Subgroups — #28407348

Order 14464.

458 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	#20299677	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299676	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	D₁₁₃⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16813686	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#90673	C₂	1	1	yes	maximal, characteristic, complemented
11	#20299394	C₂	1	1	yes	maximal, characteristic
12	#16813710	C₂²	1	2	yes	complemented
13	#16813709	—	2	1	no	—
14	#16813709	—	2	1	no	—
15	#16813709	—	2	2	no	—
16	#16813709	C₂²	1	2	yes	complemented
17	#16813709	C₂²	1	2	yes	complemented
18	#16813709	C₂²	1	2	yes	complemented
19	C₂×#14268657	—	2	1	no	—
20	C₂×#14268657	C₂²	1	1	yes	characteristic
21	D₁₁₃⋊M₁₆	—	2	1	no	—
22	D₁₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
23	D₁₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
24	C₂×#14268656	—	2	2	no	—
25	C₂×C₄×#90673	C₂²	1	1	yes	characteristic
26	#16813693	C₂²	1	2	yes	complemented
27	#16813692	C₂²	1	2	yes	complemented
28	#16813687	C₂²	1	1	yes	characteristic, complemented
29	#16813687	C₂²	1	1	yes	characteristic, complemented
30	#16813686	C₂²	1	2	yes	complemented
31	#16813686	C₂²	1	2	yes	complemented
32	#16813686	C₂²	1	1	yes	characteristic, complemented
33	#16813686	C₂²	1	1	yes	characteristic, complemented
34	#16813676	C₂²	1	2	yes	complemented
35	C₈×#90673	C₂²	1	2	yes	complemented
36	C₂×#14268646	C₂²	1	1	yes	characteristic
37	#41×D₁₁₃	C₂²	1	1	yes	characteristic
38	#16813655	C₄	1	1	yes	characteristic, complemented
39	#16813652	C₄	1	1	yes	characteristic, complemented
40	#16813651	C₄	1	2	yes	complemented
41	#16813648	C₄	1	1	yes	characteristic, complemented
42	#16813646	C₄	1	2	yes	complemented
43	#16813643	C₂²	1	2	yes	—
44	#16813641	C₂²	1	2	yes	—
45	M₁₆×D₁₁₃	C₂²	1	1	yes	characteristic
46	AGL(1,ℤ/8)×C₁₁₃	C₄	1	1	yes	characteristic, complemented
47	C₂²×#90673	—	2	1	no	—
48	#14268657	—	2	1	no	—
49	#14268657	—	2	2	no	—
50	#14268657	D₄	1	1	yes	characteristic
51	#14268657	D₄	1	1	yes	characteristic
52	#14268657	C₂³	1	2	yes	—
53	#14268657	C₂³	1	1	yes	characteristic
54	#14268657	C₂³	1	1	yes	characteristic
55	#14268656	—	2	1	no	—
56	#14268656	—	2	2	no	—
57	#14268656	—	2	2	no	—
58	#14268656	—	2	2	no	—
59	#14268656	D₄	1	2	yes	—
60	#14268655	—	2	1	no	—
61	#14268655	—	2	2	no	—
62	#14268655	C₂³	1	2	yes	—
63	C₂×#11744753	—	2	1	no	—
64	C₂×#11744753	—	2	2	no	—
65	C₂×#11744753	—	2	2	no	—
66	C₂×#11744753	—	2	2	no	—
67	C₄×#90673	—	2	1	no	—
68	C₄×#90673	C₂³	1	1	yes	characteristic
69	C₄×#90673	C₂³	1	1	yes	characteristic
70	#14268648	C₂³	1	1	yes	characteristic
71	#14268647	—	2	1	no	—
72	#14268647	C₂×C₄	1	1	yes	characteristic
73	#14268646	—	2	1	no	—
74	#14268646	—	2	1	no	—
75	#14268646	—	2	1	no	—
76	#14268646	C₂×C₄	1	2	yes	—
77	#14268646	C₂³	1	1	yes	characteristic
78	#14268646	C₂³	1	2	yes	—
79	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic
