Subgroups — #28393458

Order 14464.

504 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813580	C₂	1	1	yes	maximal, characteristic
2	#20298497	C₂	1	1	yes	maximal, characteristic, complemented
3	#20298490	C₂	1	2	yes	maximal
4	#20298488	C₂	1	2	yes	maximal
5	#20298463	C₂	1	2	yes	maximal, complemented
6	#20298455	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14268624	C₂	1	1	yes	maximal, characteristic
10	#20298443	C₂	1	2	yes	maximal, complemented
11	#389×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
13	#16813580	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
17	C₂×#14268624	C₂²	1	1	yes	characteristic
18	C₂×#14268624	C₂²	1	1	yes	characteristic
19	C₂×#14268624	C₂²	1	2	yes	—
20	C₂×#14268624	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
25	#16813529	C₂²	1	2	yes	—
26	#16813529	C₂²	1	1	yes	characteristic
27	#16813529	C₂²	1	2	yes	—
28	#16813529	C₂²	1	1	yes	characteristic
29	#16813529	C₂²	1	2	yes	—
30	#16813529	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
35	#95×C₁₁₃	C₂²	1	2	yes	—
36	C₂³×D₁₁₃	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
38	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
39	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
40	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₁₁₃	—	2	2	no	—
43	C₂×C₄×D₁₁₃	—	2	2	no	—
44	C₂×C₄×D₁₁₃	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₁₁₃	—	2	2	no	—
46	C₄⋊Dic₁₁₃	—	4	2	no	—
47	C₄⋊Dic₁₁₃	C₂×C₄	1	4	yes	—
48	#14268624	—	2	2	no	—
49	#14268624	—	2	2	no	—
50	#14268624	—	4	2	no	—
51	#14268624	C₂×C₄	1	8	yes	—
52	C₄×Dic₁₁₃	—	2	4	no	—
53	C₄×Dic₁₁₃	—	2	2	no	—
54	C₄×Dic₁₁₃	—	2	4	no	—
55	C₄×Dic₁₁₃	C₂×C₄	1	8	yes	—
56	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₁₁₃	—	2	2	no	—
62	C₄⋊C₄×C₁₁₃	—	4	2	no	—
63	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
64	C₄²×C₁₁₃	—	2	2	no	—
65	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
66	C₂²×D₁₁₃	—	2	1	no	—
67	C₂²×D₁₁₃	—	2	1	no	—
68	C₂²×D₁₁₃	—	2	1	no	—
69	C₂²×D₁₁₃	—	2	1	no	—
70	C₂²×D₁₁₃	—	2	1	no	—
71	C₂²×D₁₁₃	—	2	1	no	—
72	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₁₃	—	2	1	no	—
74	C₂×C₄×C₁₁₃	—	2	2	no	—
75	C₂×C₄×C₁₁₃	—	2	2	no	—
76	C₂×C₄×C₁₁₃	—	2	1	no	—
77	C₂×C₄×C₁₁₃	—	2	1	no	—
78	C₂×C₄×C₁₁₃	—	2	2	no	—
79	C₂×C₄×C₁₁₃	—	2	2	no	—
80	C₂×C₄×C₁₁₃	—	2	1	no	—
81	C₂×C₄×C₁₁₃	—	2	1	no	—
82	C₂×C₄×C₁₁₃	—	2	2	no	—
83	C₂×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₁₃	#41	1	2	yes	—
86	C₂×C₄×C₁₁₃	#41	1	2	yes	—
87	C₂×C₄×C₁₁₃	#41	1	2	yes	—
88	C₂×Dic₁₁₃	—	2	2	no	—
89	C₂×Dic₁₁₃	—	2	2	no	—
90	C₂×Dic₁₁₃	—	2	1	no	—
91	C₂×Dic₁₁₃	—	2	1	no	—
92	C₂×Dic₁₁₃	—	2	1	no	—
93	C₂×Dic₁₁₃	—	2	1	no	—
94	C₂×Dic₁₁₃	—	2	1	no	—
95	C₂×Dic₁₁₃	—	2	2	no	—
