Subgroups — #35297454

Order 21248.

454 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×C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	#24907620	C₂	1	1	yes	maximal, characteristic
3	#24907469	C₂	1	2	yes	maximal
4	#24905347	C₂	1	2	yes	maximal
5	C₂×#334×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₆₆	—	2	2	no	—
8	C₈×Dic₁₆₆	C₄	1	4	yes	complemented
9	C₂×#14581724	—	2	1	no	—
10	C₂×#14581724	—	2	1	no	—
11	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₈₃⋊C₈	—	2	1	no	—
13	C₂×#14581717	—	2	2	no	—
14	C₂×C₈×Dic₈₃	—	2	1	no	—
15	#19127571	C₂²	1	2	yes	—
16	#19127464	—	2	2	no	—
17	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	#334×C₈₃	—	2	2	no	—
20	C₂²×Dic₁₆₆	D₄	1	1	yes	characteristic
21	C₂²×C₈₃⋊C₈	—	2	1	no	—
22	C₄×Dic₁₆₆	—	2	1	no	—
23	C₄×Dic₁₆₆	—	2	1	no	—
24	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₆₆	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₈₃	—	2	2	no	—
27	C₂×C₄⋊Dic₈₃	—	2	2	no	—
28	C₂×C₄⋊Dic₈₃	—	2	2	no	—
29	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
30	C₂×Dic₈₃⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₈₃	—	2	1	no	—
32	#14581724	—	4	2	no	—
33	#14581724	—	4	2	no	—
34	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
35	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
36	Dic₈₃⋊C₈	—	2	4	no	—
37	#14581717	—	4	4	no	—
38	C₈×Dic₈₃	—	2	4	no	—
39	C₂²×C₈×C₈₃	—	2	2	no	—
40	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₈₃	—	2	2	no	—
44	C₄⋊C₈×C₈₃	—	2	2	no	—
45	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₈₃	D₄	1	2	yes	—
47	C₂×Dic₁₆₆	—	2	1	no	—
48	C₂×Dic₁₆₆	—	2	2	no	—
49	C₂×Dic₁₆₆	C₄⋊C₄	1	2	yes	—
50	C₂×Dic₁₆₆	SD₁₆	1	4	yes	—
51	C₂²×Dic₈₃	—	2	1	no	—
52	C₂²×Dic₈₃	—	2	1	no	—
53	C₂²×Dic₈₃	—	4	2	no	—
54	C₂×C₈₃⋊C₈	—	2	2	no	—
55	C₂×C₈₃⋊C₈	—	4	1	no	—
56	C₂×C₈₃⋊C₈	—	4	1	no	—
57	C₄⋊Dic₈₃	—	2	1	no	—
58	C₄⋊Dic₈₃	—	4	2	no	—
59	C₄⋊Dic₈₃	—	4	2	no	—
60	C₄⋊Dic₈₃	—	4	2	no	—
61	C₄⋊Dic₈₃	—	4	2	no	—
62	C₄⋊Dic₈₃	—	8	2	no	—
63	C₄⋊Dic₈₃	C₄⋊C₄	1	2	yes	—
64	Dic₈₃⋊C₄	—	2	2	no	—
65	Dic₈₃⋊C₄	—	2	2	no	—
66	C₄×Dic₈₃	—	2	2	no	—
67	C₄×Dic₈₃	—	2	2	no	—
68	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
71	C₂×C₈×C₈₃	—	2	1	no	—
72	C₂×C₈×C₈₃	—	2	1	no	—
73	C₂×C₈×C₈₃	—	4	2	no	—
74	C₂×C₈×C₈₃	—	4	2	no	—
75	C₂×C₈×C₈₃	—	4	2	no	—
76	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
79	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
80	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₈₃	#41	1	1	yes	characteristic
84	C₂³×C₈₃	#121	1	1	yes	characteristic
85	C₂×C₄×C₈₃	—	2	2	no	—
86	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
89	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
93	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#135	1	2	yes	—
96	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
98	C₂×Dic₈₃	—	2	2	no	—
99	C₂×Dic₈₃	—	2	4	no	—