80	Q₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₁₃	—	2	1	no	—
82	D₄×D₁₁₃	—	2	1	no	—
83	D₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×Dic₁₁₃	—	2	1	no	—
85	C₂×C₄×D₁₁₃	—	2	1	no	—
86	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
87	Dic₄₅₂	C₂×C₄	1	2	yes	complemented
88	#14268637	C₂×C₄	1	2	yes	complemented
89	#14268636	C₂×C₄	1	2	yes	complemented
90	#14268635	C₂×C₄	1	2	yes	complemented
91	#14268635	C₂×C₄	1	2	yes	complemented
92	#14268632	C₂×C₄	1	2	yes	complemented
93	#14268631	C₂×C₄	1	1	yes	characteristic
94	#14268630	C₂×C₄	1	2	yes	—
95	C₈×D₁₁₃	C₂³	1	2	yes	—
96	#41×C₁₁₃	C₂×C₄	1	1	yes	characteristic
97	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
98	SD₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
99	D₈×C₁₁₃	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₁₃	C₂×C₄	1	1	yes	characteristic
101	C₂³×C₁₁₃	—	2	1	no	—
102	C₂²×D₁₁₃	—	2	1	no	—
103	C₂²×D₁₁₃	#41	1	1	yes	characteristic
104	C₂×#90673	—	2	1	no	—
105	C₂×#90673	—	2	1	no	—
106	C₂×#90673	—	4	2	no	—
107	Q₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
108	D₄×C₁₁₃	—	2	1	no	—
109	D₄×C₁₁₃	—	2	1	no	—
110	D₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
111	D₄×C₁₁₃	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₁₁₃	—	2	1	no	—
113	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²⋊D₁₁₃	—	2	1	no	—
115	C₂²⋊D₁₁₃	—	2	2	no	—
116	C₂²⋊D₁₁₃	—	2	1	no	—
117	C₂²⋊D₁₁₃	—	2	2	no	—
118	C₂²⋊D₁₁₃	—	2	1	no	—
119	C₂×Dic₁₁₃	—	2	1	no	—
120	C₂×Dic₁₁₃	—	2	2	no	—
121	C₂×Dic₁₁₃	—	2	1	no	—
122	C₂×Dic₁₁₃	—	2	2	no	—
123	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
124	D₄₅₂	—	2	1	no	—
125	D₄₅₂	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₁₁₃	—	2	1	no	—
127	C₄×D₁₁₃	—	2	1	no	—
128	C₄×D₁₁₃	—	2	1	no	—
129	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
130	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
131	C₄×D₁₁₃	#41	1	1	yes	characteristic
132	C₄×D₁₁₃	C₂⁴	1	1	yes	characteristic
133	Dic₂₂₆	—	2	1	no	—
134	Dic₂₂₆	—	2	1	no	—
135	Dic₂₂₆	C₂²×C₄	1	1	yes	characteristic
136	Dic₂₂₆	C₂²×C₄	1	2	yes	—
137	#11744753	—	2	1	no	—
138	#11744753	—	2	1	no	—
139	#11744753	—	2	2	no	—
140	#11744753	C₂×D₄	1	2	yes	—
141	#11744753	C₂×D₄	1	2	yes	—
142	C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
143	#11744751	C₂²×C₄	1	2	yes	—
144	C₂²×C₁₁₃	—	2	2	no	—
145	C₂²×C₁₁₃	—	2	1	no	—
146	C₂²×C₁₁₃	—	8	1	no	—
147	C₂²×C₁₁₃	C₄×D₄	1	1	yes	characteristic
148	C₂×D₁₁₃	—	2	1	no	—
149	C₂×D₁₁₃	—	4	1	no	—
150	C₂×D₁₁₃	—	8	1	no	—
151	C₂×D₁₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂×D₁₁₃	C₂×#41	1	1	yes	characteristic
153	#90673	—	2	2	no	—
154	#90673	—	4	1	no	—
155	C₄×C₁₁₃	—	2	1	no	—
156	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
157	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
158	Dic₁₁₃	—	2	1	no	—
159	Dic₁₁₃	—	2	2	no	—
160	Dic₁₁₃	C₄×D₄	1	1	yes	characteristic
161	Dic₁₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₁₁₃	—	2	1	no	—
