Subgroups — #2252×D₈₃ #35559878

Order 21248.

558 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₈₃	C₂	1	2	yes	maximal
2	#24909738	C₂	1	1	yes	maximal, characteristic, complemented
3	#24909737	C₂	1	4	yes	maximal, complemented
4	#24909736	C₂	1	2	yes	maximal, complemented
5	#366×D₈₃	C₂	1	4	yes	maximal
6	C₄×C₁₆×D₈₃	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊Q₈×D₈₃	—	2	2	no	—
9	C₂×C₁₆×D₈₃	C₂²	1	2	yes	—
10	#19128106	C₂²	1	2	yes	—
11	#19128105	—	2	2	no	—
12	#19128105	C₂²	1	4	yes	—
13	#19128104	—	2	2	no	—
14	#107×D₈₃	—	2	2	no	—
15	#107×D₈₃	C₂²	1	4	yes	—
16	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
17	#19127527	C₂²	1	4	yes	—
18	#19127526	C₂²	1	8	yes	—
19	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₁₆×Dic₈₃	C₂²	1	2	yes	—
21	#500×C₈₃	C₂²	1	2	yes	—
22	#366×C₈₃	C₂²	1	4	yes	—
23	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
24	C₂×Q₈×D₈₃	—	4	2	no	—
25	C₂×C₈×D₈₃	D₄	1	1	yes	characteristic
26	C₂×C₈×D₈₃	C₂³	1	1	yes	characteristic
27	#14581792	—	2	2	no	—
28	#14581791	—	2	4	no	—
29	#14581791	—	4	2	no	—
30	#14581790	—	4	2	no	—
31	C₄⋊C₄×D₈₃	—	2	4	no	—
32	C₄⋊C₄×D₈₃	—	4	2	no	—
33	C₄²×D₈₃	D₄	1	1	yes	characteristic
34	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
35	C₁₆×D₈₃	Q₈	1	8	yes	complemented
36	#14581726	—	2	2	no	—
37	#14581726	C₂³	1	4	yes	—
38	#14581725	—	2	4	no	—
39	#14581725	—	2	4	no	—
40	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
41	C₈×Dic₈₃	D₄	1	1	yes	characteristic
42	C₈×Dic₈₃	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₈₃	—	2	2	no	—
44	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
45	#107×C₈₃	—	2	2	no	—
46	#107×C₈₃	C₂³	1	4	yes	—
47	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₆₆	—	4	2	no	—
49	C₂×Dic₁₆₆	—	4	4	no	—
50	Q₈×D₈₃	—	8	8	no	—
51	C₂×C₄×D₈₃	—	4	4	no	—
52	C₂×C₄×D₈₃	D₈	1	1	yes	characteristic
53	C₂×C₄×D₈₃	D₈	1	1	yes	characteristic
54	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
57	C₈×D₈₃	Q₁₆	1	4	yes	—
58	C₈×D₈₃	C₂×Q₈	1	4	yes	—
59	C₄⋊Dic₈₃	—	2	4	no	—
60	C₄⋊Dic₈₃	—	4	2	no	—
61	Dic₈₃⋊C₄	—	4	4	no	—
62	Dic₈₃⋊C₄	—	4	4	no	—
63	C₄×Dic₈₃	D₈	1	1	yes	characteristic
64	C₄×Dic₈₃	D₈	1	1	yes	characteristic
65	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
66	C₈₃⋊C₁₆	C₂×Q₈	1	4	yes	complemented
67	C₂×Q₈×C₈₃	—	4	2	no	—
68	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₈₃	—	2	4	no	—
71	C₄⋊C₄×C₈₃	—	4	2	no	—
72	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₈₃	C₂×Q₈	1	4	yes	complemented
