Subgroups — #28404263

Order 14464.

1328 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299274	C₂	1	8	yes	maximal, complemented
2	#20299238	C₂	1	8	yes	maximal, complemented
3	#20299224	C₂	1	1	yes	maximal, characteristic, complemented
4	#20299186	C₂	1	4	yes	maximal, complemented
5	#20299160	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299109	C₂	1	4	yes	maximal, complemented
7	C₂×#16813591	C₂	1	2	yes	maximal, complemented
8	#20299071	C₂	1	2	yes	maximal, complemented
9	C₂²×C₂²⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14268647	C₂²	1	4	yes	complemented
11	C₂²×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
12	C₂²×C₂²⋊D₁₁₃	C₂²	1	4	yes	complemented
13	#16813615	C₂²	1	8	yes	complemented
14	#16813608	C₂²	1	8	yes	complemented
15	#16813606	C₂²	1	4	yes	complemented
16	#16813605	C₂²	1	8	yes	complemented
17	#16813604	C₂²	1	8	yes	complemented
18	#16813601	C₂²	1	4	yes	complemented
19	#16813601	C₂²	1	4	yes	complemented
20	#16813599	C₂²	1	8	yes	complemented
21	#16813596	C₂²	1	4	yes	complemented
22	#16813596	C₂²	1	4	yes	complemented
23	#16813595	C₂²	1	8	yes	complemented
24	#16813592	C₂²	1	4	yes	complemented
25	#16813591	C₂²	1	8	yes	—
26	#16813591	C₂²	1	4	yes	complemented
27	#16813591	C₂²	1	4	yes	complemented
28	C₄×C₂²⋊D₁₁₃	C₂²	1	8	yes	complemented
29	C₄×C₂²⋊D₁₁₃	C₂²	1	8	yes	complemented
30	#16813582	C₂²	1	8	yes	complemented
31	#16813581	C₂²	1	4	yes	complemented
32	#16813578	C₂²	1	4	yes	complemented
33	C₂×C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
34	C₂×C₂²⋊Dic₁₁₃	C₂²	1	4	yes	complemented
35	C₂²⋊C₄×D₁₁₃	C₂²	1	4	yes	complemented
36	C₂⁵×C₁₁₃	C₂²	1	1	yes	characteristic
37	C₂³×C₄×C₁₁₃	C₂²	1	2	yes	complemented
38	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	8	yes	complemented
41	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	complemented
42	C₂³×D₁₁₃	C₂³	1	2	yes	—
43	#14268647	—	4	8	no	—
44	C₂×Dic₂₂₆	C₂³	1	4	yes	complemented
45	C₂×D₄₅₂	C₂³	1	4	yes	complemented
46	C₂×C₂²⋊D₁₁₃	—	2	8	no	—
47	C₂×C₂²⋊D₁₁₃	—	4	8	no	—
48	C₂×C₂²⋊D₁₁₃	C₂³	1	4	yes	—
49	C₂×C₂²⋊D₁₁₃	C₂³	1	4	yes	—
50	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
51	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
52	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
53	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
54	C₂×C₄×D₁₁₃	C₂³	1	8	yes	complemented
55	C₄⋊Dic₁₁₃	C₂³	1	4	yes	complemented
56	C₄⋊Dic₁₁₃	C₂³	1	4	yes	complemented
57	#14268626	C₂³	1	8	yes	complemented
58	#14268626	C₂³	1	8	yes	complemented
59	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
60	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
61	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
62	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
63	#14268624	C₂³	1	8	yes	complemented
