Subgroups — #13127730

Order 1664.

374 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719432	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719244	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718741	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718675	C₂	1	1	yes	maximal, characteristic
6	#11718532	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718531	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718517	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718516	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718508	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718489	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718446	C₂	1	1	yes	maximal, characteristic
13	#11718422	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718420	C₂	1	1	yes	maximal, characteristic, complemented
15	#11718417	C₂	1	1	yes	maximal, characteristic
16	C₂×Q₈×D₁₃	—	2	1	no	—
17	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
19	#88200	—	2	1	no	—
20	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
21	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×#6104	C₂²	1	1	yes	characteristic
23	#88178	—	2	1	no	—
24	C₂×#6101	—	2	1	no	—
25	#88150	—	2	1	no	—
26	#88150	—	2	1	no	—
27	#88147	C₂²	1	1	yes	characteristic
28	#88147	C₂²	1	1	yes	characteristic
29	#88144	C₂²	1	1	yes	characteristic
30	#88142	C₂²	1	1	yes	characteristic
31	#88142	C₂²	1	1	yes	characteristic
32	#88142	C₂²	1	1	yes	characteristic
33	#88132	—	2	1	no	—
34	#88132	—	2	1	no	—
35	C₄×D₅₂	C₂²	1	1	yes	characteristic
36	#88124	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
38	C₁₃×#106	C₂²	1	1	yes	characteristic
39	C₁₃×#106	C₂²	1	1	yes	characteristic
40	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₃×#102	C₂²	1	1	yes	characteristic
42	C₁₃×#102	C₂²	1	1	yes	characteristic
43	#88072	C₂²	1	1	yes	characteristic
44	#88061	C₂²	1	1	yes	characteristic
45	#88061	C₂²	1	1	yes	characteristic
46	#88058	C₂²	1	1	yes	characteristic
47	#88057	C₂²	1	1	yes	characteristic
48	#88055	C₂²	1	1	yes	characteristic
49	#88054	C₂²	1	1	yes	characteristic
50	#88054	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
52	#88049	C₂²	1	1	yes	characteristic
53	#88048	C₂²	1	1	yes	characteristic
54	#88048	C₂²	1	1	yes	characteristic
55	#88048	C₂²	1	1	yes	characteristic
56	#88047	C₂²	1	1	yes	characteristic
57	#88047	C₂²	1	1	yes	characteristic
58	#88043	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	—	2	1	no	—
60	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×Dic₁₃	—	2	1	no	—
63	C₂²×Dic₁₃	—	2	1	no	—
64	Q₈×D₁₃	—	4	4	no	—
65	C₂×D₅₂	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₃	—	2	1	no	—
67	C₂×C₄×D₁₃	—	2	1	no	—
68	C₂×C₄×D₁₃	—	2	1	no	—
69	C₂×Dic₂₆	—	2	1	no	—
70	C₂×Dic₂₆	—	2	1	no	—
71	C₂×Dic₂₆	—	2	1	no	—
72	C₂×Dic₂₆	D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	—	2	1	no	—
76	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₁₃×C₂²⋊C₄	—	2	1	no	—
80	C₄²×C₁₃	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₁₃	—	2	1	no	—
82	C₂²⋊Dic₁₃	—	2	1	no	—
83	#6104	Q₈	1	4	yes	complemented
84	#6103	—	2	1	no	—
85	#6103	—	2	1	no	—
86	#6103	—	2	1	no	—
87	C₄⋊Dic₁₃	—	2	1	no	—
88	C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
89	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