100	C₂×Dic₈₃	—	2	2	no	—
101	C₂×Dic₈₃	—	4	2	no	—
102	C₂×Dic₈₃	—	8	2	no	—
103	C₂×Dic₈₃	—	8	2	no	—
104	C₂×Dic₈₃	—	8	2	no	—
105	Dic₁₆₆	—	2	2	no	—
106	Dic₁₆₆	—	2	4	no	—
107	Dic₁₆₆	#106	1	4	yes	complemented
108	C₈×C₈₃	—	2	2	no	—
109	C₈×C₈₃	—	8	2	no	—
110	C₈×C₈₃	#103	1	4	yes	complemented
111	C₈₃⋊C₈	—	4	2	no	—
112	C₂²×C₈₃	#380	1	1	yes	characteristic
113	C₂²×C₈₃	#460	1	1	yes	characteristic
114	C₂²×C₈₃	#462	1	1	yes	characteristic
115	C₂²×C₈₃	#465	1	2	yes	—
116	C₂²×C₈₃	#471	1	2	yes	—
117	C₄×C₈₃	—	2	1	no	—
118	C₄×C₈₃	—	2	1	no	—
119	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
120	C₄×C₈₃	#417	1	1	yes	characteristic
121	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
122	C₄×C₈₃	#427	1	1	yes	characteristic
123	C₄×C₈₃	#465	1	2	yes	—
124	C₄×C₈₃	#475	1	1	yes	characteristic
125	C₄×C₈₃	#477	1	1	yes	characteristic
126	Dic₈₃	—	2	4	no	—
127	Dic₈₃	—	4	2	no	—
128	Dic₈₃	—	16	2	no	—
129	#8643	—	83	1	no	maximal, Sylow
130	C₂×C₈₃	#1666	1	2	yes	—
131	C₂×C₈₃	#1672	1	2	yes	—
132	C₂×C₈₃	#1803	1	1	yes	characteristic
133	C₂×C₈₃	#1807	1	1	yes	characteristic
134	C₂×C₈₃	#1815	1	1	yes	characteristic
135	C₂×C₈×Q₈	—	83	1	no	—
136	#1844	—	83	1	no	—
137	#1803	—	83	2	no	—
138	C₂×#334	D₈₃	1	1	yes	characteristic
139	#1296	—	83	2	no	—
140	C₈₃	#8643	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄×Q₈	—	83	1	no	—
142	C₈×Q₈	—	83	4	no	—
143	C₈×Q₈	—	166	2	no	—
144	C₂×#106	—	166	1	no	—
145	C₂×#106	—	166	1	no	—
146	C₂×C₄⋊C₈	—	83	1	no	—
147	C₂×C₄⋊C₈	—	166	1	no	—
148	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
149	C₂×#103	—	166	2	no	—
150	C₂×C₄×C₈	—	166	1	no	—
151	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
152	#384	—	83	2	no	—
153	#340	—	166	2	no	—
154	#334	—	2	2	no	—
155	C₂²×Q₈	—	83	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	166	1	no	—
158	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
159	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
160	C₄×Q₈	—	83	2	no	—
161	C₄×Q₈	—	83	2	no	—
162	C₄×Q₈	—	166	1	no	—
163	C₄×Q₈	—	166	1	no	—
164	C₂×C₄⋊C₄	—	83	1	no	—
165	C₂×C₄⋊C₄	—	166	2	no	—
166	C₂×C₄⋊C₄	—	166	2	no	—
167	C₂×C₄⋊C₄	—	166	2	no	—
168	C₂×C₄⋊C₄	—	166	1	no	—
169	C₂×C₄²	—	166	1	no	—
170	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
171	#106	—	332	2	no	—
172	#106	—	332	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	83	2	no	—
176	C₄⋊C₈	—	166	4	no	—
177	C₄⋊C₈	—	166	1	no	—
178	#103	—	332	4	no	—
179	C₄×C₈	—	166	4	no	—
180	C₄×C₈	C₄×D₈₃	1	2	yes	—
181	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
182	C₂×Q₈	—	83	4	no	—
183	C₂×Q₈	—	83	2	no	—
184	C₂×Q₈	—	166	1	no	—
185	C₂×Q₈	—	166	2	no	—
186	C₂²×C₄	—	332	2	no	—
187	C₂²×C₄	—	166	1	no	—
188	C₂²×C₄	—	166	1	no	—
189	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
191	C₂²×C₄	#13050591	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	332	1	no	—