64	#14268624	C₂³	1	8	yes	complemented
65	C₄×Dic₁₁₃	C₂³	1	8	yes	complemented
66	C₂⁴×C₁₁₃	—	2	4	no	—
67	C₂⁴×C₁₁₃	—	4	4	no	—
68	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
69	C₂⁴×C₁₁₃	C₂³	1	4	yes	—
70	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
71	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₁₃	—	4	4	no	—
73	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
74	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
76	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
79	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
80	C₂³×C₁₁₃	—	2	2	no	—
81	C₂³×C₁₁₃	—	2	4	no	—
82	C₂³×C₁₁₃	—	2	8	no	—
83	C₂³×C₁₁₃	—	2	2	no	—
84	C₂³×C₁₁₃	—	2	8	no	—
85	C₂³×C₁₁₃	—	4	2	no	—
86	C₂³×C₁₁₃	—	4	4	no	—
87	C₂³×C₁₁₃	—	4	2	no	—
88	C₂³×C₁₁₃	—	4	16	no	—
89	C₂³×C₁₁₃	#41	1	4	yes	complemented
90	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
91	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
92	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
93	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
94	C₂²×D₁₁₃	—	2	4	no	—
95	C₂²×D₁₁₃	—	4	4	no	—
96	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
97	C₂×C₄×C₁₁₃	—	2	8	no	—
98	C₂×C₄×C₁₁₃	—	4	2	no	—
99	C₂×C₄×C₁₁₃	—	4	4	no	—
100	C₂×C₄×C₁₁₃	—	4	2	no	—
101	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
102	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
103	C₂²⋊D₁₁₃	—	2	8	no	—
104	C₂²⋊D₁₁₃	—	2	8	no	—
105	C₂²⋊D₁₁₃	—	2	16	no	—
106	C₂²⋊D₁₁₃	—	4	8	no	—
107	C₂×Dic₁₁₃	—	2	4	no	—
108	C₂×Dic₁₁₃	—	4	4	no	—
109	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
110	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
111	D₄₅₂	—	4	4	no	—
112	C₄×D₁₁₃	—	4	8	no	—
113	Dic₂₂₆	—	4	4	no	—
114	C₂²×C₁₁₃	—	2	8	no	—
115	C₂²×C₁₁₃	—	2	2	no	—
116	C₂²×C₁₁₃	—	2	4	no	—
117	C₂²×C₁₁₃	—	2	2	no	—
118	C₂²×C₁₁₃	—	2	2	no	—
119	C₂²×C₁₁₃	—	2	2	no	—
120	C₂²×C₁₁₃	—	2	4	no	—
121	C₂²×C₁₁₃	—	4	4	no	—
122	C₂²×C₁₁₃	—	4	16	no	—
123	C₂²×C₁₁₃	—	4	4	no	—
124	C₂²×C₁₁₃	C₂×#41	1	4	yes	—
125	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
126	C₂²×C₁₁₃	H(5,𝔽₂)	1	4	yes	—
127	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
128	C₂×D₁₁₃	—	2	4	no	—
129	C₂×D₁₁₃	—	2	8	no	—
130	C₂×D₁₁₃	—	8	4	no	—
131	C₄×C₁₁₃	—	2	4	no	—
132	C₄×C₁₁₃	—	2	4	no	—
133	Dic₁₁₃	—	2	4	no	—
134	Dic₁₁₃	—	2	8	no	—
135	C₂×C₁₁₃	—	2	4	no	—
136	C₂×C₁₁₃	—	2	2	no	—
137	C₂×C₁₁₃	—	2	4	no	—
138	C₂×C₁₁₃	—	4	2	no	—
139	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
140	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
141	D₁₁₃	—	4	4	no	—
142	#3490	—	113	1	no	maximal, Sylow
143	C₁₁₃	#3490	1	1	yes	characteristic, complemented, Sylow
144	#551	—	113	8	no	—
145	#546	—	113	8	no	—
146	#543	—	113	1	no	—
147	#538	—	113	4	no	—
148	#534	—	113	1	no	—
