Subgroups — #28399342

Order 14464.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813642	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299066	C₂	1	1	yes	maximal, characteristic, complemented
4	#20298690	C₂	1	2	yes	maximal, complemented
5	#20298689	C₂	1	2	yes	maximal, complemented
6	#20298683	C₂	1	2	yes	maximal, complemented
7	#20298678	C₂	1	2	yes	maximal, complemented
8	#20298634	C₂	1	1	yes	maximal, characteristic, complemented
9	#20298595	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14268647	C₂²	1	1	yes	characteristic
13	#16813642	C₄	1	8	yes	complemented
14	C₂×#14268637	C₂²	1	1	yes	characteristic
15	C₂×#14268635	C₂²	1	1	yes	characteristic
16	C₂×#14268634	C₂²	1	1	yes	characteristic
17	C₂×#14268632	C₂²	1	1	yes	characteristic
18	C₂²×#11744751	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₁₃	C₂²	1	2	yes	—
20	#16813587	C₂²	1	2	yes	—
21	D₄×Dic₁₁₃	—	2	2	no	—
22	D₄×Dic₁₁₃	C₂²	1	2	yes	—
23	#16813584	C₂²	1	2	yes	—
24	#16813582	—	2	2	no	—
25	#16813581	—	2	1	no	—
26	#16813581	C₂²	1	1	yes	characteristic
27	#16813579	—	2	1	no	—
28	C₂×C₄×Dic₁₁₃	—	2	1	no	—
29	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
30	C₄²×D₁₁₃	—	2	1	no	—
31	#16813555	C₂²	1	2	yes	—
32	#16813553	C₂²	1	2	yes	—
33	#16813547	C₂²	1	2	yes	—
34	#16813544	C₂²	1	2	yes	—
35	C₈×Dic₁₁₃	C₂²	1	2	yes	—
36	C₈×Dic₁₁₃	C₂²	1	2	yes	—
37	C₂×#41×C₁₁₃	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
39	#117×C₁₁₃	C₂²	1	1	yes	characteristic
40	#103×C₁₁₃	C₂²	1	2	yes	—
41	#102×C₁₁₃	C₂²	1	2	yes	—
42	#14268647	—	2	2	no	—
43	#14268647	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₂₆	C₂³	1	1	yes	characteristic
45	C₂×D₄₅₂	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₁₃	—	2	1	no	—
47	C₂²×Dic₁₁₃	—	2	2	no	—
48	C₂²×Dic₁₁₃	D₄	1	2	yes	—
49	C₂×C₄×D₁₁₃	—	2	1	no	—
50	C₂×C₄×D₁₁₃	—	2	2	no	—
51	#14268637	C₂×C₄	1	4	yes	complemented
52	#14268635	C₂×C₄	1	4	yes	complemented
53	#14268634	C₂×C₄	1	4	yes	complemented
54	#14268632	C₂×C₄	1	4	yes	complemented
55	C₂×#11744751	C₂×C₄	1	4	yes	complemented
56	C₂×#11744751	C₂³	1	1	yes	characteristic
57	C₂×#11744751	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₁₃	—	2	2	no	—
59	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
60	#14268626	—	2	2	no	—
61	C₂²⋊Dic₁₁₃	—	2	1	no	—
62	C₂²⋊Dic₁₁₃	—	2	2	no	—
63	#14268624	—	2	2	no	—
64	C₄×Dic₁₁₃	—	2	1	no	—
65	C₄×Dic₁₁₃	—	2	2	no	—
66	C₄×Dic₁₁₃	—	2	2	no	—
67	C₄×Dic₁₁₃	D₄	1	1	yes	characteristic
68	C₄×Dic₁₁₃	D₄	1	1	yes	characteristic
69	C₄×Dic₁₁₃	C₂³	1	2	yes	—
70	#41×C₁₁₃	—	2	2	no	—
71	#41×C₁₁₃	C₂×C₄	1	4	yes	complemented
72	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₁₁₃	—	2	1	no	—
74	C₂×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₁₃	—	2	1	no	—
76	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₁₃	—	2	2	no	—
78	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
79	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₁₁₃	—	2	1	no	—
81	C₄²×C₁₁₃	—	2	1	no	—
82	C₂³×C₁₁₃	—	2	1	no	—
83	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₁₁₃	—	2	1	no	—
85	Q₈×C₁₁₃	—	2	1	no	—