96	C₂×Dic₁₁₃	—	2	1	no	—
97	C₂×Dic₁₁₃	—	2	2	no	—
98	C₂×Dic₁₁₃	—	2	2	no	—
99	C₂×Dic₁₁₃	—	2	1	no	—
100	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
101	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
102	C₂×Dic₁₁₃	#41	1	2	yes	—
103	C₄×D₁₁₃	—	2	8	no	—
104	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
106	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
107	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
108	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
109	C₂²×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×D₁₁₃	—	2	2	no	—
112	C₂×D₁₁₃	—	2	2	no	—
113	C₂×D₁₁₃	—	2	2	no	—
114	C₂×D₁₁₃	—	4	1	no	—
115	C₂×D₁₁₃	—	4	1	no	—
116	C₂×D₁₁₃	—	4	1	no	—
117	C₂×D₁₁₃	—	4	1	no	—
118	C₄×C₁₁₃	—	2	4	no	—
119	C₄×C₁₁₃	—	2	2	no	—
120	C₄×C₁₁₃	—	4	1	no	—
121	C₄×C₁₁₃	—	4	2	no	—
122	C₄×C₁₁₃	#117	1	4	yes	—
123	Dic₁₁₃	—	2	2	no	—
124	Dic₁₁₃	—	2	4	no	—
125	Dic₁₁₃	—	2	2	no	—
126	Dic₁₁₃	—	4	1	no	—
127	Dic₁₁₃	—	4	2	no	—
128	C₂×C₁₁₃	C₂×#117	1	1	yes	characteristic
129	C₂×C₁₁₃	C₄×#41	1	1	yes	characteristic
130	C₂×C₁₁₃	#520	1	1	yes	characteristic
131	C₂×C₁₁₃	#552	1	1	yes	characteristic
132	C₂×C₁₁₃	#552	1	1	yes	characteristic
133	C₂×C₁₁₃	#555	1	1	yes	characteristic
134	C₂×C₁₁₃	#555	1	1	yes	characteristic
135	D₁₁₃	—	4	2	no	—
136	#2346	—	113	1	no	maximal, Sylow
137	C₁₁₃	#2346	1	1	yes	characteristic, complemented, Sylow
138	C₂×#117	—	113	1	no	—
139	#389	—	113	1	no	—
140	#389	D₁₁₃	1	1	yes	characteristic, complemented
141	#388	—	113	2	no	—
142	#388	—	113	2	no	—
143	#382	—	113	2	no	—
144	#380	—	113	1	no	—
145	C₄×C₄⋊C₄	—	113	1	no	—
146	C₄×C₄⋊C₄	—	113	1	no	—
147	C₄×C₂²⋊C₄	—	113	1	no	—
148	#376	—	113	2	no	—
149	C₂³×C₄	—	113	1	no	—
150	#117	—	113	8	no	—
151	C₂×C₄⋊C₄	—	113	1	no	—
152	C₂×C₄⋊C₄	—	113	2	no	—
153	C₂×C₄⋊C₄	—	113	1	no	—
154	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
156	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	—	113	2	no	—
158	C₂×C₂²⋊C₄	—	113	2	no	—
159	C₂×C₂²⋊C₄	—	113	1	no	—
160	C₂×C₂²⋊C₄	—	113	1	no	—
161	C₂×C₄²	—	113	2	no	—
162	C₂×C₄²	—	113	2	no	—
163	C₂×C₄²	—	113	2	no	—
164	C₂×C₄²	—	113	1	no	—
165	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
166	#95	—	113	1	no	—
167	#95	—	113	2	no	—
168	#95	—	113	2	no	—
169	#95	—	113	1	no	—
170	#95	—	113	2	no	—
171	#95	—	113	2	no	—
172	#95	C₂×D₁₁₃	1	2	yes	—
173	C₂⁴	—	113	1	no	—
174	C₂²×C₄	—	113	1	no	—
175	C₂²×C₄	—	113	1	no	—
176	C₂²×C₄	—	113	2	no	—
177	C₂²×C₄	—	113	4	no	—
178	C₂²×C₄	—	113	1	no	—
179	C₂²×C₄	—	113	2	no	—
180	C₂²×C₄	—	226	2	no	—
181	C₂²×C₄	—	226	2	no	—
182	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
185	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	113	4	no	—
190	C₄⋊C₄	—	452	2	no	—
