Subgroups — #28407346

Order 14464.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₁₃⋊#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16813709	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299645	C₂	1	2	yes	maximal, complemented
4	#20299643	C₂	1	2	yes	maximal, complemented
5	#20299641	C₂	1	2	yes	maximal, complemented
6	#20299640	C₂	1	2	yes	maximal, complemented
7	#20299613	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299596	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299595	C₂	1	1	yes	maximal, characteristic, complemented
10	#20299564	C₂	1	1	yes	maximal, characteristic, complemented
11	#20299394	C₂	1	1	yes	maximal, characteristic
12	#16813710	C₂²	1	1	yes	characteristic, complemented
13	#16813710	C₂²	1	1	yes	characteristic, complemented
14	#16813709	—	2	1	no	—
15	#16813709	—	2	1	no	—
16	#16813709	—	2	1	no	—
17	#16813709	—	2	1	no	—
18	#16813709	C₂²	1	1	yes	characteristic, complemented
19	#16813709	C₂²	1	2	yes	complemented
20	#16813709	C₂²	1	2	yes	complemented
21	#16813709	C₂²	1	1	yes	characteristic, complemented
22	C₂×#14268657	C₂²	1	1	yes	characteristic
23	D₁₁₃⋊M₁₆	—	2	1	no	—
24	D₁₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
25	C₂×#14268656	—	2	1	no	—
26	C₂×#14268655	—	2	1	no	—
27	C₂×#14268655	C₂²	1	1	yes	characteristic
28	C₂²×#11744753	—	2	1	no	—
29	#16813701	C₂²	1	1	yes	characteristic
30	#16813693	C₂²	1	2	yes	complemented
31	#16813692	C₂²	1	2	yes	complemented
32	#16813687	C₂²	1	2	yes	complemented
33	#16813686	C₂²	1	2	yes	complemented
34	#16813686	C₂²	1	2	yes	complemented
35	#16813686	C₂²	1	2	yes	complemented
36	#16813676	C₂²	1	2	yes	complemented
37	C₈×#90673	C₂²	1	2	yes	complemented
38	C₂×#14268646	C₂²	1	1	yes	characteristic
39	#41×D₁₁₃	C₂²	1	1	yes	characteristic
40	#16813655	C₄	1	1	yes	characteristic, complemented
41	#16813652	C₄	1	1	yes	characteristic, complemented
42	#16813651	C₄	1	2	yes	complemented
43	#16813648	C₄	1	1	yes	characteristic, complemented
44	#16813646	C₄	1	2	yes	complemented
45	#16813643	C₂²	1	2	yes	—
46	#16813641	C₂²	1	2	yes	—
47	M₁₆×D₁₁₃	C₂²	1	1	yes	characteristic
48	AGL(1,ℤ/8)×C₁₁₃	C₄	1	1	yes	characteristic, complemented
49	#14268657	—	2	1	no	—
50	#14268657	—	2	1	no	—
51	#14268657	—	2	1	no	—
52	#14268657	C₂³	1	1	yes	characteristic
53	#14268657	C₂³	1	1	yes	characteristic
54	#14268657	C₂³	1	2	yes	—
55	#14268656	—	2	1	no	—
56	#14268656	—	2	2	no	—
57	#14268656	—	2	2	no	—
58	#14268656	—	2	1	no	—
59	#14268656	—	2	1	no	—
60	#14268655	—	2	1	no	—
61	#14268655	—	2	1	no	—
62	#14268655	—	2	1	no	—
63	#14268655	D₄	1	1	yes	characteristic
64	#14268655	D₄	1	1	yes	characteristic
65	#14268655	C₂³	1	1	yes	characteristic
66	#14268655	C₂³	1	1	yes	characteristic
67	C₂×#11744753	—	2	1	no	—
68	C₂×#11744753	—	2	2	no	—
69	C₂×#11744753	—	2	2	no	—
70	C₂×#11744753	—	2	1	no	—
71	C₂×#11744753	—	2	1	no	—
72	C₂×#11744753	D₄	1	1	yes	characteristic
73	C₂×#11744753	D₄	1	1	yes	characteristic
74	C₄⋊#90673	—	2	1	no	—
75	C₂²⋊#90673	—	2	1	no	—