198	C₂×C₈	—	332	1	no	—
199	C₂×C₈	—	166	2	no	—
200	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
201	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
202	C₂×C₈	#13050579	1	2	yes	—
203	C₂×C₈	Dic₃₃₂	1	2	yes	—
204	C₄⋊C₄	—	332	2	no	—
205	C₄⋊C₄	—	332	2	no	—
206	C₄⋊C₄	—	332	2	no	—
207	C₄⋊C₄	—	332	2	no	—
208	C₄⋊C₄	—	83	2	no	—
209	C₄⋊C₄	—	664	2	no	—
210	C₄⋊C₄	—	166	1	no	—
211	C₄⋊C₄	—	166	2	no	—
212	C₄⋊C₄	—	166	2	no	—
213	C₄²	—	166	2	no	—
214	C₄²	—	166	2	no	—
215	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
217	C₄²	Q₈×D₈₃	1	1	yes	characteristic
218	C₄²	#13050589	1	1	yes	characteristic
219	C₂³	#14581767	1	1	yes	characteristic
220	Q₈	—	83	4	no	—
221	Q₈	—	166	4	no	—
222	Q₈	—	166	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	332	2	no	—
225	C₂×C₄	—	664	2	no	—
226	C₂×C₄	—	664	2	no	—
227	C₂×C₄	—	664	2	no	—
228	C₂×C₄	—	166	2	no	—
229	C₂×C₄	—	166	2	no	—
230	C₂×C₄	—	166	4	no	—
231	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
233	C₂×C₄	#14581750	1	1	yes	characteristic
234	C₂×C₄	#14581763	1	1	yes	characteristic
235	C₂×C₄	#14581768	1	1	yes	characteristic
236	C₂×C₄	#14581772	1	1	yes	characteristic
237	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
238	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
239	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
240	C₂×C₄	#14581813	1	2	yes	—
241	C₂×C₄	#14581820	1	1	yes	characteristic
242	C₂×C₄	#14581825	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	8	2	no	—
245	C₈	—	332	2	no	—
246	C₈	#14581717	1	4	yes	complemented
247	C₂²	#19127558	1	1	yes	characteristic
248	C₂²	#19127902	1	1	yes	characteristic
249	C₂²	#19127912	1	1	yes	characteristic
250	C₂²	#19127931	1	2	yes	—
251	C₂²	#19127965	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	1328	2	no	—
255	C₄	—	332	2	no	—
256	C₄	—	166	4	no	—
257	C₄	C₂×#14581717	1	1	yes	characteristic
258	C₄	#19127703	1	1	yes	characteristic
259	C₄	#106×D₈₃	1	1	yes	characteristic
260	C₄	#19127736	1	1	yes	characteristic
261	C₄	#19127927	1	2	yes	—
262	C₄	#19127987	1	1	yes	characteristic
263	C₄	#19127997	1	1	yes	characteristic
264	C₂	#24906812	1	2	yes	—
265	C₂	#24906846	1	2	yes	—
266	C₂	#24907469	1	1	yes	characteristic
267	C₂	#24907485	1	1	yes	characteristic
268	C₂	#24907519	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581767
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#334×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#121
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#334	D₈₃
hypocenter	C₈₃	#8643
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₆₆	C₄	semidirect
107. Dic₁₆₆	#106	semidirect
110. C₈×C₈₃	#103	semidirect
140. C₈₃	#8643	semidirect
246. C₈	#14581717	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₆₆	2
#24907620	2
#24907469	2
#24905347	2
C₂×#334×C₈₃	2
#8643	83