemented
23	#11718260	C₂²	1	4	yes	complemented
24	#11718259	C₂²	1	1	yes	characteristic
25	#11718259	C₂²	1	1	yes	characteristic
26	#11718259	C₂²	1	2	yes	—
27	#11718259	C₂²	1	2	yes	—
28	#11718259	C₂²	1	2	yes	—
29	#11718231	C₂²	1	4	yes	complemented
30	C₄×#6092	C₂²	1	4	yes	complemented
31	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
35	#88180	C₂×C₄	1	1	yes	characteristic
36	C₂×#6099	C₂×C₄	1	1	yes	characteristic
37	C₂²×#1055	C₂³	1	1	yes	characteristic
38	M₁₆×D₁₃	—	2	4	no	—
39	C₂×#6094	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	—	2	4	no	—
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
46	#88146	C₈	1	1	yes	characteristic
47	#88146	C₈	1	1	yes	characteristic
48	C₄⋊C₄×D₁₃	C₈	1	1	yes	characteristic
49	C₄⋊C₄×D₁₃	C₈	1	1	yes	characteristic
50	C₂²⋊C₄×D₁₃	C₈	1	2	yes	—
51	#88131	C₈	1	2	yes	—
52	#88126	C₂×C₄	1	2	yes	—
53	C₄²×D₁₃	C₂×C₄	1	2	yes	—
54	#88106	—	2	4	no	—
55	C₂×#6092	—	2	4	no	—
56	C₂×#6092	C₂³	1	2	yes	—
57	C₂×#6092	C₂³	1	4	yes	—
58	C₂×#6092	C₂³	1	2	yes	—
59	C₁₃×#97	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
61	#88055	C₂×C₄	1	2	yes	—
62	#88055	C₂×C₄	1	2	yes	—
63	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
64	C₈×Dic₁₃	C₂³	1	2	yes	—
65	#88043	C₂×C₄	1	2	yes	—
66	C₄×#1055	C₂³	1	2	yes	—
67	#33422	—	13	1	no	maximal, Sylow
68	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₁₃	—	2	2	no	—
72	C₂×C₄×D₁₃	—	2	2	no	—
73	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₁₃×M₁₆	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
85	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
86	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
87	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
89	#6103	C₂×C₈	1	2	yes	—
90	#6103	C₂×C₈	1	2	yes	—
91	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
92	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
93	#6101	C₂×C₈	1	2	yes	—
94	#6101	C₂×C₈	1	2	yes	—
95	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
96	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
97	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
98	#6099	M₁₆	1	4	yes	—
99	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
100	C₂×#1055	C₂²×C₄	1	2	yes	—
101	C₂×#1055	#41	1	4	yes	—
102	C₂×#1055	C₂⁴	1	1	yes	characteristic
103	#6094	—	2	4	no	—
104	C₈×D₁₃	—	2	2	no	—
105	C₈×D₁₃	—	2	2	no	—
106	C₈×D₁₃	—	2	4	no	—
107	C₈×D₁₃	—	2	2	no	—
108	C₈×D₁₃	—	2	2	no	—
109	#6092	—	2	2	no	—
110	#6092	—	2	2	no	—
111	#6092	—	2	4	no	—
112	C₂×#405	—	13	1	no	—
113	#2164	—	13	4	no	—
114	#2163	—	13	2	no	—
115	#2162	—	13	2	no	—
116	#2151	—	13	2	no	—
117	#2112	—	13	1	no	—
118	C₂×C₂²⋊C₁₆	—	13	1	no	—
119	#2108	—	13	2	no	—
120	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
121	C₂²×D₁₃	—	2	2	no	—
122	C₂²×D₁₃	—	2	2	no	—
123	C₂²×D₁₃	—	4	1	no	—
124	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
125	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#117	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₁₃	—	2	2	no	—