76	C₄×#90673	C₂³	1	2	yes	—
77	#14268648	C₂³	1	1	yes	characteristic
78	#14268647	—	2	1	no	—
79	#14268647	C₂×C₄	1	1	yes	characteristic
80	#14268646	—	2	1	no	—
81	#14268646	—	2	1	no	—
82	#14268646	—	2	1	no	—
83	#14268646	C₂×C₄	1	2	yes	—
84	#14268646	C₂³	1	1	yes	characteristic
85	#14268646	C₂³	1	2	yes	—
86	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic
87	Q₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₁₃	—	2	1	no	—
89	D₄×D₁₁₃	—	2	1	no	—
90	D₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
91	C₂²×Dic₁₁₃	—	2	1	no	—
92	C₂×C₄×D₁₁₃	—	2	1	no	—
93	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
94	Dic₄₅₂	C₂×C₄	1	2	yes	—
95	#14268637	C₂×C₄	1	2	yes	—
96	#14268636	C₂×C₄	1	2	yes	—
97	#14268635	C₂×C₄	1	2	yes	—
98	#14268635	C₂×C₄	1	2	yes	—
99	#14268632	C₂×C₄	1	2	yes	—
100	#14268631	C₂×C₄	1	1	yes	characteristic
101	#14268630	C₂×C₄	1	2	yes	—
102	C₈×D₁₁₃	C₂³	1	2	yes	—
103	#41×C₁₁₃	C₂×C₄	1	1	yes	characteristic
104	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
105	SD₁₆×C₁₁₃	C₂×C₄	1	2	yes	—
106	D₈×C₁₁₃	C₂×C₄	1	2	yes	—
107	M₁₆×C₁₁₃	C₂×C₄	1	1	yes	characteristic
108	C₂³×C₁₁₃	—	2	1	no	—
109	C₂²×D₁₁₃	—	2	1	no	—
110	C₂²×D₁₁₃	#41	1	1	yes	characteristic
111	C₂×#90673	—	2	2	no	—
112	Q₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
113	D₄×C₁₁₃	—	2	1	no	—
114	D₄×C₁₁₃	—	2	1	no	—
115	D₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
116	D₄×C₁₁₃	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₁₁₃	—	2	1	no	—
118	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²⋊D₁₁₃	—	2	1	no	—
120	C₂²⋊D₁₁₃	—	2	2	no	—
121	C₂²⋊D₁₁₃	—	2	1	no	—
122	C₂²⋊D₁₁₃	—	2	2	no	—
123	C₂²⋊D₁₁₃	—	2	1	no	—
124	C₂×Dic₁₁₃	—	2	1	no	—
125	C₂×Dic₁₁₃	—	2	2	no	—
126	C₂×Dic₁₁₃	—	2	1	no	—
127	C₂×Dic₁₁₃	—	2	2	no	—
128	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
129	D₄₅₂	—	2	1	no	—
130	D₄₅₂	C₂²×C₄	1	1	yes	characteristic
131	C₄×D₁₁₃	—	2	1	no	—
132	C₄×D₁₁₃	—	2	1	no	—
133	C₄×D₁₁₃	—	2	1	no	—
134	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
136	C₄×D₁₁₃	#41	1	1	yes	characteristic
137	C₄×D₁₁₃	C₂⁴	1	1	yes	characteristic
138	Dic₂₂₆	—	2	1	no	—
139	Dic₂₂₆	—	2	1	no	—
140	Dic₂₂₆	C₂²×C₄	1	1	yes	characteristic
141	Dic₂₂₆	C₂²×C₄	1	2	yes	—
142	#11744753	—	2	2	no	—
143	#11744753	—	2	1	no	—
144	#11744753	—	2	1	no	—
145	#11744753	C₂×D₄	1	1	yes	characteristic
146	#11744753	C₂×D₄	1	1	yes	characteristic
147	#11744753	C₂×D₄	1	1	yes	characteristic
148	#11744753	C₂×D₄	1	1	yes	characteristic
149	C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
150	#11744751	C₂²×C₄	1	2	yes	—
151	C₂²×C₁₁₃	—	2	2	no	—
152	C₂²×C₁₁₃	—	2	1	no	—
153	C₂²×C₁₁₃	—	4	2	no	—
154	C₂²×C₁₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂×D₁₁₃	—	2	1	no	—
156	C₂×D₁₁₃	—	4	1	no	—
157	C₂×D₁₁₃	—	4	2	no	—
158	C₂×D₁₁₃	C₄×D₄	1	1	yes	characteristic
159	C₂×D₁₁₃	C₂×#41	1	1	yes	characteristic
160	#90673	—	4	2	no	—
161	C₄×C₁₁₃	—	2	1	no	—
162	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
164	Dic₁₁₃	—	2	1	no	—
165	Dic₁₁₃	—	2	2	no	—
