Subgroups — #28393207

Order 14464.

568 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298491	C₂	1	2	yes	maximal, complemented
2	#20298484	C₂	1	2	yes	maximal, complemented
3	#20298478	C₂	1	2	yes	maximal
4	#20298473	C₂	1	2	yes	maximal, complemented
5	#20298466	C₂	1	1	yes	maximal, characteristic, complemented
6	#20298464	C₂	1	1	yes	maximal, characteristic, complemented
7	#20298461	C₂	1	1	yes	maximal, characteristic, complemented
8	#20298458	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16813529	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₁₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
19	#16813529	C₂²	1	1	yes	characteristic
20	#16813529	C₂²	1	1	yes	characteristic
21	#16813529	C₂²	1	2	yes	—
22	#16813529	C₂²	1	1	yes	characteristic
23	#16813529	C₂²	1	1	yes	characteristic
24	#16813529	C₂²	1	2	yes	—
25	#16813529	C₂²	1	2	yes	—
26	#16813529	C₂²	1	2	yes	—
27	#16813529	C₂²	1	2	yes	—
28	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
29	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₁₃	C₄	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₁₃	—	2	4	no	—
37	C₂²×Dic₁₁₃	—	4	2	no	—
38	C₂²×Dic₁₁₃	—	4	2	no	—
39	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
40	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
41	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
42	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
43	C₄⋊Dic₁₁₃	—	4	2	no	—
44	C₄⋊Dic₁₁₃	—	4	2	no	—
45	C₂²⋊Dic₁₁₃	—	2	4	no	—
46	C₂²⋊Dic₁₁₃	—	4	2	no	—
47	C₂²⋊Dic₁₁₃	—	4	2	no	—
48	C₄×Dic₁₁₃	—	4	2	no	—
49	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
50	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₁₁₃	—	2	4	no	—
52	C₂×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
53	C₂²×C₄×C₁₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
56	C₂²×C₄×C₁₁₃	C₂×C₄	1	4	yes	complemented
57	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	complemented
63	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
64	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	complemented
65	C₄²×C₁₁₃	C₂×C₄	1	4	yes	complemented
66	C₂³×C₁₁₃	—	2	2	no	—
67	C₂³×C₁₁₃	—	2	1	no	—
68	C₂³×C₁₁₃	—	2	1	no	—
69	C₂³×C₁₁₃	—	2	2	no	—
70	C₂³×C₁₁₃	—	2	2	no	—
71	C₂³×C₁₁₃	—	2	1	no	—
72	C₂³×C₁₁₃	—	2	1	no	—
73	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
74	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₁₁₃	#41	1	4	yes	—
76	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
77	D₄×C₁₁₃	—	2	8	no	—
78	C₂×C₄×C₁₁₃	—	2	2	no	—
79	C₂×C₄×C₁₁₃	—	2	2	no	—
80	C₂×C₄×C₁₁₃	—	2	2	no	—
81	C₂×C₄×C₁₁₃	—	2	2	no	—
82	C₂×C₄×C₁₁₃	—	2	1	no	—
83	C₂×C₄×C₁₁₃	—	2	1	no	—
84	C₂×C₄×C₁₁₃	—	4	4	no	—
85	C₂×C₄×C₁₁₃	—	4	4	no	—
86	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
91	C₂×Dic₁₁₃	—	2	2	no	—
92	C₂×Dic₁₁₃	—	2	2	no	—