133	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
135	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
136	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
137	C₄×D₁₃	—	2	4	no	—
138	C₄×D₁₃	—	2	1	no	—
139	C₄×D₁₃	—	2	1	no	—
140	C₄×D₁₃	—	2	1	no	—
141	C₄×D₁₃	—	2	1	no	—
142	C₄×D₁₃	—	2	2	no	—
143	C₄×D₁₃	—	2	2	no	—
144	C₄×D₁₃	—	4	2	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	2	no	—
148	#1055	#117	1	2	yes	—
149	#1055	C₂×M₁₆	1	4	yes	—
150	#1055	C₂×#41	1	2	yes	—
151	C₂²×M₁₆	—	13	1	no	—
152	C₂³×C₈	—	13	1	no	—
153	C₂×#117	—	13	1	no	—
154	C₂×M₃₂	—	13	2	no	—
155	C₂²×C₁₆	—	13	2	no	—
156	#405	—	13	1	no	—
157	#405	—	13	1	no	—
158	#405	—	13	1	no	—
159	#405	—	13	2	no	—
160	#405	—	13	2	no	—
161	#405	#261	1	1	yes	characteristic
162	C₂×#97	—	13	2	no	—
163	C₂×C₄×C₈	—	13	2	no	—
164	C₄⋊C₁₆	—	13	4	no	—
165	C₄⋊C₁₆	—	13	4	no	—
166	C₂²⋊C₁₆	—	13	1	no	—
167	C₂²⋊C₁₆	—	13	1	no	—
168	C₂²⋊C₁₆	—	13	2	no	—
169	C₂²⋊C₁₆	—	13	2	no	—
170	C₂²⋊C₁₆	—	13	2	no	—
171	#346	—	13	4	no	—
172	C₄×C₁₆	—	13	4	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	#431	1	2	yes	—
176	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
177	C₂×D₁₃	—	2	1	no	—
178	C₂×D₁₃	—	2	1	no	—
179	C₂×D₁₃	—	2	1	no	—
180	C₂×D₁₃	—	2	1	no	—
181	C₂×D₁₃	—	2	1	no	—
182	C₂×D₁₃	—	4	4	no	—
183	C₂×D₁₃	#504	1	2	yes	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	#431	1	2	yes	—
187	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
188	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
189	Dic₁₃	—	2	2	no	—
190	Dic₁₃	—	2	2	no	—
191	Dic₁₃	—	2	1	no	—
192	Dic₁₃	—	2	1	no	—
193	C₂³×C₄	—	13	1	no	—
194	C₂×M₁₆	—	13	1	no	—
195	C₂×M₁₆	—	13	2	no	—
196	C₂×M₁₆	—	26	4	no	—
197	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
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	1	no	—
203	C₂²×C₈	—	13	2	no	—
204	C₂²×C₈	—	26	4	no	—
205	C₂²×C₈	C₂×#261	1	1	yes	characteristic
206	#117	—	13	1	no	—
207	#117	—	13	1	no	—
208	#117	—	13	1	no	—
209	#117	—	13	2	no	—
210	#117	—	13	2	no	—
211	#117	C₂×#261	1	1	yes	characteristic
212	C₂×C₄⋊C₄	—	13	1	no	—
213	C₂×C₄⋊C₄	—	13	1	no	—
214	C₂×C₂²⋊C₄	—	13	2	no	—
215	C₂×C₄²	—	13	2	no	—
216	M₃₂	—	26	4	no	—
217	C₂×C₁₆	—	13	2	no	—
218	C₂×C₁₆	—	13	4	no	—
219	C₂×C₁₆	—	13	2	no	—
220	C₂×C₁₆	—	26	4	no	—
221	#97	—	13	2	no	—
222	#97	—	13	2	no	—
223	#97	—	13	2	no	—
224	#97	C₂×#261	1	2	yes	—
225	C₄×C₈	—	13	2	no	—
226	C₄×C₈	—	13	2	no	—
227	C₄×C₈	—	13	2	no	—
228	C₄×C₈	C₂×#261	1	2	yes	—
229	C₂×C₁₃	—	2	2	no	—
230	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
231	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
232	C₂×C₁₃	#3407	1	1	yes	characteristic
233	D₁₃	—	2	2	no	—
234	D₁₃	—	4	1	no	—
235	C₂⁴	—	13	1	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	2	no	—
238	C₂²×C₄	—	13	1	no	—
239	C₂²×C₄	—	13	1	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	26	2	no	—
244	C₂²×C₄	—	26	2	no	—
245	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
246	M₁₆	—	2	2	no	—
247	M₁₆	—	13	4	no	—