92	#6101	—	2	1	no	—
93	#6101	—	2	1	no	—
94	#6101	—	2	1	no	—
95	#6101	—	2	1	no	—
96	#6101	—	2	1	no	—
97	#6101	—	2	1	no	—
98	#6101	D₄	1	1	yes	characteristic
99	#6101	D₄	1	1	yes	characteristic
100	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
101	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
102	C₂×#1055	C₂³	1	1	yes	characteristic
103	C₂×#1055	C₂³	1	1	yes	characteristic
104	#3384	—	13	1	no	maximal, Sylow
105	C₂³×C₁₃	—	2	1	no	—
106	C₂²×D₁₃	—	2	1	no	—
107	Q₈×C₁₃	—	4	2	no	—
108	D₄×C₁₃	—	2	1	no	—
109	D₄×C₁₃	C₂×Q₈	1	2	yes	complemented
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	8	1	no	—
114	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₁₃	—	2	1	no	—
118	C₂×Dic₁₃	—	2	1	no	—
119	C₂×Dic₁₃	—	2	1	no	—
120	C₂×Dic₁₃	—	4	2	no	—
121	C₂×Dic₁₃	—	8	1	no	—
122	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
126	D₅₂	—	2	1	no	—
127	D₅₂	C₂×Q₈	1	2	yes	—
128	C₄×D₁₃	—	4	2	no	—
129	C₄×D₁₃	—	4	2	no	—
130	C₄×D₁₃	—	4	2	no	—
131	Dic₂₆	—	2	2	no	—
132	Dic₂₆	—	4	2	no	—
133	Dic₂₆	—	4	2	no	—
134	Dic₂₆	—	8	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	1	no	—
137	#1055	—	2	1	no	—
138	#1055	C₂×Q₈	1	2	yes	complemented
139	#554	—	13	1	no	—
140	D₄×Q₈	—	13	1	no	—
141	#501	—	13	1	no	—
142	#501	—	13	1	no	—
143	#498	—	13	1	no	—
144	#478	—	13	1	no	—
145	#478	—	13	1	no	—
146	#476	—	13	1	no	—
147	#476	—	13	1	no	—
148	#476	D₁₃	1	1	yes	characteristic, complemented
149	#474	—	13	1	no	—
150	#446	—	13	1	no	—
151	#442	—	13	1	no	—
152	#442	—	13	1	no	—
153	C₄×D₈	—	13	1	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	4	1	no	—
156	C₂²×C₁₃	—	4	1	no	—
157	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₁₃	—	2	2	no	—
159	C₂×D₁₃	—	4	1	no	—
160	C₂×D₁₃	—	4	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	4	1	no	—
164	C₄×C₁₃	—	4	1	no	—
165	C₄×C₁₃	—	4	1	no	—
166	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
167	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
168	Dic₁₃	—	2	1	no	—
169	Dic₁₃	—	2	1	no	—
170	Dic₁₃	—	2	1	no	—
171	Dic₁₃	—	4	1	no	—
172	Dic₁₃	—	4	1	no	—
173	Dic₁₃	—	4	1	no	—
174	Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
175	C₂²×Q₈	—	26	1	no	—
176	C₂×D₈	—	13	1	no	—
177	C₄⋊Q₈	—	13	1	no	—
178	C₄⋊Q₈	—	13	1	no	—
179	C₄⋊Q₈	—	13	1	no	—
180	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
181	#125	—	13	1	no	—
182	#125	—	13	1	no	—
183	C₂²⋊Q₈	—	26	1	no	—
184	C₂²⋊Q₈	—	26	1	no	—
185	C₂²⋊Q₈	—	26	1	no	—
186	C₂²⋊Q₈	—	26	1	no	—
187	C₂²⋊Q₈	—	26	1	no	—
188	C₂²⋊Q₈	—	26	1	no	—
189	C₄×Q₈	—	13	1	no	—
190	C₄×D₄	—	13	1	no	—
191	C₄×D₄	—	13	1	no	—
192	C₄×D₄	—	13	1	no	—
193	C₄×D₄	—	13	1	no	—
194	C₄×D₄	—	13	1	no	—
195	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	26	1	no	—
197	#107	—	13	1	no	—
198	#107	—	13	1	no	—
199	#107	—	13	1	no	—
200	#106	—	13	1	no	—
201	#106	—	13	1	no	—
202	#106	—	13	1	no	—
203	#106	—	13	1	no	—
204	#106	C₂×D₁₃	1	1	yes	characteristic
205	#106	C₂×D₁₃	1	1	yes	characteristic
206	C₄⋊C₈	—	13	1	no	—
207	C₄⋊C₈	—	13	1	no	—
208	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
209	#102	—	13	1	no	—
210	#102	—	13	1	no	—
211	#102	—	13	1	no	—
212	#102	—	13	1	no	—
213	#102	C₂×D₁₃	1	1	yes	characteristic
214	#102	C₂×D₁₃	1	1	yes	characteristic
215	#97	—	13	1	no	—
216	#97	—	13	1	no	—
217	C₄×C₈	—	13	1	no	—