74	C₂²×D₈₃	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₈₃	—	8	4	no	—
76	C₂×C₄×C₈₃	—	4	4	no	—
77	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×Dic₈₃	—	4	4	no	—
81	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
82	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
83	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
84	C₄×D₈₃	—	8	8	no	—
85	C₄×D₈₃	D₁₆	1	4	yes	—
86	C₄×D₈₃	Q₃₂	1	4	yes	—
87	C₄×D₈₃	C₄⋊Q₈	1	2	yes	—
88	C₄×D₈₃	C₂×Q₁₆	1	2	yes	—
89	Dic₁₆₆	—	8	4	no	—
90	Dic₁₆₆	—	8	8	no	—
91	C₈×C₈₃	C₂×Q₁₆	1	2	yes	—
92	C₈×C₈₃	C₂²×Q₈	1	2	yes	—
93	C₈₃⋊C₈	C₄⋊Q₈	1	2	yes	—
94	C₈₃⋊C₈	C₂×Q₁₆	1	2	yes	—
95	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
96	C₂×D₈₃	#500	1	2	yes	—
97	C₂×D₈₃	C₂×D₁₆	1	2	yes	—
98	C₂×D₈₃	C₂×Q₃₂	1	2	yes	—
99	C₄×C₈₃	—	8	4	no	—
100	C₄×C₈₃	C₂×D₁₆	1	2	yes	—
101	C₄×C₈₃	C₂×Q₃₂	1	2	yes	—
102	C₄×C₈₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₈₃	—	8	4	no	—
105	Dic₈₃	#500	1	2	yes	—
106	Dic₈₃	C₂×D₁₆	1	2	yes	—
107	Dic₈₃	C₂×Q₃₂	1	2	yes	—
108	C₂×#2252	—	83	1	no	maximal, Sylow
109	C₂×C₈₃	C₂×#500	1	1	yes	characteristic
110	C₂×C₈₃	C₂²×D₁₆	1	1	yes	characteristic
111	C₂×C₈₃	C₂²×Q₃₂	1	1	yes	characteristic
112	D₈₃	#2252	1	4	yes	direct product factor
113	C₂×#500	—	83	2	no	—
114	#2252	—	83	2	no	—
115	#2252	—	83	4	no	—
116	#2252	—	83	1	no	—
117	#2252	D₈₃	1	1	yes	characteristic, direct product factor
118	C₂×#366	—	83	4	no	—
119	C₂×C₄×C₁₆	—	83	1	no	—
120	C₈₃	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄⋊Q₈	—	166	2	no	—
122	C₂²×C₁₆	—	83	2	no	—
123	#500	—	83	4	no	—
124	#500	—	83	2	no	—
125	#500	—	166	2	no	—
126	#500	—	166	2	no	—
127	#500	C₂×D₈₃	1	2	yes	—
128	C₂×#107	—	83	4	no	—
129	C₂×#107	—	166	2	no	—
130	C₂×C₄×C₈	—	83	1	no	—
131	#366	—	83	4	no	—
132	#366	—	83	8	no	—
133	#366	C₂×D₈₃	1	4	yes	—
134	C₄×C₁₆	—	83	2	no	—
135	C₄×C₁₆	—	83	1	no	—
136	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
137	C₂²×Q₈	—	332	2	no	—
138	C₂²×C₈	—	83	1	no	—
139	C₂²×C₈	—	83	1	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	332	2	no	—
142	C₄⋊Q₈	—	332	2	no	—
143	C₄⋊Q₈	—	166	4	no	—
144	C₄⋊Q₈	—	166	2	no	—
145	C₂×C₄⋊C₄	—	332	2	no	—
146	C₂×C₄⋊C₄	—	166	4	no	—
147	C₂×C₄²	—	83	1	no	—
148	C₂×C₁₆	—	83	2	no	—
149	C₂×C₁₆	—	83	8	no	—
150	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
151	#107	—	2	2	no	—
152	#107	—	83	4	no	—
153	#107	—	166	4	no	—
154	#107	—	166	4	no	—
155	#107	—	166	2	no	—
156	#107	C₂²×D₈₃	1	4	yes	—
157	C₄×C₈	—	83	1	no	—
158	C₄×C₈	—	83	1	no	—
159	C₄×C₈	—	83	1	no	—
