Subgroups — #28404445

Order 14464.

1232 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299331	C₂	1	4	yes	maximal, complemented
2	#20299269	C₂	1	4	yes	maximal, complemented
3	#20299238	C₂	1	8	yes	maximal, complemented
4	#20299199	C₂	1	4	yes	maximal, complemented
5	#20299167	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299165	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299151	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299145	C₂	1	4	yes	maximal, complemented
9	C₂×#16813618	C₂	1	1	yes	maximal, characteristic, complemented
10	#20299072	C₂	1	1	yes	maximal, characteristic, complemented
11	#20299066	C₂	1	1	yes	maximal, characteristic, complemented
12	#518×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14268647	C₂²	1	1	yes	characteristic
14	C₂×#14268647	C₂²	1	8	yes	complemented
15	C₂²×D₄₅₂	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₁₃	C₂²	1	4	yes	—
17	#16813621	C₂²	1	2	yes	complemented
18	#16813619	C₂²	1	2	yes	complemented
19	#16813618	C₂²	1	4	yes	—
20	#16813618	C₂²	1	2	yes	—
21	#16813618	C₂²	1	2	yes	—
22	#16813618	C₂²	1	2	yes	complemented
23	#16813613	C₂²	1	2	yes	complemented
24	#16813610	C₂²	1	2	yes	complemented
25	#16813609	C₂²	1	2	yes	complemented
26	#16813608	C₂²	1	8	yes	complemented
27	#16813605	C₂²	1	8	yes	complemented
28	#16813604	C₂²	1	8	yes	complemented
29	#16813599	C₂²	1	4	yes	—
30	#16813597	C₂²	1	4	yes	complemented
31	#16813596	C₂²	1	8	yes	complemented
32	#16813595	C₂²	1	8	yes	complemented
33	#16813595	C₂²	1	8	yes	complemented
34	#16813593	C₂²	1	4	yes	complemented
35	#16813592	C₂²	1	4	yes	—
36	#16813591	C₂²	1	4	yes	—
37	C₄×Dic₂₂₆	C₂²	1	4	yes	complemented
38	#16813587	C₂²	1	2	yes	complemented
39	C₄×D₄₅₂	C₂²	1	4	yes	complemented
40	#16813584	C₂²	1	2	yes	complemented
41	#16813584	C₂²	1	4	yes	complemented
42	C₄×C₂²⋊D₁₁₃	C₂²	1	8	yes	complemented
43	#16813582	C₂²	1	4	yes	—
44	#16813581	C₂²	1	1	yes	characteristic
45	#16813579	C₂²	1	2	yes	complemented
46	C₂²⋊C₄×D₁₁₃	C₂²	1	4	yes	—
47	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
48	C₄²×D₁₁₃	C₂²	1	2	yes	complemented
49	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
50	C₄×D₄×C₁₁₃	C₂²	1	4	yes	complemented
51	C₄×D₄×C₁₁₃	C₂²	1	4	yes	complemented
52	#117×C₁₁₃	C₂²	1	1	yes	characteristic
53	#117×C₁₁₃	C₂²	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	4	yes	—
55	C₂³×D₁₁₃	C₂³	1	2	yes	—
56	#14268647	—	2	4	no	—
57	#14268647	—	4	16	no	—
58	C₂×Dic₂₂₆	C₂³	1	1	yes	characteristic
59	C₂×Dic₂₂₆	C₂³	1	4	yes	—
60	C₂×D₄₅₂	—	2	4	no	—
61	C₂×D₄₅₂	C₂³	1	2	yes	—
62	C₂×D₄₅₂	C₂³	1	1	yes	characteristic
63	C₂×D₄₅₂	C₂³	1	2	yes	—
64	C₂×D₄₅₂	C₂³	1	8	yes	—
65	C₂×C₂²⋊D₁₁₃	—	4	8	no	—
66	C₂×C₂²⋊D₁₁₃	C₂³	1	2	yes	—
67	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
68	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
69	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
70	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
71	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
