Subgroups — #28404473

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	#20299322	C₂	1	4	yes	maximal, complemented
3	#20299271	C₂	1	8	yes	maximal, complemented
4	#20299221	C₂	1	4	yes	maximal, complemented
5	#20299162	C₂	1	1	yes	maximal, characteristic
6	#20299151	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299134	C₂	1	4	yes	maximal, complemented
8	C₂×#16813611	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299076	C₂	1	1	yes	maximal, characteristic, complemented
10	#20299072	C₂	1	1	yes	maximal, characteristic, complemented
11	#41×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#536×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₄₅₂	C₂²	1	1	yes	characteristic
14	#16813621	C₂²	1	2	yes	complemented
15	#16813618	C₂²	1	2	yes	—
16	#16813617	C₂²	1	8	yes	complemented
17	#16813617	C₂²	1	8	yes	complemented
18	#16813613	C₂²	1	2	yes	complemented
19	#16813611	C₂²	1	2	yes	—
20	#16813611	C₂²	1	4	yes	—
21	#16813611	C₂²	1	2	yes	—
22	#16813611	C₂²	1	2	yes	—
23	#16813608	C₂²	1	8	yes	complemented
24	#16813606	C₂²	1	8	yes	complemented
25	#16813604	C₂²	1	8	yes	complemented
26	#16813602	C₂²	1	8	yes	complemented
27	#16813599	C₂²	1	4	yes	—
28	#16813596	C₂²	1	4	yes	—
29	#16813590	C₂²	1	4	yes	—
30	Q₈×Dic₁₁₃	C₂²	1	2	yes	complemented
31	Q₈×Dic₁₁₃	C₂²	1	4	yes	complemented
32	Q₈×Dic₁₁₃	C₂²	1	4	yes	complemented
33	C₄×Dic₂₂₆	C₂²	1	4	yes	complemented
34	C₄×D₄₅₂	C₂²	1	4	yes	—
35	D₄×Dic₁₁₃	C₂²	1	4	yes	—
36	D₄×Dic₁₁₃	C₂²	1	2	yes	complemented
37	#16813584	C₂²	1	4	yes	—
38	#16813582	C₂²	1	8	yes	complemented
39	#16813581	C₂²	1	4	yes	—
40	#16813581	C₂²	1	1	yes	characteristic
41	#16813581	C₂²	1	1	yes	characteristic
42	#16813581	C₂²	1	2	yes	complemented
43	#16813580	C₂²	1	8	yes	complemented
44	C₂×#14268624	C₂²	1	4	yes	—
45	C₂²⋊C₄×D₁₁₃	C₂²	1	4	yes	—
46	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
47	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	complemented
48	C₂×#41×C₁₁₃	C₂²	1	1	yes	characteristic
49	C₂²×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
50	C₄⋊Q₈×C₁₁₃	C₂²	1	2	yes	complemented
51	#124×C₁₁₃	C₂²	1	2	yes	complemented
52	#123×C₁₁₃	C₂²	1	4	yes	—
53	C₂²⋊Q₈×C₁₁₃	C₂²	1	4	yes	complemented
54	#117×C₁₁₃	C₂²	1	1	yes	characteristic
55	C₂³×D₁₁₃	C₂³	1	2	yes	—
56	C₂×Dic₂₂₆	C₂³	1	4	yes	—
57	C₂×D₄₅₂	—	2	4	no	—
58	C₂×D₄₅₂	C₂³	1	2	yes	—
59	C₂×D₄₅₂	C₂³	1	2	yes	—
60	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
61	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
62	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
63	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
64	C₄⋊Dic₁₁₃	C₂³	1	4	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	4	yes	—
71	C₂²⋊Dic₁₁₃	C₂³	1	2	yes	—
72	C₂²⋊Dic₁₁₃	C₂³	1	2	yes	—
73	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
74	#14268624	C₂³	1	8	yes	—
75	#14268624	C₂³	1	8	yes	—
76	#14268624	C₂³	1	8	yes	—
77	#14268624	C₂³	1	8	yes	—
78	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
79	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
80	C₄×Dic₁₁₃	C₂³	1	2	yes	—
81	C₄×Dic₁₁₃	C₂³	1	4	yes	—
82	C₄×Dic₁₁₃	C₂³	1	2	yes	—
83	C₄×Dic₁₁₃	C₂³	1	8	yes	—
84	C₄×Dic₁₁₃	C₂³	1	2	yes	—
85	#41×C₁₁₃	—	2	4	no	—
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×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
90	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
91	C₂×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
96	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