86	Q₈×C₁₁₃	C₂²×C₄	1	2	yes	—
87	D₄×C₁₁₃	—	2	4	no	—
88	D₄×C₁₁₃	—	2	1	no	—
89	D₄×C₁₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₁₃	—	2	1	no	—
91	C₂×C₄×C₁₁₃	—	2	4	no	—
92	C₂×C₄×C₁₁₃	—	2	2	no	—
93	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₁₃	#41	1	2	yes	—
96	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₁₃	—	2	4	no	—
98	C₂×Dic₁₁₃	—	2	1	no	—
99	C₂×Dic₁₁₃	—	2	4	no	—
100	C₂×Dic₁₁₃	—	2	2	no	—
101	C₂×Dic₁₁₃	—	4	4	no	—
102	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
103	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
104	D₄₅₂	—	2	1	no	—
105	D₄₅₂	C₂²×C₄	1	2	yes	—
106	C₄×D₁₁₃	—	2	4	no	—
107	C₄×D₁₁₃	—	4	4	no	—
108	Dic₂₂₆	—	2	1	no	—
109	Dic₂₂₆	C₂²×C₄	1	2	yes	—
110	C₈×C₁₁₃	—	2	2	no	—
111	#11744751	C₂²×C₄	1	2	yes	—
112	#11744751	C₂²×C₄	1	2	yes	—
113	C₂²×C₁₁₃	—	2	2	no	—
114	C₂²×C₁₁₃	—	2	2	no	—
115	C₂²×C₁₁₃	—	4	2	no	—
116	C₂²×C₁₁₃	C₄×D₄	1	2	yes	—
117	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₁₃	—	2	2	no	—
119	C₂×D₁₁₃	—	4	2	no	—
120	C₄×C₁₁₃	—	2	2	no	—
121	C₄×C₁₁₃	—	4	2	no	—
122	C₄×C₁₁₃	C₄×D₄	1	2	yes	—
123	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
125	Dic₁₁₃	—	2	2	no	—
126	Dic₁₁₃	—	2	2	no	—
127	Dic₁₁₃	—	4	2	no	—
128	Dic₁₁₃	#135	1	4	yes	complemented
129	C₂×C₁₁₃	—	2	2	no	—
130	C₂×C₁₁₃	—	4	2	no	—
131	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₁₃	C₂×#135	1	2	yes	—
133	D₁₁₃	—	4	2	no	—
134	C₄×#135	—	113	1	no	maximal, Sylow
135	C₁₁₃	C₄×#135	1	1	yes	characteristic, complemented, Sylow
136	C₂×#135	—	113	1	no	—
137	C₄×#41	—	113	1	no	—
138	C₄×#41	—	113	1	no	—
139	C₄×Q₁₆	—	113	2	no	—
140	C₄×SD₁₆	—	113	2	no	—
141	C₄×SD₁₆	—	113	2	no	—
142	C₄×D₈	—	113	2	no	—
143	#427	—	113	1	no	—
144	#416	—	113	1	no	—
145	#416	D₁₁₃	1	1	yes	characteristic, complemented
146	C₂×C₄×C₈	—	113	1	no	—
147	C₂×#41	—	113	1	no	—
148	C₂×#41	C₂×D₁₁₃	1	1	yes	characteristic
149	#135	—	113	8	no	—
150	C₂×Q₁₆	—	113	1	no	—
151	C₂×SD₁₆	—	113	1	no	—
152	C₂×SD₁₆	—	113	1	no	—
153	C₂×D₈	—	113	1	no	—
154	C₂²×C₈	—	113	1	no	—
155	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
156	C₄×Q₈	—	113	2	no	—
157	C₄×Q₈	—	113	2	no	—
158	C₄×D₄	—	113	2	no	—
159	C₄×D₄	—	113	2	no	—
160	C₄×D₄	—	226	2	no	—
161	C₄×D₄	—	226	2	no	—
162	#117	—	113	1	no	—
163	#117	—	226	1	no	—
164	#117	—	226	1	no	—
165	#117	C₂×D₁₁₃	1	1	yes	characteristic
166	C₂×C₄²	—	113	1	no	—
167	C₂×C₄²	—	226	1	no	—
168	C₂×C₄²	—	226	1	no	—
169	#107	—	113	2	no	—
170	#106	—	113	2	no	—
171	#103	—	113	2	no	—
172	#103	C₂×D₁₁₃	1	2	yes	—
173	#102	—	113	2	no	—
174	#102	C₂×D₁₁₃	1	2	yes	—
175	C₄×C₈	—	113	2	no	—
176	C₄×C₈	—	113	2	no	—
177	#41	—	2	2	no	—
178	#41	—	113	4	no	—
179	#41	—	226	2	no	—
180	#41	C₄×D₁₁₃	1	4	yes	complemented
181	C₂×Q₈	—	113	1	no	—
182	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	113	1	no	—
185	C₂×D₄	—	226	1	no	—
186	C₂×D₄	C₂²×D₁₁₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	113	2	no	—
189	C₂²×C₄	—	226	2	no	—
190	C₂²×C₄	—	226	2	no	—
191	C₂²×C₄	—	226	1	no	—
192	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
193	Q₁₆	—	113	4	no	—