149	#525	—	113	4	no	—
150	C₂×#120	—	113	2	no	—
151	#518	—	113	2	no	—
152	C₂²×C₂²⋊C₄	D₁₁₃	1	1	yes	characteristic, complemented
153	C₂⁵	C₂×D₁₁₃	1	1	yes	characteristic
154	C₂×#41	—	113	4	no	—
155	C₂²×D₄	—	113	4	no	—
156	C₂²×D₄	—	113	2	no	—
157	C₂³×C₄	C₂×D₁₁₃	1	2	yes	complemented
158	#126	—	113	8	no	—
159	#124	—	113	8	no	—
160	#123	—	113	8	no	—
161	#123	—	113	4	no	—
162	#123	—	113	8	no	—
163	C₂²⋊Q₈	—	113	4	no	—
164	C₂²⋊Q₈	—	113	4	no	—
165	C₂²⋊Q₈	—	113	8	no	—
166	#121	—	113	4	no	—
167	#121	—	113	4	no	—
168	#121	—	113	8	no	—
169	#120	—	113	4	no	—
170	#120	—	113	8	no	—
171	#120	—	113	4	no	—
172	#120	—	113	4	no	—
173	C₄×D₄	—	113	8	no	—
174	C₄×D₄	—	113	8	no	—
175	C₄×D₄	—	113	8	no	—
176	#117	—	113	4	no	—
177	#117	—	113	4	no	—
178	C₂×C₂²⋊C₄	—	113	2	no	—
179	C₂×C₂²⋊C₄	—	113	4	no	—
180	C₂×C₂²⋊C₄	—	113	4	no	—
181	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	complemented
184	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	8	yes	complemented
185	C₂⁴	—	2	4	no	—
186	C₂⁴	—	4	4	no	—
187	C₂⁴	—	113	2	no	—
188	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
189	C₂⁴	C₂²×D₁₁₃	1	4	yes	—
190	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
191	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
192	#41	—	452	8	no	—
193	C₂×Q₈	—	113	4	no	—
194	C₂×D₄	—	113	8	no	—
195	C₂×D₄	—	113	4	no	—
196	C₂×D₄	—	113	4	no	—
197	C₂×D₄	—	113	8	no	—
198	C₂×D₄	—	113	4	no	—
199	C₂×D₄	—	452	8	no	—
200	C₂×D₄	—	226	8	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	113	4	no	—
203	C₂²×C₄	—	113	2	no	—
204	C₂²×C₄	—	113	8	no	—
205	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
206	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
207	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
208	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
209	C₄⋊C₄	—	113	8	no	—
210	C₄⋊C₄	—	113	4	no	—
211	C₄⋊C₄	—	113	4	no	—
212	C₄⋊C₄	—	113	8	no	—
213	C₂²⋊C₄	—	113	4	no	—
214	C₂²⋊C₄	—	113	8	no	—
215	C₂²⋊C₄	—	113	4	no	—
216	C₂²⋊C₄	—	113	8	no	—
217	C₂²⋊C₄	—	113	8	no	—
218	C₂²⋊C₄	—	113	8	no	—
219	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
220	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
221	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
222	C₄²	—	113	8	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	8	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	8	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	4	16	no	—
232	C₂³	—	113	4	no	—
233	C₂³	—	452	4	no	—
234	C₂³	—	226	4	no	—
235	C₂³	#14268647	1	4	yes	complemented
236	C₂³	C₂³×D₁₁₃	1	2	yes	—
237	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
238	C₂³	C₂³×D₁₁₃	1	2	yes	—
239	C₂³	C₂³×D₁₁₃	1	2	yes	—
240	Q₈	—	452	4	no	—
241	D₄	—	452	4	no	—
242	D₄	—	452	8	no	—
243	D₄	—	226	8	no	—
244	D₄	—	226	8	no	—