72	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
73	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
74	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
75	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
76	#14268626	C₂³	1	4	yes	—
77	#14268626	C₂³	1	4	yes	—
78	#14268626	C₂³	1	4	yes	—
79	C₂²⋊Dic₁₁₃	C₂³	1	2	yes	—
80	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
81	#14268624	C₂³	1	8	yes	—
82	#14268624	C₂³	1	4	yes	—
83	#14268624	C₂³	1	8	yes	—
84	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
85	C₄×Dic₁₁₃	C₂³	1	4	yes	—
86	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
87	C₄×Dic₁₁₃	C₂³	1	2	yes	—
88	C₄×Dic₁₁₃	C₂³	1	2	yes	—
89	C₂⁴×C₁₁₃	C₂³	1	2	yes	—
90	C₂×D₄×C₁₁₃	—	2	4	no	—
91	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
92	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
93	C₂²×C₄×C₁₁₃	C₂³	1	8	yes	—
94	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
96	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
97	C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
98	C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
99	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
100	C₄²×C₁₁₃	C₂³	1	2	yes	—
101	C₄²×C₁₁₃	C₂³	1	2	yes	—
102	C₂³×C₁₁₃	—	2	4	no	—
103	C₂³×C₁₁₃	—	4	4	no	—
104	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
105	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
106	C₂²×D₁₁₃	—	2	4	no	—
107	C₂²×D₁₁₃	—	4	4	no	—
108	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
109	C₂²×D₁₁₃	C₂⁴	1	2	yes	—
110	D₄×C₁₁₃	—	2	4	no	—
111	D₄×C₁₁₃	—	2	4	no	—
112	C₂×C₄×C₁₁₃	—	4	8	no	—
113	C₂×C₄×C₁₁₃	#41	1	4	yes	complemented
114	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
116	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₁₃	—	2	4	no	—
119	C₂²⋊D₁₁₃	—	2	16	no	—
120	C₂²⋊D₁₁₃	—	4	8	no	—
121	C₂²⋊D₁₁₃	—	4	8	no	—
122	C₂×Dic₁₁₃	—	2	4	no	—
123	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
124	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
125	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
126	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
127	D₄₅₂	—	2	4	no	—
128	D₄₅₂	—	2	4	no	—
129	D₄₅₂	—	2	2	no	—
130	D₄₅₂	—	4	8	no	—
131	C₄×D₁₁₃	—	2	4	no	—
132	C₄×D₁₁₃	—	2	8	no	—
133	C₄×D₁₁₃	—	4	8	no	—
134	Dic₂₂₆	—	2	2	no	—
135	Dic₂₂₆	—	2	8	no	—
136	C₂²×C₁₁₃	—	2	2	no	—
137	C₂²×C₁₁₃	—	2	4	no	—
138	C₂²×C₁₁₃	—	2	8	no	—
139	C₂²×C₁₁₃	—	8	4	no	—
140	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
141	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
142	C₂×D₁₁₃	—	2	2	no	—
143	C₂×D₁₁₃	—	2	4	no	—
144	C₂×D₁₁₃	—	2	4	no	—
145	C₂×D₁₁₃	—	2	8	no	—
146	C₂×D₁₁₃	—	8	4	no	—
147	C₄×C₁₁₃	—	2	4	no	—
148	C₄×C₁₁₃	—	2	4	no	—
149	C₄×C₁₁₃	C₂×#41	1	2	yes	—
150	C₄×C₁₁₃	C₂×#41	1	2	yes	—
151	Dic₁₁₃	—	2	2	no	—
152	Dic₁₁₃	—	2	4	no	—
153	Dic₁₁₃	—	2	2	no	—
154	Dic₁₁₃	H(5,𝔽₂)	1	4	yes	—
155	C₂×C₁₁₃	—	2	2	no	—
156	C₂×C₁₁₃	—	4	4	no	—
157	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
158	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
159	D₁₁₃	—	4	2	no	—