191	C₄⋊C₄	—	226	2	no	—
192	C₄⋊C₄	C₄×D₁₁₃	1	4	yes	—
193	C₂²⋊C₄	—	113	8	no	—
194	C₂²⋊C₄	—	452	2	no	—
195	C₂²⋊C₄	—	226	2	no	—
196	C₂²⋊C₄	—	226	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	113	8	no	—
199	C₄²	—	226	4	no	—
200	C₄²	—	226	4	no	—
201	C₄²	—	226	2	no	—
202	C₂³	—	113	2	no	—
203	C₂³	—	226	1	no	—
204	C₂³	—	226	1	no	—
205	C₂³	—	226	1	no	—
206	C₂³	—	226	1	no	—
207	C₂³	—	226	1	no	—
208	C₂³	—	226	1	no	—
209	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	113	2	no	—
221	C₂×C₄	—	113	4	no	—
222	C₂×C₄	—	113	2	no	—
223	C₂×C₄	—	226	1	no	—
224	C₂×C₄	—	226	2	no	—
225	C₂×C₄	—	226	1	no	—
226	C₂×C₄	—	226	2	no	—
227	C₂×C₄	—	226	2	no	—
228	C₂×C₄	—	226	1	no	—
229	C₂×C₄	—	226	2	no	—
230	C₂×C₄	—	226	1	no	—
231	C₂×C₄	—	226	1	no	—
232	C₂×C₄	—	226	2	no	—
233	C₂×C₄	—	226	1	no	—
234	C₂×C₄	—	226	1	no	—
235	C₂×C₄	—	226	8	no	—
236	C₂×C₄	C₂×C₄×D₁₁₃	1	4	yes	—
237	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
238	C₂×C₄	#14268646	1	2	yes	—
239	C₂×C₄	#14268648	1	2	yes	—
240	C₂×C₄	#14268648	1	2	yes	—
241	C₂²	—	452	1	no	—
242	C₂²	—	452	1	no	—
243	C₂²	—	452	1	no	—
244	C₂²	—	452	1	no	—
245	C₂²	—	226	2	no	—
246	C₂²	—	226	2	no	—
247	C₂²	—	226	2	no	—
248	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
249	C₂²	#41×D₁₁₃	1	1	yes	characteristic
250	C₂²	C₂×#14268646	1	1	yes	characteristic
251	C₂²	C₂×#14268648	1	1	yes	characteristic
252	C₂²	C₂×#14268648	1	1	yes	characteristic
253	C₂²	#16813667	1	1	yes	characteristic
254	C₂²	#16813669	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	452	1	no	—
260	C₄	—	452	2	no	—
261	C₄	—	226	2	no	—
262	C₄	—	226	2	no	—
263	C₄	—	226	4	no	—
264	C₄	#16813580	1	4	yes	—
265	C₂	—	452	2	no	—
266	C₂	C₂×#16813580	1	1	yes	characteristic
267	C₂	C₄×#14268648	1	1	yes	characteristic
268	C₂	#20299077	1	1	yes	characteristic
269	C₂	#20299286	1	1	yes	characteristic
270	C₂	#20299290	1	1	yes	characteristic
271	C₂	#20299312	1	1	yes	characteristic
272	C₂	#20299314	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁₃
commutator	C₂²×C₁₁₃	C₂³×C₄
Fitting	#389×C₁₁₃	C₂
socle, cosocle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂³×D₁₁₃	C₂³
hypercenter	#389	D₁₁₃
hypocenter	C₁₁₃	#2346
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
2. #20298497	C₂	semidirect
5. #20298463	C₂	semidirect
7. C₄⋊C₄×Dic₁₁₃	C₂	semidirect
8. C₄×C₄⋊Dic₁₁₃	C₂	semidirect
10. #20298443	C₂	semidirect
11. #389×C₁₁₃	C₂	semidirect
13. #16813580	C₄	semidirect
137. C₁₁₃	#2346	semidirect
140. #389	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813580	2
#20298497	2
#20298490	2
#20298488	2
#20298463	2
#20298455	2
C₄⋊C₄×Dic₁₁₃	2
C₄×C₄⋊Dic₁₁₃	2
C₄×#14268624	2
#20298443	2
#389×C₁₁₃	2
#2346	113