160	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	332	4	no	—
163	C₂×Q₈	—	332	2	no	—
164	C₂×Q₈	—	664	8	no	—
165	C₂²×C₄	—	332	4	no	—
166	C₂²×C₄	—	83	1	no	—
167	C₂²×C₄	—	83	1	no	—
168	C₂²×C₄	—	83	1	no	—
169	C₂×C₈	—	83	1	no	—
170	C₂×C₈	—	83	1	no	—
171	C₂×C₈	—	83	4	no	—
172	C₂×C₈	—	83	4	no	—
173	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
174	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	332	4	no	—
178	C₄⋊C₄	—	332	4	no	—
179	C₄⋊C₄	—	332	2	no	—
180	C₄⋊C₄	—	166	4	no	—
181	C₄²	—	83	1	no	—
182	C₄²	—	83	1	no	—
183	C₄²	—	83	1	no	—
184	C₄²	D₄×D₈₃	1	1	yes	characteristic
185	C₁₆	—	83	4	no	—
186	C₁₆	Q₈×D₈₃	1	4	yes	complemented
187	C₂³	—	83	1	no	—
188	Q₈	—	8	4	no	—
189	Q₈	—	664	8	no	—
190	Q₈	—	664	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	332	4	no	—
193	C₂×C₄	—	83	1	no	—
194	C₂×C₄	—	83	1	no	—
195	C₂×C₄	—	83	1	no	—
196	C₂×C₄	—	83	4	no	—
197	C₂×C₄	—	83	4	no	—
198	C₂×C₄	—	83	2	no	—
199	C₂×C₄	—	83	2	no	—
200	C₂×C₄	—	664	8	no	—
201	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
202	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
204	C₈	—	83	2	no	—
205	C₈	—	83	2	no	—
206	C₈	Q₁₆×D₈₃	1	2	yes	—
207	C₈	C₂×Q₈×D₈₃	1	2	yes	—
208	C₂²	—	83	2	no	—
209	C₂²	—	83	2	no	—
210	C₂²	—	83	2	no	—
211	C₂²	C₂×D₈×D₈₃	1	1	yes	characteristic
212	C₄	—	8	4	no	—
213	C₄	—	83	2	no	—
214	C₄	—	83	2	no	—
215	C₄	—	83	2	no	—
216	C₄	—	664	4	no	—
217	C₄	D₁₆×D₈₃	1	2	yes	—
218	C₄	Q₃₂×D₈₃	1	2	yes	—
219	C₄	C₄⋊Q₈×D₈₃	1	1	yes	characteristic
220	C₄	C₂×Q₁₆×D₈₃	1	1	yes	characteristic
221	C₂	—	83	4	no	—
222	C₂	#500×D₈₃	1	1	yes	characteristic
223	C₂	C₂×D₁₆×D₈₃	1	1	yes	characteristic
224	C₂	C₂×Q₃₂×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₃
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2252	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂²×D₈₃	C₂×D₈
hypercenter	#2252	D₈₃
hypocenter	C₈₃	C₂×#2252
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #24909738	C₂	semidirect
3. #24909737	C₂	semidirect
4. #24909736	C₂	semidirect
7. C₈₃×#2252	C₂	semidirect
35. C₁₆×D₈₃	Q₈	semidirect
66. C₈₃⋊C₁₆	C₂×Q₈	semidirect
73. C₁₆×C₈₃	C₂×Q₈	semidirect
112. D₈₃	#2252	direct (N × H)
117. #2252	D₈₃	direct (N × H)
120. C₈₃	C₂×#2252	semidirect
186. C₁₆	Q₈×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#500×D₈₃	2
#24909738	2
#24909737	2
#24909736	2
#366×D₈₃	2
C₄×C₁₆×D₈₃	2
C₈₃×#2252	2
C₂×#2252	83

Subgroups — #2252×D83 #35559878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2252×D₈₃ #35559878