Subgroups — #28404468

Order 14464.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299330	C₂	1	4	yes	maximal, complemented
2	#20299317	C₂	1	4	yes	maximal, complemented
3	#20299274	C₂	1	8	yes	maximal, complemented
4	#20299217	C₂	1	4	yes	maximal, complemented
5	#20299172	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299165	C₂	1	1	yes	maximal, characteristic
7	#20299155	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299127	C₂	1	4	yes	maximal, complemented
9	#20299076	C₂	1	1	yes	maximal, characteristic, complemented
10	#20299073	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#14268647	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#124×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14268647	C₂²	1	1	yes	characteristic
14	#16813622	C₂²	1	2	yes	complemented
15	#16813620	C₂²	1	2	yes	complemented
16	#16813618	C₂²	1	2	yes	—
17	#16813615	C₂²	1	8	yes	complemented
18	#16813615	C₂²	1	8	yes	complemented
19	#16813614	C₂²	1	2	yes	complemented
20	#16813611	C₂²	1	2	yes	complemented
21	#16813609	C₂²	1	2	yes	—
22	#16813608	C₂²	1	8	yes	complemented
23	#16813606	C₂²	1	8	yes	complemented
24	#16813605	C₂²	1	4	yes	—
25	#16813604	C₂²	1	8	yes	complemented
26	#16813601	C₂²	1	4	yes	—
27	#16813599	C₂²	1	8	yes	complemented
28	#16813598	C₂²	1	4	yes	complemented
29	#16813593	C₂²	1	4	yes	—
30	#16813591	C₂²	1	4	yes	—
31	Q₈×Dic₁₁₃	C₂²	1	4	yes	complemented
32	Q₈×Dic₁₁₃	C₂²	1	4	yes	complemented
33	C₄×Dic₂₂₆	C₂²	1	2	yes	complemented
34	C₄×Dic₂₂₆	C₂²	1	4	yes	complemented
35	C₄×D₄₅₂	C₂²	1	2	yes	complemented
36	D₄×Dic₁₁₃	C₂²	1	4	yes	—
37	D₄×Dic₁₁₃	C₂²	1	4	yes	—
38	C₄×C₂²⋊D₁₁₃	C₂²	1	4	yes	—
39	#16813582	C₂²	1	8	yes	complemented
40	#16813581	C₂²	1	4	yes	—
41	#16813581	C₂²	1	1	yes	characteristic
42	#16813581	C₂²	1	1	yes	characteristic
43	#16813581	C₂²	1	1	yes	characteristic
44	#16813579	C₂²	1	2	yes	complemented
45	#16813578	C₂²	1	8	yes	complemented
46	C₂×C₂²⋊Dic₁₁₃	C₂²	1	4	yes	—
47	C₄²×D₁₁₃	C₂²	1	2	yes	complemented
48	C₂²×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
49	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
50	#124×C₁₁₃	C₂²	1	4	yes	—
51	#124×C₁₁₃	C₂²	1	2	yes	—
52	#124×C₁₁₃	C₂²	1	2	yes	—
53	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	4	yes	—
54	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
55	#14268647	—	2	4	no	—
56	C₂×Dic₂₂₆	C₂³	1	1	yes	characteristic
57	C₂×Dic₂₂₆	C₂³	1	4	yes	—
58	C₂×D₄₅₂	C₂³	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₁₃	C₂³	1	2	yes	—
60	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
61	C₂²×Dic₁₁₃	C₂³	1	8	yes	—
62	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
63	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
64	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
65	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
66	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
67	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
68	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
69	#14268626	C₂³	1	4	yes	—
70	#14268626	C₂³	1	8	yes	—
71	#14268626	C₂³	1	4	yes	—
72	#14268626	C₂³	1	4	yes	—
73	C₂²⋊Dic₁₁₃	C₂³	1	2	yes	—
74	C₂²⋊Dic₁₁₃	C₂³	1	2	yes	—
75	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
76	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
77	C₂²⋊Dic₁₁₃	C₂³	1	2	yes	—
78	#14268624	C₂³	1	4	yes	—
79	#14268624	C₂³	1	8	yes	—
80	C₄×Dic₁₁₃	C₂³	1	4	yes	—
81	C₄×Dic₁₁₃	C₂³	1	2	yes	—
82	C₄×Dic₁₁₃	C₂³	1	8	yes	—
83	C₄×Dic₁₁₃	C₂³	1	2	yes	—
84	C₄×Dic₁₁₃	C₂³	1	2	yes	—
85	C₂⁴×C₁₁₃	C₂³	1	2	yes	—
86	C₂×Q₈×C₁₁₃	—	2	4	no	—
87	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
88	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
89	C₂×D₄×C₁₁₃	—	2	4	no	—
90	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
91	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
