Subgroups — #28404351

Order 14464.

1748 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813668	C₂	1	2	yes	maximal, complemented
2	#20299270	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299267	C₂	1	2	yes	maximal, complemented
4	D₄×D₄₅₂	C₂	1	3	yes	maximal, complemented
5	#20299233	C₂	1	3	yes	maximal, complemented
6	#20299232	C₂	1	6	yes	maximal, complemented
7	#20299164	C₂	1	6	yes	maximal, complemented
8	#20299145	C₂	1	3	yes	maximal, complemented
9	#127×D₁₁₃	C₂	1	3	yes	maximal, complemented
10	C₄×#14268648	C₂	1	1	yes	maximal, characteristic, complemented
11	#550×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16813668	—	2	16	no	—
13	C₂×#14268648	C₂²	1	1	yes	characteristic
14	C₂×#14268648	C₂²	1	2	yes	complemented
15	C₂×#14268647	C₂²	1	6	yes	complemented
16	C₂²×D₄₅₂	C₂²	1	6	yes	complemented
17	C₂×D₄×D₁₁₃	C₂²	1	6	yes	complemented
18	C₂×D₄×D₁₁₃	C₂²	1	6	yes	complemented
19	C₂×D₄×D₁₁₃	C₂²	1	6	yes	complemented
20	C₂×D₄×D₁₁₃	C₂²	1	6	yes	complemented
21	#16813621	C₂²	1	1	yes	characteristic, complemented
22	#16813619	C₂²	1	3	yes	complemented
23	#16813618	C₂²	1	6	yes	complemented
24	#16813618	C₂²	1	3	yes	complemented
25	#16813618	C₂²	1	6	yes	complemented
26	#16813608	C₂²	1	6	yes	complemented
27	#16813597	C₂²	1	6	yes	complemented
28	#16813597	C₂²	1	3	yes	complemented
29	#16813596	C₂²	1	6	yes	complemented
30	#16813595	C₂²	1	6	yes	complemented
31	#16813593	C₂²	1	6	yes	complemented
32	#16813593	C₂²	1	3	yes	complemented
33	#16813592	C₂²	1	12	yes	complemented
34	#16813590	C₂²	1	12	yes	complemented
35	Q₈×Dic₁₁₃	C₂²	1	1	yes	characteristic, complemented
36	C₄×D₄₅₂	C₂²	1	3	yes	complemented
37	C₄×D₄₅₂	C₂²	1	3	yes	complemented
38	#16813584	C₂²	1	3	yes	complemented
39	C₄×C₂²⋊D₁₁₃	C₂²	1	6	yes	complemented
40	#16813580	C₂²	1	3	yes	complemented
41	C₄²×D₁₁₃	C₂²	1	3	yes	complemented
42	C₂×#41×C₁₁₃	C₂²	1	2	yes	complemented
43	#127×C₁₁₃	C₂²	1	3	yes	complemented
44	#121×C₁₁₃	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₁₁₃	C₂²	1	3	yes	complemented
47	C₂³×D₁₁₃	C₂³	1	12	yes	—
48	#14268648	—	2	8	no	—
49	#14268648	D₄	1	8	yes	complemented
50	#14268647	—	2	24	no	—
51	C₂×Dic₂₂₆	C₂³	1	2	yes	—
52	C₂×D₄₅₂	—	2	24	no	—
