Subgroups — #15850849

Order 3328.

1050 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125756	C₂	1	2	yes	maximal, complemented
2	#13125415	C₂	1	1	yes	maximal, characteristic, complemented
3	#13125288	C₂	1	2	yes	maximal, complemented
4	C₂×#11718356	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114502	C₂	1	4	yes	maximal, complemented
6	#13114451	C₂	1	2	yes	maximal, complemented
7	#13114428	C₂	1	2	yes	maximal, complemented
8	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88162	—	2	4	no	—
10	C₂²×D₁₀₄	—	2	2	no	—
11	#11719182	C₂²	1	2	yes	—
12	C₂×#88128	C₂²	1	2	yes	—
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718776	—	2	2	no	—
17	#11718504	—	2	4	no	—
18	#11718393	—	2	4	no	—
19	#11718373	—	2	4	no	—
20	#11718356	—	2	2	no	—
21	#11718356	C₂²	1	2	yes	—
22	#11718356	C₂²	1	2	yes	—
23	#11718356	C₂²	1	1	yes	characteristic, complemented
24	#11718356	C₂²	1	4	yes	complemented
25	#11718356	C₂²	1	4	yes	complemented
26	#11718355	C₂²	1	2	yes	complemented
27	#11718355	C₂²	1	4	yes	complemented
28	#11718354	C₂²	1	4	yes	complemented
29	C₄×D₁₀₄	—	2	2	no	—
30	#11718343	C₂²	1	1	yes	characteristic, complemented
31	C₁₃×#322	C₂²	1	2	yes	complemented
32	C₈²×C₁₃	C₂²	1	2	yes	complemented
33	C₂×#6127	—	2	2	no	—
34	C₂²×D₅₂	—	2	2	no	—
35	C₂²×D₅₂	—	2	2	no	—
36	C₂²×D₅₂	—	2	2	no	—
37	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	#88184	—	2	2	no	—
41	#88184	—	2	2	no	—
42	#88162	—	4	16	no	—
43	C₂×D₁₀₄	—	2	4	no	—
44	C₂×D₁₀₄	—	2	4	no	—
45	C₂×D₁₀₄	—	2	8	no	—
46	C₂×D₁₀₄	—	2	2	no	—
47	C₂×D₁₀₄	—	2	8	no	—
48	C₂×D₁₀₄	—	4	8	no	—
49	C₂×#6095	—	2	8	no	—
50	#88129	—	2	2	no	—
51	#88128	—	2	2	no	—
52	#88128	—	2	2	no	—
53	#88128	—	2	2	no	—
54	#88128	C₂³	1	4	yes	—
55	#88128	C₂³	1	2	yes	—
56	C₄×D₅₂	—	2	2	no	—
57	C₄×D₅₂	—	2	2	no	—
58	#88123	C₂³	1	2	yes	—
59	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
60	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
61	C₁₃×#97	D₄	1	2	yes	—
62	C₄×C₈×C₁₃	D₄	1	2	yes	—
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	2	yes	—
65	C₄×C₈×C₁₃	C₂³	1	2	yes	—
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	#88061	—	2	8	no	—
68	#88061	—	2	4	no	—
69	#88058	—	2	2	no	—
70	#19629	—	13	1	no	maximal, Sylow
71	C₂³×D₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₁₃	—	4	2	no	—
76	#6127	—	4	8	no	—
77	C₂×D₅₂	—	2	2	no	—
78	C₂×D₅₂	—	2	2	no	—
79	C₂×D₅₂	—	2	4	no	—
80	C₂×D₅₂	—	2	4	no	—
81	C₂×D₅₂	—	2	4	no	—
82	C₂×D₅₂	—	2	2	no	—
83	C₂×D₅₂	—	4	8	no	—
84	C₂×D₅₂	—	4	8	no	—
85	C₂×D₅₂	—	4	2	no	—
86	C₂×D₅₂	—	4	2	no	—
87	C₂×D₅₂	—	4	2	no	—
88	C₂×D₅₂	—	8	4	no	—
89	C₂×C₄×D₁₃	—	4	2	no	—
90	C₂×Dic₂₆	—	2	2	no	—
91	C₁₃×M₁₆	D₈	1	8	yes	complemented
92	C₂×C₈×C₁₃	—	2	4	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
97	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
101	#6103	—	4	2	no	—
102	#6103	—	4	2	no	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	D₁₀₄	—	4	8	no	—
106	D₁₀₄	—	4	16	no	—
107	D₁₀₄	—	4	4	no	—
108	D₁₀₄	—	4	8	no	—
109	D₁₀₄	—	8	8	no	—
110	#6095	—	4	16	no	—
111	#3205	—	13	2	no	—
112	#3169	—	13	1	no	—
113	#3151	—	13	2	no	—
114	C₂×#493	—	13	1	no	—
115	#1721	—	13	4	no	—
116	#1712	—	13	2	no	—
117	#1706	—	13	2	no	—
118	C₈×M₁₆	D₁₃	1	1	yes	characteristic, complemented