92	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
93	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
94	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
95	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
96	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
97	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
98	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
99	C₄²×C₁₁₃	C₂³	1	2	yes	—
100	C₂³×C₁₁₃	—	2	4	no	—
101	C₂³×C₁₁₃	—	4	4	no	—
102	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
103	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
104	C₂²×D₁₁₃	C₂⁴	1	2	yes	—
105	Q₈×C₁₁₃	—	2	4	no	—
106	Q₈×C₁₁₃	—	2	4	no	—
107	D₄×C₁₁₃	—	2	4	no	—
108	D₄×C₁₁₃	—	2	4	no	—
109	C₂×C₄×C₁₁₃	—	2	4	no	—
110	C₂×C₄×C₁₁₃	—	4	2	no	—
111	C₂×C₄×C₁₁₃	#41	1	4	yes	—
112	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
113	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
114	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₁₁₃	—	2	4	no	—
118	C₂²⋊D₁₁₃	—	4	8	no	—
119	C₂×Dic₁₁₃	—	4	8	no	—
120	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
121	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
122	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
123	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
124	D₄₅₂	—	2	2	no	—
125	C₄×D₁₁₃	—	2	4	no	—
126	C₄×D₁₁₃	—	4	4	no	—
127	Dic₂₂₆	—	2	2	no	—
128	Dic₂₂₆	—	4	4	no	—
129	C₂²×C₁₁₃	—	2	2	no	—
130	C₂²×C₁₁₃	—	2	4	no	—
131	C₂²×C₁₁₃	—	2	8	no	—
132	C₂²×C₁₁₃	—	8	4	no	—
133	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
134	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
135	C₂×D₁₁₃	—	2	2	no	—
136	C₂×D₁₁₃	—	2	4	no	—
137	C₄×C₁₁₃	—	2	4	no	—
138	C₄×C₁₁₃	—	2	2	no	—
139	C₄×C₁₁₃	—	2	2	no	—
140	C₄×C₁₁₃	C₂×#41	1	2	yes	—
141	C₄×C₁₁₃	C₂×#41	1	2	yes	—
142	Dic₁₁₃	—	2	2	no	—
143	Dic₁₁₃	—	2	4	no	—
144	Dic₁₁₃	—	2	4	no	—
145	Dic₁₁₃	—	2	4	no	—
146	C₂×C₁₁₃	—	2	2	no	—
147	C₂×C₁₁₃	—	4	4	no	—
148	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
149	C₂×C₁₁₃	#585	1	2	yes	—
150	D₁₁₃	—	4	2	no	—
151	#3510	—	113	1	no	maximal, Sylow
152	C₁₁₃	#3510	1	1	yes	characteristic, complemented, Sylow
153	#559	—	113	4	no	—
154	#556	—	113	4	no	—
155	#551	—	113	8	no	—
156	#542	—	113	4	no	—
157	#536	—	113	1	no	—
158	#535	—	113	1	no	—
159	#533	—	113	1	no	—
160	#529	—	113	4	no	—
161	C₂×#124	D₁₁₃	1	1	yes	characteristic, complemented
162	#519	—	113	1	no	—
163	#518	—	113	1	no	—
164	C₄×#41	—	113	1	no	—
165	C₂×#41	—	113	1	no	—
166	C₂²×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
167	C₂²×D₄	C₂×D₁₁₃	1	1	yes	characteristic
168	C₄⋊Q₈	—	113	2	no	—
169	C₄⋊Q₈	—	113	2	no	—
170	#127	—	113	2	no	—
171	#126	—	113	8	no	—
172	#126	—	113	8	no	—
173	#125	—	113	2	no	—
174	#124	—	113	2	no	—
175	#124	—	113	2	no	—
176	#124	—	113	8	no	—
177	#124	C₂×D₁₁₃	1	2	yes	—
178	#124	C₂×D₁₁₃	1	2	yes	—
179	#124	C₂×D₁₁₃	1	4	yes	—
180	#123	—	113	4	no	—
181	#123	—	113	8	no	—
182	#123	—	113	8	no	—
183	C₂²⋊Q₈	—	113	4	no	—
184	C₂²⋊Q₈	—	113	4	no	—
185	C₂²⋊Q₈	—	113	8	no	—
186	#121	—	113	4	no	—
187	#120	—	113	4	no	—
188	C₄×Q₈	—	113	4	no	—
189	C₄×Q₈	—	113	4	no	—
190	C₄×Q₈	—	113	2	no	—
191	C₄×Q₈	—	113	4	no	—
192	C₄×D₄	—	113	4	no	—
193	C₄×D₄	—	113	4	no	—
194	C₄×D₄	—	113	4	no	—
195	C₄×D₄	—	113	8	no	—
196	C₄×D₄	—	113	2	no	—
197	#117	—	113	1	no	—
198	#117	—	113	1	no	—
199	#117	—	113	1	no	—
200	#117	—	113	4	no	—
201	#117	—	113	8	no	—
202	#117	—	113	2	no	—
203	C₂×C₂²⋊C₄	—	113	4	no	—
204	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	4	yes	—
205	C₂×C₄²	—	113	2	no	—
206	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
207	C₂⁴	C₂²×D₁₁₃	1	2	yes	—
208	#41	—	226	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	113	4	no	—
211	C₂×Q₈	—	113	1	no	—
212	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