53	C₂×D₄₅₂	C₂³	1	3	yes	—
54	C₂×D₄₅₂	C₂³	1	6	yes	—
55	C₂×D₄₅₂	C₂³	1	6	yes	—
56	C₂×D₄₅₂	C₂³	1	6	yes	—
57	C₂×D₄₅₂	C₂³	1	3	yes	—
58	C₂×C₂²⋊D₁₁₃	C₂³	1	6	yes	—
59	C₂×C₂²⋊D₁₁₃	C₂³	1	6	yes	—
60	C₂×C₂²⋊D₁₁₃	C₂³	1	6	yes	—
61	C₂×C₂²⋊D₁₁₃	C₂³	1	12	yes	—
62	D₄×D₁₁₃	—	2	24	no	—
63	D₄×D₁₁₃	—	2	24	no	—
64	D₄×D₁₁₃	—	4	12	no	—
65	D₄×D₁₁₃	—	4	12	no	—
66	C₂×C₄×D₁₁₃	C₂³	1	3	yes	—
67	C₂×C₄×D₁₁₃	C₂³	1	3	yes	—
68	C₂×C₄×D₁₁₃	C₂³	1	6	yes	—
69	C₂×C₄×D₁₁₃	C₂³	1	3	yes	—
70	C₄⋊Dic₁₁₃	C₂³	1	3	yes	—
71	C₄⋊Dic₁₁₃	C₂³	1	1	yes	characteristic
72	#14268626	C₂³	1	2	yes	—
73	C₂²⋊Dic₁₁₃	C₂³	1	6	yes	—
74	#14268624	C₂³	1	6	yes	—
75	#14268624	C₂³	1	6	yes	—
76	#14268624	C₂³	1	6	yes	—
77	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
78	C₄×Dic₁₁₃	C₂³	1	3	yes	—
79	C₄×Dic₁₁₃	C₂³	1	3	yes	—
80	#41×C₁₁₃	—	2	8	no	—
81	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₁₁₃	C₂³	1	3	yes	—
83	C₂×D₄×C₁₁₃	C₂³	1	6	yes	—
84	C₂×D₄×C₁₁₃	C₂³	1	6	yes	—
85	C₂²×C₄×C₁₁₃	C₂³	1	6	yes	—
86	C₄⋊C₄×C₁₁₃	C₂³	1	3	yes	—
87	C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₁₃	C₂³	1	6	yes	—
89	C₄²×C₁₁₃	C₂³	1	3	yes	—
90	C₂³×C₁₁₃	C₂⁴	1	6	yes	—
91	C₂²×D₁₁₃	—	2	24	no	—
92	C₂²×D₁₁₃	—	4	12	no	—
93	C₂²×D₁₁₃	—	4	12	no	—
94	C₂²×D₁₁₃	C₂⁴	1	6	yes	—
95	C₂²×D₁₁₃	C₂⁴	1	3	yes	—
96	Q₈×C₁₁₃	C₂×D₄	1	4	yes	complemented
97	D₄×C₁₁₃	—	2	12	no	—
98	D₄×C₁₁₃	—	2	6	no	—
99	D₄×C₁₁₃	—	4	6	no	—
100	C₂×C₄×C₁₁₃	—	2	12	no	—
101	C₂×C₄×C₁₁₃	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
103	C₂×C₄×C₁₁₃	C₂⁴	1	3	yes	—
104	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₁₁₃	—	2	12	no	—
106	C₂²⋊D₁₁₃	—	2	12	no	—
107	C₂²⋊D₁₁₃	—	2	12	no	—
108	C₂²⋊D₁₁₃	—	4	12	no	—
109	C₂×Dic₁₁₃	C₂⁴	1	3	yes	—
110	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
111	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
112	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
113	D₄₅₂	—	2	12	no	—
114	D₄₅₂	—	2	12	no	—
115	D₄₅₂	—	2	6	no	—
116	D₄₅₂	—	4	6	no	—
117	D₄₅₂	C₂×D₄	1	12	yes	complemented
118	C₄×D₁₁₃	—	2	12	no	—
119	C₄×D₁₁₃	—	2	6	no	—
120	C₄×D₁₁₃	—	2	6	no	—
121	C₄×D₁₁₃	C₂×D₄	1	12	yes	complemented
122	Dic₂₂₆	—	2	4	no	—
123	C₂²×C₁₁₃	—	2	6	no	—
124	C₂²×C₁₁₃	—	2	6	no	—
125	C₂²×C₁₁₃	—	2	6	no	—
