Subgroups — #15834348

Order 3328.

656 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123848	C₂	1	2	yes	maximal, complemented
2	C₄×#88158	C₂	1	1	yes	maximal, characteristic
3	#13113732	C₂	1	2	yes	maximal, complemented
4	#13113671	C₂	1	4	yes	maximal, complemented
5	#13113630	C₂	1	2	yes	maximal, complemented
6	#13113268	C₂	1	2	yes	maximal, complemented
7	C₈×#6099	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88158	C₄	1	1	yes	characteristic
10	C₄×#6127	C₂²	1	2	yes	—
11	C₁₃×#432	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718774	C₄	1	2	yes	—
14	#11718767	C₄	1	1	yes	characteristic
15	#11718586	C₂²	1	1	yes	characteristic
16	C₄×#6099	C₂²	1	1	yes	characteristic
17	#11718495	C₂²	1	4	yes	—
18	#11718492	C₂²	1	4	yes	—
19	#11718389	—	2	4	no	—
20	#11718366	—	2	2	no	—
21	D₁₃×#97	—	2	2	no	—
22	C₄×Dic₅₂	C₄	1	1	yes	characteristic
23	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
24	C₄×D₁₀₄	C₄	1	1	yes	characteristic
25	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
26	C₄×#6095	C₄	1	2	yes	—
27	C₄×#6095	C₂²	1	2	yes	—
28	C₁₃×#335	C₂²	1	2	yes	—
29	C₁₃×#334	C₂²	1	2	yes	—
30	#11718076	C₂²	1	4	yes	—
31	#11718075	C₂²	1	4	yes	—
32	#11718046	C₂²	1	2	yes	—
33	C₈×#1055	C₂²	1	2	yes	—
34	C₂×#6127	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₄×C₂²⋊D₁₃	—	2	2	no	—
39	C₄×C₂²⋊D₁₃	—	2	2	no	—
40	#88180	C₂×C₄	1	2	yes	—
41	C₂×#6099	D₄	1	2	yes	—
42	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
43	#88158	—	2	4	no	—
44	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
45	C₂×#6095	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₃	—	2	4	no	—
47	C₄×D₅₂	C₂×C₄	1	2	yes	—
48	C₄×D₅₂	C₂³	1	2	yes	—
49	#88126	—	2	2	no	—
50	C₄²×D₁₃	—	2	2	no	—
51	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
52	C₄×Dic₂₆	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
54	C₁₃×C₂²⋊C₈	—	2	2	no	—
55	C₁₃×#97	—	2	2	no	—
56	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂²⋊#1055	—	2	1	no	—
61	#88061	C₂×C₄	1	2	yes	—
62	#88061	C₂×C₄	1	2	yes	—
63	#88060	—	2	4	no	—
64	#88058	C₂×C₄	1	1	yes	characteristic
65	#88058	C₂×C₄	1	1	yes	characteristic
66	#88057	C₂×C₄	1	2	yes	—
67	#88056	C₂×C₄	1	2	yes	—
68	#88056	C₂×C₄	1	2	yes	—
69	#88055	—	2	4	no	—
70	#88054	—	2	4	no	—
71	C₄⋊#1055	—	2	1	no	—
72	#88043	D₄	1	1	yes	characteristic
73	#88043	D₄	1	1	yes	characteristic
74	C₄×#1055	C₂³	1	2	yes	—
75	C₄×#1055	C₂³	1	2	yes	—
76	#18035	—	13	1	no	maximal, Sylow
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
80	C₂×C₂²⋊D₁₃	—	2	2	no	—
81	#6127	—	2	4	no	—
82	#6127	—	4	2	no	—
83	C₂×D₅₂	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₁₃	—	2	2	no	—
85	C₂×C₄×D₁₃	—	2	2	no	—
86	C₂×C₄×D₁₃	—	2	2	no	—
87	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	4	no	—
90	C₂×C₈×C₁₃	—	4	1	no	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃	#41	1	1	yes	characteristic
98	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₂²⋊Dic₁₃	—	2	2	no	—
100	#6103	—	2	2	no	—
101	#6103	—	2	2	no	—
102	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
104	#6101	—	2	2	no	—
105	#6101	—	2	2	no	—