194	SD₁₆	—	113	4	no	—
195	SD₁₆	—	113	4	no	—
196	D₈	—	113	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	113	4	no	—
199	C₂×C₈	—	113	1	no	—
200	C₂×C₈	—	113	1	no	—
201	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
202	C₄⋊C₄	—	113	2	no	—
203	C₄⋊C₄	—	226	2	no	—
204	C₄⋊C₄	—	226	2	no	—
205	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	226	2	no	—
208	C₂²⋊C₄	—	226	2	no	—
209	C₂²⋊C₄	—	226	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	113	1	no	—
212	C₄²	—	113	1	no	—
213	C₄²	—	113	2	no	—
214	C₄²	—	226	2	no	—
215	C₄²	—	226	2	no	—
216	C₄²	—	226	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	226	1	no	—
219	C₂³	D₄×D₁₁₃	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	113	2	no	—
222	Q₈	—	226	1	no	—
223	Q₈	C₂×C₄×D₁₁₃	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	113	2	no	—
227	D₄	—	226	4	no	—
228	D₄	—	226	1	no	—
229	D₄	C₂×C₄×D₁₁₃	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	113	2	no	—
234	C₂×C₄	—	113	2	no	—
235	C₂×C₄	—	452	4	no	—
236	C₂×C₄	—	452	4	no	—
237	C₂×C₄	—	226	4	no	—
238	C₂×C₄	—	226	2	no	—
239	C₂×C₄	—	226	1	no	—
240	C₂×C₄	—	226	4	no	—
241	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
242	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
243	C₂×C₄	#14268646	1	2	yes	—
244	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	113	2	no	—
247	C₈	—	113	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	452	2	no	—
252	C₂²	—	226	2	no	—
253	C₂²	#16813582	1	2	yes	—
254	C₂²	C₂×D₄×D₁₁₃	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	113	4	no	—
258	C₄	—	452	2	no	—
259	C₄	—	226	2	no	—
260	C₄	—	226	2	no	—
261	C₄	#16813582	1	2	yes	—
262	C₄	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
263	C₄	C₂×#14268646	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	452	2	no	—
267	C₂	C₂×#16813582	1	1	yes	characteristic
268	C₂	#135×D₁₁₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₁₃
commutator	C₄×C₁₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	#416×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂²×D₄
prime radical	C₂×#16813642	C₂
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	#416	D₁₁₃
hypocenter	C₁₁₃	C₄×#135
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #41×Dic₁₁₃	C₂	semidirect
3. #20299066	C₂	semidirect
4. #20298690	C₂	semidirect
5. #20298689	C₂	semidirect
6. #20298683	C₂	semidirect
7. #20298678	C₂	semidirect
8. #20298634	C₂	semidirect
9. #20298595	C₂	semidirect
10. C₂×C₈×Dic₁₁₃	C₂	semidirect
11. #416×C₁₁₃	C₂	semidirect
13. #16813642	C₄	semidirect
43. #14268647	C₂×C₄	semidirect
51. #14268637	C₂×C₄	semidirect
52. #14268635	C₂×C₄	semidirect
53. #14268634	C₂×C₄	semidirect
54. #14268632	C₂×C₄	semidirect
55. C₂×#11744751	C₂×C₄	semidirect
71. #41×C₁₁₃	C₂×C₄	semidirect
128. Dic₁₁₃	#135	semidirect
135. C₁₁₃	C₄×#135	semidirect
145. #416	D₁₁₃	semidirect
180. #41	C₄×D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813642	2
#41×Dic₁₁₃	2
#20299066	2
#20298690	2
#20298689	2
#20298683	2
#20298678	2
#20298634	2
#20298595	2
C₂×C₈×Dic₁₁₃	2
#416×C₁₁₃	2
C₄×#135	113