126	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
127	C₂×D₁₁₃	—	2	3	no	—
128	C₂×D₁₁₃	—	2	3	no	—
129	C₂×D₁₁₃	—	2	6	no	—
130	C₂×D₁₁₃	—	2	6	no	—
131	C₂×D₁₁₃	—	2	6	no	—
132	C₂×D₁₁₃	—	8	24	no	—
133	C₂×D₁₁₃	C₂²×D₄	1	6	yes	—
134	C₄×C₁₁₃	—	2	2	no	—
135	C₄×C₁₁₃	—	2	3	no	—
136	C₄×C₁₁₃	C₂²×D₄	1	6	yes	—
137	C₄×C₁₁₃	H(5,𝔽₂)	1	2	yes	complemented
138	Dic₁₁₃	—	2	2	no	—
139	Dic₁₁₃	—	2	3	no	—
140	Dic₁₁₃	C₂²×D₄	1	2	yes	—
141	Dic₁₁₃	H(5,𝔽₂)	1	2	yes	complemented
142	C₂×C₁₁₃	—	4	6	no	—
143	C₂×C₁₁₃	C₂³×D₄	1	1	yes	characteristic
144	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	D₁₁₃	—	2	6	no	—
147	D₁₁₃	—	4	6	no	—
148	#3498	—	113	1	no	maximal, Sylow
149	C₁₁₃	#3498	1	1	yes	characteristic, complemented, Sylow
150	C₂×H(5,𝔽₂)	—	113	2	no	—
151	#550	—	113	2	no	—
152	#550	—	113	1	no	—
153	#550	D₁₁₃	1	1	yes	characteristic, complemented
154	D₄²	—	113	3	no	—
155	D₄²	—	113	6	no	—
156	D₄²	—	113	3	no	—
157	#535	—	113	6	no	—
158	#532	—	113	3	no	—
159	C₂×#127	—	113	3	no	—
160	C₄×#41	—	113	1	no	—
161	H(5,𝔽₂)	—	226	16	no	—
162	C₂×#41	—	113	1	no	—
163	C₂×#41	—	113	2	no	—
164	C₂×#41	—	113	6	no	—
165	C₂×#41	C₂×D₁₁₃	1	2	yes	complemented
166	C₂²×D₄	—	113	6	no	—
167	C₂²×D₄	—	113	6	no	—
168	C₂²×D₄	—	113	6	no	—
169	C₂²×D₄	—	113	6	no	—
170	C₂²×D₄	—	113	6	no	—
171	C₄⋊Q₈	—	113	1	no	—
172	#127	—	113	3	no	—
173	#127	—	113	6	no	—
174	#127	—	113	6	no	—
175	#127	—	113	3	no	—
176	#127	C₂×D₁₁₃	1	3	yes	complemented
177	#124	—	113	6	no	—
178	#121	—	113	3	no	—
179	#121	—	113	6	no	—
180	#121	—	113	6	no	—
181	#121	—	113	3	no	—
182	#121	—	113	6	no	—
183	#121	—	113	6	no	—
184	#121	C₂×D₁₁₃	1	6	yes	complemented
185	#120	—	113	12	no	—
186	#120	—	113	12	no	—
187	C₄×Q₈	—	113	1	no	—
188	C₄×Q₈	C₂×D₁₁₃	1	1	yes	characteristic, complemented
189	C₄×D₄	—	113	6	no	—
190	C₄×D₄	—	113	3	no	—
191	C₄×D₄	—	113	3	no	—
192	C₄×D₄	—	113	3	no	—
193	C₄×D₄	C₂×D₁₁₃	1	3	yes	complemented
194	#117	—	113	3	no	—
195	C₂×C₄²	—	113	3	no	—
196	C₂⁴	—	113	12	no	—
197	#41	—	2	8	no	—
198	#41	—	113	8	no	—
199	#41	—	226	24	no	—
200	#41	—	226	8	no	—
201	C₂×Q₈	—	113	2	no	—
202	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
203	C₂×D₄	—	113	12	no	—
204	C₂×D₄	—	113	6	no	—
205	C₂×D₄	—	113	6	no	—
206	C₂×D₄	—	113	6	no	—