97	C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
98	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
99	C₄²×C₁₁₃	C₂³	1	2	yes	—
100	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
101	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
102	C₂²×D₁₁₃	—	2	4	no	—
103	C₂²×D₁₁₃	—	4	4	no	—
104	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
105	Q₈×C₁₁₃	—	2	4	no	—
106	Q₈×C₁₁₃	—	2	4	no	—
107	Q₈×C₁₁₃	—	2	2	no	—
108	D₄×C₁₁₃	—	2	4	no	—
109	D₄×C₁₁₃	—	2	2	no	—
110	C₂×C₄×C₁₁₃	—	2	4	no	—
111	C₂×C₄×C₁₁₃	—	2	4	no	—
112	C₂×C₄×C₁₁₃	#41	1	4	yes	—
113	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
114	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
116	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₁₃	—	4	8	no	—
119	C₂×Dic₁₁₃	—	4	2	no	—
120	C₂×Dic₁₁₃	—	4	4	no	—
121	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
122	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
123	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
124	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
125	D₄₅₂	—	2	4	no	—
126	D₄₅₂	—	2	4	no	—
127	C₄×D₁₁₃	—	4	8	no	—
128	Dic₂₂₆	—	4	4	no	—
129	C₂²×C₁₁₃	—	2	2	no	—
130	C₂²×C₁₁₃	—	2	2	no	—
131	C₂²×C₁₁₃	—	2	4	no	—
132	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
133	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
134	C₂×D₁₁₃	—	2	4	no	—
135	C₂×D₁₁₃	—	2	8	no	—
136	C₂×D₁₁₃	—	8	4	no	—
137	C₄×C₁₁₃	—	2	2	no	—
138	C₄×C₁₁₃	—	2	4	no	—
139	C₄×C₁₁₃	—	2	4	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	2	no	—
146	C₂×C₁₁₃	—	2	2	no	—
147	C₂×C₁₁₃	—	4	2	no	—
148	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
149	C₂×C₁₁₃	#585	1	2	yes	—
150	D₁₁₃	—	4	4	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	D₁₁₃	1	1	yes	characteristic, complemented
158	#535	—	113	1	no	—
159	#533	—	113	1	no	—
160	#529	—	113	4	no	—
161	C₂×#124	—	113	1	no	—
162	#519	—	113	1	no	—
163	#518	—	113	1	no	—
164	C₄×#41	—	113	1	no	—
165	C₂×#41	C₂×D₁₁₃	1	1	yes	characteristic
166	C₂²×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
167	C₂²×D₄	—	113	1	no	—
168	C₄⋊Q₈	—	113	2	no	—
169	C₄⋊Q₈	C₂×D₁₁₃	1	2	yes	complemented
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	4	no	—
177	#124	—	113	2	no	—
178	#124	—	113	8	no	—
179	#124	C₂×D₁₁₃	1	2	yes	complemented
180	#123	—	113	8	no	—
181	#123	—	113	8	no	—
182	#123	C₂×D₁₁₃	1	4	yes	—
183	C₂²⋊Q₈	—	113	8	no	—
184	C₂²⋊Q₈	—	113	4	no	—
185	C₂²⋊Q₈	C₂×D₁₁₃	1	4	yes	complemented
186	#121	—	113	4	no	—
187	#120	—	113	4	no	—
188	C₄×Q₈	—	113	2	no	—
189	C₄×Q₈	—	113	4	no	—
190	C₄×Q₈	—	113	4	no	—
191	C₄×Q₈	—	113	4	no	—
192	C₄×D₄	—	113	4	no	—
193	C₄×D₄	—	113	2	no	—
194	C₄×D₄	—	113	4	no	—
195	C₄×D₄	—	113	8	no	—
196	C₄×D₄	—	113	4	no	—
197	#117	—	113	1	no	—
198	#117	—	113	2	no	—
199	#117	—	113	1	no	—
200	#117	—	113	4	no	—
201	#117	—	113	8	no	—
202	#117	C₂×D₁₁₃	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	113	4	no	—
204	C₂×C₂²⋊C₄	—	113	4	no	—
205	C₂×C₄²	—	113	1	no	—
206	C₂×C₄²	—	113	2	no	—
207	C₂⁴	—	113	2	no	—
208	#41	—	2	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	113	4	no	—
211	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
212	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
213	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
214	C₂×D₄	—	113	8	no	—
215	C₂×D₄	—	113	2	no	—
216	C₂×D₄	—	113	2	no	—
217	C₂×D₄	—	226	4	no	—
218	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