160	D₁₁₃	—	4	4	no	—
161	#3508	—	113	1	no	maximal, Sylow
162	C₁₁₃	#3508	1	1	yes	characteristic, complemented, Sylow
163	#559	—	113	4	no	—
164	#550	—	113	4	no	—
165	#546	—	113	8	no	—
166	#540	—	113	4	no	—
167	#535	—	113	1	no	—
168	#535	—	113	1	no	—
169	#533	—	113	1	no	—
170	#532	—	113	4	no	—
171	C₂×#127	—	113	1	no	—
172	#518	—	113	1	no	—
173	#518	D₁₁₃	1	1	yes	characteristic, complemented
174	C₄×#41	—	113	1	no	—
175	C₂×#41	—	113	8	no	—
176	C₂×#41	—	113	1	no	—
177	C₂²×D₄	—	113	4	no	—
178	C₂²×D₄	—	113	1	no	—
179	C₂²×D₄	C₂×D₁₁₃	1	1	yes	characteristic
180	C₄⋊Q₈	—	113	2	no	—
181	#127	—	113	2	no	—
182	#127	—	113	2	no	—
183	#127	—	113	4	no	—
184	#127	—	113	2	no	—
185	#127	—	113	2	no	—
186	#125	—	113	2	no	—
187	#124	—	113	2	no	—
188	#124	—	113	8	no	—
189	#124	—	113	2	no	—
190	#123	—	113	8	no	—
191	#123	—	113	8	no	—
192	C₂²⋊Q₈	—	113	4	no	—
193	#121	—	113	4	no	—
194	#121	—	113	8	no	—
195	#121	—	113	4	no	—
196	#121	—	113	8	no	—
197	#121	—	113	8	no	—
198	#120	—	113	4	no	—
199	#120	—	113	4	no	—
200	C₄×Q₈	—	113	2	no	—
201	C₄×Q₈	—	113	4	no	—
202	C₄×D₄	—	113	8	no	—
203	C₄×D₄	—	113	2	no	—
204	C₄×D₄	—	113	4	no	—
205	C₄×D₄	—	113	4	no	—
206	C₄×D₄	—	113	4	no	—
207	C₄×D₄	C₂×D₁₁₃	1	4	yes	complemented
208	C₄×D₄	C₂×D₁₁₃	1	4	yes	complemented
209	#117	—	113	1	no	—
210	#117	—	113	2	no	—
211	#117	C₂×D₁₁₃	1	1	yes	characteristic
212	#117	C₂×D₁₁₃	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	113	4	no	—
214	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	4	yes	—
215	C₂×C₄²	—	113	1	no	—
216	C₂×C₄²	—	113	2	no	—
217	C₂⁴	—	113	2	no	—
218	C₂⁴	C₂²×D₁₁₃	1	2	yes	—
219	#41	—	452	16	no	—
220	#41	—	226	4	no	—
221	C₂×Q₈	—	113	1	no	—
222	C₂×Q₈	—	113	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	113	8	no	—
225	C₂×D₄	—	113	8	no	—
226	C₂×D₄	—	113	2	no	—
227	C₂×D₄	—	113	8	no	—
228	C₂×D₄	—	113	8	no	—
229	C₂×D₄	—	113	2	no	—
230	C₂×D₄	—	113	2	no	—
231	C₂×D₄	—	113	1	no	—
232	C₂×D₄	—	452	8	no	—
233	C₂×D₄	—	226	4	no	—
234	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
235	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
236	C₂²×C₄	—	113	2	no	—
237	C₂²×C₄	—	113	8	no	—
238	C₂²×C₄	—	113	2	no	—
239	C₂²×C₄	—	113	4	no	—
240	C₂²×C₄	C₂²×D₁₁₃	1	8	yes	—
241	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
242	C₄⋊C₄	—	113	2	no	—
243	C₄⋊C₄	—	113	4	no	—
244	C₄⋊C₄	—	113	4	no	—
245	C₄⋊C₄	—	113	4	no	—
246	C₄⋊C₄	—	113	4	no	—
247	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
248	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
249	C₂²⋊C₄	—	113	2	no	—
250	C₂²⋊C₄	—	113	8	no	—
251	C₂²⋊C₄	—	113	8	no	—
252	C₂²⋊C₄	—	113	4	no	—
253	C₂²⋊C₄	—	113	8	no	—
254	C₂²⋊C₄	C₂²×D₁₁₃	1	2	yes	—
255	C₂²⋊C₄	C₂²×D₁₁₃	1	2	yes	—
256	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
257	C₄²	—	113	1	no	—
258	C₄²	—	113	1	no	—
259	C₄²	—	113	4	no	—
260	C₄²	—	113	2	no	—
261	C₄²	—	113	2	no	—
262	C₄²	C₂²×D₁₁₃	1	2	yes	—
263	C₄²	C₂²×D₁₁₃	1	2	yes	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	113	4	no	—
267	C₂³	—	113	2	no	—
268	C₂³	—	452	4	no	—