163	C₂×C₁₁₃	—	4	1	no	—
164	C₂×C₁₁₃	—	4	2	no	—
165	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
166	D₁₁₃	—	2	1	no	—
167	D₁₁₃	—	2	1	no	—
168	D₁₁₃	—	4	1	no	—
169	#2956	—	113	1	no	maximal, Sylow
170	C₁₁₃	#2956	1	1	yes	characteristic, complemented, Sylow
171	#575	—	113	1	no	—
172	#568	—	113	1	no	—
173	#568	—	113	1	no	—
174	#444	—	113	2	no	—
175	#444	—	113	2	no	—
176	#443	—	113	2	no	—
177	#443	—	113	2	no	—
178	#430	—	113	1	no	—
179	C₂×#104	—	113	1	no	—
180	C₂×#104	—	113	1	no	—
181	C₄×M₁₆	—	113	1	no	—
182	C₂×#41	—	113	1	no	—
183	C₂×#41	—	113	1	no	—
184	#137	—	113	1	no	—
185	#137	—	113	2	no	—
186	#137	—	113	1	no	—
187	AGL(1,ℤ/8)	—	113	1	no	—
188	AGL(1,ℤ/8)	—	113	2	no	—
189	AGL(1,ℤ/8)	#90673	1	1	yes	characteristic, complemented
190	#135	—	113	2	no	—
191	#135	—	113	2	no	—
192	#131	—	113	2	no	—
193	#131	—	113	2	no	—
194	#131	—	113	2	no	—
195	#131	—	113	2	no	—
196	#131	—	226	1	no	—
197	#131	—	226	1	no	—
198	#131	—	226	2	no	—
199	C₂×M₁₆	—	113	1	no	—
200	C₂×M₁₆	—	113	1	no	—
201	C₂×M₁₆	—	113	1	no	—
202	C₂×M₁₆	—	113	1	no	—
203	C₂×M₁₆	—	226	2	no	—
204	C₂×M₁₆	—	226	1	no	—
205	C₂×M₁₆	—	226	1	no	—
206	C₂×C₄²	—	113	1	no	—
207	#108	—	113	2	no	—
208	#108	—	113	2	no	—
209	#104	—	113	2	no	—
210	#104	—	113	1	no	—
211	#104	—	113	2	no	—
212	#104	—	113	1	no	—
213	#104	—	113	1	no	—
214	#104	—	113	1	no	—
215	#97	—	113	2	no	—
216	C₄×C₈	—	113	2	no	—
217	#41	—	113	1	no	—
218	#41	—	113	2	no	—
219	#41	—	113	1	no	—
220	#41	—	113	2	no	—
221	#41	—	113	1	no	—
222	#41	—	226	1	no	—
223	#41	—	226	1	no	—
224	#41	—	226	1	no	—
225	#41	—	226	1	no	—
226	#41	C₂×#90673	1	1	yes	characteristic
227	C₂×Q₈	—	113	1	no	—
228	C₂×Q₈	—	113	1	no	—
229	C₂×D₄	—	113	1	no	—
230	C₂×D₄	—	226	1	no	—
231	C₂×D₄	—	226	1	no	—
232	C₂×D₄	C₂×#90673	1	1	yes	characteristic
233	C₂²×C₄	—	113	1	no	—
234	C₂²×C₄	—	226	1	no	—
235	C₂²×C₄	—	226	1	no	—
236	C₂²×C₄	—	226	1	no	—
237	Q₁₆	—	113	2	no	—
238	Q₁₆	—	113	2	no	—
239	SD₁₆	—	113	2	no	—
240	SD₁₆	—	113	2	no	—
241	SD₁₆	—	113	2	no	—
242	SD₁₆	C₂×#90673	1	2	yes	complemented
243	D₈	—	113	2	no	—
244	D₈	C₂×#90673	1	2	yes	complemented
245	M₁₆	—	113	2	no	—
246	M₁₆	—	113	1	no	—
247	M₁₆	—	113	1	no	—
248	M₁₆	—	113	1	no	—
249	M₁₆	—	113	2	no	—
250	M₁₆	—	113	2	no	—
251	M₁₆	—	113	1	no	—
252	M₁₆	—	113	1	no	—
253	M₁₆	—	226	2	no	—
254	M₁₆	—	226	2	no	—
255	M₁₆	—	226	2	no	—
256	M₁₆	—	226	2	no	—
257	M₁₆	—	226	1	no	—
258	M₁₆	—	226	1	no	—
259	M₁₆	C₂×#90673	1	1	yes	characteristic
260	C₂×C₈	—	113	2	no	—
261	C₂×C₈	—	113	2	no	—
262	C₂×C₈	—	226	2	no	—
263	C₂×C₈	—	226	2	no	—
264	C₂×C₈	—	226	2	no	—
265	C₂×C₈	—	226	2	no	—
266	C₂×C₈	—	226	1	no	—
267	C₂×C₈	—	226	1	no	—
268	C₄²	—	113	1	no	—
269	C₄²	—	113	1	no	—
270	C₄²	—	226	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	113	1	no	—
273	C₂³	—	226	1	no	—
274	Q₈	—	113	1	no	—
275	Q₈	—	113	2	no	—
276	Q₈	—	226	1	no	—
277	Q₈	—	226	1	no	—
278	Q₈	C₂²×#90673	1	1	yes	characteristic
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	113	1	no	—