106	C₄×Dic₁₃	—	2	2	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	—	2	2	no	—
109	#6099	—	2	4	no	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	C₂×D₄	1	2	yes	—
112	C₂×#1055	C₂×D₄	1	2	yes	—
113	Dic₅₂	—	2	2	no	—
114	D₁₀₄	M₁₆	1	4	yes	complemented
115	#6095	—	2	4	no	—
116	C₈×D₁₃	—	4	8	no	—
117	#2970	—	13	2	no	—
118	C₄×#135	—	13	1	no	—
119	#1591	—	13	2	no	—
120	#1582	—	13	4	no	—
121	C₈×D₈	—	13	2	no	—
122	#1568	D₁₃	1	1	yes	characteristic, complemented
123	#1488	—	13	2	no	—
124	C₈×M₁₆	—	13	1	no	—
125	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
126	C₂²×D₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₂²⋊D₁₃	—	4	4	no	—
138	C₂×Dic₁₃	—	2	2	no	—
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	D₅₂	—	2	2	no	—
142	D₅₂	C₂×M₁₆	1	4	yes	—
143	C₄×D₁₃	—	4	4	no	—
144	C₄×D₁₃	—	4	4	no	—
145	C₄×D₁₃	—	8	2	no	—
146	Dic₂₆	—	2	2	no	—
147	Dic₂₆	—	2	2	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	—	4	4	no	—
152	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
153	#1055	—	2	2	no	—
154	#1055	—	4	2	no	—
155	#1055	#135	1	4	yes	complemented
156	C₂×#135	—	13	1	no	—
157	C₄×#41	—	13	2	no	—
158	#446	—	13	4	no	—
159	C₄×Q₁₆	—	13	1	no	—
160	C₄×Q₁₆	—	13	1	no	—
161	C₄×SD₁₆	—	13	2	no	—
162	C₄×SD₁₆	—	13	2	no	—
163	C₄×D₈	—	13	1	no	—
164	C₄×D₈	—	13	1	no	—
165	#435	—	26	4	no	—
166	C₈×D₄	—	13	4	no	—
167	#432	—	26	2	no	—
168	#432	C₂×D₁₃	1	2	yes	—
169	#431	—	13	1	no	—
170	#427	—	13	1	no	—
171	#416	—	13	2	no	—
172	C₄×M₁₆	—	13	1	no	—
173	C₂×#97	—	26	2	no	—
174	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
175	#335	C₂×D₁₃	1	2	yes	—
176	#334	C₂×D₁₃	1	2	yes	—
177	Q₈⋊C₈	—	13	4	no	—
178	D₄⋊C₈	—	13	4	no	—
179	#322	—	13	2	no	—
180	C₈²	—	13	2	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
185	C₂×D₁₃	—	2	4	no	—
186	C₂×D₁₃	—	4	2	no	—
187	C₂×D₁₃	—	4	2	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
193	C₄×C₁₃	#568	1	1	yes	characteristic
194	C₄×C₁₃	C₂×#135	1	2	yes	—
195	Dic₁₃	—	4	2	no	—
196	Dic₁₃	—	4	2	no	—
197	Dic₁₃	—	4	2	no	—
198	C₂×#41	—	13	2	no	—
199	#135	—	26	4	no	—
200	C₂×Q₁₆	—	13	1	no	—
201	C₂×SD₁₆	—	13	2	no	—
202	C₂×D₈	—	13	1	no	—
203	C₂×M₁₆	—	13	2	no	—
204	C₂²×C₈	—	26	4	no	—
205	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
206	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
207	C₄×Q₈	—	13	2	no	—
208	C₄×Q₈	—	13	2	no	—
209	C₄×D₄	—	13	2	no	—
210	C₄×D₄	—	13	2	no	—
211	C₄×D₄	—	26	2	no	—
212	C₄×D₄	—	26	2	no	—
213	#117	—	13	2	no	—
214	#117	—	26	2	no	—
215	C₂×C₄²	—	26	2	no	—
216	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
217	#107	—	13	1	no	—
218	#107	—	13	1	no	—
219	#106	—	13	2	no	—
220	C₄⋊C₈	—	26	4	no	—
221	C₄⋊C₈	—	26	1	no	—
222	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
223	#103	—	13	2	no	—
224	#103	—	13	2	no	—
225	#102	—	13	2	no	—
226	#102	—	13	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	—	26	4	no	—
229	C₂²⋊C₈	—	26	1	no	—
230	#97	—	2	2	no	—
231	#97	—	13	1	no	—
232	#97	—	13	1	no	—
233	#97	—	26	4	no	—
234	C₄×C₈	—	13	2	no	—
235	C₄×C₈	—	13	2	no	—
236	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