269	C₂³	—	226	4	no	—
270	C₂³	C₂³×D₁₁₃	1	4	yes	—
271	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
272	Q₈	—	226	8	no	—
273	Q₈	—	226	2	no	—
274	D₄	—	2	4	no	—
275	D₄	—	2	4	no	—
276	D₄	—	452	8	no	—
277	D₄	—	452	8	no	—
278	D₄	—	452	8	no	—
279	D₄	—	226	16	no	—
280	D₄	—	226	4	no	—
281	D₄	—	226	4	no	—
282	D₄	—	226	4	no	—
283	D₄	—	226	2	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	113	2	no	—
286	C₂×C₄	—	113	2	no	—
287	C₂×C₄	—	113	2	no	—
288	C₂×C₄	—	113	4	no	—
289	C₂×C₄	—	452	8	no	—
290	C₂×C₄	—	226	8	no	—
291	C₂×C₄	—	226	4	no	—
292	C₂×C₄	—	226	4	no	—
293	C₂×C₄	#14268647	1	4	yes	complemented
294	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
295	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
296	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	904	4	no	—
303	C₂²	—	226	2	no	—
304	C₂²	—	226	4	no	—
305	C₂²	—	226	8	no	—
306	C₂²	—	226	4	no	—
307	C₂²	C₂×#14268647	1	2	yes	—
308	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
309	C₄	—	2	4	no	—
310	C₄	—	2	4	no	—
311	C₄	—	113	4	no	—
312	C₄	—	226	2	no	—
313	C₄	—	226	4	no	—
314	C₄	—	226	2	no	—
315	C₄	C₂×#14268647	1	2	yes	—
316	C₄	C₂×#14268647	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	4	4	no	—
319	C₂	—	452	4	no	—
320	C₂	—	452	2	no	—
321	C₂	C₂²×#14268647	1	1	yes	characteristic
322	C₂	H(5,𝔽₂)×D₁₁₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	#518×C₁₁₃	C₂
hypercenter	#518	D₁₁₃
hypocenter	C₁₁₃	#3508
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
1. #20299331	C₂	semidirect
2. #20299269	C₂	semidirect
3. #20299238	C₂	semidirect
4. #20299199	C₂	semidirect
5. #20299167	C₂	semidirect
6. #20299165	C₂	semidirect
7. #20299151	C₂	semidirect
8. #20299145	C₂	semidirect
9. C₂×#16813618	C₂	semidirect
10. #20299072	C₂	semidirect
11. #20299066	C₂	semidirect
12. #518×C₁₁₃	C₂	semidirect
14. C₂×#14268647	C₂²	semidirect
17. #16813621	C₂²	semidirect
18. #16813619	C₂²	semidirect
22. #16813618	C₂²	semidirect
23. #16813613	C₂²	semidirect
24. #16813610	C₂²	semidirect
25. #16813609	C₂²	semidirect
26. #16813608	C₂²	semidirect
27. #16813605	C₂²	semidirect
28. #16813604	C₂²	semidirect
30. #16813597	C₂²	semidirect
31. #16813596	C₂²	semidirect
32. #16813595	C₂²	semidirect
33. #16813595	C₂²	semidirect
34. #16813593	C₂²	semidirect
37. C₄×Dic₂₂₆	C₂²	semidirect
38. #16813587	C₂²	semidirect
39. C₄×D₄₅₂	C₂²	semidirect
40. #16813584	C₂²	semidirect
41. #16813584	C₂²	semidirect
42. C₄×C₂²⋊D₁₁₃	C₂²	semidirect
45. #16813579	C₂²	semidirect
48. C₄²×D₁₁₃	C₂²	semidirect
50. C₄×D₄×C₁₁₃	C₂²	semidirect
51. C₄×D₄×C₁₁₃	C₂²	semidirect
113. C₂×C₄×C₁₁₃	#41	semidirect
162. C₁₁₃	#3508	semidirect
173. #518	D₁₁₃	semidirect
207. C₄×D₄	C₂×D₁₁₃	semidirect
208. C₄×D₄	C₂×D₁₁₃	semidirect
293. C₂×C₄	#14268647	semidirect

Maximal subgroups

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

subgroup	index
#20299331	2
#20299269	2
#20299238	2
#20299199	2
#20299167	2
#20299165	2
#20299151	2
#20299145	2
C₂×#16813618	2
#20299072	2
#20299066	2
#518×C₁₁₃	2
#3508	113