248	M₁₆	—	26	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	13	4	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	13	1	no	—
254	C₂×C₈	—	26	2	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	—	26	4	no	—
259	C₂×C₈	C₂²×#261	1	1	yes	characteristic
260	C₂×C₈	C₂²×#261	1	2	yes	—
261	C₂×C₈	C₂²×#261	1	1	yes	characteristic
262	C₄⋊C₄	—	13	2	no	—
263	C₄⋊C₄	—	13	2	no	—
264	C₄⋊C₄	—	13	1	no	—
265	C₄⋊C₄	—	13	1	no	—
266	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
267	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
268	C₂²⋊C₄	—	13	2	no	—
269	C₂²⋊C₄	—	13	2	no	—
270	C₂²⋊C₄	—	13	2	no	—
271	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
272	C₄²	—	13	2	no	—
273	C₄²	—	13	2	no	—
274	C₄²	—	13	2	no	—
275	C₄²	C₂²×#261	1	2	yes	—
276	C₁₆	—	26	2	no	—
277	C₁₆	—	26	2	no	—
278	C₁₆	—	26	4	no	—
279	C₁₃	#33422	1	1	yes	characteristic, complemented, Sylow
280	C₂³	—	13	1	no	—
281	C₂³	—	13	1	no	—
282	C₂³	—	26	2	no	—
283	C₂³	—	26	2	no	—
284	C₂³	—	52	1	no	—
285	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	2	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	26	4	no	—
291	C₂×C₄	—	26	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	1	no	—
295	C₂×C₄	—	26	1	no	—
296	C₂×C₄	—	26	1	no	—
297	C₂×C₄	—	26	1	no	—
298	C₂×C₄	—	52	2	no	—
299	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
300	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
301	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
302	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
303	C₂×C₄	#88237	1	2	yes	—
304	C₂×C₄	C₂³×#261	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	13	4	no	—
309	C₈	—	13	2	no	—
310	C₈	—	13	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	13	2	no	—
314	C₂²	—	26	1	no	—
315	C₂²	—	26	1	no	—
316	C₂²	—	26	1	no	—
317	C₂²	—	26	1	no	—
318	C₂²	—	26	1	no	—
319	C₂²	—	52	4	no	—
320	C₂²	#11719049	1	2	yes	—
321	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	26	2	no	—
325	C₄	—	26	2	no	—
326	C₄	—	26	1	no	—
327	C₄	—	26	1	no	—
328	C₄	#11719049	1	2	yes	—
329	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
330	C₄	C₂×#88237	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	26	2	no	—
333	C₂	—	52	1	no	—
334	C₂	C₂×#11719049	1	1	yes	characteristic
335	C₂	D₁₃⋊#504	1	1	yes	characteristic
336	C₂	#13131804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#405	C₄
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#405	#261
hypocenter	C₁₃	#33422
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #13131203	C₂	semidirect
2. #13131199	C₂	semidirect
3. #13131195	C₂	semidirect
4. #13131173	C₂	semidirect
7. #13131045	C₂	semidirect
22. #11718260	C₂²	semidirect
23. #11718260	C₂²	semidirect
29. #11718231	C₂²	semidirect
30. C₄×#6092	C₂²	semidirect
279. C₁₃	#33422	semidirect

Maximal subgroups

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

subgroup	index
#13131203	2
#13131199	2
#13131195	2
#13131173	2
#13131065	2
C₂×#11718259	2
#13131045	2
D₁₃×#405	2
#33422	13