245	D₄	—	226	16	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	113	4	no	—
251	C₂×C₄	—	113	8	no	—
252	C₂×C₄	—	452	8	no	—
253	C₂×C₄	—	452	4	no	—
254	C₂×C₄	—	226	4	no	—
255	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
256	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
257	C₂²	—	2	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	16	no	—
266	C₂²	—	4	4	no	—
267	C₂²	—	904	4	no	—
268	C₂²	—	226	4	no	—
269	C₂²	—	226	8	no	—
270	C₂²	C₂×#14268647	1	4	yes	—
271	C₂²	C₂×#14268647	1	2	yes	—
272	C₂²	#16813667	1	4	yes	—
273	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	226	4	no	—
277	C₄	—	226	8	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	4	no	—
280	C₂	—	2	4	no	—
281	C₂	—	4	2	no	—
282	C₂	—	452	4	no	—
283	C₂	C₂²×#14268647	1	1	yes	characteristic
284	C₂	C₂×#16813667	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	C₂²×C₂²⋊C₄×C₁₁₃	C₂
hypercenter	C₂²×C₂²⋊C₄	D₁₁₃
hypocenter	C₁₁₃	#3490
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #20299274	C₂	semidirect
2. #20299238	C₂	semidirect
3. #20299224	C₂	semidirect
4. #20299186	C₂	semidirect
5. #20299160	C₂	semidirect
6. #20299109	C₂	semidirect
7. C₂×#16813591	C₂	semidirect
8. #20299071	C₂	semidirect
9. C₂²×C₂²⋊C₄×C₁₁₃	C₂	semidirect
10. C₂×#14268647	C₂²	semidirect
12. C₂²×C₂²⋊D₁₁₃	C₂²	semidirect
13. #16813615	C₂²	semidirect
14. #16813608	C₂²	semidirect
15. #16813606	C₂²	semidirect
16. #16813605	C₂²	semidirect
17. #16813604	C₂²	semidirect
18. #16813601	C₂²	semidirect
19. #16813601	C₂²	semidirect
20. #16813599	C₂²	semidirect
21. #16813596	C₂²	semidirect
22. #16813596	C₂²	semidirect
23. #16813595	C₂²	semidirect
24. #16813592	C₂²	semidirect
26. #16813591	C₂²	semidirect
27. #16813591	C₂²	semidirect
28. C₄×C₂²⋊D₁₁₃	C₂²	semidirect
29. C₄×C₂²⋊D₁₁₃	C₂²	semidirect
30. #16813582	C₂²	semidirect
31. #16813581	C₂²	semidirect
32. #16813578	C₂²	semidirect
34. C₂×C₂²⋊Dic₁₁₃	C₂²	semidirect
35. C₂²⋊C₄×D₁₁₃	C₂²	semidirect
37. C₂³×C₄×C₁₁₃	C₂²	semidirect
40. C₂×C₂²⋊C₄×C₁₁₃	C₂²	semidirect
41. C₂×C₂²⋊C₄×C₁₁₃	C₂²	semidirect
44. C₂×Dic₂₂₆	C₂³	semidirect
45. C₂×D₄₅₂	C₂³	semidirect
54. C₂×C₄×D₁₁₃	C₂³	semidirect
55. C₄⋊Dic₁₁₃	C₂³	semidirect
56. C₄⋊Dic₁₁₃	C₂³	semidirect
57. #14268626	C₂³	semidirect
58. #14268626	C₂³	semidirect
63. #14268624	C₂³	semidirect
64. #14268624	C₂³	semidirect
65. C₄×Dic₁₁₃	C₂³	semidirect
89. C₂³×C₁₁₃	#41	semidirect
143. C₁₁₃	#3490	semidirect
152. C₂²×C₂²⋊C₄	D₁₁₃	semidirect
157. C₂³×C₄	C₂×D₁₁₃	semidirect
183. C₂×C₂²⋊C₄	C₂×D₁₁₃	semidirect
184. C₂×C₂²⋊C₄	C₂×D₁₁₃	semidirect
235. C₂³	#14268647	semidirect

Maximal subgroups

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

subgroup	index
#20299274	2
#20299238	2
#20299224	2
#20299186	2
#20299160	2
#20299109	2
C₂×#16813591	2
#20299071	2
C₂²×C₂²⋊C₄×C₁₁₃	2
#3490	113