166	Dic₁₁₃	C₄×D₄	1	1	yes	characteristic
167	Dic₁₁₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₁₁₃	—	2	1	no	—
169	C₂×C₁₁₃	—	4	1	no	—
170	C₂×C₁₁₃	—	4	2	no	—
171	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
172	D₁₁₃	—	2	1	no	—
173	D₁₁₃	—	2	1	no	—
174	D₁₁₃	—	4	1	no	—
175	#2955	—	113	1	no	maximal, Sylow
176	C₁₁₃	#2955	1	1	yes	characteristic, complemented, Sylow
177	#575	—	113	1	no	—
178	C₂×#131	—	113	1	no	—
179	C₂×#131	—	113	1	no	—
180	#444	—	113	2	no	—
181	#444	—	113	2	no	—
182	#443	—	113	2	no	—
183	#443	—	113	2	no	—
184	#430	—	113	1	no	—
185	#421	—	113	1	no	—
186	#421	—	113	1	no	—
187	#405	—	113	1	no	—
188	C₂×#41	—	113	1	no	—
189	C₂×#41	—	113	1	no	—
190	#137	—	113	1	no	—
191	#137	—	113	2	no	—
192	#137	—	113	1	no	—
193	AGL(1,ℤ/8)	—	113	1	no	—
194	AGL(1,ℤ/8)	—	113	2	no	—
195	AGL(1,ℤ/8)	#90673	1	1	yes	characteristic, complemented
196	#135	—	113	2	no	—
197	#135	—	113	2	no	—
198	#131	—	113	2	no	—
199	#131	—	113	1	no	—
200	#131	—	113	2	no	—
201	#131	—	113	1	no	—
202	#131	—	113	1	no	—
203	#131	—	113	1	no	—
204	#131	—	226	1	no	—
205	#131	—	226	1	no	—
206	#131	—	226	1	no	—
207	#131	—	226	1	no	—
208	C₂×M₁₆	—	113	1	no	—
209	C₂×M₁₆	—	113	1	no	—
210	C₂×M₁₆	—	113	1	no	—
211	C₂×M₁₆	—	226	1	no	—
212	C₂×M₁₆	—	226	1	no	—
213	C₂²×C₈	—	113	1	no	—
214	C₂²×C₈	—	226	1	no	—
215	C₂²×C₈	—	226	1	no	—
216	#117	—	113	1	no	—
217	#108	—	113	2	no	—
218	#108	—	113	2	no	—
219	#104	—	113	2	no	—
220	#104	—	113	2	no	—
221	#104	—	113	2	no	—
222	#104	—	113	2	no	—
223	#97	—	113	2	no	—
224	C₄×C₈	—	113	2	no	—
225	#41	—	113	1	no	—
226	#41	—	113	2	no	—
227	#41	—	113	1	no	—
228	#41	—	113	2	no	—
229	#41	—	113	1	no	—
230	#41	—	226	1	no	—
231	#41	—	226	1	no	—
232	#41	—	226	1	no	—
233	#41	—	226	1	no	—
234	#41	C₂×#90673	1	1	yes	characteristic
235	C₂×Q₈	—	113	1	no	—
236	C₂×Q₈	—	113	1	no	—
237	C₂×D₄	—	113	1	no	—
238	C₂×D₄	—	226	1	no	—
239	C₂×D₄	—	226	1	no	—
240	C₂×D₄	C₂×#90673	1	1	yes	characteristic
241	C₂²×C₄	—	113	1	no	—
242	C₂²×C₄	—	226	1	no	—
243	C₂²×C₄	—	226	1	no	—
244	Q₁₆	—	113	2	no	—
245	Q₁₆	—	113	2	no	—
246	SD₁₆	—	113	2	no	—
247	SD₁₆	—	113	2	no	—
248	SD₁₆	—	113	2	no	—
249	SD₁₆	C₂×#90673	1	2	yes	—
250	D₈	—	113	2	no	—
251	D₈	C₂×#90673	1	2	yes	—
252	M₁₆	—	113	1	no	—
253	M₁₆	—	113	2	no	—
254	M₁₆	—	113	1	no	—
255	M₁₆	—	113	1	no	—
256	M₁₆	—	113	2	no	—
257	M₁₆	—	226	2	no	—
258	M₁₆	—	226	1	no	—
259	M₁₆	—	226	2	no	—
260	M₁₆	—	226	1	no	—
261	M₁₆	—	226	1	no	—
262	M₁₆	—	226	1	no	—
263	M₁₆	—	226	1	no	—
264	M₁₆	—	226	1	no	—
265	M₁₆	C₂×#90673	1	1	yes	characteristic
266	C₂×C₈	—	113	1	no	—
267	C₂×C₈	—	113	1	no	—
268	C₂×C₈	—	113	1	no	—
269	C₂×C₈	—	113	1	no	—
270	C₂×C₈	—	113	1	no	—
271	C₂×C₈	—	113	1	no	—
272	C₂×C₈	—	113	2	no	—
273	C₂×C₈	—	226	2	no	—
274	C₂×C₈	—	226	1	no	—
275	C₂×C₈	—	226	2	no	—
276	C₂×C₈	—	226	1	no	—
277	C₂×C₈	—	226	1	no	—
278	C₂×C₈	—	226	1	no	—
279	C₂×C₈	—	226	1	no	—
280	C₂×C₈	—	226	1	no	—
281	C₄⋊C₄	—	226	1	no	—
282	C₂²⋊C₄	—	226	1	no	—
283	C₄²	—	113	2	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	113	1	no	—