93	C₂×Dic₁₁₃	—	2	2	no	—
94	C₂×Dic₁₁₃	—	2	2	no	—
95	C₂×Dic₁₁₃	—	2	2	no	—
96	C₂×Dic₁₁₃	—	2	2	no	—
97	C₂×Dic₁₁₃	—	2	2	no	—
98	C₂×Dic₁₁₃	—	2	2	no	—
99	C₂×Dic₁₁₃	—	2	2	no	—
100	C₂×Dic₁₁₃	—	2	2	no	—
101	C₂×Dic₁₁₃	—	2	2	no	—
102	C₂×Dic₁₁₃	—	2	2	no	—
103	C₂×Dic₁₁₃	—	4	8	no	—
104	C₂²×C₁₁₃	—	2	2	no	—
105	C₂²×C₁₁₃	—	2	2	no	—
106	C₂²×C₁₁₃	—	2	4	no	—
107	C₂²×C₁₁₃	—	2	4	no	—
108	C₂²×C₁₁₃	—	2	2	no	—
109	C₂²×C₁₁₃	—	2	2	no	—
110	C₂²×C₁₁₃	—	2	2	no	—
111	C₂²×C₁₁₃	—	4	2	no	—
112	C₂²×C₁₁₃	—	4	2	no	—
113	C₂²×C₁₁₃	#117	1	4	yes	—
114	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
116	C₂²×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
117	C₂²×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₄×C₁₁₃	—	2	2	no	—
120	C₄×C₁₁₃	—	2	2	no	—
121	C₄×C₁₁₃	—	2	2	no	—
122	C₄×C₁₁₃	—	2	2	no	—
123	C₄×C₁₁₃	—	2	2	no	—
124	Dic₁₁₃	—	4	2	no	—
125	Dic₁₁₃	—	4	2	no	—
126	Dic₁₁₃	—	4	2	no	—
127	Dic₁₁₃	—	4	2	no	—
128	C₂×C₁₁₃	—	2	4	no	—
129	C₂×C₁₁₃	—	4	2	no	—
130	C₂×C₁₁₃	C₂×#117	1	1	yes	characteristic
131	C₂×C₁₁₃	#518	1	1	yes	characteristic
132	C₂×C₁₁₃	#520	1	1	yes	characteristic
133	C₂×C₁₁₃	#538	1	2	yes	—
134	C₂×C₁₁₃	#539	1	2	yes	—
135	#2318	—	113	1	no	maximal, Sylow
136	C₁₁₃	#2318	1	1	yes	characteristic, complemented, Sylow
137	C₂×C₄×D₄	D₁₁₃	1	1	yes	characteristic
138	#388	—	113	2	no	—
139	#387	—	113	2	no	—
140	#386	—	113	2	no	—
141	#385	—	113	2	no	—
142	#383	—	113	1	no	—
143	#382	—	113	1	no	—
144	#381	—	113	1	no	—
145	#380	—	113	1	no	—
146	C₄×C₂²⋊C₄	—	113	1	no	—
147	C₂×#95	—	113	1	no	—
148	C₂²×D₄	C₂×D₁₁₃	1	1	yes	characteristic
149	C₂³×C₄	—	113	2	no	—
150	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
151	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
152	C₄×D₄	Dic₁₁₃	1	8	yes	complemented
153	C₂×C₄⋊C₄	—	113	2	no	—
154	C₂×C₄⋊C₄	—	113	2	no	—
155	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	—	113	2	no	—
157	C₂×C₂²⋊C₄	—	113	2	no	—
158	C₂×C₂²⋊C₄	—	113	2	no	—
159	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
161	C₂×C₄²	—	113	2	no	—
162	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
163	#95	—	113	1	no	—
164	#95	—	113	1	no	—
165	#95	—	113	2	no	—
166	#95	—	113	1	no	—
167	#95	—	113	1	no	—
168	#95	—	113	2	no	—
169	#95	—	113	2	no	—
170	#95	—	113	2	no	—
171	#95	—	113	2	no	—
172	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
173	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	C₂×Dic₁₁₃	1	4	yes	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	113	2	no	—
179	C₂²×C₄	—	113	2	no	—
180	C₂²×C₄	—	113	2	no	—
181	C₂²×C₄	—	113	2	no	—
182	C₂²×C₄	—	452	2	no	—
183	C₂²×C₄	—	452	2	no	—
184	C₂²×C₄	—	226	4	no	—
185	C₂²×C₄	C₂×Dic₁₁₃	1	4	yes	—
186	C₂²×C₄	C₂×Dic₁₁₃	1	4	yes	complemented
187	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
192	C₄⋊C₄	—	452	2	no	—
193	C₄⋊C₄	—	452	2	no	—