207	C₂×D₄	—	113	3	no	—
208	C₂×D₄	—	113	6	no	—
209	C₂×D₄	—	113	3	no	—
210	C₂×D₄	—	113	6	no	—
211	C₂×D₄	—	113	6	no	—
212	C₂×D₄	—	452	12	no	—
213	C₂×D₄	—	452	12	no	—
214	C₂×D₄	—	226	24	no	—
215	C₂×D₄	—	226	24	no	—
216	C₂×D₄	—	226	24	no	—
217	C₂×D₄	C₂²×D₁₁₃	1	3	yes	—
218	C₂×D₄	C₂²×D₁₁₃	1	6	yes	—
219	C₂×D₄	C₂²×D₁₁₃	1	6	yes	—
220	C₂²×C₄	—	113	3	no	—
221	C₂²×C₄	—	113	3	no	—
222	C₂²×C₄	—	113	3	no	—
223	C₂²×C₄	—	113	6	no	—
224	C₂²×C₄	C₂²×D₁₁₃	1	6	yes	—
225	C₄⋊C₄	—	113	3	no	—
226	C₄⋊C₄	—	113	2	no	—
227	C₄⋊C₄	—	113	1	no	—
228	C₄⋊C₄	C₂²×D₁₁₃	1	3	yes	—
229	C₄⋊C₄	C₂²×D₁₁₃	1	1	yes	characteristic
230	C₂²⋊C₄	—	113	6	no	—
231	C₂²⋊C₄	—	113	6	no	—
232	C₂²⋊C₄	—	113	6	no	—
233	C₂²⋊C₄	—	113	6	no	—
234	C₂²⋊C₄	C₂²×D₁₁₃	1	6	yes	—
235	C₄²	—	113	1	no	—
236	C₄²	—	113	3	no	—
237	C₄²	—	113	3	no	—
238	C₄²	C₂²×D₁₁₃	1	3	yes	—
239	C₂³	—	113	3	no	—
240	C₂³	—	113	6	no	—
241	C₂³	—	452	12	no	—
242	C₂³	—	452	12	no	—
243	C₂³	—	226	24	no	—
244	C₂³	C₂³×D₁₁₃	1	6	yes	—
245	Q₈	—	226	4	no	—
246	Q₈	D₄×D₁₁₃	1	4	yes	complemented
247	D₄	—	2	12	no	—
248	D₄	—	2	6	no	—
249	D₄	—	4	6	no	—
250	D₄	—	113	12	no	—
251	D₄	—	452	6	no	—
252	D₄	—	452	12	no	—
253	D₄	—	226	12	no	—
254	D₄	—	226	12	no	—
255	D₄	—	226	12	no	—
256	D₄	—	226	12	no	—
257	D₄	—	226	12	no	—
258	D₄	—	226	6	no	—
259	C₂×C₄	—	2	12	no	—
260	C₂×C₄	—	113	1	no	—
261	C₂×C₄	—	113	1	no	—
262	C₂×C₄	—	113	12	no	—
263	C₂×C₄	—	113	3	no	—
264	C₂×C₄	—	113	2	no	—
265	C₂×C₄	—	226	6	no	—
266	C₂×C₄	—	226	12	no	—
267	C₂×C₄	—	226	6	no	—
268	C₂×C₄	C₂³×D₁₁₃	1	3	yes	—
269	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
270	C₂×C₄	C₂³×D₁₁₃	1	3	yes	—
271	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
272	C₂²	—	2	6	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	2	6	no	—
275	C₂²	—	113	6	no	—
276	C₂²	—	904	24	no	—
277	C₂²	—	226	6	no	—
278	C₂²	—	226	6	no	—
279	C₂²	—	226	6	no	—
280	C₂²	—	226	3	no	—
281	C₂²	—	226	3	no	—
282	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	3	no	—
285	C₄	—	113	2	no	—
286	C₄	—	113	2	no	—
287	C₄	—	226	2	no	—
288	C₄	—	226	3	no	—
289	C₄	C₂×D₄×D₁₁₃	1	6	yes	—
290	C₄	#16813668	1	2	yes	complemented
291	C₂	—	4	6	no	—
292	C₂	—	452	6	no	—
293	C₂	—	226	6	no	—