218	C₂×C₁₃	—	4	2	no	—
219	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
220	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂×C₁₃	#577	1	1	yes	characteristic
222	D₁₃	—	4	2	no	—
223	C₂×Q₈	—	2	1	no	—
224	C₂×Q₈	—	13	1	no	—
225	C₂×Q₈	—	26	1	no	—
226	C₂×Q₈	—	26	1	no	—
227	C₂×Q₈	—	26	1	no	—
228	C₂×Q₈	—	52	4	no	—
229	C₂×D₄	—	13	1	no	—
230	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	26	1	no	—
233	C₂²×C₄	—	26	1	no	—
234	C₂²×C₄	—	26	1	no	—
235	C₂²×C₄	—	26	1	no	—
236	C₂²×C₄	—	26	1	no	—
237	D₈	—	13	4	no	—
238	C₂×C₈	—	13	1	no	—
239	C₂×C₈	—	13	1	no	—
240	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	13	1	no	—
244	C₄⋊C₄	—	13	1	no	—
245	C₄⋊C₄	—	13	1	no	—
246	C₄⋊C₄	—	13	1	no	—
247	C₄⋊C₄	—	13	1	no	—
248	C₄⋊C₄	—	13	1	no	—
249	C₄⋊C₄	—	26	1	no	—
250	C₄⋊C₄	—	26	1	no	—
251	C₄⋊C₄	—	26	1	no	—
252	C₄⋊C₄	—	26	1	no	—
253	C₄⋊C₄	—	26	1	no	—
254	C₄⋊C₄	—	26	1	no	—
255	C₄⋊C₄	—	26	1	no	—
256	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	26	1	no	—
261	C₂²⋊C₄	—	26	1	no	—
262	C₂²⋊C₄	—	26	1	no	—
263	C₂²⋊C₄	—	26	1	no	—
264	C₂²⋊C₄	—	26	1	no	—
265	C₄²	—	13	1	no	—
266	C₄²	—	13	1	no	—
267	C₄²	C₂²×D₁₃	1	1	yes	characteristic
268	C₁₃	#3384	1	1	yes	characteristic, complemented, Sylow
269	C₂³	—	2	1	no	—
270	C₂³	—	26	1	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	26	2	no	—
273	Q₈	—	52	2	no	—
274	Q₈	—	52	2	no	—
275	Q₈	—	104	1	no	—
276	D₄	—	2	1	no	—
277	D₄	—	13	2	no	—
278	D₄	—	26	1	no	—
279	D₄	Q₈×D₁₃	1	2	yes	complemented
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	1	no	—
288	C₂×C₄	—	26	1	no	—
289	C₂×C₄	—	26	1	no	—
290	C₂×C₄	—	26	1	no	—
291	C₂×C₄	—	52	2	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	52	2	no	—
295	C₂×C₄	—	104	1	no	—
296	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	13	2	no	—
302	C₈	—	26	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	26	2	no	—
307	C₂²	—	52	1	no	—
308	C₂²	—	52	1	no	—
309	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	13	2	no	—
316	C₄	—	26	1	no	—
317	C₄	—	26	1	no	—
318	C₄	—	26	1	no	—
319	C₄	—	52	1	no	—
320	C₄	—	52	1	no	—
321	C₄	—	52	1	no	—
322	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
323	C₄	#88257	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	52	2	no	—
326	C₂	#11719337	1	1	yes	characteristic
327	C₂	#11719467	1	1	yes	characteristic
328	C₂	D₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#476	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×#6104	C₂²
hypercenter	#476	D₁₃
hypocenter	C₁₃	#3384
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #11719432	C₂	semidirect
2. #11719244	C₂	semidirect
3. C₁₃×#476	C₂	semidirect
4. #11718741	C₂	semidirect
6. #11718532	C₂	semidirect
7. #11718531	C₂	semidirect
8. #11718517	C₂	semidirect
9. #11718516	C₂	semidirect
10. #11718508	C₂	semidirect
11. #11718489	C₂	semidirect
13. #11718422	C₂	semidirect
14. #11718420	C₂	semidirect
83. #6104	Q₈	semidirect
109. D₄×C₁₃	C₂×Q₈	semidirect
138. #1055	C₂×Q₈	semidirect
148. #476	D₁₃	semidirect
268. C₁₃	#3384	semidirect
279. D₄	Q₈×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
#11719432	2
#11719244	2
C₁₃×#476	2
#11718741	2
#11718675	2
#11718532	2
#11718531	2
#11718517	2
#11718516	2
#11718508	2
#11718489	2
#11718446	2
#11718422	2
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