194	C₄⋊C₄	C₂×Dic₁₁₃	1	4	yes	complemented
195	C₂²⋊C₄	—	452	2	no	—
196	C₂²⋊C₄	—	452	2	no	—
197	C₂²⋊C₄	—	226	4	no	—
198	C₂²⋊C₄	C₂×Dic₁₁₃	1	4	yes	—
199	C₂²⋊C₄	C₂×Dic₁₁₃	1	4	yes	complemented
200	C₄²	—	452	2	no	—
201	C₄²	C₂×Dic₁₁₃	1	4	yes	complemented
202	C₂³	—	2	2	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	C₂²×Dic₁₁₃	1	2	yes	—
210	C₂³	C₂²×Dic₁₁₃	1	2	yes	—
211	C₂³	#14268647	1	4	yes	—
212	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
213	D₄	—	2	8	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	452	8	no	—
223	C₂×C₄	—	226	2	no	—
224	C₂×C₄	—	226	2	no	—
225	C₂×C₄	—	226	2	no	—
226	C₂×C₄	—	226	2	no	—
227	C₂×C₄	—	226	2	no	—
228	C₂×C₄	—	226	2	no	—
229	C₂×C₄	—	226	2	no	—
230	C₂×C₄	—	226	2	no	—
231	C₂×C₄	—	226	2	no	—
232	C₂×C₄	—	226	2	no	—
233	C₂×C₄	—	226	2	no	—
234	C₂×C₄	—	226	2	no	—
235	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
236	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
237	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
238	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
239	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	#16813581	1	4	yes	—
250	C₂²	C₂³×Dic₁₁₃	1	1	yes	characteristic
251	C₂²	C₂×#14268647	1	2	yes	—
252	C₂²	#16813667	1	2	yes	—
253	C₂²	#16813668	1	1	yes	characteristic
254	C₂²	#16813669	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	452	2	no	—
261	C₄	—	452	2	no	—
262	C₄	—	452	2	no	—
263	C₄	—	452	2	no	—
264	C₂	—	2	4	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×#16813581	1	1	yes	characteristic
267	C₂	#20299073	1	1	yes	characteristic
268	C₂	#20299081	1	1	yes	characteristic
269	C₂	#20299181	1	2	yes	—
270	C₂	#20299190	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁₃
commutator	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₂×C₄×D₄×C₁₁₃	C₂
socle, cosocle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂²×D₄×C₁₁₃	C₂²
hypercenter	C₂×C₄×D₄	D₁₁₃
hypocenter	C₁₁₃	#2318
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20298491	C₂	semidirect
2. #20298484	C₂	semidirect
4. #20298473	C₂	semidirect
5. #20298466	C₂	semidirect
6. #20298464	C₂	semidirect
7. #20298461	C₂	semidirect
8. #20298458	C₂	semidirect
9. C₄×C₂²⋊Dic₁₁₃	C₂	semidirect
10. C₂×#16813529	C₂	semidirect
31. C₄×D₄×C₁₁₃	C₄	semidirect
56. C₂²×C₄×C₁₁₃	C₂×C₄	semidirect
62. C₄⋊C₄×C₁₁₃	C₂×C₄	semidirect
64. C₂²⋊C₄×C₁₁₃	C₂×C₄	semidirect
65. C₄²×C₁₁₃	C₂×C₄	semidirect
136. C₁₁₃	#2318	semidirect
152. C₄×D₄	Dic₁₁₃	semidirect
186. C₂²×C₄	C₂×Dic₁₁₃	semidirect
194. C₄⋊C₄	C₂×Dic₁₁₃	semidirect
199. C₂²⋊C₄	C₂×Dic₁₁₃	semidirect
201. C₄²	C₂×Dic₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20298491	2
#20298484	2
#20298478	2
#20298473	2
#20298466	2
#20298464	2
#20298461	2
#20298458	2
C₄×C₂²⋊Dic₁₁₃	2
C₂×#16813529	2
C₂×C₄×D₄×C₁₁₃	2
#2318	113