286	C₂³	—	226	1	no	—
287	Q₈	—	113	1	no	—
288	Q₈	—	113	2	no	—
289	Q₈	—	226	1	no	—
290	Q₈	—	226	1	no	—
291	Q₈	C₂²×#90673	1	1	yes	characteristic
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	113	1	no	—
295	D₄	—	226	1	no	—
296	D₄	—	226	1	no	—
297	D₄	—	226	1	no	—
298	D₄	—	226	2	no	—
299	D₄	—	226	2	no	—
300	D₄	—	226	1	no	—
301	D₄	C₂²×#90673	1	2	yes	—
302	D₄	C₂²×#90673	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	113	1	no	—
305	C₂×C₄	—	113	1	no	—
306	C₂×C₄	—	113	1	no	—
307	C₂×C₄	—	113	1	no	—
308	C₂×C₄	—	113	1	no	—
309	C₂×C₄	—	226	1	no	—
310	C₂×C₄	—	226	1	no	—
311	C₂×C₄	—	226	1	no	—
312	C₂×C₄	—	226	2	no	—
313	C₂×C₄	—	226	2	no	—
314	C₂×C₄	—	226	2	no	—
315	C₂×C₄	—	226	1	no	—
316	C₂×C₄	—	226	1	no	—
317	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
318	C₈	—	113	1	no	—
319	C₈	—	113	1	no	—
320	C₈	—	113	1	no	—
321	C₈	—	113	1	no	—
322	C₈	—	113	2	no	—
323	C₈	—	226	2	no	—
324	C₈	—	226	1	no	—
325	C₈	—	226	1	no	—
326	C₈	C₂²×#90673	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	113	1	no	—
331	C₂²	—	113	1	no	—
332	C₂²	—	452	1	no	—
333	C₂²	—	452	2	no	—
334	C₂²	—	226	1	no	—
335	C₂²	D₄×#90673	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	113	1	no	—
338	C₄	—	113	1	no	—
339	C₄	—	452	2	no	—
340	C₄	—	226	1	no	—
341	C₄	—	226	2	no	—
342	C₄	D₄×#90673	1	1	yes	characteristic
343	C₄	C₂³×#90673	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	452	1	no	—
348	C₂	—	226	1	no	—
349	C₂	—	226	1	no	—
350	C₂	C₂×D₄×#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#90673
commutator	C₄×C₁₁₃	C₂³×C₄
Frattini	C₄	C₂³×#90673
Fitting	AGL(1,ℤ/8)×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₂×C₄×D₄
prime radical	#20299394	C₂
cosocle	C₄×D₁₁₃	C₂⁴
hypercenter	AGL(1,ℤ/8)	#90673
hypocenter	C₁₁₃	#2955
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. D₁₁₃⋊#131	C₂	semidirect
2. C₂×#16813709	C₂	semidirect
3. #20299645	C₂	semidirect
4. #20299643	C₂	semidirect
5. #20299641	C₂	semidirect
6. #20299640	C₂	semidirect
7. #20299613	C₂	semidirect
8. #20299596	C₂	semidirect
9. #20299595	C₂	semidirect
10. #20299564	C₂	semidirect
12. #16813710	C₂²	semidirect
13. #16813710	C₂²	semidirect
18. #16813709	C₂²	semidirect
19. #16813709	C₂²	semidirect
20. #16813709	C₂²	semidirect
21. #16813709	C₂²	semidirect
30. #16813693	C₂²	semidirect
31. #16813692	C₂²	semidirect
32. #16813687	C₂²	semidirect
33. #16813686	C₂²	semidirect
34. #16813686	C₂²	semidirect
35. #16813686	C₂²	semidirect
36. #16813676	C₂²	semidirect
37. C₈×#90673	C₂²	semidirect
40. #16813655	C₄	semidirect
41. #16813652	C₄	semidirect
42. #16813651	C₄	semidirect
43. #16813648	C₄	semidirect
44. #16813646	C₄	semidirect
48. AGL(1,ℤ/8)×C₁₁₃	C₄	semidirect
176. C₁₁₃	#2955	semidirect
195. AGL(1,ℤ/8)	#90673	semidirect

Maximal subgroups

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

subgroup	index
D₁₁₃⋊#131	2
C₂×#16813709	2
#20299645	2
#20299643	2
#20299641	2
#20299640	2
#20299613	2
#20299596	2
#20299595	2
#20299564	2
#20299394	2
#2955	113