219	C₂×D₄	C₂²×D₁₁₃	1	1	yes	characteristic
220	C₂²×C₄	—	113	2	no	—
221	C₂²×C₄	—	113	4	no	—
222	C₂²×C₄	—	113	8	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	4	no	—
229	C₄⋊C₄	—	113	2	no	—
230	C₄⋊C₄	—	113	8	no	—
231	C₄⋊C₄	—	113	4	no	—
232	C₄⋊C₄	—	113	4	no	—
233	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
234	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
235	C₂²⋊C₄	—	113	2	no	—
236	C₂²⋊C₄	—	113	2	no	—
237	C₂²⋊C₄	—	113	4	no	—
238	C₂²⋊C₄	—	113	8	no	—
239	C₂²⋊C₄	—	113	8	no	—
240	C₂²⋊C₄	—	113	8	no	—
241	C₂²⋊C₄	—	113	8	no	—
242	C₂²⋊C₄	C₂²×D₁₁₃	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
244	C₄²	—	113	1	no	—
245	C₄²	—	113	2	no	—
246	C₄²	—	113	4	no	—
247	C₄²	—	113	1	no	—
248	C₄²	—	113	2	no	—
249	C₄²	—	113	8	no	—
250	C₄²	—	113	2	no	—
251	C₄²	C₂²×D₁₁₃	1	2	yes	—
252	C₂³	—	113	4	no	—
253	C₂³	—	452	4	no	—
254	C₂³	—	226	4	no	—
255	C₂³	C₂³×D₁₁₃	1	2	yes	—
256	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	—	2	2	no	—
260	Q₈	—	452	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	2	no	—
263	D₄	—	452	8	no	—
264	D₄	—	226	4	no	—
265	D₄	—	226	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	113	2	no	—
269	C₂×C₄	—	113	4	no	—
270	C₂×C₄	—	113	4	no	—
271	C₂×C₄	—	113	2	no	—
272	C₂×C₄	—	452	8	no	—
273	C₂×C₄	—	452	4	no	—
274	C₂×C₄	—	452	2	no	—
275	C₂×C₄	#14268646	1	4	yes	—
276	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
277	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
278	C₂×C₄	C₂³×D₁₁₃	1	4	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	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	904	4	no	—
285	C₂²	—	226	4	no	—
286	C₂²	—	226	8	no	—
287	C₂²	C₂×#14268646	1	2	yes	—
288	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	226	4	no	—
293	C₄	—	226	2	no	—
294	C₄	—	226	2	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	2	no	—
300	C₂	—	452	4	no	—
301	C₂	C₂²×#14268646	1	1	yes	characteristic
302	C₂	#20299435	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	#536×C₁₁₃	C₂
prime radical	#20299162	C₂
hypercenter	#536	D₁₁₃
hypocenter	C₁₁₃	#3510
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #536

Complements

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

N	H ≅ G/N	type
1. #20299330	C₂	semidirect
2. #20299322	C₂	semidirect
3. #20299271	C₂	semidirect
4. #20299221	C₂	semidirect
6. #20299151	C₂	semidirect
7. #20299134	C₂	semidirect
8. C₂×#16813611	C₂	semidirect
9. #20299076	C₂	semidirect
10. #20299072	C₂	semidirect
11. #41×Dic₁₁₃	C₂	semidirect
12. #536×C₁₁₃	C₂	semidirect
14. #16813621	C₂²	semidirect
16. #16813617	C₂²	semidirect
17. #16813617	C₂²	semidirect
18. #16813613	C₂²	semidirect
23. #16813608	C₂²	semidirect
24. #16813606	C₂²	semidirect
25. #16813604	C₂²	semidirect
26. #16813602	C₂²	semidirect
30. Q₈×Dic₁₁₃	C₂²	semidirect
31. Q₈×Dic₁₁₃	C₂²	semidirect
32. Q₈×Dic₁₁₃	C₂²	semidirect
33. C₄×Dic₂₂₆	C₂²	semidirect
36. D₄×Dic₁₁₃	C₂²	semidirect
38. #16813582	C₂²	semidirect
42. #16813581	C₂²	semidirect
43. #16813580	C₂²	semidirect
47. C₂×C₄×Dic₁₁₃	C₂²	semidirect
50. C₄⋊Q₈×C₁₁₃	C₂²	semidirect
51. #124×C₁₁₃	C₂²	semidirect
53. C₂²⋊Q₈×C₁₁₃	C₂²	semidirect
152. C₁₁₃	#3510	semidirect
157. #536	D₁₁₃	semidirect
169. C₄⋊Q₈	C₂×D₁₁₃	semidirect
179. #124	C₂×D₁₁₃	semidirect
185. C₂²⋊Q₈	C₂×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
#20299322	2
#20299271	2
#20299221	2
#20299162	2
#20299151	2
#20299134	2
C₂×#16813611	2
#20299076	2
#20299072	2
#41×Dic₁₁₃	2
#536×C₁₁₃	2
#3510	113