119	C₂³×C₁₃	#127	1	1	yes	characteristic
120	C₂²×D₁₃	—	4	2	no	—
121	C₂²×D₁₃	—	4	4	no	—
122	C₂²×D₁₃	—	8	4	no	—
123	C₂²×D₁₃	—	8	4	no	—
124	C₂²×D₁₃	—	8	4	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₁₃	—	8	4	no	—
135	C₂×Dic₁₃	—	4	2	no	—
136	D₅₂	—	4	8	no	—
137	D₅₂	—	4	4	no	—
138	D₅₂	—	4	8	no	—
139	D₅₂	—	4	4	no	—
140	D₅₂	—	4	4	no	—
141	D₅₂	—	4	8	no	—
142	D₅₂	—	8	4	no	—
143	D₅₂	—	8	4	no	—
144	D₅₂	—	8	4	no	—
145	C₄×D₁₃	—	8	4	no	—
146	Dic₂₆	—	4	4	no	—
147	C₈×C₁₃	—	2	2	no	—
148	C₈×C₁₃	—	2	4	no	—
149	C₈×C₁₃	C₂×D₈	1	8	yes	complemented
150	C₈×C₁₃	AGL(1,ℤ/8)	1	4	yes	complemented
151	C₂×AGL(1,ℤ/8)	—	26	4	no	—
152	C₂²×D₈	—	26	2	no	—
153	#532	—	13	2	no	—
154	C₂×#127	—	13	2	no	—
155	#500	—	13	1	no	—
156	#496	—	26	4	no	—
157	#493	—	13	2	no	—
158	#493	—	13	2	no	—
159	#493	—	13	4	no	—
160	#493	—	13	1	no	—
161	#493	—	13	4	no	—
162	#493	—	26	2	no	—
163	#492	—	13	4	no	—
164	#486	—	13	2	no	—
165	#486	—	13	4	no	—
166	#466	—	26	2	no	—
167	#459	—	26	4	no	—
168	#447	—	26	4	no	—
169	C₄×D₈	—	26	2	no	—
170	#431	C₂×D₁₃	1	1	yes	characteristic
171	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
173	#322	C₂×D₁₃	1	2	yes	complemented
174	C₈²	C₂×D₁₃	1	2	yes	complemented
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	#493	1	2	yes	—
178	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
179	C₂×D₁₃	—	8	2	no	—
180	C₂×D₁₃	—	8	4	no	—
181	C₂×D₁₃	—	8	4	no	—
182	C₂×D₁₃	—	8	2	no	—
183	C₂×D₁₃	—	8	4	no	—
184	C₂×D₁₃	—	8	2	no	—
185	C₂×D₁₃	—	16	8	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	#493	1	2	yes	—
189	C₄×C₁₃	C₂²×D₈	1	2	yes	—
190	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
191	C₄×C₁₃	#576	1	2	yes	—
192	Dic₁₃	—	8	2	no	—
193	C₂×#41	—	26	2	no	—
194	C₂²×D₄	—	26	2	no	—
195	C₂²×D₄	—	26	2	no	—
196	C₂²×D₄	—	26	2	no	—
197	AGL(1,ℤ/8)	—	52	16	no	—
198	C₂×SD₁₆	—	26	8	no	—
199	C₂×D₈	—	26	4	no	—
200	C₂×D₈	—	26	4	no	—
201	C₂×D₈	—	26	8	no	—
202	C₂×D₈	—	26	2	no	—
203	C₂×D₈	—	26	8	no	—
204	C₂×D₈	—	52	8	no	—
205	C₂×M₁₆	D₅₂	1	2	yes	—
206	C₂²×C₈	D₅₂	1	2	yes	—
207	C₄⋊Q₈	—	13	2	no	—
208	#127	—	13	4	no	—
209	#127	—	13	2	no	—
210	#127	—	26	2	no	—
211	#127	—	26	2	no	—
212	#127	—	26	2	no	—
213	#124	—	26	2	no	—
214	#121	—	26	2	no	—
215	#121	—	26	2	no	—
216	C₄×D₄	—	26	2	no	—
217	C₄×D₄	—	26	2	no	—
218	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
219	#107	—	26	2	no	—
220	C₄⋊C₈	D₅₂	1	2	yes	—
221	#102	—	26	4	no	—
222	#102	—	26	8	no	—
223	C₂²⋊C₈	D₅₂	1	2	yes	—
224	#97	D₅₂	1	2	yes	—
225	C₄×C₈	D₅₂	1	2	yes	—
226	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₁₃	1	2	yes	—
228	C₄×C₈	C₂²×D₁₃	1	2	yes	—
229	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
230	C₂×C₁₃	—	2	2	no	—
231	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
232	C₂×C₁₃	#3151	1	1	yes	characteristic
233	C₂×C₁₃	#3155	1	1	yes	characteristic
234	D₁₃	—	16	2	no	—
235	D₁₃	—	16	4	no	—
236	C₂⁴	—	52	2	no	—
237	#41	—	52	8	no	—
238	C₂×Q₈	—	26	2	no	—
239	C₂×D₄	—	26	2	no	—
240	C₂×D₄	—	26	4	no	—
241	C₂×D₄	—	26	2	no	—
242	C₂×D₄	—	26	2	no	—
243	C₂×D₄	—	26	4	no	—
244	C₂×D₄	—	26	4	no	—
245	C₂×D₄	—	52	8	no	—
246	C₂×D₄	—	52	2	no	—
247	C₂×D₄	—	52	2	no	—
248	C₂×D₄	—	52	2	no	—
249	C₂×D₄	—	52	2	no	—
250	C₂×D₄	—	52	8	no	—
251	C₂×D₄	—	104	4	no	—
252	C₂²×C₄	—	52	2	no	—