294	C₂	C₂²×D₄×D₁₁₃	1	1	yes	characteristic
295	C₂	H(5,𝔽₂)×D₁₁₃	1	1	yes	characteristic
296	C₂	C₂×#16813668	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	#550×C₁₁₃	C₂
hypercenter	#550	D₁₁₃
hypocenter	C₁₁₃	#3498
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. C₂×#16813668	C₂	semidirect
2. #20299270	C₂	semidirect
3. #20299267	C₂	semidirect
4. D₄×D₄₅₂	C₂	semidirect
5. #20299233	C₂	semidirect
6. #20299232	C₂	semidirect
7. #20299164	C₂	semidirect
8. #20299145	C₂	semidirect
9. #127×D₁₁₃	C₂	semidirect
10. C₄×#14268648	C₂	semidirect
11. #550×C₁₁₃	C₂	semidirect
14. C₂×#14268648	C₂²	semidirect
15. C₂×#14268647	C₂²	semidirect
16. C₂²×D₄₅₂	C₂²	semidirect
17. C₂×D₄×D₁₁₃	C₂²	semidirect
18. C₂×D₄×D₁₁₃	C₂²	semidirect
19. C₂×D₄×D₁₁₃	C₂²	semidirect
20. C₂×D₄×D₁₁₃	C₂²	semidirect
21. #16813621	C₂²	semidirect
22. #16813619	C₂²	semidirect
23. #16813618	C₂²	semidirect
24. #16813618	C₂²	semidirect
25. #16813618	C₂²	semidirect
26. #16813608	C₂²	semidirect
27. #16813597	C₂²	semidirect
28. #16813597	C₂²	semidirect
29. #16813596	C₂²	semidirect
30. #16813595	C₂²	semidirect
31. #16813593	C₂²	semidirect
32. #16813593	C₂²	semidirect
33. #16813592	C₂²	semidirect
34. #16813590	C₂²	semidirect
35. Q₈×Dic₁₁₃	C₂²	semidirect
36. C₄×D₄₅₂	C₂²	semidirect
37. C₄×D₄₅₂	C₂²	semidirect
38. #16813584	C₂²	semidirect
39. C₄×C₂²⋊D₁₁₃	C₂²	semidirect
40. #16813580	C₂²	semidirect
41. C₄²×D₁₁₃	C₂²	semidirect
42. C₂×#41×C₁₁₃	C₂²	semidirect
43. #127×C₁₁₃	C₂²	semidirect
44. #121×C₁₁₃	C₂²	semidirect
45. C₄×Q₈×C₁₁₃	C₂²	semidirect
46. C₄×D₄×C₁₁₃	C₂²	semidirect
49. #14268648	D₄	semidirect
96. Q₈×C₁₁₃	C₂×D₄	semidirect
117. D₄₅₂	C₂×D₄	semidirect
121. C₄×D₁₁₃	C₂×D₄	semidirect
137. C₄×C₁₁₃	H(5,𝔽₂)	semidirect
141. Dic₁₁₃	H(5,𝔽₂)	semidirect
149. C₁₁₃	#3498	semidirect
153. #550	D₁₁₃	semidirect
165. C₂×#41	C₂×D₁₁₃	semidirect
176. #127	C₂×D₁₁₃	semidirect
184. #121	C₂×D₁₁₃	semidirect
188. C₄×Q₈	C₂×D₁₁₃	semidirect
193. C₄×D₄	C₂×D₁₁₃	semidirect
246. Q₈	D₄×D₁₁₃	semidirect
290. C₄	#16813668	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813668	2
#20299270	2
#20299267	2
D₄×D₄₅₂	2
#20299233	2
#20299232	2
#20299164	2
#20299145	2
#127×D₁₁₃	2
C₄×#14268648	2
#550×C₁₁₃	2
#3498	113