213	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	113	8	no	—
216	C₂×D₄	—	113	2	no	—
217	C₂×D₄	—	113	1	no	—
218	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
219	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
220	C₂²×C₄	—	113	8	no	—
221	C₂²×C₄	—	113	4	no	—
222	C₂²×C₄	—	113	2	no	—
223	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
224	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
225	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
226	C₄⋊C₄	—	113	4	no	—
227	C₄⋊C₄	—	113	2	no	—
228	C₄⋊C₄	—	113	2	no	—
229	C₄⋊C₄	—	113	4	no	—
230	C₄⋊C₄	—	113	2	no	—
231	C₄⋊C₄	—	113	8	no	—
232	C₄⋊C₄	—	113	8	no	—
233	C₄⋊C₄	—	113	4	no	—
234	C₄⋊C₄	—	113	4	no	—
235	C₂²⋊C₄	—	113	2	no	—
236	C₂²⋊C₄	—	113	2	no	—
237	C₂²⋊C₄	—	113	8	no	—
238	C₂²⋊C₄	—	113	2	no	—
239	C₂²⋊C₄	—	113	8	no	—
240	C₂²⋊C₄	—	113	4	no	—
241	C₂²⋊C₄	—	113	8	no	—
242	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
243	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
244	C₄²	—	113	2	no	—
245	C₄²	—	113	4	no	—
246	C₄²	—	113	8	no	—
247	C₄²	—	113	2	no	—
248	C₄²	—	113	2	no	—
249	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
250	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
251	C₄²	C₂²×D₁₁₃	1	2	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	113	2	no	—
255	C₂³	C₂³×D₁₁₃	1	4	yes	—
256	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	—	452	4	no	—
260	Q₈	—	226	2	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	452	8	no	—
264	D₄	—	226	4	no	—
265	D₄	—	226	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	113	4	no	—
269	C₂×C₄	—	113	4	no	—
270	C₂×C₄	—	113	2	no	—
271	C₂×C₄	—	113	4	no	—
272	C₂×C₄	—	452	4	no	—
273	C₂×C₄	—	452	8	no	—
274	C₂×C₄	—	226	4	no	—
275	C₂×C₄	#14268646	1	4	yes	—
276	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
277	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
278	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
279	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	8	4	no	—
285	C₂²	—	226	2	no	—
286	C₂²	—	226	4	no	—
287	C₂²	C₂×#14268646	1	2	yes	—
288	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	226	2	no	—
293	C₄	—	226	4	no	—
294	C₄	—	226	4	no	—
295	C₄	—	226	4	no	—
296	C₄	C₂×#14268646	1	2	yes	—
297	C₄	C₂×#14268646	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	4	no	—
300	C₂	—	452	2	no	—
301	C₂	C₂²×#14268646	1	1	yes	characteristic
302	C₂	#20299434	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	C₂×#124×C₁₁₃	C₂
prime radical	#20299165	C₂
hypercenter	C₂×#124	D₁₁₃
hypocenter	C₁₁₃	#3510
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. #20299330	C₂	semidirect
2. #20299317	C₂	semidirect
3. #20299274	C₂	semidirect
4. #20299217	C₂	semidirect
5. #20299172	C₂	semidirect
7. #20299155	C₂	semidirect
8. #20299127	C₂	semidirect
9. #20299076	C₂	semidirect
10. #20299073	C₂	semidirect
11. C₄×#14268647	C₂	semidirect
12. C₂×#124×C₁₁₃	C₂	semidirect
14. #16813622	C₂²	semidirect
15. #16813620	C₂²	semidirect
17. #16813615	C₂²	semidirect
18. #16813615	C₂²	semidirect
19. #16813614	C₂²	semidirect
20. #16813611	C₂²	semidirect
22. #16813608	C₂²	semidirect
23. #16813606	C₂²	semidirect
25. #16813604	C₂²	semidirect
27. #16813599	C₂²	semidirect
28. #16813598	C₂²	semidirect
31. Q₈×Dic₁₁₃	C₂²	semidirect
32. Q₈×Dic₁₁₃	C₂²	semidirect
33. C₄×Dic₂₂₆	C₂²	semidirect
34. C₄×Dic₂₂₆	C₂²	semidirect
35. C₄×D₄₅₂	C₂²	semidirect
39. #16813582	C₂²	semidirect
44. #16813579	C₂²	semidirect
45. #16813578	C₂²	semidirect
47. C₄²×D₁₁₃	C₂²	semidirect
152. C₁₁₃	#3510	semidirect
161. C₂×#124	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20299330	2
#20299317	2
#20299274	2
#20299217	2
#20299172	2
#20299165	2
#20299155	2
#20299127	2
#20299076	2
#20299073	2
C₄×#14268647	2
C₂×#124×C₁₁₃	2
#3510	113