253	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
256	SD₁₆	—	52	16	no	—
257	D₈	—	52	8	no	—
258	D₈	—	52	16	no	—
259	D₈	—	52	8	no	—
260	D₈	—	52	4	no	—
261	D₈	—	104	8	no	—
262	M₁₆	D₁₀₄	1	8	yes	complemented
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	C₂×D₅₂	1	2	yes	—
265	C₂×C₈	C₂×D₅₂	1	4	yes	—
266	C₂×C₈	C₂×D₅₂	1	4	yes	—
267	C₂×C₈	C₂×D₅₂	1	2	yes	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	—	26	2	no	—
270	C₂²⋊C₄	—	52	2	no	—
271	C₂²⋊C₄	—	52	2	no	—
272	C₄²	C₂×D₅₂	1	1	yes	characteristic
273	C₄²	C₂×D₅₂	1	1	yes	characteristic
274	C₄²	C₂×D₅₂	1	1	yes	characteristic
275	C₄²	C₂³×D₁₃	1	1	yes	characteristic
276	C₁₃	#19629	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	52	2	no	—
278	C₂³	—	52	4	no	—
279	C₂³	—	104	4	no	—
280	C₂³	—	104	4	no	—
281	C₂³	—	104	4	no	—
282	C₂³	#88128	1	1	yes	characteristic
283	Q₈	—	52	4	no	—
284	D₄	—	52	4	no	—
285	D₄	—	52	8	no	—
286	D₄	—	52	8	no	—
287	D₄	—	52	4	no	—
288	D₄	—	52	4	no	—
289	D₄	—	52	8	no	—
290	D₄	—	104	4	no	—
291	D₄	—	104	4	no	—
292	D₄	—	104	4	no	—
293	D₄	—	104	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	52	2	no	—
297	C₂×C₄	—	104	4	no	—
298	C₂×C₄	#88128	1	1	yes	characteristic
299	C₂×C₄	#88128	1	1	yes	characteristic
300	C₂×C₄	#88128	1	1	yes	characteristic
301	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
302	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	C₂×D₁₀₄	1	8	yes	complemented
308	C₈	#88162	1	4	yes	complemented
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	104	2	no	—
312	C₂²	—	104	4	no	—
313	C₂²	—	104	4	no	—
314	C₂²	—	104	2	no	—
315	C₂²	—	104	4	no	—
316	C₂²	—	104	2	no	—
317	C₂²	—	208	8	no	—
318	C₂²	#11718356	1	2	yes	—
319	C₂²	C₂×#88128	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	104	2	no	—
323	C₄	#11718356	1	2	yes	—
324	C₄	C₂²×D₁₀₄	1	2	yes	—
325	C₄	C₂×#88162	1	2	yes	—
326	C₄	#11719457	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	208	4	no	—
329	C₂	—	208	2	no	—
330	C₂	C₂×#11718356	1	1	yes	characteristic
331	C₂	#13125288	1	1	yes	characteristic
332	C₂	#13125322	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88128
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×#127
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#19629
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #13125756	C₂	semidirect
2. #13125415	C₂	semidirect
3. #13125288	C₂	semidirect
4. C₂×#11718356	C₂	semidirect
5. #13114502	C₂	semidirect
6. #13114451	C₂	semidirect
7. #13114428	C₂	semidirect
8. C₈×C₁₃×M₁₆	C₂	semidirect
23. #11718356	C₂²	semidirect
24. #11718356	C₂²	semidirect
25. #11718356	C₂²	semidirect
26. #11718355	C₂²	semidirect
27. #11718355	C₂²	semidirect
28. #11718354	C₂²	semidirect
30. #11718343	C₂²	semidirect
31. C₁₃×#322	C₂²	semidirect
32. C₈²×C₁₃	C₂²	semidirect
91. C₁₃×M₁₆	D₈	semidirect
118. C₈×M₁₆	D₁₃	semidirect
149. C₈×C₁₃	C₂×D₈	semidirect
150. C₈×C₁₃	AGL(1,ℤ/8)	semidirect
173. #322	C₂×D₁₃	semidirect
174. C₈²	C₂×D₁₃	semidirect
262. M₁₆	D₁₀₄	semidirect
276. C₁₃	#19629	semidirect
307. C₈	C₂×D₁₀₄	semidirect
308. C₈	#88162	semidirect

Maximal subgroups

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

subgroup	index
#13125756	2
#13125415	2
#13125288	2
C₂×#11718356	2
#13114502	2
#13114451	2
#13114428	2
C₈×C₁₃×M₁₆	2
#19629	13