237	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₂×C₁₃	—	4	1	no	—
241	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
242	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
243	C₂×C₁₃	#2957	1	1	yes	characteristic
244	D₁₃	—	4	4	no	—
245	#41	—	26	4	no	—
246	#41	—	52	2	no	—
247	C₂×Q₈	—	13	2	no	—
248	C₂×D₄	—	13	2	no	—
249	C₂×D₄	—	26	2	no	—
250	C₂²×C₄	—	26	2	no	—
251	C₂²×C₄	—	26	2	no	—
252	C₂²×C₄	—	26	2	no	—
253	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	#6131	1	1	yes	characteristic
256	Q₁₆	—	26	2	no	—
257	SD₁₆	—	26	4	no	—
258	D₈	—	13	4	no	—
259	M₁₆	—	26	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	—	52	8	no	—
267	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
271	C₄⋊C₄	—	13	2	no	—
272	C₄⋊C₄	—	13	2	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	26	2	no	—
275	C₂²⋊C₄	—	26	2	no	—
276	C₂²⋊C₄	—	26	2	no	—
277	C₂²⋊C₄	—	26	2	no	—
278	C₄²	—	26	2	no	—
279	C₄²	—	26	2	no	—
280	C₄²	—	26	2	no	—
281	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
282	C₄²	D₄×D₁₃	1	1	yes	characteristic
283	C₄²	#6131	1	1	yes	characteristic
284	C₄²	C₂³×D₁₃	1	1	yes	characteristic
285	C₁₃	#18035	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	26	2	no	—
287	C₂³	#88147	1	1	yes	characteristic
288	Q₈	—	26	2	no	—
289	Q₈	—	26	2	no	—
290	D₄	—	13	4	no	—
291	D₄	—	26	2	no	—
292	D₄	—	52	4	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	26	2	no	—
298	C₂×C₄	—	26	2	no	—
299	C₂×C₄	—	26	2	no	—
300	C₂×C₄	—	52	4	no	—
301	C₂×C₄	—	52	4	no	—
302	C₂×C₄	—	104	2	no	—
303	C₂×C₄	#88147	1	1	yes	characteristic
304	C₂×C₄	#88147	1	1	yes	characteristic
305	C₂×C₄	#88147	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#6131	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	13	4	no	—
314	C₈	—	26	2	no	—
315	C₈	—	52	2	no	—
316	C₈	M₁₆×D₁₃	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	26	4	no	—
321	C₂²	—	52	2	no	—
322	C₂²	—	52	2	no	—
323	C₂²	C₂×#88147	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	52	2	no	—
329	C₄	—	52	2	no	—
330	C₄	—	52	2	no	—
331	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
332	C₄	#11719450	1	1	yes	characteristic
333	C₄	D₁₃×#135	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	52	4	no	—
336	C₂	#13123571	1	1	yes	characteristic
337	C₂	#13123607	1	1	yes	characteristic
338	C₂	#13123752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1568	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88158	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1568	D₁₃
hypocenter	C₁₃	#18035
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
1. #13123848	C₂	semidirect
3. #13113732	C₂	semidirect
4. #13113671	C₂	semidirect
5. #13113630	C₂	semidirect
6. #13113268	C₂	semidirect
7. C₈×#6099	C₂	semidirect
8. C₁₃×#1568	C₂	semidirect
114. D₁₀₄	M₁₆	semidirect
122. #1568	D₁₃	semidirect
155. #1055	#135	semidirect
285. C₁₃	#18035	semidirect

Maximal subgroups

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

subgroup	index
#13123848	2
C₄×#88158	2
#13113732	2
#13113671	2
#13113630	2
#13113268	2
C₈×#6099	2
C₁₃×#1568	2
#18035	13