282	D₄	—	226	1	no	—
283	D₄	—	226	1	no	—
284	D₄	—	226	1	no	—
285	D₄	—	226	2	no	—
286	D₄	—	226	2	no	—
287	D₄	—	226	1	no	—
288	D₄	C₂²×#90673	1	2	yes	—
289	D₄	C₂²×#90673	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	113	1	no	—
292	C₂×C₄	—	113	1	no	—
293	C₂×C₄	—	113	1	no	—
294	C₂×C₄	—	113	1	no	—
295	C₂×C₄	—	113	1	no	—
296	C₂×C₄	—	452	2	no	—
297	C₂×C₄	—	226	1	no	—
298	C₂×C₄	—	226	1	no	—
299	C₂×C₄	—	226	1	no	—
300	C₂×C₄	—	226	2	no	—
301	C₂×C₄	—	226	2	no	—
302	C₂×C₄	—	226	1	no	—
303	C₂×C₄	—	226	1	no	—
304	C₂×C₄	—	226	1	no	—
305	C₂×C₄	—	226	1	no	—
306	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
307	C₈	—	113	2	no	—
308	C₈	—	113	2	no	—
309	C₈	—	113	2	no	—
310	C₈	—	226	1	no	—
311	C₈	—	226	1	no	—
312	C₈	—	226	2	no	—
313	C₈	C₂²×#90673	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	113	1	no	—
318	C₂²	—	113	1	no	—
319	C₂²	—	904	1	no	—
320	C₂²	—	452	1	no	—
321	C₂²	—	226	1	no	—
322	C₂²	D₄×#90673	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	113	1	no	—
325	C₄	—	113	1	no	—
326	C₄	—	452	1	no	—
327	C₄	—	226	2	no	—
328	C₄	—	226	1	no	—
329	C₄	—	226	2	no	—
330	C₄	D₄×#90673	1	1	yes	characteristic
331	C₄	C₂³×#90673	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	2	no	—
334	C₂	—	4	1	no	—
335	C₂	—	452	1	no	—
336	C₂	—	226	1	no	—
337	C₂	—	226	1	no	—
338	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	#2956
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. #20299677	C₂	semidirect
2. #20299676	C₂	semidirect
3. #20299645	C₂	semidirect
4. #20299643	C₂	semidirect
5. #20299641	C₂	semidirect
6. #20299640	C₂	semidirect
7. #20299613	C₂	semidirect
8. D₁₁₃⋊#104	C₂	semidirect
9. C₂×#16813686	C₂	semidirect
10. M₁₆×#90673	C₂	semidirect
12. #16813710	C₂²	semidirect
16. #16813709	C₂²	semidirect
17. #16813709	C₂²	semidirect
18. #16813709	C₂²	semidirect
26. #16813693	C₂²	semidirect
27. #16813692	C₂²	semidirect
28. #16813687	C₂²	semidirect
29. #16813687	C₂²	semidirect
30. #16813686	C₂²	semidirect
31. #16813686	C₂²	semidirect
32. #16813686	C₂²	semidirect
33. #16813686	C₂²	semidirect
34. #16813676	C₂²	semidirect
35. C₈×#90673	C₂²	semidirect
38. #16813655	C₄	semidirect
39. #16813652	C₄	semidirect
40. #16813651	C₄	semidirect
41. #16813648	C₄	semidirect
42. #16813646	C₄	semidirect
46. AGL(1,ℤ/8)×C₁₁₃	C₄	semidirect
87. Dic₄₅₂	C₂×C₄	semidirect
88. #14268637	C₂×C₄	semidirect
89. #14268636	C₂×C₄	semidirect
90. #14268635	C₂×C₄	semidirect
91. #14268635	C₂×C₄	semidirect
92. #14268632	C₂×C₄	semidirect
98. SD₁₆×C₁₁₃	C₂×C₄	semidirect
99. D₈×C₁₁₃	C₂×C₄	semidirect
170. C₁₁₃	#2956	semidirect
189. AGL(1,ℤ/8)	#90673	semidirect
242. SD₁₆	C₂×#90673	semidirect
244. D₈	C₂×#90673	semidirect

Maximal subgroups

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

subgroup	index
#20299677	2
#20299676	2
#20299645	2
#20299643	2
#20299641	2
#20299640	2
#20299613	2
D₁₁₃⋊#104	2
C₂×#16813686	2
M₁₆×#90673	2
#20299394	2
#2956	113